Gorbunov

In this article a sufficient condition for exponential stability of a linear system with delay was proposed; a new estimating method of the degree of damping and magnitude of overregulation was developed on the basis of this condition. The exponential stability condition lies in solvability of an arbitrary non-linear matrix inequality for one matrix and two scalar unknown quantities. Explicit expressions for estimates of the lower bound of the degree of damping and the upper bound of overregulation in terms of solution to this matrix inequality were derived; a method for finding such solutions was also proposed. An example illustrating effectiveness of the proposed approach was considered.

The author considers the problem of estimating the domain of attraction for linear systems with constraints on the state. It is proved that the domain of attraction of the linear system with a convex admissible region is a convex set; the Minkowski function of the desired set and a method of its calculation are provided in this article. A method of guaranteed estimation of the required convex polyhedron is proposed, and an example is provided. The obtained results can be used to evaluate domains of attraction for linear systems with limited control.

The problem of finding a solution to the differential inclusion in the form of piecewise linear function is examined. Sufficient conditions for the local existence of linear solution to the differential inclusion and procedure for the construction of piecewise linear solution to the differential inclusion were proposed. Illustrative example is considered. The obtained results can be used for mathematical simulation of systems with an incomplete description and optimization techniques.

 The author considers the problem of a Lagrange's system equilibrium point asymptotic stabilization with delay in the control loop. For linear Lagrangian systems of a general type the author found dynamic feedback providing the solution of a task in view. The offered approach is based on the use of change of variables excluding delay from the control loop, keeping the plant equations of motion without changes. The considered change of variables represents the modified transformation known as an Artstein reduction. The energy method is applied to solving the stabilization problem for the system in the new variables. Efficiency of the offered control method is shown trough numerical simulations. The results received in this note can be used to design control algorithms for mechanical systems under conditions of limited speed of reception, processing and transfer of an operating signal.