On the JLQ problem with uncertainty

Trofino Neto et al. (1992) have shown that one can formulate an LQ problem in the presence of uncertainties in such a way that the state and input weighting matrices can be derived as the solution of a nonlinear optimization problem. Here it is shown that for linear systems with jumping parameters, the introduction of uncertainties in the state and input matrices can still be carried out and a control law derived as the solution of a similar, but more complex, problem. Moreover, both of the nonlinear optimization problems can be formulated as linear matrix inequality (LMI) problems, i.e., convex optimization problems. Hence, if an optimal solution exists, it can easily be obtained by means of some available LMI software packages. The existence of such a solution is proved under a restrictive assumption.