Finite Memory Observers and Pseudo Inverses. Application to the Parameter Sensitivity Problem.

Abstract

In this thesis finite memory observers, decomposition operators and pseudo inverses have been established for dynamic systems whose input-output relationship can be represented by linear bounded causal operators on Hilbert resolution spaces. The concepts have been applied to both the identification and sensitivity reduction of time varying parameters. Computer simulations have verified the results of the theory. For operators that admit a minimal state representation it has been shown that st and ard observability conditions are necessary and sufficient for the construction of finite memory observers. The observers act on input-output records of finite length and determine exactly the state of the given operator. The length of the records defines the memory of the observers and may be time variant. For a given linear operator, T, the observers have been used to construct finite memory linear causal maps (zeta)('t) and (upsilon)('t), defined for each time t, such that (zeta)('t)T = (upsilon)('t). These maps allow one to relate linearly a segment of the output with the corresponding segment of the input. The correspondence is independent of the past evolution of the operator T. Given n input-output pairs (v(,j), Tv(,j)) the operators (zeta)(,p) and (upsilon)(,p) are used to define a tracking function m(t,(.)) and a linear causal bounded pseudo inverse, M, for the operator T. If (alpha) = {(alpha)(,j)} in an n-vector of arbitrary functions of time and v(,(alpha)) = (SIGMA)(, )(alpha)(,j)v(,j) is considered as input to the operator T then the tracking function m(t,Tv(,(alpha))) provides an estimate of (alpha)(t). The estimate depends on the vector (zeta)(,p)Tv(,(alpha)) = (upsilon)(,p)v(,(alpha)). If (alpha) is a constant over the interval spanned by the memory length of (zeta)(,p) and (upsilon)(,p) then m(t,Tv(,(alpha))) gives the exact value of (alpha) (t). The equation MTv(,(alpha)) = v(,(alpha)) is shown to hold over a given interval {(tau),t} provided that (alpha) is a constant vector over the memory length of the operators (zeta)(,p), (upsilon)(,p), (sigma)(epsilon){(tau),t}. In applying these results to the parameter sensitivity problem it has been assumed that a system is modelled by the operator valued function S((eta)) = T + W{(.),(eta)}. The disturbance or parameter change (eta) is an n-vector valued function of time and belonging to a Hilbert resolution space. Under given conditions it is shown that it is possible to determine n functions v(,j), associated to W{u,(.)}, that can be used to construct a tracking function for (eta), and a pseudo inverse operator M. For a given input, u, it has been shown that m(t,(.)), in a linear approximation, corresponds to a pseudo inverse for the operator W{u,(.)}. If W{u,(.)} is linear and (eta) is constant, the equation TMW{u,(eta)} = W{u,(eta)} is shown to be valid. Therefore, in a first order terms analysis, the control (delta)u = - MW{u,(eta)} produces a cancellation of the effect of the disturbance. This is the principle underlying the design of the sensitivity reduction algorithm. Its stability and performance for time varying parameters have been analyzed. Computer simulations with difference systems containing up to five parameters varying r and omly in stepwise form show that the algorithm reduces significantly the effect of parameter changes when they are slow varying with respect to the memory length of the inverse. The simulations also test different factors that affect the performance of the algorithm such as characteristic of the input signal and memory length.