Robust filter design for piecewise discrete‐time systems with time‐varying delays

A novel delay‐dependent filtering design approach is developed for a class of linear piecewise discrete‐time systems with convex‐bounded parametric uncertainties and time‐varying delays. The time‐delays appear in the state as well as the output and measurement channels. The filter has a linear full‐order structure and guarantees the desired estimation accuracy over the entire uncertainty polytope. The desired accuracy is assessed in terms of either ??_∞‐performance or ??₂–??_∞ criteria. A new parametrization procedure based on a combined Finsler's Lemma and piecewise Lyapunov–Krasovskii functional is established to yield sufficient conditions for delay‐dependent filter feasibility. The filter gains are determined by solving a convex optimization problem over linear matrix inequalities. In comparison to the existing design methods, the developed methodology yields the least conservative measures since all previous overdesign limitations are almost eliminated. By means of simulation examples, the advantages of the developed technique are readily demonstrated. Copyright © 2009 John Wiley & Sons, Ltd.     

Disturbance attenuation by output feedback for linear systems subject to actuator saturation

In this paper, we study the problem of disturbance attenuation by output feedback for linear systems subject to actuator saturation. A nonlinear output feedback, expressed in the form of a quasi‐linear parameter‐varying system with state‐dependent scheduling parameter, is constructed that leads to the attenuation of the effect of the disturbance on the output of the system. The level of disturbance attenuation is measured in terms of the restricted ℒ︁₂ gain and the restricted ℒ︁₂–ℒ︁_∞ gain over a class of bounded disturbances. Copyright © 2008 John Wiley & Sons, Ltd.     

Effectiveness and limitation of circle criterion for LTI robust control systems with control input nonlinearities of sector type

This paper considers linear time invariant systems with sector type nonlinearities and proposes regional ??₂ performance analysis and synthesis methods based on the circle criterion. In particular, we consider the effect of non‐zero initial states and/or an ??₂ disturbance inputs on the ??₂ norm of a selected performance output. We show that both analysis and synthesis problems can be recast as linear matrix inequality (LMI) optimization problems, where, for synthesis, the outputs of the nonlinear elements are assumed available for control. Moreover, it is shown when the circle criterion does or does not help to improve the performance bound in robust control synthesis when compared with the existing linear analysis method. Copyright © 2005 John Wiley & Sons, Ltd.     

Performance analysis of reset control systems

In this paper we present a general linear matrix inequality‐based analysis method to determine the performance of a SISO reset control system in both the ??₂ gain and ??₂ sense. In particular, we derive convex optimization problems in terms of LMIs to compute an upperbound on the ??₂ gain performance and the ??₂ norm, using dissipativity theory with piecewise quadratic Lyapunov functions. The results are applicable to for all LTI plants and linear‐based reset controllers, thereby generalizing the available results in the literature. Furthermore, we provide simple though convincing examples to illustrate the accuracy of our proposed ??₂ gain and ??₂ norm calculations and show that, for an input constrained ??₂ problem, reset control can outperform a linear controller designed by a common nonlinear optimization method. Copyright © 2009 John Wiley & Sons, Ltd.     

Nonlinear generalization of scaled ℋ︁∞ control: global robustification against nonlinearly bounded uncertainties

This paper proposes a novel approach to the problem of ??₂ disturbance attenuation with global stability for nonlinear uncertain systems by placing great emphasis on seamless integration of linear and nonlinear controllers. This paper develops a new concept of state‐dependent scaling adapted to dynamic uncertainties and nonlinear‐gain bounded uncertainties that do not necessarily have finite linear‐gain, which is a key advance from previous scaling techniques. The proposed formulation of designing global nonlinear controllers is not only a natural extension of linear robust control, but also the approach renders the nonlinear controller identical with the linear control at the equilibrium. This paper particularly focuses on scaled ??^∞ control which is widely accepted as a powerful methodology in linear robust control, and extends it nonlinearly. If the nonlinear system belongs to a generalized class of triangular systems allowing for unmodelled dynamics, the effect of the disturbance can be attenuated to an arbitrarily small level with global asymptotic stability by partial‐state feedback control. A procedure of designing such controllers is described in the form of recursive selection of state‐dependent scaling factors. Copyright © 2004 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.     

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.     

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.     

Robust ℋ︁∞ PID control for multivariable networked control systems with disturbance/noise attenuation

In this paper, we study the design problem of PID controllers for networked control systems (NCSs) with polyhedral uncertainties. The load disturbance and measurement noise are both taken into account in the modeling to better reflect the practical scenario. By using a novel technique, the design problem of PID controllers is converted into a design problem of output feedback controllers. Our goal of this paper is two‐fold: (1) To design the robust PID tracking controllers for practical models; (2) To develop the robust ??_∞ PID control such that load and reference disturbances can be attenuated with a prescribed level. Sufficient conditions are derived by employing advanced techniques for achieving delay dependence. The proposed controller can be readily designed based on iterative suboptimal algorithms. Finally, four examples are presented to show the effectiveness of the proposed methods. Copyright © 2011 John Wiley & Sons, Ltd.     

ℒ︁2‐Gain analysis and control of uncertain nonlinear systems with bounded disturbance inputs

This paper proposes a convex approach to regional stability and ℒ︁₂‐gain analysis and control synthesis for a class of nonlinear systems subject to bounded disturbance signals, where the system matrices are allowed to be rational functions of the state and uncertain parameters. To derive sufficient conditions for analysing input‐to‐output properties, we consider polynomial Lyapunov functions of the state and uncertain parameters (assumed to be bounded) and a differential‐algebraic representation of the nonlinear system. The analysis conditions are written in terms of linear matrix inequalities determining a bound on the ℒ︁₂‐gain of the input‐to‐output operator for a class of (bounded) admissible disturbance signals. Through a suitable parametrization involving the Lyapunov and control matrices, we also propose a linear (full‐order) output feedback controller with a guaranteed bound on the ℒ︁₂‐gain. Numerical examples are used to illustrate the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

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.     

An augmented system approach to static output‐feedback stabilization with ℋ︁∞ performance for continuous‐time plants

This paper revisits the static output‐feedback stabilization problem of continuous‐time linear systems from a novel perspective. The closed‐loop system is represented in an augmented form, which facilitates the parametrization of the controller matrix. Then, new equivalent characterizations on stability and ??_∞ performance of the closed‐loop system are established in terms of matrix inequalities. On the basis of these characterizations, a necessary and sufficient condition with slack matrices for output‐feedback stabilizability is proposed, and an iteration algorithm is given to solve the condition. An extension to output‐feedback ??_∞ control is provided as well. The effectiveness and merits of the proposed approach are shown through several examples. Copyright © 2008 John Wiley & Sons, Ltd.     

Robust flow control in data‐communication networks with multiple time‐delays

Robust controller design for a flow control problem where uncertain multiple time‐varying time‐delays exist is considered. Although primarily data‐communication networks are considered, the presented approach can also be applied to other flow control problems and can even be extended to other control problems where uncertain multiple time‐varying time‐delays exist. Besides robustness, tracking and fairness requirements are also considered. To solve this problem, an ??^∞ optimization problem is set up and solved. Unlike previous approaches, where only a suboptimal solution could be found, the present approach allows to design an optimal controller. Simulation studies are carried out in order to illustrate the time‐domain performance of the designed controllers. The obtained results are also compared to the results of a suboptimal controller obtained by an earlier approach. Copyright © 2009 John Wiley & Sons, Ltd.     

State‐feedback disturbance attenuation for polytopic LPV systems with input saturation

This paper proposes a state‐feedback control law for linear parameter‐varying (LPV) systems with input saturation and disturbances. The proposed control law employs two control parts: a main control part for reducing the restricted ??₂ gain from the mismatched disturbance to the controlled output and an extra control part for eliminating the matched disturbance. Owing to this feature, the proposed control law provides a better disturbance attenuation performance than the conventional control law that deals with a unified disturbance regardless of the presence of matched and mismatched disturbances. Further, considering different forms of the feedback gain matrix K(θ(t)) and the Lyapunov function V(x(t)), three types of controllers are proposed. For each type, set invariance and the restricted ??₂ gain performance conditions are first formulated in terms of parameterized linear matrix inequalities (PLMIs) and then converted into linear matrix inequalities (LMIs) by using a parameter relaxation technique. Results from the simulation of numerical examples confirm the effectiveness of the proposed controllers. Copyright © 2009 John Wiley & Sons, Ltd.     

ℒ︁2‐Stabilization of continuous‐time linear systems with saturating actuators

This paper addresses the problem of controlling a linear system subject to actuator saturations and to ??₂‐bounded disturbances. Linear matrix inequality (LMI) conditions are proposed to design a state feedback gain in order to satisfy the closed‐loop input‐to‐state stability (ISS) and the closed‐loop finite gain ??₂ stability. By considering a quadratic candidate Lyapunov function, two particular tools are used to derive the LMI conditions: a modified sector condition, which encompasses the classical sector‐nonlinearity condition considered in some previous works, and Finsler's Lemma, which allows to derive stabilization conditions which are adapted to treat multiple objective control optimization problems in a potentially less conservative framework. Copyright © 2006 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.     

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.     

Gain‐scheduled ℋ︁2 controller synthesis for linear parameter varying systems via parameter‐dependent Lyapunov functions

This paper deals with the problem of gain‐scheduled ??₂ control for linear parameter‐varying systems. The system state–space model matrices are affinely parameterized and the admissible values of the parameters and their rate of variation are supposed to belong to a given convex bounded polyhedral domain. Based on a parameter‐dependent Lyapunov function, a linear matrix inequality methodology is proposed for designing a gain‐scheduled state feedback ??₂ controller, where the feedback gain is a matrix fraction of polynomial matrices with quadratic dependence on the scheduling parameters. Copyright © 2005 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.