New robustness bounds for discrete systems with randomperturbations

Some new stochastic stability robustness bounds for state-space models are reported. Nominally exponentially stable discrete-time systems are assumed to be subject to random parameter perturbations and novel bounds are obtained on the maximum variances of these random perturbations to maintain stability robustness. The methods employed in the analysis are the existing ones used in the deterministic framework after transforming the stochastic robustness problem to a deterministic one. The results are compared with each other and with the exact stability region in an example     

Robust tracking with unknown upper bounds on the perturbations and measurement noise

We consider the robust tracking problem for a discrete time control object with unknown upper bounds on the perturbations. The transition function in the nominal model of the controllable object is assumed to be known, while upper bounds on the external perturbation, measurement noise, and operator perturbations with respect to both output and control are assumed to be unknown or rough. Current estimates of upper bounds on the perturbations are obtained from measurement data with the quality criterion of the tracking problem as the identification criterion.     

Bounds on the solution of the algebraic Riccati equation under perturbations in the coefficients

This paper is concerned with bounds on symmetric solutions of the continuous algebraic matrix Riccati equation under perturbations on its coefficients. Special emphasis is given to the question of estimating the ‘size’ of the perturbations on the stabilizing solution of the Riccati equation when one or all its coefficients are subject to small perturbations. Upper bounds on the norm of the perturbations on the stabilizing solution are presented. Moreover, it is shown that these perturbations can be determined as a single-valued continuous function of the perturbations on the coefficient of the Riccati equation.     

线性状态空间模型的稳定鲁棒性改善

本文讨论线性定常系统的稳定鲁棒性问题。文中给出了线性定常系统的稳定鲁棒性范围的新改善界限。在结构扰动情况下,本文给出的稳定鲁棒性界限比引文[1]中的相应结果更好。用数值计算方法容易得到这一动扰界限。文中给出的两个例子证实了本文结果的优点。     

线性状态空间模型的稳定鲁棒性改善

本文讨论线性定常系统的稳定鲁棒性问题。文中给出了线性定常系统的稳定鲁棒性范围的新改善界限。在结构扰动情况下，本文给出的稳定鲁棒性界限比引文[1]中的相应结果更好。用数值计算方法容易得到这一动扰界限。文中给出的两个例子证实了本文结果的优点。     

Quantitative measures of robustness for systems including delayedperturbations

Asymptotically stable linear systems subject to delayed time-varying and nonlinear perturbations are considered. Razumikhin-type theorems are used to obtain easy-to-compute bounds on the perturbations so that the systems remain stable. Results indicate that if delayed perturbations are included, then the bound is reduced as compared to the one for nondelayed perturbations. However, in certain cases previously obtained bounds for the nondelayed perturbations guarantee stability even when delayed perturbations are in effect     

ℓ1 robust performance of discrete-time systems with structured uncertainty

This paper addresses the robust performance problem when a linear time-invariant system is subjected to both norm bounded time-varying uncertainty and worst-case external bounded input. The tightest upper bounds for steady-state robust performance measures over classes of finite memory and fading memory perturbations are computed in the regulation and tracking problems. Since the classes of finite memory and fading memory perturbations are inappropriate to the purposes of model validation and adaptive control, approximating subclasses of bounded memory perturbations and perturbations with exponentially decreasing impulse responses are considered. It is shown that the robust performance bounds obtained are nonconservative in the case of bounded memory perturbations. The worst-case steady-state robust performance measures for SISO system under coprime factor perturbations are computed for illustration.     

Reduced conservatism in stability robustness bounds by state transformation

This note addresses the issue of "conservatism" in the time domain stability robustness bounds obtained by the Lyapunov approach. A state transformation is employed to improve the upper bounds on the linear time-varying perturbation of an asymptotically stable linear time-invariant system for robust stability. This improvement is due to the variance of the conservatism of the Lyapunov approach with respect to the basis of the vector space in which the Lyapunov function is constructed. Improved bounds are obtained, using a transformation, on elemental and vector norms of perturbations (i.e., structured perturbations) as well as on a matrix norm of perturbations (i.e., unstructured perturbations). For the case of a diagonal transformation, an algorithm is proposed to find the "optimal" transformation. Several examples are presented to illustrate the proposed analysis.     

On the analysis of eigenvalue assignment robustness

The robust eigenvalue assignment of systems subject to parameter perturbations is addressed. Based on some essential properties of induced norms and matrix measures, the authors derive some sufficient conditions which ensure the assignment of the system's eigenvalues in the specified region irrespective of the system perturbations. The robustness bounds for eigenvalue assignment are obtained without the need to solve the Lyapunov equation. An example is given to illustrate the effectiveness and ease of the proposed analysis methods     

Improved upper bounds for the mixed structured singular value

In this paper, we take a new look at the mixed structured singular value problem, a problem of finding important applications in robust stability analysis. Several new upper bounds are proposed using a very simple approach which we call the multiplier approach. These new bounds are convex and computable by using linear matrix inequality (LMI) techniques. We show, most importantly, that these upper bounds are actually lower bounds of a well-known upper bound which involves the so-called D-scaling (for complex perturbations) and G-scaling (for real perturbations)     

Control-oriented model validation and errors quantification in the /spl lscr//sub 1/ setup

A priori information required for robust synthesis includes a nominal model and a model of uncertainty. The latter is typically in the form of additive exogenous disturbance and plant perturbations with assumed bounds. If these bounds are unknown or too conservative, they have to be estimated from measurement data. In this paper, the problem of errors quantification is considered in the framework of the /spl lscr//sub 1/ optimal robust control theory associated with the /spl lscr//sub /spl infin// signal space. The optimal errors quantification is to find errors bounds that are not falsified by measurement data and provide the minimum value of a given control criterion. For model with unstructured uncertainty entering the system in a linear fractional manner, the optimal errors quantification is reduced to quadratic fractional programming. For system under coprime factor perturbations, the optimal errors quantification is reduced to linear fractional programming.     

Exact bounds for robust stability of output feedback controlled fractional‐order systems with single parameter perturbations

This article investigates exact robust stability bounds of output feedback controlled fractional‐order systems with the commensurate order and single parameter perturbations in all system coefficient matrices. First, a sufficient and necessary condition for robust asymptotical stability of such systems is obtained by using the Kronecker product. Then the maximal upper bounds and minimum lower bounds for robust asymptotical stability are established, respectively, without conservatism by transforming such problems into checking whether the matrix with single parameter perturbations is nonsingular or not. Finally, two numerical examples are given to show the effectiveness of the proposed results.     

An algorithm for the robust exponential stabilization of a class of switched systems

In this paper, we consider continuous‐time switched systems whose subsystems are linear, or, more generally, homogeneous of degree one. For that class of systems, we present a control algorithm that under certain conditions generates switching signals that globally exponentially stabilizes the switched system, even in the case in which there are model uncertainties and/or measurement errors, provided that the bounds of that uncertainties and errors depend linearly on the norm of the state of the system and are small enough in a suitable sense. We also show that in the case in which the measurement errors and the model uncertainties are bounded, the algorithm globally exponentially stabilizes the system in a practical sense, with a final error which depends linearly on the bounds of both the model uncertainties and the measurement errors. In other words, the closed‐loop system is exponentially input‐to‐state‐stable if one considers the perturbations and output measurements bounds as inputs. For switched linear systems, under mild observability conditions, we design an observer whose state‐estimation drives the control algorithm to exponentially stabilize the system in absence of perturbations and to stabilize it in an ultimately bounded way when the perturbations and the output measurement errors are bounded. Finally, we illustrate the behavior of the algorithm by means of simulations. Copyright © 2014 John Wiley & Sons, Ltd.     

Perturbation bounds for root-clustering of linear discrete-time systems

Sufficient conditions are obtained for the eigenvalues of a discrete-time system to remain in a specified subregion in the unit circle with the centre at the origin in the complex plane in the presence of perturbations. The perturbations are allowed to be non-linear and time varying. Bounds are determined for linear state-space models using the Lyapunov theory. The bounds are derived for both highly and weakly structured perturbations. The results of the structured perturbations are then extended to interval matrices.     

Robustness of linear quadratic state feedback designs in the presence of system uncertainty

The well-known stabilizing property of linear quadratic state feedback (LQSF) design is used to obtain a quantitative measure of the robustness of LQSF designs in the presence of perturbations. Bounds are obtained for allowable nonlinear, time-varying perturbations such that the resulting closed-loop system remains stable. The special case of linear, time-invariant perturbations is also treated. The bounds are expressed in terms of the weighting matrices in a quadratic performance index and the corresponding positive definite solution of the algebraic matrix Riccati equation, and are easy to compute for any given LQSF design. A relationship is established between the perturbation bounds and the dominant eigenvalues of the closed-loop optimal system model. Some interesting asymptotic properties of the bounds are also discussed. An autopilot for the flare control of the Augmentor Wing Jet STOL Research Aircraft (AWJSRA) is designed, based on LQSF theory, and the results presented in this paper. The variation of the perturbation bounds to changes in the weighting matrices in the LQSF design is studied by computer simulations, and appropriate weighting matrices are chosen to obtain a reasonable bound for perturbations in the system matrix and at the same time meet the practical constraints for the flare maneuver of the AWJSRA. Results from the computer simulation of a satisfactory autopilot design for the flare control of the AWJSRA are presented.     

Robustness of discrete-time systems for unstructured stochasticperturbations

Several stochastic stability robustness measures are presented for nominally exponentially stable linear discrete-time systems with unstructured perturbations having second-moment bounds. Dependence of these measures on the stability degree of the nominal system and other parameters used in the procedure is illustrated. By using the time evolution of the second moment of the system state and stochastic Lyapunov stability results (positive super-martingale convergence theorems), the ability of nominally exponentially stable systems to maintain stability in the presence of unstructured stochastic (linear and nonlinear) perturbations is demonstrated. Quantitative results are given to determine the maximum modeling uncertainty which can be tolerated in design. Upper bounds on the second moments of stochastic perturbations to maintain the mean-square and almost sure stability of these systems in the presence of unstructured perturbations are obtained     

The stability robustness determination of state space models with real unstructured perturbations

This paper considers the robust stability of a linear time-invariant state space model subject to real parameter perturbations. The problem is to find the distance of a given stable matrix from the set of unstable matrices. A new method, based on the properties of the Kronecker sum and two other composite matrices, is developed to study this problem; this new method makes it possible to distinguish real perturbations from complex ones. Although a procedure to find the exact value of the distance is still not available, some explicit lower bounds on the distance are obtained. The bounds are applicable only for the case of real plant perturbations, and are easy to compute numerically; if the matrix is large in size, an iterative procedure is given to compute the bounds. Various examples including a 46th-order spacecraft system are given to illustrate the results obtained. The examples show that the new bounds obtained can have an arbitrary degree of improvement over previously reported ones. This work has been supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A4396.     

On robust online scheduling algorithms

While standard parallel machine scheduling is concerned with good assignments of jobs to machines, we aim to understand how the quality of an assignment is affected if the jobs’ processing times are perturbed and therefore turn out to be longer (or shorter) than declared. We focus on online scheduling with perturbations occurring at any time, such as in railway systems when trains are late. For a variety of conditions on the severity of perturbations, we present bounds on the worst case ratio of two makespans. For the first makespan, we let the online algorithm assign jobs to machines, based on the non-perturbed processing times. We compute the makespan by replacing each job’s processing time with its perturbed version while still sticking to the computed assignment. The second is an optimal offline solution for the perturbed processing times. The deviation of this ratio from the competitive ratio of the online algorithm tells us about the “price of perturbations”. We analyze this setting for Graham’s algorithm, and among other bounds show a competitive ratio of 2 for perturbations decreasing the processing time of a job arbitrarily, and a competitive ratio of less than 2.5 for perturbations doubling the processing time of a job. We complement these results by providing lower bounds for any online algorithm in this setting. Finally, we propose a risk-aware online algorithm tailored for the possible bounded increase of the processing time of one job, and we show that this algorithm can be worse than Graham’s algorithm in some cases.     

Robustness of pole-assignment in a specified region

An analysis of pole-assignment is given for systems under linear time-invariant perturbations. Based upon the Lyapunov approach, new techniques to calculate allowable element bounds for highly structured perturbations are presented. Under these allowable perturbations both stability robustness and a certain performance robustness are thus ensured. Examples are given to illustrate the proposed methods     

Stability robustness bounds for linear state-space models with structured uncertainty

In this note, we consider the robust stability analysis problem in linear state-space models. We consider systems with structured uncertainty. Some lower bounds on allowable perturbations which maintain the stability of a nominally stable system are derived. These bounds are shown to be less conservative than the existing ones.