Stability Of Uncertain Linear Systems Research Articles

Adaptive Stabilization of Uncertain Linear System With Stochastic Delay by PDE Full-State Feedback

Adaptive Stabilization of Uncertain Linear System With Stochastic Delay by PDE Full-State Feedback

Stabilization under unknown control directions and disturbed measurements via a time‐delay approach to extremum seeking

AbstractWe study stabilization of linear uncertain systems under unknown control directions using a bounded extremum seeking controller in the presence of a small time‐varying measurement delay. We assume that the measurements are subject to discontinuous disturbances. The main novelty is that these disturbances possess not only the constant part as in the existing results, but also small discontinuous part that may appear due to quantization. We consider two types of measurements: the state measurements and the state quadratic norm ones. In the latter case the constant part of the disturbances may be arbitrary large. By using the recently proposed time‐delay approach to Lie‐brackets‐based averaging, we transform the closed‐loop system to a time‐delay (neutral type) one with no terms depending on the disturbance derivative, which has a form of perturbed Lie brackets system. The input‐to‐state stability (ISS) of the time‐delay system guarantees the same for the original one. We further transform the neutral system to an ordinary differential equation (ODE) with delayed perturbations and employ variation of constants formula leading to explicit conditions in terms of simple inequalities with less conservative results in the most of numerical examples. Two numerical examples are provided to illustrate the efficiency of the method.

Lie-brackets-based averaging of affine systems via a time-delay approach

In this paper, we study input-to-state stability (ISS) of affine systems with a small parameter ɛ>0 and additive disturbances in the presence of state-delays. We present a time-delay approach to Lie-brackets-based averaging, where we transform the system to a time-delay (neutral type) one. The latter has a form of perturbed Lie brackets system. The ISS of the time-delay system guarantees the same for the original one. We present a direct Lyapunov–Krasovskii (L–K) method for the time-delay system and provide sufficient conditions for regional ISS. Further we apply the results to stabilization of linear uncertain systems under unknown control directions using the bounded extremum seeking controller with measurement delay. In contrast to the existing results that are all qualitative, we derive constructive linear matrix inequalities for finding quantitative upper bounds on ɛ and the time-delay that ensure regional ISS of the original system and on the resulting ultimate bound. Numerical examples illustrate the efficiency of our method.

A time-delay approach to stabilization by extremum seeking controller via simplified ISS analysis

We study stabilization of linear uncertain systems under unknown control directions using a bounded extremum seeking controller with a small parameter. We consider a small time-varying measurement delay. By using the recently proposed time-delay approach to Lie-Brackets-based averaging, we transform the closed-loop system to a time-delay (neutral type) one, which has a form of perturbed Lie brackets system. The input-to-state stability (ISS) of the time-delay system guarantees the same for the original one. Differently from the existing analysis via Lyapunov-Krasovskii (L-K) method, we transform the neutral system to an ordinary differential equation (ODE) with delayed perturbations and employ variation of constants formula that greatly simplifies the analysis and leads to simpler stability conditions. Two numerical examples illustrate that the proposed method allows essentially larger parameter uncertainties with bounds on the small parameter and time-delay that are not too small.

On the Separation Principle in Dynamic Output Controller Design for Uncertain Linear Systems

In this paper, the problem of designing the improper dynamic output controllers for uncertain linear multi-input/multi-output continuous-time systems is considered. In the principal part, the system state coordinate transform is outlined and matrix variable structures are found, defining the solution based on the structured linear matrix inequalities and linear matrix equality. Of principal novelty is the framework related to the system uncertainties with newly introduced inequality structures and their partial factorization. The main results are shown in detail by the example to characterize potential application of the method for quadratic stabilization of uncertain linear systems by the improper dynamic output controllers.

An LMI-Based Algorithm to Compute Robust Stabilizing Feedback Gains Directly as Optimization Variables

This article addresses the problem of robust stabilization of uncertain linear systems by static state- and output-feedback control laws. The uncertainty is supposed to belong to a polytope and both continuous- and discrete-time systems are investigated. Contrarily to the main stream of the linear matrix inequality (LMI)-based robust stabilization methods available in the literature, where the product between the Lyapunov (or slack variable) and control gain matrices is transformed in a new variable, this article proposes a change of paradigm, avoiding the standard change of variables and providing synthesis conditions that deal directly with the control gain as an optimization variable. The design procedure is formulated in terms of a locally convergent iterative algorithm based on LMIs, having as main novelties the following points: Both the Lyapunov and the closed-loop dynamic matrices appear affinely in the conditions; the iterative scheme involves the slack variables only, avoiding the classic alternation between the Lyapunov matrix and the control gain; an extra degree of freedom is created in terms of an additional scalar variable, which also appears affinely in the conditions and represents a scaling on the closed-loop dynamic matrix; a smart termination for the algorithm through a stability analysis condition. Numerical experiments based on exhaustive simulations show that all these features combined provide a robust stabilization technique capable to outperform the best available methods in the literature in terms of effectiveness, being specially suitable to address static output-feedback and decentralized control problems.

Predictor Feedback for Uncertain Linear Systems With Distributed Input Delays

While the existing literature on delay-adaptive control concentrates on uncertain plants with discrete input delays, this article proposes a predictor feedback for stabilization of uncertain linear systems with distributed input delays. A finite-dimensional linear system with distributed actuator delay may come with five types of uncertainties: unknown delay, unknown delay kernel, unknown plant parameters, unmeasurable finite-dimensional plant state, and unmeasurable infinite-dimensional actuator state. For different combinations of uncertainties, distinct control designs are developed in the article, from which the users can make selections to address a vast range of problems they face.

New delay-dependent robust exponential stability for uncertain linear systems with multiple non-differentiable time-varying delays and nonlinear perturbations

This paper studies delay-dependent robust stability analysis for uncertain linear systems with constant delay, time-varying delay and nonlinear perturbations. The restriction on the derivative of time-varying delay is removed, which means that the fast time-varying delays are allowed. Combined with Leibniz-Newton formula, integral inequalities, Wirtinger-based integral inequality, Peng-Park’s integral inequality, utilization of zero equation and new Lyapunov-Krasovskii functional have been adopted to study. New delay-dependent robust stability criteria for uncertain time-delay systems are established in terms of linear matrix inequalities (LMIs). Numerical examples and simulation suggest that the results given to illustrate the effectiveness and improvement over some existing methods.

Quantized feedback sliding-mode control: An event-triggered approach

This paper is concerned with the robust stabilization of quantized feedback sliding-mode control (SMC) for a class of uncertain linear systems via an event-triggered approach. By introducing an event-triggered mechanism, the event-triggered state instead of the state itself is quantized. And the generated finite event-triggered quantized state signal is sent through a digital network to the decoder and used for constructing a sliding mode controller to stabilize the uncertain linear systems. The relation among the lower bound of the quantized measurement saturating parameter, the upper bound of the event-triggered threshold parameter and the robust stabilization condition of linear uncertain systems, is first presented. Then, under the relation and by the combination of the proposed event-triggered mechanism for the system state and the discrete on-line adjustment policy for the quantizer sensitivity, the established quantized and event-triggered SMC laws are shown to ensure the reachability of the desired switching surface and global robust stabilization of uncertain linear systems is achieved. The theoretical result is verified via simulation illustration finally.

Combined event‐ and self‐triggered control approach with guaranteed finite‐gain stability for uncertain linear systems

A networked control strategy that maintains the finite-gain ℒ 2 stability for uncertain linear systems is proposed. The focus is placed on the reduction of the communication and energy usage that are required for the sampling and control signal updates. The approach differs from many existing approaches in that it computes the time instance of the sampling and the time instance of the control signal update separately; a self-trigger condition is used to determine the time instance of sampling, while an event-trigger condition is used at every time of sampling to decide if a new control signal should be transmitted to the actuator. This approach not only reduces the number of the control signal updates compared with the case of using only the standard self-triggered control, but also allows one to balance the cost associated with the sampling and the cost associated with the control signal update. The approach and its benefits are illustrated by simulations.

Constructive Necessary and Sufficient Condition for the Stability of Quasi-Periodic Linear Impulsive Systems

The technical note provides a computation-oriented necessary and sufficient condition for the global exponential stability of linear impulsive systems, whose impulsions are assumed to occur quasi-periodically. Based on the set-theoretic conditions for robust stability of uncertain linear systems, the existence of polyhedral Lyapunov functions is proved to be necessary and sufficient for global exponential stability of quasi-periodic linear impulsive systems. A constructive method is developed for testing the stability of the system and for computing set-induced polyhedral Lyapunov functions. The method leads to an algorithm whose complexity is similar to the standard algorithm related to discrete-time parametric uncertain systems with the state matrix belonging to a convex polytopic set.

Stabilization for linear uncertain systems with switched time-varying delays

This paper is concerned with the robust stabilization problem for a class of uncertain linear time-delay systems with large delay periods. The aim of this paper is to design a state feedback controller such that the closed-loop system is exponentially stable and preserves a desired performance in the presence of large delays. In order to deal with the large delay periods, the original system is converted into a switched delay system, in which the delays are time-varying. The delay-dependent condition is established in terms of matrix inequalities. The desired controllers can be solved by the cone complementary linearization method. An example is presented to illustrate the effectiveness of the proposed technique.

Polytopic uncertainty for linear systems: New and old complexity results

We survey the problem of deciding the stability or stabilizability of uncertain linear systems whose region of uncertainty is a polytope. This natural setting has applications in many fields of applied science, from control theory to systems engineering to biology. We focus on the algorithmic decidability of this property when one is given a particular polytope. This setting gives rise to several different algorithmic questions, depending on the nature of time (discrete/continuous), the property asked (stability/stabilizability), or the type of uncertainty (fixed/switching). Several of these questions have been answered in the literature in the last thirty years. We point out the ones that have remained open, and we answer all of them, except one which we raise as an open question. In all the cases, the results are negative in the sense that the questions are NP-hard. As a byproduct, we obtain complexity results for several other matrix problems in systems and control.

Novel Results for Global Exponential Stability of Uncertain Systems with Interval Time-varying Delay

This paper presents new results on delay-dependent global exponential stability for uncertain linear systems with interval time-varying delay. Based on Lyapunov-Krasovskii functional approach, some novel delay-dependent stability criteria are derived in terms of linear matrix inequalities (LMIs) involving the minimum and maximum delay bounds. By using delay-partitioning method and the lower bound lemma, less conservative results are obtained with fewer decision variables than the existing ones. Numerical examples are given to illustrate the usefulness and effectiveness of the proposed method.

Asymptotic stability and stabilisation of uncertain delta operator systems with time‐varying delays

This study focuses on the asymptotic stability and stabilisation of uncertain linear systems with time-varying delays via delta operator approach. By employing a new model formulation, the time-delayed delta operator system is transformed into an interconnected system for which the uncertainties can become easy to deal with. Based on a two-term approximation of delayed state and scaled small gain theorem, new delay-dependent sufficient conditions of robust asymptotic stability and state-feedback stabilisation of an uncertain delta operator time-delayed system are established by using a novel Lyapunov-Krasovskii functional. The criteria obtained unify some previously suggested relevant methods seen in literature for achieving asymptotic stability and stabilisation of both continuous and discrete systems into the delta operator framework. Numerical examples presented explicitly demonstrate the advantages and effectiveness of the proposed methods.

A new class of Lyapunov functions for the constrained stabilization of linear systems

The constrained stabilization of linear uncertain systems is investigated via the set-theoretic framework of control Lyapunov R-functions. A novel composition rule allows the design of a composite control Lyapunov function with external level set that exactly shapes the maximal controlled invariant set and inner sublevel sets arbitrarily close to any choice of smooth ones, generalizing both polyhedral and truncated ellipsoidal control Lyapunov functions. The feasibility test of the proposed smooth control Lyapunov functions can be cast into matrix inequalities conditions. The constrained linear quadratic control is addressed as an application.

Stabilization of Networked Control Systems with Uncertain Parameters

This paper addresses the problem of stabilizing linear continuous-time systems with uncertain parameters, where sensors, controllers and plants are connected by a digital communication channel. A necessary and sufficient condition for stabilization of linear uncertain systems is derived. The method to be proposed here relies on linear matrix inequalities. Simulation results show the validity of the proposed scheme.

Robust finite-time stabilisation of uncertain linear systems

This article deals with the problem of finite-time stability and stabilisation of uncertain linear systems. For linear time-varying systems subject to norm-bounded uncertainties some conditions for finite-time stability are provided. These conditions are expressed in terms of differential linear matrix inequalities. Then the problem of controller design is tackled, both for the state feedback and for the output feedback case; in both cases the controller can be found solving a suitable set of LMIs. A typical engineering case-study is included, to illustrate the applicability of the devised conditions.

LMI approach to exponential stability of linear systems with interval time-varying delays

This paper addresses exponential stability problem for a class of linear systems with time-varying delay. The time delay is assumed to be a continuous function belonging to a given interval, but not necessary to be differentiable. By constructing a set of augmented Lyapunov–Krasovskii functional combined with the Newton–Leibniz formula technique, new delay-dependent sufficient conditions for the exponential stability of the systems are first established in terms of linear matrix inequalities (LMIs). An application to exponential stability of uncertain linear systems with interval time-varying delay is given. Numerical examples are given to show the effectiveness of the obtained results.

Delay-dependent exponential stabilization for uncertain linear systems with interval non-differentiable time-varying delays

In this paper, the problem of exponential stabilization for a class of linear systems with time-varying delay is studied. The time delay is a continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, but the delay function is not necessary to be differentiable. Based on the construction of improved Lyapunov–Krasovskii functionals combined with Leibniz–Newton’s formula, new delay-dependent sufficient conditions for the exponential stabilization of the systems are first established in terms of LMIs. Numerical examples are given to demonstrate that the derived conditions are much less conservative than those given in the literature.