Synthesis of LTV Controllers for Nonlinear SISO Plants

Abstract

In Chapter 5, we saw how to design an LTI controller for an uncertain nonlinear plant, based on the assumption that the nonlinear plant equation, y = Nu, can be replaced by an LTI uncertain plant with disturbance, that is, a plant in the form of y = P _N,y u + d _N,y . Essential conditions for a successful application of the technique are that the uncertainty of P _N,y be limited such that there exists an LTI controller which stabilizes it 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. From a qualitatively point of view, 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 all t ≥0, the chosen LTI plant and disturbance set {P _N,y ,d _N,y } can be updated along the system trajectory, i.e, on progressively increasing time intervals. 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 as follows: first, choose time intervals where the plant can be well approximated by an LTI plant; then design an LTI controller for each time interval, where the initial conditions of the current time interval are the final conditions of the previous one.