Fuzzy H/sub /spl infin// output feedback control design for singularly perturbed systems with pole placement constraints: an LMI approach

This paper examines the problem of designing an H/sub /spl infin// output feedback controller with pole placement constraints for singular perturbed Takagi-Sugeno (TS) fuzzy models. We propose a fuzzy H/sub /spl infin// output feedback controller that not only guarantees the /spl Lscr//sub 2/-gain of the mapping from the exogenous input noise to the regulated output to be less than some prescribed value, but also ensures closed-loop poles of each subsystem are in a prespecified linear matrix inequality (LMI) region. In order to alleviate the numerical stiffness caused by the singular perturbation /spl epsiv/, the design technique is formulated in terms of a family of /spl epsiv/-independent linear matrix inequalities. The proposed approach can be applied both standard and nonstandard singularly perturbed nonlinear systems. A numerical example is provided to illustrate the design developed in this paper.     

A reduced order observer based controller design for H/sub /spl infin//-optimization

In this note the H/sub /spl infin// control problem with measurement feedback is investigated. It is well-known that for this problem, in general, we need controllers of the same dynamic order as the given system. However, in the case that some entries of the measurement vector are not noise-corrupted, we show that one can find dynamic compensators of a lower dynamical order. Note that this implies that the standard assumptions on the direct feedthrough matrices, as made in most papers on H/sub /spl infin// control, are not satisfied. Our result can be derived by using reduced order observer based controllers.<>     

When does the H/sup /spl infin// fixed-lag smoothing performance saturate for a finite smoothing lag?

A notable difference between the H/sup 2/ and H/sup /spl infin// smoothing is that the achievable performance in the latter problem might "saturate" as the function of the smoothing lag in the sense that there might exist a finite smoothing lag for which the achievable performance level is the same as for the infinite smoothing lag. In this note, necessary and sufficient conditions under which such a saturation occurs are derived. In particular, it is shown that the H/sup /spl infin// performance saturates only if the H/sup /spl infin// norm of the optimal error system is achieved at the infinite frequency, i.e., if the worst case disturbance for the infinite smoothing lag case can be arbitrarily fast and thus in a sense unpredictable.     

H/sub /spl infin// filtering for singular systems

This note considers the H/sub /spl infin// filtering problem for linear continuous singular systems. The purpose is the design of a linear filter such that the resulting error system is regular, impulse-free and stable while the closed-loop transfer function from the disturbance to the filtering error output satisfies a prescribed H/sub /spl infin//-norm bound constraint. Without decomposing the original system matrices, a necessary and sufficient condition for the solvability of this problem is obtained in terms of a set of linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of a desired filter is given. Finally, an illustrative example is presented to demonstrate the applicability of the proposed approach.     

Robust H/sub /spl infin// control for uncertain discrete-time-delay fuzzy systems via output feedback controllers

This work investigates the problem of robust output feedback H/sub /spl infin// control for a class of uncertain discrete-time fuzzy systems with time delays. The state-space Takagi-Sugeno fuzzy model with time delays and norm-bounded parameter uncertainties is adopted. The purpose is the design of a full-order fuzzy dynamic output feedback controller which ensures the robust asymptotic stability of the closed-loop system and guarantees an H/sub /spl infin// norm bound constraint on disturbance attenuation for all admissible uncertainties. In terms of linear matrix inequalities (LMIs), a sufficient condition for the solvability of this problem is presented. Explicit expressions of a desired output feedback controller are proposed when the given LMIs are feasible. The effectiveness and the applicability of the proposed design approach are demonstrated by applying this to the problem of robust H/sub /spl infin// control for a class of uncertain nonlinear discrete delay systems.     

Robust H/sub /spl infin// static output feedback control of fuzzy systems: an ILMI approach

This paper examines the problem of robust H/sub /spl infin// static output feedback control of a Takagi-Sugeno fuzzy system. The proposed robust H/sub /spl infin// static output feedback controller guarantees the L/sub 2/ gain of the mapping from the exogenous disturbances to the regulated output to be less than or equal to a prescribed level. The existence of a robust H/sub /spl infin// static output feedback control is given in terms of the solvability of bilinear matrix inequalities. An iterative algorithm based on the linear matrix inequality is developed to compute robust H/sub /spl infin// static output feedback gains. To reduce the conservatism of the design, the structural information of membership function characteristics is incorporated. A numerical example is used to illustrate the validity of the design methodologies.     

Control of systems with I/O delay via reduction to a one-block problem

In this paper, the standard (four-block) H/sup /spl infin// control problem for systems with a single delay in the feedback loop is studied. A simple procedure of the reduction of the problem to an equivalent one-block problem having particularly simple structure is proposed. The one-block problem is then solved by the J-spectral factorization approach, resulting in the so-called dead-time compensator (DTC) form of the controller. The advantages of the proposed procedure are its simplicity, intuitively clear derivation of the DTC form of the H/sup /spl infin// controller, and extensibility to the multiple delay case.     

Robust H/sub /spl infin// and mixed H/sub 2//H/sub /spl infin// filters for equalization designs of nonlinear communication systems: fuzzy interpolation approach

Generally, it is difficult to design equalizers for signal reconstruction of nonlinear communication channels with uncertain noises. In this paper, we propose the H/sub /spl infin// and mixed H/sub 2//H/sub /spl infin// filters for equalization/detection of nonlinear channels using fuzzy interpolation and linear matrix inequality (LMI) techniques. First, the signal transmission system is described as a state-space model and the input signal is embedded in the state vector such that the signal reconstruction (estimation) design becomes a nonlinear state estimation problem. Then, the Takagi-Sugeno fuzzy linear model is applied to interpolate the nonlinear channel at different operation points through membership functions. Since the statistics of noises are unknown, the fuzzy H/sub /spl infin// equalizer is proposed to treat the state estimation problem from the worst case (robust) point of view. When the statistics of noises are uncertain but with some nominal (or average) information available, the mixed H/sub 2//H/sub /spl infin// equalizer is employed to take the advantage of both H/sub 2/ optimal performance with nominal statistics of noises and the H/sub /spl infin// robustness performance against the statistical uncertainty of noises. Using the LMI approach, the fuzzy H/sub 2//H/sub /spl infin// equalizer/detector design problem is characterized as an eigenvalue problem (EVP). The EVP can be solved efficiently with convex optimization techniques.     

l/sup /spl infin//-bounded robustness for nonlinear systems: analysis and synthesis

The purpose of this paper is to describe systematic analysis and design tools for robust control problems with l/sup /spl infin// criteria. We first generalize the Hill-Moylan-Willems framework for dissipative systems to accommodate l/sup /spl infin// criteria, and then derive state feedback and measurement feedback synthesis procedures for l/sup /spl infin// robust control problems. The information state framework is used for the measurement feedback robust control problem. Necessary and sufficient conditions are proved, and new synthesis procedures using dynamic programming are presented.     

Distributed control design for systems interconnected over an arbitrary graph

We consider the problem of synthesizing a distributed dynamic output feedback controller achieving H/sub /spl infin// performance for a system composed of different interconnected sub-units, when the topology of the underlying graph is arbitrary. First, using tools inspired by dissipativity theory, we derive sufficient conditions in the form of finite-dimensional linear matrix inequalities when the interconnections are assumed to be ideal. These inequalities are coupled in a way that reflects the spatial structure of the problem and can be exploited to design distributed synthesis algorithms. We then investigate the case of lossy interconnection links and derive similar results for systems whose interconnection relations can be captured by a class of integral quadratic constraints that includes constant delays.     

Fault detection in a mixed H/sub 2//H/sub /spl infin// setting

In this note, we study the problem of fault detection in linear time-invariant (LTI) systems which are simultaneously effected by two classes of unknown inputs: Noises having fixed spectral densities and unknown finite energy disturbances. This problem is formulated as a mixed H/sub 2//H/sub /spl infin// filtering problem. Necessary and sufficient conditions for local optimality are presented. Moreover, it is shown that suboptimal solutions can be computed by solving a convex minimization problem with a set of linear matrix inequality (LMI) constrains. A numerical example is given to illustrate the advantage of the mixed H/sub 2//H/sub /spl infin// approach as compared to existing techniques which are based on optimization of H/sub 2/ and H/sub /spl infin// criteria.     

Sufficient LMI conditions for H/sub /spl infin// output feedback stabilization of linear discrete-time systems

In this note, sufficient conditions for H/sub /spl infin// output feedback stabilization of linear discrete-time systems are proposed via linear matrix inequalities (LMIs). In order to reduce conservatism existing in earlier LMI methods, auxiliary slack variables with structure are employed. It is shown that degree of freedoms by the introduction of auxiliary slack variables lead to more flexibility in obtaining an approximate solution of H/sub /spl infin// output feedback stabilization problems. Consequently, the proposed method can yield a less conservative result than earlier LMI methods. In particular, typical output feedback control problems, such as decentralized H/sub /spl infin// output feedback control and simultaneous H/sub /spl infin// output feedback control, can be more efficiently solved. Numerical examples are included to illustrate the advantages of the proposed LMI method.     

Robust H/sub /spl infin// control for fuzzy systems with Frobenius norm-bounded uncertainties

In this paper, we investigate the problem of robust H/sub /spl infin// performance and stabilization for a class of uncertain fuzzy systems with Frobenius norm-bounded parameter uncertainties in all system matrices. Both continuous- and discrete-time uncertain fuzzy systems are considered under a unified treatment called bounded real lemma for fuzzy systems. Unlike the bounded real lemma in the linear theory of robust H/sub /spl infin// control where necessary and sufficient conditions were obtained, only sufficient condition based on Lyapunov method is shown. Furthermore, connection between robust H/sub /spl infin// problems involving uncertainty and standard uncertainty-free H/sub /spl infin// problems is established via matrix algebra. As for controller synthesis, a state feedback fuzzy control law is designed via relaxed linear matrix inequality (LMI) formulations.     

H/sub /spl infin// fuzzy output feedback control design for nonlinear systems: an LMI approach

Addresses the problem of stabilizing a class of nonlinear systems by using an H/sub /spl infin// fuzzy output feedback controller. First, a class of nonlinear systems is approximated by a Takagi-Sugeno (TS) fuzzy model. Then, based on a well-known Lyapunov functional approach, we develop a technique for designing an H/sub /spl infin// fuzzy output feedback control law which guarantees the L/sub 2/ gain from an exogenous input to a regulated output is less or equal to a prescribed value. A design algorithm for constructing an H/sub /spl infin// fuzzy output feedback controller is given. In contrast to the existing results, the premise variables of the H/sub /spl infin// fuzzy output feedback controller are not necessarily to be the same as the premise variables of the TS fuzzy model of the plant. A numerical simulation example is presented to illustrate the theory development.     

Stabilization of singularly perturbed fuzzy systems

This paper presents some novel results for stabilizing singularly perturbed (SP) nonlinear systems with guaranteed control performance. By using Takagi-Sugeno fuzzy model, we construct the SP fuzzy (SPF) systems. The corresponding fuzzy slow and fast subsystems of the original SPF system are also obtained. Two fuzzy control designs are explored. In the first design method, we propose the composite fuzzy control to stabilize the SPF subsystem with H/sup /spl infin// control performance. Based on the Lyapunov stability theorem, the stability conditions are reduced to the linear matrix inequality (LMI) problem. The composite fuzzy control will stabilize the original SP nonlinear systems for all /spl epsiv//spl isin/(0,/spl epsiv//sup */) and the upper bound /spl epsiv//sup */ can be determined. For the second design method, we present a direct fuzzy control scheme to stabilize the SP nonlinear system with H/sup /spl infin// control performance. By utilizing the Lyapunov stability theorem, the direct fuzzy control can guarantee the stability of the original SP nonlinear systems for a given interval /spl epsiv//spl isin/[/spl epsiv/_,/spl epsiv/~]. The stability conditions are also expressed in the LMIs. Two SP nonlinear systems are adopted to demonstrate the feasibility and effectiveness of the proposed control schemes.     

An innovation approach to H/sub /spl infin// fixed-lag smoothing for continuous time-varying systems

This note studies the finite horizon H/sub /spl infin// fixed-lag smoothing problem for linear continuous time-varying systems. A technique named as reorganized innovation analysis in Krein space is developed to give a necessary and sufficient condition for the existence of an H/sub /spl infin// fixed-lag smoother. The condition is given in terms of the boundedness of two matrix functions which are derived from the solutions of two Riccati differential equations (RDEs), one standard H/sub /spl infin// filtering RDE and one H/sub 2/ type of RDE. Examples demonstrate the proposed H/sub /spl infin// fixed-lag smoother design and the fact that the existence of an H/sub /spl infin// smoother does not depend on the solvability of H/sub /spl infin// filtering.     

Output regulation of periodic signals for DPS: an infinite-dimensional signal Generator

In this note, we consider output regulation and disturbance rejection of periodic signals via state feedback in the setting of exponentially stabilizable linear infinite-dimensional systems. We show that if an infinite-dimensional exogenous system is generating periodic reference signals, solvability of the state feedback regulation problem is equivalent to solvability of the so called equations. This result allows us to consider asymptotic tracking of periodic reference signals which only have absolutely summable Fourier coefficients, while in related existing work the reference signals are confined to be infinitely smooth. We also discuss solution of the regulator equations and construct the actual feedback law to achieve output regulation in the single-input-single-output (SISO) case: The output regulation problem is solvable if the transfer function of the stabilized plant does not have zeros at the frequencies i/spl omega//sub n/ of the periodic reference signals and if the sequence ([CR(i/spl omega//sub n/, A+BK)B]/sup -1/ /spl times/(Q/spl phi//sub n/-CR(i/spl omega//sub n/, A+BK)P/spl phi//sub n/)) /sub n/spl isin/z//spl isin/l/sup n2/. A one-dimensional heat equation is used as an illustrative example.     

On the dead-time compensation from L/sup 1/ perspectives

It is known that both H/sup 2/ and H/sup /spl infin// optimization problems for dead-time systems are solved by controllers having the so-called modified Smith predictor (dead-time compensator) structure. This note shows that this is also true for the L/sup 1/ control problem. More precisely, it is demonstrated that the use of the modified Smith predictor enables one to reduce the standard L/sup 1/ problem for systems with a single loop delay to an equivalent delay-free problem. The (sub)optimal solution therefore always contains the modified Smith predictor.     

Remarks on the L/sup p/-input converging-state property

Let X /spl sub/ /spl Ropf//sup N/ and consider a system x/spl dot/ = f(x,u), f : X /spl times/ /spl Ropf//sup M/ /spl rarr/ /spl Ropf//sup N/, with the property that the associated autonomous system x/spl dot/ = f (x,0) has an asymptotically stable compactum C with region of attraction A. Assume that x is a solution of the former, defined on [0,/spl infin/), corresponding to an input function u. Assume further that, for each compact K /spl sub/ X, there exists k > 0 such that |f(z,v) - f(z,0)| /spl les/ k|v| for all (z,v) /spl isin/ /spl times/ /spl Ropf//sup M/. A simple proof is given of the following L/sup p/-input converging-state property: if u /spl isin/ L/sup p/ for some p /spl isin/ [1,/spl infin/) and x has an /spl omega/-limit point in A, then x approaches C.