The H2-optimal Control Problem of CSVIU Systems: Discounted, Counter-discounted and Long-run Solutions -- Part II: Optimal Control

TL;DR

This paper develops explicit optimal control laws for CSVIU systems under various $H_2$ performance criteria, addressing stability, convergence, and long-run behavior with novel Riccati-like and normal equation solutions.

Contribution

It introduces a new explicit form of the optimal control law for CSVIU systems, combining Riccati-like feedback and a generalized normal equation approach.

Findings

Explicit optimal control laws derived for CSVIU systems.

Stable solutions under energy and long-run average criteria.

A search method for the optimal control law based on the normal equation.

Abstract

The paper deals with stochastic control problems associated with $H_2$ performance indices such as energy or power norms or energy measurements when norms are not defined. They apply to a class of systems for which a stochastic process conveys the underlying uncertainties, known as CSVIU (Control and State Variation Increase Uncertainty). These indices allow various emphases from focusing on the transient behavior with the discounted norm to stricter conditions on stability, steady-state mean-square error and convergence rate, using the optimal overtaking criterion -- the long-run average power control stands as a midpoint in this respect. A critical advance regards the explicit form of the optimal control law, expressed in two equivalent forms. One takes a perturbed affine Riccati-like form of feedback solution; the other comes from a generalized normal equation that arises from the nondifferentiable local optimal problem. They are equivalent, but the latter allows for a search method to attain the optimal law. A detectability notion and a Riccati solution grant stochastic stability from the behavior of the norms. The energy overtaking criterion requires a further constraint on a matrix spectral radius. With these findings, the paper revisits the emerging of the inaction solution, a prominent feature of CSVIU models to deal with the uncertainty inherent to poorly known models. Besides, it provides the optimal solution and the tools to pursue it.