Robust mixed ℋ2/ℋ∞ filteringof 2-D systems

This paper deals with several challenging problems of robust filtering for two-dimensional (2-D) systems. First of all, new linear matrix inequality (LMI) characterizations for the ℋ_∞ and ℋ2 norms of 2-D systems are introduced and thoroughly established. Based on these preparatory results, convex (LMI) characterizations for robust ℋ_∞, ℋ₂, and robust mixed ℋ₂/ℋ_∞ filtering are derived. The efficiency and viability of the proposed techniques and tools are demonstrated through a set of numerical examples     

Continuous-time envelope-constrained filter design via Laguerrefilters and ℋ∞ optimization methods

The envelope-constrained filtering problem is concerned with the design of a time-invariant filter to process a given input signal such that the noiseless output of the filter is guaranteed to lie within a specified output mask while minimizing the noise gain of the filter. An algorithm is developed to solve the continuous-time envelope-constrained filter design problem with the ℋ_∞ norm of the filter as the cost and an orthonormal set of basis filters. It is shown that the problem can be reformulated and solved as a constrained ℋ _∞ model-matching problem. To illustrate the effectiveness of the design method, two numerical examples are presented that deal with the design of equalization filters for digital transmission channels     

Robust ℋ∞ filtering for continuous timevarying uncertain systems with deterministic input signals

Many dynamical systems involve not only process and measurement noise signals but also parameter uncertainty and known input signals. When ℒ₂ or ℋ_∞ filters that were designed based on a “nominal” model of the system are applied, the presence of parameter uncertainty will not only affect the noise attenuation property of the filter but also introduce a bias proportional to the known input signal, and the latter may be very appreciable. We introduce a finite-horizon robust ℋ_∞ filtering method that provides a guaranteed ℋ_∞ bound for the estimation error in the presence of both parameter uncertainty and a known input signal. This method is developed by using a game-theoretic approach, and the results generalize those obtained for cases without parameter uncertainty or without a known input signal. It is also demonstrated, via an example, that the proposed method provides significantly improved signal estimates     

Nonuniform multirate filter banks: analysis and design with anℋ∞ performance measure

A general multirate filter-bank system with multiple channels and nonuniform bands is studied with the reconstruction performance measured by the worst-case energy gain (ℋ_∞ norm) of the error system between the multirate system and a pure time-delay system. Using blocking and polyphase decomposition, we associate with the multirate system an equivalent linear time-invariant (LTI) system whose transfer matrix can be computed by a simple procedure. Based on this LTI system, the optimal design problem can be reduced to one of ℋ_∞ optimization, which can be solved by ready-made software in many cases. For illustration, optimal synthesis filters are designed for a two-channel example, and their properties are studied in detail     

Design of multirate filter banks by ℋ∞optimization

A procedure is developed to design IIR synthesis filters in a multirate filter bank. The filters minimize the l₂-induced norm of the error system between the multirate filter bank and a desired pure time-delay system. This criterion is reduced to one of ℋ_∞ optimization, for which there is ready-made software     

A mixed norm performance measure for the design of multiratefilterbanks

A mixed norm performance measure is presented to design the synthesis filters of a multirate filterbank. The mixed norm performance measure is based on the energy as well as the peak value of the error signal. Mathematically, the performance measure minimizes the l₂ -norm of the error signal subject to the l_∞-norm of the error being bounded by some positive value v (this imposes a bound on the peak value of the error signal). The design problem is shown to be that of a mixed ℋ₂/ℋ_∞ optimization problem. The theory of linear matrix inequalities (LMIs) offers a tractable solution to such multiobjective synthesis problems. The synthesis filters designed with the new performance measure are compared with those obtained by similar induced norm minimization techniques in terms of degree of reconstruction, order of the synthesis filters, SNR, and aliasing distortion     

Adaptive H∞ channel equalization for wirelesspersonal communications

In this paper, a new adaptive H_∞ filtering algorithm is developed to recursively update the tap-coefficient vector of a decision feedback equalizer (DFE) in order to adaptively equalize the time-variant dispersive fading channel of a high-rate indoor wireless personal communication system. Different from conventional L ₂ (such as the recursive least squares (RLS)) filtering algorithms which minimize the squared equalization error, the adaptive H _∞ filtering algorithm is a worst case optimization. It minimizes the effect of the worst disturbances (including input noise and modeling error) on the equalization error. Hence, the DFE with the adaptive H_∞ filtering algorithm is more robust to the disturbances than that with the RLS algorithm. Computer simulation demonstrates that better transmission performance can be achieved using the adaptive H_∞ algorithm when the received signal-to-noise ratio (SNR) is larger than 20 dB     

H∞ filtering of 2-D discrete systems

This paper deals with H_∞ filtering of two-dimensional (2-D) linear discrete systems described by a 2-D local state-space (LSS) Fornasini-Marchesini (1978) second model. Several versions of the bounded real lemma of the 2-D discrete systems are established. The 2-D bounded real lemma allows us to solve the finite horizon and infinite horizon H_∞ filtering problems using a Riccati difference equation or a Riccati inequality approach. Further a solution to the infinite horizon H_∞ filtering problem based on a linear matrix inequality (LMI) approach is developed. Our results extend existing work for one-dimensional (1-D) systems to the 2-D case and give a state-space solution to the bounded realness of 2-D discrete systems as well as 2-D H_∞ filtering for the first time. Numerical examples are given to demonstrate the Riccati difference equation approach to the 2-D finite horizon H_∞ filtering problem and the LMI approach to the 2-D infinite horizon H_∞ filtering problem     

Envelope-constrained H∞ filter design: An LMIoptimization approach

In this correspondence, we solve the envelope-constrained H_∞ filter design problem by minimizing the H_∞ norm of the filtering error transfer function subject to the constraint that the filter output is contained in a prescribed envelope. The filter design is transformed into a standard linear matrix inequality (LMI) optimization problem     

Robust H∞ filter design for uncertain linear systems withmultiple time-varying state delays

The problem of robust H∞ filtering for continuous-time uncertain linear systems with multiple time-varying delays in the state variables is investigated. The uncertain parameters are supposed to belong to a given convex bounded polyhedral domain. The aim is to design a stable linear filter assuring asymptotic stability and a prescribed H∞ performance level for the filtering error system, irrespective of the uncertainties and the time delays. Sufficient conditions for the existence of such a filter are established in terms of linear matrix inequalities, which can be efficiently solved by means of powerful convex programming tools with global convergence assured. An example illustrates the proposed methodology     

Delay-Dependent H∞ Control and Filtering for Uncertain Markovian Jump Systems With Time-Varying Delays

This paper deals with the problems of delay-dependent robust H_infin control and filtering for Markovian jump linear systems with norm-bounded parameter uncertainties and time-varying delays. In terms of linear matrix inequalities, improved delay-dependent stochastic stability and bounded real lemma (BRL) for Markovian delay systems are obtained by introducing some slack matrix variables. The conservatism caused by either model transformation or bounding techniques is reduced. Based on the proposed BRL, sufficient conditions for the solvability of the robust H_infin control and H_infin filtering problems are proposed, respectively. Dynamic output feedback controllers and full-order filters, which guarantee the resulting closed-loop system and the error system, respectively, to be stochastically stable and satisfy a prescribed H_infin performance level for all delays no larger than a given upper bound, are constructed. Numerical examples are provided to demonstrate the reduced conservatism of the proposed results in this paper.     

Dispersion-shifted dual-shape core fibers. Optimization based onspot size definitions

Design features, for very low bend and splice losses, in dispersion-shifted dual-shape core (DSC) single-mode fibers are obtained in terms of characteristic mode spot sizes W¯ responsible for splice loss, and W_∞ responsible for bend loss. Dual-shape core fiber designs are given with W¯_∞/W lying between 1.16 and 1.33 while maintaining the mode spot size between 4 and 5 μm at the operating wavelength of 1550 nm. With this design goal it is shown that bending loss would be lower in a step-index than in a graded-index DSC fiber. Further, conventional single clad step-index or triangular-index dispersion-shifted fibers are seen to have higher bending loss than well-designed DSC fibers     

Delay-dependent fault detection for switched linear systems with time-varying delays—the average dwell time approach

In this paper, the fault detection problem is investigated for a class of discrete-time switched linear systems with time-varying delays. The main purpose is to design a fault detection filter such that, for all unknown inputs, control inputs and time delays, the estimation error between the residual and fault is minimized in an exponential way. The fault detection problem is converted into an exponential H_∞ filtering problem. By using a newly constructed Lyapunov functional and the average dwell time scheme, a novel delay-dependent sufficient condition for the solvability of this problem is established in terms of linear matrix inequalities (LMIs). A numerical example is given to demonstrate the effectiveness of the developed theoretical results.     

Exponential H∞ filtering for time-varying delay systems: Markovian approach

This paper discusses the H_∞ filtering problem for a class of deterministic systems with time-varying delays, where the stochastic property of time-varying delays described by Markovian approach is taken into consideration in filter design. Firstly, the delay interval is separated into several subintervals, which can be described by Markov process. Then, a new H_∞ filtering method for deterministic system with time-varying delay is given, whose filter can switch with time delay in terms of Markov process. Sufficient conditions for the existence of H_∞ filter are obtained as linear matrix inequalities, where the mode transition rates are known exactly or inexactly. Finally, numerical examples are used to demonstrate the utility of the given methods.     

Robust H∞ Filtering for Uncertain Discrete Stochastic Systems with Time Delays

This paper considers the problem of robust H_∞ filtering for uncertain discrete-time stochastic systems with time-varying delays. The parameter uncertainties are assumed to be real time-varying norm-bounded in both the state and measurement equations. The problem to be addressed is the design of a stable filter that guarantees stochastic stability and a prescribed H_∞ performance level of the filtering error system for all admissible uncertainties and time delays. A sufficient condition for the existence of such filters is obtained in terms of a linear matrix inequality (LMI). For the case when this LMI is feasible, an explicit expression for a desired filter is given. An illustrative example is also provided to demonstrate the effectiveness and applicability of the proposed method.     

Deconvolution filter design via l1 optimizationtechnique

A new l₁ optimal deconvolution filter design approach for systems with uncertain (or unknown)-but-bounded inputs and external noises is proposed. The purpose of this deconvolution filter is to minimize the peak gain from the input signal and noise to the error by the viewpoint of the time domain. The solution consists of two steps. In the first step, the l₁ norm minimization problem is transferred to an equivalent A-norm minimization problem, and the minimum value of the peak gain is calculated. In the second step, based on the minimum peak gain, the l₁ optimal deconvolution filter is constructed by solving a set of constrained linear equations. Some techniques of inner-outer factorization, polynominal spectral factorization, linear programming, and some optimization theorems found in a book by Luenberger are applied to treat the l₁ optimal deconvolution filter design problem. Although the analysis of the algorithm seems complicated, the calculation of the proposed design algorithm for actual systems is simple. Finally, one numerical example is given to illustrate the proposed design approach. Several simulation results have confirmed that the proposed l₁ optimal deconvolution filter has more robustness than the l₂ optimal deconvolution filter under uncertain driving signals and noises     

Performance of π/4 SQAM in a hard-limited channel in thepresence of AWGN

A strategy that reduces the spectral spreading when an ideal hard-limiter is used as a first approximation to a fully saturated power amplifier is presented. This strategy combines superposed quadrature amplitude modulation (SQAM) filtering with the π/4-shift quadrature phase-shift keying (QPSK) digital transmission format adopted for the first generation of US digital cellular systems. Simulation results showed that this π/4 SQAM filtering strategy increased capacity by 35% in comparison to hard-limited π/4 QPSK. Using computer-aided design, a receive filter that would limit the degradation of E _b/N₀ to less than 1.4 dB at a bit error rate of P_e=10^-4 was selected     

H ∞ Filtering for Stochastic Systems with Markovian Switching and Partly Unknown Transition Probabilities

This paper considers the H _∞ filtering problem for stochastic systems. The systems under consideration involve Markovian switching, mode-dependent delays, Itô-type stochastic disturbance, distributed time-varying delays and partly unknown transition probabilities. Our aim is to design a full-order filter such that the corresponding filtering error system is stochastically stable and satisfies a prescribed H _∞ disturbance attenuation level. By using a new Lyapunov–Krasovskii functional, sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). A numerical example is given to illustrate the effectiveness of the proposed main results.     

Generalized sampling: stability and performance analysis

Generalized sampling provides a general mechanism for recovering an unknown input function f(x)ϵℋ from the samples of the responses of m linear shift-invariant systems sampled at 1/mth the reconstruction rate. The system can be designed to perform a projection of f(x) onto the reconstruction subspace V(φ)=span {φ(x-k)}_kϵZ; for example, the family of bandlimited signals with φ(x)=sinc(x). This implies that the reconstruction will be perfect when the input signal is included in V(φ): the traditional framework of Papoulis' (1977) generalized sampling theory. Otherwise, one recovers a signal approximation f(x)ϵV(φ) that is consistent with f(x) in the sense that it produces the same measurements. To characterize the stability of the algorithm, we prove that the dual synthesis functions that appear in the generalized sampling reconstruction formula constitute a Riesz basis of V(φ), and we use the corresponding Riesz bounds to define the condition number of the system. We then use these results to analyze the stability of various instances of interlaced and derivative sampling. Next, we consider the issue of performance, which becomes pertinent once we have extended the applicability of the method to arbitrary input functions, that is, when ℋ is considerably larger than V(φ), and the reconstruction is no longer exact. We show that the generalized sampling solution is essentially equivalent to the optimal minimum error approximation. We then perform a detailed analysis for the case in which the analysis filters are in L₂ and determine all relevant bound constants explicitly. Finally, we use an interlaced sampling example to illustrate these various calculations