Passivity-based Control of Euler-Lagrange Systems

The final objective of the research reported in this book is to contribute to the development of a system theoretic framework for control of nonlinear systems that incorporates at a fundamental level the systems physical structure and provides solutions to practical engineering problems. This is, of course, a very ambitious and somehow imprecise objective. To help delineate what we really want to accomplish we underscore the three major keywords of our work: Euler-Lagrange (EL) systems, passivity and applications. The first keyword mathematically defines the class of systems that we study, the second one the main physical property that we focus on, while the last one is our final objective. In this chapter we will develop upon this three keywords to explain the background and the contents of the book, and to motivate our approach.

It has been argued in the Introduction that a good starting point to develop a practically meaningful nonlinear control theory is to specialize the class of systems under consideration. The main reason being, of course, that the vast array of nonlinear systems renders futile the quest of a monolithic theory applicable for all systems. In particular, it defies the approach of mimicking the, by now fairly complete, linear theory. Specializing the systems, on the other hand, introduces additional constraints and structure, which may enable otherwise intractable problems to be answered. In this chapter we describe the class of systems that we will consider throughout the book and which we call Euler-Lagrange (EL) systems. The most important reason for singling out the study of EL systems is that they capture a large class of contemporary engineering problems, specially some which are intractable with linear control tools. Finally, by restricting ourselves to systems with physical constraints we believe we can contribute to reverse the tide of “find a plant for my controller” which still permeates most of the research on control of general nonlinear systems.

In the previous chapter we underlined several fundamental properties of EL systems. In particular we saw that the equilibria of an EL plant are determined by the critical points of its potential energy function, moreover the equilibrium is unique and globally stable if this function has a global and unique minimum. We also saw that this equilibrium is asymptotically stable if suitable damping is present in the system. These two fundamental properties motivated Takegaki and Arimoto in [261] to formulate the problem of set point regulation of robots in two steps, first an energy shaping stage where we modify the potential energy of the system in such a way that the “new” potential energy function has a global and unique minimum in the desired equilibrium. Second, a damping injection stage where we now modify the Rayleigh dissipation function. This seminal contribution contained the first clear exposition of the use of energy functions in robotics. (See Subsection 1.3 for a brief review of the literature). It generated a lot of interest in the robotics community since it rigorously established that computationally simple control laws, derived from energy considerations, could accomplish rather sophisticated tasks.

In this chapter we extend the passivity-based method, developed for regulation in the previous chapter, to solve trajectory tracking problems. The first main modification that we have to make is that for tracking, besides reshaping the potential energy of the EL plant, we must also shape the “kinetic energy” function. Whereas modifying the potential energy function means to relocate the equilibria of the system, the modification of the “kinetic energy” function can, roughly speaking, be rationalized as imposing a specific pattern to the transformation of potential into kinetic energy. However, the quotes here are important because the storage function that we assign to the closed loop is not an energy function in the sense that it defines the equations of motion. With an obvious abuse of notation we will still refer to this step as energy shaping, but it is better understood as passivation with a desired storage function (see Appendix A). The damping injection step is added then to make the passivity strict. The passivation objective is achieved invoking the key passive error dynamics Lemma 2.7, which states that we can always factor the workless forces in such a way that, in terms of the error signals s, the EL system behaves like a linear passive system.

¹In this chapter we illustrate one further advantage of PBC: the possibility of attenuating the effect of bounded external disturbance via high-gain feedback. Roughly speaking, this feature stems from the fundamental property of infinite gain margin of passive maps, hence stability is preserved when placed in closed-loop with high-gain operators. The simplest application of this principle is in sliding mode control, where a passifiable (e.g., minimum phase and relative degree one) system is controlled with a relay, which defines an operator with infinite gain, albeit passive. Stability is then preserved because of the fundamental property of passivity being invariant under feedback interconnection.

We start with this chapter the second part of the book which is devoted to electrical systems, and in particular to DC-to-DC power converters. The study of these devices constitutes an active area of research and development in both power electronics and control theory. Switched DC-to-DC converters have an ubiquitous variety of industrial and laboratory applications thanks to their reduced cost, simplicity and off-the-shelf availability. This part of the book consists of two chapters. In view of the presence of the switches some new considerations with respect to those made in Chapter 2 and Appendix B, are needed to formalize the mathematical modeling. This is done in Chapter 6, which is fully devoted to modeling and the exploration of the structural properties useful for PBC, which as we will see later, applies verbatim for this class of systems. This material is presented in Chapter 7.

The feedback regulation of DC-to-DC power supplies is, broadly speaking, accomplished through either PWM feedback strategies, or by inducing appropriate stabilizing sliding regimes. PWM control of these devices is treated in several books, among which we cite [117,238]. The topic has been also extensively treated, among many others, by the third author an collaborators in [244,249], where emphasis has been placed in using advanced nonlinear feedback control design techniques for the regulation of average PWM models of the various converters.

We start with this chapter the third part of the book which is dedicated to PBC of electromechanical systems. Particular emphasis will be given to AC electrical machines, to which Chapters 9–11 are devoted. Chapter 12 treats robots with AC drives, hence connecting the material of the next three chapters with our previous developments on mechanical systems of Chapter 4.

In the second part of the book we pursue our research on development of PBC for EL systems as applied to electromechanical systems. In this chapter we restrict our attention to the practically very important class of the generalized rotating electric machines [179,285]. The main contribution is the definition of a class of machines for which the output feedback torque tracking problem can be solved with PBC. Roughly speaking, the class consists of machines whose non-actuated (rotor) dynamics is suitably damped, and whose electrical and mechanical dynamics can be partially decoupled via a coordinate transformation. Machines satisfying the latter condition are known in the electric machines literature as Blondel-Park transformable [157]. In practical terms this requires that the air-gap magneto motive force can be suitably approximated by the first harmonic in a Fourier expansion. These two conditions, stemming from the construction of the machine, have clear physical interpretations in terms of the couplings between electrical, magnetic and mechanical dynamics, and are satisfied by a large number of practical machines.

The induction¹ motor, and especially the squirrel-cage induction motor, has traditionally been the workhorse of industry, due to its mechanical robustness and relatively low cost. In a wide range of servo applications with high-performance requirements it has now, due to advances in control theory and power electronics, replaced DC and synchronous drives. As a continuation of our studies on torque-control of the generalized machine in Chapter 9, we address here the problem of passivity-based speed/position control of this particularly important machine. The two phase squirrel-cage induction motor model² is first given in Section 1. Various equivalent representations, often encountered in the literature are also presented. The speed/position control problem is then formulated in Section 2.

The objective of this final chapter is twofold. First, we point out to other applications of PBC that went beyond the scope of this book. Second, we collect specific problems in PBC of EL systems on which we are currently working. They have not yet been fully resolved, and thus put forth further avenues of study.