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.     

ℋ︁∞‐filtering for singularly perturbed nonlinear systems

In this paper, we consider the ??_∞‐filtering problem for singularly perturbed (two time‐scale) nonlinear systems. Two types of filters are discussed, namely, (i) decomposition and (ii) aggregate, and sufficient conditions for the solvability of the problem in terms of Hamilton–Jacobi–Isaac's equations (HJIEs) are presented. Reduced‐order filters are also derived in each case, and the results are specialized to linear systems, in which case the HJIEs reduce to a system of linear‐matrix‐inequalities (LMIs). Based on the linearization of the nonlinear models, upper bounds ε^* of the singular parameter ε that guarantee the asymptotic stability of the nonlinear filters can also be obtained. The mixed ??₂/??_∞‐filtering problem is also discussed. Copyright © 2010 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.     

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.     

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.     

Discrete‐time mixed ℋ︁2/ℋ︁∞ nonlinear filtering

In this paper, we consider the discrete‐time mixed ??₂/??_∞ filtering problem for affine nonlinear systems. Necessary and sufficient conditions for the solvability of this problem with a finite‐dimensional filter are given in terms of a pair of coupled discrete‐time Hamilton–Jacobi‐Isaac's equations (DHJIE) with some side‐conditions. For linear systems, it is shown that these conditions reduce to a pair of coupled discrete‐time algebraic‐Riccati‐equations (DAREs) or a system of linear matrix inequalities (LMIs) similar to the ones for the control case. Both the finite‐horizon and infinite‐horizon problems are discussed. Moreover, sufficient conditions for approximate solvability of the problem are also derived. These solutions are especially useful for computational purposes, considering the difficulty of solving the coupled DHJIEs. An example is also presented to demonstrate the approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimal solutions to a two-block ℋ︁∞ problem

In this paper a generator of all optimal solutions to two-block ??_∞ problems is derived. The derivation is in frequency domain and hinges on geometrical arguments. The results involve noncanonical Wiener-Hopf factorization of rational matrices.     

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.     

2‐DOF discrete‐time nonlinear ℋ︁∞‐filters

In this paper, a new theory of two‐degrees‐of‐freedom (2‐DOF)‐??_∞ and certainty‐equivalent filters is presented. Exact and approximate solutions to the nonlinear ??_∞ filtering problem using this class of filters are derived in terms of discrete‐time Hamilton–Jacobi–Isaacs equations. The expressions for the filter gains are determined as functions of the filter state and the system's output in contrast to earlier results. Hence, it is shown that coupled with the additional degree‐of‐freedom, these filters are a substantial improvement over the earlier 1‐DOF case. The theory presented is also generalized to n‐DOF filters, which bore strong connections to linear infinite‐impulse response filters and hence are generalizations of this class of filters to the nonlinear setting. Simulation results are also given to show the usefulness of the new approach. Copyright © 2009 John Wiley & Sons, Ltd.     

State-space solutions to the ℋ︁∞/LTR design problem

The LTR design problem using an ??∞ optimality criterion is presented for two types of recovery errors, the sensitivity recovery error and the input-output recovery error. For both errors two different approaches are presented. First, following the classical LTR design philosophy, a Luenberger observer based approach is proposed, where the ??∞ part of the controller is appended to a standard full-order observer. Second, allowing for general controllers, an ??∞ state-space problem is formulated directly from the recovery errors. Both approaches lead to controller orders of at most 2n. In the minimum phase case, though, the order of the controllers can be reduced to n in all cases. The control problems corresponding to the various controller types are given as four different singular ??∞ state-space problems, and the solutions are given in terms of the relevant equations and inequalities.     

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.     

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.     

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.     

Identification in ℋ︁∞ with nonuniformly spaced frequency response measurements

In this paper, the problem of ‘system identification in ??_∞’ is investigated in the case when the given frequency response data are not necessarily on a uniformly spaced grid of frequencies. A large class of robustly convergent identification algorithms is derived. A particular algorithm is further examined and explicit worst case error bounds (in the ??_∞ norm) are derived for both discrete-time and continuous-time systems. An example is provided to illustrate the application of the algorithms.     

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.     

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.     

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.     

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.     

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.