Nonlinear Control Systems

In the previous Chapter, we have shown that a nonsingular and involutive distribution Δ induces a local partition of the state space into lower dimensional submanifolds and we have used this result to obtain local decompositions of control systems. The decompositions thus obtained are very useful to understand the behavior of control systems from the point of view of input-state and, respectively, state-output interaction. However, it must be stressed that the existence of decompositions of this type is strictly related to the assumption that the dimension of the distribution is constant at least over a neighborhood of the point around which we want to investigate the behavior of our control system.

Beginning with this Chapter, we will study — in order of increasing complexity — a series of problems concerned with the synthesis of feedback control laws for nonlinear systems of the form (1.2). We will discuss first the case of single-input single-output systems, whose simple structure lends itself to a rather elementary analysis, and then — in the next Chapter — a special class of multivariable systems, in which a straightforward extension of most of the theory developed for single-input single-output systems is possible Finally — in the last four Chapters — we will present a set of more powerful tools for the analysis and the design of more general classes of nonlinear control systems.

In this Chapter, we discuss the problem of how to control a nonlinear system in order to have its output asymptotically converging towards a prescribed steady state response. To this end, we begin by showing in what specific sense the “intuitive” notion of steady state response must be understood, in the general setup of nonlinear systems, and we identify appropriate conditions under which such a response exists. Then, beginning with the next section, we show how a prescribed steady state response can be achieved.

In Chapters 4 and 5, we have presented a number of important concepts which lead to the design of feedback laws which solve the problems of transforming a nonlinear system into an equivalent linear system (possibly after a change of coordinates in the state space), locally asymptotically stabilizing a given equilibrium point (for those nonlinear systems whose zero dynamics has an asymptotically stable equilibrium at this point), and rendering certain outputs independent of certain inputs (the problems of disturbance decoupling and noninteracting control). As pointed out several times, all the procedures illustrated in these Chapters have a local character, in the sense that they lead to the design of feedback laws which are defined only in a neighborhood of a given (equilibrium) point. We want to discuss now under what conditions and how these design methodologies can be extended so as to yield globally defined solutions to the above mentioned design problems. For the sake of simplicity, and also for reasons of space, we restrict our consideration to the case of single-input single-output systems. As seen in Chapter 5, in most cases, the analysis of the more general situation of a multi-input multi-output system is not conceptually harder and only notationally more involved.