Variance-Constrained Multi-Objective Stochastic Control and Filtering

Chapter 2Robust H_∞ Control with Variance Constraints

In stochastic control problems, the performance requirements of engineering systems are usually expressed as upper bounds on steady-state variances. Unfortunately, current control design techniques, such as LQG and H_∞ design, do not seem to give a direct solution to this kind of design problem, since they lack a convenient avenue for imposing design objectives stated in terms of upper bounds on the variance values. For example, the LQG controllers minimize a linear quadratic performance index that lacks guarantee variance constraints with respect to individual system states.

The covariance control theory developed in late 1980s has provided a more direct methodology for achieving the individual variance constraints than the LQG control theory. Covariance control theory is capable of dealing with variance-constrained control problems and, at the same time, considering other multiple performance objectives due to its design flexibility. It has been shown that the covariance control approach is capable of solving multi-objective design problems, which has found applications in dealing with transient responses, round-off errors in digital control, residence time/probability in aiming control problems, and stability robustness in the presence of parameter perturbations. Such an advantage is based on the fact that several control design objectives, such as stability, time-domain and frequency-domain performance specifications, robustness, ...