Rama K. Yedavalli

Perturbation bounds for robust stability in linear state space models

Abstract This paper addresses the aspect of stability robustness of linear systems in the time domain. Upper bounds on the linear perturbation of an asymptotically stable linear system are obtained to maintain stability, both for structured as well as unstructured perturbations using the Lyapunov approach. For structured perturbation the resulting bound is such that it garners the structural information about the nominal (as well as the perturbation) matrix into a single unified expression. In the case of unstructured perturbations, special features of the nominally stable matrix are exploited resulting in simpler expressions for the bound (without the need to solve the Lyapunov equation). Improvement of the proposed measures is illustrated with the help of examples.

Robust stability bounds on time-varying perturbations for state-space models of linear discrete-time systems

The stability robustness of linear discrete-time systems in the time domain is addressed using the Lyapunov approach. Bounds on linear time-varying perturbations that maintain the stability of an asymptotically stable linear time-invariant discrete-time nominal system are obtained for both structured and unstructured independent perturbations. Bounds are also derived assuming that various elements of the system matrix are perturbed dependently. The result for the structured perturbation case is extended to the stability analysis of interval matrices.

Robust root clustering for linear uncertain systems using generalized Lyapunov theory

In this paper, the problem of matrix root clustering in sub-regions of complex plane for linear state space models with real parameter uncertainty is considered. The nominal matrix root clustering theory of Gutman and Jury (1981, IEEE Trans. Aut. Control, AC-26, 403) using Generalized Lyapunov Equation is extended to the perturbed matrix case and bounds are derived on the perturbation to maintain root clustering inside a given region. The theory allows us to get an explicit relationship between the parameters of the root clustering region and the uncertainty range of the parameter space. The current literature available on perturbation bounds for robust stability becomes a special case of this unified theory.

Stability analysis of interval matrices: another sufficient condition

A sufficient condition for the stability of interval matrices is presented, based on a Lyapunov approach. The condition, while requiring the solution of a Lyapunov matrix equation, removes the restrictions imposed by Heinen ∥1984). Examples given illustrate the improvement of the proposed condition over the ones given by Heinen and Bialas (1983) and Daoyi (1985).

Time-domain stability robustness measures for linear regulators

The stability robustness aspect of linear systems is analyzed in the time domain. A bound on the perturbation of an asymptotically stable linear system is obtained to maintain stability using Liapunov matrix equation solution. The resulting bound is shown to be an improved upper bound over the ones recently reported in the literature. The proposed methodology is then extended to Linear Quadratic (LQ) and Linear Quadratic Gaussian (LQG) regulators. Examples given include comparison with an aircraft control problem previously analyzed.

Robust stabilizability for linear systems with both parameter variation and unstructured uncertainty

This paper considers the problem of finding one compensator which stabilizes a plant having both parameter variation and high frequency unstructured uncertainty. A new uncertain system model is proposed to characterize plants containing those two different uncertainties. Closed loop stability criterion is then derived for the new model. To obtain the desired results, number of assumptions describing the set of allowable plants are imposed. The satisfaction of these assumptions guarantees the existence of a proper stabie compensator C(s) achieving robust stabilization. The algorithm used for controller design has some most attractive features: Namely, it is recursive in nature and allows the designer to select one compensator coefficient at a time. Also, the construction of the compensator depends on the difference between the orders of numerator and denominator polynomials of the nominal plant and often leads to a lower order compensator.

Design of a LQR controller of reduced inputs for multiple spacecraft formation flying

Regarding multiple spacecraft formation flying, the observation is made that control thrust need only be applied coplanar to the local horizon to achieve complete controllability of a two-satellite formation. Without the need for zenith-nadir (radial) thrust, simplifications and reduction of the weight of the propulsion system may be accomplished. The authors focus on the validation of this radial-excluding control system on its own merits, and in comparison to a related system which does provide thrust parallel to the orbital radius. Simulations are performed using commercial ODE solvers to propagate the Keplerian dynamics of a controlled satellite, relative to an uncontrolled, leader satellite. The conclusion is drawn that, despite the exclusion of the radial thrust axis, the remaining control thrust available still provides enough control to design a gain matrix of adequate performance using linear-quadratic regulator (LQR) techniques.

Flight control application of new stability robustness bounds for linear uncertain systems

This paper addresses the issue of obtaining bounds on the real parameter perturbations of a linear state-space model for robust stability. Based on Kronecker algebra, new, easily computable sufficient bounds are derived that are much less conservative than the existing bounds since the technique is meant for only real parameter perturbations (in contrast to specializing complex variation case to real parameter case). The proposed theory is illustrated with application to several flight control examples.

Stability of matrix second-order systems: new conditions and perspectives

Modeling of many dynamic systems results in matrix second-order differential equations. In the paper, the stability issues of matrix second-order dynamical systems are discussed. In the literature, only sufficient conditions of stability and/or instability for a system in matrix second-order form are available. In this paper, necessary and sufficient conditions of asymptotic stability for time-invariant systems in matrix second-order form under different types of dynamic loadings (conservative/nonconservative) are derived and a physical interpretation is carried out. The stability conditions in the sense of Lyapunov (the jw-axis behavior of eigenvalues) are also analyzed. As the conditions are gained directly in terms of physical parameters of the system, the effect of different loadings on the system stability is made transparent by dealing with the stability issues directly in matrix second-order form.

Design of Distributed Engine Control Systems for Stability Under Communication Packet Dropouts

In this paper, we address the issue of the stability of distributed engine control systems under communication constraints and, in particular, for packet dropouts. We propose a control design procedure, labeled decentralized distributedfull-authority digitalenginecontrol andbasedonatwo-leveldecentralized controlframework. Weshow that the packet dropping margin, which is a measure of stability robustness under packet dropouts, is largely dependent on the closed-loop controller structure and that, in particular, a block-diagonal structure is more desirable. Thus, we design a controller in a decentralized framework to improve the packet dropping margin. The effectofdifferentmathematicalpartitioningonthepacketdroppingmarginisstudied.Theproposedmethodologyis applied to an F100 gas turbine engine model, which clearly demonstrates the usefulness of decentralization in improving the stability of distributed control under packet dropouts.