UBC Theses and Dissertations
Linear systems identification and optimization with application to adaptive control Hanafy, Adel Abdel Raouf
This thesis is concerned with the problem of identifying and controlling linear continuous systems. Algorithms which are feasible for real-time digital computations are developed for the design of both the controller and the identifier. The generalized equation error is shown to be applicable to a mean-square method of rapid digital estimation of linear system parameters. Due to the imposed structure of the estimator the manipulation of high order matrices is avoided. Examples illustrate the effectiveness of the estimator for a variety of cases dealing with quantization noise as well as measurements noise. In some cases, this digital identifier requires the computation of the generalized inverse of a matrix. A simple algorithm for computing the generalized inverse of a matrix is developed. This algorithm eliminates the need for Gram-Schmidt orthogonalization and its associated (in the interest of accuracy) reorthogonalization as new vectors are introduced. A two-stage estimator is developed for estimating time-invariant and time-varying parameters in linear systems. During the second stage, the time-varying parameters are considered as unknown control inputs to a linear subsystem of known dynamics. For the first stage, the digital identifier is shown to be effective in identifying the time-invariant parameters. Numerous examples illustrate the effectiveness of the method. To design a feedback controller a method of successive approximations for solving the two point boundary value problem for optimum constant gain matrices is developed. The method is shown to be comptationally equivalent to a deflected gradient method. Convergence can always be achieved by choice of a scalar step-size parameter. An online approximate method is developed which appears suitable for systems whose parameters must be identified. The two point boundary value problem is replaced by an algebraic problem, the solution of which gives a sub-optimal constant gains. The problem of trajectory sensitivity reduction by augmented state feedback is shown to be well-posed if constrained structure of the gain matrices is taken. The simple structure of constant gain matrices is considered. Based on the developed identifier and controller, one possible strategy for optimal adaptive control which is particularly attractive from an engineering point of view is studied. The strategy is to identify system parameters at the end of an observation interval and then to use the parameters to derive an "optimal" control for a subsequent control interval. Two examples are considered in order to illustrate the effectiveness of this strategy.
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