Sufficient Conditions For Robust Stability Research Articles

Kharitonov Theorem and Small-μ Methodology

In this paper, a necessary and sufficient robust stability condition for interval matrices will be established in terms of the structured singular value μ. The robust stability of interval polynomials will be then considered by choosing a special interval matrix and then recastting this problem into the standard μ-setting. The resulting μ-problem is of rank one. Hence, the μ function has an analytical expression. It is then shown that max μ < 1 if and only if the four Kharitonov polynomials are stable.

Robust stability analysis for linear uncertain time‐delay systems with delay‐dependence

Abstract This paper proposes a sufficient robust stability condition for uncertain time‐delay systems with delay‐dependence. The results obtained include information on the delay time and is less conservative than delay‐independent results which have been published when delays are small. Examples are given to illustrate the application of our results.

Comments, with reply, on "Optimal rejection of persistent disturbances, robust stability, and mixed sensitivity minimization" by M. A. Dahleh and J. B. Pearson

The commenters point out that the sufficient condition for robust stability given in the above-titled work (see ibid., vol.33, no.8, p.722-731, Aug. 1988) should be modified. In their reply, the authors agree with the commenters and describe the assumptions which led to the omission. >

Computation of stability radius for families of bivariate polynomials

In this paper we apply optimization techniques to the problem of robust stability of a family of bivariate polynomials under affine coefficient perturbations. The size of the perturbations is measured by a convex function. In this paper we concentrate onlp weighted norms, for three special casesp = 1,p = 2, andp = ∞. Necessary and sufficient conditions for robust stability are provided. Evaluation of the stability radius is reduced to a minimization problem in two-dimensional space. The results open the door to the development and implementation of reliable and efficient algorithms for the computation of the stability radius.

Robustness of sampled-data control systems having parametric uncertainty: a conic sector approach

The robustness of sampled-data control systems when the open-loop system is modeled by a family of plants having affine or interval uncertainty structure is considered. Using the conic sector approach as adapted to sampled-data systems, the authors give a sufficient condition for robust stability that depends only on edge parameters in the case of affine uncertainty and on vertex parameters when interval uncertainty is assumed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Necessary and sufficient conditions for robust stability of discrete systems with coefficients depending continuously on two interval parameters

Let a real polynomial in a complex variable, whose coefficients are any given continuous functions of two real interval parameters, be given. Necessary and sufficient conditions are derived for the polynomial to have all its zeros outside (or inside) the unit circle of the complex variable plane. In a multi-polynomial dependence of the coefficients on parameters, the conditions involve, in addition to a finite number of algebraic steps, only resolving single variable fixed coefficients real polynomials.

Robust Hurwitz l/sup p/ stability of polynomials with complex coefficients

Necessary and sufficient conditions for robust stability of a family of polynomials with complex coefficients are provided. This work generalizes a frequency-domain criteria for robust stability of continuous linear systems reported on by Y.Z. Tsypkin and B.T. Polyak (1991). >

Robustness optimization of discrete multivariate systems with non-linear time-varying unmodelled dynamics

Abstract Stability robustness in discrete-time multivariable feedback systems subject to noises and non-linear time-varying unmodelled dynamics is discussed. Based on the technique of functional analysis and the control algorithm advanced by Youla et al., we derive a necessary and sufficient condition for robust stabilization as well as a closed-form solution to the problems of synthesizing optimal controllers that maximize the excess robustness to tolerate the unmodelled dynamics and noises. A necessary and sufficient condition for the solvability of such a robust stabilization problem is also introduced by means of Nevanlinna-Pick interpolation theory. In addition, an all-pass solution for the minimum H ∞-norm problem is employed to treat our problems. Furthermore, neither the plant nor the controller has to be restricted to be stable or minimum-phase.

A necessary and sufficient condition for robust asymptotic stability of time-variant discrete systems

A necessary and sufficient condition for the stability of time-variant interval matrices is presented. This condition allows stability to be tested by checking only products of vertex (extreme) matrices. The implementation of the test in the form of an algorithm and two illustrative examples are provided. >

Robust stability of discrete multivariable systems subject to non-linear tune-varying unmodelled dynamics

Abstract Stability robustness in discrete-time multivariable feedback systems subject to noises and non-linear time-varying unmodelled dynamics is discussed. Based on the technique of functional analysis and the control algorithm advanced by Youla et al., we derive a necessary and sufficient condition for robust stabilization to tolerate the unmodelled dynamics and noises. The properties of the Schur function (Class S) is employed to synthesize the robust stabilizer. Furthermore, a necessary and sufficient condition for the solvability of such a robust stabilization problem is also presented by means of the Nevanlinna-Pick interpolation theory. In addition, neither the plant nor the controller has to be restricted to be stable or minimum-phase

Application of a New Robust Stability Margin

The aim of this paper is to derive a new robust stability margin. Known sufficient conditions for robust stability stated in gap-metric sense contain inherent conservativeness in the formulation of the various steps. In this paper conservativeness in one of the steps is removed, resulting in a new robustness margin. The key issue is that more information of the specific controller is taken into consideration. The resulting robustness margin is less conservative than the margin in directed gap-metric sense and is as easy to computer. The advantage of this robustness margin will be illustrated by an example.

Robust eigenstructure assignment with structured state-space uncertainty and unmodelled dynamics

New time domain stability robustness results have been obtained for a non-strictly proper linear time-invariant plant which is subject to simultaneous time-varying structured state-space uncertainly and norm bounded unmodelled dynamics. Sufficient conditions for robust stability are obtained both for a constant gain output feedback controller and an output feedback dynamic compensator. The robustness conditions are explicitly in terms of the closed-loop eigenvalues and eigenvectors so that these new results are especially well suited for a robust eigenstructure assignment design method. An example is presented of a robust eigenstructure assignment autopilot using dynamic compensation for the yaw/roll dynamics of a bank-to-turn missile.

How near is a stable polynomial to an unstable polynomial?

Frequency response techniques are applied to the problem of robust Hurwitz stability of a family of polynomials with complex coefficients. The distance between two polynomials is measured by a weighted l/sup p/ norm, 0>p<or= Omega . Necessary and sufficient conditions for robust stability, as well as formulas for the stability radius and a minimal norm destabilizing polynomial are provided. This work is motivated by and follows the spirit of a result reported by Y.Z. Tsypkin and B.T. Polyak (see IEEE Trans. on Autom. Control, vol.36, p.1464-9, 1991).<<ETX>>

Sensitivity and robust stability of some control systems containing a nonlinear plant

Sufficient conditions for insensitivity and robust stability of some frequently used control systems which contain a nonlinear plant and are not of a feedback type are presented. These conditions are derived from some results of the author (1991) on general input-output systems. They are stated in terms of the transmission operators that describe the blocks constituting the system.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Robust eigenstructure assignment with structured state space uncertainty

Recent sufficient conditions for robust stability and robust performance of linear time-invariant systems subject to structured state-space uncertainty are utilized to obtain a robust eigenstructure assignment design method. This new approach optimizes either the sufficient condition for stability or performance robustness while constraining the dominant eigenvalues to lie within chosen regions in the complex plane. This constrained optimization problem is solved by using the sequential unconstrained minimization technique with a quadratic extended interior penalty function. The use of constraints on certain eigenvector entries and the effect of these constraints on robustness and nominal performance are considered. Conservatism of the robustness conditions is reduced by simultaneously introducing a similarity transformation, a positive real diagonal weighting, and a unitary weighting into the design procedure. An example that illustrates the design of a robust eigenstructure assignment controller for a pitch pointing/vertical translation maneuver of the AFTIF-16 aircraft is presented.

Model matching systems: necessary and sufficient conditions for stability under plant perturbations

Necessary and sufficient conditions of robust stability are derived for multivariate perfect model matching systems under external disturbances and plant perturbations (non-linear, linear feedback perturbations and fractional perturbations). The derived results also provide an intuitive insight into controller effects on robustness. A general parametrized 2-D controller is introduced to stabilize the nominal plant.

Performance robustness of discrete-time systems with structured uncertainty

Given an interconnection of a nominal discrete-time plant and a stabilizing controller together with structured, norm-bounded, nonlinear/time-varying perturbations, necessary and sufficient conditions for robust stability and performance of the system are provided. It is shown that performance robustness is equivalent to stability robustness in the sense that both problems can be dealt with in the framework of a general stability robustness problem. The resulting stability robustness problem is shown to be equivalent to a simple algebraic one, the solution of which provides the desired necessary and sufficient conditions for performance/stability robustness. These conditions provide an effective tool for robustness analysis and can be applied to a large class of problems. In particular, it is shown that some known results can be obtained immediately as special cases of these conditions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Robust stability of control systems with polytopical uncertainty: a Nyquist approach

For technical plants, a bounded continuum of parameter values describes admissible process models. A robust controller guarantees the observance of minimal requirements for all of the allowed process models. In this paper a frequency domain approach is proposed that permits the instability of a family of polynomials to be checked. This method can be extended in an easy manner to investigate the Instability of a closed loop. The plant is thereby described by a family of models with a polytopical region of uncertainty in the parameter space. As an important special case of the general D-stability, the Hurwitz stability of the closed loop is investigated and a minimal set of necessary and sufficient conditions for robust Hurwitz stability is given. A generalization for other stability domains is possible. Besides an algebraic criterion based on a zero inclusion check for a minimal set of exposed edges of the polytopical region, a graphic criterion of Nyquist type can also be used.

In Search of Weaker Conditions to Insure Robust Stability in Linear Feedback Systems

This paper is concerned with the well-known condition for robust stability of systems with plant uncertainty. It is shown that the usual sufficient condition for robust stability, given in terms of the maximum modulus of the plant approximation error, can be relaxed to a sector condition. This sector condition, related to the phase of the uncertain portion of the plant, can increase the range of the allowed variations in the parameters of the uncertain plant sufficient for robust stability.

Robust LQG optimal controller synthesis against noise spectral uncertainties and nonlinear time-varying unmodeled dynamics

In this paper, we consider the problem of robust controller synthesis against noise spectral uncertainties and nonlinear time-varying (NLTV) unmodeled dynamics in the discrete LQG (linear-quadratic Gaussian) optimal systems. We derive a necessary and sufficient condition for robust stabilization in multivariable stochastic discrete-time systems with NLTV unmodeled dynamics. Meanwhile, an algorithm based on the robust stabilization criterion is presented for synthesizing a robust controller not only to minimize the least favorable cost functional J but also to satisfy the robust stabilization criterion by specifying an appropriate weighting scalar in the cost functional.