Cautious H2H2 optimal control using uncertain Markov parameters identified in closed loop

In this paper, we present a new cautious H2H2 optimal design approach based on the noisy and biased Markov parameters identified from a finite number of input and output samples in a closed-loop plant. This approach not only links closed-loop subspace identification with optimal control; but also directly evaluates parametric uncertainties on the identified Markov parameters. Neither a state-space model nor its stochastic uncertainty has to be realized. The effects of the parametric uncertainties on the output predictor and a quadratic cost function are explicitly analyzed. An H2H2 optimal control problem is formulated as a “minmaxminmax” problem of the expectation of the cost function with respect to the stochastic noise in the identified parameters. Analytic solution to this problem is derived in a closed form, which avoids computing the empirical mean of the quadratic cost as required by randomized algorithms. An extension of the cautious design to the solution associated with an arbitrary probability is also proposed. The solutions hence lead to easily implementable control laws, robust to the uncertainties in the identified Markov parameters from a closed-loop plant.