Andrzej W. Olbrot

Robust stability for time-delay systems: the edge theorem and graphical tests

The robust stability problem is discussed for a class of uncertain delay systems where the characteristic equations involve a polytope P of quasi-polynomials (i.e. polynomials in one complex variable and exponential powers of the variable). Given a set D in the complex plane, the goal is to find a constructive technique to verify whether all roots of every quasi-polynomial in P belong to D (that is, to verify the D-stability of P). First it is demonstrated by counterexample that Kharitonovs theorem does not hold for general delay systems. Next it is shown that under a mild assumption on the set D a polytope of quasi-polynomials is D-stable if and only if the edges of the polytope are D-stable. This extends the edge theorem for the D-stability of a polytope of polynomials. The third result gives a constructive graphical test for checking the D-stability of a polytope of quasi-polynomials which is especially simple when the set D is the open left-half plane. An application is given to demonstrate the power of the results. >

Introduction to the parametric approach to robust stability

Recent results relating to the parametric approach to robust stability are discussed. The general problem of robust stability is defined, and a brief review of Kharotinov-type, edge, and zero-exclusion results is given. Several open research directions are indicated, followed by some remarks on the potential of the approach.<<ETX>>

An overview of recent results on the parametric approach to robust stability

A discussion is presented of results relating to the parametric approach to robust stability. The general problem of robust stability is defined, and a review of Kharitonov-type and edge theorems, zero exclusion results, interval matrix stability, maximal perturbation bounds, multilinear and nonlinear perturbations, time delay systems, and numerical/graphical approaches is presented. Some open research directions are indicated, and the future of the approach is discussed.<<ETX>>

The edge theorem and graphical tests for robust stability of neutral time-delay systems

Abstract In this paper we consider the robust stability problem for a class of uncertain neutral time-delay systems where the characteristic equations involve a poly tope P of quasipoly neutrals of neutral type. Given a stability region D in the complex plane our goal is to find a constructive technique to verify whether all roots of every quasipolynomial in P belong to D (that is, to verify the D-stability of P). Our first result is that, under a mild assumption on the stability region D, P is D-stable if and only if the edges of P are D-stable. Hence, the D-stability problem of a higher dimensional polytope is reduced to the D-stability problem of a finite number of pairwise convex combinations of vertices. This extends the “Edge Theorem” developed by Bartlett, Hollot and Lin [BHL88] and Fu and Barmish [FB88] for the D-stability of a poly tope of polynomials and that of Fu, Olbrot and Polis [FOP88] for a poly tope of quasipolynomials. Our second result gives effective graphical tests for checking the D-stability of a polytope of quasipolynomials of neutral type when the set D is any open left half plane. The graphical test is based on the frequency response plots of some transfer functions associated with the vertices.

Robust stabilization: some extensions of the gain margin maximization problem

Considers robust stabilization for SISO systems with linear dependence on a real uncertain parameter which may occur either in the numerator or denominator of the plant transfer function. The authors show necessary and sufficient conditions for the existence of robust controllers. Their structure and parameterization can be obtained from the corresponding Nevanlinna-Pick interpolation problems. This work extends the previously known case of an uncertain gain parameter. >

Necessary and sufficient conditions for robust stability independent of delays and coefficient perturbations

Necessary and sufficient conditions for robust stability independent of delays for a class of delay systems with perturbed coefficients in the characteristic equation are given. Previously published results specified sufficient conditions only or necessary conditions only.

The pole placement theorem and small perturbations in parameters: the existence of an unassignable left half plane

It is proved that placing the poles of a linear time-invariant system arbitrarily far to the left of the imaginary axis is not possible if small perturbations in the model coefficients are taken into account. The results show a major difference between the behavior of nominal closed-loop system poles and the closed-loop system poles under small perturbations of the open-loop system parameters. For systems with uncertain parameters there exists an unassignable left half plane.<<ETX>>

Finite spectrum property and predictors

Abstract Three topics are discussed as related to predictors and finite spectrum property of time delay systems. First, the finite spectrum assignment (FSA) problem is greatly simplified by exploiting some particularly simple solutions to Bezout equations. Second topic is a new explicit formula for a solution to the robust control problem. The robust controller is a hybrid one. It contains an analog part solving a FSA problem and a digital part delivering minimum time deadbeat tracking with zero steady state error. The controller is robust against arbitrary input delay perturbations. Third is an application topic: Robustified Smith predictor for an Exhaust Gas Recirculation (EGR). It is shown that for this application the robustness constraints are very hard to satisfy if good nominal performance is expected. We show how to tune the Smith predictor to get better robustness and performance as available for a traditional PID.

Sufficient stability condition for a sampled-data system with digital controller

A quantitative stability condition for a sampled-data feedback system is formulated. The system is obtained from a stable continuous-time feedback system by digitalization of the controller. The condition shows how to determine a sampling frequency f/sub 0/ such that for all f larger than f/sub 0/ the discrete-time system remains stable. >

Robust parametrized controller design with an application to exhaust gas recirculation (EGR) system

An exhaust gas recirculation (EGR) control system for automotive application is briefly presented. Robustness is one of the main issues in this particular system due to nonlinearities, time delays, and the effects of complex exhaust gas dynamics. It was found that a good approximation of the system dynamics consists of a first order transfer function with a time delay where both transfer function coefficients and the time delay depend on the reference signal (desired EGR(%)). Moreover, it is sufficient to identify only four models corresponding to four different values of desired EGR (%) and assume interpolation for transfer function coefficients to characterize the system dynamics. Since the value of the reference signal is available in real time the controller design can be reduced to four controllers corresponding to four identified models and interpolation for coefficients of the controller transfer function in cases when the value of desired EGR (%) is between the assumed values. Simulation results for such designs are presented.