Aristide Halanay

A small gain theorem for linear stochastic systems

Abstract A well-known result in linear control theory is the so-called “small gain” theorem stating that if given two plants with transfer matrix functions T 1 and T 2 in H ∞ such that ‖ T 1 ‖ γ and ‖ T 2 ‖ γ , when coupling T 2 to T 1 such that u 2 = y 1 and u 1 = y 2 one obtains an internally stable closed system. The aim of the present paper is to describe a corresponding result for stochastic systems with state-dependent white noise.

Stabilization of linear systems

The general problem of stabilization of linear systems consists of designing a controller that uses information on a measurable output in order to influence the behavior of the state considered as a deviation from a desired equilibrium.

Suboptimal stabilization of linear systems with several time scales

By using the Lurie-Yakubovic-Popov equations, a suboptimal stabilizing feedback is constructed for linear systems with several time scales when only the reduced model is available.

Infinite horizon disturbance attenuation for discrete-time systems. A Popov-Yakubovich approach

It is proved that for the discrete-time linear systems with time-varying coefficients the existence of a controller which simultaneously stabilizes and provides prescribed disturbance attenuation for the resultant closed-loop system, implies the existence of global solutions to several Kalman-Szegö-Popov-Yakubovich systems. It is also proved that this fact is equivalent to the existence of the positive semidefinite stabilizing solutions to corresponding game-theoretic Riccati equations. The family of all controllers with the above mentioned properties is constructed in terms of the solutions to the cited Kalman-Szegö-Popov-Yakubovich systems. The main tool is the generalized Popov-Yakubovich theory which is essentially developed in an operator-theoretic framework.

Well-conditioned computation for H∞ controller near the optimum

The present paper gives a procedure for determining a H∞ optimal controller in the assumption that the game Riccati equations have stabilizing positive definite solutions at the optimum value. A specific feature of the construction is its first step consisting in balancing with respect to the positive definite stabilizing solutions of the Riccati equations. The justification is based on singular perturbations reduction.

Extended Nehari problem for time-variant discrete systems

Abstract In this paper a suboptimal solution to the time-variant discrete version of the extended Nehari problem is given. The main tool used to obtain such a solution, with prescribed tolerance, is the so-called anticausal stabilizing solution to the reverse discrete-time Riccati equation.

Remarks on order reduction for a robustly suboptimal controller via singular perturbations

Abstract It is shown that under a generic assumption the suboptimally robust controller constructed according to Glover and McFarlane (1989) leads to a singularly perturbed system, which may be reduced in the usual way, avoiding thus the difficulties mentioned by Habets (1991). The reduced controller obtained by this procedure is in fact optimal.

Stabilization of Linear Systems with Two Time Scales

In many situations the mathematical models used to describe the object to be regulated exhibit in a natural way several time scales. To illustrate this idea we shall consider only two examples. In the mathematical modeling of a synchronous machine, the evolution of the mechanical part is much slower than the one corresponding to the electrical one.

Adaptive Stabilization and Identification

As an alternative to high-gain stabilization to stabilize a system under incomplete information, one may also use, adaptive control, considering the fact that the gain is varying in time according to an adaptation algorithm that takes into account the measured values of the output.

Behaviour of high-gain feedback control under white-noise perturbations

It is assumed that a linear, square, invertible and minimum-phase system has been stabilized by high-gain feedback as in Dragan and Halanay (1987). It is proved that a white-noise perturbation will produce large effects unless it enters through the control channel.