# Stability criteria for some classes of nonlinear multivariable control systems

Rae, William Gifford (1966) Stability criteria for some classes of nonlinear multivariable control systems. PhD thesis, University of Glasgow.

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## Abstract

Techniques are well established for using the second method of Liapunov to determine the stability of single loop systems containing one or more nonlinear elements. In this dissertation these techniques are extended to include three separate classes of multivariable systems. In the first two classes, a linear multivariable process is controlled by many feedback loops, each containing a nonlinear sensing or actuating device. Distinction is made between linear processes which contain only feedforward interactions, and those which contain only feedback interactions. The two classes of systems considered are therefore represented by the equations y = -G(p)f (feedforward interactions in the process) and y = G1(p) (-f + G2(p) y) (feedback interactions in the process). where p is the differential operator, y is the system output vector, f is the output vector of the nonlinearities and G(p) = [gij (p);]. Generalisation of the techniques of Lure and Letov in applying Liapunov's second method provides sufficient conditions for total stability of such systems. For a given n-variable system, the criteria developed may be applied if the system is stable for all loop gains gi in the ranges ki > gi > Ki (i = l,2,...n) provided that the corresponding nonlinear characteristics fi(yi) are confined to the same sectors of the input-output plane, namely ki > fiyi / yi > Ki v. For completeness, equivalent criteria for instability of these two classes of systems are included. The third class of systems considered is one involving multiplication of system variables, and in particular a multinode representation of a nuclear reactor. Such systems, unlike the two classes discussed above, have in most cases only a finite region of stability in the state space. A stability criterion is derived by Liapunov's second method which produces such a finite region provided that the equilibrium point of the system is stable. For the examples considered, this region is entirely adequate for all realistic deviations of the system from equilibrium. In all criteria developed, the principal objective is to produce systematic stability tests which do not involve prior estimation of, for example, the Liapunov function for any given system. Although computation becomes difficult for high order systems, all criteria involve only one parameter to be chosen such that results are optimum. The criteria are not applicable to systems which are of a self-oscillatory nature, and a chapter is included on investigation of the oscillatory modes of a particular two variable system containing two functional nonlinearities.

Item Type: Thesis (PhD) Doctoral Adviser: G DS MacLellan Mathematics 1966 Enlighten Team glathesis:1966-72247 Copyright of this thesis is held by the author. 24 May 2019 15:12 24 May 2019 15:12 http://theses.gla.ac.uk/id/eprint/72247