Dynamics of model-reference and hill climbing systems

In this thesis the theory of both linear differential equations with periodic coefficients and linear differential equations with random coefficients is applied to investigate the stability and accuracy of parameter adaptation of sinusoidal perturbation and model reference adaptive control systems. Throughout dimensional analysis applied so that all the results are presented in a non-dimensional form. The first part of the thesis t s devoted to investigating the stability of such differential equations. In chapter 1 a system of linear homogeneous differential equations with periodic coefficients is considered and a numerical procedure, based on Floquet theory and well suited for use on a digital computer, is presented for obtaining necessary and sufficient conditions for asymptotic stability of the null solution. Also considered in this chapter is the so called infinite determinant method of obtaining the stability boundaries for a restricted class of linear differential equations with periodic coefficients. Chapter 2 is devoted to reviewing the current state of the stability theory of linear differential equations with random coefficients. In chapter 3 a theoretical analysis of the stability and accuracy of parameter adaptation of a single input, sinusoidal perturbation, extremum control system with output lag is considered. Using the principle of harmonic balance it is shown that various stable harmonic and sub-harmonic steady state solutions are possible in certain regions of the parameter space. By examining the domains of attraction, corresponding to the stable solutions, regions in three dimensional space are obtained within .which initial conditions will lead to a given steady state stable oscillation. It is also shown that the subharmonic steady state solutions do not correspond to the optimum solution, so that, for certain initial conditions and parameter values, it is possible for the system to reach a steady state solution which is not the optimum solution. All the theoretical results are verified by direct analogue computer simulation of the system. The remainder of the thesis is devoted to investigating the stability and accuracy of parameter adapt.at ion of model reference adaptive control systems. In order to develop a mathematical analysis, and to illustrate the difficulties involved, a stability analysis of a first order M.I.T. type system with controllable gain, when the input varies with time in both a periodic and random manner, is first carried out. Also considered are the effects of (a) random disturbances at the system output (b) and (b) periodic and random variations, with time, of the controlled process environmental parameters, on the stability of the system and the accuracy of its parameter adaptation. When the input varies sinusoidally with time stability boundaries are obtained using both a numerical implementation of Floquet theory and the infinite determinant method; the relative merits of the two methods is discussed. The theoretical results are compared with stability boundaries obtained by analogue computer simulation of the system. It is shown that the stability boundaries are complex in nature and that some knowledge of such boundaries is desirable before embarking on an analogue computer investigation of the system. When the input varies randomly with time the stability problem reduces to one of investigating the stability of a system of linear differential equations with random coefficients. Both the theory of Markov processes, involving use of the Fokker-Planck equation, and the second method of Liapunov are used to investigate the problem; limitations and difficulty of applications of the theory is discussed. The theoretical results obtained are compared with those obtained by digital simulation of the system. If the controlled process environmental parameter is allowed to become time varying then it is shown that this effects both the stability of the system and the accuracy of its parameter adaptation. Theoretical results are obtained for the cases of the parameter varying both sinusoidally and randomly with time; some of the results are compared with those obtained by digital simulation of the system. It is also shown that noise disturbance at the system output has no effect on the system stability but does effect the accuracy of the parameter adaptation. The doubts concerning the stability and the difficulty of analysis of the M.I.T. , type system have led.:researchers to think about redesigning the model reference system from the point of view of stability. In particular we have the Liapunov synthesis method where the resulting system is guaranteed stable for all possible inputs. However, in designing such systems the controlled process environmental parameters are assumed constant and, by considering the Liapunov redesign scheme of the first order M.I.T. system previously discussed, it will be shown that the effect of making such parameters time varying is to introduce a stability problem. In chapter 6 the methods developed for analysing the first order system are extended to examine the stability of a higher order M.I.T. type system. The system considered has a third order process and a second order model and a stability analysis is presented for both sinusoidal and random input. Steady state values of the adapting parameters are first obtained and the linearized variational equations , for small disturbances about such steady states, examined to answer the stability problem. Theoretical results are compared with those obtained by direct analogue computer simulation of the system. The effect, on the mathematical analysis , of replacing the system multipliers by diode switching units is also considered in this chapter. The chapter concludes by presenting a method of obtaining a Liapunov redesign scheme for the system under discussion