Induced l/sub 2/ and generalized H/sub 2/ filtering for systems with repeated scalar nonlinearities

This paper provides complete results on the filtering problem for a class of nonlinear systems described by a discrete-time state equation containing a repeated scalar nonlinearity as in recurrent neural networks. Both induced l/sub 2/ and generalized H/sub 2/ indexes are introduced to evaluate the filtering performance. For a given stable discrete-time systems with repeated scalar nonlinearities, our purpose is to design a stable full-order or reduced-order filter with the same repeated scalar nonlinearities such that the filtering error system is asymptotically stable and has a guaranteed induced l/sub 2/ or generalized H/sub 2/ performance. Sufficient conditions are obtained for the existence of admissible filters. Since these conditions involve matrix equalities, the cone complementarity linearization procedure is employed to cast the nonconvex feasibility problem into a sequential minimization problem subject to linear matrix inequalities, which can be readily solved by using standard numerical software. If these conditions are feasible, a desired filter can be easily constructed. These filtering results are further extended to discrete-time systems with both state delay and repeated scalar nonlinearities. The techniques used in this paper are very different from those used for previous controller synthesis problems, which enable us to circumvent the difficulty of dilating a positive diagonally dominant matrix. A numerical example is provided to show the applicability of the proposed theories.     

Robust filtering for 2-D state-delayed systems with NFT uncertainties

This paper is concerned with the robust filtering problem for two-dimensional (2-D) state-delayed systems with uncertainties represented by nonlinear fraction transformation. The authors first establish the stability H/sub /spl infin// performance and generalized H/sub 2/ performance criteria for the system. Based on the results, the authors propose efficient methods to solve the robust H/sub /spl infin// filtering, generalized H/sub 2/ filtering, and mixed generalized H/sub 2//H/sub /spl infin// filtering problems by using a parameter-dependent Lyapunov function approach. The methods involve solving linear matrix inequalities. Two examples are given to show the effectiveness of the proposed approach.     

Robust H/sub /spl infin// filtering for stochastic time-delay systems with missing measurements

In this paper, the robust H/sub /spl infin// filtering problem is studied for stochastic uncertain discrete time-delay systems with missing measurements. The missing measurements are described by a binary switching sequence satisfying a conditional probability distribution. We aim to design filters such that, for all possible missing observations and all admissible parameter uncertainties, the filtering error system is exponentially mean-square stable, and the prescribed H/sub /spl infin// performance constraint is met. In terms of certain linear matrix inequalities (LMIs), sufficient conditions for the solvability of the addressed problem are obtained. When these LMIs are feasible, an explicit expression of a desired robust H/sub /spl infin// filter is also given. An optimization problem is subsequently formulated by optimizing the H/sub /spl infin// filtering performances. Finally, a numerical example is provided to demonstrate the effectiveness and applicability of the proposed design approach.     

Fixed-order robust H/sub /spl infin// filter design for Markovian jump systems with uncertain switching probabilities

This paper discusses the fixed-order robust H/sub /spl infin// filtering problem for a class of Markovian jump linear systems with uncertain switching probabilities. The uncertainties under consideration are assumed to be norm-bounded in the system matrices and to be elementwise bounded in the mode transition rate matrix, respectively. First, a criterion based on linear matrix inequalities is provided for testing the H/sub /spl infin// filtering level of a filter over all the admissible uncertainties. Then, a sufficient condition for the existence of the fixed-order robust H/sub /spl infin// filters is established in terms of the solvability of a set of linear matrix inequalities with equality constraints. To determine the filter, a globally convergent algorithm involving convex optimization is suggested. Finally, a numerical example is used to illustrate that the developed theory is more effective than the existing results.     

Robust filtering for discrete-time systems with saturation and its application to transmultiplexers

This paper considers the problem of robust filtering for discrete-time linear systems subject to saturation. A generalized dynamic filter architecture is proposed, and a filter design method is developed. Our approach incorporates the conventional linear H/sub 2/ and H/sub /spl infin// filtering as well as a regional l/sub 2/ gain filtering feature developed specially for the saturation nonlinearity and is applicable to the digital transmultiplexer systems for the purpose of separating filterbank design. It turns out that our filter design can be carried out by solving a constrained optimization problem with linear matrix inequality (LMI) constraints. Simulations show that the resultant separating filters possess satisfactory reconstruction performance while working in the linear range and less degraded reconstruction performance in the presence of saturation.     

Robust H/sub /spl infin// filtering for Uncertain2-D continuous systems

This paper considers the problem of robust H/sub /spl infin// filtering for uncertain two-dimensional (2-D) continuous systems described by the Roesser state-space model. The parameter uncertainties are assumed to be norm-bounded in both the state and measurement equations. The purpose is the design of a 2-D continuous filter such that for all admissible uncertainties, the error system is asymptotically stable, and the H/sub /spl infin// norm of the transfer function, from the noise signal to the estimation error, is below a prespecified level. A sufficient condition for the existence of such filters is obtained in terms of a set of linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of a desired H/sub /spl infin// filter is given. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed method.     

H/sub /spl infin// filtering for multiple-time-delay measurements

The aim of this paper is to study the H/sub /spl infin// filtering problem for linear continuous-time systems with both current and multiple-time-delay measurements. The key technique applied for deriving the filter is the reorganized innovation analysis approach in Krein space. A necessary and sufficient condition for the existence of an H/sub /spl infin// filter is derived, and the solution to the H/sub /spl infin// filter is given in terms of solutions of Riccati and matrix differential equations. An example is finally presented to show the effectiveness of the proposed approach.     

H/sub 2/ optimal linear robust sampled-data filtering design using polynomial approach

A new frequency domain approach to robust multi-input-multi-output (MIMO) linear filter design for sampled-data systems is presented. The system and noise models are assumed to be represented by polynomial forms that are not perfectly known except that they belong to a certain set. The optimal design guarantees that the error variance is kept below an upper bound that is minimized for all admissible uncertainties. The design problem is cast in the context of H/sub 2/ via the polynomial matrix representation of systems with norm bounded unstructured uncertainties. The sampled-data mix of continuous and discrete time systems is handled by means of a lifting technique; however, it does not increase the dimensionality or alter the computational cost of the solution. The setup adopted allows dealing with several filtering problems. A simple deconvolution example illustrates the procedure.     

A delay-dependent approach to robust H/sub /spl infin// filtering for uncertain discrete-time state-delayed systems

A delay-dependent approach to robust H/sub /spl infin// filtering is proposed for linear discrete-time uncertain systems with multiple delays in the state. The uncertain parameters are supposed to reside in a polytope and the attention is focused on the design of robust filters guaranteeing a prescribed H/sub /spl infin// noise attenuation level. The proposed filter design methodology incorporates some recently appeared results, such as Moon's new version of the upper bound for the inner product of two vectors and de Oliveira's idea of parameter-dependent stability, which greatly reduce the overdesign introduced in the derivation process. In addition to the full-order filtering problem, the challenging reduced-order case is also addressed by using different linearization procedures. Both full- and reduced-order filters can be obtained from the solution of convex optimization problems in terms of linear matrix inequalities, which can be solved via efficient interior-point algorithms. Numerical examples have been presented to illustrate the feasibility and advantages of the proposed methodologies.     

An LMI approach to the H/sub /spl infin// filter design for uncertain systems with distributed delays

This paper investigates the problem of robust H/sub /spl infin// filter design for linear distributed delay systems with norm-bounded time-varying parameter uncertainties. The distributed delays are assumed to appear in both the state and measurement equations. The problem we address is the design of a filter, such that for all admissible uncertainties, the resulting error system is asymptotically stable and satisfies a prescribed H/sub /spl infin// performance level. A sufficient condition is obtained to guarantee the existence of desired H/sub /spl infin// filters, which can be constructed by solving certain linear matrix inequalities. The effectiveness of the proposed design method is demonstrated by a numerical example.     

Robust filtering for uncertain nonlinearly parameterized plants

We address the robust filtering problem for a wide class of systems whose state-space data assume a very general nonlinear dependence in the uncertain parameters. Our resolution methods rely on new linear matrix inequality characterizations of /spl Hscr//sub 2/ and /spl Hscr//sub /spl infin// performances, which, in conjunction with suitable linearization transformations of the variables, give rise to practical and computationally tractable formulations for the robust filtering problem.     

Design of reduced-order H/sub /spl infin// filter for Markovian jumping systems with time delay

This brief is concerned with the reduced-order H/sub /spl infin// filtering problem for Markovian jumping linear systems with time delay, where the jumping parameters are modeled by a discrete-time Markov process. Sufficient conditions for the existence of the reduced-order H/sub /spl infin// filter are proposed in terms of linear matrix inequalities (LMI) and a coupling nonconvex matrix rank constraint. In particular, the sufficient conditions for the existence of the zeroth-order (or static) H/sub /spl infin// filter can be expressed in terms of a set of strict LMIs. The explicit parameterization of the desired filter is also given. Finally, a numerical example is given to illustrate the proposed approach.     

Robust energy-to-peak filtering with improved LMI representations

The authors revisit the problem of robust energy-to-peak filtering for linear systems with parametric uncertainty residing in a polytope. Based on two results that appeared recently, they derive new L/sub 2/-L/sub /spl infin// performance criteria which allow the use of parameter-dependent Lyapunov functions for analysis and synthesis problems. Robust L/sub 2/-L/sub /spl infin// filters are then designed upon the new conditions and by means of the linear matrix inequality (LMI) technique, with the result that less conservativeness is achieved compared with earlier results that are based on a quadratic framework. Both continuous and discrete-time cases are considered and numerical examples illustrate the feasibility and advantage of the proposed designs.     

Suboptimal reduced-order filtering through an LMI-based method

This paper addresses the reduced-order filtering design problem for continuous-time linear systems. The H/sub 2/ and H/sub /spl infin// norms of the estimation error, used as performance criteria, are discussed and a new linear matrix inequalities (LMI)-based method for reduced-order filter design is proposed. Differently from the methods available in the literature to date, the one presented here does not solve the associated nonconvex problem by means of an appropriate optimization procedure. It is based on the a priori determination of a certain matrix that simultaneously rends convex the problem to be solved and reduces the suboptimality degree of the solution. Its efficiency is tested by means of two examples. The first one, borrowed from the literature, allows the comparison of our method, for the H/sub 2/ norm case, with three earlier techniques, one of them being the well-known balanced truncation. The second one, of higher order, consists of the estimation of the displacement of a tapered bar using an H/sub /spl infin// norm criterion.     

Robust H/sub /spl infin// control of singular impulsive systems with uncertain perturbations

This paper discusses the problem of robust H/sub /spl infin// control for singular impulsive uncertain systems. Some new fundamental properties of singular systems with impulse effect are derived. Based on the Riccati inequality approach, sufficient conditions for robust H/sub /spl infin// criteria of the corresponding singular impulsive closed-loop systems are established. A simple approach to the design of a robust impulsive controller is then presented. Finally, a design example and simulation results are given to demonstrate the effectiveness of the proposed approach.     

Gain-Constrained Extended Kalman Filtering with Stochastic Nonlinearities and Randomly Occurring Measurement Delays

In this paper, the gain-constrained extended Kalman filtering problem is studied for discrete time-varying nonlinear system with stochastic nonlinearities and randomly occurring measurement delays. Both deterministic and stochastic nonlinearities are simultaneously present in the model, where the stochastic nonlinearities are described by first moment and can encompass several classes of well-studied stochastic nonlinear functions. A diagonal matrix composed of mutually independent Bernoulli random variables is introduced to reflect the phenomenon of randomly occurring measurement delays caused by unfavorable network conditions. The aim of the addressed filtering problem is to design a finite-horizon recursive filter such that, for all stochastic nonlinearities, randomly occurring measurement delays and gain constraint, the upper bound of the cost function involving filtering error is minimized at each sampling time. It is shown that the filter gain is obtained by solving matrix equations. A numerical simulation example is provided to illustrate the effectiveness of the proposed algorithm.     

On discrete-time H/sub /spl infin// fixed-lag smoothing

The H/sub /spl infin// fixed-lag smoothing problem for discrete-time linear systems is considered. It is well known that such a problem can be reformulated as a filtering problem for a suitable augmented system. However, in order to check the solvability of the smoothing problem and compute the relevant gains, one must solve a Riccati equation involving the augmented system matrices, which is an approach that may be inefficient, especially for long lags. In this paper, efficient algorithms are worked out that permit checking of solvability and implementation of the smoother, relying only on the solution of the H/sub /spl infin// filtering Riccati equation. In particular, this provides a fast method to compute the minimum lag guaranteeing a desired attenuation level.     

Low-order H/sub /spl infin// controller design for an active suspension system via LMIs

An application of a new controller order reduction technique with stability and performance preservation based on linear matrix inequality optimization to an active suspension system is presented. In this technique, the rank of the residue matrix of a proper rational approximation of a high-order H/sub /spl infin// controller subject to the H/sub /spl infin//-norm of a frequency-weighted error between the approximated controller and the high-order H/sub /spl infin// controller is minimized. However, because solving this matrix rank minimization problem is very difficult, the rank objective function is replaced with a nuclear-norm that can be reduced to a semidefinite program so that it can be solved efficiently. Application to the active suspension system of the Automatic Laboratory of Grenoble provides a fourth-order controller. The experimental results show that the control specifications are met to a large extent.     

Minimization of L/sub 2/-sensitivity for state-space digital filters subject to L/sub 2/-dynamic-range scaling constraints

The problem of minimizing an L/sub 2/-sensitivity measure subject to L/sub 2/-norm dynamic-range scaling constraints for state-space digital filters is formulated. It is shown that the problem can be converted into an unconstrained optimization problem by using linear-algebraic techniques. Next, the unconstrained optimization problem is solved by applying an efficient quasi-Newton algorithm with closed-form formula for gradient evaluation. The coordinate transformation matrix obtained is then used to construct the optimal state-space filter structure that minimizes the L/sub 2/-sensitivity measure subject to the scaling constraints. A numerical example is presented to illustrate the utility of the proposed technique.