Brief paper: A formula for the optimal cost in the general discrete-time LEQG problem
Automatica (Journal of IFAC)
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We study the optimal control of a general class of stochastic singularly perturbed linear systems with perfect and noisy state measurements under positively and negatively exponentiated quadratic cost. The (expected) cost function to be minimized is actually taken as the long-term time average of the logarithm of the expected value of an exponentiated quadratic loss. We identify appropriate ``slow'' and ``fast'' subproblems, obtain their optimum solutions (compatible with the corresponding measurement structure), and subsequently study the performances they achieve on the full-order system as the singular perturbation parameter $\epsilon $ becomes sufficiently small, with the expressions given in all cases being exact to within $O(\sqrt{\epsilon})$. It is shown that the composite controller (obtained by appropriately combining the optimum slow and fast controllers) achieves a performance level close to the optimal one whenever the full-order problem has a solution. The slow controller, on the other hand, achieves (asymptotically, as $\epsilon \to 0$) only a finite performance level (but not necessarily optimal), provided that the fast subsystem is open-loop stable. If the intensity of the noise in the system dynamics decreases to zero, however, the slow controller also achieves a performance level close to the optimal one. The paper also presents a more direct derivation (than heretofore available) of the solution to the linear exponential quadratic Gaussian (LEQG) problem under noisy state measurements, which allows for a general quadratic cost (with cross terms) in the exponent and correlation between system and measurement noises, and obtains both necessary and sufficient conditions for existence of an optimal solution. Such a general LEQG problem is encountered in the slow-fast decomposition of the full-order problem, even if the original problem does not feature correlated noises. In this general context, the paper also establishes a complete equivalence between the LEQG problem and the \hi optimal control problem with measurement feedback, though this equivalence does not extend to the slow and fast subproblems arrived at after time-scale separation.