Synthesis of LTV Controllers for Nonlinear MIMO Plants

Abstract

In Chapter 7 we saw how to design an LTI controller for an uncertain nonlinear MIMO plant, based on the assumption that the nonlinear plant equation, y = Nu, can be replaced by an uncertain set of LTI equations with disturbances, i.e., of the type y = P _N,y u + d _N,y . Essential conditions for a successful application of the technique are that the uncertainty of the plant P _N,y be limited such that there exists an LTI controller which stabilizes the plant P _N,y , and that d _N,y , be small enough. These conditions restrict the sets of nonlinear plants to which the proposed design procedure can be applied. Thus, qualitatively we can say that if the nonlinear plant deviates too much from an LTI plant, then the existence of a solution is questionable. Clearly, since the deviation of a nonlinear plant from an LTI plant is much less on a finite time interval than for infinite time intervals, the idea of updating the chosen LTI plant and disturbance set {P _N,y , d _N,y } along the system’s trajectory, i.e, on different time slices, is useful. The result will be a piecewise LTI controller, the structure of which may not be fixed on the different time intervals. This “Scheduling” idea is well-known, see for example Becker and Packard, (1994). In summary, the design technique is: (i) time intervals where the plant can be well approximated by an LTI plant are chosen; then (ii) an LTI controller is designed for each time slice where the initial conditions of the present time interval are the final conditions of the previous one.