Romeo Ortega - PID Passivity-Based Control of Nonlinear Systems with Applications

The aforementioned integrability conditions can be obviated if the system is shifted passive, with a storage function that is positive definite with respect to a desired equilibrium. In spite of the intensive efforts to characterize the class of passive systems that are also shifted passive, the currently available results are quite restrictive – two common requirements being, for instance, that the input matrix is constant and the storage function of the original system is convex. The main results along these lines of research are reviewed in the book.

Another control scenario that we consider in the book pertains to the case when some stabilizing controller has already been added to the system, but we would like to include an additional integral action to reject the effect of additive disturbances that were neglected in the design of the aforementioned stabilizing controller. We treat the cases of constant or time‐varying disturbances as well as the scenario where the disturbances enter into the image of the input matrix – called matched disturbances – or when they are unmatched. In all cases, we give constructive solutions to the problem of designing this new integral action (IA).

Motivated by the application of PID in physical systems, we pay particular attention in the book to port‐Hamiltonian (pH) systems. It is well known that pH models describe the behavior of many physical processes and have the central feature of underscoring the importance of the energy function, the interconnection pattern, and the dissipation of the system, which are the essential ingredients of PBC. As shown throughout the book, the possibility of exploiting the latter features of pH systems in the design of a PID‐PBC or an IA allows us to obtain sharper, and in many cases, more constructive results than the ones available for general nonlinear systems.

The book is organized as follows. In Chapter 2, we give the general framework of PID‐PBC and show how they can be used to solve several stabilization and output regulation problems. In this chapter, we also discuss some obstacles for the application of PID‐PBC, in particular the dissipation obstacle, and show the relationship of PID‐PBC with the well‐known, and conceptually appealing, control‐by‐interconnection technique.

In Chapter 3, we show, via several practical examples, how the concept of passivity can be used for the analysis and tuning of PIDs.

In Chapter 4, we discuss the problem of designing of PID‐PBCs for the case when the desired value for the regulated output is different from zero. Some results on shifted passivity are presented and their use to solve this problem is discussed. This chapter is wrapped‐up with four modern practical applications.

Chapter 5is devoted to the characterization of passive outputs for pH systems. This result is then used in Chapter 6to design PID‐PBCs for pH systems that ensure Lyapunov stability. The particular case of underactuated mechanical systems is discussed in Chapter 7, where the results are illustrated with several practical examples.

In the final Chapter 8, we present the results pertaining to robustification of nonlinear controllers via the addition of IA – again, paying particular attention to pH and mechanical systems.

The appendices provide the basic definitions and background theory that is used throughout the book. In particular, some preliminaries on passivity and stability theory of state‐space systems and a brief discussion on pH systems are given. Some additional technical results and some useful lemmas needed in the book are also presented.

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).

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.

Consider the PID controller 2.1. The operator is output strictly passive. More precisely, there exists such that the following inequality holds:

Proof . To prove the lemma, we compute

Integrating the expression above, we get

The proof is completed setting .

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.