Robust Stability Of Linear Systems Research Articles

An Output-Feedback Design Approach for Robust Stabilization of Linear Systems With Uncertain Time-Delayed Dynamics in Sensors and Actuators

In this paper, we propose a control approach for the robust stabilization of linear time-invariant (LTI) systems with non-negligible sensor and actuator dynamics subject to time-delayed signals. Our proposition is based on obtaining an augmented model that encompasses the plant, sensor, and actuator dynamics and also the time-delay dynamic effect. We make use of the Padé Approximation for modeling the time-delay impact on the feedback loop. Since the actual plant state variables are not available for feedback, the sensor outputs, which represent a subset of the augmented system state variables, are used for composing a static output-feedback control law. The robust controller gains are computed by means of a two-stage strategy based on linear matrix inequalities (LMI). For obtaining less conservative conditions we consider the use of homogeneous-polynomial Lyapunov functions (HPLF)– and other decision variables– of arbitrary degree. In our proposition, we also take into account the inclusion of a minimum decay rate criterion in order to improve closed-loop system transient response. Disturbance rejection is also addressed through extensions to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathscr {H}_{2}$ </tex-math></inline-formula> guaranteed cost minimization. The effectiveness of the proposed strategy is attested in the design of a controller for the lateral axis dynamics of an aircraft and other academic examples.

Determination of the Area of Robust Stability of the System on the Basis of V. L. Kharitonov’s Theorem

The parameters of the object of regulation during operation due to various reasons may vary. These changes can lead to a change in the performance indicators of the automatic system, as well as its stability. This article proposes an approach to determine the range of acceptable values of the parameters of the control object of an automatic system with a PID controller, in which the system will remain stable. Thus, the problem arises of analyzing an automatic control system given not only by a single model with clearly defined parameters, but by a family of models belonging to a given set — the task of robust regulation. The search for ranges in which the parameters of the regulated object can change is based on the solution of the nonlinear programming problem in this paper. The conclusion of the objective function and constraint system using the theorem of V. L. Kharitonova on the robust stability of linear systems. The main idea is that each parameter of the regulatory object can be changed by some value hi1 in the direction of decrease and by hi2 — in the direction of increase. Replacing the notation used in the theorem of V. L. Kharitonov, the lower and upper boundaries of the change of parameters by the sum and difference of the nominal values of the parameters and the corresponding hi1, hi2, we get a system of restrictions. Moreover, for the stability of Kharitonov polynomials, it is most convenient to use the Lienar-Shipar criterion. The larger the values of hi1, hi2, the wider the ranges of variation of the parameters, and the smaller the inverse of the sum of these values. Based on this statement, the objective function is formed. It should be noted that the condition for the considered automatic system on which the proposed approach is based is sufficient, but not necessary, since the coefficients of the polynomial are interdependent. An example with the help of which the proposed approach is demonstrated is considered. This approach can also be applied to other linear systems for which theconditions of V. L. Kharitonova.

Determination of the area of robust stability of the system with a PID controller

An approach has been proposed to determine the area of change in the parameters of the controlled plant by the example of a system with PID controller. The search for ranges of parameter changes is based on the solution of the nonlinear programming problem, which is obtained as a conclusion of V.L. Kharitonov’s theorem on the robust stability of linear systems.

Dynamic predictor for systems with state and input delay: A time‐domain robust stability analysis

SummaryThe robust stability of linear systems with both state and input delay in closed loop with dynamic predictor‐based controller is analyzed. The problem of time‐varying matrix uncertainty is studied in the Lyapunov‐Krasovskii framework. The complete type functional with prescribed derivative expressed in terms of the delay Lyapunov matrix associated with the nominal system is a key piece of our analysis. The robust stability conditions depend on the delay Lyapunov matrix whose computation is carried out. An illustrative example is presented.

A Modeling Methodology for Robust Stability Analysis of Nonlinear Electrical Power Systems Under Parameter Uncertainties

This paper develops a modeling method for robust stability analysis of nonlinear electrical power systems over a range of operating points and under parameter uncertainties. Standard methods can guarantee stability under nominal conditions, but do not take into account any uncertainties of the model. In this study, stability is assessed by using structured singular value (SSV) analysis, also known as $\mu$ analysis. This method provides a measure of stability robustness of linear systems against all considered sources of structured uncertainties. The aim of this study is to apply the SSV method for robust small-signal analysis of nonlinear systems over a range of operating points and parameter variations. To that end, a modeling methodology is developed to represent any such system with an equivalent linear model that contains all system variability, in addition to being suitable for $\mu$ analysis. The method employs symbolic linearization around an arbitrary operating point. Furthermore, in order to reduce conservativeness in the stability assessment of the nonlinear system, the approach takes into account dependences of operating points on parameter variations. The methodology is verified through $\mu$ analysis of the equivalent linear model of a 4-kW permanent magnet machine drive, which successfully predicts the destabilizing torque over a range of different operating points and under parameter variations. Further, the predictions from $\mu$ analysis are validated against experimental results.

Necessary Conditions for Robust Stability of Linear Systems

In this paper we consider parameterized Lyapunov inequalities arising in robust stability analysis of linear systems with structured uncertainty. A necessary condition of their feasibility is presented. This condition is based on an optimization method developed by the author. Its properties include being reasonably easy to check, and its coverage being adjustable via the algorithm’s settings.

Chattering in sliding mode control systems with boundary layer approximation of discontinuous control

It has been a widely accepted notion that approximation of discontinuous control by certain continuous function in a boundary layer results in chattering elimination in sliding mode control systems. It is shown through three different types of analysis that in the presence of parasitic dynamics, this approach to chattering elimination would work only if the slope of the continuous nonlinear function within the boundary layer is low enough, which may result in the deterioration of performance of the system. A few examples are provided. An approach to robust stability of linear systems from the consideration of the saturating control is proposed.

Robust Stabilization of Linear Systems with Delayed State and Control

Robust stabilization of linear systems with delays on both the state and control input is studied in this paper. Using an improved Lyapunov-Krasovskii functional, we establish new criteria that ensure the robust stability of the closed-loop system with memoryless state feedback controls. The generalized conditions are derived in terms of linear matrix inequalities (LMIs), allowing us to compute simultaneously the two bounds that characterize the exponential stability rate of the solution and can be easily solved by numerical algorithms.

Robust Stability of Polytopic Systems via Affine Parameter-Dependent Lyapunov Functions

This paper studies robust stability of linear systems with polytopic uncertainty. New necessary and sufficient conditions for the existence of an affine parameter-dependent Lyapunov function assuring the Hurwitz or the Schur stability of a polytopic system are presented. These conditions are composed of a family of linear matrix inequality conditions of increasing precision. At each step, a set of linear matrix inequalities provides sufficient conditions for the existence of the affine parameter-dependent Lyapunov function, and necessity is asymptotically attained. Compared with the existing results in the literature, it is shown that the new stability conditions provide less conservative tests at each step. Numerical examples are given to illustrate the effectiveness of the new results.

Application of semi-definite programming to robust stability of delay systems

We study here robust stability of linear systems with several uncertain incommensurate delays, more precisely the property usually called delay-dependent stability. The main result of this paper consists in establishing that the latter is equivalent to the feasibility of some Linear Matrix Inequality (LMI), a convex optimization problem whose numerical solution is well documented. The method is based on two main techniques: • use of Padé approximation to transform the system into some singularly perturbed finite-dimensional system, for which robust dichotomy has to be checked; • recursive applications of Generalized Kalman–Yakubovich–Popov (KYP) lemma to characterise by an LMI the previous property.

Delay-dependent robust stability of linear systems with non-commensurate time varying delays

In this article we investigate the robust stability of uncertain systems with non-commensurate time varying delays. Two types of perturbations are considered. First, we consider time varying and unknown but norm bounded uncertainties. Next systems with polytopic-type uncertainties are studied. Based on Finsler's lemma we propose a new method for achieving less restrictive sufficient conditions for delay-dependent robust stability. The results are presented in terms of linear matrix inequalities. Three examples are given for illustration and comparison with previous results.

New delay-dependent stability criteria of uncertain linear systems with multiple time-varying state delays

Abstract This paper deals with the problem of delay-dependent robust stability for linear systems with time-varying structured uncertainties and multiple time-varying state delays. Based on Lyapunov stability theory combined with linear matrix inequalities (LMIs) techniques, some new delay-dependent stability criteria in terms of LMIs are derived by taking the relationship between the terms in the Leibniz–Newton formula into account. Since free weighting matrices are used to express this relationship and appropriate ones are selected by means of LMIs, the new improved criteria are much less conservative and more general. Numerical examples and simulation suggest that the results are effective and are an improvement over previous ones.

A two-operator approach to robust stabilization of linear systems on

Robust non-fragile feedback stabilization is studied for causal linear time-invariant (LTI) input–output systems defined in an L 2() signal space setup, or equivalently in an L 2(j) setup in the frequency domain. Two-operator descriptions for the plant and the controller are used with bounded operators defined on the full space L 2(). Basic properties of causal LTI operators on weighted L 2 spaces, L 2(, w), which are relevant to stabilization theory, are also established for different choices of the weighting function w. Weighted L 2 spaces allow the inclusion of persistent signals such as steps and thus provide the potential of important generalizations of the robust stabilization theory on L 2(). In particular, it is shown that there is a large class of weights w for which the L 2(, w) setup leads to H ∞ optimization. This implies that robust H ∞ design can be applied to a large class of weighted L 2 spaces on . Problems of closability of unstable causal LTI convolution operators are discussed. This strengthens the importance of the two-operator model Ay = Bu as the basic plant input–output model on the full time axis. It is discussed how to obtain such models with A and B bounded causal coprime operators defined on the full signal space.

Robust stabilization of linear systems with time-varying point delays via a delay-free dynamic controller

Robust stabilization of linear systems with time-varying point delays via a delay-free dynamic controller

Optimal NPID Stabilization of Linear Systems

We consider the problem of finding the optimal, robust stabilization of linear systems within a family of nonlinear feedback laws. Investigation of the efficiency of full-state based and partial-state based so-called NPID feedback schemes proposed for the stabilization of systems in robotic applications has provided the motivation for our work. We prove that, for a given quadratic Lyapunov function and a given family of nonlinear feedback laws, there exist optimal piecewise linear feedbacks that make the generalized Lyapunov derivative of the closed-loop system minimal. The result provides improved stabilization over the nonlinear stabilizing feedback law proposed in Ref. 1 as demonstrated in simulations of the Sarcos Dextrous Manipulator.

ROBUST STABILITY ANALYSIS OF LINEAR SYSTEMS WITH TIME-VARYING DELAYS

Robust stability of linear systems in presence of bounded uncertain time-varying time delays is studied. The time delay robustness problem is treated in the Integral Quadratic Constraint (IQC) framework. The stability criterion is formulated as frequency dependent linear matrix inequalities. The criterion can be equivalently formulated as a Semi-Definite Program (SDP) by applying Kalman-Yakubovich-Popov (KYP) lemma. Therefore, checking the criterion can be done efficiently by using various SDP solvers.

A Convex Approach to Robust Stability for Linear Systems with Uncertain Scalar Parameters

In this paper, robust stability for linear systems with several uncertain (complex and/or real) scalar parameters is studied. A countable family of conditions sufficient for robust stability is given, in terms of solvability of some simple linear matrix inequalities (LMIs). These conditions are of increasing precision, and it is shown conversely that robust stability implies solvability of these LMIs from a certain rank and beyond. This result constitutes an extension of the characterization by solvability of Lyapunov inequality of the asymptotic stability for usual linear systems. It is based on the search of parameter-dependent quadratic Lyapunov functions, polynomial of increasing degree in the parameters.

Robust control of state delayed systems with polytopic type uncertainties via parameter-dependent Lyapunov functionals

This paper considers the problem of robust stability and robust stabilization for linear systems with a constant time-delay in the state and subject to real convex polytopic uncertainty. First of all, for robust stability problem, we exploit new matrix inequalities characterization of delay-dependent quadratic stability results, demonstrate that it allows the use of parameter-dependent Lyapunov functionals, and develop control design methods based on linear matrix inequalities (LMIs) for solving the robust control problem. Next, the problem of determining the maximum time-delay under which the system remains stable is cast into a generalized eigenvalue problem and thus solved by LMI techniques. Finally, illustrative examples are given to demonstrate the advantage of these new representations.

LMI condition for robust stability of linear systems with both time‐varying elemental and norm‐bounded uncertainties

In this paper, the stability robustness of linear systems with both time‐varying elemental (structured) and norm‐bounded (unstructured) uncertainties is investigated. A sufficient condition is proposed in terms of linear matrix inequalities (LMIs) for ensuring that the linear systems with both time‐varying elemental and norm‐bounded uncertainties are asymptotically stable. Two examples are given to show that the proposed sufficient condition is less conservative than the existing one reported recently.

Robust stability functionals of state delayed systems with polytopic type uncertainties via parameterdependent Lyapunov functions

This paper considers the problem of robust stability for linear systems with a constant time-delay in the state and subject to real convex polytopic uncertainty. Both delay-independent and delay-dependent stability conditions are characterized by linear matrix inequalities which allow the use of parameter-dependent Lyapunov functionals to analyse the system stability against parameter perturbations, and therefore improve the conservativeness resulting from the single Lyapunov functional. In order to determine the maximum of time-delay within which the system remains stable, the problem can be cast into a generalized eigenvalue problem and solved by standard LMI solvers. Two examples are included to illustrate the proposed method.