UBC Theses and Dissertations

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UBC Theses and Dissertations

Numerical computation of nearly-optimal feedback control laws and optimal control programs Longmuir, Alan Gordon


An investigation is made into the approximate synthesis of optimal feedback controllers from the maximum principle necessary conditions. The overall synthesis can be separated into two phases: the computation of optimal open-loop controls (control programs) and trajectories from the necessary conditions, and the processing of this data to obtain an approximate representation of the optimal control as a state function. A particular technique for approximating the optimal feedback control from the optimal open-loop controls and trajectories is proposed and examined in Part I of the thesis. Parameters in a prechosen suboptimal controller structure are computed such that a sum of integral square deviations between the suboptimal and optimal feedback controls is minimized. The deviations are computed and summed over a certain set of trajectories which "cover" the system operating region. Experimentation with various controller structures is quite feasible since the controller parameters are computed by solving linear algebraic equations. Examples are given to illustrate the application of the technique and ways in which suitable controller structures may be found. If general purpose functions are to be used for this purpose, piecewise polynomial functions are recommended and techniques for their use are discussed. The synthesis method advocated is evaluated with respect to control sensitivity and instrumentation and compared to alternative procedures. Part II is concerned with the computation of optimal control programs, the most time consuming numerical task in the synthesis procedure. A new numerical optimization technique is presented which extends the function space Newton-Raphson method (quasilinearization) to a more general terminal condition. More significantly, a generalized Ricatti transformation is employed, and as a consequence, the integration of the unstable coupled canonical system is eliminated. Examples are given as evidence of the improved numerical qualities of the new algorithm. This method is one example of a class of algorithms, defined and developed in the thesis, called second variation methods. Some methods in this class have previously appeared in the literature but they are developed in the thesis from a unified point of view. The recognition of this class allows the relationships between the various methods to be seen more clearly as well as allowing techniques developed for use in one algorithm to be used in others.

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