DESIGN AND APPLICATION OF SECOND ORDER
SLIDING MODE CONTROL ALGORITHMS
Thesis submitted for the degree of
Doctor of Philosophy
at the University of Leicester
by
Mohammad Khalid Khan
Department of Engineering
University of Leicester
N o v e m b e r 2003 UMI Number: U180131
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ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 To my grandfather IBRAHEEM KHAN for his efforts in my education A b st r a c t
DESIGN AND APPLICATION OF SECOND ORDER SLIDING MODE CONTROL ALGORITHMS by Mohammad Khalid Khan
The thesis considers the development and application of second order sliding mode control algorithms. Second order sliding mode control keeps the main advantages of standard slid ing modes and has the additional advantage that it can be used to remove chattering effect, providing smooth or at least piecewise smooth control. The method also provides better accuracy with respect to switching delays.
A comparison has been made between dynamic sliding modes and higher order sliding modes. The application of dynamic sliding mode control has been demonstrated for sys tems not affine in the control in a MIMO case study for the control of an IC engine. The super-twisting algorithm has been implemented for robust speed control of a diesel engine available in the laboratory where the sliding variable has relative degree one with respect to the control input. A theoretical case has been made for the application of the algorithm and bounds on the controller parameters have been generated. The implementation results demonstrate the practical importance of higher order sliding mode control.
A new second order-sliding algorithm has been developed to stabilize systems where the sliding variable has relative degree two with respect to the control input. More over, it does not require the derivative of the sliding variable to be measured or observed and hence reduces the number of sensors required for control implementation. Closed loop simulation of various systems has been carried out to validate the theory. The algorithm has been applied using dSPACE for position and speed control of a DC motor in SISO configuration.
The algorithm has also been extended for a class of nonlinear uncertain MIMO systems. A MIMO case study for water level control in coupled twin-tanks system has been presented. The controller has been implemented in the laboratory to validate the theoretical assertions made in the thesis. A cknowledgements
First of all I thank to Almighty ALLAH for blessing me with ability, courage and strength to complete my studies.
I am grateful to my supervisor Professor Sarah K. Spurgeon for her support, supervision and invaluable advice throughout my research work. Her positive criticism and feedback were especially a great asset to me. I would like to thank Dr. Christopher Edwards for useful suggestions to improve the readability of the thesis and Dr. Matthew Turner for the valuable directives on anti-windup schemes.
Further, I would also like to thank Dr. Arie Levant for discussions and critical suggestions over e-mails throughout my research. I further extend my regards to Professor Elbert Hen dricks for providing me with the Mean Value Engine Model (MVEM). I am grateful to Keng Boon for giving me introduction on dSPACE, and to Mr. Peter Barwell for helping me in debugging problems with the ‘twin-tank’ rig.
I wish to thank to all my colleagues in the Control & Instrumentation Group at the Depart ment of Engineering, University of Leicester for their friendship and lively environment they provided. Dr. Liqun Yao was always available for consultation throughout my PhD studies.
How can I forget to mention Mr. Bilal Haveliwala, Mr. Mukhtar Ahmad and their families for the help and friendship, which provided an enjoyable and comfortable homely stay at Leicester.
I am grateful to my parents for their prayers, guidance and support throughout my educa tion. Their inspiration and encouragement has been invaluable. Finally, thanks to my wife, Zainab, for her love, support and patience especially during the write-up and viva preparation when she has to handle our daughter, Hiba, all day alone.
Mohammad Khalid Khan Ta b l e o f C o n t e n t s
Table of Contents 1
1 Introduction 5 1.1 Thesis contribution ...... 7 1.2 Thesis stru ctu re ...... 9
2 Sliding Mode Control 11 2.1 Introduction ...... 11 2.2 Motivation for sliding mode co n tro l ...... 11 2.3 Sliding mode control: literature survey ...... 12 2.4 Sliding mode control: Basic concepts ...... 14 2.4.1 Sliding surface d e s i g n ...... 15 2.4.2 Control law design ...... 16 2.5 An illustrative exam ple ...... 18 2.5.1 Sliding surface d e s i g n ...... 19 2.5.2 Control law design ...... 19 2.5.3 Equivalent control ...... 20 2.5.4 Robustness property ...... 21 2.5.5 Order reduction ...... 22 2.6 Chattering avoidance ...... 23 2.7 Concluding Rem arks ...... 26
3 Higher Order Sliding Modes 29 3.1 Introduction ...... 29 3.2 Literature survey ...... 30 3.3 Higher order sliding modes (HOSM) ...... 32 3.3.1 Sliding motion and sliding s e t ...... 33 3.3.2 Real sliding ...... 35 3.4 Dynamic sliding modes (DSM ) ...... 36 3.4.1 Equivalent control m e th o d ...... 38 3.4.2 Indirect sliding method ...... 38
1 CONTENTS 2
3.5 Finite time converging HOSM algorithm s ...... 41 3.5.1 Terminal sliding mode c o n tro l ...... 42 3.5.2 Second order sliding control algorithms ...... 44 3.6 HOSM vs. DSM ...... 49 3.6.1 An illustrative exam ple ...... 50 3.7 HOSM reachability condition ...... 53 3.8 C o n c lu sio n ...... 54
4 HOSM Case Studies 56 4.1 Introduction ...... 56 4.2 Super-twisting algorithm ...... 57 4.3 Diesel engine speed control ...... 58 4.3.1 Controller design and tuning ...... 59 4.3.2 Speed response to load change ...... 60 4.4 HOSM and unmodelled dynam ics ...... 61 4.5 IC engine speed control ...... 64 4.5.1 The engine model ...... 64 4.5.2 Controller design ...... 67 4.5.3 Simulation results ...... 68 4.6 Engine control using DSM control ...... 71 4.6.1 Mean value engine model (M VEM ) ...... 72 4.6.2 Input-output (I-O) representation ...... 74 4.6.3 Controller design ...... 75 4.6.4 Simulation results ...... 77 4.7 Conclusions ...... 79
5 A New Second Order Sliding Algorithm 81 5.1 Introduction ...... 81 5.2 Problem formulation ...... 83 5.2.1 Chattering avoidance ...... 84 5.2.2 Limited state availability ...... 85 5.2.3 The problem statement ...... 86 5.3 The proposed algorithm ...... 87 5.4 Selection of controller coefficients ...... 88 5.4.1 Choice of u0 ...... 88 5.4.2 Case 1: A < u0 ...... 90 5.4.3 Case 2: A = u0 ...... 90 CONTENTS 3
5.4.4 Case 3: A > u0 ...... 91 5.4.5 Selection of A; ...... 91 5.4.6 Summary of observations ...... 92 5.5 Some simulation exam ples ...... 95 5.5.1 Underwater v ehicle ...... 95 5.5.2 Single axis jet-controlled aircraft ...... 97 5.5.3 Nonlinear tunnel diode circuit ...... 99 5.6 Anti-lock braking system ...... 101 5.7 Conclusion ...... 105
6 A Multi-Input 2-Sliding Control 106 6.1 Introduction ...... 106 6.2 Background ...... 106 6.3 A MIMO 2-sliding control ...... 108 6.3.1 Sliding surface d e s i g n ...... 109 6.3.2 Controller Design ...... I l l 6.4 Illustrative exam ples ...... 112 6.5 Conclusions ...... 117
7 Case Studies 119 7.1 Introduction ...... 119 7.2 DC motor position control ...... 119 7.2.1 DC motor m o d e l ...... 119 7.2.2 Sliding surface d esig n ...... 120 7.2.3 Uncertainty bounds ...... 121 7.2.4 Simulation results ...... 122 7.3 DC motor speed control ...... 123 7.3.1 Sliding surface d esign ...... 124 7.3.2 Uncertainty bounds ...... 125 7.3.3 Simulation results ...... 125 7.4 Controller implementation ...... 126 7.4.1 Experimental S etup ...... 126 7.4.2 Position Control ...... 128 7.4.3 Speed Control ...... 130 7.4.4 Robustness to lo a d ...... 132 7.5 Liquid level control in coupled-tanks ...... 134 7.5.1 System modelling ...... 134 CONTENTS 4
7.5.2 System constraints ...... 135 7.5.3 Simulation results ...... 137 7.5.4 Anti-windup s c h e m e ...... 143 7.6 The twin-tanks laboratory rig ...... 146 7.6.1 System description ...... 146 7.6.2 The rig model ...... 148 7.6.3 Admittance coefficients of pipes ...... 150 7.6.4 Simulation study of the rig m o d el ...... 152 7.6.5 Controller implementation on the rig ...... 154 7.6.6 Robustness to leakage ...... 156 7.7 Conclusion ...... 157
8 Conclusions And Future Research 159 8.1 Introduction ...... 159 8.2 Concluding remarks ...... 159 8.3 Future research directions ...... 161
References 163 Ch a p t e r 1
I ntroduction
A mathematical model of any physical system is always approximate. The difference be tween the system model and the actual system may be due to plant parameter variations or due to neglected high frequency system dynamics. Most practical systems are nonlin ear in nature, therefore error may be introduced due to the linearization of system non- linearities. Controllers designed on the basis of imperfect system models are required to be robust to such perturbations to maintain stability and performance. Therefore, to en hance the performance and safety of such systems, robust control strategies have to be applied. Sliding mode control is a nonlinear control methodology which is inherently ro bust [27, 102, 104, 106, 116]. It operates by keeping a predefined variable, the so-called sliding variable, zero. The robustness properties result from the application of a discontin uous control signal. The method has been successfully applied to a class of systems which can tolerate such high frequency signals to the input such as electrical systems [106]. Other authors approximate the discontinuous control signal for application [113], but in this case robustness is compromised.
The recently developed Higher Order Sliding Mode (HOSM) control methodology, which is the topic of investigation of this thesis, has removed some of the theoretical limitations of classical sliding mode control and therefore increased the scope of application for such methods. The higher order sliding mode control has been successfully applied to mechanical systems of great importance such as robotic systems and aircraft dynamics.
All higher order sliding mode controllers available in the literature are based on state feed back because the sliding surface is based on the full system state [4, 5, 11, 56]. Furthermore, the control not only requires the sliding variable but also requires the knowledge of a cer tain number of derivatives of the sliding variable which are either provided by designing an observer [53, 91] or by designing robust real time differentiators [61, 62].
To date very few practical applications of higher order sliding mode control have been re-
5 Chapter 1. Introduction 6 ported in the literature [6, 15, 28, 64]. The robustness of higher order sliding mode control has been demonstrated by the implementation of a super twisting algorithm [56] for speed control of a diesel engine-generator set under uncertain load conditions using dSPACE. The results demonstrate that the algorithm is not only robust to unknown load variations but also to the variation in the system states which are not directly controlled but remain bounded. Despite some robustness properties being demonstrated, it has been shown in the thesis that when there is an unmodelled dynamics, say an actuator dynamics present, the same algo rithm does not converge into the 2-sliding set and develops limit cycle behaviour. By way of a simulation study of IC engine speed control, it has been suggested that if the parameters are selected differently, the algorithm again stabilises the plant robustly. A MIMO case study for speed, manifold pressure and air-fuel ratio control of a car engine using dynamic sliding mode control is also carried out. However, for the application for dynamic sliding mode control, the system model needs to be known accurately.
The main topic of this thesis is to devise a new robust higher order sliding mode controller which does not require the derivatives of the sliding variable and provides a method for tuning controller parameters during actual implementation on the Genset rig. The thesis presents a second order sliding mode control which does not require the knowledge of the derivative of the sliding variable. In the general sense, the algorithm requires only partial state feedback and in the case of second order systems, it actually requires only the output to be measured. The algorithm has been applied to various systems of industrial importance such as stabilisation of an under water vehicle and single axis aircraft and wheel slip control of an anti-lock braking system. The algorithm has also been implemented for position control of a DC motor in the dSPACE environment.
The higher order sliding mode control algorithms developed so far are SISO in nature. Bartolini et al. [10] have extended the sub-optimal algorithm [3] for the diagonally domi nant class of MIMO systems and for diagonally non-dominant MIMO systems a hierarchal methodology was adopted. The thesis extends the algorithm for a class of nonlinear MIMO systems which may not be affine in the control. The algorithm has been simulated for stabi lization of angular velocities of an spacecraft dynamics and other mathematically important systems. The actual implementation of the control is carried out for water level control in a coupled twin-tank system. The nonlinear model of the plant has been derived and different plant parameters identified by experimentation. The controller is coded in C++ and imple mented through a Windows 98 PC with 166MHz CPU speed. The thesis contributions are detailed in the following section. The structure of the thesis is then described in Section 1.2. Chapter 1. Introduction 1
1.1 Thesis contribution
The thesis has contributed towards design and implementation of second order sliding mode control algorithms with the following individual contributions:
• Theoretical developments
1. It is demonstrated that the super-twisting algorithm exhibits limit cycle behaviour when applied to systems where the sliding variable has relative degree greater than one.
2. The work of Lu and Spurgeon [68] has been extended by applying it for MIMO control of an IC engine which also extends the work of Weihua et al. [110] where tracking control is done using feedback linearization.
3. A new second order-sliding mode control algorithm has been developed to sta bilize a second order nonlinear system. It has thus been shown that the method is applicable to systems with relative degree two with respect to the sliding vari able. The method does not require the derivative of the sliding variable and hence reduces the number of sensors required for control implementation.
4. The algorithm developed here provides dynamic control if the full state vector is available, even if the sliding surface is not control dependent. This is in contrast to the work of Sira-Ramirez [91] where dynamic control arises only if the sliding surface is control dependent. This is also different to the work of Lu and Spur geon [70] where the sliding surface is a nonlinear function of the derivative of the nth—state (where n is the system order), which is not measurable.
5. A higher order sliding mode algorithm has been developed for a class of nonlinear multi-input multi-output system.
• Simulation studies
1. A Simulation study for speed control of an IC engine without acceleration mea surement has been presented. The selected controller parameters do not conform to the original selection criteria and therefore it was suggested that the modified algorithm has the potential to deal with systems with relative degree two with respect to the sliding variable for which it was not initially designed.
2. Application of the dynamic sliding mode control algorithm for robust control of nonlinear MIMO systems which are not affine in the control has been shown Chapter 1. Introduction 8
by the simulation study for MIMO control of an IC engine model (MVEM). The dynamic sliding mode control algorithm requires the system model together with uncertainties to be differentiated at least once. This requirement restricts the applicability of dynamic sliding mode control only to systems for which the mathematical model is completely known.
3. The novel 2-sliding control algorithm developed in Chapter 5 has been employed to simulate the robust control of an under water vehicle model,single axis jet controlled aircraft model and the stabilisation of a tunnel diode circuit. In all these examples, the system has a nonlinear second order uncertain dynamics to be controlled which is first converted into generalised controller canonical form (GCCF). The controller requires only the output to be measured. The derivative of the output is neither measured nor estimated.
4. The anti-lock braking system (ABS) model has also been investigated. It is a third order system with wheel slip (A) as the output. The model when converted into GCCF has a stable zero dynamics. The simulation study shows that the proposed controller stabilises the wheel-slip without the knowledge of the derivative of wheel slip.
5. DC motors are widely used as actuators in control industry. The simulation stud ies for angular position and speed control have been carried out to assess the nominal performance of the DC motor. The angular position as system output has relative degree three. However, after the selection a suitable sliding vari able,which has relative degree two with respect to the control input, the control algorithm developed in Chapter 5 has been applied. The algorithm does not re quire the derivative of the sliding variable. The simulation results show very good set point tracking despite poor knowledge of model parameters. The algo rithm also tracks the set point speed when a load is applied to the shaft which is unknown to the controller.
6. The algorithm developed for the robust control of uncertain nonlinear MIMO systems has been simulated for the stabilisation of angular velocities in a rigid body MIMO spacecraft model and for water level control in the MIMO twin- tank system to assess their nominal performances. Both the models are coupled in nature. The results show good set-point tracking and provide smooth or at least piecewise linear control input. Chapter 1. Introduction 9
• Implementation of developed control algorithms
1. The super-twisting algorithm has been implemented for robust speed control of a diesel engine available in the laboratory using dSPACE.
2. The new 2-sliding algorithm developed in Chapter 5 has been implemented for servomotor control using dSPACE.
3. The 2-sliding algorithm has also been implemented for water level control in a twin-tanks system in MIMO configuration. The controller code is written in C++.
1.2 Thesis structure
The thesis has been organised in eight Chapters. The first Chapter introduces the thesis itself.
Chapter 2 introduces basic sliding mode theory. The literature survey is presented with illustrative design examples. Sliding mode control design for systems in nonlinear canonical form is also introduced and the methods to remove chattering are discussed.
Chapter 3 introduces the concept of higher order sliding mode control. The HOSM concept and related definitions are discussed with illustrations. An extensive literature survey in the area of HOSM is presented with a brief review of the properties of various HOSM algorithms available in the literature. Dynamical sliding mode (DSM) control is introduced and the relative advantages and disadvantages are discussed for HOSM and DSM.
In Chapter 4 the super-twisting algorithm is applied for robust speed control of a diesel engine-generator set in the dSPACE environment. A mathematical case is made for the selection of controller parameters based on an identified model. It is also shown in the same chapter that the super-twisting algorithm shows limit cycle behaviour when implemented on a system where the sliding variable has relative degree two with respect to the control input. However, by applying this algorithm to a second order IC engine model, it is suggested that there is scope for the algorithm to be modified to suit second order systems. Chapter 4 also presents a simulation case study for MIMO control of an IC engine using a dynamic sliding mode strategy developed in [68, 72] which requires exact knowledge of the system model.
Chapter 5 presents a novel second order sliding mode control algorithm for systems where the sliding variable has relative degree two with respect to the system input. The algorithm does not require the derivative of the sliding variable to be measured or estimated. The conditions for stability of the algorithm are proposed and justified through various simulation studies. The application of this new control algorithm is demonstrated by applying it to Chapter 1. Introduction 10 various systems of interest such as an anti-lock braking system, under water vehicle position control and single axis aircraft control.
The algorithm developed in Chapter 5 is extended in Chapter 6 to a class of nonlinear MIMO systems. The theory developed in this chapter is further illustrated through examples such as the control of rigid body spacecraft dynamics.
In Chapter 7, two case studies are presented. One SISO case study involves position and speed control of a DC servo motor. The linear model of the servo motor is used and uncer tainty bounds calculated by maximising various parameters. The controller is implemented in the dSPACE environment. The second MIMO case study involves water level control in an interconnected twin-tank system. The nonlinear model of the twin-tanks is developed and different parameters identified. The controller is designed and simulated to assess the nominal performance. Finally, the controller is implemented on the rig by writing code in C++ which is run on a Windows 98 PC at 166 MHz CPU speed.
Finally, the thesis concludes in Chapter 8 and further directions for future work are sug gested. Ch a p t e r 2
S l i d i n g M o d e C o n t r o l
2.1 Introduction
This Chapter introduces the basic concepts of sliding mode control theory. Section 2.2 presents motivation for sliding mode control and then a thorough literature survey of lin ear and nonlinear sliding mode strategies is presented. The fundamentals of sliding mode theory are explained in Section 2.3 and made clear by an illustrative example in Section 2.5. The sliding mode design issues pertaining to robustness and performance will be discussed in Section 2.5.4. The concepts introduced in this Chapter will form the necessary back ground needed to understand the concept of higher order sliding modes to be introduced in next Chapter.
2.2 Motivation for sliding mode control
The representation of any physical plant by a mathematical model is always approximate. The unavoidable mismatch between the model used for controller design and the actual plant may be due to insufficient knowledge of system parameters called parametric uncertainties or due to imperfect modelling of system behaviour called unmodelled dynamics. Such poorly known quantities may be constant or time varying. The uncertainties may enter into the plant dynamics through the input or output channel. A feedback control system designed using an imperfect model is required to be robust to these uncertainties in order to maintain stability and performance otherwise the system may become unstable and/or performance may degrade.
From the many different methods used to control uncertain systems, Sliding Mode Control is a simple and effective method. It is a particular type of Variable Structure Control. In sliding mode control, the states of the system are driven and then constrained in a manifold, called the sliding manifold or sliding surface, by a control law that changes value depending upon a predefined switching rule. The dynamical behaviour when the system is constrained
11 Chapter 2. Sliding Mode Control 12
in the sliding manifold is described as the ideal sliding mode. From a design point of view, it is attractive in a sense that the dynamic behaviour of the system during the sliding mo tion can be tailored by suitably choosing the sliding surface. Sliding mode control converts a n —dimensional single-input single-output (SISO) tracking problem into the stabilization problem of first order which can be easier to control. The controlled plant in the sliding mode is totally invariant to disturbances and parameter variations with known bounds which are implicit in the control channels called matched uncertainties. Stability and consistent performance is thus maintained. This property makes sliding mode control an appropriate candidate for robust control.
2.3 Sliding mode control: literature survey
The term ‘sliding mode control’ first appeared in the context of variable-structure systems, specifically speaking relay systems, and became the principal operational mode for this class of systems. Practically all design methods for variable-structure systems are based on the deliberate introduction of sliding modes. The concept of sliding mode control appeared in the Russian literature in the late fifties. However, the vibration control of a DC generator of an aircraft by V. Kulebakin (1932) and the use of relays for controlling the course of a ship by Nikolski (1934) can also be considered as contemporary ‘sliding mode control’ [106]. It was Emel’yanov who first observed that due to altering the structure in the course of controlling a process, the properties could be attained which were not inherent in any of the individual structures [105]. The survey paper by Utkin [102] introduced this concept in the English lit erature. After this, the theory has been extended in various directions. The technique became popular because of its application to a wide class of systems containing discontinuous control elements such as relays. Young [116] applied variable structure system theory to controller design for manipulators. Sliding mode control has subsequently been applied and developed for under water vehicles [96], robust regulators, adaptive control, tracking problems in the control of electrical motors, chemical processes, robotic manipulators and in simulation of automatic flight control, helicopter stability and space systems. A generic sliding mode con troller for linear systems with bounded uncertainties was proposed by Ryan and Corless [85]. Spurgeon and Davies [99] extended this proposition to controller design for output tracking. Burton and Zinober [18] introduced a smoothness into the control scheme using continu ous approximations of the bang-bang control. Many such sliding mode controllers need full state information which sometimes is difficult to obtain, if not impossible. An observer is thus required to predict the system states [16, 73, 94, 95, 103, 109]. Bondarev et al. [16] Chapter 2. Sliding Mode Control 13 presented the idea of chattering removal using observers by generating a sliding mode in the observer states and closing the feedback loop through an asymptotic observer of the plant. Thus chattering is localised inside a high frequency loop which bypasses the plant.
Sliding mode control strategies have been well researched in the past from the theoretical point of view for SISO and MIMO systems in regular form and linear in the control [27, 104, 106]. A tutorial dealing with multi-variable nonlinear systems was written by DeCarlo et al. [24]. The outstanding developments for nonlinear control systems based on differential geometric ideas [46] have found immediate applications, and extensions to sliding mode control and closely related areas.
Sliding mode control is inherently a nonlinear methodology. Therefore, its applications are not limited to linear systems. It provides a framework for controller design for a wide class of nonlinear systems. A controller can be expected to perform better if it is based on a nonlinear model rather than a linear approximation. However, generic nonlinear controller design methods are rarely available. For controller design, the nonlinear model needs to be in specific forms such as canonical form or normal form. If a system model is not in the desired form, conversion of any nonlinear system to the desired type of form is neither straightforward nor generic in nature. Different types of system representations have been considered for controller design in the literature. Slotine and Sastry [97] and Slotine and Hedrick [98] considered sliding mode controller design based on an input-output decoupled form where the sliding surface consists of a Hurwitz polynomial of the output error and its derivatives. A boundary layer in the presence of bounded uncertainties in the model was also introduced. The system does not remain on the sliding surface but remains confined within boundaries around the sliding surface. Boundary layer control smoothes the control and thus reduces chattering.
A differential algebraic representation of system dynamics first proposed by Fliess [35, 37] has opened new horizons in the control of nonlinear systems using sliding mode control. He proposed that a general nonlinear system can be represented by a generalised controllable canonical form (GCCF) using dynamic state and output feedback. Sira-Ramirez proposed [89, 91, 93] a sliding mode controller synthesis scheme based on the GCCF and a particular way for designing the sliding surface was introduced. Lu and Spurgeon [68] have proposed a generic strategy for sliding mode control of nonlinear systems based upon the GCCF. Differential 1-0 system models [46, 80] which represent a wide class of practical systems, were used. Only SISO systems, for the sake of simplicity, will be discussed in the sequel. Chapter 2. Sliding Mode Control 14
2.4 Sliding mode control: Basic concepts
Sliding Mode Control utilises high speed switching feedback control i.e. the control law switches between different values according to some defined rules called a switching func tion, which depend upon the system states. The sliding mode controller synthesis consists of two stages: the sliding surface design and design of the control action so that the sliding surface is reached. When the system dynamics is constrained in the sliding manifold, this is called a sliding motion. During sliding, the system dynamics is independent of the control and is governed by the properties of the chosen sliding surface.
Consider a nonlinear SISO system affine in the control
X\ = x2
x 2 = X3
Xn = f(x, t) + g(x, t)u (2.1)
y = x i
where x = {xi,x2, - ■ • ,xn) G 5Rn is the plant state vector, u, y G 5R are the scaler input and output of the plant, f(x , t) : 5Rn x and g(x, t) : 3^^ x 3ftn are not exactly known. The control problem is to ensure tracking of the given time varying reference trajectory yd by the output y.
Consider also that the system can be written in 1-0 form as follows
y{n) = f(y,t) + g(y,t)u (2.2) where y — (y,y, • • • and the function f(y,t) is not known however, its estimate f(y, t) is known and the control gain g(y, t) is a positive function such that
\f{y,t) - f(y,t)\ < F(y,t), Vy, t (2.3)
0 < Gmin(y,t) < g(y,t) < Gmax(y,t), Vy,£ (2.4) where F(y, t) is a positive known function. The control gain g(y, t) is uncertain but with known bounds and its estimate can be taken as the geometric mean of the bounds i.e.
y(i)■> f) VGminiffi ' Gmaxiy> f)-
Define g = yjGmax(y,t)/Gmin(y, t) then the following inequality holds
- i ^ g(y,t) ^ ~~ g{y,t) ~ ^ which will be used for calculating one of the control parameters in Section 2.4.2. Chapter 2. Sliding Mode Control 15
2.4.1 Sliding surface design
The sliding variable is a fictitious system output such that if it is regulated to zero, the control task is achieved. Once the system states are sliding on the sliding surface, the system char acteristics are governed by the characteristics of the sliding surface only. Therefore, sliding surface design is performed to achieve the characteristics required from the system when sliding. For example, if the tracking error is to be kept at zero, then the sliding variable may be selected as a stable differential equation of tracking error. Generally, sliding variable is selected as a linear equation of the system states
s = S(x,t) (2.6)
where S(-, •) : 5?n x 5R+ is a differentiable function. The system is said to be in the sliding mode when S(x, t) = 0 (2.7)
If S(x, t) gives rise to a stable linear differential equation by design, the system states will automatically converge to the origin asymptotically. The condition on the sliding surface is that it should have relative degree one with respect to the system input i.e., the first time derivative of the sliding variable is a function of the control input. The sliding surface can also be a nonlinear function of the states [75]. While linear sliding surfaces provide only asymptotic convergence of system states to the equilibrium point, nonlinear sliding surfaces can provide finite time convergence [13, 100, 114].
Consider a linear sliding surface satisfying the above stated condition as follows
s = e^n ^ + • • • + C2 C + cie -j- Cge (2.8) where e = y — yd and the c* coefficients are selected such that the polynomial n—2 A"-1 + ^CjA* = 0 (2.9)
i=0 is Hurwitz. If the sliding surface, s, is made zero the tracking error (e) will automatically converge to zero asymptotically by virtue of the polynomial being Hurwitz. Moreover, the sliding variable (2.8) satisfies the relative degree-one-condition because the first derivative of the sliding variable, s, is a function of the control input, as follows
n —2 s = f(x, t) - y(d\t) + Y 2 Cie{l+1\ x , t) + g(x, t)u(t) (2.10)
i = 0 The second step is to design a control law which can stabilise the sliding surface dynamics in equation (2.10). Thus, the nth order SISO tracking problem (2.2) is now converted to the Chapter 2. Sliding Mode Control 16 stabilization of the first order dynamics (2.10) to zero, which is one of the advantages of sliding mode control design.
2.4.2 Control law design
The purpose of the control law is to drive the system trajectories into the sliding mani fold, and thereafter to maintain them within the sliding manifold despite uncertainties being present. In other words, the controller should regulate the sliding variable. This is achieved by setting conditions on the control law which makes the sliding surface attractive. Such a condition is called the reachability condition. Lyapunov’s direct method provides a stability analysis tool that does not involve solving the differential equations. Loosely speaking, it states that if one can find a positive definite function V such that the time derivative along the system trajectories is negative, then the dynamic system is stable. The method is usually utilised to design a control law which makes the sliding surface attractive.
For regulating the sliding variable, consider a positive definite Lyapunov function
V = \s2 (2.11) which implies that V = ss
Hence, the sliding variable s will converge to zero only if
ss < 0 (2.12)
The equation (2.12) represents a condition on the sliding variable s and its first derivative s, and is a reachability condition. A general definition of a reachability condition is given in Lu and Spurgeon [67]. If s satisfies any of the following equations, where ki and k2 are positive constants, then s satisfies the reachability condition (2.12).
s = —k\s (2.13)
s = -k2sign(s) (2.14)
s = —kis — k2sign(s) (2.15)
All the above conditions will ensure that ss < 0 and hence make the sliding manifold attrac tive. Linear reachability conditions similar to that defined in equation (2.13) only guarantee asymptotic convergence of the sliding variable and the convergence rate is controlled by the design constant ki. Therefore, they are termed asymptotic reachability conditions. Nonlinear sliding reachability conditions such as those defined in equations (2.14) and (2.15) provide Chapter 2. Sliding Mode Control 17 finite time regulation of the sliding variable where the reaching time is upper bounded by a function of the design parameters (ki, k2) and the initial conditions of the system states.
Consider again the equation (2.10). The approximate feedback control law to achieve s = 0 is given by
u = g(x,t)~l ^-f(x,t) + ^n) - (2.16) where e^+1^ = xi+2 — Ud+1\t)- The robustness against uncertainties in f(x,t) and g(x,t) can be taken care of by the discontinuous control. Assume that the control structure is given by u = u — g(x, t)~lk2 sign(s); k2 > 0 (2.17)
Considering the finite time quadratic stability criterion
~ s 2 = ss < - 77|s| (2.18)
where 77 is a positive design parameter which is inversely proportional to the reaching time. The condition on k2 can be given by [96]
k2 > ti(F + rj) +g(x,t)(fj, - l)|u| (2.19)
Hence, the control law (2.17) satisfying equation (2.19) ensures that the system trajectories will reach the sliding surface in finite time given by
tT < (2.20) V
Once the system states are on the sliding surface, the tracking error will go to zero asymptot ically according to equation (2.9). The error dynamics, when the system is in sliding mode, only depend upon the coefficients of the equation (2.9) and are independent of the control. The control only acts to push the system states on the sliding surface. The system behaviour can be divided into two parts:
Reaching mode: The system motion for 0 < t < tr, during which time the system trajecto ries are not inside the sliding manifold but approaching it, is called the reaching mode. The time in reaching mode should be minimised as robustness to matched uncertainties is not present in the reaching mode.
Sliding mode: The system motion for t > tr, when the system trajectories are confined within the sliding manifold, is termed the sliding mode or sliding motion. Chapter 2. Sliding Mode Control 18
The general condition on the time derivative of the Lyapunov function [13, 17] to provide finite time convergence is
V < —cVa, c>0, 0 < a < 1 (2.21) with the solution V(t) < (c( 1 - a)t + V ^ ) ^ , Fo = V(0) which vanishes within a time tr which is upper bounded by
t < ____ ^____V l~a Lr — c(l — a)\ 0 For example, the following finite time reachability conditions (2.22) which can be derived from equation (2.21) provide finite time convergence of the sliding variable s.
s = — A|s|5 sign(s). (2.22)
The convergence time is given by 2 tr < ^Vlsof, S0 = S ( 0 ) .
The sliding surface must be reachable in finite time as stated previously, the advantages of siding motion can not be realised by asymptotic convergence of the sliding variable towards zero. The problem is compounded in the multi-input case when ‘hierarchical control’ is applied where a chain of sliding surfaces is considered and it is presumed that the trajectory actually lies in the intersection of preceding the sliding surfaces. In the case of asymptotic convergence, this assumption may be invalid.
2.5 An illustrative example
To illustrate different aspects of sliding mode control design, consider the double integrator given by y(t) = u(t) (2.23) which, when written in the regular form, is:
X\ = x2 (2.24a)
x2 = u (2.24b)
y = x i (2.24c)
The objective is to move system states (xi,x2) = (y, y) to the origin. The double integrator in a simple sense represents many second order systems of practical importance including the Ball & Beam, simple pendulum [27], under water vehicle [84] and a satellite [96]. Chapter 2. Sliding Mode Control 19
2.5.1 Sliding surface design
The objective is to make both y and y equal to zero. The sliding surface cannot be y as making it zero does not necessarily make y = 0. The fact that once y is zero, y will au tomatically become zero, makes y a good candidate for the sliding surface. However, the relative degree of y is two which contradicts the relative-degree-one condition. Consider the following candidate sliding surface function,
s = m y + y (2.25)
where ra is a positive design parameter. The sliding surface in equation (2.25) has relative degree one because the first time derivative of the equation (2.25) gives s, as a function of u as follows s = m y + u (2.26)
and once s is made zero, y and y will automatically become zero asymptotically according to the linear differential equation y + m y = 0.
Therefore, s defined in equation (2.25) is an appropriate sliding variable. The design pa rameter ra can be selected appropriately to decide the convergence speed when in sliding motion.
2.5.2 Control law design
After selecting the sliding variable, a control law must be designed so that the sliding surface is reached, i.e., made attractive. Consider a Lyapunov function V = |s 2 which implies that
V ss
s(my + y)
s(my + u)
provided u + my = —psign(s). (2.27)
From this the control action, u can be calculated as
u = —my — psign(s) (2.28) or if | ray | < p, the control action u = — psign(s) (2.29) Chapter 2. Sliding Mode Control 20 will be sufficient. Consider p = 1. The control in equation (2.29) can also be written as
— 1 if 5 > 0, (2.30) 1+1 if s < 0 The phase portrait for the double integrator (2.23) is shown in Fig. 2.1 with m = 1. From the phase portrait of the closed loop system, the two-stage dynamics are clearly visible. The initial phase, occurring whilst the states are driven towards the sliding surface, the reaching mode followed by the sliding mode in which the trajectories are moving towards the origin constrained in the sliding surface along the line: y = —my.
5*0
0
TJ
0
Figure 2.1: Closed loop phase portrait for small y
2.5.3 Equivalent control
Assume that the ideal sliding motion is established for t > tr, which is guaranteed by the selection of the control in (2.28) satisfying the finite time reachability condition in (2.14). During sliding motion, the trajectories are constrained in the sliding surface, i.e., s(t) = 0 => s(t) = 0, t > tr. Using (2.23) and (2.25) we have
s = my + u (2.31)
Since s = 0 for t > tr, the control law which maintains the sliding motion is calculated by equating (2.31) to zero, therefore
ueq = —my, t > tr (2.32)
This control law is referred to as the equivalent control. This is not the actual control applied but represents the average sense of the applied control. Chapter 2. Sliding Mode Control 21
(a) Discontinuous control input,u
0.5
lime (sec)
(b) Equivalent control input, ueq
Figure 2.2: Control action
2.5.4 Robustness property
Now, a nonlinear term asin(y) is added in the input channel. One can consider it as a bounded uncertainty or disturbance. The uncertainties affecting the system through the input channel are referred to as matched uncertainties. The system model thus becomes
V\ = V2 (2.33a)
y2 = u(t) + asin(y) (2.33b)
y = y i with a — —0.25.
The same controller (2.29) is applied to the new model (2.33) and the phase portrait is shown in Fig. 2.3. From the phase portrait it is evident that once the sliding mode is reached, the system behaves like the ideal double integrator. Therefore, we can conclude that the disturbance term asin(y) has been completely rejected once the sliding mode is established and hence the closed loop system is robust, i.e., insensitive to the mismatches between the model used for control design and the actual plant. However, the reaching phase, when the trajectories are heading towards the sliding surface, is affected by the disturbance term and that is why the phase portrait in the reaching mode is different from that of the nominal plant. Thus the control system should be designed such that the reaching mode is short Chapter 2. Sliding Mode Control 22
Ideal C ase — With Disturbance s -0
0
0
Figure 2.3: Phase portrait with disturbance term and establishment of a sliding mode is guaranteed [see, 76, 115, for elimination of reaching phase]. The sufficient conditions which guarantee that the ideal sliding motion will take place and make the sliding surface locally attractive, ensuring the trajectories are directed towards the sliding surface at least within a domain enclosing the sliding surface, can be derived from Lyapunov’s stability theorem. Consider the reachability condition in equation (2.12)
In the case of the double integrator, ss = s(my + y) = s(my — sign(s)) < |s|(m|y| — 1) (2.35) This implies that the trajectories will be directed towards the sliding surface within the region \y\ < 1/m. A sliding mode is globally reachable if the domain of attraction is the entire state space. Otherwise the domain of attraction is a subset of the state space. Once the value of y is within the region \y\ < 1/m, sliding motion is guaranteed. 2.5.5 Order reduction An ideal sliding motion exists only when the state trajectories of the controlled plant satisfy s(y> y) — 0- If infinite frequency switching of the control was possible, the motion would be constrained to remain on the sliding surface, s. The motion when confined to the sliding surface, s follows the relation: y + my = 0 => V = -my (2.36) Chapter 2. Sliding Mode Control 23 which represents a first order differential equation (one order less than the system order) decaying exponentially to the origin depending upon the parameter m, i.e., the choice of sliding surface controls the dynamic behaviour in the sliding mode. During ideal sliding motion, the state trajectories belong to a vector space whose dimension is one less than the dimension of the original state vector in the original system (2.36) which is also independent of the control. This can be seen as the zero dynamics of the system with respect to the sliding variable, s, as an output [see, 46]. Thus, in terms of sliding mode design, the performance is governed by the sliding surface while the control law guarantees that the reachability condition is satisfied. Thus, apart from the inherent robustness properties shown by sliding mode control, the state trajectories also belong to a lower dimensional space than that of the system state space. The order of the differential equations describing the system in the sliding motion is reduced by one. In the more general case of a multi-input system, the order of the differential equation describing the system in the sliding motion is reduced by an integer equal to the number of inputs. 2.6 Chattering avoidance The ideal sliding motion requires infinitely fast switching of the control action. In actual plants, switching is possible only at a finite frequency causing the trajectories to oscillate within a neighbourhood of the sliding surface. This high frequency motion is termed as chattering. The two main causes of chattering are [119] Delays: The basic assumption of sliding mode theory is that the switching frequency is infinite which is not possible practically. Therefore, practical considerations cause delays which leads the system dynamics to oscillate around the sliding manifold. Parasitic dynamics: Even though it is assumed that the switching device is switching ide ally at infinite frequency, the presence of parasitic dynamics in series with the plant causes a small amplitude high frequency oscillation to appear in the neighbourhood of the sliding manifold. The parasitic dynamics represent the fast actuator and sensor dynamics which are often neglected in controller design if the associated poles are well damped, and outside the desired bandwidth of the closed loop system. The mathematical basis for this neglect comes from singular perturbation theory, where the real system behaviour is close to that of the ideal system and the difference between them on account of neglecting parasitic dynamics, decays rapidly. However, the theory is not applicable to VSS since they are governed by Chapter 2. Sliding Mode Control 24 differential equations with discontinuous right hand sides. The interactions between the par asitic dynamics and VSS generate a non-decaying oscillatory component of finite amplitude and frequency, which is generally referred to as chattering. The chattering phenomenon makes sliding mode control systems unacceptable for mechan ical systems where it may excite the unmodelled high frequency dynamics and cause in creased wear and tear of the actuator. A number of direct ways to tackle the chattering phenomenon from a theoretical view point have been developed. Some approaches to avoid chattering create a boundary layer in the vicinity of the discon tinuity surface or replace the discontinuous control by an arbitrarily close but continuous approximation [18, 104] where the states are no longer required to remain on the sliding sur face s but arbitrary close to it. The main idea is to avoid the real discontinuity by following other smooth dynamics within the boundary layer. Thus, the concept of pseudo-sliding came into existence. The natural solution is to use a high-gain saturation function such as (2.37) where 28 is the boundary layer thickness (see Figure 2.4 (b) for 5 = 0.1). Another approxi mation is the power law interpolation structure. sign(s) if |s| > 8 v(s,8) = < (£/|s|)^ ^ sign(s) 0 < |s| < 8 q G [0, 1) (2.38) 0 s = 0 Figure 2.5 (b) shows the plot for q = 0.5 and 8 = 0.1. It should be noted that saturation (a) Signum function (b) Saturation function W lO w0] -1 -1 0 0 S S Figure 2.4: Signum and saturation functions functions are not differentiable. The following differentiable alternatives can be used to obtain an arbitrary close approximation of the signum function. Chapter 2. Sliding Mode Control 25 Signum-like function: (2.39) where 8 is a small positive scaler and is not the boundary layer. It can be visualised that as 8 —> 0, the function is(-) tends point-wise to the signum function. The variable 8 can be used to trade off the requirement of maintaining ideal performance with that of ensuring a smooth control action. Figure 2.5 (a) depicts the plot for equation (2.39) for 8 = 0.005. Arctan function: For sufficiently small values of 8, the following function is a good differ entiable approximation of the signum function. (2.40) The lower the value of 8, the better the approximation. Figure 2.5 (c) has been drawn for 8 = 0.02. Hyperbolic tan function: Another smooth approximation of sign(-) is the tanh function given as follows: (2.41) where 8 < 1 is a small positive number which defines the slope of the curve. Fig ure 2.5 (d) shows the curve for 8 = 0.1. It should be stressed that with the approximations above, ideal sliding no longer takes place and the total invariance property with respect to matched uncertainty will be lost. The con tinuous control action drives the states to a neighbourhood of the switching surface. How ever, arbitrary close approximations to ideal sliding can be obtained by making (5 small. In the literature this is called pseudo-sliding. An experimental comparison of these different approaches to eliminate chattering is made in deJager [25] where it is reported that no signif icant difference arises between the various methods when steady state tracking error in the presence of a constant disturbance is considered. With this kind of control law, the chattering is removed but it is difficult to show stability inside the boundary layer. The closed loop system response with the same initial conditions and control law but with sign(s) replaced by the signum-like function v(s, <)) as in equation (2.39) with 8 = 0.005 is shown in Fig. 2.6. The control input is smoothed but ideal sliding no longer takes place and hence robustness is compromised as the system becomes sensitive to matched disturbances in the sliding motion. The continuous control only drives the trajectories in a neighbourhood of sliding surface. Chapter 2. Sliding Mode Control 26 (a) Signum -like function (b) Power law interpolation 0 > -1 -1 0 s (c) tan-1 (s/8) function (d) tanh(s/8) function 1 1 0 ro 1 1 0 0 Figure 2.5: Different smoothing techniques Other methods include chattering avoidance by the use of an observer [see, 104] where the sliding motion is established on a manifold in the observer state space. This method localizes high frequency phenomenon in the feedback loop by closing it through an asymptotic ob server of the plant. However, this approach assumes that an asymptotic observer can indeed be designed such that the observation error converges to zero asymptotically. If the system is perfectly known, the asymptotic stabilization method by Sira-Ramirez [89, 91] does not produce chattering. For uncertain systems with known bounds, the algorithms by Bartolini et al. [4, 9] featuring second order sliding modes and dynamic sliding mode control strategies by Lu and Spurgeon [67] also produce a chatter free response. 2.7 Concluding Remarks It has been observed that sliding mode control is a nonlinear robust methodology. Once the sliding motion is established, the plant becomes insensitive to matched disturbances and parameter variations with known bounds and at the same time the plant obeys a reduced order differential equation. In short, sliding mode control has the following advantageous characteristics: Chapter 2. Sliding Mode Control 27 = 0.5 c 8 £ 8 (a) Smooth control action,u X10-5 2 0 •2 ■4 -6 -8 ■10 12 3 4 5 6 7 (b) Trajectories in the neighbourhood of sliding surface Figure 2.6: Effect of pseudo-sliding • The dynamic behaviour of the system may be tailored by the particular choice of slid ing surface. • An n —dimensional tracking problem is converted to a lower order stabilisation prob lem. • The order of the system dynamics under a sliding motion reduces by the number of inputs. • The closed loop system response becomes totally insensitive to a particular class of uncertainties called matched uncertainties. The first property enables the designer to specify system performance and hence is attractive from a design point of view. The second property makes it easier to design the control as it is easier to stabilise a first order system involving s rather than an nth—order system. The invariance property clearly makes sliding mode control an appropriate candidate for robust control. In spite of proven robustness properties, sliding modes have the main drawback of chattering, i.e., the high frequency oscillations of the controlled systems. Moreover, the system to be controlled must often be converted to the so called regular form and then the sliding surface Chapter 2. Sliding Mode Control 28 should be selected such that the plant has relative degree one with respect to it. Chattering avoidance by smooth approximation of sign(-) function loses the robustness prop erties to some extent. The higher order sliding due to Emel’yanov et al. [29, 31], Levant [56] and Sira-Ramirez [89] proved to be an effective tool for hiding chattering in the higher derivatives of the sliding surface. In the next chapter higher order sliding modes will be discussed in detail and it will be seen that the basic limitations considered to be inherent properties of sliding mode control, for example, need of regular form and relative degree one constraints etc. will no longer be necessary for controller design. Further the chattering phenomenon will be hidden in the higher derivatives of the sliding variable. Ch a pte r 3 H i g h e r O r d e r S l i d i n g M o d e s 3.1 Introduction In Chapter 2, it has been seen that sliding mode control has many advantages including ro bustness with respect to internal and external matched disturbances with known bounds and specification of system behaviour by selection of a suitable sliding manifold which results in reduced order system dynamics in the sliding motion. However, it has the main drawback of chattering. High frequency oscillations of the controlled system may overshadow its advan tages. Moreover, the sliding surface should be selected such that the plant has relative degree one with respect to it, which restricts the choice. The chattering problem has been tackled by creating a boundary layer in the vicinity of the discontinuity surface or by replacing the discontinuous control by an arbitrarily close but continuous approximation of the signum function as detailed in Section 2.6. The idea was to avoid the real discontinuity by following other smooth dynamics within the boundary layer. With this kind of control law, the chattering is removed but ideal sliding no longer takes place. Observer based techniques generate a sliding motion in the observer state space which tends to coincide with the ideal sliding modes. But designing an observer for a system poses its own limitations particularly in the presence of noise. The performance becomes a function of the observer gain which affects the tracking error. It has been shown that an increase in observer order is counter productive in minimising the tracking error [11]. Thus observer order becomes part of the cost function in observer based chattering avoidance techniques. This Chapter presents a generalised sliding mode control termed as higher order sliding mode control. Higher order sliding mode control not only removes some of the fundamental limitations of the traditional approach but also provides improved tracking accuracy under sliding motion. Section 3.2 presents a thorough literature survey for higher order sliding mode control. Theoretical concepts and definitions will be presented in Section 3.3. Dy- 29 Chapter 3. Higher Order Sliding Modes 30 namic sliding mode control will be discussed in Section 3.4 and its relation with higher order sliding mode will be presented in Section 3.6. Some recently developed second order sliding controllers will be reviewed in Section 3.5. 3.2 Literature survey The discontinuous control which became synonymous with sliding mode control makes the sliding mode control robust with respect to matched uncertainties. However, the same dis continuous control is the main cause for chattering as well. Creation of a boundary layer around the sliding manifold and the use of observers were attempted for chattering removal. However, the control still acts on the derivative of the sliding variable. Emel’yanov et al. [31] initially presented the idea of acting on the higher derivatives of the sliding variable and provided second order sliding algorithms such as the twisting algorithm, and algorithm with a prescribed law of convergence [29]. Another so called real second order drift algorithm was presented in Emel’yanov et al. [30], The term real means that the algorithm has a mean ing only when there is a switching delay. In the ideal case of infinite frequency switching, it ceases to converge. The so called super twisting algorithm, which is applicable to systems with relative degree one with respect to the sliding variable, [32] completely removes chat tering. Levant [56] reported that in second order sliding, the sliding accuracy is proportional to the square of the switching time delay which turns out to be another advantage of higher order sliding modes. Since then, a number of such controllers have been developed in the literature and are cur rently finding useful applications. Bartolini et al. presented a sub-optimal second order sliding algorithm in [3, 7, 9]. The higher order sliding mode concept was applied to the control of constrained manipulators in Bartolini et al. [ 6 ]. Levant et al. [64] have applied it to aircraft pitch control and the same idea has been implemented for robust exact differentia tion in [59]. Levant [60] demonstrated that the twisting algorithm with a variable integration step size performs better in the presence of measurement errors. A collection of higher order sliding mode algorithms has been presented in [40, 57]. An exact differentiator has been designed in Levant [59] and utilised in Levant [61] to observe the derivative of the sliding variable. The second order sliding mode control has been extended for MIMO systems in Bartolini et al. [3, 6 , 10], Ferrara and Giacomini [33] and Chang [20]. Stability of sliding modes of higher order using simple relay systems has been discussed in [40, 47, 48]. The case of first order sliding modes is trivial and second order sliding motion Chapter 3. Higher Order Sliding Modes 31 can be stabilised under certain conditions but sliding order higher than two is unstable [40] or produces a stable periodic motion [47]. Alternatively, developments due to Fliess [35, 37], Martin et al. [78], [see also, 39] in nonlin ear control theory propose the use of differential algebra for the conceptual formulation, clear understanding and definitive solution of long standing problems in automatic control. Re classification of nonlinear systems into flat and non-flat systems is the natural consequence of the differential algebraic approach. The set of flat systems not only contains all control lable linear systems but also some of the nonlinear systems which can be made equivalent to linear systems via input dependent mapping using state or output dynamic feedback. Formalization of sliding mode control theory within the framework of differential algebra by Fliess [34, 35, 36, 37], Fliess and Hassler [38], Levine [65], Conte et al. [22] and Sira- Ramirez [92] has lead to another independent direction termed as ‘Dynamic Sliding Mode Control’ (DSM). All the basic notions of the theory are recovered from this viewpoint and some fundamental limitations of the traditional approach are removed. For instance, input dependent sliding surfaces are seen to arise naturally from this approach. Input dependent sliding manifolds are shown to lead to continuous, rather than bang-bang, control and chatter free sliding regimes in Sira-Ramirez [91]. Higher order sliding modes is also an immediate consequence of the differential algebraic approach [34, 8 8 , 92]. Application of the sliding mode technique is then not restricted to systems linear or affine in the control. The case studies of a chemical reactor by Sira-Ramirez [91] and an automotive engine model in Khan and Spurgeon [52] are not affine in the control. Application of dynamic sliding modes for asymptotic linearization of nonlinear systems has been studied by Lu and Spurgeon [67, 70] and finds application in the linearization of flat and non-flat SISO systems [69, 71, 8 6 , 90]. The same concept has been extended for MIMO systems in Lu and Spurgeon [ 6 8 , 72] and has been applied for the control of a MIMO, IC engine model in Khan and Spurgeon [52]. The ball and beam case study in Lu and Spurgeon [69], which is a singular non-flat system, has shown that HOSM ideas have the potential to deal with non-flat problems. There are various strategies to deal with the chattering problem in sliding mode control as discussed in Section 2.6 and they have their own domains of application. Yet, the most comprehensive technique is the higher order sliding mode approach [29, 56], which allows all restrictions to be removed, while preserving the main sliding mode features and improving accuracy. Dynamic sliding mode control is similar to higher order sliding mode control Chapter 3. Higher Order Sliding Modes 32 although it has been developed independently. Little work has been done to extend these higher order sliding mode strategies developed for SISO systems to MIMO systems which are generally coupled. Suitable higher order reach ability conditions for SISO as well as for MIMO systems still need to be investigated. All higher order sliding mode algorithms presently developed are dependent upon the availabil ity of the derivatives of sliding variable or at least its sign. 3.3 Higher order sliding modes (HOSM) As discussed in Section 2.5.1, in selecting a sliding surface in classical sliding mode control, one of the conditions that must be fulfilled is that it must have relative degree one with respect to the control. An obvious question is “can this condition be relaxed?” The answer “yes” generalises the concept. Thus the concept of higher order sliding modes is the generalization of classical sliding modes in the sense of generalising the condition on the relative degree of the sliding surface. The driving idea behind higher order sliding is to act on the higher order derivatives of the sliding variable rather than the first derivative as in classical sliding modes. It keeps the main advantages of classical sliding modes and has the additional advantage of increased accuracy with respect to switching delay ( t ) [56]. Recall Filippov’s solution to differential equations with discontinuous right hand sides x = v(x) (3.1) where x E 5Rn, and v is a locally bounded measurable vector function. Definition 3.1 An absolutely continuous function x(t) : [0, r] —> 3Rn is said to be a solution of equation (3.1) in the Filippov sense if for almost all t E [0, r], r > 0, it is the solution of the differential inclusion x E lF(x), x(0) = x0. (3.2) where, fF(x) is the closed convex hull of all limits of v(x). Therefore, any solution of equation (3.1) is defined as an absolutely continuous function x(t), satisfying the differential inclusion (3.2) almost everywhere. In the sequel, the solution of differential equations with discontinuous right hand side will be assumed in Flippov’s sense. Chapter 3. Higher Order Sliding Modes 33 3.3.1 Sliding motion and sliding set Consider a nonlinear system x = f(x)+g(x)u (3.3) where, the state vector x e 5ftn, the control u £ 3ft and /(•), g(-) are smooth vector functions of proper dimension. The control u is determined by feedback control u = U(t,x), where U is possibly a discontinuous function. Define a candidate sliding surface s £ 3ft as s = S(t, x) (3.4) such that by making it zero, the control objective is fulfilled. S(-) is a sufficiently smooth constraint function. If the system in equation (3.3) has a globally defined relative degree r [46] with respect to the sliding variable s defined in equation (3.4), then the sliding surface dynamics can be expressed in 1 - 0 form as s(r) = LrfS(x)-hLgLrf- 1S{x)u = 4>(t, x) + 7 (t, x)u (3.5) where Lg,Lf are Lie derivatives, LgL y l S{x) ^ 0 holds globally by assumption and the discontinuity does not appear in the first (r — 1 ) total time derivatives of the constraint variable s along the trajectories, i.e., s, s, s, ■ • • , exist and are single valued. The resulting zero dynamics of order n — r [46] can be written as rj = Z(s, r]) (3.6) where, 77 satisfy the condition LgT]i(x) = 0; i = 1, • • • , (n - r) and s = [5, 5, • • • , s(r-1)] = [5, L/s, L'js, • • • , Lrj~ls}. Assumption: The zero dynamics 77 = Z(s, 77) (3.7) is input-to-state stable i.e., for any bounded s(t), the internal states r)(t) remain bounded. For r = n, there are no zero dynamics and the system is said to be fully linearizable. Now the task is to find a feedback control u such that the nonlinear uncertain dynamics s (r) = (j)(t, x) 4- 7 (t, x)u (3.8) Chapter 3. Higher Order Sliding Modes 34 can be stabilized on the basis of suitable known upper bounds to the uncertainties appearing in (3.8). In general, any sliding mode controller that keeps s = 0 needs s,s,s,• • • ,s^r~^ to be made available. It must be stressed that while the simple discontinuous relay control on s = 0 is effective if r = 1, it is possibly unstable if r = 2 and always unstable for r > 2 [40] or at most produces a stable periodic motion [47]. Definition 3.2 (rth—order sliding set) Given the constraint (3.4), its rth—order sliding set (Sr) is defined by r equalities s = s = s = • • • = s(r_1) = 0 (3.9) which constitute an r—dimensional condition on the system states. According to this definition, if the only imposed condition on the sliding motion is that of s = 0, then the sliding set Definition 3.3 (rth—order sliding motion) Assume the rth—order sliding set (3.9) is non empty and assume that it consists of Filippov's trajectories of the discontinuous dynamical system (3.3). Then the corresponding motion satisfying (3.9) is termed anrth— order sliding motion with respect to the constraint function s. t d s s= d s= 0 Figure 3.1: second order sliding mode trajectory The sliding order is a measure of the degree of smoothness of the sliding variable in the vicinity of the sliding mode. The standard sliding, referred to as first order sliding, features Chapter 3. Higher Order Sliding Modes 35 finite time convergence while higher order sliding modes may be asymptotic as well as fi nite time converging. Asymptotically stable HOSM are easily found in many classical VSS examples [31, 89]. Stable HOSMs also exist in systems with fast actuators [63] and reveal themselves in this case by the spontaneous disappearance of the chattering effect. Consider a trivial two-dimensional example: x = u with sliding variable s = x. The system can be written as s = u. Both, first and second order sliding algorithms can be used to stabilise it. A finite time converging second order sliding algorithms with discontinuous input u can stabilise s and s to zero in finite time. The sliding variables, and its first time derivative, s, will be smooth but s will be discontinuous. The system converges into 2- dimensional sliding set s = s = 0. Now, consider another definition of the sliding variable s = cx + x, c > 0 , which has relative degree one with respect to the discontinuous control input u. This leads to s being discontinuous therefore, the sliding mode established is only of first order. 3.3.2 Real sliding In r th—order sliding mode control, the system trajectories are theoretically kept within the rth—order sliding set ( Any ideal sliding motion should be understood as a limiting motion when switching imper fections (delays) vanish i.e., the switching frequency tends to infinity. Let e be some measure of these switching imperfections. Then the sliding mode may be defined by a sliding preci sion asymptotic as e —+ 0 . Definition 3.4 (Levant [56]) Let 7 (e) be a real valued function such that 7 (s) —> 0 as £ —> 0. A real sliding algorithm, on the constraints = 0, is said to be of order r(r > 0) with respect to 7 (c) if for any compact set of initial conditions and for any time interval [t\, tf\ there exists a constant C, such that the steady state process for t E [ti, tf[ satisfies M < C\y(e)\r. In the case of the minimal switching time interval being r > 0, the r th—order sliding condi tion can be stated as \s\ < C\r\r. Thus, to achieve r th—order accuracy in a discrete realization, the sliding order has to be at least r. First order real sliding is achieved by practical application of standard sliding mode Chapter 3. Higher Order Sliding Modes 36 control. The second order real sliding motion is achieved by discrete implementation of the second order sliding algorithms [56]. In fact an arbitrary order real sliding can be achieved by discretization of the same order sliding algorithm. Figure 3.2 provides a graphical illustration of the accuracy between classical (first order) and 2 nd order real sliding mode control. Classical sliding mode 2nd order sliding mode |s| ~ 0(t) |5| ^ 0 ( r 2) Figure 3.2: Standard vs. 2nd order sliding modes 3.4 Dynamic sliding modes (DSM) Independently developed dynamic sliding mode control is a kind of higher order sliding mode control in the sense that it tries to develop higher orders by involving integrals of the control action. Second order sliding modes can be established by using derivatives of the actual control as the input. The concept has been generalised using the differential algebraic approach [35, 37, 39, 78] to nonlinear control theory and classifying nonlinear systems into flat and non-flat systems. A large set of nonlinear systems can be classified as differentially flat systems, which can be made equivalent to linear systems via an input dependent map ping using state or output based dynamic feedback. This input dependent mapping proved instrumental in the definition of input dependent sliding surfaces [91]. In the traditional approach, the sliding surface (s) is a function of the system states only; and the derivative of the sliding variable (s) is an algebraic function of the states and input only. There is no scope for the input or its derivatives to be part of the sliding surface. Hence, all resulting sliding mode controllers were, necessarily, of static nature. The differential algebraic definition removes this unnecessary restriction on the sliding surface and naturally points to the possibility of a dynamical sliding mode controller [89]. Chapter 3. Higher Order Sliding Modes 37 If a system has relative degree p with respect to its sliding surface s there are two possibilities for r th—order sliding mode control: Case 1: r = p A system can be written as in the 1-0 form using output s as: S{ — •Si+i i — 1 ■ ■ ■ p 1 (3.10) sp = f ( s ) -f g(s)u where s = (s1, s2, • • • , sp) In this case, it is possible to stabilise the system in finite time by selecting the actual control to be discontinuous. Case 2: r > p This is a more general case. Here an rth—order sliding mode control is being applied to a system with relative degree p, where r > p. It becomes necessary to increase the system dimension by adding the input u and its (r — p — 1 ) derivatives as state variables. Then one gets a new extended system Si = si+1 i = 1 • • • (r — 1) sr = f(s,u,u, • • • ,u(r~P~^) + g(s,u,u, • • • ,U^r~P~^)v (3.11) u(r-p) = v where, s == (si, s 2 , • • • , sr ) and u = (it, ii, • • • , vSr~p~l\ In this case, it is also possible to stabilise the system in finite time by selecting a discontinuous control signal which is the (r — p)th time derivative of the actual control and therefore, always gives a smooth control. An approximate actuator dynamics can be considered at the design stage to give an implementable control signal. Assume that the extended system has stable zero dynamics with respect to s, then the rth—order sliding mode with respect to s is equivalent to finite time stabilisation of the fol lowing system Si = si+1 i = 1 • • • r - 1 sr = f(s,t) + g(s,t)v(t) (3.12) V = si where v — u represents Case 1 and v = u^r~p^ represents Case 2. Input dependent sliding surfaces come under this second category. A sliding surface which is a function of the control input ensures that even classical sliding mode control provides smooth chatter free control with uasa virtual control. In simple terms, input dependent sliding surfaces naturally give rise to second order sliding motion by using the same classical sliding mode control methods. In dynamic sliding mode control, the total time derivative of the control always appears Chapter 3. Higher Order Sliding Modes 38 linearly and application of sliding mode control is no longer restricted to systems linear or affine in the control [52, 91]. The obvious questions which arise here are: how to select an input dependent sliding sur face? Is there any natural design formulation? There are two ways to accomplish this in the literature. First, the equivalent control method described in [91] and second the indirect method due to Lu and Spurgeon [70] which leads to input-dependent sliding surfaces. Both methods are described below. 3.4.1 Equivalent control method Recall again the system (3.3) x = f(x) + g(x)u The equivalent control to stabilise the system states to zero is given by Ueq = g~1(x)f(x) (3.13) Select an input dependent sliding surface s = u - Ueq (3.14) which ensures that zeroing of the sliding variable will enforce the equivalent control and hence the system will remain in the sliding motion. Differentiating the sliding variable, s = u — Ueq (3.15) The first order finite time reachability condition s = — fcsign(s) yields u = Ueq — k sign(-s) (3.16) k is selected such that the closed loop system is stable. The auxiliary control u which is dis continuous in (3.16) ensures that the actual control is smooth and chatter free. This method requires the system to be written in regular form so that the equivalent control can be calcu lated. 3.4.2 Indirect sliding method The indirect sliding method employed by Lu and Spurgeon [70] applies to systems written in 1-0 form. Consider a locally observable system in state space form X = f(x,u,t) (3.17) y = h(x, u, t) Chapter 3. Higher Order Sliding Modes 39 where x G 5Rn, u G 5R, y G 3ft and f(x , it, £) and /i(x, it, t) are smooth vector functions. The following locally equivalent differential 1 - 0 form exists involving the control input and its certain number of derivatives, say (3. j/<"> = (a) (ft is a C 1 function, (b) the regularity condition ( 3 ' 1 9 ) is satisfied. The corresponding zero dynamics of the system model (3.18) is defined as 0(0, u,t) = 0 The system (3.18) is called minimum phase if the zero dynamics is uniformly asymptotically stable. The usual choice of the sliding surface as a linear function of the states, i.e., s = S(x) is termed a static sliding surface and produces static feedback [24, 104]. A sliding surface which is aset of differential equations and has dynamics is termed a dynamic sliding surface and produces dynamic feedback [ 6 8 , 89] which may be control independent or control de pendent and the method is termed indirect or dynamic sliding mode control. In this case, a sliding surface in general can be represented as a function of states and certain derivatives of the control as s = S(t,y, • • • , y {n\ u, • • • , u{/3)) (3.20) One possible dynamic sliding surface is n * = $ > y{l~1] + 0(y, y, • ■ ■ , y(n-1\ u, • • • , u{p\t) (3.21) 2—1 where is a Hurwitz polynomial with an + 1 = 1. It is to be noted that this sliding surface is control dependent and makes the system equivalent to an asymptotically stable linear system in the ideal sliding mode. Chapter 3. Higher Order Sliding Modes 40 After selecting the sliding surface, a sliding reachability condition must be chosen which ensures that the sliding mode will be attained and the system trajectories will remain confined in the sliding mode once it is reached. The popular choices are given in Section 2.4.2 where the derivative is taken along the system trajectories. Take derivatives of the sliding surface along the system trajectories and use the reachability condition to close the loop. If the regularity condition (3.19) is satisfied, using implicit func tion theorem, explicit differential equation for the highest order time derivative of control, u ^ +1) which always appears linearly, can be solved as (3.22) and then the dynamic controller can be realized by a chain of integrators. This explicit repre sentation is locally valid only in a domain where the regularity condition (3.19) is satisfied. To make the concept more clear, a SISO example, which is feedback linearizable, is pre sented as follows. Suppose that n = r for the system (3.17). The locally equivalent differen tial 1 - 0 system is as follows Vi Vi+1 > (3.23) yr = 4>(y, u, t), i = 1 , - • ■ ,r - 1 The system state is feedback linearisable because of the assumption r = n otherwise certain higher time derivatives of the control will appear in equation (3.23) which would help in further smoothing the control designed by this method. The zero dynamics 0 (0 , u, t) = 0 is assumed to be stable. Now consider a candidate sliding surface r r (3.24) with { 1 j j fly*— 1 3 * * * 3 } being a set of coefficients of a Hurwitz polynomial. Differentiat ing the sliding variable will yield Chapter 3. Higher Order Sliding Modes 41 The derivative of the sliding variable, s, will always involve certain derivatives of the control which appear linearly. Therefore, the system (3.23) is not required to be affine in the control. The first order finite time discontinuous reachability condition s = —k sign(s) yields -1 r— 1 d(f) dcf) . . . / y a iVi+ 1 + + ’Q^ y i+ 1 S18n (S) 2 = 1,... ,r - 1 — \ £ ) i= 1 (3.26) The designed coefficient k can be selected using Lyapunov theory such that the sliding sur face is reachable under uncertainties. From equation (3.26) it is evident that the actual control will be smooth and chatter free because of the fact that the control derivative is discontinuous. In dynamic sliding mode control methods, the switching occurs in the highest derivatives of the control and thus the actual control signals are continuous [ 6 8 , 70, 72, 91]. This is one of the advantages of the dynamic sliding mode technique. These two input dependent sliding surfaces which eventually lead to dynamical sliding mode control are similar in nature. However, in the latter case the system is not required to be affine in the control as the derivative of the control always appears linearly in the first order sliding surface dynamics. Both methods require the derivative of the corresponding functions, i.e., /, g or 4> which restricts the applicability of these methods only to systems with smooth drift and gain functions. There is sometimes confusion in the literature about the number of successive derivations required of the sliding variable (s) for an r th—order sliding motion. The notion adopted here is from Levant [56]. What ever number of derivatives of s are considered, once in the sliding mode, all of the derivatives will be nullified. Therefore, the number of successive continuous total derivatives of s in a vicinity of the mode is taken as the number defining the order of the sliding mode. In other words r th—order sliding means that the first (r — 1) derivatives of 5 will be continuous and the r th—derivative, s^r\ will be discontinuous. 3.5 Finite time converging HOSM algorithms Generally, the sliding surface is designed as a linear dynamic equation of the error to be made zero e.g. s = e + ce. However, a linear sliding surface can only guarantee asymptotic error convergence once the system states are in the sliding mode. Dynamic sliding mode control leads to asymptotically stable higher order sliding modes i.e., the system takes infinite time to reach the sliding manifold. The benefits of sliding mode control can only be utilised when the system trajectories are constrained within the sliding surface or manifold. In this section, some finite time reaching algorithms will be reviewed. Chapter 3. Higher Order Sliding Modes 42 3.5.1 Terminal sliding mode control In terminal sliding mode control [74, 75, 118] the sliding surface is selected as a nonlinear function of error of the form s = e + cea//b, a = 2i + 1 , b > a where, i is a positive integer. The error converges to zero in finite time. However, these methods have a singularity problem [83]. The singular points are located around the origin in the state space. In other words, if the system error dynamics is within the set {(e, e)|e = 0 , e 7^ 0 } and the system is still not on the sliding surface, the control signal will explode. Consider the double integrator system e = u where the error e has to be made zero in finite time. Select the sliding surface s = e + ce b, a = 2i + 1, b > a and c > 0 (3.27) Taking the first time derivative of equation (3.27) gives CLC a — b * s = u + -r-ee b (3.28) b Now using the reachability condition s = —Asign(s) (3.29) equations (3.28) and (3.29) yield —A sign(s) = u -f ^ e e ~ (3.30) b The control u, can be written as \ . f X CLC . a - b u = — Asign(s ) — —ee b b \ / \ CLC a-b . a . . _ = —Asign(s) — —e b (s — ceb), using equation (3.27) b . . . CCL a — b CL l a — b = —Asign(s)— — se b + —— e b b b The term seV 5 has a singularity on e = 0, s^0=>e^0 and this will cause the control action to explode. However, the control action u will be bounded if the system initial conditions are in the domain s = e = 0 , which is not always the case. To avoid this phenomena, conventional terminal sliding mode control generally uses a satu ration function instead of the sign function. This causes the problem of drifting of the system Chapter 3. Higher Order Sliding Modes 43 states around the origin (e = e = 0 ) where the motion is bounded by the boundary layer thickness. On the axis, however, there is only one point, the origin, which is not a singular point. Thus conventional terminal sliding mode control may generate a very large control signal in the steady state. It has to be emphasised that the problem of singularity does not occur if the system states are at the sliding surface initially and there is no reaching phase, which is not generally the case. Terminal sliding mode control utilises the properties of a finite time stable nonlinear differ ential equation of the following type. Lemma 3.5 Assume that a positive definite function V (t ) satisfies the differential inequality V(t) < — cVa(t), 'it > t0, V(t0) > 0 (3.31) where c>0, 0 < a < 1 are constants. Then, for any given t0, V (t ) satisfies the inequality Vl~a{t) < ^ 1_a(to) - c(l - a)(t - t0), t0 < t < ti (3.32) and V(t) = 0, i t > h (3.33) where t\ is given by tt, i = t.n t 0+ 4- V' °(to} (3.34) c(l — a) Proof Tang [100]: Rewriting equation (3.31) < _c f V~aVdt < ~ c f dt Jtn j Jtnto to < c{t - to) =*V'-a{t) - V1~°'(t0) < —c(l - a)(t - t0) => - V l~a{to) < - c ( l - a)(t - t0)\ as K 1_“(ti) = 0 . = > t\ < to H— —------c(l — a) Hence proved. ■ In a modified terminal sliding mode control [84], the above problem is addressed and the reaching phase is totally eliminated by using a sliding surface augmented by a function. It is guaranteed that the error converges to zero in finite time. The augmentation function is required to be sufficiently differentiable and satisfy certain boundary conditions dependent Chapter 3. Higher Order Sliding Modes 44 upon the initial condition of the system state. The requirement on the initial state information restricts the applicability of the control method. Moreover, the saturation function is used to avoid the chattering phenomena which has obvious drawbacks. 3.5.2 Second order sliding control algorithms Several finite time convergence higher order algorithms are due to Bartolini et al. [7], Emel’yanov et al. [29], Levant [56] and a collection is reported in Levant [57]. In practice, because of switching imperfections, a delay is introduced which causes real sliding rather than ideal sliding. In real sliding modes, the system states deviate from the sliding surface s = 0 and sup|s| = O(r) where r is a measure of the delay in switching. However, Levant has shown that in the real second order sliding mode, the relationship sup|s| = 0 (r2) holds and hence the performance can be improved. These algorithms are developed for uncertain SISO nonlinear systems (3.35) where, x G X C 5Rn is a state vector, u £ U C 3ft is a bounded input, t an independent vari able time and T : 5Rn + 2 —> is a sufficiently smooth vector function. The sliding variable (s) is selected such that the derivative of the sliding variable can be written as follows: s f(t,x) +g{t,x)u (3.36) The classical VSS control techniques can be applied to stabilise the system dynamics (3.36). Nevertheless second order sliding mode control can be used for smoothing out the input signal as the control derivative is used as an input. The system dynamics with the new input then become s = f(t,x) + g(t,x)u (3.37) u V which can be written in 1 - 0 form as s 4>{t, x, u ) -I- 7 (t, x , u)v (3.38) u V where, d d \ d d (f)(t,x,u) = — f(t,x) + — f(t,x)T(t,x,u) + —g(t,x) + — g(t,x)T{t,x,u) u and ~l{t,x,u) = g(t,x) Chapter 3. Higher Order Sliding Modes 45 It is assumed that the systems dynamics (3.38) satisfy the following bounding conditions 0 < Tm < 7(t,x,u) < r M |,s| < s 0 (3.39) \(t,x,u)\ < $ where, Tm, Tm, s0, $ are some appropriate positive constants. Some second order sliding finite time converging algorithms for stabilization of the systems of type (3.38) are now described. Twisting algorithm This algorithm [56] is characterized by a twisting of the phase portrait around the origin, see Fig. 3.3. The finite time convergence to the origin is due to switching between two different control amplitudes as the trajectory comes nearer to the origin. The sign of the derivative of the sliding variable is required for decision making. — u, \u\ > Uq v(t) = < — V m sign(s) s s > 0, |u| < uq (3.40) —Vm sign(s) ss < 0 , |u| < uq and the corresponding sufficient conditions for the finite time convergence to the sliding manifold are [29, 56]: 4Tm $ V m > max , , So I j vm > (3.41) i m and r rnVM - $ > TMVm + $ Suboptimal algorithm This algorithm is actually the optimized twisting algorithm [3]. v(t) = ~oc(t) VM s i g n [s(t) - 0 . 5 s ( t M )] (3.42) a*, if [s(0 - 0.5s(^m )][s(^m) - s(t)] > 0 1, if [s(0 - 0.5 s ( £ m ) M ^ m ) - s(0] ^ 0 where tM is such that s(tM) = 0 and s(tM) represents the last stationary value of the s ( 0 function. The corresponding sufficient conditions for finite time convergence to the sliding Chapter 3. Higher Order Sliding Modes 46 0 0 S x 10"3 Figure 3.3: Twisting phase plot manifold are: a* 6 (0 1] n (o, (3.43) VM > max —— , — ------— 1 m 3i m Qi 1 M J o o S Figure 3.4: Phase plot of suboptimal algorithm Algorithm with prescribed law of convergence ofs In this algorithm [29], the sliding variable proceeds to the origin according to a pre-specified guiding function, g(s). Chapter 3. Higher Order Sliding Modes 47 where V > 0, the function g(s) is continuous everywhere except s = 0 and it is assumed that all the solutions of the equation s = g(s) vanish in finite time. For example g(s) = —Alsl1/ 2 sign(s), with A > 0 will serve the purpose. The phase plot is shown in Fig. 3.5. 0 Figure 3.5: Phase plot for prescribed law of variation algorithm Drift algorithm The drift algorithm [30] is different from the other second order sliding algorithms in the sense that the sign of s is constant until a certain vicinity of the origin is reached. This eliminates overshoot for s, see Fig. 3.6. A very important point with regard to the drift algorithm is that it ceases to perform its function if the switching delay is zero and sliding becomes ideal. Thus, the drift algorithm has only a real sliding interpretation. —u, \u\ > Uq v{t) = ^ —Vm sign(s) ss > 0, \u\ < u0 (3.45) —Vrn sign(s) ss < 0 , |u| < Super-twisting algorithm The super twisting algorithm [56] has only been designed for systems which have relative degree one with respect to the sliding variable and hence the sliding variable dynamics can be written as s = 0 (£,x) + 7 (2 , x)u Chapter 3. Higher Order Sliding Modes 48 0 0 5 Figure 3.6: Drift algorithm phase plot with the bounding conditions \(j>(t,x)\ < $ o < r m < y(t,x) < Tm (3.46) |s| < So where Tm, TM, s0 and The super twisting algorithm when applied to system (3.46) converges in finite time. This defines a smooth control law, u(t), as the combination of two terms. The first is defined in terms of an integral of a discontinuous function of sliding variable, while the second is the continuous function of the sliding variable. The trajectories of the super twisting algorithm are characterized by twisting around the origin on the phase portrait of sliding variable, Fig. 3.7. The super twisting algorithm converges in finite time and is defined by the following control law: u(t) = ui(t) + u2{t) (3.47) where Chapter 3. Higher Order Sliding Modes 49 and the corresponding sufficient conditions for finite time convergence are [4]: $ W > > 0 A > (3.48) TUW - 4») 0 < p < 0.5 The algorithm does not need the sign of the time derivative of the sliding variable to provide 0 0 S Figure 3.7: Super-twisting phase plot smooth control. For p = 1, the algorithm converges to the origin exponentially. For systems where s 0 = °o and there is no bound on the control, the algorithm can be simplified as: u(t) = —A|s|p sign(s) +u\ (3.49) iii = — W sign(s) 3.6 HOSM vs. DSM The dynamic sliding mode control strategy is very similar to higher order sliding mode con trol. However, the main emphasis in dynamic sliding mode control is to utilise the classical first order sliding mode technique in a way that it eventually gives a smooth input signal by considering extended systems containing control derivatives as the new control. In higher order sliding mode control, the emphasis is on designing new control algorithms which pro vide a smooth control signal. These algorithms may or may not utilise the control derivative as a new control as in the super twisting algorithm [56]. Higher order sliding mode control as developed by Levant originates from a differential geometric understanding of systems which is in contrast to the differential algebraic point of view employed by Sira-Ramirez. Moreover, Levant’s synthesis of the controller is not straightforward. It relies upon intuitive methods. Chapter 3. Higher Order Sliding Modes 50 The dynamic sliding mode approach requires all derivatives of s to be available to the con troller and uses the derivative of the actual control as an auxiliary control which increases the closed loop system order. In other words, this approach is good for systems which are perfectly known. The convergence of the system states to the dynamic sliding manifold is generally asymptotic while Levant’s algorithms are finite time converging. Thus, a dynamic sliding mode strategy generates a higher order sliding mode by selecting sliding surfaces that involve integral action on the control which not only provide smooth chatter free control but also provide smooth sliding dynamics which are at least differentiable once. The algorithms presented by Levant and Bartolini provide smooth sliding surface dy namics but do not provide a smooth control action and hence do not remove chattering except in the case of the super twisting algorithm. To avoid chattering using higher order sliding mode algorithms, the dynamic sliding mode control philosophy needs to be employed. The following example clarifies this point. 3.6.1 An illustrative example Recall the double integrator example in equation (2.23) in Section 2.5.1 y = u where the sliding surface s = m y+ y was reached in finite time with the discontinuous input u = — sign(s). Dynamic sliding mode design Suppose that the following asymptotically stable closed loop dynamics are imposed on the controlled system: y = -miy - m2y (3.50) where (l,mi,m2) are Hurwitz coefficients. One ideally obtains the required closed loop behaviour whenever u = —miy - m2y hence s(y,y,u) = u + rai2/ + ra2y (3.51) qualifies as a candidate for the sliding variable and requires all the system states. To make it locally attractive, the reachability condition ss < 0 is satisfied by imposing on s a dis continuous dynamics of s = — W sign(s), W > 0. The following discontinuous differential Chapter 3. Higher Order Sliding Modes 51 equation is obtained u = —W sign(u + miy + m2y) — m iu — m 2y (3.52) whose solution will give the required input, u. The simulation results with m i = 2 and m 2 — 1 are shown in Fig. 3.8 to Fig. 3.11 for regulation of the double integrator plant. The output variable (y) goes to zero asymptotically i.e., not in finite time. The input signal is not only smoothed out but also its amplitude reduced considerably. This is particularly important in situations where the control amplitude is restricted. However, the response is slow when compared to the static sliding mode. This example also shows the possible application of sliding modes for systems of relative degree zero, where the output explicitly depends upon input. Ui 0.5 —0.5 Figure 3.8: System states 0.6 0.4 0.2 0 - 0.2 -0.4 0 1 2 3 4 5 6 7 8 9 10 Figure 3.9: Sliding variable Figure 3.10: Smooth input Chapter 3. Higher Order Sliding Modes 52 0 1 2 3 4 5 6 7 8 9 10 Figure 3.11: Auxiliary input Design using the second order sliding algorithm Now consider the same double integrator problem and apply one of the finite time converging second order sliding mode algorithms presented in Section 3.5.2. The twisting algorithm (3.40) will be employed with the parameters u0 = 2, Vm = 2, Vm = 0.8. As the double integrator dynamics is already in 1-0 form, the output variable, y, itself quali fies as a sliding variable. It is evident in Figure 3.12 that the system states converge to zero 0.5 to -0.5 -1.5 time, sec Figure 3.12: The system states: converging in finite time 01 2 3 4 5 6 7 8 9 10 time, sec Figure 3.13: The discontinuous input in finite time. The control action ( u) is discontinuous and it can not be said that chattering is removed. In conclusion, higher order sliding mode control does provide a smooth sliding dynamics but remains short of providing chatter free control except in the case of the super twisting Chapter 3. Higher Order Sliding Modes 53 control. Of course, chattering can be removed by using the control derivative as an auxiliary input similar as in the dynamic sliding mode control. Thus chattering removal is not the property of higher order sliding mode but it is due to the control integration involved. In other words, higher order sliding mode control involving control integration provides a chatter free response and nothing can be said about simple higher order sliding mode control with respect to chattering removal. 3.7 HOSM reachability condition In classical sliding mode control, after selecting a suitable sliding surface s, it is necessary to select a reachability condition so that the sliding surface is reached. One choice of reachabil ity condition is ss < 0 , for s^O, where the derivative is taken along the system trajectories. This condition only allows the system to reach the sliding surface asymptotically, i.e., in infinite time, which is not desirable. Another, stronger reachability condition is the so called 77—reachability condition which is given by S S < —77|s|, 77 > 0 (3.53) This condition allow the system to reach sliding surface in finite time t r In the case of higher order sliding mode control, the above mentioned reachability condi tions do not hold and need to be appropriately modified. In Sira-Ramirez [91], the con cept of higher order sliding reachability conditions essentially means that a new sliding sur face containing the derivatives of the first surface is reached in finite time using equation (3.53). Consider a continuous sliding surface s which has r — 1 continuous time deriva tives (s, s, • • • , s(r_1)). This actually means that r th—order sliding is possible. Now another variable o is defined such that r —1 G ~ 'Yh aiSi (3.54) i = 0 where, a* > 0 and the vector [ar_i, • • • ,a2,ai,a0] is Hurwitz. Then the 77—reachability condition (3.53) when applied on < 7 , a new sliding manifold, provides a finite time to reach the manifold a — 0. However, the actual sliding variable s will go to zero asymptotically, governed by the stable linear differential equation r — 1 (3.55) A terminal sliding mode based reachability condition, presented here first time, converges in finite time in contrast to the above. Chapter 3. Higher Order Sliding Modes 54 Let s be a sliding surface with (r — 1) smooth time derivatives. Consider also the state transformation [s, s, • • • , s^ ] = [s0, s i, • • • , sr]. An r th—order sliding reachability condition is defined as si+1 = ^ + ki\si\pl sign(s^) i = 0, • • • , r — 1; pi G (0,1) (3.56) sr = — kr\sr\pr sign(sr) (3.57) In fact the condition on p for differentiability of the term \s\p sign(s) is given in [87] as r < p < 1 (3.58) r + 1 which helps to define a condition of each pi as follows r — i < pi < 1 ; i = 1 , • • • ,r. (3.59) r — i + 1 The coefficients ki are appropriate positive constants. One can easily see that sr reaches zero in finite time following equation (3.57). Once si+1 = 0, the dynamics become Si + ki\si\p sign(sj) = 0 (3.60) It is proved in [17, 117] that the solution S i ( t ) of the equation (3.60) will reach zero in finite time, i.e., the sliding surface s and its r time derivatives will be zeroed in finite time. If sr = 0 is reached in finite time t r given by the sliding surface s = 0 will be reached at r-1 1 t s = t r + 'S ' ------—------r | s r_ i ( i r + i _ i ) | 1 Pr_l k r —i\ 1 P r —i) where ti is the reaching time of s* = 0 . 3.8 Conclusion In this Chapter, basic notions related to higher order sliding mode control have been pre sented. The approach has removed some of the basic limitations of classical sliding mode design such as the requirement that the system model should be in the regular form which is affine in the control and the relative degree of the sliding variable must be one etc. General ization of the sliding mode concept to higher order sliding modes appears to be natural. The sliding surface is no more restricted to be a function of states but input dependent sliding surfaces proved to be advantageous with respect to the elimination of chattering. Chapter 3. Higher Order Sliding Modes 55 Recently developed 2-sliding, finite time converging, algorithms for SISO systems have been reviewed. As nonlinear systems which are differentially flat can be linearised by input de pendent transformation and dynamic state/output feedback, therefore control algorithms de veloped for SISO systems can be extended for MIMO flat systems with some effort. Indeed, Bartolini et al. [ 6 , 10] have extended the suboptimal 2-sliding algorithm for second order systems and applied it for control of manipulators. For standard sliding mode control, the reachability condition ss < 0 which is due to Lya punov’s stability criterion is utilised for designing the control. However, such a reachability condition is not available for a control problem involving higher order sliding modes. A finite time higher order reachability condition based on terminal sliding mode control has been presented first time. Second order sliding mode control is the simplest case of higher order sliding mode control. The algorithms reviewed in the Chapter require the knowledge of certain time derivatives of the sliding variable ( s) to implement higher order sliding mode controllers. In the case of second order sliding mode control, the knowledge of s or at least its sign is required. The super-twisting 2-sliding algorithm which has been designed for systems with relative degree one with respect to the input produces smooth control. In the next Chapter, its ap plication for the robust speed control of a diesel engine is presented and some suggestions made to extend it for relative degree two systems. Ch a p t e r 4 HOSM Case Studies 4.1 Introduction In Chapter 3, different higher order sliding algorithms, particularly second order sliding algo rithms, were reviewed. All algorithms designed for second order systems require the knowl edge of the derivatives of the sliding variable to implement the resulting higher order sliding mode control. The super-twisting algorithm by Levant [56] does not require the derivative of the sliding variable to be measured but it was developed and analysed for systems with relative degree one with respect to the input. In [41], the super twisting algorithm has been applied for robust speed control of a diesel engine. The algorithm parameters are selected using a new tuning algorithm rather than by using the uncertainty bounds of the system. However, the work presented in this chapter incorporates the system bounds for selecting controller parameters. In this Chapter, the super-twisting algorithm is discussed in detail. Section 4.3 presents the results of speed control of a diesel engine-generator set. The implementation results show robustness to various step loads. However, if the super twisting algorithm is applied to systems of relative degree two, it develops limit cycle behaviour and does not converge into the second order sliding set s = s = 0 as shown in Section 4.4. Section 4.5 presents a simulation case study carried out for the speed control of a second order internal combustion (IC) engine model. It will be shown that the limit cycle behaviour can be minimised if the controller parameters are selected outside the previously published permissible limits. Section 4.6 presents the application of dynamic sliding mode control for MIMO engine con trol. Dynamic sliding mode control produces asymptotically stable higher order sliding mode control. In the case of engine control, where the first derivative of the control input is in volved, dynamic sliding mode is equivalent to asymptotically stable second order sliding mode. 56 Chapter 4. HOSM Case Studies 57 4.2 Super-twisting algorithm The super-twisting algorithm as discussed in Section 3.5.2 has been developed for uncertain SISO nonlinear systems, affine in the control u. x = f{t,x{t)) + g(t,x(t))u(t)\ where x E X C is a state vector, u £ U C 3ft is a bounded input and t is the independent time variable. The sliding surface s(t, x) is selected such that by zeroing it, the control objective is achieved. The system dynamics can thus be written as s = 0 (£, s) + 7 (t, s)u (4.1) The dynamics in equation (4.1) are assumed to satisfy the bounding conditions. \(f)(t,x)| < $ 0 < Tm < 7 (t, x) < Tm (4.2) |s| < s0 where Tm, TM, s 0 and $ are some positive constants. The following super twisting algorithm stabilises the system dynamics (4.1) to zero in finite time u(t) = ui(t) + u2(t) (4.3) where —u, \u\ > 1 iii — < —VKsign(s), \u\ < 1 —A|s0 |psign(s), |s| > so u2 = < —A|s|psign(s), \s\ < so The sufficient conditions for finite time convergence of equation (4.1) are given in [56] as $ W > — > 0 -1* m 4$Tm(^ + $) A2> (4.4) - *) 0 < p < 0.5 In the following Section, this algorithm has been applied for speed control of a diesel engine- generator set, which has first order dynamics, and is subjected to various electrical loading conditions. Chapter 4. HOSM Case Studies 58 4.3 Diesel engine speed control The speed control of a diesel engine acting as an electrical generator is required to be robust as the electrical load applied varies depending upon the user demand. The electrical load on the engine is not known although the limits on the load are known and engine speed is available for control. The engine speed can be robustly controlled using classical sliding mode control but high frequency chattering at the input makes it unsuitable. The super twisting controller produces a smooth input and is thus applied to robustly control the speed in such a diesel Genset [51]. Moreover, the algorithm does not need an exact model of the system but requires only uncertainty bounds. The system to be controlled is a 1000 series Perkins four cylinder, four litre, tubo-charged diesel powered engine which is coupled to an electrical generator. The maximum generated power is 65 kW and this is dissipated via an electrical resistor load bank. The nominal engine speed is 1500 rpm. A schematic diagram of the hardware configuration for the implementation is shown in Fig ure 4.1. The speed information is obtained from the magnetic pickup pulse of the teeth on the generator flywheel and then converted to revolutions per minute (rpm). The control sig nal is converted into a PWM signal at 400 Hz before it is amplified to 12 Volts which is the required current level to drive the electrical actuator. Reference Super-twisting Signal speed EngineActuator controller processing unit speed Speed conversion Speed sensor in rpm Figure 4.1: The schematic diagram of the hardware setup for controller implementation The controller is designed in the MALAB/SIMULINK environment and is then converted into C-code for proper dSPACE implementation using a Texas Instrument TMS 320F240 DSP micro-controller. This micro-controller is connected with the diesel engine to control the speed. The model of the diesel engine experimentally identified in [41] is adopted- -0.0793 -0.2271 -98.3846 X = x + u (4.5) 0.1327 -0.0145 -22.3471 0.4119 0.2625 x where y represents the engine speed in rpm. The state vector, x, does not represent any Chapter 4. HOSM Case Studies 59 physical engine variable. The control, u C U G [0 1] represents the duty cycle of the PWM signal (12 Volts, 400 Hz) which drives the electrical actuator. The system is BIBO stable i.e., the bounded input will produce bounded output because both poles are on the left hand side of the s—plane and hence for a bounded input, the states are upper bounded. The system states are transformed such that the speed becomes one of the states using the linear transformation z = Tx, where z is the new state vector and T is given as follows 0.4119 0.2625 T = 1 0 The transformed system can be written as i = Az -f Bu (4.6) y = Cz where 0.0053 -0.0987 -46.3941 r -i A = B = and C = 1 0 0.3222 -0.0990 -22.3471 The state transformation makes the measured output one of the states and hence leaves only one state to be observed. This reduces the size of the inclusion enveloping the uncertain system. The parametric uncertainties in the system may be understood to be represented in terms of the variation in the coefficients of the matrices A, B and C over a suitable range. 4.3.1 Controller design and tuning To apply the super twisting algorithm (4.3), the engine model has to be written in the format given in (4.1) as follows y — CA(:, 1 )y + C*A(:, 2)z2 + C Bu + 77 (4.7) As y is measurable, the uncertainties lie only in the evolution of the second state z2. Here the additional term, denoted 77, represents the effect of electrical load on the engine speed. This representation requires one state to be observed and the bounding value of 77 to be known. As the states are bounded, instead of observing the state, its bounding value is calculated. Once the bounding value of the state z2 is known, the system can be written in an inclusion form y G CA(:, l)y + [ - $ 4>] + [Tm VM]u (4.8) where 4>, Tm and TM are constants and can be calculated using interval mathematics [1] such that | CA(:, 2) + 771 < and Tm < \CB\ < T M- Chapter 4. HOSM Case Studies 60 It is assumed that the error in the engine speed does not increase more than 700 rpm i.e. s0 = 700. The coefficients of the vector B are allowed to vary within a range of ±5%. The effect of various step load disturbances on the speed has been practically evaluated and it has been found that the bounding value of rj is 700 rpm/sec. The bounding values calculated by maximising possible uncertainties are $ = 1.49 x 103, Tm = 93.46 and TM = 103.30. The controller selected on the basis of these bounds will be conservative as all uncertainties generally do not occur together and if that situation occurs, they do not simultaneously attain their maximum values. The actual controller applied to the system is as follows u = — Ls — A|s|°'5sign(s) + U\ (4.9) iii = — V^sign(s) The additional term — Ls is to cancel speed dependent term in the dynamics (4.8). However, final value of L is selected during implementation. The sufficient values of controller coef ficients selected using (4.4) are A = 0.87 and W = 14.45. The estimate of controller gains by equation (4.4) usually turns out to be very conservative and the controller stabilises the system for lesser gains. Exact values of the controller gains are obtained by suitable tuning. There are only two controller gains available for tuning, namely A and W. Before tuning starts, both A and W are set to a small value. The tuning starts with consideration of W. The gain W is increased gradually until oscillations appear in the speed signal. The value of W is then decreased until the speed reaches steady state. The A is tuned in the same way. The values of W and A were found to be 0.01 and 0.0064 respectively. L is tuned in such a way that acceptable performance is achieved. The gain is set to a small value such as 0.0001 and gradually increased until the required performance is achieved at 0.0005. 4.3.2 Speed response to load change The dynamic behaviour of interest is the speed change with load. An electrical load (i.e. re sistive load) is applied to the system during steady state conditions and the resulting speed response is studied. The tests are carried out in the presence of two different loading condi tions; first a large step load of 60 kW is applied at once and then the 60 kW load is applied in steps of 20 kW and then reduced in the same step-wise fashion. The large step load of 60 kW corresponds to a sudden high power demand by the consumer and the smaller loads of 20 kW simulates the situation of varying demands by the consumer. The applied load is limited to 92.3% (i.e. 60 kW) of maximum load (i.e. 65 kW). Figure 4.2 Chapter 4. HOSM Case Studies 61 shows the results at 1500 rpm to a step load change of 60 kW. The super twisting controller settles down in about 5 seconds. The speed drop is of the order of 30%. In Fig. 4.2(b), it is seen that the control signal reaches its maximum value to recover the engine speed. During the recovery phase, the engine speed shows overshoot of about 3.4%. 1600 1500 1300 1200 1100 1000 Tim e, s (a) Speed response to 60 kW step load H 0.6 0.4 ">i» mi,|ir"n (b) Control signal at 60 kW step load Figure 4.2: Speed and control signal response to step load of 60 kW at 1500 rpm Figure 4.3 shows the engine speed response when the load is increased to 60 kW in steps of 20 kW and then decreases in the same fashion. The speed recovers within two to four seconds. The speed recovers quickly with a maximum overshoot of 75 rpm. The engine runs continuously without exceeding the allowable upper speed limit during speed overshoot. Overall, the super-twisting controller performs well when subjected to large and small load disturbances. The speed recovers promptly with no steady-state error in the speed signal. 4.4 HOSM and unmodelled dynamics Higher order sliding mode control is robust against bounded matched uncertainties. The robustness of second order sliding mode control has been studied in Blom and de Jager [15] by implementing a second order control for tracking trajectories on an XY-table. It was reported that robustness to unmodelled dynamics is not better than with PD-control. However, the second order sliding mode controller performs slightly better in the case of Chapter 4. HOSM Case Studies 62 I , . , T - — , , 1700 1600 ...... 1500 V ._ ____ K ...... r I T : ' 1400 .1300 1200 1100 1000 Time, s Figure 4.3: Speed response with step load of ±20 kW parameter errors. In Van de Wall et al. [108], it has been reported that a classical sliding mode and a dynamical sliding mode controller give similar results as far as errors in the model of the manipulator is concerned; neglected flexibilities were included. However, classical sliding mode control gave better nominal performance. In case of measurement noise, the sliding variable does not converge into origin but in a neighbourhood it. In Brown et al. [17], it has been reported that for smooth second order sliding mode control this region is bigger than classical sliding mode control but with reduced control magnitude. To investigate the effect of unmodelled dynamics on the super-twisting control algorithm, consider the following first order system to be controlled x = (f>(x) + r(x )u with an actuator having first order dynamics v = 7 (u) + rj(v)u + d{t) where d{t ) represents a bounded external disturbance. The complete dynamics of the system with actuator can be written as d4> o r x = dx + ~dxv * + r^)^) + r^bM + v(v)r(x)u which may be represented by a second order SISO system of the following type X\ = x2 = f(xi,x2,t) + g(x1,x2,t)u (4.10) y = x i Chapter 4. HOSM Case Studies 63 where 0 < |/()| < F and 0 < < g(-) < Gmax are uncertain, bounded functions with known bounds. The time derivatives of / and g, namely / and g, are also assumed bounded. This type of system appears naturally due to the presence of actuator or sensor dynamics [119]. It is required to stabilize the output y of this system using a sliding mode control with the condition that neither measured nor observed x2 is available to the controller. The system output y can be considered as a suitable sliding variable s. It has been noticed that the super-twisting algorithm is not robust against such unmodelled dynamics. A simple example is simulated here to support this claim. Three cases for the term f(x, t) in equation (4.10) are considered: f(x, t) = 0: The system model becomes similar to that of double integrator. f(x, t) = a constant: The system has a constant matched uncertainty of f(x, t) = 2 with its derivative equal to zero. f(x, t) = a time varying function: The system has a matched time varying uncertainty of f(x, t) = 2 sin(i) which has a bounded derivative. The controller parameters are selected as A = 7, W = 3, k = 1 and p = 0.5 respecting the sufficient values. The simulation is carried out using a step size of 0.1 millisecond. The simulation results show limit cycle behaviour for all three cases in Fig. 4.4(a), 4.4(b) and Fig. 4.5. This limit cycle is due to the increased system order due to the unmodelled dynamics of order one. It will be seen in the Section 5.4.6 that the modified algorithm stabilise the second order systems to origin and does not show this limit cycle behaviour. Figure 4.5: Response for f(x, t) = 2 sin(£) Chapter 4. HOSM Case Studies 64 (a) for f(x,t) = 0 (b) for f(x,t) = 2 Figure 4.4: Responses for various drift terms In the next Section, a case study for IC engine speed control is presented to demonstrate that the algorithm (4.3) can be modified to suit systems where the sliding variable has relative degree two with respect to the input. 4.5 IC engine speed control In this case study, super-twisting 2-sliding control algorithm has been applied to an IC engine model. It is verified that the engine speed is an appropriate flat output and thus stabilizing this output alone will effectively stabilize the closed loop system. Error in speed is selected as an appropriate sliding variable and it has relative degree two with respect to the control input. Though the super-twisting algorithm has been developed for a system where sliding variable has relative degree one with respect to the control input, it will be applied here for a relative degree two problem. It is possible to minimise the steady state oscillations by selecting controller parameters which do not satisfy the original stability conditions given in equation (4.4). The engine model considered for simulation of the controller performance is adapted from Crossley and Cook [23] and the Mathworks [79] with the assumption that the stoichiometric air-fuel ratio (AFR) is controlled by another independent controller. 4.5.1 The engine model The model of an automotive engine presented by Crossley and Cook [23] is considered. All the assumptions made here are from the original source. The two main subsystems of the engine model are 1. The crankshaft dynamics Chapter 4. HOSM Case Studies 65 2. The manifold dynamics The crankshaft speed state equation can be written as Jhj 'Teng where n is the engine speed, reng is the torque produced by the engine, t\ is the variable load torque and J is the effective engine inertia. The torque reng is described by the following empirical function Ten g = ^e0 + &elm a + ^e2 (AF R) + ke3(AF R)2 + ke±6 + ke$Q2 + ke6n + ke7n2 + ke$n6 + keg6ma + kei092ma where kei, i = 0, • • • ,10; are constant coefficients, AFR is the air-fuel ratio and 6 is the spark advance i.e. The instant at which fuel mixture inside the cylinder is ignited or sparked. The variable ma is the air mass charged in the cylinder during the intake stroke, which takes place in the first tt radians crankshaft rotation of the four-stroke cycle. Thus, in the model, ma was obtained by integrating the air mass flow from the manifold and resetting the integrator at the end of each ingestion stroke. This results in a variable reset period depending on the rotational speed ( treset = ir/ri). Finally, it is known that in the actual engine, a delay exists between the ingestion of the air-fuel and the related torque production. Therefore, an induction to power lag of 7r radians was assumed and, consequently, a variable delay (it/n) was included in the model. However, for the purpose of controller design, the output of the integrator block with variable reset can be closely approximated by m ao7r rria = ------n The nominal load torque, comprising of rolling friction, engine friction and aerodynamic drag torques, can be expressed as a function of the speed, n as follows: Ti = Tr + Tf + Ta with rr = kn, Tf = ki2 + ki3n, ra = ki±n2 where, ku, (i = 1,- • •, 4) are constant coefficients. The intake manifold dynamics, modelled as a first order differential equation, is given by . _ RTm . . P m t / ( F l ai -f- Tfla o ) * m where R is the gas constant, Vrn is the manifold volume and Trn is the manifold temperature. RTrn/Vm is assumed to be constant. The air mass flow rate into the cylinders from the manifold rhai, is given by the following function of manifold pressure pmt and speed n Flao kjjioQ kmolTiPm T" k m o 2 T ip rn T k m o 3 P m Chapter 4. HOSM Case Studies 66 The air mass flow rate into the manifold rhai, is the function of manifold pressure prn and the throttle angle a, as follows rnai = f(a)g(Pm) where /(a ) = k t ho + kth iO t 4- k t h20t2 4- k th z o ? (4.11) , N I 1 Prn 5; 9 \p rn) = <______\ Patm \ / PatmPm Prn Prn --> 0•^Pa.tm The coefficients k moi and k th i, i = 0,- • •, 3 are constants determined by experimental data and p atm is the atmospheric pressure. Let the speed n, and the manifold pressure pm, define the state vector ,= M = (n \x 2) \p m and the control variable u = f(a). (4.12) Then the state space description of the system can be written as xi = x(x) (4-13) x2 = £(x)+x(x)u (4.14) where Ki . K2 ( krnoO . > 2 X(^) — 4“ T~ ( 4“ kmol'^2 4- kmo2 ^ 2 4- kmo3xlx2 u J ^ oc^ + J (K 3xi + ke7x\ - ki 1 - kn - ki3xi - kiAx\) t r ( kmo0 kmolx lx2 kmo2xlx2 ^,mo3x\x2) ^ rn i{x) = g{x2) with K\ = keo 4- ke2(AFR) 4- keg(AF R)2 4- ke±Q 4- ke^02 K 2 = 7r(kei 4- keq6 4- keio92) = keQ 4- keg6 Differentiating the first state equation (4.13) yields Xi = \kelxx - ki3 - 2ktAxi I x(x) j\ 2 H ~j~ (kmo\ “I- <^‘kmo2x2 ~t~ krno2,X\)£(x) K 2 7~ (kmol “I” 2/cmo2^2 “1“ kmo3xl>) r)(x) Remark 1: The engine model considered is differentially flat with speed as the flat output [39, 78]. From equation (4.13), it is possible to write x2 as a function of x\ and xi as follows. k m o 2 ^ 2 x \ + (k m o l + k rno3x l ) ^ 2 x 2 + K\ + The equation has two solutions but only the smaller one is physically acceptable. Similarly, eliminating x2 from equation (4.15) and using equation (4.14), u can also be written as a function of x\, x\ and x\ . Remark 2: Due to the flatness of the system demonstrated above, it is sufficient to stabilize only the flat output which in this case is engine speed. The other variables are automatically fixed provided the reference trajectory has been designed keeping the constraint in mind and does not pass through any singularities, such as zero speed in this case [39]. 4.5.2 Controller design Engine control is a real-life control problem. The existence of unknown torque disturbances and parameter variations, which in turn affect the speed, is an important issue to be consid ered. Increase in load torque results in a dip in the speed and vice-versa. Thus, the problem is to stabilize the engine speed, at the desired speed level, x\d- The engine model has been given in equations (4.13), (4.14) and (4.15). Consider the sliding surface as the speed error, s = x 1 - xld which has relative degree two with respect to the control input and satisfies the following second order differential equation s = (f)(x) + rj(x)u (4.16) where Although the super-twisting controller is not designed for the kind of problem in equa tion (4.16), it has been implemented here to see if further tuning of the controller parameters, Chapter 4. HOSM Case Studies 68 which do not satisfy the stability conditions, will stabilise the system. The control law does not require any information on the time derivatives of the sliding variable, s, and no explicit knowledge of other system parameters. This minimises the number of sensors required for implementation. 4.5.3 Simulation results This section illustrates the performance of the controller through simulation results. The model considered here is the The Mathworks benchmark model [79]. Simulations are carried out for the speed starting from 300 rad/sec to 450 rad/sec and then back to 300 rad/sec. Within the range of operation the system bounding values are calculated as 4> = 1.17 x 104, Tm = 219.17 and Tm = 1.15xl03. According to the conditions of stability in equation (4.4), W and A should be greater than 53.38 and 2.27 respectively and p = 0.5. Figure 4.6 shows the simulation results for W = 54,A = 2.3 and p = 0.5. The actual input a is calculated online by solving the third order algebraic equation (4.12) which has only one practically acceptable solution. Different initial conditions chosen are: n(0) = 300, Pm(0) = 0.6 and «i(0) = 16. It can be seen that there are oscillations around both the speed set points and the severe oscillations at the 400 rad/sec are unacceptable. Moreover, when a load of 50 Nm is applied, the engine become unstable when it is removed as shown by dotted line. 500 410 2 300 400 ,200 390 14.5 100 lime, sec Figure 4.6: The engine response To obtain good simulation results without oscillations, the controller coefficients are further tuned as W = 6, A = 2 and p = 0.5 which do not satisfy the conditions given in (4.4). The improved simulation results shown in Fig. 4.7 and 4.8. In the simulation it can be seen that the controller shows robustness in the presence of initial condition errors, parameter variations and unknown external disturbances. Chapter 4. HOSM Case Studies 69 500 450 § 4 0 0 350 300 250 Tim e (sec) Figure 4.7: Rotational speed, n(rad/sec 200 V) -100 -200 20 Tim e (sec) Figure 4.8: Sliding surface, s Uncertainties and disturbances have been taken in to consideration in the simulation, firstly, by modifying open loop parameters such as rotational moment of inertia J, spark advance 6 and optimal stoichiometric air-fuel ratio AFR, up to 20% from their nominal values (see Fig. 4.12), secondly, by introducing unknown load torque variations (see Fig. 4.9), i.e. step change of constant power load at different speeds representing activation and deactivation of constant power electrical appliances such as the compressor of the air-conditioning unit and filtered white noise representing additional random torque disturbances. 2 .5 1------1------1------1------1------ I 2 - ...... I ^...... "I...... ;...|...... - *o<0 o t 1-5 - ...... - ofo-5-...... - 0I ----1------1------1 ------1------0 5 10 15 20 25 Tim e (sec) Figure 4.9: Unknown disturbances due to de/activation of car appliances (kW) Variations in the nominal value of the manifold temperature (Fig. 4.11) were also included. The step change of constant power load (2.2 kW) is at £ = 3.5,12,16 and 23 sec (see Fig. 4.9). As can be seen in Fig. 4.7, there is no significant jump or dip at these points. Chapter 4. HOSM Case Studies 70 ".60 -J 40 Tim e (sec) Figure 4.10: Load Torque, T} (Nm) -20 Tim e (sec) Figure 4.11: Unknown % variations in manifold temperature, T„ 3a 25 si • | 20 & u 15 g wwwww^ £ 10 * 10 15 25 Time (sec) Figure 4.13: Throttle angle, a (deg) In the next section dynamic sliding mode control is applied to a MIMO engine control prob lem which produces asymptotically stable second order sliding modes. This dynamic sliding mode control requires an exact model of the system to be controlled and be available at the design stage, but it does not need the system model to be affine in the control. c -20 c -30 -40 Tim e (sec) Figure 4.12: Unknown % variations in air-fuel-ratio ,AFR Chapter 4. HOSM Case Studies 71 g 0.4------1------'------'------i------2 0 5 10 15 20 25 Tim e (sec) Figure 4.14: Manifold pressure, Pm 4.6 Engine control using DSM control The objectives of a typical control system for an internal combustion (IC) engine are not only to maintain the desired engine speed but also to satisfy regulations relating to exhaust emissions despite the ever-present torque disturbances. Moreover, the control should be robust for engines operating with different aging history of the components under vastly different environmental conditions. Sliding mode control, being a robust technique, is a natural candidate for engine control. This case study uses dynamic sliding mode control for MIMO control of an IC engine and produces asymptotically stable second order sliding modes. Throttle opening (a), spark advanced) and injected fuel mass flow m are the three control variables. The throttle provides large authority but is relatively slow while the spark advance provides a much faster actuation path but with limited authority. A good controller has to utilize both judiciously. The injected fuel mass flow rate has been used to stabilize the air- fuel ratio at the stoichiometric value where catalytic converter has high efficiency. A three stage Mean Value Engine Model (MVEM) [45] of an IC engine is used, as the model used in the previous section [23, 79] is only a SISO model for engine speed control. The MVEM model is not affine in control. It is first represented in Local Generalized Controller Canonical Form (LGCCF) [35, 89] for the design purpose and then an input dependent dy namic sliding surface approach [70] is applied, which produces asymptotically stable second order sliding modes. The designed controller is valid over the entire operating envelope. The air-fuel ratio is stabilized to its ideal stoichiometric value of 14.67, even during rapid throttle transients, and speed and manifold pressure track given trajectories. The MATLAB® simula tion of the controller presented includes noise in the air-fuel ratio measurement and unknown load torque disturbances. A sinusoidal speed demand of varying frequency and amplitude has been used to test the controller performance across a range of different operating regions. Chapter 4. HOSM Case Studies 72 4.6.1 Mean value engine model (MVEM) There are a number of IC engine dynamic models in the literature but the Mean Value En gine Model (MVEM) developed by Hendricks et al. [44], Hendricks and Sorenson [45] is mathematically compact and can be parameterised for different engines easily. Therefore, the engine model adopted in this case study is the one developed in [44]. The MVEM en gine model consists of three subsystems: the fuelling system, the air flow system and the crankshaft dynamics. These subsystems are described briefly as follows. Fluid film flow model Fluid mass flow through the manifold has two components namely fuel vapour mass flow rhfv and fuel film mass flow rhff and the total mass flow m / is not measurable. The dynam ical sub-model is given as rhff = — (-m ff + Xmfi) (4.17) Tf rhfv = (l-X)rhfi (4.18) rhf = ihff + rrify (4.19) where r h is the injected fuel mass flow and X is the fraction of mfi which is deposited on the manifold as fuel film mass flow, rriff. The portion of fuel which evaporates is termed as fuel vapour mass flow and is denoted by rhfv. The fuel evaporation time constant is denoted by Tf and various mass flows are in kg/s. The measurable quantity in this sub-model is the air-fuel ratio at the exhaust. The normalised air-fuel ratio (A) is a nonlinear function of system states and is given by following equation A = h{x) m'' vap (4-2°) \hhif where Xth = 14.67 is the stoichiometric air-fuel ratio. Crankshaft speed state equation This sub-model is derived using energy conservation laws. The dynamic equation governing the crankshaft dynamics is given by h = j^ (Hu Vihif(t - Td) - Pi - Pb) (4.21) where n, the crank shaft speed, is in krpm, pif the manifold pressure, is in bar, Hu, the fuel heating value, is in kJ/kg and I is the total moment of inertia which can be converted suitably Chapter 4. HOSM Case Studies 73 so that the speed equation (4.21) is given in rpm as follows where Iac is the total (engine + load) moment of inertia in kg-m2. Loss power (Pi) and load power ( Pb), both in kW, can be expressed as nonlinear functions of speed and manifold pressure as Pi(n,pi) = n(ki + k2n + k3n2) + n(-k4 + k5n)pi (4.22) Pb = kbn3 (4.23) The indicated efficiency (r?*) is the product of various efficiencies and is given by Vi = PinPipPixme. (4.24) The various efficiencies in (4.24) are nonlinear functions of the system states and input as given below Vin(n) = k6( 1 - k7n~0-36) ViP(Pi) = k8 + k9pi - kw p2 ViX — —^ii + ki2X — &i3 A2 Pie = e~d2/26™bt where 0 is the spark advance in degrees and 0mbt is the maximum break torque spark timing which is given by 6mbt = min(min(0i,02),45) 01 = k\4Pi + &15 + 7I47 02 = &16 Pi + ktf + n 47 4.7 n, if n < 4.8 7747 — ^ 4.7 x 4.8, if 77, > 4.8. with ki as constants. Manifold pressure state equation This subsystem governs the air mass flow flowing through the manifold and is derived from the gas equation. The corresponding dynamic equation is given by Chapter 4. HOSM Case Studies 74 where rhap, the air mass flow rate into cylinder (kg/sec), is a nonlinear function of crank-shaft speed (n) and manifold pressure (pt) as follows Vd mn„ = ■(kisPi + k19)n ap 120 RTi The air mass flow rate past the throttle plate, rnat, is a function of manifold pressure (jpi) and throttle opening (a:) as P a 'That = mati—£=/32(pr) Pi (<*) J\ / CL = k2Qp2{Pr)Pl(0t) where Pi{a) = 1 — cos(a — a 0) i f P r > P c *<»>-< else where R is the Gas Constant, a0 is the closed throttle angle in degrees and pr{= Pi/Pa) is the pressure ratio. The atmospheric pressure (bar) and temperature (°K) are denoted by pa, Ta respectively and p c , K are constants. 4.6.2 Input-output (I-O) representation The IC engine model is a nonlinear, MIMO, coupled system having three states x E 5ft3 and three control inputs u G 5ft3 and three outputs y G 5ft3 given as follows A h /\ rhfi ' x = n u = n a J \ P i j w The system model can be written in the following controller canonical form x = f(x) + 9(?)u{t) (4.26) where, ( ( - X i / T f + U i / t^ Ui \ / w = u{t) e~e2/2 e2mbt J \ l — cos(a — o;0) j V (gi o o \ l \ - x 0 0 g(x) 0 92 o 0 ~pP^HuVinVipPi\x l 0 \0 0 g3J \ 0 0 ^Fgk,2Qp2{x?>)) Chapter 4. HOSM Case Studies 75 The error dynamics associated with the trajectory tracking problem can be written in 1-0 form as 3 ei = 0i = Lfh + ^TLg-hui (4.27) i— 1 &2 = 02 ~ 02 — %2d + g^U2 (4.28) ^3 = 03 — 03 ~ %3d + / A — 1 \ e = ^2 — %2 ~ %2d \e3J \X3 - X3dJ where x2d and x3d are the desired trajectories of speed, x2 and manifold pressure, x3. The system model is dynamic feedback linearizable [110] and in proper 1-0 form because • The number of inputs and outputs are equal; • 0i, 0 2 , 0 3 are C1 functions; The regularity condition <9(01, 02, 03) det 7^0 d{iLi,U2,U3)_ is satisfied in the operating range. Standard sliding mode control can be applied and the control can be obtained but chattering will occur which is unacceptable in mechanical systems like the engine. The first equation involves the input derivative while the other two do not. The idea here is that standard sliding mode control will be applied to the air-fuel channel while a dynamic sliding mode control strategy will be used for the other two channels to get smooth control. 4.6.3 Controller design The sliding surfaces chosen for the three channels in equations (4.27),(4.28) and (4.29) are Si = ei, (4.30) S 2 = <22^2 + 02, (4.31) s3 — a 3e3 + 03 (4.32) and the following decoupled reachability conditions have been selected Sj — TCjSj fcotSign(si), i — 1,2,3. (4.33) Chapter 4. HOSM Case Studies 76 A/F-ratio channel Taking the derivative of equation (4.30) along the trajectories of (4.27) yields, 3 Si |(4 .27 ) ~ Lfh + E L9ihui + Lgihui (4.34) i—2 The zero dynamics Lgih iii = 0; Lgih = (1 — X) > 0 is asymptotically stable. From equations (4.33) and (4.34), the derivative of the first control is obtained as = T~T I - faisi - Lfh - V ' Lgihui ) (4.35) L« h \ s / Speed and manifold pressure channel The zero dynamics associated with the remaining two channels are g2u2 = 0 and g3u3 = 0 which is stable in the range of operation. Taking derivatives of equation (4.31) and (4.32) along system trajectories yields Sf |(4 .28 ,4 .29) — (-^/i T 'U/iLgi') From equation (4.33) and (4.36), the derivative of the other two control variables (auxiliary controls) are solved as iii — ( fcoisign(s^) {^Lji~\~'iLiLgi^(f)i fi giUi~\~Xid~\~Xid}lgi'LL^.f i 2,3. (4.37) where, M g h df2 9 x i 9x2 9 x 3 9 /3 M djs _ 9 x i 9 x 2 9 x 3 _ 9 g 2 9.92 9.92 9 x i 9X2 9 x 3 9,93 §33. 9.93 _ 9 x i 9x2 9X3 _ 96? du2 9 m 3 _9 u 3_ The controller is implemented by integrating the auxiliary control obtained in equations (4.35) and (4.37). The closed-loop system dimensions are equal to the open loop system dimension plus the dimension of the controller, i.e., 6 in this case. Chapter 4. HOSM Case Studies 77 4.6.4 Simulation results The simulation results presented here are for a 1275 cc British Leyland engine for which data is published by Hendricks et al. [44]. The injection-time torque delay rd has not been considered. Normalized air-fuel ratio has been stabilized to 1 and the desired trajectories of speed and manifold pressure are as follows: rid — 1.5 + 0.1 sin(£) + 0.05 cos(1.5£) Pid = 0.9 + 0.06 sin(1.5£) The initial conditions for the system states are (1.2 x 10-3, 1.55,0.9) and that of the dynamic controller are (2.1 x 10~3,25.5,35). The design constants a2 = a3 = 29, K = 225 and k0 = 5. From the simulation results, it is evident that the inputs are smooth and well with in the effective range of operation. 1 .1 ------1------1------1------1------1------1------1------1------1------ 1.05 - g o 13 1 — “ —1 ~ “ 1 “ 1 u. 0.95 - 0.9 L______i______i______i______i------1------1------1------1------1------J 0 2 4 6 8 10 12 14 16 18 20 time (sec) Figure 4.15: Tracking air-fuel ratio A 1800 1700 1500 1400 1300 0 2 4 6 8 1012 14 16 18 20 time (sec) Figure 4.16: Tracking speed n 1.05 0.95 (3 ■Q 0-' 0.85 0 2 4 6 8 10 12 14 16 18 20 time (sec) Figure 4.17: Tracking manifold pressure pm Chapter 4. HOSM Case Studies 2.2 0 2 4 6 8 10 12 1416 18 20 tim e (sec) Figure 4.18: Injected fuel mass flow, u\ = rhft (kg/sec) 40 35 30 25 20 0 2 4 6 8 1012 14 16 18 20 tim e (sec) Figure 4.19: Spark advance angle, 9, (degree) 44 42 40 38 a 36 34 32 30 0 2 4 6 8 10 12 14 16 18 20 tim e (sec) Figure 4.20: Throttle plate angle, a (degree) 18 10 0 2 4 6 8 10 12 14 16 18 20 tim e (sec) Figure 4.21: Max. Brake Torque angle, Qmbt(^ g-) Chapter 4. HOSM Case Studies 19 time (sec) (a) Sliding surface, si 8 10 12 tim e (sec) (b) Sliding surface, S2 “I------r~ ”i------1” 10 12 14 16 18 20 time (sec) (c) Sliding surface, S3 Figure 4.22: Sliding surfaces 4.7 Conclusions In this Chapter, the super-twisting algorithm has been applied for speed control of a diesel engine-generator set where the sliding variable has relative degree one with respect to the input. The controller shows robustness to unknown load changes. It has been demonstrated, however, that if it is applied to systems where the sliding variable has relative degree two with respect to the input, the system does not converge to the 2-sliding set s = s = 0 but develops limit cycle behaviour. It has also been demonstrated by a simulation study that further tuning of the parameters suppresses the limit cycle behaviour. However, the parameters finally selected do not satisfy Chapter 4. HOSM Case Studies 80 the original stability conditions. This provides an opportunity for redefining the conditions imposed on the controller parameters for stability and changing the controller structure if possible. A MIMO case study involving the application of sliding mode techniques to a multi input multi output IC engine model has been performed. The MVEM model of an IC engine is first represented in Local Generalized Controller Canonical Form (LGCCF) for the design purpose and then an input dependent dynamic sliding surface approach is applied. Undesir able chattering effects at the actuator are avoided by dynamic sliding surface design. The particular choice of the sliding surface is input dependent which produces a dynamic con troller. Such dynamic policies are desirable in sliding mode control as they effectively filter out the chattering of the control signal. This work extends the work of Weihua et al. [110] by designing a dynamic sliding mode controller for this MVEM. The designed controller is valid over the entire operating envelope. However, the method requires exact knowledge of the system model which is not practically possible. A sliding mode controller designed by neglecting the actuator dynamics causes chattering when implemented using the actuator. Most actuators can be modelled as a second order stable systems, even though the exact ac tuator dynamics are not known. The combination of dynamic sliding mode techniques and the super-twisting algorithm may have the potential to deal with this problem. In the next Chapter, a new second order sliding mode algorithm is presented to stabilise systems where the sliding variable, s, has relative degree two with respect to the input. The algorithm does not require knowledge of s and hence is simple to implement on an actual system. In the next Chapter, a new second order sliding mode algorithm is presented to stabilise the systems where sliding variable, s, has relative degree two with respect to the input. The algorithm does not require the knowledge of s and hence simple to implement on an actual system. Ch a p t e r 5 A N e w S e c o n d O r d e r S l i d i n g A l g o r i t h m 5.1 Introduction In Chapter 3, different higher order sliding algorithms, particularly second order sliding algorithms, have been reviewed. All except super-twisting algorithm require the knowledge of the derivatives of the sliding variable to implement the resulting higher order sliding mode control. Levant [56, 57] presented second order sliding algorithms to stabilise second order uncertain nonlinear systems but these require the knowledge of the derivative of the sliding variable, s, to implement them [58-60, 64]. The super twisting algorithm by Levant [56], however, does not require the derivative of the sliding variable to be measured but it has been developed and analysed for systems with relative degree one with respect to the input. However, this super twisting algorithm, if applied to systems of relative degree two, developed limit cycle behaviour and does not converge into the second order sliding set s = s = 0 as shown in Section 4.4. In addition to this, the conditions for controller parameter selection are no longer valid. Bartolini et al. [3, 5, 7-10] presented and applied an optimised version of the twisting algorithm. The algorithm still requires knowledge of the sign of the output-derivative however. Consider the following nonlinear system ±i = xi+i, i = l ... n - 1 (5.1) X n = f(x) + g(x)u where x = (a?i, • • • , xn) is the state vector and f(x) and g(x) are smooth vector functions of appropriate dimensions. The state vector is assumed measurable. The problem of stabilizing the system (5.1) without chattering has been addressed by Sira-Ramirez [89] who augmented 81 Chapter 5. A New Second Order Sliding Algorithm 82 the system in the following way ±i = xi+i i = 1, • • • ,n — l % n + 1 (5-2) ±U+1 = + J t9^ U + The sliding surface in the augmented space is selected as: n S = X n+1 + ^ 2 ° iXi (5-3) 2=1 where [1, cn, cn_i, • • • , c2, ci] are coefficients of a Hurwitz polynomial. As xn+i is not mea surable, an observer is designed to estimate it. The method requires the system to be fully known which is not the case when there are uncertainties in the drift function, f(x), and gain, g(x). Krupp and Shtessel [54], Krupp et al. [53] and Shkolnikov and Shtessel [86] have also pro posed a sliding mode control for this class of problem and assumed that the input is applied through an actuator. The following actuator dynamics are assumed = a(z) + b(z)v (5.4) u = z where a( ) and &(•) are some unknown smooth functions and r is the known order of the actuator dynamics. The sliding variable, a linear function of the error in the systems states, is selected as n s ^ ^ Cj Xi Xfa, (5.5) i= 1 where Xdi is the desired value of X{. The system dynamics (5.1) together with actuator dynamics (5.4) is then written in terms of the sliding variable dynamics as follows = E(t,x, z)+ F((t,x, z))v (5.6) where r is the order of the actuator dynamics. Equation (5.6) contains all the uncertainties of both models i.e., the system and the actuator. A sliding mode observer must be designed to estimate the derivatives of the sliding variable. Classical sliding mode control is then applied to stabilize the sliding variable s to zero asymptotically. The method does not avoid chattering at the actuator. Furthermore, asymptotic convergence of s does not guarantee asymptotic convergence of the system error (e). Levant has applied the super-twisting algorithm for the same class of system. As the super twisting algorithm produces a smooth input, chattering at the actuator is avoided. However, Chapter 5. A New Second Order Sliding Algorithm 83 an observer to predict the derivatives of the sliding variable must be designed. The method is generalised in [58]. Sira-Ramirez and Hernandez [93] have proposed the use of integral re-constructors for sta bilisation of such systems. The main idea is to predict the system derivatives by integrating the system output. The method is only applicable to differentially-flat-systems where the system input (or its derivative to any order) can be represented as a function of the system output and a certain number of its derivatives. As system (5.1) is an example of a differ entially flat system, the integral re-constructors can be used to estimate the system states. However, only a limited class of functions can be considered for the drift term, f(x), in the system (5.1) such as constants, ramps and parabolic ramp etc. Bartolini et al. [9] presented an improved sub-optimal algorithm [7] applicable to second order systems. The algorithm uses the time derivative of the control as an auxiliary control and does not require the derivative of the sliding variable and hence does not require the use of an observer. It acts on the knowledge of the stationary values (local minima or maxima) of the sliding variable. It is an improved version of the former twisting algorithm [56, 57] proposed by Levant in the sense that the sign of the sliding variable derivative (s) is detected by finding stationary values of the sliding variable (5). It is implemented by incorporating a peak detector, involving a memory element, into the controller [7]. A new second order sliding algorithm to stabilise systems of type (5.1) is presented in this Chapter. The algorithm neither requires the output derivative to be measured or observed nor the knowledge of its sign, which makes it simple to implement. Sufficient conditions for stability and certain guidelines for selecting controller coefficients are given. If the upper bounds on the system states are known, which is the case for most practical systems, the sufficient conditions for stability of the control loop can be calculated. The results are then applied to various examples in Section 5.5; control of an anti-lock braking system (ABS) is considered in Section 5.6. 5.2 Problem formulation Consider a nonlinear second order uncertain system of the type 2 / i = 2/2 2/2 = ^ ( 2/ 1, 2/2, t) + Q{yuV2,t)u (5.7) y = 2/1 Chapter 5. A New Second Order Sliding Algorithm 84 where .F(-) and Q(-) are smooth functions and the control u is possibly discontinuous. The control objective is to steer the system output y to zero using a sliding mode control with the condition that measured or observed y2 is not available to the controller. This latter condition is not straightforward. Previous attempts to solve this problem have already been discussed in Section 5.1 where it has been seen that some knowledge of y2 is always required. Either y2 is predicted by an observer or at least its sign is detected using a peak detector. Dynamics of the form (5.7) arise in situations where chattering at the actuator should be avoided and the formulation is particularly relevant when the complete state vector is not available for measurement, as will be seen in the subsection below. 5.2.1 Chattering avoidance Consider the nonlinear system (5.1) and assume that all the system states are available for measurement and chattering has to be avoided at the actuator. Choose the sliding surface n — 1 s = x n + ^ CiXi (5.8) i—1 where, [1, cn_i, • • • , c2, ci] are coefficients of a Hurwitz polynomial. Consider the first and second derivative of the sliding variable, s(x) 71—1 s = f(x) + g(x)u + ^2 cixi+1 (5-9) i= 1 n —2 s = ±u + cn-i(f(x) + g(x)u) + y;CjXj +2 + g{x)u (5.10) i= 1 Using equation (5.8) with the dynamics in equation (5.9) and (5.10), the system (5.1) can be written as 0 I n - 2 x = X + y i (5.11) ~C X n = —Cx + yi (5.12) and Vi = V2 (5.13) y2 = Fi(x,u) + Qi(x)v (5.14) u = v (5.15) where, x = [xu x2, • • • , xn_i]T, C = [c2, c2, • • • , cn_i], yi = s and y2 = s. The first two equations (5.11) and (5.12) form a linear system controlled by yx, which is stable by the Chapter 5. A New Second Order Sliding Algorithm 85 choice of c's. The other two equations, (5.13) and (5.14), represent a nonlinear uncertain second order system with control v. If the control v (possibly discontinuous) steers both yi and y2 to zero in finite time, then the linear system becomes an autonomous system governed by the following linear differential equation n —1 Xr + ^ 0 ^ = 0. (5.16) i= 1 If it is assumed that the drift and gain functions of the nonlinear system (5.14) satisfy, at least locally, the following bounding condition 0 < |^i(-)l < ^ (5.17) 0 < Gmin < Q\{') < Gr then the nonlinear subsystem y i = V2 V2 = fi{yi,y2) + Gi{yi,y2)v (5.18) y = y i can be dealt with as if it is independent of the linear subsystem. Thus an n—dimensional nonlinear problem has been reduced to the stabilization of a nonlinear second order uncertain system, the solution of which is similar to that of the system (5.7). 5.2.2 Limited state availability Now, if the state variable xn in equation (5.1) is not available for measurement, then select the sliding surface n —2 S = Xrx-i + ^CiXi (5.19) 1 = 1 The sliding variable (5.19) has relative degree two and its first and second time derivatives can be written as n —2 Xr + C{Xi+1 (5.20) 1—1 n —2 S = f(x) + CiXi+2 + 9 u (5.21) 1=1 The system (5.1) then can be written similarly to the previous case as follows 0 j I n - 3 x = x + y i (5.22) -C xn_i = -C x + yi (5.23) Chapter 5. A New Second Order Sliding Algorithm 86 and 2/i = 2/2 (5.24) 2/2 = F2{x) + Q2(x)u (5.25) where, x — [xi, x2, • • • ,xn~2]T, C = [ci, c2, • • • , cn_2], yi = s and y2 = s. If the control u (possibly discontinuous) steers both y\ and y2 to zero in finite time, then the linear system becomes an autonomous system governed by the following linear differential equation n —2 Xr.»-i . + ^2 CiXi = 0. (5.26) i= 1 If the drift and gain terms of the nonlinear subsystem satisfy the bounding conditions 0 < \F2(-)\ < F (5.27) & ( • ) < G max the nonlinear system can be solved independently and the problem is reduced to stabilizing the system described in equation (5.7). 5.2.3 The problem statement As discussed above, a nonlinear system can be represented by two interconnected systems, one of them a linear system and the other a nonlinear uncertain second order system, by selecting a suitable sliding variable s, which has relative degree two with respect to the input. The linear subsystem is driven by the output of the nonlinear subsystem. The two subsys tems are coupled together. However, if the drift and gain functions satisfy certain bounding conditions then the nonlinear subsystem can be stabilised independently. Finally, the problem of stabilizing an n—dimensional nonlinear system (5.1) has been re duced to stabilizing a nonlinear uncertain second order system 2/i = 2/2 2/2 = F{yi ,2/2,0 + 2 / = 2 / i with the condition that the drift and gain function satisfy the following bounding condition 0 < |H-)| < F (5.29) 0 ^ Gmin — Q{ ) < G m a x * and the state variable y2 is neither available for measurement nor is being observed using an observer. Furthermore, even the sign of y2 is not assumed available to the controller. Chapter 5. A New Second Order Sliding Algorithm 87 5.3 The proposed algorithm Consider the nonlinear system (5.28) and select the system output y as a suitable sliding variable. The following 2-sliding algorithm steers y to zero and achieves the control task. For tracking problems, the controller can be defined in a similar manner where yi(e) in the error space is used as the sliding variable. The algorithm is defined by the following control law. u(t) = ui(t) + u2(t) (5.30) uY = —Asign(y) —ku, |u| > uq U2 = < —VFsign(y), otherwise, where A ,W,k,u0 are positive constants and their values depend upon the bounding values described in equation (5.29). Sufficient conditions for selecting the controller parameters will be derived by applying the algorithm to the system (5.28) for various drift and gain functions. The system responses for critical controller parameter combinations will be studied. The observations from these simulation studies can then be used to infer an upper bound on the envelope of the output when under closed loop control. The following combinations of drift, F{yi,y2,t), and gain, G{yi,V2 , t), functions are used for initial investigation: Drift term Three types of drift functions are considered: • T{y\ , y2, t) — 0; the system model becomes similar to that of a double integrator. • !F(yi, 2/2 j t) = 1; the system has constant matched uncertainty. • Jr(yi,y2,t) = asin(£), a > 0; the system has variable but bounded and matched uncertainty. Gain term As G(yi,y2 ,t) > 0 is required for the system to be controllable, the following two cases are considered: • G(yuV2,t) = l • G(yi, y2, t) = a time varying function with known sign. The case when F(yi,y2 ,t) = 0 and G(yi,y2 ,t) = 1 will be referred to as the ‘double integrator’ system in this Chapter. Chapter 5. A New Second Order Sliding Algorithm 88 5.4 Selection of controller coefficients A series of simulation studies have been carried out so that the different controller parameters can be selected to ensure system stability. The controller parameters u0 and A will be seen to be the most important for stability. Selection of k will be seen to help in reducing oscillations and provide damping to the system. 5.4.1 Choice of u0 The critical constant, which determines the switching condition for the controller is the value of u0. This plays a central role in the stability of the algorithm. Consider the system as follows: 2/1 = 2/25 y2 = l + u (5.31) Now, IF{y,t)\max = F = 1, Gmin = Gmax = G = 1, which gives F/G = 1. Fig. 5.1 shows the system response for different values of u0. It is evident that the only condition for convergence of the system output to zero is u0 > F/G provided A > u0. The system output does not converge for u0 < F/G A=u„=2>F/G X=2,u =1.1>F/G 0.6 0.5 A A : A 0.4 0.3 0.2 0.1 - 0.1 5 10 15 20 tim e time /U2,u_=1=F/G \=2,u.=0.5 >» -0 .5 time time Figure 5.1: System response Consider another case when the control gain is variable. The system can then be written as 2/i = 2/25 2/2 = 1 + (2 + sin(f)) u (5.32) It follows that F = 1, Gmin = 1 and F/Gmin — 1. Here the important point to be observed is that G is a sinusoidal function varying between 1 and 3 giving F/Gmin = 1- Consider, Chapter 5. A New Second Order Sliding Algorithm 89 for example, u0 = 0.9. Except for a small region where F/G > 0.9, the system output converges to zero in the appropriate region as shown in Fig. 5.2. X=2,u0=0.9 0.15 =- 0.05 -0.05 25 30 40 CT> ^ 0 . 6 0.3 Figure 5.2: System response with variable g(y, t) Now a third case is considered where both drift and gain terms are time varying functions. Consider the system V\ — V2 \ V2 = cos (t) + (2 + cos(*)) u (5.33) In this case | cos(t) | < 1 and 1 < 2 + cos (t) < 3. Therefore, F = 1, Gmin = 1 and Gmax = 3, which implies that F/Gmin = 1- Figure 5.3 shows the system response for controller parameters u0 = 0.9, A = 2, k = 5 and W = 0.5. The negative peaks in Fig. 5.3 represent the point where u0 < F/Gmin- 0.1 0.08 0.06 0.04 0.02 0 - 0.02 -0.04 0 5 10 15 20 25 30 35 40 time, sec 0.4 O- 0.2 LL time, sec Figure 5.3: System response with variable F(y, t) and Q(y, t) Chapter 5. A New Second Order Sliding Algorithm 90 5.4.2 Case 1: A < uQ The controller, whose coefficients are selected such that the condition A < u0 is satisfied, is applied to the first case which resembles the double integrator model. The output does not converge to the origin for A < u0 as shown in Fig. 5.4 2 1.5 >. 1 E 0-5 3 o 0 -0 .5 -1 0 1 23456789 10 lime Figure 5.4: Unbounded Oscillatory response for A < uQ 5.4.3 Case 2: A = u0 It has been observed that for A = u0 the output does not converge to the origin but rather oscillates with a constant peak value that is dependent upon the initial states of the system. The Fig. 5.5 shows the oscillations in the case of the double integrator. It can be observed from Fig. 5.5 that the frequency of the output oscillations is a function of the controller coefficients. In Fig. 5.5 the responses are shown for the same set of controller coefficients. It can be inferred that the condition A = u0 corresponds to the case of marginal stability. However, it has been noticed that for a given system, increase in the A and uQ values 10 10 10 Figure 5.5: Frequency of oscillations vary with A and u0 increases the frequency of oscillations in the output. Fig. 5.5 also shows the increase in the Chapter 5. A New Second Order Sliding Algorithm 91 frequency of the output oscillations with respect to the increase in the values of the controller coefficients, A and u 0. 5.4.4 Case 3: A > u 0 For A > u 0, the output converges to zero. Fig. 5.6 and Fig. 5.7 show the effect of an increase in A keeping u0 and other parameters constant. To analyse whether the settling time is a function of (A — it0) or (A /u 0), system responses are plotted for constant (A — uq) in Fig. 5.6 and for constant A / u 0 ratio in Fig. 5.7. Analysing the responses in Fig. 5.6 and Fig. 5.7, it is evident that the settling time is a function of both (A — u0) and A/ u 0. It seems that selection of A and u 0 that give larger (A — uQ) and A/ u 0 will be more suitable. 0.5 ■ ■ ■ X=5.5, u q=5 - X=2. uq =1.5 X=2.5, u q=2 -0 .5 time Figure 5.6: Systems responses for constant (A — -u0) 0.6 0.4 0.2 - 0.2 -0.4 time Figure 5.7: Systems responses for constant A / u 0 5.4.5 Selection of k The constant k in the controller helps to damp the system response and reduces the overshoot values. For stability, k needs to be positive. Extensive simulation study shows that after a certain point, increase in k is not helpful, sometimes even counterproductive. Fig. 5.8 shows the effect of k on the time taken to settle the system into the 2-sliding set. During the tuning process, k = u0 has been found to be a suitable starting point. Chapter 5. A New Second Order Sliding Algorithm 92 | (a) A - w0 = 1 (b) A = 2uq | | ? 1 k (c ) A — wq = 2 (d ) A - 3w q I I k (e ) A — wq = 3 (f) A = 4w0 Figure 5.8: Effect of k on settling time 5.4.6 Summary of observations In the previous sections, it has been observed that the response of the closed loop system is oscillatory of constant peak value for A = u0 > F/Gmin. The peak amplitude of Chapter 5. A New Second Order Sliding Algorithm 93 oscillations depend upon the initial states. However, the frequency of oscillation is system dependent. • damped oscillatory for A > u0 > F/Gmin. • unbounded for u0 < F/Gmin- • unbounded oscillatory for A < u0. In digital implementation, the sequence {s*} of the intersection points with the axis s = 0, is a convergent series because it satisfies ®i+l < q < 1 (5.34) before it settles into the real second order sliding set. The settling time can be estimated as Sj+l the sum of the encircling time sequence using a geometric series. A typical plot of IS shown in the Figure 5.9. 0.95 0.9 -0 8 5 0.75 0.7 time, sec Figure 5.9: A typical plot for Sj+l The term u2 is bounded by ±(A — u0) and the control effort u remains bounded by ±(2A — u0) with uq < \u\ < (2A — u0). Hence, the optimal values of A and u0 are constrained by the permissible control. Increased values of (A — u0) reduce the settling time by shifting the only real pole towards the left but increase the amplitude of oscillations in y2. It has been observed that the optimal value of k depends upon A and u0. A suitable value of k to start tuning can be taken as u0. Chapter 5. A New Second Order Sliding Algorithm 94 From the above observations, it is proposed that the algorithm (5.30) stabilizes the sys tem (5.28) in a finite time which can be calculated as the sum of a geometric series. The sufficient conditions for stability are: A > > su p (F/G), k > 0 and W > 0 (5.35) Furthermore, it has been observed that if there is a norm bounded noise disturbance y (y) in the first channel of the system i.e. unmatched uncertainty, which makes the first equation of the system iji = y2 4- rj(y) then the algorithm (5.30) still stabilizes yi to zero as y2 converges to —r)(y). It ceases to be a second order sliding algorithm. However, the algorithm is useful for systems where the output yi is strictly required to be zero but a boundedness condition on the second state y2 is sufficient. In the Section 4.4, it has already been seen that for second order systems the super-twisting algorithm show limit cycle behaviour. However, the same drift terms new algorithm (5.30) does not show limit cycle behaviour rather stabilise it to the origin as shown in the following Fig. 5.10. (a) for f(x, t) = 0 (b) for f(x, t) = 2 (c) for f(x, t) = 2 sin(t) Figure 5.10: Responses for various drift terms Chapter 5. A New Second Order Sliding Algorithm 95 Example 5.1 Consider the following nonlinear second order system model which was sug gested by Dr. Arie Levant to test the algorithm, as follows s = + g(t,s,s)u (5.36) where, - sign(/(£, s, s)) if |/(f, s, s)| > F d_ —/Lisign(s) if ss < 0; |/(£, s, s)| < F (5.37) dt 0 otherwise where p = 0.1 and F = 0.5 are positive constants; and g(t,s,s) = 1 and consider the following two sets of initial states of the system 1. s = 0, s — 5; 2. s = 0, s = 0.001 Controller coefficients are selected as u0 = 1, A = 6, W = 2 and k = 1. The controller sta bilises the system states to zero as claimed. The simulation results are presented in Fig. 5.11 and Fig. 5.12. 0.5 -0 .5 20 time, se c 6 4 2 -o 0 -2 -4 0 2 4 68 10 12 14 16 18 20 Figure 5.11: Convergence with large initial conditions 5.5 Some simulation examples 5.5.1 Underwater vehicle Consider an underwater vehicle model [2] as used by Slotine and Li [96]. mx -1- aria;I = u (5.38) Chapter 5. A New Second Order Sliding Algorithm 96 0.5 -0 .5 0.5 2.5 3.5 4.5 time Figure 5.12: Convergence with small initial conditions where x is the position of the vehicle and the input u is the propeller force. The vehicle mass m and drag coefficient c are assumed to be bounded as: 1 < m < 5 and 0.5 < c < 1.5. The actual values of m and c used in the simulation are m = 3 + 1.5sin(|x|t) and c = 1.2 T- 0.2sin(|±|£) The underwater system (5.38) can be represented in the standard form (5.28) as X\ = x2 (5.39) x 2 = ~ —x\ sign(rr2) + —u m m The propeller of the underwater vehicle can provide a bounded force which in turn can produce bounded speed (x2). It is assumed here that motion remains within the operational range |x2| < 1. Taking into account the uncertainties, system bounds can be calculated as: Q fix , t) = Xo sin (^ 2 ) m m (5.40) 1.2 + 0.2sin(|i:|£) -Xr 3 -f 1.5 sin(|i;|£) J_ < L5 Gmin = 1/4.5 and Gmax = 1/1.5. Therefore, ^ 0 > sup \f(-)\/g(-) = 1 will suffice. For simulation, u0 = 3, A = 8, k = 5 and W = 0.5 are taken. The initial conditions for the system states are [0.5,0] and for the controller are [0]. The simulation results are presented in Figures 5.13, 5.14 and 5.15. Chapter 5. A New Second Order Sliding Algorithm 97 0.4 0.2 - 0.2 -0 .4 time Figure 5.13: The vehicle position, x 0.5 -0 .5 12 time Figure 5.14: The vehicle speed, x I 0 T 1 1 F I i ! - ■ ■ r r 10 ...... :...... i...... ]...... j...... '•*: ...... | ...... | ...... 5 - \ 0 - -5 F -10 1 c i - t ... i . 1 I 1 .1 ------1------1 0 2 4 6 8 10 12 14 16 18 20 time Figure 5.15: The input, u 5.5.2 Single axis jet-controlled aircraft Consider the second order plant representing the kinematic and dynamic equations of a single axis jet-controlled aircraft where the attitude variable is measured with respect to a skewed axis and specified in terms of the Cayley-Rodrigues parametrisation [91]. x = 0.5(1 + x2)uj Cj = -u (5.41) u y = x - X where x represents the Cayley-Rodrigues orientation parameter, cu is the main axis angular velocity and u is an externally applied input torque. J is the moment of inertia of the aircraft about its principal axis. The set point X = 1.5 rad and J = 70 Nms-2 Chapter 5. A New Second Order Sliding Algorithm 98 The system can be written in the standard form (5.18) as follows: y\ 1/2 y\ 2 , 0.5, 2\ V2 2 (5.42) r r ^ + T (1+yi)u y y i = x x It is readily established that 2 l / i 2 :l/2 < y \ i + i/?' Assuming that y2 is upper bounded by 0.2, the bounding values for the drift term can be calculated as F = 0.08- \Vi 1 + 1/?' The gain term is a function of the measurable variable yi, which yields that — = 0 16J__ — __ G (1 + yl)2' The controller parameters u0 = 2 + F/G, A = 4uQ, k = 4 and W = 1 are selected for simulation. The simulation results are presented in Fig. 5.16. 1.6 - 0 1 2 3 4 5 6 78 9 10 (a) The attitude variable, x 0.06 0.04 0.02 3 - 0.02 -0 .0 4 -0 .0 6 -0 .0 8 time, sec (b) The angular velocity, cj Figure 5.16: Single axis jet-controlled aircraft simulations Chapter 5. A New Second Order Sliding Algorithm 99 lime, sec Figure 5.17: The input, u 5.5.3 Nonlinear tunnel diode circuit Consider a tunnel diode circuit [14, 50] as shown in Fig 5.18. The state space model of the Tunnel Diode Figure 5.18: Tunnel diode circuit tunnel diode circuit is given by xi = i)~x2\ X2 = 7 [—#1 —Rx 2 + u] Lj y = x i Assume the circuit parameters are R = 1.5 kO, C = 2 pF = 2 x 10~12 F, and L = 5 x 10-6 H. The input u = E and g(xi) is the current flowing through the tunnel diode and it is a nonlinear function of the potential difference across it. If the time is measured in nanoseconds and currents x2 and g(x\) in mA, the state space model can be written as: xi = —0.5g(xi) — 0.5x2 (5.43) x2 = —0.2a:! — 0.3:r2 + 0.2 u Similar to [14], it is assumed that g(x i) = sin(zi) (5.44) Chapter 5. A New Second Order Sliding Algorithm 100 The circuit model (5.43) together with (5.44) can be written in the standard 1-0 form as Vi = 2/2 1 1 1 y2 = To ^Vl ~ L5sin2/i) + Yo ^ cosyi _ 3)^ - (5-45) yi = Assuming that 2/2 < 1, a bound over the uncertain part of the drift term can be given as F = ^ 5cosyi “ 3I‘ The control task is to stabilise the output of the system (5.45) to origin. Therefore, the controller parameter u0 can be selected as u0 > F/G > |5cos?/i - 3\\y2\ > |5 cos 2/1 — 3| The value of u0 for simulation is taken as |5 cos 2/1 — 3| + 2. The other controller parameters are selected as A = 4u0, k = 3 and W = 3. The part of the drift term dependent upon the measured output i.e. 0.1 ( 2/1 — 1.5 sin 2/1) is applied as negative feedback. The simulation results are presented in Fig. 5.19 0.6 0.4 0.2 -0.2 time, sec (a) Potential across the tunnel diode x\, V 0.5 -0.5 -1.5 time, sec (b) Current x2, mA Figure 5.19: The tunnel diode circuit simulation results Chapter 5. A New Second Order Sliding Algorithm 1 0 1 Figure 5.20: The control input u, Volts 5.6 Anti-lock braking system The conditions established in the previous section will be utilised for controller coefficient selection for stabilising the dynamics of an ABS system. The ABS system consists of three subsystems; namely the vehicle dynamics, wheel dynamics and brake dynamics. They can be summarised in the following dynamic equations [82]. i, . ££„(*) RwNv fw , 1 X2 = J— M 'V + ~ r x 3 (5.46) j V ) J w 003 . h x3 = ------1-----u T T A = (x2- x-^/xi (5.47) where 0.25fcizi /%N 0 x A h = ( l + fc2xf+fc3xffarf+fa.f V \ 2’ m( ) “ IXp pXI + A2 y l+fc4Xl+fc5^f J The system states are the vehicle angular velocity (calculated over wheel radius), xif wheel angular velocity, x2 and brake torque, x3. The input u is the brake pedal pressure. The system output is the wheel slip, A and /i(A) is the adhesion coefficient of the road. This will vary depending upon the road conditions. A typical p — A graph is shown in Fig. 5.21 for different road conditions. The desired value of output A^ = —0.12 is selected because from p — A curve, 0.1 < A < 0.2 gives optimal performance by stoping at shorter distance. So the performance of ABS controller depends upon how well it can stabilise A in that optimal range during various road conditions. Chapter 5. A New Second Order Sliding Algorithm 102 0.8 Concrete road Acceleration Normal road c o Slippery road Deceleration -0 .8 ■1 -0 .2 0 0.2 1 Wheel slip, A Figure 5.21: p — A curves for different roads. 8.8 8.6 8.4 8.2 7.8 7.6 x,‘1 Figure 5.22: Evolution of control gain g(xi) The system coefficients are given in Table 5.1. The ABS system can be transformed into a second order system when written in 1-0 form. Differentiating A once yields A = 0 .5 9 5 ^ (1 + A)*! - ( + A)) ^ + -p- (5.48) Mv \ Jw MVRW J xi Jwi and differentiating A twice yields A = f(xi,x3,x3)+g(xi)u Chapter 5. A New Second Order Sliding Algorithm 103 Table 5.1: ABS model parameters Vehicle Coefficients Wheel Coefficients Road Coefficients M v N v Rw N w j W Ap Pp 1000 kg 2287 N 0.31 m 4 0.65 Kg-m2 -0.175 -0.5 Electromagnetic Brake Coefficients h k2 h &4 k§ 1.0507 0.0666 -7.0876036 x 10~7 -5.45875 -0.00093 where p/ \ r\ c n c /i W^w . Rw^vp NVNW{\ -j- A)/X X3 X\ = -0.595(1 + A )- + W + ------J - 5 +0.595^Arc1 + 1.19^(1 + A)ij - ^ ^ A p . - (1 + ^ M v M y Jwxi MVRW x\ x3 JivTX i kf) 9{x i) = JWTX\ Therefore, the system output A in equation (5.47) has relative degree two in 9R3 with respect to the input. The first order zero dynamics of the system can be characterized by restricting x to the following subspace 2* = {x 6 5R3|A = Xd, A = 0} (5.49) and equating u to zero, the process yields ^3 x3 = ----- (5.50) T which is globally asymptotically stable and shows that the system is minimum phase. Within the range of operation, the function /(•)/ 200 150 § 1 0 0 0.02 0.04 0.06 0.08 0.12 0.14 0.16 0.18 0.2 time, s Figure 5.23: Evolution o f/(•)/#(•) Chapter 5. A New Second Order Sliding Algorithm 104 To stabilize the system, uq > sup /(• ) /g(-) = 200 should be selected. This has been found to be quite conservative. A smaller value u0 = 35 is selected for simulation. Other parameters are selected asA = 7 5 > u 0,£; = 200 and = 15. The simulated system response for 0.2 seconds is shown in Fig. 5.24-5.26. 66 > 64 0.02 0.04 0.06 0.08 0.12 0.14 0.16 0.18 0.2 time, s -0 .0 1 5 A ffi I 0 0.02 0.04 0.08 0.1 0.12 0.14 0.160.06 0.18 0.2 Figure 5.24: Vehicle speed x\, Wheel speeds x2 and Wheel slip, A 600 500 400 200 100 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 time, s Figure 5.25: Brake torque x3 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Figure 5.26: The control u Chapter 5. A New Second Order Sliding Algorithm 105 5.7 Conclusion A new 2-sliding algorithm has been presented for systems with relative degree two with respect to the input. The only input to the controller is the measured output; the algorithm does not require measurement or estimation of the output derivative. Sufficient conditions for stability have been proposed. Simulation results for stabilisation of various SISO nonlinear models including an ABS system using the algorithm have been presented to validate the assertions made. The algorithm will be extended for a class of MIMO systems and simulation studies will be presented in the next chapter. C h a p t e r 6 A M u l t i -In put 2-S lid in g Contro l 6.1 Introduction In Chapter 5, a new second order sliding mode control algorithm has been proposed for nonlinear SISO systems and some simulation examples presented. In this chapter, the algo rithm is extended for a class of nonlinear MEMO systems which can be represented in 1-0 form. The case of decoupled systems and the case of diagonally dominant systems are both considered. In Section 6.2, basic theory about the 1-0 representation of MEMO systems is presented. In Section 6.3 the SISO algorithm presented in the Chapter 5 is extended to a class of nonlinear MIMO systems which can be represented in 1-0 form. Some examples with simulation results are presented in Section 6.4 and conclusions are made in Section 6.5. 6.2 Background Sliding mode control for multivariable linear systems as well as nonlinear systems is well documented in the literature [24, 27, 104]. Most methods rely on the regular form of the nominal system. It is not straight forward to convert nonlinear systems in to regular form [50, 104]. However, if an appropriate canonical form for nonlinear systems is adopted, it will have a zero dynamics associated with it. The stability of the system not only depends upon the reduced order system but also on the stability of the corresponding zero dynam ics [28, 35, 46]. Various SISO control techniques can also be extended to MIMO systems by linearising the system and then designing linear controllers [20, 28, 98, 110]. Proper differential input output representations are selected as the appropriate canonical form for sliding mode controller design in this Chapter. It follows that dynamic sliding mode con trol will always result when a sliding surface having relative degree two with respect to the control is selected. Such dynamic sliding policies are desirable from the point of view of chattering removal. Dynamic sliding regimes are considered to naturally provide a dynamic 106 Chapter 6. A Multi-Input 2-Sliding Control 107 control [10, 11, 20, 68, 72, 91]. However, in [91] dynamic surfaces are considered only in the traditional sliding mode sense and produce a dynamic controller only if the system contains certain control derivatives (u^\f3 > 1) in the 1-0 representation for removal of chattering. This is not the case for all systems which are linearisable by state feedback. The work in [68, 70, 72] asymptotically linearises the plant and uses a control dependent sliding surface which always produces a dynamic controller. A second order sliding mode control is proposed for MIMO systems in [20] but the resulting controller requires the derivative of the sliding variable for implementation. Bartolini et al. [10] have also extended their SISO algorithm for a class of nonlinear MIMO systems. However, the algorithm still requires sta tionary values of the sliding variable (local maxima or minima) to be detected online. The method developed in this chapter uses second order sliding mode control where the sliding surface may or may not be control dependent and always produces a dynamic controller in conjunction with the canonical form representation, contrary to the work in [68] and [91]. The proposed method combines the elements from [91] and [68] together with [10, 20] and is applicable to a wider class of system which are not necessarily affine in the control. More over, the controller does not require the derivative of the sliding surface vector. Consider a locally observable general MIMO nonlinear system in state space form x = tp(x, u , t) (6.1) y = h(x, u, t) where x E 3ftn, u E 3ftm, y E $RP and ip : x x 3?+ -> and h : Un x x 3R+ — are smooth vector functions. The following locally equivalent differential 1-0 form exists [68] y[ni) = (6.2) y{pp) = y ti] = My,u,t), z = i,•••,£. (6.3) where Uj = (Uj,Uj,••• j = 1 ,••• ,m and u = (xti, - • - ,um)- Similarly yi = (2/»,yu • • • , 2/ini_1)) and y - (yu • • • , yp) with Y7i ni — n • This representation is the same as the Local Generalized Controller Canonical Form (LGCCF) of Fliess [35]. A differential 1-0 system is called proper if Chapter 6. A Multi-Input 2-Sliding Control 108 (b) all (c) the following regularity condition is satisfied ^(011 ' ' ' i 0m) det A large class of nonlinear systems, especially mechanical systems, are naturally in this form. Additionally, a wider class of nonlinear systems, termed as ‘differentially flat systems’, can be written in this form together with dynamic compensators which may be a chain of inte grators in their simplest sense [34, 37]. Definition 6.1 (Zero Dynamics) The corresponding zero dynamics of the system model (6.3) is defined as 0i(O,u, t) = 0 (6.4) 0P(O ,u,t) = 0 The system (6.3) is called minimum phase if the zero dynamics (6.4) is uniformly asymptoti cally stable. This zero dynamics is the dynamics of the control and is a generalisation of definitions in [34]. It is different from the zero dynamics defined in [46], which is the dynamics of the uncontrolled states. Only proper minimum phase systems are considered in this Chapter. 6.3 A MIMO 2-sliding control A sliding surface will be selected such that it has relative degree two with respect to the auxiliary control and the second order sliding algorithm developed in Chapter 5 will be extended for proper minimum phase nonlinear MIMO systems. The system (6.3) can be written in the following generalised canonical form 3 = (6.5) S>n,P —1 — Sf1>rii Chapter 6. A Multi-Input 2-Sliding Control 109 The system (6.5) can also be written in expanded form as fS 1n i - l = S»ni \ (6.6) cm _ cm SI — S2 crn _ cm S n m -1 s rim in m =4>m(Lu,t) where f = (fj, • • • , = (yi} • • • , y^ni~1}), i = 1, • • • , m and f = (£ \ • • • , £m). 6.3.1 Sliding surface design The sliding surface, s, is selected such that when it is made zero the control objective is achieved. In the case of MIMO systems, the sliding surface is a vector which has the same dimension as that of the control vector u i.e., s G SRm. The sliding surface, in the general sense, can be selected as a nonlinear function of the system sates such as s = *(£,«) However, only linear sliding surfaces are considered here as follows U i — 1 = 4 + 4 5 - *=!,--• (6.7) 3 = 1 where the vector of constants {c\, cl2, • • • , x) are such that the polynomials H i — 1 + ^2 C)K = °> i = 1, • • • , m. (6.8) j= 1 are Hurwitz. It is to be noted that the choice of sliding surfaces in equation (6.7) is naturally control dependent if $ > 0 for i = 1, • • • ,m. However, as will be seen in this Section, for a dynamic controller which is required for chattering removal, no condition on fa is imposed here, contrary to that proposed in [91] where Pi > 1 is the condition for the controller to be dynamic, and thus chattering is avoided. Moreover, the definition (6.7) does not assume that the sliding surface is control dependent. Therefore, the methodology is different from Chapter 6. A Multi-Input 2-Sliding Control 110 what is proposed in [68], where the sliding surface is always control dependent. The first two successive time derivatives of the sliding variables (6.7) along the system trajectories can be given by ni-2 Si = u, t) + + Y 4 4 + 1 ’ i = (6.9) j=i (Jt ^ Si = U, t) + <._!<&(£, U, t) + Cln._ 2Cni + 4 4 + 2 3 = 1 o , m nk c\ i m Pk o / _ o r \ / 777. 77. a / 777. /?fc 1 q / 777. Q JL _ G*0i , ..(j+1) I ^ „.(&+!) dt 2^/2-*t ggk**j +1 2-*/2-*t q U) k 2 ^ Qyifa) k k= 1 j= l fc=l j=0 ^ “fc fc=l U k k^i Tlj 3 rj i +4-i0i(?. *, *) + 4-2^ + E 4Q+2 + (6-10) i=i Let z* = (zj, 4 , • • • , zj.) = (ui, Ui, • • • , 4 /?l))’ * = (^4' • • »zm) and 4 A+1) = for i = 1 , • • • , m. The controlu can be implemented using a chain of integrators with auxiliary input which can be written in Brunovsky canonical form as z% = GiZ1 + HjVi, i = 1, • • • , m Using equation (6.7), the system (6.5) can be written as 4 = A if + BiSi (6.11) §i =fi(€,z,t) + gi(€,z,t)vi (6.12) z 1 = GiZ1 + HiVi, i = (6.13) where 4 — (4 >' ‘' ,& ._!) and O I m nk 02 JTi /3fc —1 02 m 02 Jfe=l J = 1 ^ k= 1 j=0 ^ /c fc=l VUk k^i rij—3 + Cni-l^tKj 4 + Chi-2^ni + X y 4 4 + 2 J = 1 (t +\ &<$)i ' 1 9i{tz,t) = — Tpr; * = V " ,771 ou) ’ Chapter 6. A Multi-Input 2-Sliding Control 111 and I i 1 O o Ifii—2 QQ ll -c * 1 1 i 1 o o £ b i-i ii 0 1 where Cl = [1 c\ • • • , cln._^\. The complete MIMO system (6.1) can be written as £ = A£ + B s (6.14) s = / ( £ > M ) (6.15) i = Gz + Hv (6.16) where £ = (I 1, • ■ • , £m) G s == (su • • • , sm) and A = diag(Au • • • , Am) G sr("-"0* B = diag{Bu • • • , Bm) G G = diag(G1, • • • , Gm) G 3^+lx^+1, i/ = ^ ( ^ , • • • , Hm), f = (fu ■ ■ ■ , fm)T, g = (Pi j ' ' ‘ 5 ^m) • Therefore, a MIMO system is converted into three subsystems. The first subsystem is an (n — m) dimensional linear system driven by the sliding variable vector, s. When s = 0, the linear subsystem is a stable autonomous system as the polynomials in (6.7) are Hurwitz by design. The second subsystem represents the second order nonlinear uncertain dynamics of the m sliding variables and the third subsystem is a chain of integrators, the output of which provides the actual control. In the case of the system (6.1) being linearizable by change of coordinates and state feedback, the MIMO system representation in 1-0 form (6.3) will be independent of any control derivative, i.e., P j= 0 , j = 1, • • • ,771 and therefore, the auxiliary control vector, v, which drives the sliding surface dynamics (6.15) is simply the time derivative of the actual control vector u. Even if v is chosen to be discon tinuous, the actual control input, u, to the system will be smooth because of the integrator dynamics which can also be an actuator dynamics. 6.3.2 Controller Design Once the sliding surface dynamics in (6.15) is stabilised to s = 0, the MIMO system dynam ics under sliding motion is given by £ = M (6.17) Chapter 6. A Multi-Input 2-Sliding Control 112 which is stable because A is a block diagonal matrix with all diagonal blocks (A{, i = 1, • • • , m) representing the stable dynamics (6.8). Though the system in (6.15) is coupled with the states of the system in (6.14), it can be decoupled if the vector field /(£, z, t) and the gain matrix #(£, z, t), even if uncertain, satisfy the following bounding conditions in any bounded domain | MS,z,t)\ Vli = V2t (6.19) V2i = fi(€,z,t) + gi(£,z,t)vi This representation contains all those uncertainties which do not violate the bounding con ditions (6.18). The algorithm discussed in the previous Chapter can be applied because the control Vi always appears linearly. The controller parameters are selected as follows u0i > sup Fi/Gm, Ai > u0i (6.20) k{ > 0, and > 0 Sometimes, the controller (6.20) may be conservative. To reduce the conservatism, the con ditions (6.18) imposed on the sliding surface dynamics can be relaxed as \fM,z,t)\ < Fil+F i2\yli\ (6 .21) Gmu + Gm2i\yii\ ^ ^ Gmu + GM2i\yii\ and the controller parameters can be selected as follows > Fil + Fiilvul Gmii + Gmzilyi^ A\ i Ugi (6-22) ki > 0, and W{ > 0 6.4 Illustrative examples In this Section, simulation studies of two multi-input system models are presented. The controller parameters are selected according to the conditions given in (6.20) and (6.22). Example 6.1 Consider the following mathematical model with two control inputs. The model does not represent any physical system. x\ = cos X2 + sin t + u\ (6.23) X2 = x\+ sin t -f u2 Chapter 6. A Multi-Input 2-Sliding Control 113 The control task is to stabilise the system to the origin. The system (6.23) can be written as Xi = — (xi + sin£ + u2) sin(^2) + cost + Ui x2 = cos x2 + sin t + u\ + cos t + u2 Using the transformation ui = xi — cos x2 — sin t u2 — x2 — X\ — sin t the system can be written in proper 1-0 form as Xi = — x2 ■ sina;2 + cost + iii x2 = x\ 4- cos t + u2 Let yi = (xi, x2) and y2 = (ii, x2). The system can be represented in second order form as follows 2/i = 2/2 (6.24) 2/2 = /(2/i> 2/2 ) + v (6.25) where the auxiliary control vector v = u and If 12/2!| < 2 and \y22\ < 3, the control parameters can be selected as u0l2 = (3,4), Ai>2 = (5,6), klj2 = (4,4) and Wlj2 = (5,6). The simulation results are presented in the following Figures 6.1 and 6.2. In Fig 6.1, the system output x\ and x2 take different times to settle and exhibit different transient behaviour. This can be controlled by selecting suitable parameters. In the steady state, the control input u\ converges to —(1 + sin t) and u2 converges to — sin t which is the steady state solution of the system equations (6.23). Chapter 6. A Multi-Input 2-Sliding Control 114 -0 .5 lime, se c Figure 6.1: The plant outputs 0.5 -0 .5 -1 .5 -2 .5 -4 time, sec Figure 6.2: The plant inputs Example 6.2 (Rigid Spacecraft Model) Consider a first order MIMO model of a rigid space craft. The control objective is to stabilise the angular velocities. The dynamics of the rigid spacecraft [77] can be described as J2 — J3 (j i = ^2^3 T Ui J l J3 — J l U>2 — CU3CUi + u2 (6.26) J2 Jl — J2 U 3 = CJ1CJ2 "h u 3 Ja Chapter 6. A Multi-Input 2-Sliding Control 115 where u = [oji,u)2 ,ujz]t is the angular velocity vector and J\ = 2500 kg.m2, J2 = 6500 kg.m2 and J3 = 8500 kg.m2 are the inertia momentum of the three axes of the body. Let the sliding surface vector s = to. The equations (6.26) can be written as si = ai(a2S3 + a3S2 ) + ai(s3u2 + s2u3) + s2 = tt2(a3Si + &i«s3) + o2(s3Ui + siu 3) + v2 (6.27) S3 ~ U3(a l s2 "I-a 2sl) + a3(s l'li2 + s2'^l) + ^3 Ui Vi i = 1,2,3. where J2-J3 J3 — J1 , J i ~ J2 ai = a2 and a3 = Ji J2 J3 using the transformation ui = S i — a is 2s 3, u2 = s2 — a2s3si and u3 = s 3 - a3s i s 2 The system dynamics can be written as s'i = ai(l-si)(a2s2 + a3s2) + a i(s3s2 + s2s3) + ^i s 2 = a2( l - s 2)(a 3s? + a is 2) + a2(s3s i + s i s 3) + v2 (6.28) S 3 — ^3(1 — s3)(als2 + a2Si) + ^3(SiS2 + S2Si) + V3 Let y1 = (si, s 2, S 3 ) 6 3£3, y2 = (si, s2, s3) e 5ft3 and y = (yu y2), then the system can be written in the following second order form which is suitable for controller design. y\ = 2/2 y2 = fi (2/1) + h (2/1, 2 / 2 ) + V (6.29) where ^ai(l - 5i)(a2s| + a3s2)^ /i(-) = a2( 1 - ^a3( l - 53) ( a is 2 + a2s 2)y ai 0 0 1 - si 0 0 0 a3 a2 'si 0 a2 0 0 1 - s2 0 a2 0 ai S2 0 0 <23 0 0 1 - s3 a2 ai 0 A is the known part of the system which will be used as feedback to the controller and ^ a i ( s 3 S2 + S2S3)^ C Li 0 0 0 2/ i 3 2/ i 2 v n — / > ( • ) = a2(s35l + -S1S3) 0 «2 0 2/ i 3 0 2/ii 2/22 ^ ^ 3 (si S2 + S2Si) O J 0 0 Sr a 3 10 _2/23_ 1 1 — 1 Chapter 6. A Multi-Input 2-Sliding Control 116 which involves the vector y2 which is not available for measurement. However, if it is as sumed that there is an upper bound on each component of y2 within the operating domain i.e., |y2i | < Y2i where Y2 is a vector of upper bounds on y2 then bounds on the f 2(i) can be given by l/2(l)| < laiKlyull^l + lyi.ll^l) l/2(2)| < kKlj/ullyj.l + ly iJ^ I) l/ 2(3)|<|a 2|(|y il||y22| + |yl2||y 2 l|) Therefore, the controller parameters can be selected as A* > u0i > / 2(i). For the simulation study, it is assumed that |V^(i) | < 1 and the controller parameters are selected as follows CLl 0 0 |yi3| + \yu\ CO + II O 0 a2 0 l y j + \yiA 0 0 «3 J 2/1J + \vu\_ Ai 3 UQi, VFj — 2, and ki = 10, i = 1,2,3. The simulation results are presented in the Figures 6.3, 6.4 and 6.5. The angular velocities converge to zero with an acceptable amount overshoot and the control inputs are zero in the steady state as expected from the system equation (6.26). The auxiliary controls in Fig. 6.5 show high frequency chattering but this is confined to the software variables and does not appear on any physical states. 0.4 3" 0.2 -0 .2 0.5 2.5 3.5 0.5 -0.5 0.5 2.5 3.5 0.6 0.4 -0 .2 0.5 2.5 3.5 time, sec Figure 6.3: The angular velocities Chapter 6. A Multi-Input 2-Sliding Control 117 0.51------r _1 ______I______I______I______I______I______I______0 0.5 1 1.5 2 2.5 3 3.5 0.5 -0.5 0.5 2.5 3.5 0.5 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 time, sec Figure 6.4: The plant control inputs 20 : ...... i 1 10 >~ 0 _ / -10 / -20 - _____ i______1_ I 0.5 1 1.5 2 2.5 3 3.5 20 10 -10 -20 I 0.5 1 1.5 2 2.5 3 3.5 20 1 ' 1 1 1 - 1 1 ...... ;...... \ 10 \ h ; .... , >” 0 1...... -10 / ...... :...... ; V -20 1 1 1 1 1 1 ( 0.5 1 1.5 2 2.5 3 3.5 time, sec Figure 6.5: The auxiliary inputs 6.5 Conclusions A novel second order sliding mode control algorithm has been presented for a class of MIMO nonlinear systems. The nonlinear system does not have to be expressed in regular form; rather an 1-0 representation is used. A large class of nonlinear systems may be modelled by square differential input-output equations. Higher order sliding mode control together with this 1-0 representation has been adopted in this Chapter. This produces a dynamic control irrespective of whether the sliding surface is control dependent or not. The auxiliary control, the highest derivative of the actual control, always appears linearly which facilitates control Chapter 6. A Multi-Input 2-Sliding Control 118 design. The algorithm does not require the derivative of the sliding surfaces, thus eliminating the requirement of designing an observer or peak detector, contrary to many other sliding mode control strategies. The resulting control is dynamic and eliminates the chattering at the system input. To illustrate the design, two nonlinear models have been stabilised using the controller designed in this Chapter; one of the studies relates to a rigid spacecraft model with three controls. Simulation results have been presented which validate the methodology. In the next Chapter, a MIMO case study for water level control in a two-coupled-tank system will be presented with the actual implementation results on the rig which utilise the control strategy developed in this Chapter. Chapter 7 Case Stu d ies 7.1 Introduction In the last two Chapters, theory for the control of SISO and MIMO systems via a novel HOSM control has been presented. This Chapter presents two case studies; one is SISO and the other is MIMO. The SISO case study in Section 7.2 considers the robust angular position and speed control of a DC motor. Section 7.4 presents the DC motor implementation results using the dSPACE environment. The second MIMO case study in Section 7.5 considers water level control in a coupled twin-tanks system using the control algorithm developed in Chapter 6. The simulation results are presented in Section 7.5.3 and the actual results of implementation on the rig are presented in Section 7.6. 7.2 DC motor position control DC motors are commonly used as an actuator in control system applications. Their excel lent dynamic performance and ease of control makes them a popular choice for many servo applications. In this case study, the design of the 2-sliding controller presented in Chapter 5 is considered for robust control of angular position and speed control of a DC motor in a SISO configuration. The controller parameter selection is based on the method discussed in Section 5.4. A simulation study for position control is presented in Section 7.2.4 and the implementation results are presented and discussed in Section 7.4. 7.2.1 DC motor model The motor was excited using a PRBS voltage signal and the output data collected using the dSPACE setup. The generated data is then mapped into a parameter vector. It is not required to have a very accurate model to implement this algorithm. However, the relative degree of the output must be known. The following parametric linear system model has been 119 Chapter 7. Case Studies 120 identified [43]. x = Ax + bu (7.1) y = Cx where 0 1.8665 0 0 A = 0 -2.0652 25.989 b = 0 0 -63.5268 -141.0943 2000 C- 1 0 0 where xi is the angular position of the shaft ( 0), x2 is the angular speed of the shaft ( lu) and x3 is the armature current (/) of the motor. The angular position control of the DC motor model (7.1) is carried out in two sates- first a sliding variable is selected such that it has relative degree two with respect to the control input and then control algorithm (5.30) is applied to stabilise the second order sliding variable dynamics to zero. The algorithm (5.30) does not require derivative of the sliding variable therefore, it saves one sensor used for measuring motor current. 7.2.2 Sliding surface design The system (7.1) has three states, namely angular position, 6, angular speed u and the motor current (/). The input is the applied voltage and the output is the angular position, 6. Let the set point for the angular position of the shaft be denoted by 6r. Incorporating the set point error, ee = (6 — 9r), into a new state vector, z = [zi, z2, z3]T — [e#,x2, x^]T, the system (7.1) can also be written as i = Az + bu (7.2) Consider the following sliding variable, sq, which has relative degree two with respect to the control input, u. Sq = s\ee + &e (7.3) = sizi + z2 (7.4) = Sz (7.5) where S = [si 1 0] and sx is a positive design constant. It should be noted that the slid ing surface is a function of two states only. Therefore, the control algorithm based on the knowledge of this sliding surface will be a partial state feedback law rather than a full state Chapter 7. Case Studies 121 feedback law. From equation (7.5), the first time derivative of the sliding variable, sd, can be written as se = Sz - S(Az + bu) = SAz as Sb = 0 — {s\A\ 2 -\-A22) Z2 + A22,Zz (7.6) Using equations (7.4) and (7.6), the following transformation which transforms the states from (zi, z2 ,2 3 ) 6 ^ into the (zi, sq, sq) G 3ft3 space can be written as Zi Z2 z3 = T zi se se (7.7) where 1 T = -S i 0 ~A23 {s1A12 4- A22) A23 The second time derivative of sq can be given by se = SAz = SA(Az + bu) = SA2z + SAbu = Lese + f 6(zi,so)+ge(zi,so,so)u (7.8) In the case of a nonlinear system, the /,$>(•) and ge(-) are nonlinear vector fields. However, in the case of a linear model, these will be matrices which can be calculated using equation (7.6) as follows Le = Si7li2^422 + A\2 + 7 I237I32 + T32(siA2zAi2 + 7I227I23 + 7 I237I33 ) fe{zlj Se) = —S i ( s i 7l i 27 l 22 + 7^22 + 7^237132)^1 + ( s i 7l i 2 + 7I22 + 7133)50 9e(zi,so,se) = SAb 7.2.3 Uncertainty bounds For the case of angular position control of the motor, the transformed system dynamics involving the sliding surface can be written as zi = —S\Z\ T se (7.9a) se = Lese + fe(z 1, s9) + ge(z 1 , se, se)u (7.9b) Chapter 7. Case Studies 122 where z\ = 9 — 9r is the set point error. These are two coupled subsystems. The first subsystem (7.9a) is a linear known system driven by the sliding variable sq. With sq = 0, this is a stable subsystem because si is a positive design parameter. The second subsystem (7.9b) is an uncertain system and contains all the nonlinearities and uncertainties present in the original system. The drift term in equation (7.9b) depends upon the solution of (7.9a) which makes it coupled. However if fg(-) and go(-) can be represented by their bounding values, then the system (7.9b) can be separately stabilised. Stabilisation of the sliding variable, sg to zero will fulfill the control task by eventually stabilising the subsystem (7.9a) to the origin. The term Lgsg where Lg < 0 naturally provides asymptotic convergence so it need not to be cancelled. Within the operating range \fe(-)\ < Fe = 34672.27 and G6min < ge{-) < Ggmax where G vq m tn = 49379.10 and G vmq ax = 54576.9. It has been considered that there is ±5% variation in ge due to uncertain parameters in the system matrices A and b. The simultaneous maximisation of various uncertainties yields Fe/Gemin = 0.63. This value will certainly be conservative because it is unlikely that the uncertainties will have their maximal values at the same time. Therefore, the controller parameters will have to be tuned at the time of simulation and/or implementation. 7.2.4 Simulation results The system model (7.1) together with the sliding surface definition (7.3) for angular position control has been simulated. The controller parameter u0 when selected as u0 > F/Gmin = 0.63 gives a very fast response with large overshoot. Being u0 > F/Gmin a sufficient con dition, the controller is further tuned for good response by reducing the value of u0 = 0.21. Other controller parameters are selected as A = 1.4u0, k = 0.41 and W = 0.1. The angular position, 6, tracks the reference well as shown in Fig. 7.1. The sliding surface is shown in Fig. 7.2 and the corresponding angular speed u is depicted in Fig. 7.3. The motor current is shown in Fig. 7.4. 200 !- 150 9 ; 100 2.50.5 3.5 time (sec) Figure 7.1: Angular position, 6 (deg) Chapter 7. Case Studies 123 ~i------r J_____ L 0.5 1 2 2.5 time (sec) Figure 7.2: Sliding variable, s 3000 2000 1000 -3 0 0 0 0.5 2.5 3.5 time (sec) Figure 7.3: Angular speed, u (rpm) 4 2 £c E o -2 -4 0 0.5 1 1.5 2 2.5 3 3.5 4 time (sec) Figure 7.4: Motor current, I 7.3 DC motor speed control The DC motor model for speed control is a subsystem of the model (7.1) and it can be rewritten as x = Ax -f bu (7.10) y = Cx where -2.0652 25.989 0 A = b = -63.5268 -141.0943 2000 C = 1 0 where x\ is the angular speed of the shaft {uf) and x2 is the armature current (/) of the motor. Chapter 7. Case Studies 124 7.3.1 Sliding surface design Let the set point error angular speed of the motor shaft be ur and define the set point error ew = uj — ujr. Incorporating the set point error into a new state vector, 2 = [zi z2]T = [eu x2]T, the system (7.10) can also be written as Z = Az + bu (7.11) Now, consider the following sliding variable, sw, for speed control Zi — S„,z (7.12) where Su — [1 0]. The sliding variable has relative degree two with respect to the control input; it only depends upon the motor speed and does not depend upon the motor current. Equation (7.12) together with the system dynamics (7.10) yields — S^) z (7.13) — Su(Az + bu) — SuAx as SJb = 0 — ^11^1 + Ai2Z2 (7.14) Using equations (7.12) and (7.14), the transformation which transforms the states from z 6 ■ft2 into the [su can be given as Z = f[su s J T (7.15) where r 1 0 T = —A12 An A12 The second derivative of the sliding variable can be written using equation (7.13) as SuAz — SuA{Az 4- bu) — SuA2z 4- S^Abu = SUA2T [sw So;]71 4- SuAbu fui(Su') 4“ 9uj(su>) S(jj)u (7.16) Chapter 7. Case Studies 125 where Lw — ~SU}A/‘ 1 -AJA12 ^11 T fw(sj) = S^A2 0 -A Quji^Sun sj) S^Ab 7.3.2 Uncertainty bounds The system (7.10) together with equation (7.12), (7.14) and (7.16) can be written as yi = V2 y2 = - L uyi + fu(y2) + gu(yi,y2 )u where y \ = s^, y 2 — sw. The term —L^s^ where K > o need not be cancelled as it naturally provides asymptotic convergence. Within the operating range |/w(-)| < Fu — 1431.595 and GUmin < &,(■) < GUmax where GUmin = 49379.10 and GUmax = 54576.9. It has been assumed that there is ±5% variation in gu{-) due to uncertain parameters in the system matrices A and b. The maximisation of various uncertainties yields Fu/GUmin — 0.029. 7.3.3 Simulation results The system model (7.1) has been simulated for angular speed control. The motor speed has relative degree two with respect to the control input. In this case therefore the sliding surface is nothing but the speed error. The same controller used for position control has been used here for speed control. The controller parameters are u0 = 0.21, A = 1.4u0, k = 0.41 and W = 0.1. The angular speed uj tracks the reference as shown in Fig. 7.5 and the corresponding angular position, 6, is depicted in Fig. 7.7 and the motor current has been shown in Fig. 7.4. The abrupt change in 9 between 1.5 and 2 sec. is due to the change in the angular position of the motor from 360 to zero degrees. 2000 1500 F S 1000 / 3 s 500 » f 0 if -5 0 0 .. 2 2.5 time (sec) Figure 7.5: Angular speed, uj (rpm) Chapter 7. Case Studies 126 r , ... 1 1 lr 1 1 .4 1______1______1______1______1______1______1______1______0 0.5 1 1.5 2 2.5 3 3.5 4 time (sec) Figure 7.6: Sliding variable, s 400 ; 200 100 2.5 3.50.5 time (sec) Figure 7.7: Angular position, 9 (deg) 4 1------1------1------1------1------1------1------r _6I------1------1------1------1------1------1------0 0.5 1 1.5 2 2.5 3 3.5 4 time (sec) Figure 7.8: Motor current, / 7.4 Controller implementation This section presents the results obtained by actual implementation of the controllers on the DC motor rig. Two separate experiments, one for angular position control and a second for angular speed control, have been carried out. 7.4.1 Experimental Setup A schematic diagram of the hardware configuration for the Modular Servo System MS 150 Mk2 produced by Feedback Instruments Ltd. is shown in Fig. 7.9. The system comprises a DC motor and other peripherals such as amplifiers and potentiometers which are required to run the controller using dSPACE. The data transmission and reception is through the dSPACE Chapter 7. Case Studies 127 real time control environment. The dSPACE Real-time Control Environment allows easy prototyping of controllers within the SIMULINK environment and downloading of these controllers to dedicated Digital Signal Processor (DSP) based systems for implementation, In SIMULINK and dSPACE Hardware of MS150Mk2 Unit OP-amp [reference 2-sliding DC controller Pre-Amp Signal M otor Conditioning Servo-Arap (Power Amp ) Unit Conversion Unit Potentiometers Figure 7.9: Schematic diagram of the experimental setup Figure 7.10 shows the SIMULINK setup of the 2-sliding control algorithm. The SIMULINK model of the designed controller is converted into C-code for proper dSPACE implementa tion using a specific function in the MATLAB/RealTime Toolbox. This C code is further cross compiled into an executable file downloaded to the DSP and run in real time on a Texas Instrument TMS 320F240 DSP microcontroller. This microcontroller is connected with the DC motor. Once the model is implemented, the simulation runs completely on the real-time hardware DS1102. The graphical user interface (GUI) of the dSPACE software enables various aspects of the operation of the control system to be viewed and enables the controller parameters to be further tuned while the control system is operating. The DC motor is a permanent magnet type. It has a maximum speed of 3000 rpm. Its torque at 2A is of the order of 0.1 Nm and the rotor inertia is 3.9 x 10_5Kg m2. The input is applied in terms of a voltage which passes through a pre-amplifier, servo-amplifier and power amplifier chain before being applied to the motor. The servo amplifier connected to the motor has two inputs to enable the motor to run in both directions. The position and speed are read by precision servo mounted type potentiometers fitted with calibrated position and speed indicating dials. There is the facility to provide +1V step disturbances at the output. The calibration graphs of angular position, 9 and angular speed, u are shown in the Fig 7.11 and Fig. 7.18 respectively. Chapter 7. Case Studies 128 K3 .... Figure 7.10: SIMULINK setup of the controller 7.4.2 Position Control This section presents the implementation results for angular position control of a DC motor. The sliding surface definition (7.3) used for simulation in the previous section is used here as well. The controller parameters selected are based on the uncertainty bounds calculated using the system model (7.1). The controller parameters are selected as in simulation u0 — 0.21, A = 1.4u0, k = 0.41 and W = 0.1. The same controller has been used for position as well as speed control purpose. 150 100 Slopes 33.8836 Offset= 39.5011 -5 0 -100 -1 5 0 -200 -6 -5 -3 - 2 8 (Volts) Figure 7.11: Calibration of angular position (i 9) Chapter 7. Case Studies 129 The angular position, 0, together with the reference position is shown in Fig. 7.12. Fig 7.13 shows the zoomed rising and falling edges of position tracking which are otherwise not clearly visible due to the large time scale. The sliding surface is shown in Fig. 7.14. The corresponding angular speed, uj, is shown in Fig. 7.15 and the motor current is shown in Fig. 7.16. The control input u is shown in Fig. 7.17. 200 150 -50 20 40 time (sec) Figure 7.12: Position Control 160 150 140 140 "§ 130 O) § 120 ■E 100 110 100 28.5 29.5 36.5 37 37.5 time (sec) time (sec) Figure 7.13: Zoomed view for position tracking 1 j i II i : I f : ' ...... i...... 1 ...... i i i i 0 10 20 30 40 50 60 time (sec) Figure 7.14: Sliding variable, s Chapter 7. Case Studies 130 — 1 ...... 1...... 1 8 0 . L ____ 1 1 -2 1 1 I 10 20 30 40 50 60 time (sec) Figure 7.15: Speed I I I I I 0.5 y | g | | | | j ^ O -0.5 ...... -1 10 20 30 40 50 60 time (sec) Figure 7.16: Current 30 40 time (sec) Figure 7.17: The control voltage, u 7.4.3 Speed Control This section presents the implementation results for angular speed control of the DC motor. The error in speed is similar to that used for simulation in the previous section. The controller parameters selected are u0 = 0.21, A = 1.4u0, k = 0.41 and W = 0.1. The angular speed u together with the reference speed is shown in Fig. 7.19. Fig 7.20 shows the zoomed rising and falling edges of speed tracking which are otherwise not clearly visible due to the large time scale. The sliding surface (speed error) is shown in Fig. 7.21. The corresponding angular position, 6, is shown in Fig. 7.23 and the motor current is shown in Fig. 7.22. The control input, u, is shown in Fig. 7.24. Chapter 7. Case Studies 2500 2000 Slope= 352.0334 Offset- -9.8518 1500 2 ccCL 1000 500 -5 0 0 S p eed (Volts) Figure 7.18: Calibration of angular speed (a;) 2000 1500 1000 r~ -500 25 30 time (sec) Figure 7.19: Tracking the reference speed 800 1200 1000 600 8 > 800 T3o ® 400 g> 600 S2 400 200 200 12 12.2 12.4 12.6 30.3 30.4 30.5 30.6 30.7 30.8 time (sec) time (sec) Figure 7.20: Zoomed view for speed tracking The DC motor during speed control was subjected to a load at the motor shaft. The control control The shaft. motor the at load a to subjected was control speed during motor DC The system shows robustness to load variations of varying degrees. Loading is provided by eddy- by provided is Loading degrees. varying of variations load to shows robustness system load to Robustness 7.4.4 Chapter Current _2 -1 - 2 - 4 - 7. 4 2 n u 4 5 5 0 5 0 5 0 5 0 45 40 35 30 25 20 15 10 5 0 5 0 5 0 5 0 5 0 45 40 35 30 25 20 15 10 5 0 ______1 ------CaseStudies — i “ — I— i i i i i i i i i > i i I ______1 ------i ______I ______...... 1 ------Figure 7.24: The control voltage, u voltage, control The 7.24: Figure I Figure 7.21: Sliding variable, variable, Sliding 7.21: Figure Figure 7.23: Angular position Angular 7.23: Figure : : ...... i i r~ i i i i i ...... i ______I ______1 ------iue72: Current 7.22: Figure » ...... - ; - ; i______i i ______1 ------time(sec) time (sec) time time (sec) time time(sec) .1 ...... I i ______1 ------1 : 1 I ...... i i ______s 1 ------...... i i ______1 ------...... i I ______r ______... 132 Chapter 7. Case Studies 133 current braking using a permanent magnet which has a scale to indicate the amount of load applied. The following load is applied to the motor shaft: Loading Scale Motor Speed (rpm) Load Torque (Nmm) 5 686 60 3.5 1037 22 1.5 1733 22 The load has been applied manually and removed at each speed shown in the figure. It can be seen from Figures 7.25 and 7.26 that the motor does not deviate from the speed set point due to the activation and deactivation of load. Other motor variables shown in Fig. 7.22, Fig. 7.23 and Fig. 7.24 are well within the system limits. 2000 1500 1000 . 500 -500 20 time (sec) Figure 7.25: Tracking the reference speed 4 ------1------1------1------1------r 1 1 ' I .4------1------1------1------1------1------0 10 20 30 40 50 60 time (sec) Figure 7.26: Sliding variable 2 1------1------1------r time (sec) Figure 7.27: Current Chapter 7. Case Studies 134 -5 -10 _L_ 1 _l_ 20 30 4050 60 time (sec) Figure 7.28: Angular position 0 10 20 30 40 50 60 time (sec) Figure 7.29: The control voltage, u 7.5 Liquid level control in coupled-tanks In this case study, a 2-sliding controller has been designed for liquid level control of two connected water tanks as shown in Fig. 7.30. The controller has been designed based on the theory developed in Chapter 6 and simulation results are presented in Section 7.5.3. The actual results from the implementation of the controller on the twin-tanks rig are presented in Section 7.6. 7.5.1 System modelling Consider the two connected tanks [20] as shown in Fig. 7.30. The total water heads in Figure 7.30: The plant of a liquid-level control system Tank 1 and Tank 2 are hi and h2 respectively, which are the two outputs of interest and Chapter 7. Case Studies 135 qi, q2, are the two inflows into the tanks. It is assumed that the capacities, C\ and C2, of Tank-1 and Tank-2 respectively are nonlinear functions of the heads, i.e., the cross sectional areas of both the tanks are varying in a nonlinear manner along their heights. Moreover, the interconnecting and outflow pipes have a square-root relationship to the water head. The admittance coefficients ki and k2 for valve 1 and valve 2 respectively are assumed to be constants. The system is a nonlinear dynamical system and the governing dynamical equations can be written as C\hi — — kiVKh + qi (7.17) C2h2 = k\ \/ Ah — k2yj~h2 + q2 (7.18) where (7.19) The inflow rates, Ui and u2 are the control inputs. The tank capacities Ci} i = 1,2 are nonlinear functions of the water head in the tanks but remain bounded as follows Cim < C{ < CiM, i — 1,2. where C*m and CiM are some positive constants. 7.5.2 System constraints The fluid flow into the tanks (qi and q2) cannot be negative because the pumps can only pump in. Therefore constraints on the inflow are given by qi > 0 and (7.20) q2 > 0 (7.21) In the steady state, for constant water level set points, the respective derivatives must be zero separately i.e., hi = h2 = 0. Therefore, in the steady state, the following algebraic relationship holds. 0 — —k\ y/~Ah -j- (7.22) 0 = ki V Ah — k2\fh2 + Q2 (7.23) where the steady state inflows (Qi, Q2) are given by Qi = kiV Ah (7.24) Q2 = —ki V~Ah + k2 yj~h2 (7.25) Chapter 7. Case Studies 136 Using equation (7.24), the constraint (7.20) on the input can be can be reformulated as a constraint on the output set point as follows V Ah > 0 =»/ii > h2 (7.26) Similarly, constraint (7.21) can be reformulated using (7.25) in the following inequality k2 y/hv > k\ V Ah klh2 > k\{h\ - h2) or ( k\ + k\)h2 > k\h\ ^ h 2 > -A C _ h t (7.27) n / j r 1/2 Therefore, for given values of the plant parameters ki, k2, in order to satisfy the constraints (7.20) and (7.21) on the inflows, the desired liquid levels in the tanks must satisfy the constraints (7.26) and (7.27) which can be combined as follows ^ ^ < 1 (7-28) kj + k% hi For plant parameters ki = 2, k2 = 1, the equilibrium point h2 must be greater than or equal to 80% of hi. For the equilibrium point hi = 1 and h2 = 1, the steady state inflows are Qi = 0, Q2 = 1; and for the equilibrium position hi = 1, h2 = 0.8, the steady state inflows are Qi = 0.89 and Q2 = 0. It has been seen in [21, 49, 81] that in chemical plants, selecting the flow rate as an input is more effective than using flow as the input. Thus, if flow rates (qi and q2) are considered as the inputs, the system dynamics can be written in 1-0 form as: C ,/i, = -c[h1-k 1^ ~ + q1 (7.29) or hi = % { - h V A h + qif + - kl[Ciqi c l 2 C C i 2 C1C2 VM fc]fc2V^2 , + Ui 2 y/AhC Ah . h2 C2h2 = -(Vfo + k i-y ^ -h —^ + q, (7.30) 2 \/ Ah 2\fh2 v C2n nrr 7 rr~ \2 (Ci 4 -C2 ) kik2{h2 — AK) or ft, = VAft - + q2f - + - ^ 2 hqi _ / ki J f c N 72 + u2 2C2 2C i\JAh \ y/Ah \fh2 J 2C2 Chapter 7. Case Studies 137 where, C[ — dCi/dhi, i = 1,2 and Ah can be calculated as per equation (7.19). It seems from equations (7.29) and (7.30) that there is a discontinuity in the right hand side at Ah = 0. In fact, for Ah = 0, there is no discontinuity and the system model is governed by the following dynamic equations: Ci hi = —^kiVAh — — (?i + <71 Cl Oi QI I ^ OF ^ = C f “ Q2qi + Q~Ul c*2 ^2 = —C'2h2 — k2—j=-\-q2 2y h2 f _ ( rii , ^ 2 \ ( ki\/Ah ( q2 ^ /~q , ^ 2 , 1 or h2 - -(C2 + ^ = ) |_ _ _ + _ j+ _ V k + _ + _ lla 7.5.3 Simulation results The numerical values of the plant parameters are taken from Chang [20] as k\ = 2 and k2 = 1. The bounding values of the tank capacities are 3.2 < C\ < 4.8 and 0.8 < C2 < 1.2. For the simulation purpose, the actual tank capacities are given by the following nonlinear functions Ci = 4 + 0.8sin C2 = 1 + 0 . 2 sin ( 7^ 2) The system model in (7.30) and (7.30) can be simply written as hi = fi(hi, h2, hi, h2) + -p^-ui (7.31) Ci h2 = f2 (hi, h2,, hi, Ii2 ) + — u2 (7.32) 0 2 For the given plant, the functions Cifi(-) and (Fig. 7.31) can be plotted by max imising all possible variations. The maximal values calculated are C i|/i(-)|max = 2.9 and C2\f2(-)\maX = 12.3. The controller parameters u0l and u02 should be greater than Ci\fi(-)\rnax and C2\f2 (-)\max respectively. However, because all possible uncertainties may not occur simultaneously, the controller parameter values will be fine tuned for different cases separately. Chapter 7. Case Studies 138 o (a) Plot of CM-) (b) Plot ofC2/ 2(-) Figure 7.31: Plots of drift functions Constant heights Equal set point values The desired values of the water level set point for both the tanks are /i1 = h2 = 1. The controller coefficients selected on the basis of the bounding values are u0l 2 = [4,5], Jclt2 = 20, Ai;2 = [35,25] and Wli2 = 2. 0.8 .-0 .6 0.4 0.2 time (seconds) 0.8 0.4 0.2 time (seconds) Figure 7.32: Regulation of equal liquid levels Chapter 7. Case Studies 139 4 3 £c 1 0 ■1 1 2 3 4 5 67 8 9 100 time (seconds) 3.5 2.5 0.5 time (seconds) Figure 7.33: Inlet flows q\ and q2 The inflows (<71 and q2) settle at the desired equilibrium values which are Q\ = 0 and Q2 = 1 as can be seen in Fig. 7.33. Unequal set point values The desired values of the water level set points for both the tanks are selected as hi = 1 and h2 = 0 . 8 which satisfy the constraint given in equa tion (7.28) for the flows to be positive. The simulation results using the same controller are presented in the following figures. 0.8 0.4 0.2 time (seconds) 0.8 0.6 0.4 0.2 time (seconds) Figure 7.34: Regulation for different liquid levels Chapter 7. Case Studies 140 4 3 1 c 0 0 1 2 3 4 5 6 78 9 10 lime (seconds) 2.5 lime (seconds) Figure 7.35: Inlet flows It is evident from Fig. 7.34 that the set point is appropriately tracked. Moreover, the inflows (qi and q2) settle at the desired equilibrium values which are Qi = 0.89 and Q2 = 0 as can be seen in Fig. 7.35. Tracking a gradient Case 1: In this case the water levels in both the tanks rise independently at a given rate to the final level which is the same for both the tanks. For Tank 1, the level rises to hi = 1 at a gradient of 1/3 and in Tank 2 it rises to the same level h2 = I but with a gradient 2/7. The controller coefficients for the simulation study are k\%2 = 20, Ai)2 = [35,25] and Wi>2 = 2 and u 0l 2 = [1, 2]. 0.2 time (seconds) 1 - 0.8 - ^fi.6 - 0123456789 time (seconds) Figure 7.36: Tracking the same liquid levels with different gradients Chapter 7. Case Studies 141 x 10"3 -2 - 4 0 1 2 3 4 5 6 7 8 9 time (seconds) x 10"4 u ____ “ T" ' ■ ** 7 f 1 ...... i i i i 1 0 1 23456789 time (seconds) Figure 7.37: Sliding variable 2.5 0.5 0 1 2 3 4 5 6 7 8 9 time (seconds) 0.8 0.6 0.4 0.2 time (seconds) Figure 7.38: The inlet flows Case 2: This case is similar to the previous one, but the desired values of the water levels in both the tanks are not the same i.e., hi = 1 and h2 = 0.8 with a rising slope of 1/4. The same controller has been used to simulate this case as well. Chapter 7. Case Studies 1 0.8 0.4 0.2 0 0 1 2 3 4 5 6 7 8 9 time (seconds) 1 0.8 0.4 0.2 0 0 1 2 3 4 56 7 8 9 time (seconds) Figure 7.39: Tracking different liquid levels with different slopes x 10-3 8 6 4 2 * 0 -2 - 4 -6 0123456789 time (seconds) x 10"3 2 1 ■n" 0 -1 -2 0123456789 time (seconds) Figure 7.40: Sliding variable Chapter 7. Case Studies 143 2.5 0.5 1 2 3 4 5 6 7 8 90 time (seconds) 0.6 S 0.4 £ 0.2 0 1 2 3 4 56 7 8 9 time (seconds) Figure 7.41: The inlet flows Figures 7.33 shows that inlet flow is negative for a certain time period, which is not possible in the physical system. The possible cause for this behaviour may be a windup effect at the input integrator and one of the solutions for this is to apply an anti-windup scheme [26, 42, 101]. This is discussed in the next section. 7.5.4 Anti-windup scheme The actuators on the tank system do not allow reverse flow. This establishes a lower limit for the inlet flow. Theoretically, there is no upper limit for saturation. However, there may be an upper saturation limit for each actuator which varies according to the design type and manufacturer. If integral action is combined with such actuators, integrator windup may occur whenever the actuator saturates. In the presence of a large error signal the integrator continues to increase and thereby forces the actuator to remain saturated for an extended period of time. The actuator saturation degrades the system performance and produces large overshoots. Edwards and Postlethwaite [26] present a comparative study of anti-windup schemes for a plant being controlled by a linear controller. The following anti-windup scheme is different in the sense that it does not assume that the controller is linear. The scheme hence does not utilise controller parameters for tuning the anti-windup loop. The actual control input is obtained by integrating the output of the nonlinear 2-sliding controller. This integrator has been shown separately for implementing the anti-windup scheme. Chapter 7. Case Studies 144 NLC Plant Figure 7.42: Anti-windup Scheme for input saturation For the scheme to be stable, the anti-windup loop between v, the integrator input and d, the saturated output, must be stable. The input signal, v can be written using the Laplace transform as v = —k ( —d + - v \ 5 where s is the Laplace transform variable and the transfer function from dtov can be written as: v(s) ks d(s) = J + k ( } which is stable for all positive values of k. Higher values of k should give better performance. In other words, for k » 1, anti-windup scheme must satisfy u = d. Consider C(s) = 1/s, the actuator input in Laplace transform form, u(s), can be written as u(s) = C (s)v(s) = C (s) (v(s) — kue(s)) = C (s) (v(s) — ku(s) + kd(s)) (1 + kC(s)) u(s) = C(s)v(s) + k C (s)d (s) ° ( s) -/ n kC (s) ^ \ or u{s) = TTkCiT)v{s) + J T k c & d{s) = d(s) (7.34) as for large positive value of k, C(s) .. n kC(s) 1 + kC(s) ~ 0 a i + kC(s) ~ The anti-windup scheme is now applied on the equal and constant height case in Section 7.5.3 with a condition that the flows (< 71,2 ) must be positive and limited by an amplitude of 3 units. The only anti-windup parameter k is selected equal to 5. All other controller parameters are kept the same. The simulation results are as shown in Fig. 7.43 and 7.44 Chapter 7. Case Studies 145 0.4 0.2 time (seconds) 0.4 0.2 time (seconds) Figure 7.43: Liquid levels control with anti-windup scheme 2.5 0.5 -0 .5 time (seconds) 2.5 0.5 -0 .5 time (seconds) Figure 7.44: The inlet flows The results achieved with the anti-windup scheme are similar to the previous results but actuator constraints are respected. This can also be achieved by applying a saturation limit to the input. Chapter 7. Case Studies 146 7.6 The twin-tanks laboratory rig 7.6.1 System description The twin-tanks system consists of two small tanks mounted above a reservoir which provides storage for the water. Water is pumped into the bottom of each tank by two independent pumps. The head of water in each tank is clearly visible on the scale attached at the front of the tanks. Each tank is fitted with two outlets of different size at the base. The amount of water returning to the reservoir through the outlet pipes is approximately proportional to the square root of the head of water in the tank. The separating wall between the two tanks has two circular holes of diameter 5mm each to provide coupling between the tanks. Two separate potentiometers are used to measure the liquid levels in both tanks. Each poten tiometer is interfaced with an A/D converter via a signal conditioning unit to comply with the input requirements of the A/D. The potentiometer output signals are in the voltage range 0 — 2.2V and the signal conditioning unit converts this to the range of 0 — 10V. The poten tiometers are calibrated such that the zero level represents the rest point of the water level i.e., when the tank is nearly empty, while the full state (2.2Volts) is calibrated at the level of the opening of the rear overflow stand-pipe. The D/A converter produces an analog output from —10 to +10V. However, only the positive range is utilised here due to a positive voltage constraint on the pump input from 0 to +10V. The D/A converter is connected to the pump via a power amplifier. The controller code is written in C++ and runs on a Windows 98 PC with CPU speed 166 MHz. The pump interfaces between the plant and the computer. It pumps water from the storage tank to the water tank when a voltage is applied. The pump only increases the liquid level and is not responsible for pumping the water out of the tank. It is assumed that the back pressure created by the water-head does not affect the flow rate of the pump significantly. A flowmeter is connected in series with each pump to measure the flow. The flow rate of the pump is plotted against the digital control signal in Figure 7.46(a) and 7.46(b). It has been noticed that the pump characteristics are not linear for the whole digital input range of 0 — 255. Moreover the pump saturates when the digital input reaches 255. It has been calculated that the gradient of Fig. 7.45(a) is 17.1340 digits/cm and that of Fig. 7.45(b) is 17.2543 digits/cm. The A/D converter is calibrated by reading the digital values for known liquid levels. Similarly, the D/A converter is calibrated by giving known digital commands to the pump and reading the corresponding flow from the flowmeter. The gradient of Fig. 7.46(a) is 0.7913 cc/sec/digit and that of Fig. 7.46(b) is 0.9042 cc/sec/digit. Chapter 7. Case Studies Table 7.1: Calibration data for potentiometers and pumps Data for potentiometers Data for pumps Water Level hi h2 Input flow Qi 92 in cm in digits in digits in digits in litres/min in litres/min 0 0 4 40 1.25 2.4 0.5 3 6 50 1.5 3.2 1 12 14 60 2.5 3.8 2 33 32 70 3.45 4.4 3 50 50 80 4.1 4.75 4 68 68 90 4.5 5.2 5 87 85 100 5 5.6 6 100 100 110 5.6 6.2 7 118 119 120 6 6.5 8 136 137 130 6.4 6.8 9 153 155 140 6.75 7.1 10 170 170 150 7.1 7.45 11 187 189 160 7.4 7.6 12 202 206 170 7.5 7.75 13 219 224 180 7.6 7.95 190 7.95 8 200 8.1 8.1 225 8.5 8.4 300 300 250 g 250 200 ~ 200 0> 5 150 3 150 to 2; 100 -5 0 -5 0 - 5 0 5 10 15 20 - 5 0 5 10 15 20 Water level hi (cm) Water level h2 (cm) (a) Potentiometer 1 (b) Potentiometer 2 Figure 7.45: Calibration of potentiometers Chapter 7. Case Studies 148 200 200 150 pP 150 100 100 a- 50 0 100 200 0 100 200 Digital value of q1 Digital value of q2 (a) Pump 1 (b) Pump 2 Figure 7.46: Calibration of pumps The basic block diagram of the twin tank system is shown in Figure 7.47. The digital value of the liquid level ( y) is compared with the desired digital level (r). From the block diagram in Fig 7.47, it seems that the controller should be digital in nature. However, if the sampling interval is very small, then a continuous controller can be implemented in this setup. In this particular case the sampling step is of 2.66 x 10~4 which is quite small and hence the controller can be implemented in continuous sense. Controller D/A Pump Process A/D Sensor Figure 7.47: Block diagram of the twin tank system 7.6.2 The rig model The twin-tanks rig contains two rectangular tanks separated by a wall. The separating wall between the two tanks has two circular holes which together form the connecting pipe with admittance coefficient fa; this will be referred to as pipe fa. Each tank in the twin tanks configuration is equipped with two outlet pipes of different radius. These two pipes have admittance coefficients of fa and fa where fa > k2 and will be referred to as pipe k2 and pipe fa, respectively. The pump cannot fill the tank if both pipes k2 and fa are opened. Therefore, only pipe fa is used to design the controller. A leak can be simulated by opening pipe fa which allows more outflow than pipe fa. The dynamic equations of the rig can be Chapter 7. Case Studies 149 Tnriirr nuipii ■ .... Figure 7.48: The twin tank rig in the Control Laboratory formulated using (7.17) and (7.18) k = - — s/Ah + jq!(7.35) ft? — —V Ah — \Zfi2 + —Q2 (7.36) where y/Ah is same as defined in equation (7.19). Both tanks have the same area of 11.6 x 13.4 = 155.44 cm2 and ki = 23.45 and k2 = 17.62. The method for calculation of the admittance coefficient is detailed in the following section. For the plant available in laboratory, the water level in Tank 2, h2, must not be theoretically less than 63.89% that of h\ to satisfy positive flow constraints. If the flow rates are considered as the inputs,i.e. q\ = U\ and q2 = u2, the system dynamics can be written in 1-0 form as: y ki hi — h2 1 h' = ~2A^KT^ + A U'(73?) y h\ — h2 k2 • 1 h2 = — — — h2 4—jU2 (7.38) 2A y/lfn-fal 2 Vh2 2 A 2 It seems that at hi = h2 the right hand sides of equations (7.37) and (7.38) are discontinuous. At hi = h2, the system model is decoupled and follows the following dynamic equations hx = l 9l (7.39) h* = - ^ V ^ + ^ 9 2 (7.40) Chapter 7. Case Studies 150 which can be written as the following form as required by the following controller synthesis. /»! = JU! (7.41) ko ko 1 h2 = 2A*-2Aiq2 + AU2 ( 7 ' 4 2 ) 7.6.3 Admittance coefficients of pipes Consider a gravity drained tank as shown in Fig. 7.49. The dynamics of a gravity drained tank is given by a square-root-relationship [66] as in equation (7.43) where k is the admittance coefficient of the outflow pipe connected to the bottom of the tank. Figure 7.49: Gravity drained liquid tank Ah = - k V h (7.43) Let h0 represent the initial liquid level and T denote the time taken in draining that volume through the pipe. Integration of equation (7.43) yields f° dh k r Jho h ~Jo A 0 k T 2 V h ~ -J ho A 0 =* 2 = j T (7.44) From equation (7.44), it is evident that the 2V^o vs. T plot has a constant gradient of k/A. The pipe admittance coefficient k can be calculated by multiplying the gradient of the plot with the tank cross section area, A. To calculate the admittance coefficient of a particular pipe connected into the tanks, the liq uid from various levels in the tank was allowed to fall freely under gravity through the pipe separately and data collected. This was then plotted in Figures 7.50(a), 7.50(b) and 7.50(c). Chapter 7. Case Studies 151 Table 7.2: Experimental data for pipes Water Level Draining time T (sec) h0 (cm) Pipe 1 Pipe 2 Pipe 3 5 32.45 6 33.36 38.01 27.65 7 45.13 31.8 8 51.37 36.5 9 39.68 57.42 39.55 10 63.08 43.14 11 69.04 47.12 12 47.75 75.07 50.82 13 80.45 55.71 14 86.11 15 51.78 91.88 The draining experiment was carried out three times for every liquid level h0. For the in terconnection link between the two pipes, tank 1 is drained through tank 2. This does not directly correspond to ‘drained under gravity’ and therefore the value of ki is likely to be in error. Moreover, both the tanks have some pipes lying inside them so the cross section area was not constant as assumed during the experimentation. Error may also be introduced in estimating the value of k\ and k2 because the time taken in draining out the water filled up to various heights in the tank is not measured to a great accuracy. It is difficult to judge when the water height has become zero because the water keeps trickling out for some time. 10 5 5 8 4 4 © 6 3 3 4 2 2 2 1 1 0 0 0 0 20 40 60 0 20 40 60 80 100 20 40 60 (a) The connecting pipe (b) The pipe No. 2 (c) The pipe No. 3 Figure 7.50: Calibration of pipes Chapter 7. Case Studies 152 Online water level measurement is inaccurate because the thread to which the float is con nected does not remain vertical when the water level in the tank is rising. 7.6.4 Simulation study of the rig model The twin-tanks model given in equations (7.37) and (7.38) can be simply written as hi = fi{hi,h2,hi,h2) + —ui (7.45) h2 — f2(hi, h2, hi, h2) + ~^u2 (7.46) where ki hi - h2 /i(-) = - 2A y/\hi — h2\ f•_ r \ _ ^ 1 — 2 k 2 • ~ 2Ay/\hi-h2\ ~ The set point water level is selected as hi = h2 = 12cm. From Table 7.1, correspond ing digital values for both channels can be selected as 202 and 206 respectively. For the given plant the functions A fi(-) and A f2(-) are plotted in Fig. 7.51 and the bounding val ues are calculated by maximising all the possible variations as A\fi(-)\max = 23.4 and A |/2(-)La* = 33.7. The controller parameters it0l and u02 should be greater than A\fi(-)\max and A \f2(-)\max respectively. The controller parameters selected for the simulation study shown in the fol lowing figures are u0l2 = [25,35], Ai)2 = [100,150], &i>2 = [1,1] and W ij2 = [10,10], which satisfy the selection criteria discussed in the previous Chapter. The inflows at equilibrium, (Q i ,Q 2), are calculated using the following equations 0 = sign(/iI - h2) + I q , (7.47) 0 = ^n/|Ai - ft2|sign(ft, - ft2) (7.48) which yields Qi = 0 and Q2 = 61.07. The simulation results in Fig. 7.52 and 7.53 show good stabilisation of the water levels in both tanks. The value of inflow in both tanks settles to the steady state values shown in Fig. 7.53. Chapter 7. Case Studies 153 250 200 250 150 200 150 100 100 (a) The bounding value of the function -A i/i(-) 250 250 200 150 100 (b) The bounding value of the functionA 2 h(-) Figure 7.51: Bounding values Chapter 7. Case Studies 154 14 12 10 8 6 4 0 510 15 20 25 30 time (seconds) 14 12 10 8 6 4 0 5 10 15 20 25 30 time (seconds) Figure 7.52: Stabilisation of water level in both tanks 0 5 10 15 20 25 30 time (seconds) 300 250 5 150 - 100 30 time (seconds) Figure 7.53: The flows into both tanks 7.6.5 Controller implementation on the rig To implement the controller on the rig, the code has been written in C++ and runs on a PC with 166 MHz CPU speed. The experimental rig available in the laboratory has already been described in Section 7.6. The implementation results are shown in the following figures. It has been noticed that the pumps do not respond to digital inputs less that 25 in the case of Tank 1 and 30 in case of Tank 2. To counter this dead zone, a lower saturation limit of 25 and 30 is applied to both motor inputs respectively. During practical implementation of the algorithm it is noticed that the controller used for simulation studies does not provide good Chapter 7. Case Studies 155 results as expected due to the presence of second order actuator dynamics. Therefore con troller is further tuned to such that it provides acceptable results. The controller parameters finally selected are A = [150,200], u0 = [75,100], k = [37,37] and W = [300,300]. In the case where both tank levels are the same, the input flow ( qi) should settle to zero according to the simulation results. However, in Fig. 7.54, it settles at 25 which means zero inflow because a lower saturation limit of 25. In case of unequal water levels in both tanks maintain the different levels robustly, the effective q\ settles to a constant value and the flow into the Tank 2, q2, settle to zero. Stabilisation of same levels in both the tanks 150 100 40 time (sec) 150 100 40 time (sec) Figure 7.54: Implementation results for stabilisation of same water level in both tanks 250 j 200 2 150 100 lime (sec) 250 s 200 u 150 100 32 time (sec) Figure 7.55: The control input to the pumps Chapter 7. Case Studies 156 Stabilisation of different levels in both the tanks 100 - 50 - 0 U------1------1------1------1------1------L. 0 16 32 48 64 80 96 time (sec) 150 100 time (sec) Figure 7.56: Implementation results for stabilisation of different levels in both tanks 0 16 32 48 64 80 96 time (sec) 250 FT * 20 0 1 I o H G I £ 150 H c I I 100II 50 1 ' 0 — 0 16 32 48 64 80 96 time (sec) Figure 7.57: The control input to the pumps 7.6.6 Robustness to leakage To simulate a leak in Tank 1, the outlet pipe with admittance coefficient equal to kx is opened. In the case of Tank 2, a leak condition has been simulated by opening the outlet pipe with wider cross sectional area (admittance coefficient k3 = 26.26) than that of the pipe which has been used to model the system (ki = 23.45). The controller robustly stabilise the water levels to the desired level in both cases. Chapter 7. Case Studies 157 It can be seen in Fig. 7.59 that drop in water level due to the leak in Tank 1 is compensated by increase in the inlet flow, qi. 150 100 time (sec) 150 100 time (sec) Figure 7.58: Stabilisation of levels with leak in Tank 1 0 ______I______I______I______I______1______I______I______I______I______L- 0 8 16 24 32 40 48 56 64 72 80 time (sec) q I______I______I______I______I______1______1______I______I______I______L- 0 8 16 24 32 40 48 56 64 72 80 time (sec) Figure 7.59: The control input to the pumps 7.7 Conclusion In this chapter, two case studies involving the 2-sliding controller, developed in Chapter 5, have been presented. The first case study involves robust control of the angular position and speed of a DC motor in SISO configuration. The results obtained by implementation using dSPACE have been shown to be robust. The uncertainty bounds are calculated using interval Chapter 7. Case Studies 158 mathematics by maximising the uncertainty terms. The results validate the theory presented in Chapter 5. In the second case study, liquid level control in interconnected twin-tanks is considered. This is a MIMO case study. The system has been modelled and a simulation study carried out. Actual implementation is then performed on the rig. The controller is designed based on the theory developed in Chapter 6. The algorithm is run on a PC. The implementation results show robustness to parameter variations such as tank area and the admittance coefficients of various pipes. The pump dynamics were also ignored during controller design. The implementation results verify the theory proposed in Chapter 6. The next chapter concludes the thesis. A great deal of research is still required in this emerg ing area of higher order sliding mode control. Therefore, directions for further research are identified and are also presented in the next chapter. C h a p t e r 8 C o n c l u s i o n s A n d F u t u r e Re s e a r c h 8.1 Introduction After discussing basic theory and definitions pertaining to higher order sliding mode control in Chapter 3, case studies were presented in Chapter 4 and motivation made for the work carried out in this thesis. A new two sliding algorithm was presented in Chapter 5 and it was extended for a class of MIMO systems in Chapter 6. The algorithm has been implemented for DC motor control and water level control and the results are presented in Chapter 7. This chapter concludes the thesis in Section 8.2 and suggests future directions for the re search emanating from the thesis in Section 8.3. 8.2 Concluding remarks The thesis considered the development of higher order sliding mode based control algo rithms. Higher order sliding mode control keeps the main advantages of standard sliding modes and has the additional advantage that it can be used to remove chattering effects, providing smooth or at least piece wise smooth, control. The method also provides better accuracy with respect to switching delays. The practical importance of higher order sliding mode control has been demonstrated. A theoretical case has been made for the application of the super-twisting algorithm to ro bust speed control of a diesel engine and bounds on the controller parameters have been generated. Robustness to unmodelled dynamics has also been discussed in the context of the super-twisting algorithm and it has been shown that the method exhibits limit cycle be haviour in the case of unmodelled actuator dynamics of first order. In the area of dynamic sliding mode control, MIMO control of an IC engine in which speed, manifold pressure and air-fuel-ratio track the desired trajectory has been presented. This demonstrates the application of dynamic sliding mode control for known systems, which are 159 Chapter 8. Conclusions And Future Research 160 not affine in the control. A new second order sliding algorithm has been developed to stabilize a second order nonlin ear system. This method is thus applicable to systems where the sliding variable has relative degree two with respect to the control input. Moreover, it does not require the derivative of the sliding variable to be measured or estimated and hence reduces the number of sensors re quired for control implementation. The algorithm produces a smooth input via the dynamic control even though the sliding variable is not control dependent. This is in contrast to other algorithms available in the literature. The following simulation studies validate the theory developed in the thesis. • The speed control of an IC engine without acceleration measurement has been pre sented. The selected controller parameters do not conform to the original selection criteria for the super-twisting algorithm and therefore it was suggested that a mod ified algorithm has the potential to deal with systems with relative degree two with respect to the sliding variable for which the super-twisting algorithm was not designed initially. • The simulation studies for position control of an underwater vehicle, stabilisation of a single axis jet controlled aircraft model and stabilisation of a tunnel diode circuit model have been presented. All these systems have uncertain nonlinear dynamics for which the bounds are calculated using interval mathematics. The controller achieves the control task without measuring or estimating the derivative of the system output. • An anti-lock braking system (ABS) is a third order system. The system output i.e., the wheel-slip has relative degree two with respect to the control and the resulting zero dynamics is stable. The 2-sliding control algorithm has been applied to control the wheel-slip without measuring or estimating the derivative of the wheel slip. • A simulation study for angular speed and position control of a DC motor is carried out to assess the nominal performance of the system. The angular position of the motor shaft has relative degree three and therefore a suitable sliding variable is selected such that it has relative degree two with respect to the system input. Once the sliding variable is stabilised to zero, the motor shaft settles to the desired angular position asymptotically. The algorithm has also been physically implemented on an experiment for angular position and speed control of a DC motor in the dSPACE environment. The results show robustness to Chapter 8. Conclusions And Future Research 161 unknown loads and other nonlinearities such as motor stiction and poor knowledge of motor parameters. The newly developed algorithm has been extended for a class of MIMO nonlinear systems and implemented to the coupled MIMO twin-tanks system for water level control. The fol lowing simulation studies have been presented to validate the theoretical results. • The simulation study for the stabilisation of angular velocities in a rigid body MIMO spacecraft model has been presented. The simulation results shows the smooth control input which has been provided via the dynamic control. • The simulation study for water level control in the MIMO twin-tank system has been presented to assess the nominal performance. The algorithm has again been used via the dynamic control which provides smooth or at least piecewise linear system input. The implementation of the control algorithm has been considered for water level control on the twin-tanks. The control algorithm was coded in C++ and run on a Windows 98 PC running at 166 MHz CPU. The uncertainties were contributed by poor knowledge of the various system parameters and the neglect of pump dynamics. Water leakage has been introduced by opening different water outlet pipes and the controller is found to be robust. 8.3 Future research directions The algorithm developed in Chapter 5 has local stability because of the absolute bounds required over the uncertainties. It can be improved by providing uncertainty bounds as a function of the system output and further improved by parametrisation of controller param eters in terms of system specifications. The fully developed theoretical tools in other areas of control theory can be utilised to expand the higher order sliding mode control research. Two broad areas are discussed below. Optimised HOSM Control Higher order sliding mode control involves zeroing of not only the sliding variable but also certain derivatives of the sliding variable. The problem of higher order sliding mode control of a nonlinear minimum phase system can be transformed into input-output (I-O) form by considering sliding variable and a finite number of its derivatives; the system then becomes equivalent to that of a chain of integrators with nonlinear bounded uncertainties. Finally the Chapter 8. Conclusions And Future Research 162 problem becomes that of finite time stabilisation of a linear system with nonlinear bounded non-structured parametric uncertainties. This problem can be solved using optimal linear control mythologies [12]. The time vary ing sliding manifold can be selected by optimizing a quadratic cost function [107] involving derivatives of the sliding variable and hence will provide extra design freedom. The prelimi nary work done by Laghrouche et al. [55] has the potential for development to a novel theory for optimal higher order sliding mode control. Discrete HOSM control Controllers are frequently implemented by a computer using measurements at discrete in stants. The problems due to switching of the control signal are made worse in general by discrete implementation; sometimes the system being controlled may even become unsta ble. Hence there is a need to design discrete higher order sliding mode controllers using discretised system models and taking care of sampling time. Fast output sampling is a technique to observe the discrete system states by sampling the system output sufficiently frequently. To estimate the derivatives of the sliding variable, fast output sampling can be utilised. Werner and Furuta [111], Werner and Meister [112], who developed the fast output sampling method for controlling sampled data systems via output feedback, has successfully applied the technique to a laboratory helicopter model. Chakravarthini et al. [19] has used fast output sampling for controlling nonlinear systems using only output measurements by the so called traditional, or first order sliding mode control. The use of fast output sampling can be explored to implement output feedback higher order sliding mode control in discrete time. 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