DOCSLIB.ORG

Lur'e Problem and Absolute Stability

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

MCE693/793: Analysis and Control of Nonlinear Systems Classical Lur’e Problem and Absolute Stability Hanz Richter Mechanical Engineering Department Cleveland State University Introduction Many nonlinear control problems involve a feedback loop involving a linear system and a static nonlinearity: 0 e G(s) y − φ(y) A widely-studied case is to consider G(s) without direct transmission term: x˙ = Ax + Be y = CT x (D =0 in the state-space representation). −1 G(s) = C(sI A) B − e = Φ(y) − 2 / 25 Introduction Suppose Φ(y) = k, some constant gain. Recalling the Nyquist criterion, if G(s) is stable and the Nyquist graph does not enclose the critical point 1/k, − we conclude asymptotic stability. When Φ(y) is a nonlinearity, we have used the describing function method as an approximation and the “critical point” 1/N(A, w). − This method is approximate and aimed at limit cycle analysis, but does not give definitive results when applied to stability analysis. The general (exact) stability problem for the above systems with stable G(s) and static nonlinearity is known as the Lur’e Problem after the paper by Lur’e and Postnikov, Stability theory of regulating systems, (Russian) Prikladnaya Matematika i Mekhanika (1944). 3 / 25 Sector-Bounded Nonlinearities A sector (also sectoric, sector-bounded) nonlinearity is a continuous function Φ: R R satisfying 7→ Φ(y) k1 k2 ≤ y ≤ for some k1 0, k2 0 and for all y =0. The definition implies Φ(0) = 0 ≥ ≥ 6 and yΦ(y) 0 (first and third quadrants). ≥ Φ(y) k2y k1y y 4 / 25 Absolute Stability and Aizerman Conjecture The Nyquist criterion gives a conclusive answer about stability whenever Φ(y) = k (squashed sector). The Absolute Stability question (Lur’e problem) is: If Φ is a sector nonlinearity, what conditions must G(s) satisfy (in terms of the sector bounds) so that the system is U.A.S.I.L.? In late-40’s Soviet research they thought that for a sector nonlinearity, stability could be ensured if the Nyquist criterion held for all k [k1,k2]. This was ∈ known as the Aizerman Conjecture. 5 / 25 Falsehood of the Aizerman Conjecture - Taylor Example The Aizerman conjecture is false, because limit cycles can exist. Take 10s2 G(s) = s4 +0.04s3 +2.021s2 +0.0404s +0.9803 3 and k1 =0, k2 = , with Φ(y) = y . ∞ However, it has been shown that the Aizerman conjecture is true for all first and second-order systems and higher-order systems of special forms. 6 / 25 The Kalman Conjecture Kalman’s conjecture (1957) uses the maximum and minimum slopes of Φ(y) instead of the sector bounds. Suppose the slope of Φ satisfies dΦ M M ≤ dy ≤ Kalman conjecture: The closed-loop system is asymptotically stable if the Nyquist plot of G(s) does not enclose the critical point 1/k for all values − M k M. ≤ ≤ The Kalman conjecture is also false, as a counterexample by Taylor shows. Taylor also showed that replacing the nonlinearity by its describing function and using the Nyquist criterion is not a valid method in general. 7 / 25 The Popov Criterion Popov’s criterion is only a sufficient condition. Assumptions: ■ G(s) is asymptotically stable. In a state-space realization (A,B,C), we require A to be Hurwitz and (A, B) controllable (a minimal realization, no pole-zero cancellations). ■ The nonlinearity is in the sector [0,k] Popov’s criterion: If there exists a constant q such that for all w 0 ≥ 1 Re [(1 + jwq)G(jw)] + > 0 k then the closed-loop system is asymptotically stable. If we separate G(jw) = G1(w) + jG2(w), we obtain the equivalent form: 1 G1(w) wqG2(w) + > 0 − k 8 / 25 Popov Criterion- Graphical Interpretation 1 The criterion requires G1(w) > + qwG2(w). Call x = G1(w) =Re(G(jw)) − k and y = wG2(w) = wIm(G(jw)). Also define W (jw) = x + jy. Then the Re-Im plot of W (jw) needs to be on one side of the Popov line 1 x qy + =0 − k This is called a Popov plot (note that w is taken in [0, )). Im(W (jw)) = wIm(G(jw)) ∞ 1 q Re(W (jw)) = Re(G(jw)) − 1 k W (jw) Popov line 9 / 25 Ranges for q and k Asymptotic stability is obtained when the inequality holds for the given k and some q. In some cases (next example), q =0 makes the inequality hold for the sector [0, ]. This means that the system is asymptotically stable for any ∞ nonlinearity in that sector, including time-varying nonlinearities. More precise information about the feasible intervals for q has been obtained (Narendra and Taylor, Frequency Domain Criteria for Absolute Stability, Hsu and Meyer, Modern Control Principles and Applications). For instance: ■ Single-valued, time-invariant nonlinearity in (0, ): q R. But if k = ∞ ∈ ∞ (relay): q [0, ) ∈ ∞ ■ Time-independent hysteresis in (0, ): q ( , 0] ∞ ∈ −∞ ■ General, time-dependent nonlinearity in (0, ]: q =0. That is, if no ∞ information about Φ is available, the inequality must hold with q =0. 10 / 25 Examples Take s +3 G(s) = s2 +7s + 10 We check that the system will be asymptotically stable for any nonlinearity Φ (0, ] ∈ ∞ Now take 1 G(s) = (s + 1)(s + 4)(s + 5) We see that for Φ = k, the Routh-Hurwitz criterion gives the range (0, 270) for k. For other nonlinearities, we formulate the problem of finding the maximum sector bound k for which there is a feasible q and solve it graphically. 11 / 25 Pole-Shifting Method The Popov criterion holds only for G(s) asymptotically stable. When this fails, we can use a loop transformation: ′ α¯ G (s) − 0 + + y Φ G(s) − + − ′ Φ α¯ Constant α¯ must be chosen so that G′(s) = G(s)/(1+αG ¯ (s)) becomes asymptotically stable and Φ′ = Φ α¯ is still in the first and third quadrants. − Note that this requires α α¯, where α is the low sector bound of Φ. ≥ 12 / 25 Example Take 1 G(s) = s(s + 1) Use pole shifting and the Popov criterion to analyze the absolute stability of the corresponding Lur’e system. 13 / 25 The Circle Criterion The Popov criterion does not take advantage of information about the low sector bound when available. The circle criterion includes information about both sector bounds [kl,kh] for Φ. Let D(cl,cr) denote a circle centered on the real axis such that 1/cl and − 1/cr are the real axis intersections. − For G(s) asymptotically stable and kl 0, the criterion has a simple ≥ statement: The Lur’e system is absolutely stable (asymptotically stable for all nonlinearities Φ [kl,kh]) if Φ(u, t) satisfies: ∈ Φ kl <cl cr <kr ≤ u ≤ and the Nyquist plot of G(jw) does not touch or enclose D(cl,cr). Recall that the Nyquist plot is traced with −∞ < w < ∞. 14 / 25 Circle Criterion - Graphical Interpretation Im (G(jw)) − 1 − 1 cl cr Re (G(jw)) The circle is not unique in general, and circles close to the imaginary axis yield wider sectors. Example: Saturation nonlinearity. The slope of the function within the 1 saturation band determines , and we can take 1/cl . − cr − → −∞ 15 / 25 Circle Criterion - More General Systems The pole shifting method can be used for unstable systems, but a more general version of the criterion is available (see S&L). Assumptions: ■ G(s) has no purely imaginary poles and ρ poles with positive real parts. ■ The nonlinearity is in [k1,k2] Then there are four cases, according to the location of [k1,k2] relative to zero: 1. 0 <k1 k2 ≤ 2. 0 = k1 <k2 3. k1 < 0 <k2 4. k1 <k2 < 0 16 / 25 Circle Criterion - More General Systems The system is asymptotically stable if the condition matching the previous cases holds: ■ Case 1: The Nyquist plot of G does not enter the circle C(k1,k2) and encircles it ρ times counterclockwise. ■ Case 2: The Nyquist plot of G is contained in the half-space Re(z) > 1/k2. − ■ Case 3: The Nyquist plot of G is in the interior of C(k1,k2) ■ Case 4: The Nyquist plot of G does not enter the circle C( k1, k2) − − − and encircles it ρ times counterclockwise. 17 / 25 Example Again, take 1 G(s) = s(s + 1) Use pole shifting and the circle criterion to analyze the absolute stability of the corresponding Lur’e system. 18 / 25 Positive-Real Transfer Functions and the Lur’e Problem Suppose the linear system in the Lur’e problem is asymptotically stable. What additional conditions can be imposed on the system so that the closed-loop system remains stable for any nonlinearities in the sector [0, 1]? Note can use pole shifting and/or scaling to find a more general sector. Let G(s) have state-space realization (A,B,C, 0) and use the quadratic Lyapunov function: V (x) = xT Px with P = P T > 0. We show in class that 2 V˙ xT (AT P + P A)x +2xT (CT PB)Φ 2Φ ≤ − − For this, note that the sector condition is equivalent to φ(y)(φ(y) ky) 0 − ≤ 19 / 25 Lur’e Problem and Kalman-Yakubovich-Popov Lemma We found 2 V˙ xT (AT P + P A)x +2xT (CT PB)Φ 2Φ ≤ − − If we can find L, ǫ > 0 so that AT P + P A = LT L ǫP − − BT P = C √2L − we can show that 2 V˙ ǫxT Px xT LT Lx +2√2xT LT Φ 2Φ ≤ − − − so 2 V˙ ǫxT Px Lx √2Φ ǫxT Px ≤ − − || − || ≤ − which shows global asymptotic stability.
Recommended publications
  • Basic Nonlinear Control Systems - D

    Basic Nonlinear Control Systems - D

    CONTROL SYSTEMS, ROBOTICS, AND AUTOMATION – Vol. III – Basic Nonlinear Control Systems - D. P. Atherton BASIC NONLINEAR CONTROL SYSTEMS D. P. Atherton University of Sussex, UK Keywords: Nonlinearity, friction, saturation, relay, backlash, frequency response, describing function, limit cycle, stability, phase plane, subharmonics, chaos, synchronization, compensation. Contents 1. Introduction 2. Forms of nonlinearity 3. Structure and behavior 4. Stability 5. Aspects of design 6. Conclusions Glossary Bibliography Biographical Sketch Summary This section is written to introduce the reader to nonlinearity and its effect in control systems, and also to ‘set the stage’ for the subsequent articles. Nonlinearity is first defined and then some of the physical effects which cause nonlinear behavior are discussed. This is followed by some comments on the stability of nonlinear systems, the existence of limit cycles, and the unique behavioral aspects which nonlinear feedback systems may possess. Finally a few brief comments are made on designing controllers for nonlinear systems. 1. Introduction Linear systems have the important property that they satisfy the superposition principle. This leads to many important advantages in methods for their analysis. For example, in circuit theory when an RLC circuit has both d.c and a.c input voltages, the voltage or current elsewhereUNESCO in the circuit can be– found EOLSS by summing the results of separate analyses for the d.c. and a.c. inputs taken individually, also if the magnitude of the a.c. voltage is doubledSAMPLE then the a.c. voltages or CHAPTERScurrents elsewhere in the circuit will be doubled. Thus, mathematically a linear system with input x()t and output yt() satisfies the property that the output for an input ax12() t+ bx () t is ay12() t+ by () t , if y1()t and y2 ()t are the outputs in response to the inputs x1()t and x2 ()t , respectively, and a and b are constants.
  • Neural Control for Nonlinear Dynamic Systems

    Neural Control for Nonlinear Dynamic Systems

    Neural Control for Nonlinear Dynamic Systems Ssu-Hsin Yu Anuradha M. Annaswamy Department of Mechanical Engineering Department of Mechanical Engineering Massachusetts Institute of Technology Massachusetts Institute of Technology Cambridge, MA 02139 Cambridge, MA 02139 Email: [email protected] Email: [email protected] Abstract A neural network based approach is presented for controlling two distinct types of nonlinear systems. The first corresponds to nonlinear systems with parametric uncertainties where the parameters occur nonlinearly. The second corresponds to systems for which stabilizing control struc­ tures cannot be determined. The proposed neural controllers are shown to result in closed-loop system stability under certain conditions. 1 INTRODUCTION The problem that we address here is the control of general nonlinear dynamic systems in the presence of uncertainties. Suppose the nonlinear dynamic system is described as x= f(x , u , 0) , y = h(x, u, 0) where u denotes an external input, y is the output, x is the state, and 0 is the parameter which represents constant quantities in the system. The control objectives are to stabilize the system in the presence of disturbances and to ensure that reference trajectories can be tracked accurately, with minimum delay. While uncertainties can be classified in many different ways, we focus here on two scenarios. One occurs because the changes in the environment and operating conditions introduce uncertainties in the system parameter O. As a result, control objectives such as regulation and tracking, which may be realizable using a continuous function u = J'(x, 0) cannot be achieved since o is unknown. Another class of problems arises due to the complexity of the nonlinear function f.
  • Control Theory

    Control Theory

    Control theory S. Simrock DESY, Hamburg, Germany Abstract In engineering and mathematics, control theory deals with the behaviour of dynamical systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain ref- erence over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. Rapid advances in digital system technology have radically altered the control design options. It has become routinely practicable to design very complicated digital controllers and to carry out the extensive calculations required for their design. These advances in im- plementation and design capability can be obtained at low cost because of the widespread availability of inexpensive and powerful digital processing plat- forms and high-speed analog IO devices. 1 Introduction The emphasis of this tutorial on control theory is on the design of digital controls to achieve good dy- namic response and small errors while using signals that are sampled in time and quantized in amplitude. Both transform (classical control) and state-space (modern control) methods are described and applied to illustrative examples. The transform methods emphasized are the root-locus method of Evans and fre- quency response. The state-space methods developed are the technique of pole assignment augmented by an estimator (observer) and optimal quadratic-loss control. The optimal control problems use the steady-state constant gain solution. Other topics covered are system identification and non-linear control. System identification is a general term to describe mathematical tools and algorithms that build dynamical models from measured data.
  • Fundamentals of Backstepping Control

    Fundamentals of Backstepping Control

    Appendix Fundamentals of Backstepping Control This Appendix provides the fundamentals of the recursive backstepping control method. The presented material is a summary of more detailed descriptions that may be found in [43, 49]. Lyapunov-based controller design may be systematically tack- led by a recursive design procedure called backstepping. Backstepping is suitable for strict-feedback systems that are also known as “lower triangular”. An example of a strict-feedback systems is: ˙ ξ1 = f1(ξ1) + g1(ξ1)ξ2 ˙ ξ2 = f2(ξ1,ξ2) + g2(ξ1,ξ2)ξ3 . (A.1) ˙ ξr−1 = fr−1(ξ1,ξ2,...,ξr−1) + gr−1(ξ1,ξ2,...,ξr−1)ξr ˙ ξr = fr (ξ1,ξ2,...,ξr ) + gr (ξ1,ξ2,...,ξr )u where ξ1,...,ξr ∈ R, u ∈ R is the control input and fi , gi for i = 1,...,r are known functions. A typical feedback linearization approach in most cases leads to cancellation of useful nonlinearities. The backstepping design exhibits more flexi- bility compared to feedback linearization since it does not require that the resulting input–output dynamics be linear. Cancellation of potentially useful nonlinearities can be avoided resulting in less complex controllers. The main idea is to use some of the state variables of (A.1) as “virtual controls” or “pseudo controls”, and de- pending on the dynamics of each state, design intermediate control laws. The back- stepping design is a recursive procedure where a Lyapunov function is derived for the entire system. The recursive procedure can be easily expanded from the nominal case of a system augmented by an integrator. This is also referred to as integra- tor backstepping.
  • Nonlinear Control - an Overview

    Nonlinear Control - an Overview

    1 NONLINEAR CONTROL - AN OVERVIEW Fernando Lobo Pereira, fl[email protected] Porto University, Portugal Nonlinear Control, 2008, FEUP References 1 M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall, Englewood Cli®s, NJ, 1978. 2 A. Isidori, Nonlinear Control Systems, Springer-Verlag, 199X. 3 E. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer-Verlag, 1998. 4 C. T. Chen, Introduction to Linear Systems Theory, Van Nostrand Reinhold Company, New York, NY, 1970. 5 C. A. Desoer, Notes for a Second Course on Linear Systems, Holt Rhinehart, Winston, 1970. 6 K. Ogata, State Space Analysis of Control Systems, Prentice-Hall, Englewood Cli®s, NJ, 1965. 7 J. M. Carvalho, Dynamical Systems and Automatic Control, Prentice-Hall, New York, NY, 1994. Fernando Lobo Pereira, fl[email protected] 1 Nonlinear Control, 2008, FEUP Introduction Problem description - system components, objectives and control issues. Example 1: telescope mirrors control PICTURE Most of real systems involve nonlinearities in one way or another. However, many concepts for linear systems play an important role since some of the techniques to deal with nonlinear systems are: ² Change of variable in phase space so that the resulting system is linear. ² Nonlinearities may be cancelled by picking up appropriate feedback control laws. ² Pervasive analogy ... Fernando Lobo Pereira, fl[email protected] 2 Nonlinear Control, 2008, FEUP Introduction Example 2: Pendulum Modeling PICTURE Dynamics (Newton's law for rotating objects): mθÄ(t) + mgsin(θ(t)) = u(t) State variables: θ; θ_ Control/Input variable: u (external applied torque) Let us assume m = 1; l = 1; g = 1. Fernando Lobo Pereira, fl[email protected] 3 Nonlinear Control, 2008, FEUP Introduction Control Design ½ (0; 0) stable equilibrium Stationary positions: (1) (¼; 0) unstable equilibrium.
  • Modeling, Control and Simulation of Control-Affine Nonlinear Systems with State-Dependent Transfer Functions

    Modeling, Control and Simulation of Control-Affine Nonlinear Systems with State-Dependent Transfer Functions

    New Jersey Institute of Technology Digital Commons @ NJIT Dissertations Electronic Theses and Dissertations Spring 5-31-2014 Modeling, control and simulation of control-affine nonlinear systems with state-dependent transfer functions Roger Kobla Kwadzogah New Jersey Institute of Technology Follow this and additional works at: https://digitalcommons.njit.edu/dissertations Part of the Electrical and Electronics Commons Recommended Citation Kwadzogah, Roger Kobla, "Modeling, control and simulation of control-affine nonlinear systems with state-dependent transfer functions" (2014). Dissertations. 164. https://digitalcommons.njit.edu/dissertations/164 This Dissertation is brought to you for free and open access by the Electronic Theses and Dissertations at Digital Commons @ NJIT. It has been accepted for inclusion in Dissertations by an authorized administrator of Digital Commons @ NJIT. For more information, please contact [email protected]. Copyright Warning & Restrictions The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or reproduction is not to be “used for any purpose other than private study, scholarship, or research.” If a, user makes a request for, or later uses, a photocopy or reproduction for purposes in excess of “fair use” that user may be
  • Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles

    Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles

    Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles by Reza Olfati-Saber Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering and Computer Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2001 © Massachusetts Institute of Technology 2001. All rights reserved. Author ................................................... Department of Electrical Engineering and Computer Science January 15, 2000 C ertified by.......................................................... Alexandre Megretski, Associate Professor of Electrical Engineering Thesis Supervisor Accepted by............................................ Arthur C. Smith Chairman, Department Committee on Graduate Students Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles by Reza Olfati-Saber Submitted to the Department of Electrical Engineering and Computer Science on January 15, 2000, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering and Computer Science Abstract This thesis is devoted to nonlinear control, reduction, and classification of underac- tuated mechanical systems. Underactuated systems are mechanical control systems with fewer controls than the number of configuration variables. Control of underactu- ated systems is currently an active field of research due to their broad applications in Robotics, Aerospace Vehicles, and Marine Vehicles. The examples of underactuated systems include flexible-link robots, nobile robots, walking robots, robots on mo- bile platforms, cars, locomotive systems, snake-type and swimming robots, acrobatic robots, aircraft, spacecraft, helicopters, satellites, surface vessels, and underwater ve- hicles. Based on recent surveys, control of general underactuated systems is a major open problem. Almost all real-life mechanical systems possess kinetic symmetry properties, i.e.
  • Nonlinear Control Via Input-Output Feedback Linearization of a Robot Manipulator

    Nonlinear Control Via Input-Output Feedback Linearization of a Robot Manipulator

    Advances in Science, Technology and Engineering Systems Journal Vol. 3, No. 5, 374-381(2018) ASTESJ www.astesj.com ISSN: 2415-6698 NonLinear Control via Input-Output Feedback Linearization of a Robot Manipulator Wafa Ghozlane*, Jilani Knani Laboratory of Automatic Research LA.R.A, Department of Electrical Engineering, National School of Engineers of Tunis ENIT, University of Tunis ElManar, 1002 Tunis, Tunisia. A R T I C L E I N F O A B S T R A C T Article history: This paper presents the input-output feedback linearization and decoupling algorithm for Received: 08 September, 2018 control of nonlinear Multi-input Multi-output MIMO systems. The studied analysis was Accepted: 13 October, 2018 motivated through its application to a robot manipulator with six degrees of freedom. The Online: 18 October, 2018 nonlinear MIMO system was transformed into six independent single-input single-output SISO linear local systems. We added PD linear controller to each subsystem for purposes Keywords: of stabilization and tracking reference trajectories, the obtained results in different Input-Output Feedback simulations shown that this technique has been successfully implemented. linearization Nonlinear system Robot manipulator MIMO nonlinear system SISO linear systems PD controller 1. Introduction the angular position of each joint of this robot arm for stabilization and tracking purposes. The obtained results in different In recent years, Feedback linearization has been attracted a simulations shown the efficiency of the derived approach [7]. great deal of interesting research. It's an approach designed to the nonlinear control systems, which based on the idea of This paper is organized as follows: It is divided into five transforming nonlinear dynamics into a linear form.
  • Design of Terminal Sliding Mode Controllers for Disturbed Non-Linear Systems Described by Matrix Differential Equations of the Second and First Orders

    Design of Terminal Sliding Mode Controllers for Disturbed Non-Linear Systems Described by Matrix Differential Equations of the Second and First Orders

    applied sciences Article Design of Terminal Sliding Mode Controllers for Disturbed Non-Linear Systems Described by Matrix Differential Equations of the Second and First Orders Paweł Skruch * and Marek Długosz Department of Automatic Control and Robotics, AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Krakow, Poland; [email protected] * Correspondence: [email protected] Received: 2 February 2019; Accepted: 1 June 2019; Published: 6 June 2019 Abstract: This paper describes a design scheme for terminal sliding mode controllers of certain types of non-linear dynamical systems. Two classes of such systems are considered: the dynamic behavior of the first class of systems is described by non-linear second-order matrix differential equations, and the other class is described by non-linear first-order matrix differential equations. These two classes of non-linear systems are not completely disjointed, and are, therefore, investigated together; however, they are certainly not equivalent. In both cases, the systems experience unknown disturbances which are considered bounded. Sliding surfaces are defined by equations combining the state of the system and the expected trajectory. The control laws are drawn to force the system trajectory from an initial condition to the defined sliding surface in finite time. After reaching the sliding surface, the system trajectory remains on it. The effectiveness of the approaches proposed is verified by a few computer simulation examples. Keywords: non-linear dynamical system; disturbance; sliding mode control; sliding surface 1. Introduction 1.1. Motivation Stabilization of non-linear systems finds applications in many areas of engineering, particularly in the fields of mechanics, robotics, electronics, and so on [1,2].
  • Neural Networks for Control

    Neural Networks for Control

    Neural Networks for Control Martin T. Hagan School of Electrical & Computer Engineering Oklahoma State University [email protected] Howard B. Demuth Electrical Engineering Department University of Idaho [email protected] Abstract the inverse of a system we are trying to control, in which case the neural network can be used to imple- The purpose of this tutorial is to provide a quick ment the controller. At the end of this tutorial we overview of neural networks and to explain how they will present several control architectures demon- can be used in control systems. We introduce the strating a variety of uses for function approximator multilayer perceptron neural network and describe neural networks. how it can be used for function approximation. The backpropagation algorithm (including its variations) Output is the principal procedure for training multilayer Unknown perceptrons; it is briefly described here. Care must Function - be taken, when training perceptron networks, to en- Input Error sure that they do not overfit the training data and then fail to generalize well in new situations. Several Neural + techniques for improving generalization are dis- Network cused. The tutorial also presents several control ar- Predicted Adaptation chitectures, such as model reference adaptive Output control, model predictive control, and internal model control, in which multilayer perceptron neural net- works can be used as basic building blocks. Figure 1 Neural Network as Function Approximator In the next section we will present the multilayer 1. Introduction perceptron neural network, and will demonstrate In this tutorial we want to give a brief introduction how it can be used as a function approximator.
  • Sliding Mode Control of Constrained Nonlinear Systems

    Sliding Mode Control of Constrained Nonlinear Systems

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Archivio istituzionale della ricerca - Politecnico di Milano 1 Sliding Mode Control of Constrained Nonlinear Systems chattering reduction is not the only possible purpose of HOSM, since this latter allows the use of a relative degree greater than one between Gian Paolo Incremona, Matteo Rubagotti, and Antonella Ferrara the discontinuous control input and the sliding variable. In the first-order SMC formulation, input saturations are imme- diately satisfied if the control variable switches between values Abstract—This technical note introduces the design of sliding mode control algorithms for nonlinear systems in the presence of hard inequal- that are inside the imposed boundaries. When HOSM is used for ity constraints on both control and state variables. Relying on general chattering reduction, specific solutions have been proposed to achieve results on minimum-time higher-order sliding mode for unconstrained the satisfaction of saturation constraints (see, e.g., [19], [20] for the systems, a general order control law is formulated to robustly steer the second-order case). On the other hand, the possible presence of state state to the origin, while satisfying all the imposed constraints. Results on minimum-time convergence to the sliding manifold, as well as on the constraints is usually not taken into account in SMC formulations. maximization of the domain of attraction, are analytically proved for Recently, few solutions have been proposed in order to merge SMC the first-order and second-order sliding mode cases. A general result is and MPC, and combine the constraints satisfaction property of MPC presented regarding the domain of attraction in the general order case, with the robustness of SMC [21]–[25].
  • Slotine • Li APPLIED NONLINEAR CONTROL

    Slotine • Li APPLIED NONLINEAR CONTROL

    Slotine • Li APPLIED NONLINEAR CONTROL ! i APPLIED NONLINEAR CONTROL Jean-Jacques E Slotine Weiping Li Applied Nonlinear Control JEAN-JACQUES E. SLOTINE Massachusetts Institute of Technology WEIPING LI Massachusetts Institute of Technology' Prentice Hall Englewood Cliffs, New Jersey 07632 Library of Congress Cataloging-in-Publication Data Slotine, J.-J. E. (Jean-Jacques E.) Applied nonlinear control / Jean-Jacques E. Slotine, Weiping Li p. cm. Includes bibliographical references. ISBN 0-13-040890-5 1, Nonlinear control theory. I. Li, Weiping. II. Title. QA402.35.S56 1991 90-33365 629.8'312-dc20 C1P Editorial/production supervision and interior design: JENNIFER WENZEL Cover design: KAREN STEPHENS Manufacturing Buyer: LORI BULWIN =^= © 1991 by Prentice-Hall, Inc. ^=&= A Division of Simon & Schuster T k Englewood Cliffs, New Jersey 07632 All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Printed in the United States of America 20 19 18 17 16 15 14 13 12 1] ISBN D-13-DHDfiTa-S Prentice-Hall International (UK) Limited, London Prentice-Hall of Australia Pty. Limited, Sydney Prentice-Hall Canada Inc., Toronto Prentice-Hail Hispanoamericana, S.A., Mexico Prentice-Hall of India Private Limited, New Delhi Prentice-Hall of Japan, Inc., Tokyo Simon & Schuster Asia Pte. Ltd., Singapore Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro To Our Parents Contents Preface xi 1. Introduction 1 1.1 Why Nonlinear Control ? 1 1.2 Nonlinear System Behavior 4 1.3 An Overview of the Book 12 1.4 Notes and References 13 Part I: Nonlinear Systems Analysis 14 Introduction to Part I 14 2.