PID Passivity-Based Control of Nonlinear Systems with Applications
Description
In PID Passivity-Based Control of Nonlinear Systems with Applications, expert researchers and authors Drs. Romeo Ortega, Jose Guadalupe Romero, Pablo Borja, and Alejandro Donaire deliver a comprehensive and detailed discussion of the most crucial and relevant concepts in the analysis and design of proportional-integral-derivative controllers using passivity techniques. The accomplished authors present a formal treatment of the recent research in the area and offer readers practical applications of the developed methods to physical systems, including electrical, mechanical, electromechanical, power electronics, and process control.
The book offers the material with minimal mathematical background, making it relevant to a wide audience. Familiarity with the theoretical tools reported in the control systems literature is not necessary to understand the concepts contained within. You'll learn about a wide range of concepts, including disturbance rejection via PID control, PID control of mechanical systems, and Lyapunov stability of PID controllers.
Readers will also benefit from the inclusion of:
* A thorough introduction to a class of physical systems described in the port-Hamiltonian form and a presentation of the systematic procedures to design PID-PBC for them
* An exploration of the applications to electrical, electromechanical, and process control systems of Lyapunov stability of PID controllers
* Practical discussions of the regulation and tracking of bilinear systems via PID control and their application to power electronics and thermal process control
* A concise treatment of the characterization of passive outputs, incremental models, and Port Hamiltonian and Euler-Lagrange systems
Perfect for senior undergraduate and graduate students studying control systems, PID Passivity-Based Control will also earn a place in the libraries of engineers who practice in this area and seek a one-stop and fully updated reference on the subject.
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Persons
ROMEO ORTEGA, PhD, is a full-time professor and researcher at the Mexico Autonomous Institute of Technology, Mexico. He is a Fellow Member of the IEEE since 1999. He has served as chairman on several IFAC and IEEE committees and participated in various editorial boards of international journals.
JOSÉ GUADALUPE ROMERO, PhD, is a full-time professor and researcher at the Mexico Autonomous Institute of Technology, Mexico. His research interests are focused on nonlinear and adaptive control, stability analysis, and the state estimation problem.
PABLO BORJA, PhD, is a Postdoctoral researcher at the University of Groningen, Netherlands. His research interests encompass nonlinear systems, passivity-based control, and model reduction.
ALEJANDRO DONAIRE, PhD, is a full-time academic at the University of Newcastle, Australia. His research interests include nonlinear systems, passivity, and control theory.
Content
Author Biographies xv
Preface xix
Acknowledgments xxiii
Acronyms xxv
Notation xxix
1 Introduction 1
2 Motivation and Basic Construction of PID Passivity-based Control 5
2.1 L2-Stability and Output Regulation to Zero 6
2.2 Well-Posedness Conditions 9
2.3 PID-PBC and the Dissipation Obstacle 10
2.3.1 Passive systems and the dissipation obstacle 11
2.3.2 Steady-state operation and the dissipation obstacle 12
2.4 PI-PBC with y0 and Control by Interconnection 14
3 Use of Passivity for Analysis and Tuning of PIDs: Two Practical Examples 19
3.1 Tuning of the PI Gains for Control of Induction Motors 21
3.1.1 Problem formulation 23
3.1.2 Change of coordinates 27
3.1.3 Tuning rules and performance intervals 30
3.1.4 Concluding remarks 35
3.2 PI-PBC of a Fuel Cell System 36
3.2.1 Control problem formulation 41
3.2.2 Limitations of current controllers and the role of passivity 46
3.2.3 Model linearization and useful properties 48
3.2.4 Main result 50
3.2.5 An asymptotically stable PI-PBC 54
3.2.6 Simulation results 57
3.2.7 Concluding remarks and future work 58
4 PID-PBC for Nonzero Regulated Output Reference 61
4.1 PI-PBC for Global Tracking 63
4.1.1 PI global tracking problem 63
4.1.2 Construction of a shifted passive output 65
4.1.3 A PI global tracking controller 67
4.2 Conditions for Shifted Passivity of General Nonlinear Systems 68
4.2.1 Shifted passivity definition 69
4.2.2 Main results 70
4.3 Conditions for Shifted Passivity of port-Hamiltonian Systems 73
4.3.1 Problems formulation 74
4.3.2 Shifted passivity 75
4.3.3 Shifted passifiability via output-feedback 77
4.3.4 Stability of the forced equilibria 78
4.3.5 Application to quadratic pH systems 79
4.4 PI-PBC of Power Converters 81
4.4.1 Model of the power converters 81
4.4.2 Construction of a shifted passive output 82
4.4.3 PI stabilization 85
4.4.4 Application to a quadratic boost converter 86
4.5 PI-PBC of HVDC Power Systems 89
4.5.1 Background 89
4.5.2 Port-Hamiltonian model of the system 91
4.5.3 Main result 93
4.5.4 Relation of PI-PBC with Akagi's PQ method 95
4.6 PI-PBC of Wind Energy Systems 96
4.6.1 Background 96
4.6.2 System model 98
4.6.3 Control problem formulation 102
4.6.4 Proposed PI-PBC 104
4.7 Shifted Passivity of PI-Controlled Permanent Magnet Synchronous Motors 107
4.7.1 Background 107
4.7.2 Motor models 108
4.7.3 Problem formulation 111
4.7.4 Main result 113
4.7.5 Conclusions and future research 114
5 Parameterization of All Passive Outputs for port-Hamiltonian Systems 115
5.1 Parameterization of all Passive Outputs 116
5.2 Some Particular Cases 118
5.3 Two Additional Remarks 120
5.4 Examples 121
5.4.1 A level control system 121
5.4.2 A microelectromechanical optical switch 123
6 Lyapunov Stabilization of port-Hamiltonian Systems 125
6.1 Generation of Lyapunov Functions 127
6.1.1 Basic PDE 128
6.1.2 Lyapunov stability analysis 129
6.2 Explicit Solution of the PDE 131
6.2.1 The power shaping output 132
6.2.2 A more general solution 133
6.2.3 On the use of multipliers 135
6.3 Derivative Action on Relative Degree Zero Outputs 137
6.3.1 Preservation of the port-Hamiltonian Structure of I-PBC 138
6.3.2 Projection of the new passive output 140
6.3.3 Lyapunov stabilization with the new PID-PBC 141
6.4 Examples 142
6.4.1 A microelectromechanical optical switch (continued) 143
6.4.2 Boost converter 144
6.4.3 2-dimensional controllable LTI systems 146
6.4.4 Control by Interconnection vs PI-PBC 148
6.4.5 The use of the derivative action 150
7 Underactuated Mechanical Systems 153
7.1 Historical Review and Chapter Contents 153
7.1.1 Potential energy shaping of fully actuated systems 154
7.1.2 Total energy shaping of underactuated systems 156
7.1.3 Two formulations of PID-PBC 157
7.2 Shaping the Energy with a PID 158
7.3 PID-PBC of port-Hamiltonian Systems 161
7.3.1 Assumptions on the system 161
7.3.2 A suitable change of coordinates 163
7.3.3 Generating new passive outputs 165
7.3.4 Projection of the total storage function 167
7.3.5 Main stability result 169
7.4 PID-PBC of Euler-Lagrange Systems 172
7.4.1 Passive outputs for Euler-Lagrange systems 173
7.4.2 Passive outputs for Euler-Lagrange systems in Spong's normal form 175
7.5 Extensions 176
7.5.1 Tracking constant speed trajectories 176
7.5.2 Removing the cancellation of Va(qa) 178
7.5.3 Enlarging the class of integral actions 179
7.6 Examples 180
7.6.1 Tracking for inverted pendulum on a cart 180
7.6.2 Cart-pendulum on an inclined plane 182
7.7 PID-PBC of Constrained Euler-Lagrange Systems 190
7.7.1 System model and problem formulation 191
7.7.2 Reduced purely differential model 195
7.7.3 Design of the PID-PBC 196
7.7.4 Main stability result 199
7.7.5 Simulation Results 200
7.7.6 Experimental Results 202
8 Disturbance Rejection in port-Hamiltonian Systems 207
8.1 Some Remarks On Notation and Assignable Equilibria 209
8.1.1 Notational simplifications 209
8.1.2 Assignable equilibria for constant d 210
8.2 Integral Action on the Passive Output 211
8.3 Solution Using Coordinate Changes 214
8.3.1 A feedback equivalence problem 214
8.3.2 Local solutions of the feedback equivalent problem 217
8.3.3 Stability of the closed-loop 219
8.4 Solution Using Nonseparable Energy Functions 221
8.4.1 Matched and unmatched disturbances 222
8.4.2 Robust matched disturbance rejection 225
8.5 Robust Integral Action for Fully Actuated Mechanical Systems 230
8.6 Robust Integral Action for Underactuated Mechanical Systems 237
8.6.1 Standard interconnection and damping assignment PBC 239
8.6.2 Main result 241
8.7 A New Robust Integral Action for Underactuated Mechanical Systems 244
8.7.1 System model 244
8.7.2 Coordinate transformation 245
8.7.3 Verification of requisites 246
8.7.4 Robust integral action controller 248
8.8 Examples 248
8.8.1 Mechanical systems with constant inertia matrix 249
8.8.2 Prismatic robot 250
8.8.3 The Acrobot system 255
8.8.4 Disk on disk system 260
8.8.5 Damped vertical take-off and landing aircraft 265
A Passivity and Stability Theory for State-Space Systems 269
A.1 Characterization of Passive Systems 269
A.2 Passivity Theorem 271
A.3 Lyapunov Stability of Passive Systems 273
B Two Stability Results and Assignable Equilibria 275
B.1 Two Stability Results 275
B.2 Assignable Equilibria 276
C Some Differential Geometric Results 279
C.1 Invariant Manifolds 279
C.2 Gradient Vector Fields 280
C.3 A Technical Lemma 281
D Port-Hamiltonian Systems 283
D.1 Definition of port-Hamiltonian Systems and Passivity Property 283
D.2 Physical Examples 284
D.3 Euler-Lagrange Models 286
D.4 Port-Hamiltonian Representation of GAS Systems 288
Index 309
2
Motivation and Basic Construction of PID Passivity-Based Control
In this chapter, we present the PID controller structure considered throughout the book and discuss the motivation to wrap the controller around a passive output - that we call in the sequel PID-PBC. We present the basic construction of PID-PBCs using the so-called natural passive output. We discuss the technical issue of well posedness of the feedback interconnection and discuss the role of the dissipation structure of the system on the feasibility of using this kind of PID-PBC. Finally, we briefly discuss the connection of PID-PBC with the formally appealing method of control by interconnection (CbI).
2.1 -Stability and Output Regulation to Zero
Throughout the book, we consider the nonlinear system described in (1) wrapped around a PID controller described by
(2.1)where with , and are the PID tuning gains. The key property of PID controllers that we exploit in the book is that it defines an output strictly passive map . This well-known property (Ortega and García-Canseco, 2004; van der Schaft, 2016) is summarized in the lemma below.
The main idea of PID-PBC is to exploit the passivity property of PIDs and, invoking the Passivity Theorem, see Section A.2 of Appendix A, wrap the PID around a passive output of the system to ensure -stability of the closed-loop system. This result is summarized in the proposition below, whose proof follows directly from the Passivity Theorem, passivity of the mapping and output strict passivity of the mapping defined by the PID-PBC.
Figure 2.1 Block diagram representation of the closed-loop system of Proposition 2.1.
2.2 Well-Posedness Conditions
As discussed earlier, it is necessary to ensure that the control law 2.1 can be computed without differentiation nor singularities. The latter may arise due to the presence of the derivative term . Clearly, this term can be added only when the output has relative degree one, that is, when .2 But even for systems without a derivative action, a singularity may appear for systems with relative degree zero, that is with .
The required well-posedness assumptions in both cases are stated in the lemma below, whose proof follows immediately computing the expressions of for the closed-loop system. As will become clear below, the assumptions are technical only, and rather weak, but they are given for the sake of completeness.
2.3 PID-PBC and the Dissipation Obstacle
In this section, we reveal a subtle aspect of the practical application of PID-PBC, namely that for passive systems of relative degree one, there exists a steady state only if the energy extracted from the controller is zero at the equilibrium. The latter condition is known in PBC as dissipation obstacle and is present in many physical systems, for instance, all electrical circuits with leaky energy storing elements operating in nonzero equilibria - i.e. capacitors in parallel, or inductors in series, with resistors. Interestingly, this obstacle is absent in position regulation of mechanical systems since dissipation (due to Coulomb friction) is zero at standstill.
After briefly recalling the nature and mathematical definition of the dissipation obstacle, we prove the claim of inexistence of equilibria stated above in a more general context than just PID-PBC, namely for all dynamic controllers incorporating an integral action on a passive output of relative degree one.
2.3.1 Passive Systems and the Dissipation Obstacle
To mathematically define the dissipation obstacle of a passive system with storage function , let us compute its derivative
(2.2)As a corollary of Hill-Moylan's theorem, see Theorem A.1, we see that the only passive output of relative degree one is the so-called natural output, that we identify with the subindex , and is given by
(2.3)Substituting the definition above in 2.2, we can give to it the interpretation of power-balance equation, where is the energy stored by the system, is the supplied power and is the system's dissipation. In passivity theory, it is said that the system does not suffer from the dissipation obstacle - at an assignable equilibrium - if
(2.4)Notice that for pH systems, see Definition D.1, the dissipation obstacle translates into
(2.5)where is the dissipation matrix and is a bona fide energy function - yielding a clear physical interpretation.
The dissipation obstacle is a phenomenon whose origin is the existence of pervasive dissipation, that is, dissipation that is present even at the equilibrium state. It is a multifaceted phenomenon that has been discussed at length in the PBC literature, where it is shown that the key energy shaping step of PBC (Ortega et al., 2008, Proposition 1), the generation of Casimir functions for CbI (van der Schaft, 2016, Remark 7.1.9) and the assignment of a minimum at the desired point to the shaped energy function (Zhang et al., 2015, Proposition 2) are all stymied by the dissipation obstacle.
2.3.2 Steady-State Operation and the Dissipation Obstacle
The proposition below shows that the application of PID-PBC with the natural output is severely stymied by the dissipation obstacle. Actually, we will prove a much more general result that contains, as a particular case, the PID-PBC scenario.
2.4 PI-PBC with and Control by Interconnection
In this section, we give an interpretation of proportional-integral (PI) PBC with the natural output as a particular case of CbI, which is a physically (and conceptually) appealing method to stabilize equilibria of nonlinear systems widely studied in the literature, cf, Duindam et al. (2009), Ortega et al. (2008), and van der Schaft (2016).
CbI has been mainly studied for pH systems, where the physical properties can be fully exploited to give a nice interpretation to the control action, viewed not with the standard signal-processing viewpoint, but as an energy exchange process. Here, we present CbI in the more general case of the -system , which we assume passive with storage function , and the controller
with , which is also passive with storage function . Clearly, the integral action of the PI-PBC is a particular case of this controller with the choices , , and .
These systems are coupled via an interconnection that preserves power, that is which satisfies . For instance, the classical negative feedback interconnection
The proportional action of the PI-PBC may be assimilated as a preliminary damping injection to the plant giving rise to the new process model
In view of the passivity properties, the storage function of the overall system
(2.6)is nonincreasing, alas, not necessarily positive definite - with respect to the desired equilibrium . To construct a bona-fide Lyapunov function, it is proposed in CbI to prove the existence of an invariant foliation
with a smooth mapping and . In CbI, a cross-term of the form , with a free differentiable function, is added to the function given in 2.6 to create the function
that, due to the invariance property of , satisfies , hence, is still nonincreasing. If we manage to prove that is positive definite, the desired equilibrium will be stable. However, the asymptotic stability requirement, and the fact that is invariant, imposes the constraint on the initial conditions
That is, the trajectory should start on the leaf of that contains the desired equilibrium - fixing the initial conditions of the controller. Invoking Sard's theorem (Spivak, 1995), we see that is a nowhere dense set, hence, the asymptotic stability claim is nonrobust (Ortega, 2021). Two solutions to alleviate this problem - estimation of the constant or breaking the invariance of via damping injection - have been reported in Castaños et al. (2009), but this adds significant complications to the scheme.
In Chapter 6, we give a solution to the...
