Nonlinear Systems

Why do we need to have a nonlinear theory and why bother to study a qualitative nonlinear theory? After all, most models that are currently available are linear, and if a nonlinear model is to be used, computers are getting to be ever more powerful at simulating them. Do we really need a nonlinear theory? This is not a naive question, since linear models are so much more tractable than nonlinear ones and we can analyze quite sophisticated and high dimensional linear systems. Further, if one uses linear models with some possibly time-varying parameters, one may model real systems surprisingly well. Moreover, although nonlinear models may be conceptually more satisfying and elegant, they are of little use if one cannot learn anything from their behavior. Certainly, many practitioners in industry claim that they can do quite well with linear time varying models. Of course, an opposing argument is that we may use the ever increasing power of the computer to qualitatively understand the behavior of systems more completely and not have to approximate their behavior by linear systems.

In the previous chapter, we saw several classical examples of planar (or 2 dimensional) nonlinear dynamical systems. We also saw that nonlinear dynamical systems can show interesting and subtle behavior and that it is important to be careful when talking about solutions of nonlinear differential equations. Before dedicating ourselves to the task of building up in detail the requisite mathematical machinery we will first study in semi-rigorous but detailed fashion the dynamics of planar dynamical systems; that is, systems with two state variables. We say semi-rigorous because we have not yet given precise mathematical conditions under which a system of differential equations has a unique solution and have not yet built up some necessary mathematical prerequisites. These mathematical prerequisites are deferred to Chapter 3.

In this chapter we will review very briefly some definitions from algebra and analysis that we will periodically use in the book. We will spend a little more time on existence, and uniqueness theorems for differential equations, fixed point theorems and some introductory concepts from degree theory. We end the chapter with an introduction to differential topology, especially the basics of the definitions and properties of manifolds. More details about differential geometry may be found in Chapter 8.

In this chapter we introduce the reader to various methods for the input-output analysis of nonlinear systems. The methods are divided into three categories:

The study of the stability of dynamical systems has a very rich history. Many famous mathematicians, physicists, and astronomers worked on axiomatizing the concepts of stability. A problem, which attracted a great deal of early interest was the problem of stability of the solar system, generalized under the title “the N-body stability problem.” One of the first to state formally what he called the principle of “least total energy” was Torricelli (1608–1647), who said that a system of bodies was at a stable equilibrium point if it was a point of (locally) minimal total energy. In the middle of the eighteenth century, Laplace and Lagrange took the Torricelli principle one step further: They showed that if the system is conservative (that is, it conserves total energy—kinetic plus potential), then a state corresponding to zero kinetic energy and minimum potential energy is a stable equilibrium point. In turn, several others showed that Torricelli’s principle also holds when the systems are dissipative, i.e., total energy decreases along trajectories of the system. However, the abstract definition of stability for a dynamical system not necessarily derived for a conservative or dissipative system and a characterization of stability were not made till 1892 by a Russian mathematician/engineer, Lyapunov, in response to certain open problems in determining stable configurations of rotating bodies of fluids posed by Poincaré. The original paper of Lyapunov of 1892, was translated into French very shortly there after, but its English translation appeared only recently in [193].

In this chapter we will give the reader an idea of the many different ways that Lyapunov theory can be utilized in applications. We start with some very classical examples of stabilization of nonlinear systems through their Jacobian linearization, and the circle and Popov criteria (which we have visited already in Chapter 4) for linear feedback systems with a nonlinearity in the feedback loop. A very considerable amount of effort has been spent in the control community in applying Lyapunov techniques to adaptive identification and control. We give the reader an introduction to adaptive identification techniques. Adaptive control is far too detailed an undertaking for this book, and we refer to Sastry and Bodson [259] for a more detailed treatment of this topic. Here, we study multiple time scale systems and singular perturbation and averaging from the Lyapunov standpoint here. A more geometric view of singular perturbation is given in the next chapter.

In this section we develop the rudiments of the modern theory of dynamical systems. To give the reader a bird’s eye view of this exciting field, we will for the most part skip the proofs. Some results are identical in ℝ ⁿ to those stated in ℝ² in Chapter 2, but the center manifold theorem and bifurcation calculations are new in this chapter. We will also use machinery that we have built up enroute to this chapter to sharpen statements made in Chapter 2. Readers who wish to consult more detailed books are referred to Guckenheimer and Holmes [122], Ruelle [247], or Wiggins [329].

The material on matrix Lie groups in this chapter is based on notes written by Claire Tomlin, of Stanford University and Yi Ma.

In this chapter we begin with a study of the modern geometric theory of nonlinear control. The theory began with early attempts to extend results from linear control theory to the nonlinear case, such as results on controllability and observability. This work was pioneered by Brockett, Hermann, Krener, Fliess, Sussmann and others in the 1970s. Later, in the 1980s in a seminal paper by Isidori, Krener, Gori-Giorgi, and Monaco [150] it was shown that not only could the results on controllability and observability be extended but that large amounts of the linear geometric control theory, as represented, say, in Wonham [331] had a nonlinear counterpart. This paper, in turn, spurred a tremendous growth of results in nonlinear control in the 1980s. On a parallel course with this one, was a program begun by Brockett and Fliess on embedding linear systems in nonlinear ones. This program can be thought of as one for linearizing systems by state feedback and change of coordinates. Several breakthroughs, beginning with [45; 154; 148; 67], and continuing with the work of Byrnes and Isidori [56; 54; 55], the contents are summarized are in Isidori’s book [149], yielded a fantastic set of new tools for designing control laws for large classes of nonlinear systems (see also a recent survey by Krener [170]). This theory is what we refer to in the chapter title.

This chapter has been heavily based on the research work and joint papers with John Hauser, of the University of Colorado, George Meyer, of NASA Ames, Petar Kokotović of the University of California, Santa Barbara, and Claire Tomlin, of Stanford University.

In this chapter, we give the reader a brief introduction to controllability and observability of nonlinear systems. We discuss topics of constructive controllability for nonlinear systems, that is, algorithms for steering the system from a specified initial state to a specified final state. This topic has been very much at the forefront of research in nonlinear systems recently because of the interest in non-holonomic systems. We give a reader of the power of differential geometric methods for nonlinear control when we discuss input-output expansions for nonlinear systems and disturbance decoupling problems using appropriately defined invariant distributions.

This chapter has been extensively based on the research work of (and papers written with) Richard Murray, Dawn Tilbury, and Linda Bushneil. The text of the chapter follows notes written with George Pappas, John Lygeros, and Dawn Tilbury [241].

Nonlinear control is very much a growing endeavor, with many new results and techniques being introduced. In the summary sections of each of the preceding chapters we have given the reader a sense of the excitement surrounding each of the new directions. In this chapter, we talk about another area of tremendous recent excitement: hybrid systems. The growth of this area is almost directly attributable to advances in computation, communication, and new methods of distributed sensing and actuation, which makes it critical to have methods for designing and analyzing systems which involve interaction between software and electro-mechanical systems. In this chapter, we give a sense of the research agenda in two areas where this activity is especially current: embedded control and multi-agent distributed control systems.