This paper deals with a state feedback ${\mathcal{H}}_{\mathrm{\infty }}$ control problem for linear time-invariant discrete-time descriptor systems with norm-bounded parametric uncertainties. To this end, bounded real lemma (BRL) is extended on the class of uncertain descriptor systems. The control design procedure based on the conditions of BRL for uncertain descriptor systems is proposed. Numerical examples are included to illustrate the effectiveness of the present result.

The design problems of robust static controllers for discrete-time systems with norm-bounded parametric uncertainties and random input disturbances are considered. The controllers under consideration stabilize the plant for all possible values of uncertainty from a given set of parameters and also guarantee a desired suppression level for random exogenous disturbances. A numerical example is given.

For linear discrete-time descriptor systems anisotropy-based bounded real lemma in terms LMI was formulated and proved. The algorithm of anisotropic norm computation using convex optimization is given. A numerical example is considered.

In this survey, we discuss various approaches to control theory that have arisen in the recent decades and reflect the desire to reach a trade-off between the \(LQG /H_2 \)-control theory and the \(H_{\infty } \)-control theory. The theories of the kind include the theory of risk-sensitive controllers, the theory of suboptimal control with a constraint on the \(H_{\infty } \)-entropy functional, the mixed \(H_2/H_{\infty } \)-control theory, the minimax \(LQG \)-control theory, the anisotropy-based theory, and some others. The survey discusses in more detail the anisotropy-based control theory, which includes both the \( LQG/H_2\)- and the \(H_{\infty } \)-theory within a single statement of the problem.