<Emphasis Type="Italic">l</Emphasis><Subscript>∞</Subscript> Fuzzy filter design for nonlinear systems with missing measurements: Fuzzy basis-dependent Lyapunov function approach

In this paper, l _∞ fuzzy filtering problem is dealt for nonlinear systems with both persistent bounded disturbances and missing probabilistic sensor information. The Takagi–Sugeno (T–S) fuzzy model is adopted to represent a nonlinear dynamic system. The measurement output is assumed to contain randomly missing data, which is modeled by a Bernoulli distributed with a known conditional probability. To design the l _∞ fuzzy filter and guarantee tracking performance, the effect of the perturbation against persistent bounded disturbances is reduced by using the minimum l _∞ performance. By using the fuzzy basis-dependent Lyapunov function approach, a sufficient condition is established that ensure the mean square exponential stability of the filtering error. The proposed sufficient condition is represented as some linear matrix inequalities (LMIs), and the filter gain is obtained by the solution to a set of LMIs. Finally, the effectiveness of the proposed design method is shown via an example.     

Exponential L 2-L ∞ filtering for a class of stochastic system with Markovian jump parameters and mixed mode-dependent time-delays

This paper discusses the exponential L ₂-L _∞ filtering problem of a class of nonlinear stochastic systems with Markovian jumping parameters and mixed mode-dependent time-varying delays. By introducing a new multiple mode-dependent Lyapunov-Krasovskii functional, stochastic analysis is conducted. The condition for the existence of mode-dependent L ₂-L _∞ filter, in which the filtering error is guaranteed to be exponentially stable with prescribed L ₂-L _∞ performance, is developed. The developed criterion is delay-range-dependent, mode-dependent and decay-rate-dependent. Based on the derived criterion, the L ₂-L _∞ filtering problems are solved. The mode-dependent filter coefficients can be obtained by solving a set of linear matrix inequalities (LMIs). Numerical simulations are presented to illustrate the effectiveness of the proposed approach.     

H∞ filtering for discrete-time systems with randomly varying sensor delays

This paper investigates an H_∞ filtering problem for discrete-time systems with randomly varying sensor delays. The stochastic variable involved is a Bernoulli distributed white sequence appearing in measured outputs. This measurement mode can be used to characterize the effect of communication delays and/or data-loss in information transmissions across limited bandwidth communication channels over a wide area. H_∞ filtering of this class of systems is used to design a filter using the measurements with random delays to ensure the mean-square stochastic stability of the filtering error system and to guarantee a prescribed H_∞ filtering performance. A sufficient condition for the existence of such a filter is presented in terms of the feasibility of a linear matrix inequality (LMI). Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.     

Robust H∞ sliding mode observer‐based fault‐tolerant control for One‐sided Lipschitz nonlinear systems

This paper presents fault tolerant controllers for a class of one‐sided Lipschitz nonlinear systems with external disturbances. A sliding mode observer (SMO) is integrated with the H_∞ filtering approach as the fault detection and isolation module. The problem is investigated in the presence of faults and disturbances simultaneously. The H_∞ ‐SMO is capable of approximating faults accurately, while reducing the effect of disturbances in the estimation of the state vector and occurred faults. Accordingly, using only a single SMO, the estimation error of the state vector and faults can be made simultaneously arbitrarily small. In addition, to deal with the weighted bilinear form appearing in the one‐sided Lipschitz condition, the quadratically inner bounded condition presented in the literature is employed in this paper as a useful solution. The proposed method guarantees the stability of the overall closed‐loop system, and after a short transient time, the estimation errors for state vector and fault signal converge to a small neighborhood of the origin. The effectiveness of the presented algorithm is confirmed in two examples including a single arm robot with a flexible joint and a numerical simulation.     

Error variance-constrained ℋ∞ filtering for a class of nonlinear stochastic systems with degraded measurements: the finite horizon case

This article is concerned with the robust ?_∞ filtering problem for a class of time-varying nonlinear stochastic systems with error variance constraint. The stochastic nonlinearities considered are quite general, which contain several well-studied stochastic nonlinear systems as special cases. The purpose of the filtering problem is to design a filter which is capable of achieving the pre-specified ?_∞ performance and meanwhile guaranteeing a minimised upper-bounded on the filtering error variance. By means of the adjoint system method, a necessary and sufficient condition for satisfying the ?_∞ constraint is first given, expressed as a forward Riccati-like difference equation. Then an upper-bound on the variance of filtering error system is given, guaranteeing the error variance is not more than a certain value at each sampling instant. The existence condition for the desired filter is established, in terms of the feasibility of a set of difference Riccati-like equations, which can be solved forward in time, hence is suitable for online computation. A numerical example is presented finally to show the effectiveness and applicability of the proposed method.     

Robust H2/H∞ global linearization filter design for nonlinear stochastic time-varying delay systems

One can design a robust H_∞ filter for a general nonlinear stochastic system with external disturbance by solving a second-order nonlinear stochastic partial Hamilton-Jacobi inequality (HJI), which is difficult to be solved. In this paper, the robust mixed H₂/H_∞ globally linearized filter design problem is investigated for a general nonlinear stochastic time-varying delay system with external disturbance, where the state is governed by a stochastic Itô-type equation. Based on a globally linearized model, a stochastic bounded real lemma is established by the Lyapunov–Krasovskii functional theory, and the robust H_∞ globally linearized filter is designed by solving the simultaneous linear matrix inequalities instead of solving an HJI. For a given attenuation level, the H₂ globally linearized filtering problem with the worst case disturbance in the H_∞ filter case is known as the mixed H₂/H_∞ globally linearized filtering problem, which can be formulated as a linear programming problem with simultaneous LMI constraints. Therefore, this method is applicable for state estimation in nonlinear stochastic time-varying delay systems with unknown exogenous disturbance when state variables are unavailable. A simulation example is provided to illustrate the effectiveness of the proposed method.     

Output-feedback control for switched linear systems subject to actuator saturation

This article is devoted to the output-feedback ?_∞ control problem for switched linear systems subject to actuator saturation. We consider both continuous- and discrete-time switched systems. Using the minimal switching rule, nonlinear output feedbacks expressed in the form of quasi-linear parameter varying system are designed to satisfy a pre-specified disturbance attenuation level defined by the regional ?₂ (?₂)-gains over a class of energy-bounded disturbances. The conditions are expressed in bilinear matrix inequalities and can be solved by line search coupled with linear matrix inequalities optimisation. A spherical inverted pendulum example is used to illustrate the effectiveness of the proposed approach.     

State-feedback ℋ∞ control for stochastic time-delay nonlinear systems with state and disturbance-dependent noise

This article focuses on the state-feedback ?_∞ control problem for the stochastic nonlinear systems with state and disturbance-dependent noise and time-varying state delays. Based on the maxmin optimisation approach, both the delay-independent and the delay-dependent Hamilton–Jacobi-inequalities (HJIs) are developed for synthesising the state-feedback ?_∞ controller for a general type of stochastic nonlinear systems. It is shown that the resulting control system achieves stochastic stability in probability and the prescribed disturbance attenuation level. For a class of stochastic affine nonlinear systems, the delay-independent as well as delay-dependent matrix-valued inequalities are proposed; the resulting control system satisfies global asymptotic stability in the mean-square sense and the required disturbance attenuation level. By modelling the nonlinearities as uncertainties in corresponding stochastic time-delay systems, the sufficient conditions in terms of a linear matrix inequality (LMI) and a bilinear matrix inequality (BMI) are derived to facilitate the design of the state-feedback ?_∞ controller. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed methods.     

On error-free filtering of finite-state singular processes under dependent distortions

We consider the problem of finding some sufficient conditions under which causal error-free filtering for a singular stationary stochastic process X = {X _n} with a finite number of states from noisy observations is possible. For a rather general model of observations where the observable stationary process is absolutely regular with respect to the estimated process X, it is proved (using an information-theoretic approach) that under a natural additional condition, the causal error-free (with probability one) filtering is possible.     

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.     

Observers for a class of Lipschitz systems with extension to performance analysis

In this paper, observer design for a class of Lipschitz nonlinear dynamical systems is investigated. One of the main contributions lies in the use of the differential mean value theorem (DMVT) which allows transforming the nonlinear error dynamics into a linear parameter varying (LPV) system. This has the advantage of introducing a general Lipschitz-like condition on the Jacobian matrix for differentiable systems. To ensure asymptotic convergence, in both continuous and discrete time systems, such sufficient conditions expressed in terms of linear matrix inequalities (LMIs) are established. An extension to H_∞ filtering design is obtained also for systems with nonlinear outputs. A comparison with respect to the observer method of Gauthier et al. [A simple observer for nonlinear systems. Applications to bioreactors, IEEE Trans. Automat. Control 37(6) (1992) 875–880] is presented to show that the proposed approach avoids high gain for a class of triangular globally Lipschitz systems. In the last section, academic examples are given to show the performances and some limits of the proposed approach. The last example is introduced with the goal to illustrate good performances on robustness to measurement errors by avoiding high gain.     

On H2 model reduction of bilinear systems

The H₂ model reduction problem for continuous-time bilinear systems is studied in this paper. By defining the H₂ norm of bilinear systems in terms of the state-space matrices, the H₂ model reduction error is computed via the reachability or observability gramian. Necessary conditions for the reduced order bilinear models to be H₂ optimal are given. The gradient flow approach is used to obtain the solution of the H₂ model reduction problem. The formulation allows certain properties of the original models to be preserved in the reduced order models. The model reduction procedure developed can also be applied to finite-dimensional linear time-invariant systems. A numerical example is employed to illustrate the effectiveness of the proposed method.     

Set Membership identification of nonlinear systems

In the paper the problem of identifying nonlinear dynamic systems, described in nonlinear regression form, is considered, using finite and noise-corrupted measurements. Most methods in the literature are based on the estimation of a model within a finitely parametrized model class describing the functional form of involved nonlinearities. A key problem in these methods is the proper choice of the model class, typically realized by a search, from the simplest to more complex ones (linear, bilinear, polynomial, neural networks, etc.). In this paper an alternative approach, based on a Set Membership framework is presented, not requiring assumptions on the functional form of the regression function describing the relations between measured input and output, but assuming only some information on its regularity, given by bounds on its gradient. In this way, the problem of considering approximate functional forms is circumvented. Moreover, noise is assumed to be bounded, in contrast with statistical methods, which rely on assumptions such as stationarity, ergodicity, uncorrelation, type of distribution, etc., whose validity may be difficult to test reliably and is lost in presence of approximate modeling. In this paper, necessary and sufficient conditions are given for the validation of the considered assumptions. An optimal interval estimate of the regression function is obtained, providing its uncertainty range for any assigned regressor values. The set estimate allows to derive an optimal identification algorithm, giving estimates with minimal guaranteed L_p error on the assigned domain of the regressors. The properties of the optimal estimate are investigated and its worst-case L_p identification error is evaluated. The presented approach is tested and compared with other nonlinear methods on the identification of a water heater, a mechanical system with input saturation and a vehicle with controlled suspensions.     

Robust H ∞ filter design of uncertain T-S fuzzy neutral systems with time-varying delays

This article addresses the problem of robust H _∞ filter design of a class of Takagi–Sugeno fuzzy neutral systems with time-varying delays and norm-bounded parameter uncertainties. A fuzzy filter is constructed, which ensures both the robust stability and a prescribed H _∞ performance of the filtering error system. A linear matrix inequality approach is developed, and a delay-dependent sufficient condition is obtained. A simulation example is provided to demonstrate the effectiveness of the proposed approach.     

An innovation approach to H∞ prediction for continuous-time systems with application to systems with delayed measurements

This paper studies the problem of H_∞ prediction for linear continuous-time systems. By developing a new method of characterizing the innovation process and applying a novel innovation analysis approach in Krein space, a necessary and sufficient condition for the existence of a finite horizon H_∞ predictor is derived. The solution to the H_∞ prediction is given in terms of solutions of Riccati and matrix differential equations. We further extend our study to give a necessary and sufficient condition for the H_∞ filtering of linear continuous-time systems with both instantaneous and delayed measurements.     

Networked H∞ filtering for linear discrete-time systems

This paper studies the problem of networked H_∞ filtering for linear discrete-time systems. A new model is proposed as the filtering error system to simultaneously capture the communication constraint, random packet dropout and quantization effects in the networked systems. A sufficient condition is presented for the filtering error system to be mean square exponentially stable with a prescribed H_∞ performance by employing the multiple Lyapunov function method. The obtained condition depends on some parameters of the networked systems, such as the access sequence of nodes, packet dropout rate and quantization density. With these parameters fixed, a design procedure for the desired H_∞ filter is also presented based on the derived condition. Finally, an illustrative example is utilized to show the effectiveness of the proposed method.     

Robust H∞ control for Van de Vusse reactor via T–S fuzzy bilinear scheme

This paper proposes a fuzzy bilinear scheme including a fuzzy modeling method and a fuzzy controller for a class of uncertain nonlinear systems with an additive disturbance input. Firstly, this procedure of modeling describes how to transform a nonlinear system into a fuzzy bilinear system (FBS). For controller design problem, the parallel distributed compensation (PDC) method is adopted to design a fuzzy controller which ensures the robust asymptotic stability of the uncertain T–S FBS with an additive disturbance input and guarantees an H_∞ norm bound constraint on disturbance attenuation. Besides, some sufficient conditions are derived to guarantee the robust stabilization of the overall fuzzy control system via linear matrix inequalities (LMIs). Finally, the Van de Vusse reactor (Chen, Lin, Lin, & Tong, 2008; Chong & Zak, 1996; Kashiwagi & Rong, 2002; Li, Tsai, Lee, Hsiao, & Chao, 2008; Perez, Ogunnaike, & Devasia, 2000) with additive disturbance input is utilized to demonstrate the validity and feasibility of the proposed control scheme.     

Event‐triggered filter design for Markovian jump delay systems with nonlinear perturbation using quantized measurement

This paper deals with the problem of mixed passivity and H_∞ filter design for a class of Markovian jump delay systems with nonlinear perturbation under event‐triggered scheme and quantization. Firstly, based on an integral inequality, a new sufficient condition for the stochastic stability and performance analysis of the filtering error system is proposed. Secondly, a mode‐dependent condition for the solvability of the filter design problem is given in terms of linear matrix inequalities (LMIs). The filter parameters can be derived using feasible solutions of the presented LMIs. Finally, three numerical examples are given to illustrate the effectiveness and advantages of the proposed filter design method.     

Robust H∞ filtering for uncertain impulsive stochastic systems under sampled measurements

This paper is concerned with the problem of robust H_∞ filtering for uncertain impulsive stochastic systems under sampled measurements. The parameter uncertainties are assumed to be time-varying norm-bounded. The aim is to design a stochastically stable filter, using the locally sampled measurements, which ensures both the robust stochastic stability and a prescribed level of H_∞ performance for the filtering error dynamics for all admissible uncertainties. A sufficient condition for the existence of such a filter is proposed in terms of certain linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of a desired filter is given. An example is provided to demonstrate the effectiveness of the proposed approach.