Per Johan Nicklasson

Trajectory tracking control

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.

Nested-loop passivity-based control: An illustrative example

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.

Modeling of switched DC-to-DC power converters

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.

Feedback interconnected systems: Robots with AC drives

Throughout the book we have stressed the fact that PBC is compatible with one of the important viewpoints of systems theory that complicated systems are best thought of as being interconnections of simpler subsystems, each one of them being characterized by its dissipation properties. This aggregation procedure has three important implications. First, it is consistent with the dominating approaches for modeling and simulation based on some kind of network representation and energy flow. Second, it help us to think in terms of the structure of the system and to realize that sometimes the pattern of the interconnections is more important than the detailed behaviour of the components. Finally, it is indeed a design-oriented methodology which allows us to isolate the “free subsystems” — sensors and actuators.

Other applications and current research

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.

Voltage-fed induction motors

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.

Passivity-based control of DC-to-DC power converters

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.

Generalized AC motor

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.

Set-point regulation

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.

Current-fed induction motors

In Chapter 3 we proved that PBC of mechanical systems reduces, in regulation tasks with full state feedback, to the classical PD controller used in most robotic applications. Furthermore, when velocities are not available for measurement the PBC methodology suggests to replace the velocities by their approximate derivatives, which is also a standard procedure in applications. This “downward compatibility” of PBC with current engineering practice is a remarkable feature whose importance can hardly be overestimated. On one hand, it provides a solid system-theoretic foundation to popular control strategies which enhances their understanding and paves the way for subsequent improvements. On the other hand, viewing the new controllers as “upgrades” of the existing ones, it facilitates the transfer of these developments to practitioners. In this chapter we will show that, under some simplifying assumptions on the machine model, the PBC for electrical machines presented in previous chapters also has a “downward compatibility” property with the industry standard field-oriented controller (FOC).