Gain‐scheduled dynamic output feedback control for discrete‐time LPV systems

This paper presents synthesis conditions for the design of gain‐scheduled dynamic output feedback controllers for discrete‐time linear parameter‐varying systems. The state‐space matrix representation of the plant and of the controller can have a homogeneous polynomial dependency of arbitrary degree on the scheduling parameter. As an immediate extension, conditions for the synthesis of a multiobjective ??_∞ and ??₂ gain‐scheduled dynamic feedback controller are also provided. The scheduling parameters vary inside a polytope and are assumed to be a priori unknown, but measured in real‐time. If bounds on the rate of parameter variation are known, they can be taken into account, providing less conservative results. The geometric properties of the uncertainty domain are exploited to derive finite sets of linear matrix inequalities based on the existence of a homogeneous polynomially parameter‐dependent Lyapunov function. An application of the control design to a realistic engineering problem illustrates the benefits of the proposed approach. Copyright © 2011 John Wiley & Sons, Ltd.     

ℋ︁∞ and l2–l∞ filtering for two‐dimensional linear parameter‐varying systems

In this paper, the ??_∞ and l₂–l_∞ filtering problem is investigated for two‐dimensional (2‐D) discrete‐time linear parameter‐varying (LPV) systems. Based on the well‐known Fornasini–Marchesini local state‐space (FMLSS) model, the mathematical model of 2‐D systems under consideration is established by incorporating the parameter‐varying phenomenon. The purpose of the problem addressed is to design full‐order ??_∞ and l₂–l_∞ filters such that the filtering error dynamics is asymptotic stable and the prescribed noise attenuation levels in ??_∞ and l₂–l_∞ senses can be achieved, respectively. Sufficient conditions are derived for existence of such filters in terms of parameterized linear matrix inequalities (PLMIs), and the corresponding filter synthesis problem is then transformed into a convex optimization problem that can be efficiently solved by using standard software packages. A simulation example is exploited to demonstrate the usefulness and effectiveness of the proposed design method. Copyright © 2007 John Wiley & Sons, Ltd.     

ℋ︁2/ℋ︁∞ output information‐based disturbance attenuation for differential linear repetitive processes

Repetitive processes propagate information in two independent directions where the duration of one is finite. They pose control problems that cannot be solved by application of results for other classes of 2D systems. This paper develops controller design algorithms for differential linear processes, where information in one direction is governed by a matrix differential equation and in the other by a matrix discrete equation, in an ??₂/??_∞ setting. The objectives are stabilization and disturbance attenuation, and the controller used is actuated by the process output and hence the use of a state observer is avoided. Copyright © 2010 John Wiley & Sons, Ltd.     

Simultaneous ℋ︁2/ℋ︁∞ control of uncertain jump systems with functional time‐delays

This paper presents new results pertaining to the control design of a class of linear uncertain systems with Markovian jump parameters. An integral part of the system dynamics is a delayed state in which the time‐delays are mode dependent. The jumping parameters are modelled as a continuous‐time, discrete‐state Markov process and the uncertainties are norm‐bounded. We construct an appropriate Lyapunov–Krasovskii functional and design a simultaneous ℋ︁₂/ℋ︁_∞ controller which minimizes a quadratic ℋ︁₂ performance measure while satisfying a prescribed ℋ︁_∞‐norm bound on the closed‐loop system. It is established that sufficient conditions for the existence of the simultaneous ℋ︁₂/ℋ︁_∞ controller and the associated performance upper bound are cast in the form of linear matrix inequalities. Simulation results are provided and extension to the case where the jumping rates are subject to uncertainties is presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Mixed ℋ︁2/ℋ︁∞ nonlinear filtering

In this paper, we consider the mixed ??₂/??_∞ filtering problem for affine nonlinear systems. Sufficient conditions for the solvability of this problem with a finite‐dimensional filter are given in terms of a pair of coupled Hamilton–Jacobi–Isaacs equations (HJIEs). For linear systems, it is shown that these conditions reduce to a pair of coupled Riccati equations similar to the ones for the control case. Both the finite‐horizon and the infinite‐horizon problems are discussed. Simulation results are presented to show the usefulness of the scheme, and the results are generalized to include other classes of nonlinear systems. Copyright © 2008 John Wiley & Sons, Ltd.     

Correction to ‘Mixed ℋ︁2/ℋ︁∞ Nonlinear Filtering’

In Section 5 of Aliyu and Boukas (Int. J. Robust Nonlinear Control 2009; 19 :394–417), the authors have presented certainty‐equivalent filters for the mixed ??₂/??_∞ filtering problem for affine nonlinear systems. Sufficient conditions for the solvability of the problem with a finite‐dimensional filter are given in terms of a pair of coupled Hamilton–Jacobi–Isaacs equations (HJIEs). In this note, we supply a correction to these HJIEs. Moreover, for linear systems this correction is not necessary. Copyright © 2011 John Wiley & Sons, Ltd.     

A BMI approach for ℋ︁∞ gain scheduling of discrete time‐varying systems

The problem of gain‐scheduled state feedback control for discrete‐time linear systems with time‐varying parameters is considered in this paper. The time‐varying parameters are assumed to belong to the unit simplex and to have bounded rates of variation, which depend on the values of the parameters and can vary from slow to arbitrarily fast. An augmented state vector is defined to take into account possible time‐delayed inputs, allowing a simplified closed‐loop analysis by means of parameter‐dependent Lyapunov functions. A gain‐scheduled state feedback controller that minimizes an upper bound to the ??_∞ performance of the closed‐loop system is proposed. No grids in the parametric space are used. The design conditions are expressed in terms of bilinear matrix inequalities (BMIs) due to the use of extra variables introduced by the Finsler's lemma. By fixing some of the extra variables, the BMIs reduce to a convex optimization problem, providing an alternate semi‐definite programming algorithm to solve the problem. Robust controllers for time‐invariant uncertain parameters, as well as gain‐scheduled controllers for arbitrarily time‐varying parameters, can be obtained as particular cases of the proposed conditions. As illustrated by numerical examples, the extra variables in the BMIs can provide better results in terms of the closed‐loop ??_∞ performance. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust ℋ︁∞ filtering for uncertain Markovian jump linear systems

This paper investigates the problem of ??_∞ filtering for a class of uncertain Markovian jump linear systems. The uncertainty is assumed to be norm‐bounded and appears in all the matrices of the system state‐space model, including the coefficient matrices of the noise signals. It is also assumed that the jumping parameter is available. We develop a methodology for designing a Markovian jump linear filter that ensures a prescribed bound on the ??₂‐induced gain from the noise signals to the estimation error, irrespective of the uncertainty. The proposed design is given in terms of linear matrix inequalities. Copyright © 2002 John Wiley & Sons, Ltd.     

State–space solution to the ℋ︁−/ℋ︁∞ fault‐detection problem

In this paper we give an optimal state–space solution to the ??_?/??_∞ fault‐detection (FD) problem for linear time invariant dynamic systems. An optimal ??_?/??_∞ FD filter minimizes the sensitivity of the residual signal to disturbances while maintaining a minimum level of sensitivity to faults. We provide a state–space realization of the optimal filter in an observer form using the solution of a linear matrix inequalities optimization problem. We also show that, through the use of weighting filters, the detection performance can be enhanced and some assumptions can be removed. Two numerical examples are given to illustrate the algorithm. Copyright © 2011 John Wiley & Sons, Ltd.     

A new design of delay‐dependent robust ℋ︁∞ filtering for continuous‐time polytopic systems with time‐varying delay

This paper revisits the problem of delay‐dependent robust ?_∞ filtering design for a class of continuous‐time polytopic linear systems with a time‐varying state delay. Based on a newly developed parameter‐dependent Lyapunov–Krasovskii functional combined with Projection Lemma and an improved free‐weighting matrix technique for delay‐dependent criteria, a new sufficient condition for robust ?_∞ performance analysis is first derived and then the filter synthesis is developed by using a simple matrix inequality linearization technique. It is shown that the desired filters can be constructed by solving a set of linear matrix inequalities. Finally, two simulation examples are given to show the effectiveness and less conservatism of the proposed method in comparison with the existing approaches. Copyright © 2009 John Wiley & Sons, Ltd.     

ℋ︁∞ filtering for discrete‐time linear systems with Markovian jumping parameters†

This paper investigates the problem of ??_∞ filtering for discrete‐time linear systems with Markovian jumping parameters. It is assumed that the jumping parameter is available. This paper develops necessary and sufficient conditions for designing a discrete‐time Markovian jump linear filter which ensures a prescribed bound on the ?₂‐induced gain from the noise signals to the estimation error. The proposed filter design is given in terms of linear matrix inequalities. Copyright © 2003 John Wiley & Sons, Ltd.     

CLOSED FORMULAE FOR THE GENERALIZED ℋ︁∞ SENSITIVITY MINIMIZATION PROBLEM

The paper presents a complete solution for the multivariable, continuous-time Generalized ℋ︁_∞ (𝒢ℋ︁_∞) sensitivity minimization problem. In contrast with existing solutions, derived via polynomial methods, the state-space solution given here is essentially non-iterative. Closed formulae for the minimum and a particular optimal controller are derived in terms of a real Schur decomposition, the solution of two Lyapunov equations and a single, well-conditioned eigenvalue problem. © 1997 by John Wiley & Sons, Ltd.     

On ℋ︁∞ model reduction for discrete‐time linear time‐invariant systems using linear matrix inequalities

In this paper, we address the ??_∞ model reduction problem for linear time‐invariant discrete‐time systems. We revisit this problem by means of linear matrix inequality (LMI) approaches and first show a concise proof for the well‐known lower bounds on the approximation error, which is given in terms of the Hankel singular values of the system to be reduced. In addition, when we reduce the system order by the multiplicity of the smallest Hankel singular value, we show that the ??_∞ optimal reduced‐order model can readily be constructed via LMI optimization. These results can be regarded as complete counterparts of those recently obtained in the continuous‐time system setting.     

Gain‐scheduled H∞ filtering of parameter‐varying systems

This paper deals with the gain‐scheduled H_∞ filtering problem for a class of parameter‐varying systems. A sufficient condition for the existence of a gain‐scheduled filter, which guarantees the asymptotic stability with an H_∞ noise attenuation level bound for the filtering error system, is given in terms of a finite number of linear matrix inequalities (LMIs). The filter is designed to be parameter‐varying and have a nonlinear fractional transformation structure. A numerical example is presented to demonstrate the application of the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.     

ℒ︁2 gain analysis of linear systems with a single switching

For analysis and design of fault‐tolerant control systems, it is very important to evaluate the effects of failures, especially the transient responses caused by failures. This paper considers the ??₂ gain analysis of linear systems with a single switching, and gives necessary and sufficient conditions for the analysis. Also, the worst disturbance corresponding to the switching ??₂ gain is explicitly characterized. Copyright © 2010 John Wiley & Sons, Ltd.     

Robust delay‐dependent ℋ︁∞ control of time‐delay systems with state and input delays

This paper studies the design problem of robust delay‐dependent ??_∞ controller for a class of time‐delay control systems with time‐varying state and input delays, which are assumed to be noncoincident. The system is subject to norm‐bounded uncertainties and ??₂ disturbances. Based on the selection of an augmented form of Lyapunov–Krasovskii (L‐K) functional, first a Bounded Real Lemma (BRL) is obtained in terms of linear matrix inequalities (LMIs) such that the nominal, unforced time‐delay system is guaranteed to be globally asymptotically stable with minimum allowable disturbance attenuation level. Extending BRL, sufficient delay‐dependent criteria are developed for a stabilizing ??_∞ controller synthesis involving a matrix inequality for which a nonlinear optimization algorithm with LMIs is proposed to get feasible solution to the problem. Moreover, for the case of existence of norm‐bounded uncertainties, both the BRL and ??_∞ stabilization criteria are easily extended by employing a well‐known bounding technique. A plenty of numerical examples are given to illustrate the application of the proposed methodology of this note. The achieved numerical results on the maximum allowable delay bound and minimum allowable disturbance attenuation level are exhibited to be less conservative in comparison to those of existing methods in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

Gain‐scheduled leader‐follower tracking control for interconnected parameter varying systems

This paper considers the gain‐scheduled leader‐follower tracking control problem for a parameter varying complex interconnected system with directed communication topology and uncertain norm‐bounded coupling between the agents. A gain‐scheduled consensus‐type control protocol is proposed and a sufficient condition is obtained, which guarantees a suboptimal bound on the system tracking performance under this protocol. An interpolation technique is used to obtain a protocol schedule, which is continuous in the scheduling parameter. The effectiveness of the proposed method is demonstrated using a simulation example. Copyright © 2015 John Wiley & Sons, Ltd.     

Generalized nonlinear ℋ︁∞ synthesis condition with its numerically efficient solution

In this paper, we will first derive a general synthesis condition for the output‐feedback ??_∞ control of smooth nonlinear systems. Computationally efficient ??_∞ control design procedure for a subclass of smooth nonlinear systems with polynomial vector field is then proposed by converting the resulting Hamilton‐Jacobi‐Isaacs inequalities from rational forms to their equivalent polynomial forms. Using quadratic Lyapunov functions, both the state‐feedback and output‐feedback problems will be reformulated as semi‐definite optimization conditions and locally tractable solutions can be obtained through sum‐of‐squares (SOS) programming. The proposed nonlinear ??_∞ design approach achieves significant relaxations on the plant structure compared with existing results in the literature. Moreover, the SOS‐based solution algorithm provides an effective computational scheme to break the bottleneck in solving nonlinear ??_∞ and optimal control problems. The proposed nonlinear ??_∞ control approach has been applied to several examples to demonstrate its advantages over existing nonlinear control techniques and its usefulness to engineering problems. Copyright © 2010 John Wiley & Sons, Ltd.     

ℋ︁∞ sliding mode observers for uncertain nonlinear Lipschitz systems with fault estimation synthesis

This paper presents a scheme to design robust sliding mode observers(SMO) with ??_∞ performance for uncertain nonlinear Lipschitz systems where both faults and disturbances are considered. We study the necessary conditions to achieve insensitivity of the proposed sliding mode observer to the unknown input(fault). The objective is to derive a sufficient condition using linear matrix inequality(LMI) optimization for minimizing the ??_∞ gain between the estimation error and disturbances, while at the same time the design method guarantees that the solution of the LMI optimization satisfies the so‐called structural matching condition. The sliding motion affects only a part of the system through a novel reduced‐order sliding mode controller. Furthermore, the so‐called equivalent control concept is discussed for fault estimation. Finally, a numerical example of MCK chaos demonstrates the high performance of the results compared with a pure SMO. Copyright © 2010 John Wiley & Sons, Ltd.     

Gain‐scheduled filtering for linear parameter‐varying systems using inexact scheduling parameters with bounded variation rates

This paper is concerned with the problem of full‐order H₂ linear parameter‐varying filter design for continuous‐time systems with bounded rate of variations under the condition that the scheduling parameters do not exactly fit the real ones. The scheduling parameters and their derivatives are supposed to belong to polytopes with known vertices. The synthesis conditions are formulated in terms of parameter‐dependent bilinear matrix inequalities by means of parameter‐dependent Lyapunov function and introducing some extra variables for the filter design. An iterative procedure is presented to cast the bilinear matrix inequalities problem into a linear matrix inequality optimization problem. The design of robust filters for both time‐varying and time‐invariant systems can be viewed as particular cases of the proposed method. The merit of the method presented in this paper lies in two fields. The first pertains to dealing with the measurement uncertainty in a less conservative manner than available approaches in the gain‐scheduled filtering problem. The second is to provide more efficient methods than the existing ones in the literature for the robust filter design. Copyright © 2015 John Wiley & Sons, Ltd.