# 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.

## Keywords

Time Slice Taylor Series Expansion Plant Output Nonlinear Plant Closed Loop Simulation## Preview

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