UNIVERSITY OF CINCINNATI
______, 20 _____
I,______, hereby submit this as part of the requirements for the degree of:
______in: ______It is entitled: ______
Approved by: ______
Stability Analysis and Design of Servo-Hydraulic Systems
- A Bifurcation Study
A dissertation submitted to the
Division of Research and Advanced Studies of the University of Cincinnati
in partial fulfillment of the requirements for the degree of
DOCTORATE OF PHILOSOPHY (Ph.D.)
in the Department of Mechanical, Industrial and Nuclear Engineering of the College of Engineering
2002
by
Amit Shukla
B.E.M.E., MNR Engineering College, India 1996 M.S.M.E., University of Cincinnati, 1998
Committee Chair: Dr. David F. Thompson
Abstract
Design for robust stability is one of the most important issues in nonlinear systems theory. The validity of linear system design in a small neighborhood is not a sufficient criterion for systems that undergo parametric variations and have strong nonlinear characteristics. With rapid growth in the systems theory, the design of nonlinear systems using bifurcation theory- based procedures has been one of the key developments. Servo-hydraulic systems are one of the most commonly used actuation and control devices, due to their force to weight ratio. They also are highly nonlinear in nature and hence provide considerable difficulty in the design and analysis of these systems and their control algorithms.
The goal of this dissertation is to tackle some of the issues of the nonlinear systems theory with applications to servo-hydraulic systems. The use of bifurcation theory for the design and analysis of a nonlinear system is illustrated, and a detailed investigation into the dynamics associated with the servo-hydraulic systems is done. Further, the model decomposition/reduction strategy for parametric study in the nonlinear system is suggested. The idea of control-induced bifurcation is introduced and explained in light of servo-hydraulic systems. The servo-hydraulic system nonlinearities are explained and their effects on the robust stability are highlighted. This numerical work is also complemented with the experimental results on the servo-hydraulic circuits. This general procedure for robust stability design and control design, under the influence of nonlinearities, presented in this work can be used for any nonlinear system. The limitations of bifurcation theory based tools are also highlighted.
Acknowledgements
This work has been possible due to motivation and support of various individuals who gave me sound advice and guidance at numerous occasions throughout my years as a graduate student at the University of Cincinnati. I express my deepest gratitude to my academic advisor Dr. David F.
Thompson, whose excellent guidance and critical comments paved the way for this work as well as contributed to my development as an academician. He has been a true guide, a mentor and a friend. I also wish to acknowledge Dr. Randall J. Allemang, Dr. Ronald Huston, and Dr. Edward
J. Berger for being on my dissertation committee and for providing valuable help and guidance.
All of them have shaped my graduate education and life very significantly by their great teaching and thoughtful concerns.
I am also grateful to the members of UC SDRL, specially, Dr. David Brown, Dr. Jay Kim, Dr.
Allyn Phillips, Dr. Gregory Kremer, Dr. Doug Adams, Dr. Bill Fladung, Dan Lazor, Srinivas
Kowta, Tom Terrell, Jeff Hylok and Bruce Fouts who made my stay at UC most enjoyable. I also would like to thank Rhonda Christman for her exceptional secretarial help during my 6 years at
UC.
I am also thankful to the National Science Foundation for providing monetary support for this work and Ford Motor Company (Dr. Gregory M. Pietron) for support and fruitful discussions on hydraulic system modeling.
This has been a project which would be impossible without unflinching support of my family. I dedicate this work to my parents. They provided an atmosphere and everything else in my upbringing for me to reach this point in my life. The support and understanding that my wife gave me provided extra boost for completion of this work. Her love has been my strength for the past few years. Her parents were also very cooperative and patient regarding my tenure as a graduate student.
Table of Contents
Abstract
List of Figures 4
List of Symbols 9
1. Introduction 11
1.1. Background/Motivation 11
1.2. Applications 16
1.3. Organization of the dissertation 17
2. Servo hydraulic drives 20
2.1. Previous work 20
2.2. Fundamentals of hydraulic system modeling 23
2.2.1. Orifice flows 23
2.2.2. Continuity equations 24
2.2.3. Example 25
2.2.4. Valve modeling 26
2.3. Modeling assumptions 33
2.4. Physically realistic models of servo hydraulic system 34
2.5. Modeling of servo-pump actuator system 37
2.6. Modeling of servo-valve actuator system 39
2.7. Servo-valve actuator model with line dynamics 42
3. Linear and nonlinear analysis of servo-hydraulic systems 45
3.1. Linear analysis of servo-hydraulic system –a review example 45
3.2. Experimental results for servo-hydraulic systems – a review 47
3.3. Nonlinearities in the servo-hydraulic systems and their effect on stability 48
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3.3.1. Effect of flow nonlinearity on the stability 48
3.3.2. Effect of saturation on the stability 50
3.4. Large scale coupled nonlinear systems 50
3.5. Why nonlinear analysis is needed? 52
4. Nonlinear systems analysis and bifurcation theory 53
4.1. Dynamical systems and equilibrium points 54
4.2. Generic bifurcations 57
4.3. Global bifurcations, jumps and non-local behavior 62
4.4. Stability of forced systems-feedback loop nonlinearities and nonlinear feedback 63
4.5. Nonlinear systems analysis tools 64
5. Nonlinear systems analysis –a bifurcation theory based approach 66
5.1. Previous work 66
5.2. Multi-parameter multi-space bifurcation theory 71
5.2.1. Parameter space, eigen-space and state space 74
5.2.2. Constant velocity solutions 76
5.2.3. Robust bifurcation stability analysis 77
5.3. Decomposition of the system 80
5.3.1. Fastest unstable mode with associated parameters 81
5.3.2. Transformations for parametric model decomposition 82
5.4. Control induced bifurcations 83
5.5. Control of bifurcation instabilities 85
5.6. Measure of nonlinearity 86
5.7. Numerical aspects of bifurcation analysis 87
5.7.1. Data noise and nonlinear effects 89
6. Preliminary bifurcation results: numerical 90
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6.1. Characterization of nonlinear dynamics- computational results 90
6.2. Results for servo-pump actuator system 90
6.2.1. Simulation studies 91
6.2.2. Parametric bifurcation studies 94
6.3. Results for the servo-valve actuator model 101
6.4. Chapter summary 104
7. Development of experimental apparatus 125
7.1. Development of the test stand 125
7.2. Servo-valve model development and refinement 127
7.2.1. Pressure-flow-voltage nonlinear static characteristics 129
7.2.2. Linear transfer function model of the servo-valve 131
7.2.3. Servo-valve actuator model and real-time control schematic 133
7.3. Effect of accumulators 137
7.4. Chapter summary 139
8. Control Studies- Introduction 140
8.1. PD controller 140
8.2. Effect of PD controller on the servo-valve actuator system 141
8.3. Effect of 5th order linear controller on the servo-valve actuator system 161
8.4. Chapter summary 177
9. Conclusions and suggestions for future research 180
10. Bibliography 184
11. Appendix 198
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List of Figures
2.1 A sample hydraulic system model 25
2.2 A valve schematic 27
2.3 Port flow area schematic of a spool valve 28
2.4 Open valve port area configuration 29
2.5 Closed valve configuration 30
2.6 Port flow area as a function of the valve displacement 30
2.7 A common four way spool valve 31
2.8 Axial and lateral components of flow forces in a valve port 32
2.9 Model 1, servo-pump actuator system 38
2.10 Block diagram of model 1 38
2.11 Servo-valve actuator model schematic with external loads 41
2.12 Pipe line model schematic 43
3.1 Proportional feedback for the linearized servo-pump actuator model 46
3.2 Root locus for varying the proportional feedback gain 46
3.3 Dead zone and saturation nonlinearity-input and response 50
3.4 Parameter space investigation using simulation 51
4.1 Time response, phase plane and eigenplane 55
4.2 Invariant subspaces and manifolds 56
4.3 Saddle node bifurcation plot representing the behavior of equilibrium point 58
4.4 Trans critical bifurcation 59
4.5 Pitch fork bifurcation 59
4.6 Super critical Hopf bifurcation 60
4.7 Soft and hard generation of limit cycles 60
4.8 Transition from stationary to periodic solution, via Hopf Bifurcations 62
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4.9 Nonlinear system decomposition- nonlinear feedback 63
5.1 Parameter space, state space and eigenspace 75
5.2 Block diagram of the state space system with proportional output feedback law 78
5.3 Decomposition of system for robust stability analysis 80
5.4 Block diagram of plant and controller depicting extended parameter vector 84
5.5 Control induced bifurcation and distance to bifurcation 84
5.6 Parametric step size: large steps may skip some details 87
6.1 Eigenvalues of the nominal open loop servo-pump actuator model 92
6.2 Open loop simulation response of the servo-pump actuator system 93
6.3 Closed loop eigenvalue plot 94
6.4 Unstable response of the servo-pump actuator system for a high feedback gain 95
6.5 Effect of varying orifice diameter on the stability of the system 96
6.6 Effect of varying the constant velocity of the solution 97
6.7 Effect of varying the proportional gain and load mass 99
6.8 Effect of varying proportional feedback gain and orifice diameter 100
6.9 Effect of varying controller parameter 101
6.10 Servo-valve actuator model 102
6.11 Block diagram of closed loop servo-valve actuator model 102
6.12 Eigenvalues of the nominal system with force input and feedback PD 105
6.13 Displacement and velocity of the cylinder for nominal closed loop case – 106
step input
6.14 Displacement of the valve spool for nominal closed loop case – step input 107
6.15 Displacement and velocity of the cylinder for closed loop – ramp input 108
6.16 Displacement of the valve spool for nominal closed loop case – step input 109
Eigenvalues of the nominal PD control system with ramp input 6.17 110
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6.18 Effect of varying Kp 111
6.19 Effect of varying Kd 112
6.20 Effect of varying line pressure 113
6.21 Effect of varying load mass 114
6.22 Effect of varying proportional gain and constant velocity 115
6.23 Stability boundary in the parameter space of proportional gain and constant 116
velocity
6.24 Time response for parameter values below critical 117
6.25 Time response for parameter values above critical 118
6.26 Effect of proportional gain and load mass 119
6.27 Stability boundary in the parameter space (proportional gain and load mass) 120
6.28 Effect of proportional gain and valve spool diameter 121
6.29 Stability boundary in the parameter space (proportional gain and valve dia.) 122
6.30 Stability boundary in the parameter space (proportional and derivative gain) 123
6.31 Effect of varying position feedback gain and velocity feedback gain 124
7.1 Test stand with real time data acquisition and control hardware 125
7.2 Flapper nozzle servo proportional valve 127
7.3 Servo-valve reduced order model 128
7.4 Pressure differential-voltage input-flow thorough put characteristics 129
7.5 Effective kvalve as a function of the pressure differential across the valve 130
7.6 Servo-valve actuator area characteristic model 131
7.7 Line pressure to valve voltage transfer function for a long hose 132
7.8 Line pressure to valve voltage transfer function for a short hose 133
7.9 Open loop Bode plot of the valve transfer function 134
7.10 Servo-valve actuator model schematic 135
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7.11 Servo-valve actuator system with real time control hardware/software 135
7.12 Experimental transfer function of the closed loop (PD controller) servo-valve 136
actuator system
7.13 Open loop Bode magnitude – actuator position/valve voltage 138
7.14 Open loop Bode phase – actuator position/valve position 138
7.15 Effect of accumulator size on the frequency of load mass mode 139
8.1 Bode diagram of a PD controller 141
8.2 Eigenvalues of the nominal closed loop system 142
8.3 Effect of variation in proportional gain 145
8.4 Effect of variation in derivative gain 146
8.5 Effect of variation in load mass 147
8.6 Effect of variation of constant velocity of the system 148
8.7 Simulation response of the system for controller parameter just below critical 149
8.8 Simulation response of the system for controller parameter just below critical 150
8.9 Effect of variation on the grid of controller parameters 151
8.10 Effect of parameter variation on a grid of parameters (Kp, M) 152
8.11 Effect of parameter variation on a grid of parameters (Kp, V) 153
8.12 Stability boundary in two-dimensional parameter space (Kp, Kd) 155
8.13 Effect of load mass on the stability boundary in two-dimensional parameter 156
space (Kp, Kd)
8.14 Stability boundary in two-dimensional parameter space (Mload, Kp) 157
8.15 Effect of varying constant velocity on the stability boundary in two- 158
dimensional parameter space (Mload, Kp)
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8.16 Experimental data of cylinder position with reference for parameter values 159
above critical
8.17 Experimental data of cylinder position with reference for parameter values 160 above critical
8.18 Bode magnitude plot of the compensator 162
8.19 Bode phase plot of the compensator 162
8.20 Nominal closed loop system is stable 163
8.21 Effect of varying Bode gain of the controller 165
8.22 Effect of varying load mass 166
8.23 Effect of varying the constant velocity 168
8.24 Effect of varying compensator gain (100-250) and load mass (50-500lb) 169
8.25 Effect of varying line pressure and load mass 170
8.26 Simulation response of cylinder position and velocity for compensator gain 170
below critical
8.27 Simulation response of cylinder position and velocity for compensator gain 172
above critical
8.28 Experimental response of the cylinder for parameters above critical 172
8.29 Experimental response of the cylinder pressure for parameters above critical 174
8.30 Bifurcation stability boundary in the 2-dimensional parameter space of load 175
mass and Bode gain
8.31 A comparison of the two controllers 176
8.32 Measure of nonlinearity for varying load mass 179
8.33 Measure of nonlinearity for varying Bode gain 179
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List of Symbols
Symbol Description
x State vector
• State derivative (with respect to time) vector x f (,x tu,,p) State equations
Equilibrium point xo
Jacobian matrix Dx
N Measure of bifurcation nonlinearity
Φ Volume flow rate of flow through pipeline
ν Kinematic viscosity
ℜ Real space
air Percentage air in the working fluid
Aport Port area
c Damping
Cd Coefficient of discharge
cr Clearance radius
cv Velocity
D Diameter of spool
d Orifice diameter
det determinant
dline Line diameter
F Force
G(s) Compensator
Jx Jacobian matrix with respect to x
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k stiffness
Kgain Feedback gain
Kv Bode gain kvalve Effective discharge coefficient for the servo-valve
M mass
P Pressure
p Parameters
P(s) Plant
Pline Line pressure
Q Flow qport Flow thorough port
R Reference signal
t time
u Input
V Volume
x Position
Κd Derivative gain
Κp Proportional gain
α Natural frequency modification factor of pipe line modes
β Damping modification factor of pipe line modes
βεff Effective Bulk modulus
λ Eigenvalue
ρ Fluid density
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Chapter 1- Introduction
1. Introduction
1.1. Background and motivation
Hydraulic systems and their uses are mentioned in the ancient Roman texts, where they was used to gain mechanical advantage. Since then there have been immense advances in the field of hydraulic systems, most importantly during the period of World War II. Servo-hydraulic systems are used because of their mechanical advantage, response accuracy, self-lubricating and heat transfer properties of the fluid, relatively large torques, large torque-to-inertia ratios, high loop gains, relatively high stiffness and small position error. Although the high cost of hydraulic components and hydraulic power, loss of power due to leakage, inflexibility, nonlinear response and error-prone low-power operation tends to limit the use of servo-hydraulic drives, they nevertheless constitute a large subset of all industrial drives and are extensively used in the transportation and manufacturing industries.
Design of servo-hydraulic systems is a challenge because of the ever-increasing demands on the performance and economics of the intended systems. Further, these systems have been optimized in design over so many years that further improvement is harder to achieve using linear system theory. The current practice places stringent demands on the system by requiring it to be more reliable and robust in a larger design space. Servo-hydraulic systems may fail due to loss of power due to wear and/or leakage, clogging of the fluid flow due to contamination, inaccurate or undesired response and loss of stability. The loss of dynamic stability is in many cases the most important causes of system failure. In this case, the servo-hydraulic systems undergo large pressure oscillations under certain operating conditions, which may result in catastrophic failure.
The stability of these systems is their most important asset. Any deviation from the optimal response is not affordable in the current state of the art industry applications since the performance dictates the economics of the whole operation and can also result in serious loss of
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efficiency. These large amplitude oscillations are noticed in various applications of servo- hydraulic systems, which range from the aircraft and car industry to heavy machinery to fluid power and manufacturing industry.
With the advent of cheap computational power, it is possible to perform design studies which were simply cost prohibitive some time ago. Designers in the hydraulic systems area, and more generally in mechanical systems, are going for more detailed and accurate designs than ever before. Design considerations now include static as well as dynamic attributes. System designs now include the dynamic response and robust stability behavior for better performance. It is easy to study and design a system under linearity assumptions, but as the size of the design space increases these assumptions do not hold true. Further, all realistic models of the real world are nonlinear, involving a large set of forcing and boundary conditions. The system characteristic with the boundary conditions and initial conditions together make the system exhibit nonlinear response. These servo-hydraulic systems have inherent nonlinearities and exhibit nonlinear response and complex dynamic behavior. This nonlinear nature makes such systems difficult to analyze and design. It is this nonlinear nature which results in unexpected dynamic response under very similar yet different operating conditions. These dynamic responses can range from a simple overshoot to large transients to undamped responses to bifurcations to complex chaotic responses. It is possible to observe some or all of these responses in the servo-hydraulic systems depending upon the design space and parameter space. These nonlinearities can be asymmetric which adds to the problem of design.
The servo-hydraulic drives also provide for interesting dynamical behavior due to the presence of square root nonlinearity and coupling between the hydraulic and mechanical response. Also, the drives have complex dynamics associated with them due to the presence of friction, hard limits, complicated port flows, and intricate coupling between various components in the circuit. A typical servo-hydraulic circuit consists of servo valves, orifices, tubes, volumes, pumps, solenoid valves, etc. These drives exhibit modes, which range from very slow (drift) to very fast ones (as associated with the valve spool and flapper dynamics).
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Presence of nonlinearities in the systems provides for the intricate dynamics exhibited by most of the mechanical systems. Most of the static nonlinearities can be classified as state saturation/cut-off, state jump, polynomial nonlinearity, delay discontinuity, motion limiting stops/ impact nonlinearity, or backlash. The dynamic nonlinear behavior due to coupling and interaction of system states is another area of prime interest as far as dynamic performance is concerned.
These interactions can result in significant modifications in qualitative behavior of the dynamical system under small perturbations in parameter space. The design of systems with significant nonlinear behavior for dynamic performance is still not very well developed and documented.
The use of hybrid system models and bifurcation theory as design tool is a reasonable choice for the design of systems with nonlinearities.
Servo-hydraulic systems are generally designed by linearizing the nonlinearities around a nominal operating point in conjunction with “rules of thumb” and other established practices. It is essential to simulate the detailed design, as simulation with a detailed model is one possible mode of verification of stability and performance. This simulation is very cost prohibitive in the case of nonlinear systems, even with all the available computing power, since it is impossible to investigate all the possible cases. For a nonlinear system, the linear superposition principle does not hold true. This entails that each and every combination of boundary condition and initial conditions can have a dramatically different response in a nonlinear system.
In nonlinear dynamical systems theory, bifurcation theory is a very powerful methodology to study structural stability of the system. Multi-parameter bifurcation theory exists for continuous time systems as well as for discrete time systems. The importance of the bifurcation theory can be realized by the fact that it provides a tool for the study of dynamical system without extensive simulations.
Nonlinearities in the system behavior pose problems in understanding, modeling and identification of the system, but the effect of nonlinearities on the system performance is not always devastating. Many times, nonlinearities result in increased stability margin and bandwidth, although they make it hard to predict the response under some new conditions. The most
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important difference in the linear models and nonlinear models is the presence of multiple equilibrium solutions and their interactions as the system evolves or modifies itself under the influence of various attributes – internal or external. Use of bifurcation theory as a tool for design for nonlinearities is a very significant development because it provides a relatively inexpensive tool to characterize and classify the dynamical behavior of the system involved. This characterization is important in the understanding of the nonlinearity and its effect on the total system, which may be highly coupled in nature.
Most realistic systems have many parameters and it is important to understand the effect of the each one of those on the qualitative behavior of system dynamics. Multiple parameter bifurcation theory provides a tool for mapping this effect in a nonlinear sense and hence is very handy. Further, for efficient parametric study, it is often required to do a decomposition of the system and reduce the parameter set to a smaller dimension. This is important for all systems and is of interest in case of servo-hydraulic systems. Perturbation methods provide a good tool for bifurcation and continuation studies, although the most fundamental restriction of these methods is that they do not work across a discontinuity. This limits their use in case of mechanical systems, including servo-hydraulic systems, which can be most easily modeled with hard limits or discontinuities. It is important to understand that any model is just an imitation of the reality and hence it is important to determine if the dynamic response obtained is a manifestation of the model or true nature of the physical system. Servo-hydraulic systems are easily modeled using a hybrid system model, which couples a number of low order models with a state change criterion to imitate the complex dynamics accurately. Hybrid system models can generate dynamic behavior is equivalent to the effect of discontinuity in the higher order model. This gives rise to the issue of accurately depicting the state transitions as well as distinguishing between the real and model dynamics.
Further, this design for nonlinearity is to be robust so that parametric uncertainties do not result in catastrophic failure of the system. This requires the analysis of the system models for the distance to dynamic failure, which is equivalent to distance to bifurcation (usually measured in
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parameter space) or rate of bifurcation. Ideally, all the complex dynamics involved in systems with nonlinearities should be identified and cataloged for use by future designers. For servo- hydraulic systems, this study has not been reported in any detail in the literature and hence is of importance as the use of servo-hydraulic systems in industrial settings is tremendous with demands for more compact and high performance systems every day. Linear models have been studied in great detail, but the nonlinearities involved are so dominant in the routine operation of the servo-hydraulic systems that it is inaccurate to ignore them.
This design using a bifurcation theory-based approach is suitable for the analysis, identification, and control of any undesirable dynamics. This would include any dynamic response which is a result of lower and higher order bifurcations to any chaotic dynamics. The ability to control any undesirable modes is important in pushing the envelope of the design space.
The feedback control of any such modes is of interest in the case of servo-hydraulic systems since these systems are used mostly in closed-loop operation. This gives rise to the idea of control- induced bifurcations and their stabilizability and controllability. These bifurcations are a result of feedback and can be avoided by suitably designing the system and the control loop. Moreover, properly designed feedback control can often stabilize an inherently unstable open loop system.
Feedback coupled with nonlinearities in the system can result in complex dynamics even for simple systems. The very fact that simple systems with nonlinear coupling can exhibit complex response is one of the major tenet of chaos theory.
These aspects of mechanical system design for nonlinearities are of importance under current design practice. This dissertation aims to highlight issues mentioned above and also provides a comprehensive study of the dynamics of servo-hydraulic mechanisms. The nature of nonlinearity, control-induced bifurcations, model and dynamical behavior of servo-hydraulic systems are all important aspects of this work. This research aims to provide significant insights and concrete design methodologies in these areas. The idea of control of bifurcations as well as control-induced bifurcation is an important aspect of feedback design and control. It is this idea that also distinguishes the dynamical behavior of the systems. The system dynamics can then be
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classified as either an inherent system property or an artifact of modeling or one resulting from the feedback. Once the dynamics have been characterized then it is possible to modify them by design or control. The goal of this dissertation is to develop a framework for the nonlinear design and analysis of servo-hydraulic systems with emphasis on using bifurcation based procedures for the study of possible qualitative dynamical behavior under various operating conditions including nonlinear loads and boundary conditions. It is also a goal to develop a framework for decomposition of large-scale systems for analysis and diagnosis in a nonlinear sense.
Furthermore, the classification of the behavior of nonlinearity -static and dynamic- according to its effect on the stability of the system dynamics, using the notion of distance to bifurcation, is an important aspect of this work.
1.2. Applications
Servo-hydraulic systems are concerned with the transmission and control of energy for the purpose of moving and imposing forces on the machine elements with certain mechanical efficiency. It is a means of converting, transmitting, controlling, and applying fluid energy to perform useful work. Throughout the history of mankind, a natural evolution of energy and power utilization has taken place. The mode of transmission and control of power has also changed from manual to mechanical to hydraulic and electrical. Combinations of these modes are used in the industry these days. Torricelli discovered orifice flow laws in 1664, which started the study of the laws of hydraulics; and more recently hydraulic component standards were developed which provide a guideline in designing these systems. Servo-hydraulic systems are often preferred over mechanical systems because they can rotate, push, pull, oscillate and regulate mechanisms in modern industry. They can transmit power with fast response and can multiply force simply and efficiently. They can provide a constant force or torque irrespective of the speed changes and are more reliable.
Servo-hydraulic systems are used in various industries, some of which are agriculture, automobiles, aircrafts, chemical processing plants, construction, lumber industry, material
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handling, mining, metal working, military, manufacturing, and transportation industries. This would include industrial robots and control of machine tools, suspension, brakes and clutches for automotive vehicles, flight control actuation, and stabilization of wheel set in rail transport, and active stabilization of large civil engineering structures. Basically, servo-hydraulic systems are used where the requirement of power is large with compact size. With the advent of good control algorithms, accurate and precise use of servo-hydraulic mechanisms is very common and control authority is increased with the use of embedding the control functions in the software with accompanying sensing and actuation mechanisms. It must be noted that all these systems, when designed using ad-hoc or trial and error methods, can nevertheless result in unexpected dynamics due to the highly nonlinear nature of servo-hydraulic systems.
1.3. Organization of the dissertation
The goal of this research is to understand the dynamical behavior of the servo-hydraulic drives under the effects of various nonlinear operating conditions and system dynamics. Further, it provides a detailed methodology for design and analysis of systems with nonlinearities using bifurcation theory-based approaches. It investigates the effect of controller parameters and plant parameters on the qualitative behavior of the system dynamics. The effect of controller parameters is very important distinction since the feedback provides an extended envelope for the design of systems. This research also aims to characterize the nature of nonlinearities for their effect on the stability behavior of the system. To achieve this, first the effect of various parameters has to be understood, and then a decomposition performed on the parameter space to reduce the problem for efficient handling with the bifurcation-continuation based analysis and design schemes. Since the bifurcation-continuation based procedures do not work across the discontinuities, it limits the analysis to only the smooth-continuous regimes of operation. Thus it is important to understand how a discontinuity, either induced due to modeling or inherent in the system characteristics, affects the dynamical behavior, especially robust stability.
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System nonlinearities are either static or dynamic in nature and they tend to be coupled together to provide intricate dynamics, which is a delight to the theoretician but could present great difficulty for the design engineer. These nonlinearities result in various system behaviors including stability bifurcations, which could be as simple as saddle-node or Hopf bifurcations or as complex as a chaotic response. A detailed study of these behaviors as shown by the servo- hydraulic mechanisms is one of the goals of this dissertation. To study the contribution of the nonlinearities, the idea of constant velocity bifurcation analysis is proposed. This helps in characterizing nonlinearities for their effect on stability behavior. Further, the control of complex dynamics is also important in the feedback sense, which means it is to be understood when a sub- critical bifurcation can be changed to a supercritical bifurcation by a smooth state feedback, which is equivalent to the local feedback stabilization of the system at the bifurcation point. This local feedback stabilization becomes nontrivial only when the unstabilizable eigenvalues of the linearization have zero real parts. This research provides qualitative as well as quantitative estimates on the regions of stability for the servo-hydraulic systems under various feedback and boundary conditions. This parameter space investigation provides crucial information for future analysis and design of these systems. The methodology proposed in this work can be extended to any system which shows complex dynamics due to the presence of nonlinearities.
This dissertation is organized into nine chapters. Chapter 2 provides a review of the servo-hydraulic drives and reviews the previous work done in the area. It also provides details of general modeling issues for servo-hydraulic drives, and then physically realistic models of the two test systems (servo-pump actuator system and servo-valve actuator system) is given. Chapter
3 compares the linear vs. nonlinear dynamics of the servo-hydraulic systems with emphasis on the qualitative aspects and hence provides motivation for the need of nonlinear analysis and design of servo-hydraulic systems. Chapter 4 introduces the basic theory of dynamical systems, with emphasis on multi-parameter bifurcation theory for use as a design and analysis tool.
Chapter 5 deals with the bifurcation theory and its various details for use as a multi-parameter design and analysis tool. This chapter focuses on issues including distance to bifurcation and
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robust stability, model decomposition for analysis, control induced bifurcations, and control of bifurcation instabilities. It also discusses various numerical issues related to bifurcation analysis.
Chapter 6 provides preliminary results obtained on the servo-pump actuator model and servo- valve actuator model. Chapter 7 describes the experimental apparatus development and validation of the model. Chapter 8 provides some experimental control studies done on the system for stability analysis. Chapter 9 provides summary and conclusions. Suggestions for future research are also outlined in this chapter.
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Chapter 2 – Servo hydraulic drives
2. Servo hydraulic drives
Servo-hydraulic systems are in many instances the most important class of motion and force control systems used in various industries like manufacturing, transportation and heavy equipment. They offer advantages from the point of high force and power capacities with relatively compact actuator sizes. However, hydraulic actuation comes with a penalty in the form of significant nonlinear dynamics, and basic nonlinearities associated with the servo-valve and actuators remain even after removing the complicated control loop nonlinearities. This chapter includes a review of previous work in this area, hydraulic system modeling fundamentals and the details of two test system models.
2.1. Previous work
Servo-hydraulic systems have been a subject of extensive study. The stability of the servo-hydraulic systems is a topic of great practical interest, especially with regards to nonlinear oscillations. The servo-hydraulic system modeling can be found in various references including
Blackburn et. al. (1960), Lewis and Stern (1962), McCloy and Martin (1980), Merritt (1967),
Nightingale (1956). The text by Blackburn (1960) et.al. also provides the nonlinear dynamics of a hydraulic system with a single stage pressure relief valve. Lewis and Stern (1962) also have an interesting comparison of the steady state forces and transient forces. Merritt (1980) is a compilation of practical and empirical information on the hydraulic control system design. Most of these references give a linearized stability analysis of a single stage pressure relief valve, although Merritt notes the requirement of a damping orifice in the system for stability reasons.
Some other works on hydraulic system dynamics are Stringer (1976), Viersma (1980), Skaistis
(1988), and Watton (1989). The importance of flow forces in the stability of the servo-hydraulic system is discussed in most of these works. It is noted that the strength of these flow forces as
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compared to system stiffness and damping would determine the possibility of limit cycle oscillations. But most of these discuss the linearized analysis of the systems. An extensive survey reveals that the most published work in this area deals with the linearized or quasi-linearized analysis or simulations. Linear analysis tends to overlook the complex dynamics, which is only possible in the system with nonlinear interaction between the states, and simulation is cost- prohibitive in a system that does not satisfy the linear superposition principle.
The study of dynamic stability of servo-hydraulic systems has been an area of interest to various investigators, which includes Noton and Turnbull (1958), who used a graphical method of isoclines to obtain approximation to a describing function and predict the possibility of continuous oscillations in a four-way spool valve. Foster and Kulkarni (1968) did a quasi-linear analysis; Burton (1975) used the describing function approximation to the nonlinearities using the binomial expansion for a parameter plane technique. Ray (1978) did extensive simulation of the relief valve to investigate the chattering and limit cycle oscillations due to input pressure variation. Dokainish and Elemadany (1978) used a harmonic balance method to study the nonlinear response of the relief valve. Taft and Twill (1978) did a linearized analysis, which they compared to experiments and simulation results for a three-way under lapped valve. Jen-Kou
Chou (1983), McClamroch (1984), and Cox and French (1986) used describing function-type methods for limit cycle analysis. McLeod (1985) did a linearized Routh-Hurwitz analysis for several operating conditions. Karmel (1990) and Margolis (1996) did simulation studies for hydraulic systems and demonstrated the ability of the systems to exhibit oscillations. Kremer
(1998) did extensive bifurcation based studies of the hydraulic control systems and showed the emergence of limit cycle oscillation and their parameter dependence. Some recent work in the area of analysis of servo-hydraulic systems would certainly include Alleyne and Liu (1999) in which they studied the limitations of force tracking control of the servo-hydraulic systems.
Hwang (1996) investigated the sliding mode control of an electro-hydraulic system. Plummer and
Vaughan (1996) investigated the robust adaptive control of servo-hydraulic systems. Tsao and
Tomizuka (1994) studied the robust, adaptive control of hydraulic-servo for noncircular
21
machining. Yao, Bu, Reedy, and Chiu (1999) verified theoretical results with experimental data for a robust control of the single-rod hydraulic actuator.
The study of servo-hydraulic systems would consist of the actuator dynamics, valve dynamics, and any line and load dynamics. Investigators in this area have used various tools and methodologies available at times to investigate dynamical behavior. This includes Fourier analysis of the response, investigation of the effect of port flow on stability, estimation of limit cycle amplitude and frequency, effect of feedback on the valve dynamics, valve chatter and its causes, effect of high frequency oscillations in pressure at valve input, effect of leakage, study of super-harmonic and sub-harmonic responses in valve dynamics, effect of actuator seal friction on stability, effect of saturation in valve travel, impedance matching between load dynamics and valve dynamics, effect of damping on the stability, and effect of fluid bulk modulus.
Further, the study of servo-hydraulic systems is also done to investigate various control- related issues. This would include studies in the area of adaptive control of actuators, neural networks and Lyapunov-based techniques. This would certainly include Alleyne, Neuhaus, and
Hedrick (1993), Alleyne and Hedrick (1995), Cho and Hedrick (1991), Hwang (1996). Also, application of feedback linearization techniques has been demonstrated by a number of authors including Del Re and Isidori (1995) and Vossoughi and Donath (1992), and Richard and
Scavarda (1996) and Thompson, Pryun and Shukla (1999). Kremer and Thompson (1998) developed a bifurcation-based procedure for designing and analyzing stability of nonlinear hydraulic control circuits. This work uses the concept of the shortest distance to instability to investigate robust stability of the hydraulic models. It uses the concept of bifurcation surface to analyze the robust stability of the nonlinear systems with smooth nonlinearities.
Loads acting on the servo-hydraulic systems can be highly nonlinear in nature. This is of utmost practical interest. Notable examples of such loads are force tracking control applications, civil engineering structural testing applications, and system loads with significant friction effects.
These three classes of applications are a very short list of possible cases which may cause significant nonlinear behavior, but does cover a range of all possible cases that are of interest to
22
industry. Another example would include hardware-in-the-loop simulation of the nonlinear structural testing specimens, which are of considerable importance for earthquake resistant designs. McClamroch (1985) outlines stability issues involving hydraulic actuators with coupled nonlinear structural loads.
2.2. Fundamentals of hydraulic system modeling
For the purpose of dynamic analysis and control design, hydraulic systems are modeled using the approximation that the systems can be represented as a collection of lumped volumes separated by sharp edged orifices. Such low order models are widely used in industrial practice
(such as in the automotive industry) and, with sufficient detail, have been found to yield good correlation with experimental data. However, model updating/validation is needed in instances where a highly accurate representation of the physical system is required. The orifice flow is classified as either being laminar or turbulent, depending on the Reynolds number. Each volume has to satisfy the continuity equations as laws of conservation of mass and energy holds true.
2.2.1. Orifice flows
To define the dynamics of fluid flow through an orifice, it is important to note that whenever the pressure differential is large for all operating points of interest, it can be safely assumed that the flow always has a large enough Reynolds number so that it can be calculated using the turbulent flow equation (Merritt, 1967). The magnitude of the volume rate of flow (in3/sec)
qport , through the orifice of area Aport , is given by Equation 2.1.
2 qC= A P (2.1) port d port ρ differential
This equation is obtained by applying the Bernoulli’s equation to the streamline flow, for
the turbulent flow case where Cd is the discharge coefficient, and Pdifferential is the differential
pressure across the two chambers connected by the orifice. The sign of the Pdifferential determines
23
the direction of the flow and ρ is the density of the fluid. It should be noted that although the assumption of compressible fluid volumes is made, effective change in density of the fluid is small. If the orifice is assumed to be sharp edged then the discharge coefficient can be assumed constant. The value 0.61 is normally used for the discharge coefficient (Merritt, 1967). The flow through the orifice is normally turbulent, but becomes laminar when the pressure difference across the orifice is close to zero, corresponding to the low flow rate and hence low Reynolds number. The laminar and turbulent regimes must be both modeled, as the turbulent flow equation has an infinite gain (i.e. infinite slope), at zero flow and hence causes numerical problems in analysis. Generally the transition point is selected around 1 Psi pressure drop and hence the flow equation can be given as:
2 CAd port ()Pdifferential , 0≤≤Pdifferential 1psi ρ qport = (2.2) 2 CA (P ) sgn(P ), P >1psi d port differential differential differential ρ
This combined laminar and turbulent flow equation is continuous but not smooth at the transition point, which can be easily demonstrated. This flow equation can be further used to compute the pressure derivative relationship for each volume element in the circuit.
2.2.2. Continuity equation
For a fixed volume system, the continuity equation is a manifestation of conservation of mass; total flow into the system must be equal to the total flow out of the system. Generally for hydraulic systems this equation can be given as
dV V • QQ=++P (2.3) ∑∑in out dt β
24
where V is the volume, dV dt is the flow consumed by expansion of the control volume, and
• VP β is the compressibility flow resulting from pressure changes, and β is the effective bulk modulus of the fluid. The effective bulk modulus (Merritt, 1967) of the fluid is given by
1 β = (2.4) eff 1%air + βairfree 1.4(PPvolume + atm )
where βairfree is a constant equal to the bulk modulus of the fluid with no air entrainment, and
air% is the entrained air percentage by volume in the fluid at the atmospheric pressure Patm .
2.2.3. Example
An example of a basic hydraulic system, which illustrates use of the preceding laws, is given in the Figure 2.1.
Mass M k Qin P1 P2
Q12 x c
Figure 2.1: A sample hydraulic system model.
The continuity equations for the two volumes can be written as follows
V QQ=+1 P in 12 β 1 1 (2.5) dV22V QP12 =+2 dt β2
25
dV 2 is due to the fact that the volume changes with time as the piston and load mass moves, and dt
is equal to Aload x . The flow Qin is the input flow for this system, and the flow Q12 is the flow through the orifice due to the pressure differential between the two chambers, given as
2 CAdorifice ()P12−≤P, 0P12−P≤1psi ρ Q12 = (2.6) 2 C A ()P −−P sgn()P P , P −P >1psi d orifice 12 12 12 ρ where Aorifice, is the area of the sharp edged orifice separating the two chambers. The load dynamics can be defined by considering the force balance using the Newton’s Law of motion and is given as
mx + cx +=kx (P12−P )Aload (2.7)
Thus, the state space representation of this hydraulic system will have following states:
x x (2.8) P1 P2
It is easily seen that the above hydraulic system, while being modeled as a lumped volumes interconnected with the sharp edged orifices, uses the fundamental concepts of continuity and orifice flows with Newton’s Law of motion to capture the dynamics of the system. It should be noted that in this system the nonlinearity due to square root nature of the orifice flow is the dominant static nonlinearity.
2.2.4. Valve modeling A valve, as shown in Figure 2.2, has two major components, a valve body housing and the actuation unit. The valve body houses the valve seat, and the clearance between it and the valve plug determine the amount of flow through the valve. Thus, the equations of motion can be
26
derived for the valve position and velocity using the force balance. The various forces that act on the valve body are pressure, inertia, spring, and flow forces. Using Newton’s Law, the equation of motion can be written as follows:
Figure 2.2: A valve schematic (Courtesy: Bosch Corp. www.bosch.com).
∑∑FFpressure ++spring ∑Fflow =Mvalve−spool xvalve−spool (2.9)
The mass term M valve−spool represents the mass of valve spool and that of the fluid between the valve lands in the pressure control chamber. The force due to the pressure differential
Fpressure consists of the pressure differential on the valve lands which includes the effect of variable area on both sides of the valve spool (which are exposed to the line pressure) and since sump pressure is assumed to be approximately equal to zero. The magnitude of the variable spring force Fspring depends on the spring rate and the valve spool position. There is also a constant preload spring force due to the difference between the spring’s free length and the installed length. The spring rate could be constant or a function of the valve spool displacement, which makes the total spring force a nonlinear function. The flow forces Fflow are the forces
27
which act on the valve spool due to the fluid movement in the various chambers/lands and through the ports. They are also called flow-induced forces or Bernoulli’s forces (Merritt, 1967).
Pline>Po X
D
Po
D
X
Figure 2.3: Port flow area schematic of a spool valve.
Port Area: Port area Aport (Figure 2.3), is a nonlinear function of the valve position as given below: π Dc ,0x ≤ r 2 2 ADport =+π xcr ,0≤x≤xmax (2.10) 2 dline Axmax =≤π , max x 2
When the valve is closed ( x ≤ 0 ), the flow limiting area is the annular area created by the radial clearance between the valve spool and the sleeve. As the valve starts to regulate, as in
28
Figure 2.4, the flow area becomes a function of the valve diameter D , valve position x and radial clearance cr . Since the radial clearance is small relative to the valve diameter, the flow is approximated as the product of the valve circumference (Figure 2.5) and the length of the port opening. Finally, a flow saturation limit is imposed by the size of the hydraulic line connecting the port to the discharge area. The maximum area Amax, (Figure 2.6) is given by the flow saturation in the line or the hard limit on the valve travel.
Port Flow: To completely define the valve dynamics, a relationship for the net flow analysis using the continuity equations has to be developed for a control volume. This should take into account any leakage across the valve spool and the compressible nature of the fluid. The conservation of mass or continuity equation is given as dV V • QQ=++P (2.11) ∑∑in out dt β
Pline >Po
x
Cr
X 22+ C P r D o
Figure 2.4: Open valve port area configuration.
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Pline>Po
Cr
Po D
Figure 2.5: Closed (overlapped) valve port configuration.
Effective Flow Area Aport Wide Open
Overlapped π drspool c _ spool x Laminar Leakage max xspool
Valve Displacement
Figure 2.6: Port flow area as the function of valve displacement.
30
where V is the volume, dV dt is the flow consumed by expansion of the control volume, and
• VP β is the compressibility flow resulting from pressure changes, and β is the effective bulk modulus of the fluid. The valve port is considered as an orifice, whose area is given by Equation
2.10. The flow equation (Equation 2.2) can be further used to compute the pressure derivative relationship for each volume element in the circuit. Thus the state space description of a servo- hydraulic system consists of the two position-velocity states for each valve spool and a pressure state for each independent fluid volume. Further, the relationships that define the mass conservation and force balance are important to derive the intended state space description of the servo-hydraulic system.
Pline L X X
Input Force or Position
Pa Pb
Figure 2.7: A common four-way spool-valve.
Flow forces: The equations for static and dynamic flow forces are derived using the general equation of momentum to a control volume in the valve. The static and steady-state flow forces are a result of the inherent fluid accelerating property of the valve port and are present during the flow out of the valve chamber through the port. For a common four-way spool valve, as in Figure
2.7, (with inflow and outflow through the same port, depending on valve position) the static flow
31
force is analogous to a centering spring since it always acts in a direction to close the valve. The static flow force is due to a pressure gradient in the chamber caused by a higher velocity near the port.
Faxial θ
F jet
Figure 2.8: Axial and lateral components of flow forces in a valve port.
The dynamic flow forces are present only if the fluid is accelerating in the chamber. To determine the static flow force, the jet force due to the flow needs to be calculated and is given by the mass of the fluid flowing through the flow limiting passage multiplied by the local acceleration of the fluid near the port. According to the Newton’s third Law, the jet force has an equal and opposite reaction force. This reaction can be projected into an axial and a lateral component (Figure 2.8). When the port is circumferential, the net lateral force is negligible. The jet angle is a function of the valve spool position and radial clearance (Merritt, 1967). This relationship between the jet angle and the valve spool position and radial clearance is complex and Kremer (1998) used a polynomial curve-fit to approximate the exact relationship, which will be used in this work. 2 ρqport Fjet = (2.13a) CAc port
FFaxial = − jet cosθ (2.13b)
The dynamic or transient flow is present only when the flow is accelerating through the chamber and the magnitude of the axial fluid reaction force is given by Newton’s second Law
(Blackburn et.al., 1960). dq F= ρLport (2.14) dynamicflow dt
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where L is the damping length (Figure 2.7) or the distance between the centers of incoming and outgoing flows. As the flow is a function of position and pressure differential, the resulting dynamic flow force has a primary damping term and a secondary term that is proportional to the pressure derivative and the total dynamic flow force is evaluated as follows: dq FL= ρ port dynamicflow dt
≥00xx≥max π dx spool Pline x (2.15) 0 ≤≤xxmax 2 2 = LC 2ρ xc+ d r A + port P 2 P differential differential
The direction of the flow force is determined by the direction of the fluid acceleration.
The net reaction force on the valve will be opposite the direction of pressure difference driving the flow. If the valve position is defined so that a positive direction corresponds to opening valve port, then a positive valve velocity means the valve port is opening and the flow rate is increasing. The positively damped and negatively damped valve corresponds to whether the dynamic flow force is in the same direction or opposite to the valve velocity. It is easy to show that a valve is positively damped if the fluid flows out through the metering orifice and is negatively damped if the inflow is metered.
2.3. Modeling assumptions
Servo-hydraulic systems generally consist of an actuator piston and ram, connected to a load. The actuator is supplied with the pressurized hydraulic fluid metered by the spool valve.
The spool is driven by an electrical or hydraulic or mechanical input. In other instances, notably flight control actuation, the fluid supply to the actuator may be driven directly by a reversible servo pump. Several simplifying assumptions are made to help transform the physical system into
33
a mathematical model. Some of those assumptions, which apply to most of the servo-hydraulic systems, are as follows:
• Energy losses due to bends, fittings and sudden changes in flow cross sections are negligible.
• Sump pressure is negligible and cavitation is not modeled.
• Ports are symmetric circumferentially.
• Servo-hydraulic systems can be accurately modeled using lumped parameters.
• The transition from laminar to turbulent flow in orifice occurs at a pressure difference of 1
Psi.
The assumption on the transition pressure eliminates the need for calculating flow transition based on the Reynolds number calculations (Kremer, 1998). Leakage flows are assumed negligible except for the annular flow for a closed bypass port. Further, this research also includes models of short tubes and pipelines for including flow dynamics and their effects on the model stabilities. Saturation in valve travel and flows is also considered in this study. For a physically realistic model of servo-hydraulic systems, the effect of variable bulk modulus, discharge coefficients, transition grooves, short tubes and other hydraulic subsystem models should also be added as required. It is to be noted that there are other ways to model the systems behavior, for example the transition from laminar to turbulent zone could be substituted by a variable discharge coefficient (Viall and Zhang, 2000).
These models are restricted such that :
• Flow behavior is calculated in a macro sense
• Cavitation is not modeled
• Supply reservoirs, for example providing Pline in servo-valve actuator model, are
constant pressure and infinite capacity.
2.4. Physically realistic models of general servo-hydraulic system The servo-hydraulic system can be modeled as a set of coupled nonlinear ordinary differential equations, as suggested by the example given in Equations (2.5)- (2.8). The servo-hydraulic
34
system when modeled as a set of lumped volumes connected through a set of sharp-edged orifices result in two position states (i.e. position and velocity) for each valve spool, two position states for each load (i.e. position and velocity) and one pressure state for each volume. The servo- hydraulic systems have two primary units, a control valve and an actuation unit. The dynamics of the servo-hydraulic system are captured using a set of ordinary differential equations and involve the valve dynamics and the actuator dynamics coupled with the flow-pressure dynamics of the system subject to the external input and other operating conditions.
The state space description of the nonlinear system is helpful as it facilitates the analysis using various nonlinear dynamics techniques. It requires the system to be modeled as a finite number of coupled, first order ordinary differential equations which can be written as follows in the vector notation: dx ≡=xx f (,tu,,p) (2.16) dt where x∈ℜn is an n-dimensional state vector and f (,x tu,,p)is a smooth function defined as
nnu np n f :ℜ×ℜ×ℜ ×ℜ →ℜ and depends on time t, inputs u, and parameters p, where nu is the number of inputs and np is the number of parameters. If f (,x tu,,p)does not depend explicitly on the input it is called unforced, and if it does not depend upon the time then it is time invariant.
A set of equations is autonomous if they are both unforced and time invariant. It is obvious that higher order differential equations can be decomposed into more than one, first-order differential equations. Thus the state space vector for a servo-hydraulic system with n-valves and m-pressure volumes is as follows:
35
x1
xnv_ alve Position _____ xf = (x,...., x) v 111n 1 x = Velocity (2.17) xf = (x,...., x) v nn1 n nv_ alve _____ P Pressure 1 Pmv_ ol
The typical equation of motion for valve consist of a force balance as given below:
• 1 vFnv− alve=+()∑∑pressure Fspring +∑Fshear +∑Fflow +∑Fbias (2.18) M v and the typical continuity equation for a pressure state equation is given as:
βvolume dV()volume PQvolume =−∑∑volume,,in Qvolume out − (2.19) Vdvolume t where volume refers to the control volume being analyzed. Calculation of the flow estimates for a given control volume is the most difficult task at hand in most cases but can be accomplished by defining the flows with respect to the metering entities in the circuit. The models used in this research were selected specifically to highlight the effect of servo-hydraulic system nonlinearity and classify the dynamics related to actuator, valve, pipeline and load. This helps in breaking down the interconnection between the different components of any typical servo-hydraulic system, and their individual contributions to the total system dynamics. It is important to realize the interconnection between these components and the effect of coupling between the hydraulic and mechanical modes. The primary idea of selecting these systems was to isolate the nonlinearities in orifice, flow forces, feedback, loadings, and line dynamics including vibro- acoustic interactions in the system.
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2.5. Modeling of servo-pump actuator system
Figure 2.9 illustrates is a model of the valveless servo-hydraulic system being driven by a reversible servo-pump. This is an “academic” model, developed for study of fundamental nonlinear dynamic characteristics. This model, denoted as Model 1, is thought to be of lowest practical order and complexity so as to yield meaningful dynamic behavior. The parameter choices are basically arbitrary but include variables that are easily controllable and can be varied numerically and physically for study. Similar models for flight control actuation are developed by
Raymond and Chenoweth (1993). It is assumed that the flows are such that the inlet pressure does not go below zero, which is assumed to be the sump pressure. The line dynamics are represented in an approximate sense as the volumes separated by an orifice, which is the primary nonlinearity in the system, apart from the loading. The other assumptions as highlighted in the previous section hold true. The nominal system parameters (in English units) are given in Table
1.
Table 1: Nominal system parameters for servo-pump model (Model 1).
Symbol Description Nominal Value Units kl Load spring stiffness 1000 lbf/in Cl Load damping 4 lbfsec/in M Load mass 200 lbf 2 Aload Piston area 7 in β Bulk modulus 22000 psi air Percentage air 0.05 3 Vanm Volume A, nominal 35 in 3 Vbnm Volume B, nominal 35 in 3 V1nm Volume 1, nominal 10 in 3 V2nm Volume 2, nominal 10 in d1 Orifice 1 (diameter) 0.1 in d2 Orifice 2 (diameter) 0.1 in 2 Aleak Leakage area 0.001 in 2 2 ρ Fluid density 0.0000795 lbf in /sec
37
qp
Figure 2.9: Model 1, servo-pump actuator model with two orifices.
u=pump flow x(position) Reference input (position) kgain xx = f (,u) -
Figure 2.10: Block diagram of Model 1.
The open loop system representation in the state space form is given as follows:
M xc+ x +=kx A(Pa−Pb) (2.20)
β ()Pa Pqa =−[]a1 qleak −Aload x Vxa () β ()P Pq =−b +q+Ax bbVx() 2 leakp b (2.21) β ()P1 Pq11=−[]a +q V1
β ()P2 Pq22=−[]b −q V2
38
qCleak = ( (2 / ρ)) d Aleak ( ( Pa - Pb ))sgn(Pa - Pb ) (2.22)
qCad_1 = ( (2/ ρ)) Aa_1( Pa-P1 ))sgn(Pa-P1) (2.23)
qC2_bd= ((2/ρ)) A2_b((P2 -Pb))sgn(P2 -Pb)
Flows are very important functions of the states and inputs in servo-hydraulic systems and must be accurately modeled for the precise analysis and design. This representation does not show laminar to turbulent transition region for flows, which is used in the model. Generally this system (servo-pump actuator model) has an equilibrium solution at zero flow, which is of not much interest as far as the dynamic stability analysis is concerned. As such the use of a constant velocity solution modeled in the perturbation coordinates with respect to a reference trajectory is exploited to highlight fully the effect of orifice nonlinearity in the system. This model is expected to show simple bifurcation behavior in open and closed loop dynamics.
2.6. Modeling of servo-valve actuator system
In the following section, the more common arrangement for a servo-hydraulic drive, consisting of a servo valve and a cylinder (i.e., actuator) is discussed. The open loop system input is valve spool position, or force, depending on whether the spool valve dynamics are modeled or not. This model, referred to as Model 2 (Figure 2.11) has combined dynamics of a servo-valve with the actuator that are connected to each other by a set of orifice restrictions and volumes.
These volumes represent the associated volumes in the system apart from the actuator cylinder and valve land volumes. The assumptions defined previously hold true and the model is analyzed for stability behavior. In a closed loop sense, the system has proportional feedback with position or force input. For nonlinear system characterization purposes it is essential to investigate other points on the trajectory apart from the zero flow conditions and hence the analysis is done at the constant velocity solution perturbations. The servo-valve actuator system can be modeled as follows using a set of nonlinear differential equations with nominal parameters as given in Table
2.
39
Mx ++cx kx =A()Pab−P +fd
mx vv++cx kxv=fv
β ()P1 Pq11=−[]qa1 V1
β ()P2 Pq22=−[]b q2 (2.24) V2 β ()P a Pqaa=−1 qleak−Apx,0Pa≥ Vxa () β ()P b Pqbb=−2 +qleak+Apx,0Pb≥ Vxb ()
qCleak =−(2 ρ) d Aleak Pb Pa sgn(Pa −Pb ) and for
xv > 0
qC1=2 ρ d Aport Pline −P1sgn(Pline −P1) (2.25)
qC22= 2 ρ dpAort P
qCad11=−2 ρ AaP1Pasgn(P1−Pa) (2.26)
qC22bd=−2 ρ AbPbP2sgn(Pb−P2)
The equations can be suitably modified for the negative valve spool travel. At the maximum travel of the valve spool the system is reduced to one with fixed position and zero valve spool velocity and acceleration. State variables for the problem include position and velocity states for the valve spool and the piston/actuator as well as the pressure states for the four volume chambers. Hence the state vector can be written as:
xv x v x x (2.27) P 1 P2 P a Pb
40
Figure 2.11: Servo-valve actuator model (Model 2) schematic with external restoring and dissipative loads.
Table 2: Nominal system parameters for the servo-valve actuator model.
Symbol Description Nominal Value Units k Load spring stiffness 1000 lbf/in c Load damping 4 lbfsec/in M Load mass 200 lbf 2 Aload Piston area 5 in β Bulk modulus 22000 psi air Percentage air 0.05 3 Vanm Volume A, nominal 50 in 3 Vbnm Volume B, nominal 50 in 3 V1nm Volume 1, nominal 20 in 3 V2nm Volume 2, nominal 20 in d1 Orifice 1 (diameter) 0.1 in d2 Orifice 2 (diameter) 0.1 in 2 Aleak Leakage area 0.001 in 2 2 ρ Fluid density 0.0000795 lbf in /sec Pline Line pressure 450 psi cv Valve damping 5 lbfsec/in kv Valve spring stiffness 3500 lbfsec/in mv Valve mass 0.2 lbf
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2.7. Servo-valve actuator model with line dynamics
The line dynamics in the servo-hydraulic systems are important, as it is the leading cause of limiting the system bandwidth. The effect of line dynamics could be serious on the stability of the system as it gives a water hammer type response every time the flow direction in the circuit changes. To analyze this, the system needs to include appropriate dynamic models of the transmission lines and their effect on the system dynamics should be computed. The line dynamics in lower accuracy models presented in Sec. 2.4 and 2.5 are modeled using the volumes
(V1 and V2) (Figure 2.9, 2.11) with no internal dynamic properties.
Theoretical models of a single transmission line carrying compressible fluid assume that the wall of the transmission line is rigid, flow through the line is laminar, temperature is constant, and flow is one-dimensional. The following model can be given by using mass balance and momentum balance (Yang and Tobler, 1991), respectively:
∂∂Pu∂vv + c2 ρ ++=0 ∂∂tx∂rr (2.28) ∂∂uP11∂2u∂u +=µ 2 + ∂∂txρ ∂rr∂r where c is the velocity of sound. This model takes into account the velocity profile with time t, and position x. v is the radial fluid velocity, and u is the axial fluid velocity. The volume rate of flow is given as follows:
r Φ=(,x t) ∫ 2π rudr (2.29) 0 These models can be approximated by modal approximations with first two modes accounted for in each subsystem. The modal approximation (Yang and Tobler, 1991) for the transmission line
(Figure 2.12) between valve and actuator is given as follows:
42
Pa Pb Qa D Qb
Figure 2.12: Pipe line model schematic.
i+1 2(−1) λci 8β 2 s + DZα 2 no−() α ββλλ22D ss22++88ci n ss++ci 22 PPbi αα ααa = (2.30) QQi+1 ai 2s 2(−1) λci b ZDαα22D on n ββλλ22 ss22++88ci ss++ci αα22αα
2Z • i+1 o 0(−1)Zocλ i 0 − PP D P bi i+1 bi n a =(1− ) λβ8 + (2.31) • −−ci QQ2 Q 2 ai 0 b ai Zoαα 2 ZDonα
where Equation 2.30 can be reduced to its state space realization as given in Equation 2.31 and Pa and Qa denote pressure and flow at the inlet of the pipeline. Pb and Qb denotes the pressure and flow at the outlet of the pipeline. The subscript i in the pressure and flow states denotes the
th contribution of the i mode related to that state. Also, dissipation parameter, Dn and line impedance constant Zo are given in Equation 2.32.
43
lν ρooc DZno==2 crooA
λπci =−(iD0.5) / n i=1,2,3...
βe 2 where coo==A π r ρo
βe =effective bulk modulus (2.32) l=line length ν =kinematic viscosity
ρo = density
Also, the values of natural frequency modification factor (α), and damping modification factor
(β) are given in Table 3. All other parameters are associated with the hydraulic oil (Mobil DTE
24) and a 3/8 in. diameter pipeline is used.
Table 3. Natural frequency (α) and damping modification factor (β) for the two-mode approximation of pipe line dynamics.
Mode # α β
1 1.06 2.31
2 1.05 3.38
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Chapter 3- Linear and nonlinear analysis of servo-hydraulic systems
3. Linear and nonlinear analysis of servo-hydraulic systems
The analysis and design of servo-hydraulic system requires full understanding of the impact of the nonlinearities present in the system and their effect on the system response. It is essential to know the deviations that result in response from its true nonlinear behavior, due to linear approximation of the full nonlinear system. The nonlinearities such as saturation, and flow nonlinearities cause significant changes in the stability behavior of the system. This chapter includes a comparison of linear and nonlinear analysis of servo-hydraulic systems with a review of the effects of nonlinearities on the stability properties of the system.
3.1. Linear analysis of servo hydraulic systems –a review example
An a priori (i.e., fixed parameter) linearized model of the servo-pump actuator system, as given by Model 1, (Figure 2.10) with pump flow as input and the actuator position as the output, at zero flow, has the following open loop transfer function:
Xs() N()s Ps()==65432 (3.1) qspump () s+ c54s++csc3s++c2sc1s+c0 where Ns( ) =+1.932s433.914*10 s3+2.053*106s2+7.165*107s−6.2*10−5 and
3 c5 = 2.0335*10 6 c4 =1.095*10 c = 7.7475*107 3 (3.2) 6 c2 =1.30189*10 −4 c1 = 2.0*10
c0 = 0
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After closing the loop around the plant (as defined by Equation 3.1) using a proportional feedback gain, the resultant system is as shown in Figure 3.1 and the closed loop transfer function is given is Equation 3.3.
kP()s kN()s = 65432 (3.3) 1(+kP s)s +c54s ++c s c3s ++c2s c1s +c0+kN(s)
qpump x kP(s) -
Figure 3.1: Proportional feedback for the linearized servo-pump actuator model.
Figure 3.2: Root locus for varying proportional feedback gain.
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The root locus obtained by varying the proportional negative feedback gain (Figure 3.1) is shown in Figure 3.2.
This linearized analysis shows that the value of feedback gain at which this system becomes unstable is k=687. However, the true value as obtained by the full nonlinear analysis of the system via the bifurcation theory-based methods reveal that feedback gain at which the system loses stability at a zero flow case is k=480. Details of this analysis are given in following chapters. This clearly demonstrates that linear models fail to capture some of the critical dynamics and stability behavior as compared to the true nonlinear system behavior. This is even more evident in the global behavior of the nonlinear systems as linear approximations are only valid in a local sense.
3.2 Experimental results for servo-hydraulic system – a review Experimental results (Merritt, 1967) show that the effect of flow forces is to reduce the open loop gain of the servos and also affect the stability of the system. The gain of the servo is a function of the forcing frequency and it is largest near the natural frequency; this is the worst possible condition with regards to loop stability, and the flow forces tend to increase the resonant amplitude in the region of the natural frequency. With smaller inertia loads, viscous damping has a pronounced effect. If the natural frequency lies within the operational bandwidth, it may be necessary, from considerations of stability, to reduce the flow forces. If the load is predominantly of the viscous or spring nature, the stability of the system is not so drastically affected as then the inertia load predominates. Destabilizing effect of the flow force can be greatly reduced by introduction of a viscous load, perhaps in the form of a dashpot or by the use of an orifice. The presence of a leak tends to reduce the positional accuracy of the system, but the increase in gain thus made possible may more than compensate for this. If the flow forces are eliminated, then the electro-hydraulic gain is very high; this is undesirable as most of the nonlinearities are in this part of the feedback loop.
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The analysis of a linear servo system, excluding leakage effects, indicates that there are no threshold conditions (McCloy and Martin, 1980). The system responds to the smallest input signals and disturbances, while the time taken for similar movement is constant irrespective of the overall amplitude scale of movement. In practice, however, a hydraulic servo has nonlinearities which detract from this ideal, and experimental results show that as the input amplitude is reduced, the output waveform is not in exact proportion to the input signal.
It is thus noted that the true experimental result does not match with the linear system model behavior and hence the servo-hydraulic system should be modeled as a full nonlinear system. It is thus assumed that the stability results obtained using the linear second order model can differ in some cases from the actual behavior of the nonlinear system depending upon the contribution of the nonlinearity to the stability behavior of the system at any given operating point. It is the goal of this research that the trends observed in the experimental results are replicated using the bifurcation theory-based approach. Thus model refinement to match the experimental behavior is a critical and important step of this research.
3.3 Nonlinearities in servo-hydraulic systems and their effect on stability
A servo-hydraulic system has strong nonlinear characteristics which include friction, saturation, flow nonlinearity, and various dynamic nonlinearities like hydro-mechanical coupling.
The state space representation of a servo-hydraulic model clearly shows the coupling between mechanical (displacement and velocity) states to hydraulic (pressure) states. Beside the square law relationship between the pressure and flow, in reality a servo-hydraulic system shows other significant non-ideal behaviors such as flow deadband, saturation, nonlinear opening of valve orifice, and friction (Ziaei and Sepehri, 2000). These nonlinear effects tend to cause the actual response of the servo-hydraulic systems to deviate from the idealized linear response. This does not mean that all of these nonlinearities are “bad”; some of them tend to provide stability to the system response, especially if they are dissipative in nature. Thus, the difference between the linear approximation and the actual nonlinear system response and stability is most visible for very low or very high end of the performance bandwidth. Various other details which add to the
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nonlinearity of the problem are leakage flows, flow forces – static and dynamic, viscous shear forces, variable bulk modulus, flow transitions – laminar to turbulent, effect of hydraulic volumes and the flexible line dynamics.
These nonlinearities provide difficulties in designing globally robust control algorithms since they are only valid in a local sense, especially if any linear approximations are used.
Further, these nonlinearities can result in complex dynamics like limit cycle oscillations which are not possible in linear systems. These systems are known to undergo Hopf bifurcations and resultant pressure oscillations can be damaging to the successful operation of the pump and/or other components in the servo-hydraulic circuit.
3.3.1 Effect of flow nonlinearity on the stability of servo-hydraulic system
Not all of these nonlinearities are harmful; actually some of them provide stability to the system by dissipating energy. Friction is one of them and helps in increasing the size of the stability region by dissipating energy (Feeney and Moon, 1993), but also results in the wear of the system, which changes the characteristics of the system and hence can result in the loss of control efficiency. Similarly, the variable bulk modulus results in increased stability of the system when very small volume, such as those related to the valve lands, are dominating in the dynamics.
On the other hand, the flow characteristic (Equation 2.2) results in numerical instabilities
(Chou, 1983), as transition between laminar and turbulent regimes of flow is only a numerical approximation.
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Response Response
Input Input
Deadzone Nonlinearity Saturation Nonlinearity
Figure 3.3: Dead zone and saturation nonlinearity – input and response.
3.3.2 Effect of saturation on the stability of servo-hydraulic system
Practical hydraulic control systems have a limit on the flow (e.g., due to pump capacity) and the motion of the valve in the bore. These saturations present significant nonlinear effects in the response of the system. Flow saturations can decide the bandwidth of the whole system, while the saturation in position will result in an impact type response in the system (Merritt, 1967). This causes a very high frequency oscillation (Wang, 2000) and provides a little uncertainty in the behavior after the saturation (Figure 3.3) has occurred. Both of these nonlinearities result in state transitions and change the model completely. In fact, even the current bifurcation analysis tools also fail when the continuous system goes through state transitions due to saturation or due to other effects (e.g., state-event systems with a variable number of degrees of freedom). Further, the dead zone or saturation type nonlinearities, which have neutral stable mode in which the escape condition is not unique, do not have unique equilibrium and hence cannot be analyzed using the standard techniques from passivity and Lyapunov theory.
3.4 Large-scale coupled nonlinear systems
The servo-hydraulic systems are a typical example of a large-scale coupled nonlinear system with nominal operation such that the nonlinearities are excited and hence dominantly contribute to the dynamical behavior exhibited. These systems exhibit a range of dynamical
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behavior from simple harmonic/sub-harmonic response to frequency distortion (an early indication of the violation of the linearity assumptions), to parametric oscillations as a sign of
Hopf-bifurcations, to grazing bifurcations (as seen when there is a hard limit), to a chaotic response or catastrophe which is simply an extreme case of operational failure. The details of these various dynamical effects will be discussed in the later chapters of this work and can be found in many standard texts on dynamical systems like Seydel (1992), Wiggins (1991),
Guckenheimer and Holmes (1991). The nonlinearity is a natural occurrence and provides for the existence of the multiple equilibriums, which interact with each other in a local sense due to parametric variation and hence cause these interesting dynamics. Dynamic behavior similar to servo-hydraulic mechanisms can be found in the other natural or man-made systems which have strongly nonlinear characteristics or coupling between various attributes of the system.
Response 3 P2 Response 2
New Parameter
Response 1 ? Response 4
P1
Figure 3.4: Parameter space investigation using simulation.
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3.5 Why nonlinear analysis is needed?
In the case of a nonlinear system, an exhaustive simulation or experiment may uncover some modes of failure but still may not uncover all the possible unstable modes, as the parametric space investigation required for any such study is simply not practical due to computation cost and time involved (Figure 3.4). So, a methodology for the nonlinear system analysis for robust stability analysis is of utmost importance. In the linear case, various parametric analysis tools are available and are used regularly by the scientific community. For the systems which have strong nonlinearities guiding their dynamic behavior, linearity assumptions do not hold true even for a small neighborhood of parameter space. The use of bifurcation-based procedures for the analysis of systems is very important in designing for nonlinearities.
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Chapter 4- Nonlinear systems analysis and bifurcation theory
4 Nonlinear systems analysis and bifurcation theory A nonlinear-coupled system is mathematically represented by the set of differential equations xx = f (,tu,,p) where f can depend on time t, input u, and parameters p and is continuous in the time domain. This chapter reviews the fundamental theory of dynamical systems and generic local bifurcations. This chapter further explores the concept of stability of forced systems and nonlinear feedback. A review of the nonlinear system analysis tools is also given in this chapter.
Equivalent representation of the discrete system can be done using maps. The flow of the state is evolved as defined by the above set of differential equations by simply integrating over time. The fundamental theorem of existence-uniqueness provides the basis of the uniqueness of these flows, or solutions or trajectories. According to the theorem, let E be an open subset of
n 1 ℜ containing xo and assume that fC∈ (E), then there exists an a>0 such that the initial value problem x= f ()x (4.1) x0()=xo has a unique solution x(t)on the closed interval [-a,a]. The proof of this theorem can be seen in any standard text on dynamical systems or differential equations including Perko (1996). Further, if there exists a δ > 0 such that for all xo∈ Nδ (xo), where Nδ is compact set in the neighborhood of the initial condition, the above initial value problem has a unique solution
1 n+1 xx(,t o ) with x∈CG() where Ga= [,−×a]Nδ (xo )⊂ℜ and xx(,t o )is a twice-
n+m continuous differentiable function of t for each value of xo . Also, if E is an open subset ofℜ
n m 1 containing the point (,xopo)where xo ∈ℜ and po ∈ℜ and assume that f ∈C(E), then there exists an a>0 and δ > 0 such that for all x∈ Nδ (xo ) and p ∈ Npδ (o ), the above initial
1 value problem has a unique solution xx(,to,po) with x∈CG() where
Ga=−[,a]×Nδ(xo)×Nδ(po). These relations define the uniqueness and existence of the solution for each and every combination of initial condition, boundary condition, input and
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parameter values. The flow of the vector field f (x) is a set of mappings φt such that
φto()x= φ(t,xo). The first step in the analysis of nonlinear systems is to compute the equilibrium points and to describe the dynamics of the system near those points.
4.1 Dynamical systems and equilibrium points
n A point xo ∈ℜ is called an equilibrium point or critical point of the initial value
problem (4.1) if f ()xo = 0. Physically, equilibrium represents a situation without life. It may mean no motion of a pendulum, no reaction in a reactor, no nerve activity, no flutter in airfoil, no laser operation, or no circadian rhythms of biological clocks. This equilibrium point or critical point or singular point of the vector field is called a sink, source or a saddle (Figure 4.1)
depending on the sign and values of eigenvalues of the matrix Dfx(xo), which is the Jacobian
∂f of the system, and is defined as Dfxo()x = . The Jacobian is also the coefficient of the ∂x x=xo first order approximation of the full nonlinear system. Using Taylor’s theorem, the first order approximation of the full nonlinear system can be written as:
fD()xx= xof( )(x-xo)+ (4.2)
The state space of this system can be divided into stable, unstable and center subspaces
(Figure 4.2) that define the existence of the stable, unstable and center manifolds of appropriate dimensions as given by the Stable Manifold Theorem which is a very important result for
dynamical systems theory. It states that if the matrix Dfx (xo )has k eigenvalues with negative real parts and n-k eigenvalues with positive real parts, then there exists a k-dimensional differentiable manifold S tangent to the stable subspace E s of the first order linear approximation
of the nonlinear system at xo such that t≥0,φt (S) ⊂S and for all xo ∈ S , limφt (x0 ) = 0 and t→∞
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Asymptotically Stable Limit Cycle Unstable
x x x x x
t t t x x t x x
x x
x x x
Im Im Im X X X Re Re Re X X X
Figure 4.1: Time response, phase plane and eigenplane.
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u there exists an n-k dimensional differentiable manifold U tangent to the unstable E at xo such
that tU≤0,φt ( ) ⊂U and for all xo ∈U , lim φt (x0 ) = 0 . t→−∞
Wc Ec Eu
Wu
Ws
Es
Figure 4.2: Invariant Subspaces and Manifolds: Stable (s), Unstable (u) and Center (c).
The Center Manifold Theorem (Perko, 1996) states that if there are m eigenvalues with the zero real parts then there exists an m-dimensional center manifold, which is tangent to the center
c subspace E which is invariant under the flow φt . The Hartman-Grobman theorem (Guckenheimer and Holmes, 1983) is another very important result in the qualitative theory of ordinary differential equations. A fixed point is hyperbolic if none of the eigenvalues of the system Jacobian matrix have zero real part. Similarly, a non-hyperbolic fixed point has at least one eigenvalue with the zero real part. The theorem shows that near a hyperbolic fixed point, x0, the nonlinear system x = f (x) has the same qualitative structure as the linear system xx = A with AD= x f(x0 ). This allows us to use advanced techniques of stability analysis like continuation-type bifurcation methods. The Hartman-Grobman theorem shows that in the neighborhood of the hyperbolic critical point, the qualitative behavior of the nonlinear system is determined by its linear part. The Local Center Manifold theorem (Guckenheimer and Holmes,
1983) extends this result for nonhyperbolic critical points. It states that the qualitative behavior of the nonlinear system around non-hyperbolic critical points can be approximated by the qualitative
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behavior of the nonlinear system on the center manifold. Since the dimension of the center manifold is less than n, this simplifies the problem of determining the qualitative behavior of the system near a nonhyperbolic critical point. This complexity in analysis is further reduced by the use of normal forms (Nayfeh, 1998) to approximate the nonlinear component of the system.
Peixoto’s theorem (Perko, 1996) implies that typically a two-dimensional vector field will contain only sinks, saddles, sources, and repelling and attracting closed orbits in its invariant sets.
4.2 Generic bifurcations
Systems of physical interest have parameters that appear in the defining equations. As these parameters are varied, changes may occur in the qualitative nature of the solutions for those parameter values (Guckenheimer and Holmes, 1996). These changes are called bifurcations and the parameter values are called bifurcation values. In parameter regions consisting of structurally unstable systems, the detailed changes in the topological equivalence class of a flow can be exceedingly complicated. Since the analysis of such bifurcations in second order systems is generally performed by studying the vector field near the degenerate (bifurcating) equilibrium point or closed orbit, and bifurcating solutions are also found in a neighborhood of that limit set, these bifurcations are referred to as local. Local bifurcations result due to changes in the qualitative dynamics close to the critical points.
The term bifurcation was originally used by Poincare to describe the “splitting” of equilibrium solutions in a family of differential equations. If xx = f (,tu,,p)is the system of differential equations describing the system depending on the m-dimensional parameter vector p, then as p varies, the implicit function theorem (Wiggins, 1990) implies that these equilibria are described by smooth functions of p away from those points at which the Jacobian derivative of
f (,x tu,,p)with respect to x has a zero eigenvalue.
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x Stable branch Unstable branch
p
Figure 4.3: Saddle node bifurcation plot representing the behavior of equilibrium point x0 with respect to a parametric variation p.
It is of particular interest that there are identifiable classes of bifurcations which occur repeatedly in nature. Ideally it would be nice to have a classifications of bifurcations which produced a specific list of possibilities for each model depending on the size of state space and parameter space and any symmetries or similar attributes it might have. There has been much work done in the area of these classification schemes, which are based on the transversality in differential topology. The transversality theorem (Seydel, 1997) implies that when two manifolds of dimensions k and l meet in an n-dimensional space, then in general, their intersection would be a manifold of dimension (k+l-n). It should be noted that non-transversal intersections can be perturbed to transversal ones, but transversal intersections retain their topology under perturbations. The codimension of an l-dimensional sub-manifold of n-space is (n-l).
Codimension of the bifurcation is the smallest dimension of a parameter space which contains the bifurcations in the persistent way. At the bifurcation value, some of these transversality conditions for structural stability are violated, and these determine the type of bifurcations that occur. The simplest bifurcations of equilibrium are those with codimension one (Guckenheimer and Holmes, 1983). These are generically called saddle node (Figure 4.3), trans-critical (Figure
4.4), pitchfork (Figure 4.5) and Hopf bifurcations (Figure 4.6).
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x
p
Stable branch Unstable branch
Figure 4.4: Trans critical bifurcation.
A generic saddle node bifurcation (Kuznetsov, 1995) is qualitatively like a family of equations x =−px2 in the direction of the zero eigenvector, with hyperbolic behavior in the complementary directions. The transversality conditions control the non-degeneracy of the behavior with respect to the parameter and the dominant effect of the quadratic nonlinear term.
There may be global changes in a phase portrait associated with a saddle node bifurcation. The bifurcation of one-parameter families at equilibrium with a zero eigenvalue can be perturbed to saddle-node bifurcations (Wiggins, 1990). Thus, one expects that the zero eigenvalue bifurcations encountered, in applications, will be saddle-nodes. If the exchange of stability occurs, then the transcritical bifurcation or pitchfork bifurcations happen depending on the transversality condition. The direction of bifurcation and the stability of the branches are determined by the sign of the transversality conditions, and the bifurcations are called subcritical and supercritical depending on the sign.
x
Stable branch Unstable branch
p
Figure 4.5: Pitchfork bifurcation (super critical).
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x 2
Periodic Orbits Equilibria µp
x µ=µ 1 p=p0 o
Figure 4.6: Super critical Hopf bifurcation, with bifurcation parameter p.
x x
Periodic Oscillations Periodic Oscillations
p p
Figure 4.7: Soft and hard generation of limit cycles.
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The generic transcritical bifurcation (Kuznetsov, 1995) is given as x =px −x2 and a generic pitchfork bifurcation is given as x = px − x3 . A stationary bifurcation does not give rise to such interesting phenomenon (Figure 4.6,4.7); the branching solutions show no more life than the previously known solutions. The full richness of the nonlinear world is not found at the equilibrium. The type of bifurcation, which connects the equilibrium point with the periodic motion, is called Hopf bifurcation (Figure 4.8). Near Hopf bifurcations, there is only one periodic solution for each value of the parameter vector.
The dimensions of the stable and unstable manifolds of parameter space do change if the eigenvalues of the Jacobian of f (,x tu,,p)cross the imaginary axis at the equilibrium solution.
This qualitative change in the local flow near the equilibrium point must be marked by some other local changes in the phase portraits not involving fixed points. The Hopf bifurcation theorem states that the contents, of the qualitative nature of the system, remain unchanged if the higher order term are added to the system near origin. The theorem (Hassard et. al., 1981) is given below:
n Suppose that the system xx =∈fp(, ),xℜ,p∈ℜ has an equilibrium point (,xopo)at which the following properties are satisfied: Dfxo(,x po), the Jacobian at the equilibrium point in state space and parameter space, has a simple pair of pure imaginary eigenvalues and no other eigenvalues with zero real parts. This implies that there is a smooth curve of equilibria
((x p),p)with x()po= xo. The eigenvalues λ()p ,λ(p) of Dfxo(,x po)which are imaginary d at p = p vary smoothly with p . If moreover, (Reλ( p)) | = d≠ 0 then there is a unique o dt pp= o n three-dimensional manifold passing through (,xopo) in ℜ ×ℜand a smooth system of coordinates for which the Taylor expansion of degree three on the center manifold is given by:
x =+(dp a(x22+y ))x −(ω +cp +b(x22+y ))y (4.3) yc =+(ω p+b(x22+y))x+(dp+a(x2+y2))y
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If a ≠ 0 then there is a surface of periodic solutions in the center manifold which has quadratic tangency with the eigenspace of λ()p ,λ(p) agreeing to second order with the paraboloid p =−(/ad)(x2+y2). If a < 0 , then these periodic solutions are stable limit cycles, while if a > 0 the periodic solutions are repelling.
x ≠ 0 Periodic
x = 0 Hopf x ≠ 0
Figure 4.8: Transition from stationary to periodic solutions, via Hopf bifurcation.
In practice, computations of the bifurcations of periodic orbits from a defining system of equations are substantially more difficult than those of equilibria because the equations must be integrated near the periodic orbit to find the Poincare return map before further analysis can be done.
4.3 Global bifurcations, jumps and non-local behavior
Global bifurcations are such that they result in a qualitative change in the orbit structure of an extended region of the phase space. The complete theory of global bifurcations is far from known, because techniques for the global analysis of the orbit structure of dynamical systems are under development (Wiggins, 1991).
There are instances when systems suddenly change to a different qualitative behavior and this does not include the local bifurcations. An example would be Lorenz attractor, which undergoes phase switching. It is not possible to study these jumps in the phase space by using
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continuation type methods. These non-local dynamics results in global bifurcations thus sometimes leading to failure of the system.
4.4 Stability of forced systems-feedback loop nonlinearities and nonlinear feedback Nonlinear systems can, sometimes, be decomposed as a linear part and a nonlinear part
(Figure 4.9). This decomposition can be restrictive in a sense that many nonlinear interactions cannot be decoupled as such. This would appear to be true in servo hydraulic systems since it is not thought possible to decouple the hydro-mechanical interaction in a meaningful way.
In general, nonlinearities in systems take many different forms, one possible feature is that they produce more than one frequency in the system response while being excited by a single frequency. This is also known as frequency distortion. External harmonic forces generate higher and lower harmonics in the total response of nonlinear systems. The internal feedback due to the nonlinearity is equivalent to an internal force. The forced response is at the excitation frequency and the free response is at the higher and/or lower harmonic of that frequency. This could be explained by the idea of feedback of the nonlinear force to the linear system. Natural frequencies of vibration change with the amplitude of the response. The ideas of feedback of nonlinear force have been used for system analysis and design for a very long time, and were used in the identification and experimental analysis by Adams and Allemang (1999).
SYSTEM
Input Linear Output Plant
Nonlinear Plant
Controller
Figure 4.9: Nonlinear system decomposition: Nonlinear feedback.
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The nonlinear systems like servo-hydraulic systems are most often used in a feedback control loop for better operational bandwidth, control and performance. This results in an additional feedback loop being closed around the nonlinear plant (Figure 4.9). This transforms the state space and parameter space, and does result in the qualitative change in the stability behavior of the system in closed loop; generally, it is used to increase the bandwidth and performance and hence increase the stability margin. As is often the case in feedback control systems, feedback could also be a cause of unstable dynamics in the closed loop; thus feedback gain should be treated as a parameter for bifurcation analysis. This treatment would help identify the rate of bifurcation in state space or equivalently, the closest distance to bifurcation in parameter space, which includes the feedback gain as a parameter hence providing an estimate of the rate with respect to the feedback control. It is just a matter of looking at the projections on the respective direction in the extended parameter space for studying the effect of various parameters on the bifurcation stability of the system. This idea does not require the decoupling of the linear and nonlinear part of the system model and hence is not restrictive as the idea of nonlinear feedback force analysis.
4.5 Nonlinear system analysis tools
The nonlinear system analysis has been a focus of theoreticians, mathematicians, physicists and engineers alike for a long time. Early development was due to the interest in celestial dynamics. Linearized analysis in time domain or frequency domain was the most convenient tool to study the dynamics of the system. Frequency domain analysis techniques are commonly used for the stability analysis. The servo-hydraulic systems analyzed in this dissertation are complex multiple degree-of-freedom systems that contain multiple nonlinearities.
Most of the literature deals with specialized cases rather than the design and analysis of general nonlinear systems because there is no general theory available for the analysis of nonlinear systems. The harmonic balance method (Kremer, 1998) is a frequency domain approach for
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calculating the steady state response of a system. Assuming that there is a certain periodic solution to the system, the time domain equations are converted into the frequency domain algebraic equations. This can be used to handle multiple degrees of freedom with multiple nonlinearities. The describing function technique is a special form of harmonic balance method and is a graphical first order technique to handle a single static nonlinearity. Various generalization of describing function theory has been developed by Swern (1981), Tamura (1981) and Mees (1972), Jones and Billings (1991) including many others. General harmonic balance methods are required to analyze general nonlinear systems with multiple nonlinearities, multi- variable systems and multiple harmonic responses. All harmonic balance methods are special cases of Galerkin’s method. Nonlinearities are converted into polynomial approximations or describing function approximations and then studied in the time domain.
Absolute stability theory guarantees the global asymptotic stability of feedback control systems with dynamic linear time-invariant system in the forward path and a memoryless nonlinearity in the feedback path. Common methods are Lyapunov’s direct method (Showronski,
1990), Popov’s criterion and the circle criterion. There are differences between the analysis methods depending upon the allowable class of nonlinearities.
Continuation methods are used to generate the branch tracing and path following.
Haselgrove (1960) and Klopfenstein (1961) contributed basic work in the area of continuation. In
1960s, continuation method were introduced in the analysis of engineering and scientific problems of civil engineering, flow research, chemical reactions, solidification, and combustion.
An important class of continuation methods is homotopy for the computation of one solution.
Other predictor-corrector methods are available and are in use widely. Continuation methods differ in predictor, parameterization strategy, corrector and step-length control.
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Chapter 5-Nonlinear systems analysis - a bifurcation theory based approach
5 Nonlinear systems analysis - a bifurcation theory based approach
Nonlinear systems exhibit various dynamic behaviors, which includes limit cycles and other types of bifurcation behavior. It is important to catalogue the types of behavior a system like servo-hydraulic systems can undergo for better analysis and design. Further, to fully categorize the dynamics of a nonlinear system, the effect of parameters on the states and stability properties of the system should be studied. The effect of parametric variations along the directions on parameter space on the qualitative change in stability behavior is an important aspect of bifurcation analysis. This provides insight into the stability robustness of the system.
This chapter reviews the previous work in the area of nonlinear systems analysis and then provides insights into the multi-parameter multi-space bifurcation theory. This chapter also introduces the notion of constant velocity solutions and control induced bifurcations and explains the use of eigenspace, parameter space and state space in the analysis. This chapter imparts contributions to the areas of system decomposition, robust stability analysis and numerical bifurcation analysis.
5.1 Previous work
The area of nonlinear system analysis and design has been investigated for many years and the techniques have evolved from time to time depending upon the availability of resources and the need of the technical community. The list of previous work which is of greatest relevance would start with Venkatasubramanian et al. (1995), which highlights the idea of stability boundaries or bifurcation surfaces in the parameter space and methods of constructing them.
Further, Dobson (1993) developed a method for computation of the locally closest distance and direction to bifurcation surface using the idea of intersection of the hyper-surface with the direction vector in the parameter space. Kremer and Thompson (1998) demonstrated the
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effectiveness of bifurcation theory in the design of hydraulic control systems using this technique.
The use of the graphical Hopf theorem for limit cycle analysis was demonstrated by various investigators including Mees and Chua (1979), Mees (1981), and Moiola and Chen (1993, 1996).
The application of the QFT (Quantitative Feedback Theory) technique to a fluid-elastic system for suppression of nonlinear dynamics was considered by Yu, Bajaj, and Nwokah (1993). Berns,
Moiola and Chen (1998) developed a feedback control of limit cycle amplitudes using higher order harmonic balance methods using degenerate Hopf bifurcation theory. Feng and Liew
(2000) studied modal interactions in parametrically excited systems with zero-one internal resonance using the Melnikov functions.
Scheidl and Manhartsgruber (1998) analyze the dynamic behavior of partially singularly perturbed servo-hydraulic system of 10th order is investigated using the center manifold theorem.
This work uses the center manifold theory in Fenichel’s format is used to study the behavior of the drive system in the case of periodic motions. The phase space is decomposed into sub-spaces with sufficient differentiability properties. Approximate analytical solutions are given and transition layers are estimated by asymptotic expansion. The stability of these transition layers is also investigated and the theoretical results compared with the numerical solutions. The local analysis of continuous bifurcations in the n-dimensional piecewise smooth dynamical systems are explored in Bernardo et al. (1999). The classical development in the area of border collision bifurcations for piecewise smooth systems was done by Feigin (1994).
The models of hybrid systems, which undergo state transitions, and their bifurcation stability behavior, have been studied by Chen and Aihara (1998). Michel and Hu (1999) proposed the stability theory of a general hybrid dynamical system, which used the idea of
Lyapunov stability and Lagrange stability theorems. Chellaboina et al. (2000) developed an invariance principle for nonlinear hybrid and impulsive dynamical systems, which have left - continuous flows. This idea is of importance since the bifurcation theory analysis across the discontinuities or singularities is still not well developed. Zhai et al. (2000) studied the stability analysis of hybrid linear systems, which are composed of stable and unstable subsystems using
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average dwell time approach. Schinkel, Wang and Hunt (2000) proposed a design of robust and stable controller for hybrid systems. Demir (2000) developed theory and efficient numerical methods for nonlinear perturbation and noise analysis of oscillators described by a system of differential-algebraic equations. Olivar et al. (2000) studied discontinuous converter dynamics models and generated geometrical and topological features related to smooth and non-smooth bifurcations.
In the analysis of Hopf bifurcations, the work of Hassard et al. (1981) gives a detailed analysis procedure for Hopf bifurcations. Banerjee et al. (2000) studied power electronic having piecewise smooth structure for bifurcation stability. Mohrenschildt (2000) used automata theory for the analysis of stability of hybrid systems and extended the notion of controllability and observability to hybrid systems. Guckenhiemer and Williams (2000) studied asymptotic analysis of subcritical Hopf-homoclinic bifurcation in the modified Hodgkin-Huxley neuron model, which exhibited existence of Hopf bifurcation close to the homoclinic bifurcation and an asymptotic theory for the scaling properties of the interspike intervals in singularly perturbed systems. He
(2000) developed a method coupling homotopy techniques and perturbation techniques to develop a systematic approach for defining the slowly varying perturbation parameter. Basso et al. (1997) derived frequency domain conditions for the existence of limit cycle for an unforced feedback interconnection of linear and nonlinear systems. This idea was again derived on the principles of harmonic balance and control theory.
Various investigators have studied specific cases of the nonlinearity using one or more techniques at hand. Natsiavas and Verros (1999) investigated the dynamics of a strongly nonlinear oscillator with asymmetric damping, which is equivalent to Coulomb friction. Salenger et al. (1999) studied the transition of a damped nonlinear oscillator using Pade approximations.
Goncalves et al. (2000) studied the global stability analysis of relay feedback systems using quadratic Lyapunov functions for the associated Poincare map. Kayihan and Doyle (2000) studied the effect of friction on the process control valve in a distributed control system and fault diagnosis and tracking techniques are presented. This work provides insights into the effect of
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friction on the control valve. Andreaus and Casini (2000) provided a nonlinear dynamical study of a single degree-of-freedom oscillator with a hysteretic motion limiting stop. This work included a comparison of response for soft and hard stops and demonstrated the existence of stable orbit and periodic doubling routes to chaos. Khinast and Luss (2000) developed qualitative characterization of the parameter space using continuation procedures for adiabatic reverse-flow reactor in which the loci of the singular points for forced periodic features is computed. The concept of nonlinearity measure (Helbig et al. 2000) and its computation (Harris et al. 2000) provide a quantification of open loop and closed loop nonlinearity of the system. The methodology is demonstrated on an operating point analysis of the continuously operated stirred tank reactor. A slightly different approach to measuring the control-law nonlinearity is to define an optimal degree of nonlinear compensation in the controller, a system property distinct from open loop nonlinearity (Stack and Doyle, 1997). The idea of interpolated cell mapping, a technique for parameter space investigation, to obtain initial condition map corresponding to multiple existing periodic motions, is also used to study the bifurcation in escape equation models
(Raghothama and Narayanan, 2000). The possibility of studying the nonlinear behavior of the physical systems by small feedback action require bounds on the possible energy changes of the system, for control of the forced response in case of nonlinear response (Fradkov, 1999). Orsi et al. (1998) developed sufficient conditions for a nonlinear dynamical system to posses an unbounded solution. The stability of systems with hysteresis nonlinearities, parametric uncertainty and finite dimension unmodeled dynamics is such that equilibrium is generally not unique (Jonsson, 1998).
Nayfeh (1999) and others have studied the dynamics of cubic nonlinear vibration absorbers and demonstrated the existence of nonlinear resonance and bifurcations. The patterns of bifurcations governing the escape of periodically forced oscillations from a potential well over a smooth potential barrier with varying system parameter can help decide the relationship between the optimal escape and nonlinear resonance (Stewart et al., 1995). Contraction analysis
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(Lohmiller and Slotine, 1998) is another tool which is used for the analysis of nonlinear systems.
This generalizes the idea of eigenvalue analysis to the nonlinear systems.
Analysis of the control of chaos has been another area of significant research. Aston and
Bird (1997) demonstrated a methodology to extend the basin of attraction related to the maximum parameter perturbation using the idea developed by Ott et al. (1990). Perturbation methods have been used extensively for the analysis of nonlinear systems; more recently, the work has been focused towards the analysis of the singularly perturbed nonlinear systems, including discrete systems (Bouyekhf and Moudni, 1997). Hayase et al. (2000) studied the micro-stick-slip vibrations in the servo-hydraulic systems, and suggested that by compensating for the servo-valve nonlinearity using feedback, some micro-stick-slip vibrations could be eliminated. The addition of dither frequency to eliminate stick and reduce the dead-band is well known. Lee and Kim
(1999) developed a hydraulic servo cylinder with mechanical feedback and also studied its stability.
Actuator saturation is a cause of nonlinear behavior in the servo-hydraulic systems.
Gokcek et al. (2000) addressed the issue of disturbance rejection in control systems with saturating actuators using the method of stochastic linearization. Herman and Franchek (2000) studied the design and experimental validation of the engine idle speed controller for a V-8 fuel- injected engine using actuator saturation. Existence of saddle-node bifurcations in a relief valve is well known (Maccari, 2000) and is often studied using the asymptotic perturbations. The existence of sub-harmonic and combination resonance is also demonstrated. Magnusson (2000) demonstrated a methodology to analyze bifurcation points by asymptotic expansion in the bifurcation equations, which are obtained using the Lyapunov-Schmidt decomposition. Feckan and Gruendler (2000) provided a functional analytic approach together with Lyapunov-Schmidt method to study bifurcations from homoclinic and periodic orbits for singular differential equations. The effect of magnitude saturation in the control of bifurcations is motivated by the problems in the area of rotating stall in the compression stages of gas turbines (Wang and
Murray, 1999). The design of feedback control laws to achieve desirable size of the region of
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attraction when the noise is modeled as a closed set of initial conditions in the phase space is proposed by Murray (1999). The dynamics of the actuator is sufficiently fast and the output feedback controller tends to stabilize the origin of the actual system, and the actuator dynamics for output feedback stability purpose is not required to be faster than the system dynamics
(Aldhaheri and Khalil, 1996; Khalil, 2000). The effect of actuator saturation on the performance of PD-controlled servo systems is to result in the drop in performance at the maximum gain
(Goldfrab and Sirithanapipat, 1999).
Numerical continuation is a very important methodology for analysis of nonlinear systems for bifurcation stability and significant work certainly includes that of Allgower and
Georg (1990) and Seydel (1994). The application of numerical continuation is found in various areas of engineering including the aeroelastic stability problems (Gee, 2000). Continuous wavelet transforms are also used to perform Hopf bifurcation analysis on the aeroelastic problems
(Mastrodddi and Bettoli, 1999). Analyses of systems for aeroelastic stability includes multiple nonlinearities including a free play nonlinearity (Kim and Lee, 1996) and cubic restoring force
(Lee and Jiang, 1999). A detailed study of nonlinear aeroelastic analysis of airfoils for bifurcations and chaos was done by Lee et al. (1999) and provides an extensive detail on the nonlinearities, structural and aerodynamic, found in aeroelastic systems, their solution techniques and bifurcation associated with them including the effect of gust in longitudinal atmospheric turbulence.
5.2 Multi-parameter multi-space bifurcation theory
The results on co-dimension one bifurcations for dynamical systems can be extended to n-dimensional systems as a result of the center manifold theorem (Wiggins, 1990). The dependence of the system on the parameters is of interest and the direction of the bifurcation study in the parameter space is of importance as it guides the existence of various types of bifurcations in the system. It is important to note that higher-order dynamics do not affect the qualitative behavior of the phase space in a local sense, but this cannot be guaranteed globally.
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For a nonlinear system dependent on a parameter xx = fp(, ),x∈ℜnm,p∈ℜ , this could be extended into a system with following state equations xx = fp(, ),p = 0 and then the center manifold theorem can be applied. The center manifold is tangent to the parameter plane as well as the eigenspace (Kuznetsov, 1995). All essential events near the bifurcation parameter value occur on the invariant manifold and are captured by the projection of the system on the center manifold. Essential dynamics include any and all qualitative changes associated around the equilibrium point. The saddle-node and Hopf bifurcations occur for real eigenvalues and complex conjugate eigenvalues crossing the real axis, respectively. The center manifold is not unique in either the fold or Hopf cases, but the bifurcating equilibrium or the cycle belong to any of the center manifolds. In the fold bifurcation case, the manifold is unique near the saddle and coincides with its unstable manifold if it exists. The uniqueness is lost at the stable node.
Similarly, in the Hopf bifurcation case (Hassard et al. 1981), the manifold is unique and coincides with the unstable manifold of the saddle-focus until the stable limit cycle, where the uniqueness breaks down. Periodic solutions or limit cycles also go through the fold, flip and Naimark-Sacker bifurcation depending upon the multiplier value (Berns et al. 1998). If two limit cycles (stable and unstable), denoted by two points on the Poincare map, collide and disappear on an attracting invariant manifold, this results in a fold bifurcation of cycles. Further, in a period doubling bifurcation, a cycle of period two appears for the map, while the fixed point changes stability.
One-parameter systems undergo saddle-node and saddle-saddle bifurcations on the plane with one or more homoclinic orbits. Several other exotic bifurcations like, nontransversal homoclinic orbit to a hyperbolic cycle or homoclinic orbit to a nonhyperbolic limit cycle or blue-sky catastrophe occur in multi-parameter systems, and is beyond the scope of the present work
(Wiggins, 1990). In engineering, our interest primarily lies in the study of loss of stability of a system via the co-dimension one bifurcations in the applications presented here.
In a multi-parameter system, there is a bifurcation curve/plane in the parameter space along which the system has an equilibrium point exhibiting the same bifurcation (Seydel, 1994).
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A smooth scalar function ψ =ψ (,x p)can be constructed in terms of Jacobian matrix ∂f ∂x .
This results in a system:
fp(,x )= 0 (5.1) ψ (,x p)= 0
n+m which, generically defines a curve Γ passing through the equilibrium point (,xopo) in ℜ . Γ defines the equilibria satisfying the defining bifurcation condition and the standard projection of Γ onto the parameter space p-plane results in the corresponding bifurcation boundary. This function ψ =ψ (,x p) for fold bifurcations is, defining the curve of equilibria having at least one zero eigenvalue, given below:
∂f (,x p) ψψ==(,x p) det( ) (5.2) ∂x and for the Hopf bifurcation is:
∂fp(,x ) ψψ==(,x p) det(2 I) (5.3) ∂x
where denotes the bi-alternate product (Wiggins, 1990) of two matrices. If more than one parameter is varied simultaneously to track a bifurcation curve Γ , then the following events might occur to the monitored non-hyperbolic equilibrium at some parameter values:
• Extra eigenvalues can approach the imaginary axis and thus change the dimension of the
center manifold.
• Some of the nondegeneracy conditions for the co-dimension one bifurcation can be
violated.
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This essentially means that, in higher dimension parameter space any combination of lower-order bifurcation can be seen like, Hopf-Hopf, Bogdanov-Takens (double zero) or fold-
Hopf bifurcations (Seydel, 1994).
Singularity theory is concerned with the local properties of smooth functions near a zero of the function. It provides a classification of various cases of bifurcations, based on codimension, since codimension k-manifolds in a smooth function space can be algebraically described by imposing conditions on derivatives of the function. However, problems arise in the study of degenerate local bifurcations. Fundamental work of Takens (1974), Langford (1979) and
Guckenheimer (1981) has shown that that arbitrarily near these degenerate bifurcation points, complicated global dynamics such as invariant tori and Smale horseshoe may arise. These phenomena cannot be described and/or detected using the singularity theory technique.
5.2.1 Parameter space, eigenspace and state space
The dynamics of the nonlinear problem that can be represented in the form xx = f (,pu,,t) has three spaces: state space, eigenspace, and parameter space (Figure 5.1), which allow us to study the system. The state space consists of all the states required to represent the system and is of n-dimensions. The parameter vector includes all system constants that are used to define the system dynamics and may vary quasi-statically. Parameters define the system properties to help model the system and do not include system states. The dimensionality of the parameter space is equal to the number of parameters in the system. It is important to note that the dimensionality of both of these systems is a result of the modeling and hence are up to designer to decide. In the case of closed loop systems with feedback control parameters to be determined, the feedback gain and other controller parameters can be included in the parameter vector to generate
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x x 2 Domain of Attraction
x (p )=0 x1 o o t
Parameter Space Eigen Space p2 Im
x x x X Stability boundary
Re local to xo(po) X x x x
X x x x po p1 λ(po)
Figure 5.1: Parameter space, state space and eigen space.
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an extended parameter space. This extended parameter space helps in the study of control of bifurcations and control-induced bifurcations.
5.2.2 Constant velocity solutions
Considering a general forced system with known input u*(t), xx = f (,pu,,t) the stability of solution x(t) in the neighborhood of the known solution x∗ (t) can be reduced to consideration of a related unforced system. Let xx()tt=∗()+ξ(t), then ξ (t) represents a perturbation of the original solution. It is sufficient to investigate the stability behavior of the perturbed solution to discuss the stability of the original solution in a local sense.
• ∗ i xx+=ξf (∗∗,pu,,t)+{f(x+ξ,pu,,t)−f(x∗,pu,,t)} (5.4)
This could be reduced to i ξξ=+{f (xx∗ ,pu,,t)−f(* ,pu,,t)}=h(ξ,pu,,t) (5.5) which could be approximated to i ξ =hp(,ξ,u,t)≈A(t)ξ (5.6) when higher order terms are negligible. This perturbation is effective in capturing the dynamic changes, due to parameter variation, as the Jacobian approximation is calculated at each point in the parameter space. If the solution x∗ is on a trajectory or flow where the piston or the valve spool is moving at a constant velocity for a given servo-hydraulic system, that is defined as the constant velocity solutions. The advantage of analysis at a point on the trajectory other than zero velocity solution is in the excitation of the significant nonlinear response of the system (i.e, zero input corresponds to the zero flow/zero velocity condition). Further, the zero flow case for hydraulic systems is not of interest. For successful identification, detection and design for any nonlinear effects, it is essential for the model to show full nonlinear behavior. The above methodology provides for analyzing stability in terms of bifurcation of an equilibrium point, since constant velocity does not qualify as an equilibrium condition in the original state space
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coordinates, as described in Chapter 2. Then the nonlinear effects could be properly accounted for and design could include the robustness towards the nonlinear effects of parametric variations.
The analysis along the direction of changing velocity gives an insight into the effect of nonlinearity on the bifurcation stability of the system. This coupled with the feedback gain, can provide significant insight into the area of closed loop stability boundary in the parameter space.
The mapping of stability boundary in the parameter space assists in the robustness analysis, as a given distance to instability in parameter space is an equivalent measure of the fastest loss of stability mode in eigenvalue space.
It should be noted that although the loss of stability of a periodic orbit is of theoretical interest, the design engineer is predominantly interested in the mechanisms by which a stable system loses stability via a co-dimension one bifurcation, like saddle-node or Hopf bifurcation.
First emergence of bifurcation phenomenon along a direction in parameter space is a sign of failure of the nonlinear system under study, and hence needs redesign.
5.2.3 Robust bifurcation stability analysis
The robust stability analysis involves the study of parametric variation due to uncertainty or due to wear and use. The effect of parametric variation could be severe in case of large coupled nonlinear systems as it can initiate a bifurcation; subsequent bifurcations even lead to a chaotic response and dramatic failure of the system. Robust stability analysis tools in linear systems are well understood and quite well developed. However, the understanding of nonlinearity and their effects on the robustness is still a matter of prime investigation, as nonlinear response is hard to predict unless it is already characterized and included into the design (Bhattacharya, et.al. 1995).
It is important to realize that nonlinear response could be dramatically different at various points in the parameter space and state space, and to design for all points in parameter space is nearly impossible. However, in a closed loop operation, the design could be modified to provide desired response locally, and could even be kept safely far-off from the stability boundary, if the bifurcation surface in the extended parameter space is known. Thus the robustness could be
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maintained and analyzed along the various directions in parameter space. The robustness analysis problem can be formulated as a study of effect of perturbation around a nominal point with unknown direction of perturbation. Thus, the importance of bifurcation theory to study stability robustness in a highly nonlinear system like servo-hydraulic systems is apparent.
The nonlinear system can be written as follows using the Taylor series approximation
xx ==fp(, ,u,t) Dx f(xoo)(x-x)+ (5.8)
Then in the closed loop sense, as shown in Figure 5.2, with a linear output feedback and proportional gain k, the nonlinear state space decomposition becomes, where xo is assumed to be an equilibrium point:
xx = Dfx()o (x-xo)+ (5.9)
xx = AB+u; x =x-xo (5.10)
0 AD=x f(,x u) + (5.11a) x oo,u
∂f 0 Bu=()−u+ (5.11b) ∂u 0 x 00,u
yC= x (5.11c) uk= y
u B x ∫ x C y +
A
k
Figure 5.2: Block diagram of the state space system with proportional output feedback law.
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The equation (5.11c) is the output equation, in conjugation with the proportional feedback law. This has appropriate assumptions about the feedback law and the input and output observability. In a sufficiently small neighborhood of the operating point, the stability of the nonlinear system can be understood by looking at the corresponding linear approximations, provided it is calculated for each parametric perturbation. Then the closed-loop stability can be investigated locally by the analysis of stability of the matrix M (Bhattacharya, et.al., 1995 and
Lungo and Paolone, 1997): M AB+ kC (5.12)
The parametric perturbation due to uncertainty or any other reason is p =+ppo ∆, where po is the nominal parametric vector and ∆p is the perturbation. Then:
M ()pM=+(poo∆p)=M(p)+∆M(po,∆p) (5.13)
The parameter space stability robustness radius in a linear sense is a measure of the size of resulting perturbation on the effective system under the given parametric perturbation. Thus the estimate of ∆M (,po ∆p) provides an important measure on the robustness of the system via the eigenvalue analysis. This is equivalent to the gain margin in a one-parameter space of feedback gain. This is directly related to the distance from the bifurcation surface and not just the stability boundary on the eigenspace. It is important to realize that in a local sense, any additive or multiplicative parametric perturbation results in a nonlinear closed loop transfer function, which has the same number of unstable eigenvalues as the unperturbed system, meaning that stability is dependent on the size of the perturbation given that the unperturbed system is stable.
The perturbed system is stable given the fact that the perturbation is small enough so as not to cross the imaginary axis on the nonlinear eigenlocus plots, developed using the bifurcation theory methods.
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5.3 Decomposition of the system
Any real-life mechanical system is “large” in scale, which means it can only be modeled using both a huge state vector and a long list of parameters, or as a group of small, coupled subsystems. The idea of using low-order interconnected models to represent complex dynamics is very important since it gives the designer more control in understanding the “troublesome” dynamics of the large system and thus helping in parametric reduction for efficient parameter space investigation of stability robustness properties.
The dimensionality of the parameter vector used for bifurcation analysis can complicate the possibility of finding all the global modes of instability (Figure 5.3). Thus, the smaller the parameter vector is, the easier it is to detect and characterize the dynamic behavior associated with the system. This decomposition of the large-scale system is based on two assumptions: firstly, the low-order models can represent all the relevant dynamics and secondly, certain parameters would contribute to certain nonlinear modes of the system more than the other parameters. This parametric contribution can be assessed using modal decomposition in the nonlinear system. Modal decomposition can numerically be done via singular value decomposition of the Jacobian matrix to study the participation of each parameter to every modes exhibited by the system. The idea of stability robustness gives rise to the issue of fastest unstable mode; there could be more than one fast unstable mode, which then results in higher- order bifurcations.
Bifurcation Parameter Space Robust Stability in
Extended Parameter Parameter
Space Parameters Space Associated with Fast Dynamics
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Figure 5.3: Decomposition of system for robust stability analysis.
It is also important to decouple the fast modes from the slow modes for ease of analysis, as then perturbation or asymptotic analysis could be easily applied to the system. The fast dynamics of the system are generally responsible for the associated loss of qualitative stability; on the other hand, slow modes tend to provide sluggish nature to the system dynamics and hence create numerical problems (i.e., in the use of continuation-based procedure for the analysis of nonlinear dynamics).
5.3.1 Fastest unstable mode with associated parameters
The rate of change of stability behavior is more important in the parametric variation studies than the stability at a particular operating point. The easiest measure of the rate of stability bifurcation is the distance to the bifurcation surface (Dobson, 1996) in the associated parameter space. This distance to bifurcation surface is computed on the relevant parameter space as a norm of the distance between the operating point and the closest point on the bifurcation surface. The possibility of computing the distance to instability is complicated by the fact that the bifurcation surface could be composed of many bifurcating modes/hyper surfaces.
The nonlinear system could be approximated as follows:
xx = Dfx()o (x-xo)+ (5.14a)
xx = AB+u; x =x-xo (5.14b) 0 AD=x f(,x u) + (5.14c) x oo,u ∂f 0 Bu=()−u+ (5.14d) ∂u 0 x 00,u
Or, for a very small parametric variation, hence for computing the parametric sensitivity of the eigenvalues, this nonlinear system can be approximated as x = Ax, with system matrix A (n × n matrix) being approximated by the system Jacobian. If λi and Ui are the eigenvalues and the corresponding eigenvectors, where i=1,2,3,4...,n, it is well known that:
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AUii= λUi;i =1,2,3,..,n T (5.15) AVjj==λ Vj; j 1,2,..,n
TT Also, VUii=1;VjUi=1; i≠j as the left and right eigenvectors are orthogonal. Differentiating with respect to parameter p, it is easy to show that the first order eigenvalue parametric sensitivity is: ∂λ ∂A = VT U (5.16) ∂∂p ip i
This is used to estimate the local direction of fastest eigenvalue change and hence is useful in parametric decomposition of the model for bifurcation stability analysis. Similarly, second order sensitivity equations can be derived. But since the eigenvalue calculations are valid in a local sense, the second order eigenvalue sensitivity calculations are prone to errors and are not recommended.
5.3.2 Transformations for parametric model decomposition
The full nonlinear system can exhibit dynamics dominated by certain modes and related parameters. Thus the identification of the fastest unstable mode for a direction in extended parameter space, and mapping of the corresponding parameters affecting the bifurcation stability of those modes, is an important step in efficient analysis of the nonlinear system. The system stability in open or closed loop is given by the eigenvalues of the Jacobian matrix.
xx = fp(, ,t,u)
Jfx = ∂(,xp,t,u)/∂x (5.17)
Jfp = ∂∂(,x p,t,u)/p
Using singular value decomposition ' Jfx=∂ (,xxp,t,u)/∂ =Ux*S*Vx (5.18) ' Jfpp=∂ (,x p,t,u)/∂p=U*S*Vp where Ui and Vi are corresponding left and right eigenvectors, which provide information about the contribution of each mode to the dynamics of the system at that operating point. This gives insight into the modal composition of the system for a given set of parameters, initial conditions,
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boundary conditions and inputs. For analyzing the effect of parameter variation on the system dynamics, specifically modal contributions of the parameter, so that model decomposition can be performed, let the extended parameter list be perturbed on a suitable scale.
p = po +∆p (5.19a)
Then the Jacobians ' Jfx=∂ (,xxp,t,u)/∂ =Ux*S*Vx (5.19b) ' Jfpp=∂ (,x p,t,u)/∂p=U*S*Vp
are estimated again for the new (perturbed) parameter values. By comparing the value of Ui and
Vi for the original and perturbed solutions, the effect of each parameter on a particular mode is understood. This is a local estimate for decomposition of the model so as to reduce the dimensionality of the parameter space for the bifurcation analysis. This decomposition should be performed at each perturbation of the parameter vector to effectively capture all the significant dynamics of the system. Similar reduction of system dynamics has been investigated by Lall and
Marsden (1999).
5.4 Control induced bifurcations
In the parameter space, the computation of closest distance and direction to the bifurcation provides insight into the stability robustness of the operating point (Dobson, 1996). In the closed loop, this robustness measure provides a handle on the effect of the feedback gain and other control loop parameters on the stability of the bifurcation (Figure 5.4). The projection of this direction along which the shortest distance to bifurcation lies along the individual parameter axes provides comparative measure of the components of rate of bifurcation. The system stability is more robust with respect to perturbations of certain parameters of the system if the projection of the distance to instability is larger for that parameter in comparison to the others. This idea is one of the important contributions of this work.
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u x( = f x,pu,,t) y
yC= x
Controller
Figure 5.4: Block diagram of the plant and controller depicting extended parameter vector composed of plant parameters and controller parameters.
Feedback Gain
d=d12+d +d3 po d Parameter 1 d2 d3
d1 Bifurcation Surface Parameter 2
Figure 5.5: Control induced bifurcation and distance to bifurcation.
The distance to the stability boundary in the parameter space is a measure of the robust bifurcation stability. If a parameter is varied along a direction in parameter space (Figure 5.5) and results in the loss of stability via a bifurcation, then stability behavior is sensitive to that parameter. If the emergence of bifurcation is due to variation in controller parameter, then the resulting bifurcation is a control induced bifurcation. Robustness is one of the most important issues in control design. There are perturbations, which do not change the qualitative nature of local bifurcation behavior of the closed loop system, but only change the amplitude of the bifurcation equilibrium. In practice, perturbation might not be mathematically generic so uncertainties must be quantified to analyze the bifurcation of closed loop system under those
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perturbations. If the perturbations are small enough, then the perturbed bifurcation equilibrium and the nominal bifurcation equilibrium are close, although the qualitative bifurcation behavior may be altered by the perturbation.
5.5 Control of bifurcation instabilities
The geometric structure of stabilizability of finite dimensional linear time invariant systems has been well known (Alvarez, 1996). Many techniques, such as pole placement and solving linear algebraic Lyapunov functions, can be used to design a stabilizing feedback. For a finite dimensional nonlinear systems, the question of feedback stabilization becomes much more difficult and subtle. The two basic ideas in the literature are derived from Lyapunov stability theory.
The first idea is based on Lyapunov’s second method that uses control Lyapunov functions. The advantage of this technique is that global stabilizability can be tackled. The drawback is that the method is too general and solving for control Lyapunov functions can be difficult for many systems. Artstein’s theorem states that the necessary and sufficient condition for smooth feedback stabilization is the existence of a smooth control Lyapunov function. For nonlinear systems, stabilizability via continuous feedback is not guaranteed even if for each point in a neighborhood of the equilibrium there exists a smooth control law steering the system to equilibrium (Brockett, 1983).
The second idea is based on the Lyapunov direct method and Poincare normal form theory. The basic observation is that if the linearized system is stabilizable, then the full nonlinear system is also stabilizable by the same linear feedback. If the linearization is unstabilizable and there is an uncontrollable eigenvalue with positive real part, then the full nonlinear system is not stabilizable via a continuous feedback (Tesi et.al., 1995). So the only challenging problem in local feedback stabilization is the critical case: all the linearly unstabilizable eigenvalues are on the imaginary axis. In this case, the system can be reduced to the local center manifold that is tangent to the eigenspace associated with the unstabilizable critical eigenvalues. Normal form
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transformation is then used to analyze the stability of the reduced system. The advantage of this method is that local stabilizability for a fairly large class of systems can be analyzed. The drawback is that it is a local technique so it is not applicable to issues of global stabilizability
(Seron, et.al., 1999) .
The local feedback stabilization is closely related to control of bifurcations in that if the system is locally stabilizable at the bifurcation point, then the bifurcation for the closed loop system is supercritical. Although the qualitative local bifurcation behavior for the closed loop system is robust to parametric uncertainties, the location for the bifurcation, the center manifold, the nominal equilibrium and the periodic orbits are all perturbed away. If the system is unstabilizable, then the Hopf bifurcation for the closed loop system is also robust, i.e., the bifurcation remains sub-critical under perturbations in the control gains and parametric variations.
5.6 Measure of bifurcation nonlinearity
The deviation of the system from its linear approximation close to an operating point has
been a focus of study. This would include works of several researchers including Nayfeh and
Balachandran (2000), Khalil (1994), Guckenhiemer and Holmes (1986) and Helbig, Marquardt
and Allgower (2000). The primary disadvantage of linearized analysis is its lack of capturing the
full nonlinear behavior even locally. The nonlinear effects tend to dominate the system dynamics
as the system starts to operate close to the edge of the stability boundary. These nonlinearites can
be a manifestation of the inherent system dynamics as well as the control law. Considering the
open loop system only can be misleading to estimate the effect of nonlinear effects of feedback
control law. It is important to include the effect of all parameters in the extended parameter
space to fully capture the effects of nonlinearity. It is important to quantify the difference
between a linear system model and a nonlinear system model so as to capture the effect of
nonlinearity on the stability boundary. To incorporate this idea of a quantitative estimate of the
nonlinear behavior of the system, a notion of the measure of bifurcation nonlinearity is
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proposed. The measure of bifurcation nonlinearity, N of a nonlinear system (Equation 5.8),
which depends on parameters p and states x is defined as follows:
N =−Dp(x ( ), p) D(x ( p), p) (5.20) xoxoo2
This is a two-norm of the difference in Jacobians (DX) evaluated along a direction in the parameter space. This gives an estimate of the nonlinear nature of the full system as any parametric studies are conducted to estimate the stability boundary. This measure (Equation
5.20) is proposed to capture the essence of difference between the dynamics of a nonlinear system as compared to the dynamics of its linearized approximation about an equilibrium point. This measure effectively captures the effect of local parametric variations on the equilibrium point and the overall system dynamics.
5.7 Numerical aspects of bifurcation analysis
The numerical bifurcation analysis primarily involves location of equilibrium, their continuation with respect to parameters, and for the detection, analysis, and continuation of limit cycles and their associated bifurcations (Fujii and Ramm, 1997). Numerical issues related to path following are many and of concern to investigate the stability behavior of the system. One frequent problem is undesired jumping between the bifurcation branches. This jump is an indication of the closed paths on the state space and is in a sense a “failure”.
x x
p p
Figure 5.6: Parametric step-size: large steps (left) may skip some details (right).
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Another question is, if the two or more branches in the bifurcation curve intersect. In such situation, the issue can be clarified by the increasing the resolution near the bifurcation point on each bifurcation branch, in the eigen-locus. This resolution or step-size is important (Figure 5.6) since an overly big step-size could result in overlooking the bifurcation, which might originate and terminate in a very small region of the parameter space.
The calculation of equilibrium point requires a solution of a nonlinear algebraic set of equations of the form fp(,x ,t,u)= 0. For stable equilibrium this solution can be obtained by integrating or by solving the above set of equations. In general for an equilibrium that is neither stable nor repelling, the location problem can only be solved provided the position of equilibrium is known approximately, because one typically starts with an equilibrium found analytically or by integration at fixed parameter values and then “continues” it by small stepwise variations of parameters. The standard procedure that generates the sequence of points that converge to equilibrium in very general conditions is Newton’s method (Seydel, 1997). This in its most elementary form has a recurrence relation given by:
x=j+1 xj+η j (5.21) and the displacement is given by the solution of the following algebraic equation:
J()x jjη =−f(x j) (5.22)
The continuation of equilibrium is the process of estimating the curve in the space of (x, p). Computation of this equilibrium curve defines the dependence of the equilibrium on the parameter. Most of the continuation algorithms for bifurcation analysis implement predictor- corrector methods and include three basic steps performed (Allgower, 1990) repeatedly:
• Prediction of the next point
• Correction
• Step-size control
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5.7.1 Data noise and nonlinear effects
In the experimental or numerical analysis of the nonlinear systems the data noise can be a significant hindrance, it could distort the qualitative behavior significantly by introducing frequency distortions (Skaistis, 1988). Some nonlinear effects have a similar feel so it provides additional difficulty in the characterization and analysis, since all real systems are limited in some sense and at certain accuracy level it is impossible to distinguish between the data noise and the true system response. The broadband excitation response looks very similar to chaotic response.
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Chapter 6- Preliminary Results: numerical investigation
6 Results: numerical investigation
This chapter provides a summary of the results obtained via bifurcation stability analysis of the two test systems for various parametric variations. This provides initial proof of concept as well as gives better insight into the stability behavior of the two test systems.
6.1 Characterization of nonlinear dynamics – computational results
The simulation and bifurcation study is done on the two test servo-hydraulic systems
(servo-pump actuator system (Figure 2.9) and servo-valve actuator system (Figure 2.11)). These hydraulic systems are defined in the previous chapters. This preliminary study reveals the inherent dynamics of the system in open and closed loop configurations. This section discusses a detailed insight into the models as obtained by the preliminary stability analysis and hence identifies the various influential parameter sets for extended investigations. The study of these preliminary models is essential to facilitate the identification of key parameters as well as appropriate sizing of the various components for experimental design. This includes studies in the control-induced bifurcations and the analysis at constant velocity solutions.
6.2 Results for servo-pump actuator system
The servo-pump actuator is characterized by significant nonlinear behavior due to two sharp-edged orifices, which restrict the flow to the actuator. The other parameters of significant interest are load mass, load stiffness, stroke length of piston, operating or line pressure, piston dimensions, pump flow, and feedback proportional gains.
This system is modeled in perturbation coordinates for bifurcation analysis to study the effect of change of parameters. The effect of parametric variation on the stability of the system is analyzed by studying the eigenvalues of the Jacobian matrix. Also, the effect of zeros on the
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dynamics of the system should not be overlooked in this approach. Zeros of the system affect the controller design and closed loop performance.
6.2.1 Simulation studies
The open loop eigenvalue plot of the nominal system is shown in Figure 6.1. This shows that the nominal system is stable, as expected, for low enough velocity and pump flow. Also, the nominal system has two eigenvalues at origin in open loop configuration. Simulation response of open loop model is type 1, which is physically realistic. A type 1 system is defined to be one which has a one free integrator, and correspondingly one zero eigenvalue. A type 1 system (open loop) has a ramp response to a step input. This system has two eigenvalues at the origin and one of them is cancelled by a system zero. The system dynamics is fourth order while the system is modeled as sixth order. Eigenvalues of the system are :
-320.32, -330.60, 0.0000, -0.0000, -21.39 + 87.32i, -21.39 – 87.32i
Zero of the nonlinear system are:
-8.4959e+012, -3.4134e+002, -3.2276e+002, -4.5749e+001, 0.00
The rank of the observability matrix is 5 and that of the controllability matrix is 4, while the rank of the system matrix, A, or Jacobian is 4. A system is said to be minimal (Brogan, 1990) linearly if it is both controllable and observable. A system is controllable if the controllability matrix is full rank and is observable if the observability matrix is full rank too. The zeros of the system can be computed via the full nonlinear model by assigning pump flow to be the input and position to be the output with proportional feedback. A reduced-order representation of the nonlinear system model is not thought to be possible globally. Locally the linearized model is uncontrollable and unobservable. The nonminimal nature of the system model results in a singular Jacobian and thus leads to numerical problems in the computation of the equilibrium point. A methodology is developed for state space reduction of the models near the steady state conditions, based on the physics of the system. This model also shows the two complex conjugate
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modes associated with the load mass. These modes are stable for this operating condition and nominal point. The frequency of the complex conjugate mode is of particular interest during experimental design of the test apparatus such that the system dynamics is within the control bandwidth of the servo-valve and can be effectively analyzed experimentally. Slow real modes are associated with the hydraulic fluid drain. This hydraulic fluid drain is estimated by analyzing a decoupled piston draining the pressurized fluid to exhaust through an orifice. Simulation response of the open loop nominal system is given in Figure 6.2. This response is stable as demonstrated by the eigenvalue analysis. The operating condition used for these studies is a constant velocity of 1 in./sec.
Figure 6.1: Eigenvalues of the nominal open loop servo-pump actuator model.
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1 6 250
0.8 4 200 n 0.6 y 2 t o i i t c o P1 si 0.4 l 0 ve Po 150 0.2 -2
0 -4 100 0 0.5 1 0 0.5 1 0 0.5 1 time time time 150 250 250
100 200 200 P2 Pa Pb 50 150 150
0 100 100 0 0.5 1 0 0.5 1 0 0.5 1 time time time
Figure 6.2: The open loop simulation response of the servo-pump actuator system at the nominal shows a type 1 response for a step input.
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6.2.2 Parametric bifurcation studies
This preliminary model is studied for parametric variations and their effect on the system stability. The closed loop response is stable for small gains as shown in Figure 6.3. At a high proportional feedback gain the system response starts to become unstable. This result is shown in the Figure 6.3. The simulation response demonstrates instability of the system (Figure 6.4).
80
60
40
20
y
nar 0
agi m I -20
-40
-60
-80 -80 -60 -40 -20 0 20 40 60 80 Real
Figure 6.3: Closed loop (proportional position feedback with kgain=20.0) shows a stable response.
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0.5
0.04 0.4 6000 n 0.03 y 0.3 t o i i
t 4000 c i o P1 s
l 0.2 0.02 ve Po 0.1 2000 0.01 0
0 0.05 0.1 0.15 0.05 0.1 0.15 0.05 0.1 0.15 time time time
80 170 160
60 165 155
P2 Pa Pb 150 40 160
145 20 155 140 0 0.05 0.05 0.1 0.15 0 0.1 time time time
Figure 6.4: Unstable response of the servo-pump actuator system for a high feedback gain (kgain=500).
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150 130
110
90 70
50
30 y
ar 10 n i
ag -10
m I -30 -50 -70
-90 -110 -130
-150 -200-180-160-140-120-100-80 -60 -40 -20 0 20 40 60 80 100 Real
Figure 6.5: Effect of varying orifice diameter (0.2 in to 0.5 in).
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120
100 80
60
40 20 y nar 0 agi m I -20
-40
-60
-80
-100
-120 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120
Real
Figure 6.6: Effect of varying the constant velocity (0.5 to 3.0 in/sec) of the solution.
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It is possible to drive the system unstable by increasing the orifice diameter. The emergence of Hopf bifurcation and the saddle node bifurcation are evident (Figure 6.5). Effect of varying the constant velocity of the system at a constant feedback gain value also demonstrates the sensitivity of the load mass mode to become unstable via a Hopf bifurcation. As expected the system tends to become unstable for a higher velocity of piston and hence high servo pump flow case (Figure 6.6). The effect of varying the proportional gain and the load mass on the stability behavior of the system is seen in Figure 6.7. It is again visible that this parametric variation can result in Hopf bifurcation of the system and the load mass mode is the one that becomes unstable via a Hopf bifurcation. This two dimensional grid is useful in demonstrating the sensitivity variation in the parameter space.
The effect of varying proportional feedback gain and the orifice diameter on the stability of the system is shown in Figure 6.8. The emergence of complex dynamics is associated with the interaction of the hydraulic and mechanical modes. This necessitates the use of advanced continuation type techniques for the study of the stability behavior. The sensitivity of parametric variation to the stability of the system may be different for various directions in the parameter space. It is a very important to study this sensitivity to help reduce the parameter space for further analysis in any nonlinear system. The effect of parametric variations along a direction in the parameter space are shown in Figure 6.9.
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Figure 6.7: Effect of varying the proportional gain (20 to 400) and the load mass (20 to 220 lbf) on the stability behavior of the system.
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Figure 6.8: Effect of varying proportional feedback gain (20 to 400) and the orifice diameter (0.2 to 0.5 in) on the stability of the system.
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Figure 6.9: Effect of parameter variation along a direction (Kp+Kd*0.01+M).
6.3 Results for servo-valve actuator model
The study of the servo-valve actuator model (Figure 6.10) involves the study of following issues:
1. Model without valve dynamics (i.e., valve position input)
2. Model with valve dynamics (i.e., valve force input)
3. Effect of parametric variation on the stability of the system
These results will be used to further characterize the model and investigate the dynamics associated with this system. The servo-valve actuator model with valve dynamics is predominantly used in this study.
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Figure 6.10: Servo-valve actuator model.
+ Controller Valve Plant Reference Error Input xx = f (,pu,,t) Kp+Kd s -
Figure 6.11: Block diagram of closed loop servo-valve actuator model. The plant input is valve spool force.
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In this work the bifurcation analysis is done for a fixed proportional controller and for a fixed PD controller. The operating condition used is a constant velocity of 1 in./sec. This nominal closed loop servo-valve actuator model with a PD controller (Kp= 100, Kd= 10) shows (Figure 6.12) four complex conjugate modes associated with the load mass and the servo-proportional valve. These modes are stable for this operating condition. Real axis eigenvalues are associated with hydraulic fluid modes. The simulation response of the cylinder with step input shows a stable response
(Figure 6.13). The valve response (Figure 6.14) is much faster than the cylinder response. Further it is possible with the nominal closed loop system to drive the cylinder at constant velocity
(Figure 6.15) by means of a reference input consisting of ramp function.
As indicated previously, the analysis of nonzero constant velocity/flow operating points is conducted to evaluate the effect of parameters on the bifurcation behavior, which would not be evident at zero velocity flow conditions. The response of the valve is shown in Figure 6.16 and the corresponding eigenvalues are shown in Figure 6.17. The effect of varying proportional gain
(Kp) is to drive the system unstable via a Hopf bifurcation (Figure 6.18). The effect of changing the derivative gain (Kd) is shown in Figure 6.19. The effect of changing line pressure is to lower the frequency of the load mass mode and close to the unstable condition (Figure 6.20). Figure
6.21 shows the effect of the variation of load mass. The frequency of the load mass mode is lowered as the load mass is increased.
The effect of varying two parameter sets is studied to gain further insight into the nature of the stability boundary in the parameter space. The distance from the stability boundary is a measure of robust bifurcation stability of the system. The effect of varying Kp (200-400) and constant velocity of the cylinder motion (0.5 to 1.5 in/sec) is shown in Figure 6.22. Figure 6.23 shows the stability boundary in the space of Kp and constant velocity. Figure 6.24 shows the simulation response of the stable system with parameters below critical value. The response undergoes a limit cycle oscillation for parameter values above a critical value (Figure 6.25).
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The effect of varying proportional gain (200-400) and load mass (100 – 700 lbf) is shown in Figure 6.26. The stability boundary in the parameter space (proportional gain and load mass) is shown in Figure 6.27. The effect of varying proportional gain (200-400) and valve spool diameter
(0.4-1.5 in) is shown in Figure 6.28. The stability boundary in the parameter space (Kp and valve spool diameter) is shown in Figure 6.29. The effect of varying Kp and Kd as parameters in the extended parameter space is shown in Figure 6.30. This demonstrates the fact that variation in the controller parameters can also result in loss of stability via bifurcation. The effect of these parametric studies is to explore the stability boundary and hence understand the robust bifurcation stability of the system. Figure 6.31 shows the behavior of the system for variation along Kp-
+Kd*0.1 in the parameter space.
6.4 Chapter summary
The effect of various parameters on the stability of the system is studied to demonstrate the existence of Hopf and saddle-node bifurcations for the two test systems (servo-pump actuator and servo-valve actuator system). The aim is to understand the complicated quantitative and qualitative behavior of the two systems. These studies demonstrate the nature and sensitivity of the stability boundary in the parameter space. The idea is to conduct initial exploration of the parameter space for the servo-hydraulic systems and to identify the parameters which affect the stability boundary. Moreover, this is an attempt to catalog effect of these parameters on the stability boundary. The servo-valve actuator system is subsequently implemented as an experimental test stand, and these preliminary studies are useful in the design of the experimental apparatus. This development is the topic of the next chapter.
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Figure 6.12: Eigenvalues of the nominal system with force input and feedback PD controller for servo-valve actuator model. One high frequency real mode at 2530 is not shown.
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Figure 6.13: Displacement and velocity of the cylinder for nominal closed loop PD controlled stable case –step input.
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Figure 6.14: Displacement of the valve for closed loop PD controlled stable case.
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Figure 6.15: Displacement and velocity of the cylinder for closed loop PD controller with ramp input.
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Figure 6.16: Displacement of the valve for closed loop PD controller constant velocity case with ramp input.
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Figure 6.17: Eigenvalues of nominal system (PD control) with ramp input.
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Figure 6.18: Effect of varying Kp (50 to 500). Hopf bifurcation occurs at approx. Kp=500.
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Figure 6.19: Effect of varying Kd (10 to 200). Saddle node bifurcation occurs at approx. Kd=200 as shown in the magnified portion of the plot (below).
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Effect of varying line pressure 300
Fixed PD-controller 200
100 y r a n i 0 g a Im
-100
-200
-300 -600 -500 -400 -300 -200 -100 0 Real
Figure 6.20: Effect of varying line pressure (300 to 1200 Psi).
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Figure 6.21: Effect of varying load mass (200 to 600 lbf).
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Figure 6.22: Effect of varying proportional gain and constant velocity.
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Figure 6.23: Stability boundary in the parameter space of proportional gain and constant velocity.
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Figure 6.24: Time response for parameters values (proportional gain and constant velocity) below critical.
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Figure 6.25: Time response for parameters values (proportional gain and constant velocity) above critical.
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Figure 6.26: Effect of proportional gain and load mass.
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Figure 6.27: Stability boundary in the parameter space (proportional gain and load mass).
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300
200
100
0
-100
-200
-300
-400 -600 -500 -400 -300 -200 -100 0 100
Figure 6.28: Effect of varying proportional gain and valve spool diameter.
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Figure 6.29: Stability boundary in the parameter space (proportional gain and valve spool diameter).
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Figure 6.30: Stability boundary in the parameter space (proportional and derivative gain).
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Figure 6.31: Effect of varying position feedback gain and velocity feedback gain.
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Chapter 7- Experimental apparatus development
7 Introduction
The development of the experimental apparatus and the methodology for model validation and analysis with appropriate real-time control hardware/software is described in this chapter.
Further, a full nonlinear model is developed for the experimental hardware implemented system, and is refined and validated using various methodologies. The various experimental issues related to control bandwidth and signal to noise ratio are investigated. Individual components as well as the full servo-valve actuator system are both validated. This chapter describes the details of this development and validation.
7.1 Development of test stand
An experimental test stand (Figure 7.1) is developed for verification of model and numerical results obtained for the stability analysis of the servo-valve actuator system. The test stand will also be used to conduct experimental investigations of the geometric effects of the various control structures. The test stand consists of a reaction stand for supporting the electro-hydraulic cylinder.
The electro-hydraulic cylinder used in this test stand is made by Parker Hannifin Corporation
(model number 3LXLTS34A7). These cylinders are equipped with MTS Temposonics linear differential transducer (LDT) (model number LHTRB00U00701V0) to measure cylinder position.
The servo-valve used for this analysis is a flapper-nozzle type servo-proportional valve (Merritt,
1967) made by Parker Hannifin Corporation (model number DY05AFCNA5). A sample schematic of a similar valve is shown in Figure 7.2. The hydraulic power supply consists of the gear pump installed on the Parker electro-hydraulic trainers. To reduce the effect of pump noise in the line pressure, a diaphragm type accumulator (charged to 400 Psi) is used. A high-pressure inline filter is used to guarantee the cleanliness of the hydraulic fluid. A Tektronix PS280 power supply is used power the servo-valve current driver card (Parker BD101-24) as well as the LDT on the cylinders. A Matlab based real time data acquisition and control hardware and software
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(WINCON 3.2), developed by Quanser Computing (www.quanser.com) is used to provide closed loop control with various control structures which can be implemented in Simulink via the Matlab real-time workshop. The pressure signal is measured by means of PCB Piezotronics static pressure sensor (model number 1502A02FJ1 KPSIS). The data acquisition and real time control system is setup in a client-server (two PC) configuration to provide maximum computational power. The overall test stand setup is shown in Figure 7.1. The reaction support for the cylinder was designed to withstand any static as well as dynamic loads (Krutz, 2001). This test stand was also designed to have no significant structural modes within the anticipated control bandwidth (up to 40 Hz).
Figure 7.1: Test stand with real time data acquisition and control hardware.
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Figure 7.2: Flapper nozzle servo-proportional valve (www.moog.com).
7.2 Servo-valve model validation and refinement
A flapper-nozzle servo valve (Figure 7.2) consists of a flapper-nozzle stage and a cylindrical spool stage. The electromagnetic torque motor drives the flapper, which is connected to the housing by a spring. The flapper deflection controls the oil flow through the nozzles, thus controlling the pressures on both sides of the spool. This creates a pressure differential across the spool and results in movement of the spool. This movement results in deflection of the mechanical feedback spring. The nonlinearities in this valve can be categorized as follows:
• Nonlinear torque motor
• Flapper Dynamics
• Nonlinear flow through nozzles
• Nonlinear flow forces on flapper
• Pressure-flow dynamics
• Spool dynamics
• Clearance in feedback spring
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• Nonlinear port flow in spool
A flapper-nozzle servo-valve can be modeled by first principles (Merritt, 1967), but preliminary simulation studies showed that flapper dynamics had no significant effect on the system dynamics below 800 Hz. A reduced order model of the servo-valve is developed. This reduction is based on the physics of the system and is developed iteratively based on the experimental data. The aim is to approximate the full nonlinear dynamic characteristics of the servo-valve by decomposing it in two parts (Figure 7.3). Based on the simulation studies, it was decided that the two parts consist of a linear transfer function from command voltage (V(s)) given to the valve amplifier card to the spool position (X(s)) of the valve and a nonlinear pressure-flow-voltage static (i.e., algebraic)
characteristic of the form qq=(,xvalve ∆P), where q is the flow with a valve position input of xvalve and pressure differential ∆P . The servo-valve has a flow throughput based on the command voltage and the pressure differential across the servo-valve. This is referred to as pressure-flow- voltage characteristics. As described in the previous chapters, this is a highly nonlinear function.
A command voltage generates a position of the valve spool, which results in a flow through the servo-valve.
Main Spool
Position
Valve Command X(s) Voltage Linear Transfer Function Pressure-Flow-Voltage Flow V(s) Model Model q
Valve Position/Voltage Input Static Characteristics
Figure 7.3: Servo-valve reduced order model.
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7.2.1 Pressure-flow-voltage nonlinear static characteristics
In order to capture the nonlinear nature of the pressure-flow-voltage characteristics of the servo-valve several, static tests were conducted. These static tests involved studying response of the servo-valve for static input under a dummy load (needle valve). The data obtained is shown in
Figure 7.4.
Figure 7.4: Pressure differential-voltage input-flow throughput characteristics of the servo- valve.
It is observed from the experimental data that the valve flow, q (gal./min.) characteristics can be modeled (Kowta, 2002) as follows:
qk= valveV∆P (7.1) where V is the valve input voltage, kvalve is a function of valve discharge coefficient, hydraulic fluid density, nominal valve spool diameter and valve spool position. To model these flow
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characteristics, kvalve is assumed to be a function of valve input voltage (thus a function of valve spool position). This approach is similar to previous studies in which the variable discharge coefficient of the servo-valve based on the valve position was used (Viall and Zhang, 2000).
Figure 7.5: Effective kvalve as a function of the pressure differential across the valve.
The variation of kvalve with the valve opening is shown in Figure 7.6. The valve is symmetrical and has no measurable dead-band. The data is limited by the lower limit of the flow meter available, which has a range of 0.2 to 2 gal./min.. Based on the experimental observations (Figure
7.4), following fifth-order polynomial model for the effective kvalve is proposed:
54321 kVvalve =+a54V aV+a3V+a2V+a1V+a0 aa==0; 0.00298482488548 01 (7.2) aa23==0.00048272373205; 0.00009829412559
aa45==-0.00001264772476; 0.00000038527429
The resultant model matches well with the experimental data as shown in Figure 7.6.
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Figure 7.6: The servo-valve area characteristics model.
7.2.2 Linear transfer function model of the servo valve
The transfer function for the valve input voltage to the valve spool position could be obtained from the flapper-nozzle servo valve model from first principles, the details of which are given in many references including Schothorst (1997). The servo-valve (Parker DYO5AFCNA5) does not have a measurable spool position output signal so an experiment was designed to evaluate the frequency response of the servo-valve in terms of valve voltage input to load pressure output. An orifice load (using a needle valve) is connected to the servo-valve with a short hose. Then sine input tests were performed to capture the transfer function between the servo-valve voltage input and load pressure output. The effect of the length of hose is clearly visible (Figure 7.7, 7.8) in terms of the decrease in the corner frequency of the system as the hose length is increased. No other dynamics are observed in the range of 0-60 Hz. At higher frequencies, the signal to noise ratio is very low.
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___ Model -*- Experiment
Figure 7.7 Load pressure/valve voltage transfer function for a long (12 ft) hose.
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___ Model -*- Experiment
Figure 7.8: Load pressure/valve voltage transfer function for a short (2 ft) hose.
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The linear transfer function (voltage input to spool position output) representation of the valve
(based on the Parker catalog information) is thus chosen as follows:
k X ()s= V(s) (7.3) ss (1 ++) 2 (1 ) 2ππ80 2 900 where k is 0.005 in./V. and is shown in Figure 7.9.
Figure 7.9: The open loop Bode plot of the valve transfer function model.
7.2.3 Servo-valve actuator model and real-time control schematic
The servo-valve actuator model schematic is given in Figure 7.10. The system construction is also given in Figure 7.11. The closed loop transfer function of the servo-valve actuator system as observed experimentally is given in Figure 7.12.
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PLANT
Controller Valve Dynamics U(s) Xv(s) Reference E(s) Xsool(s) Fluid-structure G(s) Pvalve(s) + interaction and R(s) - actuator dynamics
Figure 7.10: Servo-valve actuator model schematic.
System
Hydraulic Power Unit D/A Converter Servo-valve
Hydraulic Cylinder Load Mass PC with Real Time Control Hardware/Software
Pressure Sensor Position Sensor
Figure 7.11:Servo-valve actuator system with the real-time control hardware/software.
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Figure 7.12: Experimental transfer function of the closed loop (PD controlled) servo-valve actuator system.
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Figure 7.12 shows two modes (8Hz, and 25.5 Hz) related to the model. The 8 Hz mode is the load mass mode and the 25.5 Hz mode is the first pipeline mode. The servo-valve actuator model with first two modes of the pipeline dynamics also shows the similar closed loop Bode plot. Thus the model seems to match the experimental observations. The details of this model are given in the Appendix. A linearization of this model for a nominal system is also presented in the
Appendix. The open loop response of the numerical model of the servo-valve actuator system is shown in Figure 7.13 and 7.14. This will be used in the next chapter to design higher order controllers.
7.3 Effect of accumulators
The analysis of the servo-valve actuator system shows that the frequency of the complex load mass mode is around 47 Hz. Depending upon the control bandwidth of the servo-valve actuator system, this frequency is very high for the experimental stability analysis and control. Thus accumulators are added to reduce the effective stiffness of the system (Figure 7.1). These are diaphragm type accumulators (0.5 liters volume, charged to 150 Psi). The effect of adding accumulators is to create additional volume in the cylinder and to modify the effective bulk modulus of the electro-hydraulic cylinder. The effect of size of the accumulator used to reduce the frequency of the load mass mode is shown in Figure 7.15. The temperature effects and other nonlinear effects of the dynamics induced due to the diaphragm are neglected.
It is clearly visible that the by adding the accumulators if is possible to reduce the frequency of the load mass mode so that it is sufficiently with in the control bandwidth of the system. The control bandwidth of the servo-valve is significantly high (up to 80 Hz, per valve specifications as given by Parker (Parker Catalog: HY14-1483-B1/USA)), but due to the pipelines in between the servo-valve and the actuator (cylinder), the effective system control bandwidth is around 25 Hz.
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Figure 7.13: Open loop Bode magnitude: Actuator position/valve voltage.
Figure 7.14: Open loop Bode phase plot: Actuator position/valve voltage.
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7.4 Chapter summary
In this chapter, the design of a servo-hydraulic system test-stand is presented with details about selected components. A detailed experimental study is also presented for development of a physically-based model for this test stand. A reduced-order model of the flapper nozzle valve is developed. This model is then validated via the experiments described. The effect of accumulators on the load mass mode is studied; these are useful in decreasing the frequency of the load mass mode so as to place this mode within the system control bandwidth requirements.
After this model development and refinement, a detailed study of stability properties, effect of feedback, and experimental evaluation of the stability boundaries of the system in parameter space is to be conducted. This is given in Chapter 8.
Figure 7.15: Effect of the size of accumulator on the frequency of the load mass mode.
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Chapter 8- Control studies
8 Introduction
A model for the experimental test stand is developed and validated in the previous chapter.
The aim of this chapter is to study the stability behavior of the servo-valve actuator system (with pipe line model) under the influence of different linear control structures. In this chapter a controller is designed based upon a linearized plant model. Two types of controllers are investigated; a PD (proportional plus derivative) controller and a fifth-order linear controller are used in this study. Then the nonlinear bifurcation stability of the closed loop system under the effect of each of these controllers is investigated. All of the studies done in this chapter are with feedback of cylinder position signal measured by the LVDT. The effect of PD control on the stability of the system is studied under constant velocity operation. Further, a higher order controller is designed via loop shaping methods. This controller is then implemented in the system and stability analysis is conducted. It is shown that the stable region of the parameter space may be enlarged by the use of an higher order controller as compared to the simple PD controller. It is the principal aim of this work to characterize the nonlinear bifurcation stability of the servo-valve actuator system under the linear feedback structures. Both experimental and numerical results are presented.
8.1 PD controller
A PD controller is very frequently used in industrial control systems. This is a logical first step in analyzing the effect of feedback on the system stability. It is a non-model-based controller which requires very little prior knowledge of the system plant; as such these controllers are often tuned online. Most systems benefit from a phase lead in some suitable frequency range. The
transfer function of the PD controller is Gsc()= Kp+ Kds where Kp is the proportional gain and
Kd is the derivative gain. The PD controller may be tuned using Ziegler-Nichols rules (D’azzo and Houpis, 1966). If the model of the plant is available then further mathematical analysis can be
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done to tune the controller parameters. A Bode diagram of a sample PD controller is given in
Figure 8.1. The effect of a PD controller on the plant is to add a phase lead as seen from its Bode diagram. These controllers are universally used because of the ease of implementation and variability.
Figure 8.1: Bode diagram of a PD controller given by Gc=(s+10)
8.2 Effect of PD controller on the servo-valve actuator system
The effect of the PD controller on the stability behavior of the servo valve actuator system is studied, both experimentally and numerically. The controller parameters (Kp and Kd) are part of the extended parameter space and hence can be used in design and analysis of the closed loop operation of the system. The control structure is implemented both numerically and experimentally according to the schematic shown in Figure 7.11. The reference signal used for all of these studies is a constant velocity ramp (for constant velocity operation of the system) for reasons discussed in Sec. 5.2.2. The nominal closed loop eigenvalues of the servo-valve
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Figure 8.2: Eigenvalues of the nominal closed loop system (Kp = 100 and Kd =1). Some stable high frequency modes are not shown.
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Table 4: Eigenvalues and damping of the nominal closed loop system.
Eigenvalue Damping (%) 0 - -1.48 - -1.23e+001 + 1.17e+002i 1.05e-001 -1.23e+001 - 1.17e+002i 1.05e-001 -1.24e+001 + 1.18e+002i 1.05e-001 -1.24e+001 - 1.18e+002i 1.05e-001 -1.83e+001 + 3.53e+002i 5.19e-002 -1.83e+001 - 3.53e+002i 5.19e-002 -1.83e+001 + 3.53e+002i 5.18e-002 -1.83e+001 - 3.53e+002i 5.18e-002 -2.56e+001 + 4.58e+001i 4.88e-001 -2.56e+001 - 4.58e+001i 4.88e-001 -5.02e+002 - -5.03e+002 - -5.65e+003 -
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actuator system for a constant velocity of 0.2 in/sec with Kp=100 and Kd=0.1 and a load mass (M) of 100 lbs shown in Figure 8.2. The nominal closed loop system is stable. The complex modes (as shown in Figure 8.2) are contributed by the load mass and line dynamics. A list of nominal eigenvalues and damping is given in Table 4.
The system has a non-minimal state space representation as discussed previously in Sec.
6.2 and has a pole-zero cancellation at the origin. The effect of increase in the controller parameters on the stability of the system is to result in Hopf bifurcation. The effect of variation in the proportional gain (Kp) is shown in Figure 8.3. This shows the emergence of Hopf bifurcation in the system at Kp=150 for a derivative gain Kd of 1. This is also experimentally verified and the critical frequency of this system is 78 radians/sec. This critical frequency is also observed experimentally. The effect of variation in Kd is shown in Figure 8.4. Hopf bifurcation occurs at a critical value of Kd=63. The effect of variation in the load mass is to reduce the frequency of the load mass mode, as seen in Figure 8.5. The Hopf bifurcation occurs at 400 lb for the nominal controller parameters. The effect of increasing the constant steady state velocity of the system is to increase the flow gain and hence is similar to increasing the proportional gain as seen from the
Figure 8.6. The Hopf bifurcation occurs at a constant velocity of 2.0 in/sec. The simulation responses of the closed loop system for two sets of controller parameters, below and above critical value, are shown in Figure 8.7, 8.8. This is in agreement with the numerical bifurcation result. Also, the critical frequency (12.4 Hz) of the limit cycle oscillations is as predicted by the bifurcation analysis. The effect of variation of parameter on a two dimensional grid is studied.
This gives insight into the bifurcation stability boundary. The effect of variation of controller parameters (Kp, Kd) is shown in Figure 8.9. The effect of variation of mass and proportional gain is shown in Figure 8.10. The effect of increasing the load mass is to reduce the critical frequency of the bifurcating mode. The effect of variation of constant velocity and the proportional gain of the system is shown in Figure 8.11.
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Figure 8.3: The effect of variation of proportional gain. Hopf bifurcation occurs at Kp=150, Kd=1.
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Figure 8.4: The effect of variation of derivative gain. Hopf bifurcation occurs at Kp=100, Kd=63.
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Figure 8.5: The effect of variation of load mass. Hopf bifurcation occurs at M=400 lb.
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Figure 8.6: The effect of variation of the constant velocity of the system. Hopf bifurcation occurs at V=2.0 in/sec.
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Figure 8.7: Simulation response of the system for controller parameters just below critical (Kp=140, Kd=1).
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Figure 8.8: Simulation response of the system for controller parameters just above critical (Kp=155, Kd=1).
150
Figure 8.9: Effect of parameter variation on a grid of controller parameters (Kp, Kd).
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Figure 8.10: Effect of parameter variation on a grid of parameters (Kp, M)
152
.
Figure 8.11: Effect of parameter variation on a grid of parameters (Kp, V).
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The stability boundary is sensitive to the change in parameters. This sensitivity is also a nonlinear function of the parameters. Some insight into this behavior can be obtained by looking at the bifurcation boundary in a multi parameter space. Figure 8.12 shows the Hopf bifurcation surface as the control parameters (Kp, Kd) are varied. Experimental results are also shown on the same figure. The experimental studies were started at a stable nominal in the parameter space and then directional searches were conducted for estimating the stability boundary. The experimental data is at an offset because the limit cycle oscillation amplitude has to be above noise level (5 Psi oscillations –peak to peak) to be detected during the experiment. Thus it is seen that the experimental data matches the bifurcation result very satisfactorily. Figure 8.13 shows the relative movement of the stability boundary in the parameter space of the controller as load mass is changed. The nonlinear sensitivity of the stability boundary is clearly evident from the figure.
Figure 8.14 shows the stability boundary in the two-dimensional parameter space of load mass and proportional gain (Kp). The experimental observations are also depicted in the figure, which match the numerical results. Figure 8.15 demonstrates the level cuts of stability boundary in the two-dimensional parameter space of load mass and proportional gain (Kp) as the constant velocity of the solution is varied. Figure 8.16, 8.17 show the experimental data (cylinder position and cylinder chamber pressure) for a set of parameters above the critical value (Kp=200, M=100,
Kd=1, V=0.2 in/sec). It is clearly visible that small oscillations in the position of the cylinder cause large oscillations in the cylinder pressure chambers. These large pressure oscillations are one of the leading causes of system failure.
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Figure 8.12: Stability boundary in two-dimensional parameter space (Kp, Kd) for load mass=370lb - experimental and numerical result.
155
Figure 8.13: Effect of load mass on the stability boundary in two-dimensional parameter space (Kp, Kd) - numerical result .
156
Figure 8.14: Stability boundary in two-dimensional parameter space (Mload, Kp) for constant velocity =0.2 in/sec and Kd=0.1- experimental and numerical result.
157
Figure 8.15: Effect of varying constant velocity on the stability boundary in two-dimensional parameter space (Mload, Kp) for Kd=0.1- numerical result.
158
.
Figure 8.16: Experimental data of cylinder position with reference for parameter values above critical (Kp=200 Kd=1 M=100lb).
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Figure 8.17: Experimental data of cylinder pressure for parameter values above critical (Kp=200 Kd=1 M=100lb)-pressure in both chambers of the cylinder is presented.
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8.3 Effect of fifth-order linear controller on the servo-valve actuator system
The study of the PD controller in the preceding section showed that some control of the stability boundary is possible by a simple feedback structure, and it also demonstrated the effect of feedback on the stability of the system. A higher order (fifth-order) controller for a linear plant model based upon an operating condition of a 2 volt (10% of full scale) command input to the servo-valve with a load mass of 100 lbs and a line pressure of 450 Psi is designed to improve the stability of the servo-valve actuator system. Based upon preliminary studies, it was decided that the compensated system should have at least 30o phase margin (D’azzo and Houpis, 1966) and a gain margin (D’azzo and Houpis, 1966) of 3 db or more with an open loop bandwidth of 8 Hz.
Classical compensator design is a trial-and-error process which is discussed in detail by several authors including D’azzo and Houpis (1966). Using Bode analysis and design (D’azzo and
Houpis, 1966) the following fifth-order compensator is developed:
1.433e015 s432 + 1.054e017 s + 5.396e019 s + 2.391e021 s + 1.445e023 Cs()= (8.1) (s98 + 5615 s + 1.042e007 s7 + 9.839e009 s6 + 5.478e012 s5 + 1.909e015 s4 + 4.221e017 s32 + 5.751e019 s + 4.402e021 s + 1.445e023)
The poles of the compensator are as follows:
-3.1280e+003 -5.6497e+002 -3.9273e+002 +7.8390e+001i -3.9273e+002 -7.8390e+001i -3.9502e+002 -2.1400e+002 +8.3589e+001i -2.1400e+002 -8.3589e+001i -1.6932e+002 -1.4444e+002
The zeros of the compensator are as follows:
-1.2946e+001 +1.8229e+002i -1.2946e+001 -1.8229e+002i -2.3811e+001 +4.9526e+001i -2.3811e+001 -4.9526e+001i
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Figure 8.18: Bode magnitude plot of the compensator.
Figure 8.19: Bode phase plot of the compensator.
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Figure 8.20: Nominal closed loop system is stable (Kv=100), some high frequency stable eigenvalues are not shown.
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Table 5: Eigenvalue and damping of the closed loop nominal system.
Eigenvalues Damping (%) -5.82e-005 - -5.15e+000 - -1.22e+001 + 3.43e+002i 3.57e-002 -1.22e+001 - 3.43e+002i 3.57e-002 -1.23e+001 + 3.43e+002i 3.58e-002 -1.23e+001 - 3.43e+002i 3.58e-002 -1.83e+001 + 1.05e+003i 1.75e-002 -1.83e+001 - 1.05e+003i 1.75e-002 -1.83e+001 + 1.05e+003i 1.75e-002 -1.83e+001 - 1.05e+003i 1.75e-002 -2.63e+001 + 4.73e+001i 4.86e-001 -2.63e+001 - 4.73e+001i 4.86e-001 -6.56e+001 - -1.40e+002 + 1.17e+002i 7.67e-001 -1.40e+002 - 1.17e+002i 7.67e-001 -2.82e+002 + 1.75e+002i 8.50e-001 -2.82e+002 - 1.75e+002i 8.50e-001 -4.56e+002 + 1.52e+002i 9.48e-001 -4.56e+002 - 1.52e+002i 9.48e-001 -5.10e+002 - -5.77e+002 + 5.10e+001i 9.96e-001 -5.77e+002 - 5.10e+001i 9.96e-001 -3.13e+003 - -5.65e+003 -
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Figure 8.21: Effect of varying Bode gain of the controller (Kv). Hopf Bifurcation occurs at Kv=220, M=100 lb, V=0.2 in/sec.
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Figure 8.22: Effect of varying load mass. Hopf Bifurcation occurs at Kv=100, M=600 lb, V=0.2 in/sec.
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Bode gain of a system transfer function of type -m (with m poles at s=0) is defined as the
DC gain of sm P(s), when the transfer function is represented in Bode form. Critical Bode gain is the value of the Bode gain when the system loses stability via bifurcation. This fifth-order controller is designed to increase the critical Bode gain of the servo-valve actuator system as compared to the PD controlled case. The aim is to increase the stable region of the extended parameter space by shifting the stability boundary further out and thus extending the envelope of the design in closed loop. The Bode magnitude and phase of the compensator is given in Figure
8.18 and 8.19. Nominal eigenvalues of the closed loop system are stable (Figure 8.20) as shown in Table 5. The effect of varying Bode gain is to drive the system unstable (Figure 8.21) with a
Hopf bifurcation occurring at Kv=220. The effects of increasing load mass is to reduce the frequency of the associated mode, and eventually result in loss of stability via Hopf bifurcation
(Figure 8.22). Effect of varying the constant velocity of the system is shown in Figure 8.23. Hopf bifurcation occurs at a velocity of 2.8 in/sec. Two-dimensional grids in parameter space are investigated for stability behavior. Figure 8.24 shows the effect of variation in load mass and compensator gain. Figure 8.25 shows the effect of line pressure and load mass. Simulation of the nonlinear model also verifies the bifurcation stability result. Figure 8.26 and 8.27 are the simulation results obtained for compensator gain below critical value and compensator gain above critical value, respectively. Figure 8.28 shows the experimental cylinder position response for compensator gain above critical. The oscillation in cylinder chamber pressure (Figure 8.29) has an amplitude of nearly 80 Psi, which can be very detrimental to the system performance and could eventually lead to failure. Under this unstable operating condition, the flexible pipelines also vibrate violently causing large acoustic emissions. The critical frequency of the system is around 11 Hz. It is found that the load mass mode is the bifurcating mode. Figure 8.30 shows level cuts of the stability boundary of the system as velocity is varied.
167
Figure 8.23: Effect of varying the constant velocity.
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Figure 8.24: Effect of varying compensator gain (100-250) and load mass (50-500 lb).
169
Figure 8.25: Effect of varying line pressure and load mass.
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Figure 8.26: Simulation response of cylinder position and velocity for compensator gain below critical (Kv=100 M=370 lb).
171
8.27: Simulation response of cylinder position and velocity for compensator gain above critical (Kv=220 M=370 lb).
172
Figure 8.28: Experimental response of the cylinder for parameters above critical (Kv=220).
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Figure 8.29: Experimental pressure response in the cylinder chambers for parameter above critical (Kv=220).
174
Figure 8.30: Bifurcation stability boundary in the 2-dimensional parameter space of load mass and Bode gain. Level cuts of constant velocity solution are shown. Experimental data matches closely with the numerical result.
175
Figure 8.31: A comparison of the two controllers – PD and fifth-order for a constant velocity case (0.2 in/sec).
176
Table 6: Comparison of critical parameter values
Parameter PD controller case Fifth-order controller case Bode gain 190 220
Constant velocity (in/sec) 2.0 2.2
Load mass (lbs) 400 560
Table 7: Comparison of critical parameter values for linear analysis and bifurcation analysis for a fifth –order controller case. Critical parameter Linear analysis Bifurcation analysis
Bode gain 68 220
8.4 Comparison of control studies
It is noted that as the constant velocity of the solution is increased, the system becomes less stable. The experimental observation of the critical gains is in agreement with the numerical result obtained via the bifurcation analysis.
The fifth-order compensator was designed to improve the stability margin of the servo- valve actuator system over the PD controlled system in closed loop operation. Figure 8.31 shows that the higher-order controller indeed results in a larger stability margin as compared to the PD controlled case. A similar comparison of a few parameters is presented in the Table 6. Critical parameter values of the system for the two controllers (PD and fifth-order) are compared and it is clearly visible from this comparison (Table 6) that the fifth-order controller manages to enlarge the stable region of the servo-valve actuator system for better closed-loop stability margin.
The measure of bifurcation nonlinearity as defined is Sec. 5.6 is estimated for the servo- valve actuator system with the fifth-order compensator. It is seen from Figure 8.32 and 8.33 that the increase in the nonlinear nature of the system is accurately captured by this metric. It should
177
be noted even further that the linear Bode analysis of the servo-valve actuator system suggests that the fifth-order controller for a gain margin of 3 db in conjunction with the nominal Kv of 1 will have a critical gain of at least 68 for load mass of 100 lb. But in actual experimental set up it is observed that the critical gain of the fifth-order controller is 220 for the same operating condition (Table 7). This opens a further range of stability margin which can be utilized for operating the system at the edge of the stability boundary. This range will not be known if the designer was to trust only the linearized analysis. Thus not only the nonlinear analysis provides better estimates of the stability boundary but also provides increased robustness and control bandwidth. Thus it is clear that with a design of higher order controllers stability margin can be increased even with simple linear controllers.
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Figure 8.32: Measure of bifurcation nonlinearity as the load mass is varied for the servo-valve actuator system with the fifth-order compensator.
Figure 8.33: Measure of bifurcation nonlinearity as the load mass is varied for the servo-valve actuator system with the fifth-order compensator.
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Chapter 9- Conclusions and suggestions for future research
The stability analysis and design of nonlinear systems is studied in this research with an emphasis on the nonlinear dynamic behavior of servo-hydraulic systems and their components.
The concept of constant velocity solutions is proposed and demonstrated. The idea of extended parameter space is introduced. A mathematical model is developed. This model includes modal approximation of the pipe line dynamics between the servo-valve and actuator. It was found that the inherent nature of servo-hydraulic system models is such that near a stationary point it becomes non-minimal. This non-minimal nature of the model is highlighted. Bifurcation theory is used to analyze nonlinear stability of the servo-hydraulic system. The three test cases (servo- pump actuator system, servo-valve actuator system, physically reduced flapper nozzle servo- valve actuator system) are investigated in this dissertation. Bifurcation stability results for all the three cases are given in this dissertation. An experimental test stand is developed to implement a servo-valve actuator system. The refinement of the model is an iterative process limited by the measurement sensors available and the signal to noise ratio characteristics of the system. Once the mathematical model validation is conducted, further control and stability studies can be accomplished. The existence of saddle-node and Hopf bifurcation in the servo-hydraulic systems is demonstrated both numerically and experimentally in this work.
The effect of feedback and different control structures is evaluated. The effect of controllers on the stability of the servo-hydraulic system is also an important aspect of this work.
The idea of control-induced bifurcations is proposed and demonstrated in this work. A stabilizing controller and a stable nominal point is calculated for each specified control structure. Then the bifurcation stability boundary for the system with the control structure is investigated. The stability behavior of a simple PD controller is first studied both numerically and experimentally.
The primary output feedback structure used in this study is position feedback. A fifth-order
180
controller is also designed for a nominal point using Bode design methods to increase the critical
Bode gain for the overall system. It is demonstrated that the fifth-order controller does extend the stability boundary out thus giving more stability margin for exploitation. This leads to extension of the stability region as compared to the PD controller. The results obtained are validated both by nonlinear simulation and via the experimental observations on the test stand. This work is a qualitative and quantitative study of the effect of feedback on the stability of the servo-hydraulic systems. Following are the primary conceptual contributions of this work:
1. Development of the idea of constant velocity solutions for servo-hydraulic system
analysis.
2. Development of the concept of control induced bifurcations.
3. Introduction of the notion of extended parameter space.
4. Proposed a measure of bifurcation nonlinearity.
The development of the test stand and its analytical model paves the way for future research in this area. This test stand and model can be used in several future studies. These studies can be related to stability analysis, closed loop performance, robust control design, optimal control studies and/or evaluation of various other feedback control structures. The effect of other boundary conditions and configurations can also be evaluated in future. Following are some of the proposed directions of future research:
1. The effect of various hardware modifications such as hard plumbing, larger hydraulic
power supply, and different valve and actuator configurations can be investigated for
its effect on the system stability.
2. The effect of pressure feedback can be studied for bifurcation stability and control.
3. The effect of other control structures can be investigated which could include
feedback linearization.
4. A QFT based robust controller design can be used to investigate its effect on the
stability of the system.
181
5. Other nonlinear controllers (neural-network based controllers) can be investigated.
6. Codimension-1 bifurcations are the subject of this study, however further bifurcations
of higher co-dimensions can be investigated.
7. Development of hybrid system models of the servo-hydraulic systems to include hard
nonlinearities can be investigated. The effect of these hard nonlinearities on the
system performance can be investigated.
8. A development of the comprehensive methodology for bifurcation analysis of the
hybrid systems is very highly desirable and a very significant direction of future
research.
9. A global methodology for parametric space and state space decomposition for the
nonlinear systems can be investigated. Any development along these lines would be
useful to the entire nonlinear systems community.
10. Optimization based strategies can be investigated to develop control structures with
multiple objectives such as performance, bifurcation stability margin, and robustness.
This would require computation of optimally minimum distance to instability as well
as the robustness bounds.
This work provides foundation for further research in the area of nonlinear systems and more specifically in the area of servo-hydraulic systems by providing a basic ground work using experimental and numerical investigations. These studies, when employed for design for nonlinearity, are useful in optimal use of resources available to the design and controls engineer.
This leads to new explorations and also, opens further vistas for the efficient use of control effort as well as the stability region. Nonlinearity, when exploited properly, can be used to increase the system performance and stability margin. This design for nonlinearity is the general direction of future research. Nonlinearity in the systems should be evaluated as a design parameter. The nonlinear nature of the system should be quantified (e.g., by using the measure for bifurcation
182
nonlinearity). These estimates can then be exploited for their potential during the design as well as operation of the intended system.
183
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Appendix-1: Servo-valve actuator mode with pipeline dynamics and fifth order complex controller – Matlab implementation
function [xdot] =SVA_TLD(t,x) global x_valve global param; global omega;
%Kd=param(1); %Kd=param(2); vref=param(3); mload=param(4)/32.2/12; Pline=param(2); %450;
Kv=param(1);
% xload=x(4); dxload=x(5); % % % P1=x(6); Pa=x(6); Pb=x(7); xref=vref*t; %0.2*t+0.05*sin(2*pi*omega*t);%0.2*t+ vref=vref; %0.2+(0.05*(cos(2*pi*omega*t))*2*pi*omega);%0.2; %
% CONTROLLER STATES con_num =[0,0,0,0,0,1.433055185448249e+015,1.053510289988146e+017,5.395545716638774e+ 019,2.391325547474389e+021,1.445311090374986e+023]; con_den =[1,5.615220397075729e+003,1.042386415118273e+007,9.839272314838625e+009,5.4 78011475094872e+012,1.909445984590581e+015,4.220774524445812e+017,5.750710 724984177e+019,4.401930654600736e+021,1.445311090374986e+023];
[A_con,B_con,C_con,D_con]=tf2ss(con_num,con_den); x_con=x(end-8:end); con_input=Kv*(xref-xload); dx_con=A_con*x_con+B_con*con_input; u_volt=C_con*x_con+D_con*con_input;
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% ************************************** % SERVO VALVE % ************************************** % states.... % x1v=x(1); % x2v=x(2); % x3v=x(3);
% transfer function.. % num= 7.143846147141077e+006; % den=conv([1 2*pi*80],[1 2*pi*80]); % den=conv(den,[1 2*pi*900]); num= 1;%7.143846147141077e+006; den=conv([1/(2*pi*80) 1],[1/(2*pi*80) 1]); den=conv(den,[1/(2*pi*900) 1]);
% state space model.... [A,B,C,D]=tf2ss(num,den); xdot(1:3,1)=A*[x(1);x(2);x(3)]+B*[u_volt]; x_valve=C*[x(1);x(2);x(3)]+D*u_volt ;
% line dynamics output...... P1qa=[1 0 1 0; 0 1 0 1]*[x(8);x(9);x(10);x(11)]; P1=P1qa(1); qa=P1qa(2); qa=-qa;
P2qb=[1 0 1 0; 0 1 0 1]*[x(12);x(13);x(14);x(15)]; P2=P2qb(1); qb=P2qb(2);
% area characteristic.. and flow calculations......
%valve volt for area ch V_volt= x_valve; %area,cd characteristics area_curve=[0.00000038527429 -0.00001264772476 0.00009829412559 0.00048272373205 0.00298482488548 0] *3.85; %0.00003070588999]; if abs(V_volt)<=10.0, Cd_area_rho=polyval(area_curve,abs(V_volt)); else Cd_area_rho=polyval(area_curve,10); end if V_volt>=0.0, if abs(Pline-P1)<=1, qv1=Cd_area_rho*(abs(Pline-P1))*sign(Pline-P1); else qv1=Cd_area_rho*sqrt((abs(Pline-P1)))*sign(Pline-P1);
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end if abs(P2)<=1, q2v=Cd_area_rho*(abs(P2))*sign(P2); else q2v=Cd_area_rho*sqrt((abs(P2)))*sign(P2); end else if abs(P1)<=1, qv1=-Cd_area_rho*(abs(P1))*sign(P1); else qv1=-Cd_area_rho*sqrt((abs(P1)))*sign(P1); end if (Pline-P2)<=1, q2v=Cd_area_rho*(abs(Pline-P2))*sign(P2-Pline); else q2v=Cd_area_rho*sqrt((abs(Pline-P2)))*sign(P2-Pline); end end
% ************************************** % PIPE LINE DYNAMICS % ************************************** % state space rho=0.0000795;; % density ro=3/8/2; %E =4.716260998743000e+003; E=22000; % E=250; co=sqrt(E/rho);%sound velocity of fluid mu=0.0496000992001984 ;%2*1.55*0.001/0.78e-4; % mu =0.02388; L=6*12; om=mu/ro^2; Dn=L*mu/(co*ro*ro); Zo=rho*co/(pi*ro*ro); lambda1=pi/(2*Dn); lambda2=3*pi/(2*Dn);
alpha1=1.06; alpha2=1.04; beta1=2.31; beta2=3.38;
% alpha1=1.07; % alpha2=1.05; % beta1=1.76; % beta2=2.94;
200
%inlet side A10=[0 Zo*lambda1 ;... -lambda1/(Zo*alpha1*alpha1) -8*beta1/alpha1];
B10=[0 -2*Zo/Dn; 2/(Zo*Dn*alpha1*alpha1) 0];
A20=[0 -Zo*lambda2 ;... lambda2/(Zo*alpha2*alpha2) -8*beta2/alpha2]; B20=[0 -2*Zo/Dn; 2/(Zo*Dn*alpha2*alpha2) 0];
A_tl0=[A10 zeros(2,2);zeros(2,2) A20]; B_tl0=[B10' B20']'; diff_mat=[1 -8*Zo*Dn; 0 1];
Hn=-1*(inv(A10)*B10+inv(A20)*B20); G=inv(Hn)*diff_mat; xdot(8:11,1)=om*(A_tl0*[x(8);x(9);x(10);x(11)]+B_tl0*G*[Pa;-qv1]); xdot(12:15,1)=om*(A_tl0*[x(12);x(13);x(14);x(15)]+B_tl0*G*[Pb;q2v]); % ************************************** % **************************************
% M C K of piston Cd=0.61; k=0; c=10;
Aleak=pi*0.001*0.001/4; if abs(Pa-Pb)<=1, qleak=Cd*Aleak*sqrt(2/rho)*(abs(Pa-Pb))*sign(Pa-Pb); else qleak=Cd*Aleak*sqrt(2/rho*abs(Pa-Pb))*sign(Pa-Pb); end
%cross sectional area of piston Ap1=pi/4*2.5^2;% lower volume inch^2 Ap2=pi/4*(2.5^2-1); % upper volume inch^2
% Volumes of each chamber, in^3 Va=3.5*Ap1;% lower volume Vb=3.5*Ap2;% upper volume
% Variable volume Va=Va+Ap1*x(4); Vb=Vb-Ap2*x(4); beta_airfree=22000; percentage_air=0.05; P_atm= 14.4;
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beta_airfree=22000; percentage_air=0.05; P_atm= 14.4;
% % % Calculate the variable Beta. % beta_eff1=1/(1/beta_airfree + percentage_air/(1.4*(Pa+P_atm))); % beta_eff2=1/(1/beta_airfree + percentage_air/(1.4*(Pb+P_atm))); % % beta_eff1=1/(1/beta_eff1a + 1/(1.4*(150+P_atm))); % % beta_eff2=1/(1/beta_eff2a + 1/(1.4*(150+P_atm)));
beta_eff1a=1/(1/beta_airfree + percentage_air/(1.4*(Pa+P_atm))); beta_eff2a=1/(1/beta_airfree + percentage_air/(1.4*(Pb+P_atm))); beta_eff1=1/(1/beta_eff1a + 1/(1.4*(150+P_atm))); beta_eff2=1/(1/beta_eff2a + 1/(1.4*(150+P_atm))); % %load states xdot(4)=x(5); xdot(5)=(-k*x(4)-c*x(5)+Ap1*Pa-Ap2*Pb-mload*32.2*12)/mload;
xdot(6)=beta_eff1/Va*(qa-qleak-Ap1*x(5)); xdot(7)=beta_eff2/Vb*(qleak+Ap2*x(5)-qb); xdot=[xdot;dx_con]; % ************************************* % change the state vector...... % ************************************* %xdot=[dxv;dxload;ddxload;dPa;dPb;dlined1;dlined2];
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