Erik I. Verriest

Stability of stochastic systems with uncertain time delays

Abstract For linear delay systems with bilinear noise sufficient conditions are given for the global asymptotic stochastic stability independent of the length of the delay(s). For linear stochastic noise terms, sufficient conditions for the existence of an invariant distribution, for all values of the delay are given. It is shown that the gaussian distribution is the unique invariant distribution. The covariance and correlation matrix function of the resulting stationary process are completely characterized by a Lyapunov-type equation. All these sufficient conditions are obtained in the form of the existence of some positive definite matrices satisfying certain Riccati-type equations.

Robust stability of delay-difference equations

Some issues in the stability of difference-delay in the linear and the nonlinear case are investigated. In particular, sufficient conditions are derived under which a system remains stable or unstable, independent of the length of the delay(s). Connections are made to certain Riccati equations and to singular perturbations of discrete maps.

Stability of Systems with Distributed Delays

Abstract For linear systems with distributed delays, sufficient conditions are given for the stability and asymptotic stability independent of the matrix valued weight function on the delayed state. All these sufficient conditions, obtained. via the Lyapunov-Krasovskii theory, are obtained in the form of the existence of some positive definite matrix functions satisfying certain Riccati-type differential equations. Connections are made with the theory of robust control, and its frequency domain criteria. New results on stabilizabilty with distributed feedback are derived.

Robust stability of time varying systems with unknown bounded delays

For linear time-varying delay-differential systems sufficient conditions are given for the stability and asymptotic stability independent of the length of the delay(s). Coefficients as well as delay times are assumed to be time-varing. All these sufficient conditions are obtained in the form of the existence of some positive definite matrix functions satisfying certain Riccati-type differential equations. Connections are made with the theory of robust control.<<ETX>>

Stabilization of deterministic and stochastic systems with uncertain time delays

Sufficient conditions for stability of deterministic and stochastic delay systems are exploited to derive stabilizability conditions under state and under output feedback for such systems. The existence proofs are constructive, and hence lead to the prescription of a whole class of stabilizing gains.<<ETX>>

Stability and stabilization of stochastic systems with distributed delays

For linear systems with distributed delays and bilinear stochastic perturbations, sufficient conditions are given for the stability and asymptotic stability independent of the matrix valued weight functions on the delayed state. These sufficient conditions, obtained via the Lyapunov-Krasovskii theory, revolve around the existence of some positive definite matrix functions satisfying certain Riccati-type differential equations. Connections are made with the theory of robust control and its frequency domain criteria. New results on stabilizability with distributed feedback are derived.

On the sensitivity of generalized state-space systems

The synthesis of minimum-sensitivity state-space realizations of linear time-invariant systems is a well understood problem. Such realizations have been linked to balanced realizations. The theory is extended to the synthesis of minimum-sensitivity generalized state-space models for singular linear systems. A scalar sensitivity measure is defined, and the minimization of the measure over all admissible realizations is considered. Since minimal realizations are not required to be related by a similarity transformation, the optimization problem is more complex. A criterion is given for determining optimal sensitivity structures. In the nonsingular case, the criterion reduces to the familiar result. The simple example of the right-shift operator is considered.<<ETX>>

Periodic Systems Realization Theory with applications

Abstract A general framework based on operators on sequences is presented. It is shown that it is very suitable for the description of periodic system. With it two liftings are derived which enable to represent a periodic system by a time invariant one. Conections are also made to the notion of operational transfer functions, and more generally the operational transfer inclusion. The realization of periodic systems from their impulse response data is given. Finally, this framework is shown to be advantageous in the computation of robust periodic controllers for time-invariant systems.

A Hankel matrix approach to singular system realization theory

A Hankel matrix approach to singular system realization theory is presented. The focus is exclusively on discrete-time or descriptor systems. Based on definitions of reachability and observability, the notion of a system Hankel matrix is first defined. The system Hankel matrix is then shown to have reachability-observability matrix factorizations which can be used to solve the realization synthesis problem. The notion of a balanced realization for singular systems is derived using the singular value decomposition of the system Hankel matrix. The theory is applied to synthesizing realizations which have minimum parametric sensitivity properties. It is demonstrated that such realizations are related to the notion of balancing.<<ETX>>

Optimal control for maximal accuracy with an arbitrary control space metric

The finite dimensional theory of minimal sensitivity design is extended to infinite dimensions. The high accuracy control of the state vector of a system is a practical application of this problem. First the discrete time high accuracy control problem is solved for a single input system with fixed bound on the relative error of the control. The optimal steering is characterized as one that is zero for the longest possible time. The continuous time problem is solved via the maximum principle and the example of the rocket car with relative control error is solved in detail. The maximum accuracy and the accuracy/time problem have a solution of bang-zero-bang type. The accuracy/energy problem also exhibits a coasting period.