Explore the foundational and advanced subjects associated with proportional-integral-derivative controllers from leading authors in the field
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.
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.
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...