Gain-scheduled H ∞ filtering of parameter-varying discrete-time systems via parameter-dependent Lyapunov functions

This paper is concerned with the problem of gain-scheduled H _∞ filter design for a class of parameter-varying discrete-time systems. A new LMI-based design approach is proposed by using parameter-dependent Lyapunov functions. Recommended by Editorial Board member Huanshui Zhang under the direction of Editor Jae Weon Choi. This work was supported in part by the National Natural Science Foundation of P. R. China under Grants 60874058, by 973 program No 2009CB320600, but also the National Natural Science Foundation of Province of Zhejiang under Grants Y107056, and in part by a Research Grant from the Australian Research Council. Shaosheng Zhou received the B.S. degree in Applied Mathematics and the M.Sc. and Ph.D. degrees in Electrical Engineering, in January 1992, July 1996 and October 2001, from Qufu Normal University and Southeast University. His research interests include nonlinear control and stochastic systems. Baoyong Zhang received the B.S. and M.Sc. degrees in Applied Mathematics, in July 2003 and July 2006, all from Qufu Normal University. His research interests include and nonlinear systems, robust control and filtering. Wei Xing Zheng received the B.Sc. degree in Applied Mathematics and the M.Sc. and Ph.D. degrees in Electrical Engineering, in January 1982, July 1984 and February 1989, respectively, all from the Southeast University, Nanjing, China. His research interests include signal processing and system identification.     

A new recurslve method for the analysis of linear time invariant dynamic systems via double-term triangular functions （DTTF） in state space environment

This paper presents a new recursive method for system analysis via double-term triangular functions (DTTF) in state space environment. The proposed method uses orthogonal triangular function sets and proves to be more accurate as compared to single term Walsh series (STWS) method with respect to mean integral square error (MISE). This has been established theoretically and comparison of error with respect to MISE is presented for clarity. A numerical example is treated to establish the proposed method. Relevant curves for the solutions of states of the dynamic system are also presented with plots of percentage error for DTTF-based analysis.     

A stabilizing control for a class of uncertain linear delayed systems via state feedback—Development of the results

Stabilization of uncertain delayed systems via linear state feedback including no delays is considered. The present report is a development of the previous results. It is shown that the obtained condition is fundamentally equivalent to Wei's results called antisymmetric stepwise configuration. He constructed his results for the stabilization of a linear system including no delays.     

On classical state space realisability of quadratic input–output differential equations

This article studies the realisability property of continuous-time quadratic input–output (i/o) equations in the classical state space form. Constraints on the parameters of the quadratic i/o model are suggested that lead to realisable models. The complete list of second- and third-order realisable i/o quadratic models is given and two subclasses of the n-th order realisable i/o quadratic systems are suggested. Our conditions rely basically upon the property that certain combinations of coefficients of the i/o equations are zero or not zero. We provide explicit state equations for realisable second-order quadratic i/o equations, and for one realisable subclass of quadratic i/o equations of arbitrary order.     

Analysis of hybrid systems’ dynamics using the common Lyapunov functions and multiple homomorphisms

Consideration was given to the hybrid systems obeying the nonlinear common differential equations with switched right-hand sides and state jumps (pulses). Conditions for availability of dynamic characteristics like stability, attraction, invariance, and boundedness were formulated in terms of the common Lyapunov functions or multiple homomorphisms.     

H∞ control of parameter-dependent state-delayed systems using polynomial parameter-dependent quadratic functions

This paper presents the issue of robust disturbance attenuation and robust asymptotic stability problem for finite-dimensional linear parameter-dependent state-delayed systems. The use of polynomial parameter-dependent quadratic Lyapunov functions and linear matrix inequalities (LMIs) formulations for robust H _∞ control are considered. It is shown that the state feedback control can be determined to guarantee the stability of the closed-loop system independently of the time-delay. We present an illustrative example to demonstrate the applicability of the proposed design approach.     

Delay-dependent H-infinity control for continuous time-delay systems via state feedback

The delay-dependent H-infinity analysis and H-infinity control problems for continuous time-delay systems are studied. By introducing an equality with some free weighting matrices, an improved criterion of delay-dependent stability with H-infinity performance for such systems is presented, and a criterion of existence and some design methods of delay-dependent H-infinity controller for such systems are proposed in term of a set of matrix inequalities, which is solved efficiently by an iterative algorithm. Further, the corresponding results for the delay-dependent robust H-infinity analysis and robust H-infinity control problems for continuous time-delay uncertain systems are given. Finally, two numerical examples are given to illustrate the efficiency of the proposed method by comparing with the other existing results.     

Multiple Lyapunov functions approach to observer-based H ∞ control for switched systems

This article investigates the issue of H _∞ control for a class of continuous-time switched Lipschitz nonlinear systems. None of the individual subsystems is assumed to be stabilisable with H _∞ disturbance attenuation. Based on a generalised multiple Lyapunov functions (GMLFs) approach, which removes the nonincreasing requirement at switching points, a sufficient condition for the solvability of the H _∞ control problem under a state estimation-dependent switching law is presented. Observers, controllers and a switching law are simultaneously designed. As an extension, a sufficient condition for exponential stabilisability is also given.     

Sensitivity analysis of systems involving λ-variation via block-pulse functions

The problem of the sensitivity analysis of systems involving λ-variations may be treated by a new method based on the idea of solving the sensitivity equation of a system using orthogonal functions. Though the presented method is based on the set of block-pulse functions it can also be used for other sets of orthogonal functions. The approach can also be applied for solving other problems of a similar nature (inversion of Laplace transform involving Dirac impulses, analysis of singular perturbed systems etc.).     

Computation of the state transition matrix for a class of delay-differential systems†

Control systems can drift into instability or, less catastrophically, exhibit resonance behaviour. One role for adaptive controllers is to learn sufficient information concerning the dominant closed-loop system mode so as to apply effective feedback to dampen these modes. In such situations the adaptive loop augments the fixed controller feedback loop. This paper presents an algorithm for adaptive resonance suppression and provides simulation results to study its behaviour in the presence of high-order unmodelled dynamics. The algorithm appears particularly useful for enhancing existing fixed controller designs.     

LMI conditions for quadratic and robust D-stability of interval systems

Sufficient conditions for the quadratic D-stability and further robust D-stability of interval systems are presented in this paper. This robust D-stability condition is based on a parameter-dependent Lyapunov function obtained from the feasibility of a set of linear matrix inequalities （LMIs） defined at a series of partial-vertex-based interval matrices other than the total vertex matrices as in previous results. The results contain the usual quadratic and robust stability of continuous-time and discrete-time interval systems as particular cases. The illustrative example shows that this method is effective and less conservative for checking the quadratic and robust D-stability of interval systems.     

A method of determining the crosscorrelation functions for a class of non-linear systems†

A use is made of the method of finding output autocorrelation functions of continuous time-invariant nonlinear systems, when they are represented by their functional expansion. This enables the input-output crosscorrelation functions to be calculated by the use of multi-dimensional Laplace transforms.

A simple example for a pseudo random binary sequence input is presented.     

Concerning Liapunov functions for linear systems— I†

A method of constructing Liapunov functions for linear time-varying (LTV) systems to conclude asymptotic stability in the large of motions is proposed. The way different Liapunov functions may give different sets of sufficient conditions is analysed. Liapunov functions are formed for certain types of LTV systems which are analogous to non-asymptotically stable linear time-invariant systems, A transformation matrix is found which is useful in the evaluation of Liapunov functions for nth order systems in phase variable form where the coefficient an(t) is zero.     

On the state estimation for non-linear dynamic systems †

The object of this paper is to present an approximate technique for state estimation of non-linear dynamical systems under noisy observations. The conditional cumulant is introduced, by which the conditional probability density can be characterized. A set of dynamical equations satisfied by conditional cumulants is derived, and an approximate method is proposed for computing the cumulants. The relation of the cumulant method to the stochastic linearization technique is also discussed. Finally the state estimation problem for linear stochaatic system with state-dependent disturbance is solved to illustrate the use of the Gaussian approximation.     

Application of state space statistical linearization to optimal stochastic control of non-linear systems†

The formulas for different methods of state space version of statistical linearization are given and applied to sub-optimal stochastic control of non-linear systems, as recently suggested by Wonham and Cashman (1968). Comparison with their optimal results shows that there is no immediate reason to prefer one linearization method to another. Especially the conjecture of Kazakov does not hold in general, i.e. the mean coefficients of two different linearization methods must provide best results.     

A pascal program for the solution of sparse linear systems – Part I

A Pascal program that implements the Gaussian elimination strategy for the solution of (very) sparse linear systems is presented.     

Computer generation of sensitivity functions for sampled-data control systems†

Methods have been presented previously (Suryanarayanan and Soudack 1970) for the generation of sensitivity functions for general non-linear sampled-data systems. In this paper, it is shown that economy of simulator components (for hybrid computer generation) or of computational time (for digital computer generation) is achieved for simultaneously generating a number of sensitivity functions in a class of linear sampled-data control systems. Extensive use of signal flow graph algebra is made for this purpose and comprehensive examples are presented.