Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions

This paper deals with robust stability analysis of linear state space systems affected by time-varying uncertainties with bounded variation rate. A new class of parameter-dependent Lyapunov functions is introduced, whose main feature is that the dependence on the uncertain parameters and the state variables are both expressed as polynomial homogeneous forms. This class of Lyapunov functions generalizes those successfully employed in the special cases of unbounded variation rates and time-invariant perturbations. The main result of the paper is a sufficient condition to determine the sought Lyapunov function, which amounts to solving an LMI feasibility problem, derived via a suitable parameterization of polynomial homogeneous forms. Moreover, lower bounds on the maximum variation rate for which robust stability of the system is preserved, are shown to be computable in terms of generalized eigenvalue problems. Numerical examples are provided to illustrate how the proposed approach compares with other techniques available in the literature.     

Robust H-infinity filter for a class of polytopic Delta operator systems

This study deals with the robust H-infinity filtering for a class of Delta operator systems with polytopic uncertainties. By the aid of introducing two slack matrices to eliminate the coupling between systems matrices and Lyapunov matrices, an improved version of the bounded real lemma is given via linear matrix inequality formulation, which shows a close correspondence between the continuous-and discrete-time H-infinity performance criterion. Based on it, the existence condition of the desired filter is obtained such that the corresponding filtering error system is asymptotically stable with a guaranteed performance index. A numerical example is employed to illustrate the feasibility and advantages of the proposed design.     

Robust H-infmity estimation for continuous-time polytopic uncertain systems

1IntroductionAs a fundamental problem in control area,linearestimation has been extensively investigated since Kalmanpresented the optimal filteringtheory[1].It is well knownthat the Kalmanfilteringtheoryis based onthe assumptionsthat the accurate mathematical models of the consideredsystems and the statistics features of the noise inputs areknown.Inthis setting,the Kalmanfilteringis also called H2filtering,which minimizes the H2norm of the operatorfromthe external input to the estimation er…     

Robust H-infinity estimation for continuous-time polytopic uncertain systems

The design of full_order robust H_infinity estimators is investigated for continuous_time polytopic uncertain systems. The main purpose is to obtain a stable and proper linear estimator such that the estimation error system remains robustly stable with a prescribed H_infinity attenuation level. Based on a recently proposed H_infinity performance criterion which exhibits a kind of decoupling between the Lyapunov matrix and the system dynamic matrices, a sufficient condition for the existence of the robust estimator is provided in terms of linear matrix inequalities. It is shown that the proposed design strategy allows the use of parameter_dependent Lyapunov functions and hence it is less conservative than earlier results. A numerical example is employed to illustrate the feasibility and advantage of the proposed design.     

An ellipsoidal off-line MPC scheme for uncertain polytopic discrete-time systems

An off-line Model Predictive Control (MPC) method based on ellipsoidal calculus and viability theory is described in order to address control problems in the presence of state and input constraints for uncertain polytopic linear plants subject to persistent disturbances. In order to reduce the computational burdens and conservativeness of traditional polytopic MPC schemes, the present approach carries out off-line most of the computations and it makes use of closed-loop predictions to improve the control performance. This is done by recursively pre-computing suitable ellipsoidal inner approximations of the exact controllable sets and solving on-line a simple and numerically low-demanding optimization problem subject to a set-membership constraint. Comparisons with three other recent off-line MPC approaches are also provided in the final example.     

Robust analysis and control for a class of uncertain nonlinear discrete-time systems

This paper deals with the problem of robust analysis and control of a class of nonlinear discrete-time systems with (constant) uncertain parameters. For the analysis problem we use a polynomial Lyapunov function and we generalize, for nonlinear systems, the “extended stability” notion proposed by Oliveira et al. (1999) in the context of linear discrete-time uncertain systems. As a result, we propose an LMI optimization problem to maximize an estimate of the domain of attraction, and also extend this approach to the synthesis problem by considering parameter-dependent Lyapunov functions and nonlinear multipliers. Numerical examples illustrate the approach and show its potential for solving analysis and control problems of nonlinear discrete-time systems.     

Robust fault detection of uncertain linear systems via quasi-LMIs

A novel H_∞ deconvolution filter design method is discussed for fault detection of uncertain polytopic linear systems subject to unknown inputs. The enhancement of fault sensitivity is characterized in terms of an H_- index. By means of the Projection Lemma and Congruence Transformations, a quasi-convex LMI formulation of the design problem is obtained. The effectiveness of the filter is illustrated via a numerical example.     

Robust control of constrained systems via convex optimization

A successful controller design paradigm must take into account both model uncertainty and design specifications. Model uncertainty can be successfully addressed using ??^∞ robust control theory. However, this framework cannot directly accommodate the realistic case where in addition to robustness considerations the system is subject to both time- and frequency-domain specifications, such as bounds on the control action. In this paper we propose a design procedure, based upon the use of convex optimization, that takes explicitly into account both time- and frequency-domain specifications. The main result of the paper is a new framework to address problems having both control and output constraints and model uncertainty. Additionally, the paper serves as a brief tutorial on the issues involved in addressing design problems with multiple design specifications via convex optimization.     

Nonlinear robust H-infinity filtering for a class of uncertain systems via convex optimization

A new approach for robust H-infinity filtering for a class of Lipschitz nonlinear systems with time-varying uncertainties both in the linear and nonlinear parts of the system is proposed in an LMI framework. The admissible Lipschitz constant of the system and the disturbance attenuation level are maximized simultaneously through convex multi-objective optimization. The resulting H-infinity filter guarantees asymptotic stability of the estimation error dynamics with exponential convergence and is robust against nonlinear additive uncertainty and time-varying parametric uncertainties. Explicit bounds on the nonlinear uncertainty are derived based on norm-wise and element-wise robustness analysis.     

Sliding mode control for polytopic differential inclusion systems

The stabilization problem of polytopic differential inclusion (PDI) systems is investigated by using sliding mode control. Sliding surface is designed and sufficient conditions for asymptotic stability of sliding mode dynamics are derived. A novel feedback law is established to make the state of system reach the sliding surface in a finite time. Finally, an example is given to illustrate the validity of the proposed design.     

Simplified adaptive control via improved robust positive real conditions

A new method is derived for embedding plants in a robust state-feedback scheme, to achieve strictly positive realness of the resulting augmented plants. A state-feedback gain is derived that guarantees the strictly positive realness of the closed-loop in presence of polytopic type, possibly time-varying, parameter uncertainties in the model that describes the plant. This is achieved by assigning different Lyapunov functions to each of the vertices of the uncertainty polytope. The obtained feedback gain is used to apply existing methods for robust simplified adaptive control on systems with possibly time-varying polytopic uncertainties.     

Robust output-feedback control of linear discrete-time systems

The problem of designing H_∞ dynamic output-feedback controllers for linear discrete-time systems with polytopic type parameter uncertainties is considered. Given a transfer function matrix of a system with uncertain real parameters that reside in some known ranges, an appropriate, not necessarily minimal, state-space model of the system is described which permits reconstruction of all its states via the delayed inputs and outputs of the plant. The resulting model incorporates the uncertain parameters of the transfer function matrix in the state-space matrices. A recently developed linear parameter-dependent LMI approach to state-feedback H_∞ control of uncertain polytopic systems is then used to design a robust output-feedback controllers that are of order comparable to the one of the plant. These controllers ensure the stability and guarantee a prescribed performance level within the uncertainty polytope.     

电梯群控系统的一种鲁棒离散优化调度策略

针对电梯群控调度过程中交通不确定的问题,提出一种电梯群控调度的鲁棒离散优化方法.将交通流作为不确定参数,考虑当前时刻和下一时刻交通流状况,将下一时刻交通流预测的不准确性作为不确定因素,建立电梯群控调度的鲁棒离散优化模型.针对该模型的特点,进一步提出了解决方法.仿真结果表明,该方法在电梯群控调度性能和交通模式的适应性方面得到较大改善.     

On the non-conservatism of a novel LMI relaxation for robust analysis of polytopic systems

It is known that robust analysis of polytopic systems, i.e. linear system with polytopic state space uncertainty, is a difficult problem. Recently we have proposed for this problem, in both continuous-time and discrete-time cases, the use of homogeneous polynomially parameter-dependent quadratic Lyapunov functions (HPD-QLFs) by introducing a novel LMI relaxation which provides a sufficient condition via a convex optimization without requiring the use of auxiliary multipliers of unknown degree. In this paper we show that this condition is not only sufficient but it is also necessary.     

Robust performance analysis and synthesis of linear polytopic discrete-time periodic systems via LMIs

A particular class of uncertain linear discrete-time periodic systems is considered. The problem of robust stabilization of real polytopic linear discrete-time periodic systems via a periodic state-feedback control law is tackled here, along with performance optimization. Using additional slack variables and the periodic Lyapunov lemma, an extended sufficient condition of robust stabilization is proposed. Based on periodic parameter-dependent Lyapunov functions, this last condition is shown to be always less conservative than the more classic one based on the quadratic stability framework. This is illustrated on a numerical example from the literature.     

Robust exponential stability and stabilization of linear uncertain polytopic time-delay systems

This paper proposes new sufficient conditions for the exponential stability and stabilization of linear uncertain polytopic time-delay systems. The conditions for exponential stability are expressed in terms of Kharitonov-type linear matrix inequalities (LMIs) and we develop control design methods based on LMIs for solving stabilization problem. Our method consists of a combination of the LMI approach and the use of parameter-dependent Lyapunov functionals, which allows to compute simultaneously the two bounds that characterize the exponetial stability rate of the solution. Numerical examples illustrating the conditions are given.     

Robust exponential stability and stabilization of linear uncertain polytopic time-delay systems

This paper proposes new sufficient conditions for the exponential stability and stabilization.of linear uncertain polytopic time-delay systems. The conditions for exponential stability are expressed in terms of Kharitonov-type linear matrix inequalities （LMIs） and we develop control design methods based on LMIs for solving stabilization problem. Our method consists of a combination of the LMI approach and the use of parameter-dependent Lyapunov functionals, which allows to compute simultaneously the two bounds that characterize the exponetial stability rate of the solution. Numerical examples illustrating the conditions are given.     

Robust V stabilization of singular systems with polytopic uncertainties

The robust D stabilization problem is considered for singular systems with polytopic uncertainties in this paper. Both the derivative matrix E and the state matrix A are with uncertainties, which were not considered before. First, with the introduction of some free matrices, a necessary and sufficient condition for the singular system to be D stable is proposed, based on which, the robust D stable problem is solved, and a sufficient condition for the closed system to be robust D stabilizable is obtained. The desired state feedback controller is given in an explicit expression. Numerical examples show the efficiency of the proposed approach.     

Robust D stabilization of singular systems with polytopic uncertainties

The robust D stabilization problem is considered for singular systems with polytopic uncertainties in this paper. Both the derivative matrix E and the state matrix A are with uncertainties, which were not considered before. First, with the introduction of some free matrices, a necessary and sufficient condition for the singular system to be D stable is proposed, based on which, the robust D stable problem is solved, and a sufficient condition for the closed system to be robust D stabilizable is obtained. The desired state feedback controller is given in an explicit expression. Numerical examples show the efficiency of the proposed approach.     

Hankel norm of polytopic systems with time-varying delay

This paper addresses the Hankel norm of polytopic systems with time-varying delay. Delay-dependent sufficient conditions are presented such that the polytopic system is stable and the Hankel norm of the polytopic system is less than a prescribed scalar in terms of linear matrix inequalities. A numerical example is presented.