A Gradient Flow Approach to Optimal Model Reduction of Discrete-Time Periodic Systems

This paper is concerned with the optimal model reduction for linear discrete periodic time-varying systems and digital filters. Specifically, for a given stable periodic time-varying model, we shall seek a lower order periodic time-varying model to approximate the original model in an optimal H ₂ norm sense. By orthogonal projections of the original model, we convert the optimal periodic model reduction problem into an unconstrained optimization problem. Two effective algorithms are then developed to solve the optimization problem. The algorithms ensure that the H ₂ cost decreases monotonically and converges to an optimal (local) solution. Numerical examples are given to demonstrate the computational efficiency of the proposed method. The present paper extends the optimal model reduction for linear time invariant systems to linear periodic discrete time-varying systems.     

LMI Approach to Robust Delay-Dependent Mixed H 2/H ∞ Controller of Uncertain Neutral Systems with Discrete and Distributed Time-Varying Delays

The design of a robust mixed H ₂/H _∞ controller for a class of uncertain neutral systems with discrete, distributed, and input time-varying delays is considered. More precisely, the proposed robust mixed H ₂/H _∞ controller minimizes an upper bound of the H ₂ performance measure, while guaranteeing an H _∞ norm bound constraint. Based on the Lyapunov-Krasovskii functional theory, a delay-dependent criterion is derived for the existence of a desired mixed H ₂/H _∞ controller, which can be constructed easily via feasible linear matrix inequalities (LMIs). Furthermore, a convex optimization problem satisfying some LMI constraints is formulated to obtain a suboptimal robust mixed H ₂/H _∞ controller achieving the minimization of an upper bound of the closed-loop H ₂ performance measure. Finally, a numerical example is illustrated to show the usefulness of the obtained design method.The research reported here was supported by the National Science Council of Taiwan, ROC under Grant NSC 94-2213-E-507-002.     

Robust stochastic stability and H∞ performance for a class of uncertain impulsive stochastic systems

In this paper, we discuss the problem of robust stochastic stability and H_∞ performance for a class of uncertain impulsive stochastic systems under sampled measurements. The parameter uncertainties are assumed to be time-varying and value-bounded. We give a sufficient condition in terms of certain linear matrix inequalities (LMIs) to guarantee the uncertain impulsive stochastic system to be robustly stochastically stable. Furthermore, we discuss a stochastically stable filter, using the locally sampled measurements, which ensures both the stochastic stability and a prescribed level of H_∞ performance for the filtering error system for all admissible uncertainties. We give a sufficient condition for the existence of such a filter and an explicit expression of a desired filter if relevant conditions are satisfied.     

Robust H ∞ Filtering for a Class of Nonlinear Discrete-Time Markovian Jump Systems

This paper considers the problem of the robust H filtering for a class of nonlinear discrete-time Markovian jump systems with real time-varying norm-bounded parameter uncertainty. For each mode, the nonlinearity is assumed to satisfy the global Lipschitz conditions and appears in both the state and measured output equations. The problem that we address is the design of a nonlinear filter which ensures robust stochastic stability and a prescribed H performance level of the filtering error system for all admissible uncertainties. A sufficient condition for the solvability of this problem is obtained in terms of a set of linear matrix inequalities; an explicit expression of a desired nonlinear H filter is also given. Finally, an example is provided to demonstrate the effectiveness of the proposed approach.     

Direct Method of Constructing H 2-Suboptimal Controllers: Discrete-Time Systems

For discrete-time systems, an H ₂-suboptimal control problem is defined and analyzed. Then, an algorithm called H ₂-suboptimal state feedback gain sequence (Algorithm A1) is developed. Rather than utilizing a perturbation method, which is numerically stiff and computationally prohibitive, Algorithm A1 utilizes a direct eigenvalue assignment method to come up with a sequence of H ₂-suboptimal state feedback gains. Also, although the sequence of H ₂-suboptimal state feedback gains constructed by Algorithm A1 depends on a parameter , the construction procedure itself does not require explicitly the value of the parameter . Next, attention is focused on constructing a sequence of H ₂-suboptimal estimator-based measurement feedback controllers. Three different estimator structures (prediction, current, and reduced-order estimators) are considered. For a given H ₂-suboptimal state feedback gain, a sequence of estimator gains for any of the three estimators considered can be constructed by merely dualizing Algorithm A1. The direct method of constructing H ₂-suboptimal controllers developed here has a number of advantages over the perturbation method, e.g., it has the ability to design all three types of estimator-based controllers while still maintaining throughout the design the computational simplicity of it. This paper is the discrete-time version of Ref. 1. There are some conceptual similarities as well as fundamental differences between the H ₂-suboptimal control problems for continuous-time and discrete-time systems. The fundamental differences arise mainly from the fact that, in contrast to continuous-time systems, for discrete-time systems the infimum of the H ₂-norm over the class of strictly proper controllers is in general different from the infimum of the H ₂-norm over the class of proper controllers.     

H ∞-Control for Markovian Jumping Linear Systems with Parametric Uncertainty

This paper studies the problem of H -control for linear systems with Markovian jumping parameters. The jumping parameters considered here are two separable continuous-time, discrete-state Markov processes, one appearing in the system matrices and one appearing in the control variable. Our attention is focused on the design of linear state feedback controllers such that both stochastic stability and a prescribed H -performance are achieved. We also deal with the robust H -control problem for linear systems with both Markovian jumping parameters and parameter uncertainties. The parameter uncertainties are assumed to be real, time-varying, norm-bounded, appearing in the state matrix. Both the finite-horizon and infinite-horizon cases are analyzed. We show that the control problems for linear Markovian jumping systems with and without parameter uncertainties can be solved in terms of the solutions to a set of coupled differential Riccati equations for the finite-horizon case or algebraic Riccati equations for the infinite-horizon case. Particularly, robust H -controllers are also designed when the jumping rates have parameter uncertainties.     

Robust finite-time H∞ control of singular stochastic systems via static output feedback

This paper addresses the problem of robust finite-time stabilization of singular stochastic systems via static output feedback. Firstly, sufficient conditions of singular stochastic finite-time boundedness on static output feedback are obtained for the family of singular stochastic systems with parametric uncertainties and time-varying norm-bounded disturbance. Then the results are extended to singular stochastic H_∞ finite-time boundedness for the class of singular stochastic systems. Designed algorithm for static output feedback controller is provided to guarantee that the underlying closed-loop singular stochastic system is singular stochastic H_∞ finite-time boundedness in terms of strict linear matrix equalities with a fixed parameter. Finally, an illustrative example is presented to show the validity of the developed methodology.     

Observer-based finite-time control of time-delayed jump systems

This paper provides the observer-based finite-time control problem of time-delayed Markov jump systems that possess randomly jumping parameters. The transition of the jumping parameters is governed by a finite-state Markov process. The observer-based finite-time H_∞ controller via state feedback is proposed to guarantee the stochastic finite-time boundedness and stochastic finite-time stabilization of the resulting closed-loop system for all admissible disturbances and unknown time-delays. Based on stochastic finite-time stability analysis, sufficient conditions that ensure stochastic robust control performance of time-delay jump systems are derived. The control criterion is formulated in the form of linear matrix inequalities and the designed finite-time stabilization controller is described as an optimization one. The presented results are extended to time-varying delayed MJSs. Simulation results illustrate the effectiveness of the developed approaches.     

Exponential H∞ synchronization for discrete‐time chaotic neural networks with time delays and stochastic perturbations via output feedback control

This paper investigates the problem of exponential H_∞ synchronization of discrete‐time chaotic neural networks with time delays and stochastic perturbations. First, by using the Lyapunov‐Krasovskii (Lyapunov) functional and output feedback controller, we establish the H_∞ performance of exponential synchronization in the mean square of master‐slave systems, which is analyzed using a matrix inequality approach. Second, the parameters of a desired output feedback controller can be achieved by solving a linear matrix inequality. Finally, 2 simulated examples are presented to show the effectiveness of the theoretical results.     

Robust H ∞ control for singular systems with time-varying uncertainties

In this paper, we investigate the problem of robust H _∞ control for singular systems with polytopic time-varying parameter uncertainties. By introducing the notion of generalized quadratic H _∞ performance, the relationship between the existence of a robust H _∞ dynamic state feedback controller and that of a robust H _∞ static state feedback controller is given. By using matrix inequalities, the existence conditions of robust H _∞ static state feedback and dynamic output feedback controllers are derived. Moreover, the design methods for such controllers are provided in terms of the solutions of matrix inequalities. An example is also presented to demonstrate the validity of the proposed methods. __________ Translated from Journal of Northeastern University (Natural Science), 2004, 25(2): 110–113     

Resilient and robust finite-time H∞ control for uncertain discrete-time jump nonlinear systems

This paper studies the robust and resilient finite-time H_∞ control problem for uncertain discrete-time nonlinear systems with Markovian jump parameters. With the help of linear matrix inequalities and stochastic analysis techniques, the criteria concerning stochastic finite-time boundedness and stochastic H_∞ finite-time boundedness are initially established for the nonlinear stochastic model. We then turn to stochastic finite-time controller analysis and design to guarantee that the stochastic model is stochastically H_∞ finite-time bounded by employing matrix decomposition method. Applying resilient control schemes, the resilient and robust finite-time controllers are further designed to ensure stochastic H_∞ finite-time boundedness of the derived stochastic nonlinear systems. Moreover, the results concerning stochastic finite-time stability and stochastic finite-time boundedness are addressed. All derived criteria are expressed in terms of linear matrix inequalities, which can be solved by utilizing the available convex optimal method. Finally, the validity of obtained methods is illustrated by numerical examples.     

Transfer Function Optimization Procedure for the H 2/H ∞ Problem

In this paper, the H ₂/H problem is considered in a transfer-function setting, i.e., without a priori chosen bounds on the controller order. An optimization procedure is described which is based on a parametrization of all feasible descending directions stemming from a given point of the feasible transfer-function set. A search direction at each such point can be obtained on the basis of the solution of a convex finite-dimensional problem which can be converted into a LMI problem. Moving along the chosen direction in each step, the procedure in question generates a sequence of feasible points whose cost functional values converge to the optimal value of the H ₂/H problem. Moreover, this sequence of feasible points is shown to converge in the sense of a weighted H ₂ norm; and it does so to the solution of the H ₂/H problem whenever such a solution exists.     

Robust <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Output Dynamic Observer-Based Control Design of Uncertain Neutral Systems

In this paper, the robust H _∞ control problem of output dynamic observer-based control for a class of uncertain neutral systems is considered. The linear matrix inequality optimization approach is used to design the new H _∞ output dynamic controls. Three classes of H _∞ observer-based controls are proposed. The minimal H _∞-norm bound and the maximal perturbed bound are given. Based on the result of this paper, the constraint of matrix equality is not necessary for designing the H _∞ observer-based controls. A numerical example is given to stress the usefulness of the proposed results. Communicated by C. T. Leondes The research reported here was supported by the National Science Council of Taiwan, ROC under Grant NSC 94-2213-E-507-002.     

Robust H ∞ Filtering for Uncertain Time-Delay Systems: Matrix Inequality Approach

In this paper, a new robust H _∞ filtering problem for uncertain time-delay systems is considered. Based on the Lyapunov method, a design criterion of the robust H _∞ filter, in which the filtering process remains asymptotically stable for all admissible uncertainties and the transfer function from the disturbance inputs to error state outputs satisfies the prespecified H _∞ norm upper bound constraint, is derived in terms of matrix inequalities. The inequalities can be solved easily by efficient convex optimization algorithms. A numerical example is included to illustrate the validity of the proposed design approach.     

A cone complementarity linearization approach to robust controller design for continuous-time piecewise linear systems with linear fractional uncertainties

This paper investigates the robust H_∞ control problem for uncertain continuous-time piecewise systems by using the piecewise continuous Lyapunov function. The uncertainties of the systems under consideration are expressed in a linear fractional form. A strict linear matrix inequality approach is developed to obtain stability condition and H_∞ performance. The H_∞ controller design problem is solved by exploiting the cone complementarity linearization (CCL) method, which can be cast into an iterative minimization problem subject to LMI constraints. Finally two examples are given to illustrate the application of the proposed approach.     

Output Feedback H∞ Control for a Class of Nonlinear Stochastic Systems

This paper deals with the output feedback H∞ control problem for a class of nonlinear stochastic systems. Based on the latest developed theory of stochastic dissipation, a notable result about the nonlinear H∞ output feedback control of deterministic system is generalized to the stochastic case. Finally, in the cases of state feedback and output feedback, two families of controllers are provided respectively.     

Complex structured singular value analysis using fixed-structure dynamic D-scales

Models of systems are always inexact. Hence, to better predict the performance of a system it is necessary to take into account uncertainty in a nominal model of a system. The structured singular value was developed to nonconservatively analyze robust stability and performance for systems with multiple-block uncertainty. In practice, optimization techniques are used to compute an upper bound on the structured singular value. For dynamic uncertainty with bounded magnitude and arbitrary phase (i.e., "complex uncertainty"), the standard approach to computing an upper bound involves finding diagonal scaling matrices D(jω) that minimize σ_max (D(jω)G(jω)D^-1(jω)) over a (theoretically) infinite number of frequencies. The order of the corresponding stable, minimum phase, rational function D(s) (if it exists) is hence arbitrary, which can lead to very high order controllers when D(s) is used for controller synthesis. This paper develops a fixed-structure approach to computing an upper bound for the complex structured singular value. In particular, by relying on results from mixed-norm H₂/H_∞ analysisD(s) is a priori constrained to be a rational matrix function of a chosen order and a new approach to computing an upper bound on the structured singular value is developed. The results are illustrated using two examples which clearly demonstrate the suboptimality of standard curve fitting. The proposed approach can be extended to mixed uncertainty and structured singular value controller synthesis without D — K type iteration.     

Delay-dependent robust control for uncertain discrete-time singular systems with time-delays

The robust memoryless state feedback H_∞ control problem for uncertain time-delay discrete-time singular systems is discussed. Under a series of equivalent transformation, the equivalence of this problem and the robust state feedback H_∞ control problem for standard state-space uncertain time-delay discrete-time systems is presented. In terms of matrix inequality, the delay-dependent sufficient condition for the solution of this problem is given, the design method of the memoryless state feedback controller and the controller are also given.     

Robust H∞ control for a class of switched nonlinear cascade systems via multiple Lyapunov functions approach

This paper addresses the problem of robust H_∞ control for a class of switched nonlinear cascade systems with parameter uncertainty using the multiple Lyapunov functions (MLFs) approach. Each subsystem under consideration is composed of two cascade-connected parts. The uncertain parameters are assumed to be in a known compact set and are allowed to enter the system nonlinearly. Based on the explicit construction of Lyapunov functions, which avoids solving the Hamilton-Jacobi equations, sufficient conditions for the solvability of the robust H_∞ control problem are presented. As an application, the hybrid robust H_∞ control problem for a class of uncertain non-switched nonlinear cascade systems is solved when no single continuous controller is effective. Finally, a numerical example is provided to demonstrate the feasibility of the proposed method.