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By Peter S. Maybeck

This quantity builds upon the principles set in Volumes 1 and a pair of. bankruptcy thirteen introduces the fundamental techniques of stochastic keep watch over and dynamic programming because the primary technique of synthesizing optimum stochastic regulate legislation.

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Stochastic models, estimation and control. Volume 3

This quantity builds upon the rules set in Volumes 1 and a couple of. bankruptcy thirteen introduces the elemental recommendations of stochastic regulate and dynamic programming because the basic technique of synthesizing optimum stochastic regulate legislation.

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24 13. DYNAMIC PROGRAMMING AND STOCHASTIC CONTROL Note that the state variance P ( r ) satisfies the forward equation dP/dr = 2FP + G2Q for this problem, whereas K , satisfies the backward equation dK,/dr’ = - 2 F K , . These are adjoint differential equations and will be discussed further in later sections. , if the system is stable, dK,/dt’ = - 2 F K , assures us of a cost that grows as we go backward in time. 5 OPTIMAL STOCHASTIC CONTROL WITH PERFECT KNOWLEDGE OF THE STATE This section considers the dynamic programming solution for optimal control of systems described by stochastic models, assuming that perfect knowledge of the entire state is available at discrete sample times [l, 3,7-9,12,17,18,24, 25,28,29,31,37].

4 diagrams the structure of the optimal stochastic controller for the LQG problem. As seen in part (a), it is an algorithm that accepts sampleddata measurements z ( t i ,wk)= zi as inputs, and outputs the optimal control u*(ti) to apply as constant values to the system over the next sample period, from ti to t i + l . As shown in part (b), the Kalman filter is an algorithm that accepts knowledge of previously applied control u*(ti - and current measurement zi and outputs the optimal state estimate %(ti').

For the LQG controller problem, this density is Gaussian and completely specified by its mean %(ti+)and covariance P(ti+):if these are available, we need not remember all of the measurement values Zi explicitly. Thus, we can equivalently solve and P(ti+),the latter of which the optimization problem as a function of %(ti+) does not depend on Zi, as provided by a Kalman filter. The great advantage is that we do not have to remember an ever-increasing number of values in order to generate the optimal control.

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