### Article

## Robust state-feedback H_\infty control for discrete-time descriptor systems with norm-bounded parametric uncertainties

This paper deals with a state feedback H ∞ 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.

An optimal control problem is formulated for a class of nonlinear systems for which there exists a coordinate representation (diffeomorphism) transforming the original system into a system with a linear main part and a nonlinear feedback. In this case the coordinate transformation significantly changes the form of original quadratic functional. The penalty matrices become dependent on the system state. The linearity of the structure of the transformed system and the quadratic functional make it possible to pass over from the Hamilton–Jacoby–Bellman equation to the Riccati type equation with state-dependent parameters upon the control synthesis. Note that it is impossible to solve the obtained form of Riccati equation analytically in the general case. It is necessary to approximate the solution; this approximation is realized by numerical methods using symbolic computer packages or interpolation methods. In the latter case, it is possible to obtain the suboptimal control. The presented example illustrates the application of the proposed control method for the feedback linearizable nonlinear system.

This paper is dedicated to optimal state-feedback control problem for discrete-time descriptor systems in presence of “colored” noise with known mean anisotropy level. Here “colored” noise stands for a stationary Gaussian sequence, generated by a linear shaping filter from the Gaussian white noise sequence. The control goal is to find a state feedback control law which makes the closed-loop system admissible and minimizes its a-anisotropic norm (mean anisotropy level a is known).

The theoretical fundamentals for solving the linear quadratic problems may be some times used to design the optimal control actions for the nonlinear systems. The method relying on the Riccati equation with state-dependent coefficients is promising and rapidly developing tools for design of the nonlinear controllers. The set of possible suboptimal solutions is generated by ambiguous representation of the nonlinear system as a linearly structured system with state-depended coefficients and the lack of sufficiently universal algorithms to solve the Riccati equation also having state-depended coefficients. The paper proposed a method to design a guaranteed control for the uncertain nonlinear plant with state-depended parameters. An example of design the controller for an uncertain nonlinear system was presented.

In this paper, anisotropy-based control problem with regional pole assignment for descriptor systems is investigated. The purpose is to find a state-feedback control law, which guar- antees desirable disturbance attenuation level from stochastic input with unknown covariance to controllable output of the closed-loop system, and ensures, that all finite eigenvalues of the closed-loop system belong to the given region inside the unit disk. The proposed control design procedure is based on solving convex optimization problem with strict constraints. The numerical effectiveness is illustrated by numerical example.

In this paper, linear discrete-time descriptor systems with norm-bounded parametric uncertainties are under consider- ation. The input signal is supposed to be a “colored” noise with bounded mean anisotropy. Sufficient conditions of anisotropic norm boundedness for such class of systems are given.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.