I. E. Zuber

Invariant stabilization of classes of uncertain systems with delays

This paper deals with uncertain systems with delayed argument and having the property that the entries of the state matrix are functionals of arbitrary nature with the only available information being the bounds on their variations. Using quadratic Lyapunov-Krasovskii functionals of the special form, the control is designed such that it is robust against the variations in the plant matrix, the system output decays exponentially no matter what the persistent exogenous disturbance is, and the state vector remains bounded.

Vector control design for robust stabilization of a class of uncertain systems

We consider a state-space control system such that the elements of the system matrix and the matrix of the control term are functionals having arbitrary nature. By using a quadratic Lyapunov function, a stabilizing vector control is designed such that it does not depend on the form of the elements of the system matrix, but rather on the bounds of their possible variations.

Using the direct and indirect control to stabilize some classes of uncertain systems. II. Pulse and discrete systems

Consideration was given to several classes of the uncertain pulse systems with a sufficiently high impulsing frequency. Direct and indirect stabilizing controls that are robust to the plant matrix were constructed using the quadratic Lyapunov functions and the method of averaging. Direct stabilizing control was constructed for some classes of the uncertain discrete systems.

Robust stabilization of some class of uncertain systems

AbstractConsider an uncertain system

Using the Direct and Indirect Control to Stabilize Some Classes of Uncertain Systems. I. Continuous Systems

\dot x = A( \cdot )x + B( \cdot )u,

Global stabilization of nonlinear systems by quadratic Lyapunov functions

where A(·) ∈ ℝn × n, B(·) ∈ ℝn × m, and the elements of matrices A(·) and B(·) are arbitrary functionals. It is assumed that all elements are uniformly bounded, and that the first r elements counted from above and situated on a certain fixed upper superdiagonal are alternating. It is also assumed that m = n − r, and that a matrix formed by the last m rows of matrix B(·) is nonsingular. The control u = S(·)x is synthesized, and conditions on the admissible matrix B(·) ensuring the global asymptotical stability of the system are obtained. We consider the case when modulation of the components of the vector u is realized by means of synchronous amplitude-frequency pulse modulators of the first kind. A lower estimate for the pulse frequency under which the pulse system is globally asymptotically stable is obtained.

Synthesis of invariantly stable discrete uncertain systems

The system of an arbitrary order with one scalar control is considered. It is assumed that coefficients of the system and vectors of distribution of control are bounded and are functionals of an arbitrary character. With the aid of the Lyapunov function in a quadratic form with the constant Jacobian matrix of coefficients of a special form, the conditions for bounds of a change of the coefficients are obtained, in the fulfillment of which the linear stationary control by the state is synthesized. In this control, the system becomes exponentially stable on the whole.

Stability of indefinite systems

Consideration was given to some classes of uncertain systems where the elements of the matrix of the controlled plant are arbitrary functionals about which only the variation boundaries are known. The direct and indirect (dynamic) controls that are robust to the plant elements are constructed using the quadratic Lyapunov functions.