Discrete Delay Systems Research Articles

Synthesis of Controllers Which Explicitly Contain Plant States Generators

This paper deals with multiobjective control problems for systems whose partial states can be generated by a stable system. For given problems we propose a method to synthesize controllers which explicitly contain the stable system. The synthesis method is proposed as a consequence of the following results : (a) there exists an equivalent problem where partial states of a given generalized plant are exactly available; (b) the resultant controller is constructed by the stable system and another controller for the equivalent problem. The proposing method is advantageous owing to the following reasons : (1) it is possible to synthesize, based on LMIs, controllers which explicitly contain stable systems; those stable systems generate partial states of generalized plants; (2) computational complexity in LMI synthesis can be reduced; (3) computational time to produce control input can be shorten. Moreover, numerical experiments clarify the effectiveness of the proposed method in the case of discrete input delay systems.

Frequency domain stability criterion for discrete delay systems

This paper extends earlier work on stability of discrete time delay systems, and derives an analog to the known frequency sweep for continuous time delay systems.

Delay dependent stability of scalar discrete delay systems

Necessary and sufficient conditions for the delay dependent stability of a linear difference-delay equation are derived. In particular, the delay dependent stability of a scalar delay delay system is related to the existence of a positive Markov parameter sequence for an associated fixed second order system.

The numerical solution of discrete-delay systems

In this note we consider the numerical solution of initial-value discrete-delay systems. An interpolation procedure is introduced to compute the numerical solution. The convergence and stability of the interpolation procedure for the discrete-delay systems is discussed.

The numerical solution of discrete-delay systems

In this note, we consider the numerical solution of initial-value discrete-delay systems. An interpolation procedure is introduced to compute the numerical solution. The stability of the interpolation procedure for linear discrete-delay systems is discussed.

Feedback Stabilization over Commutative Rings: Further Study of the Coordinate-Free Approach

This paper is concerned with the coordinate-free approach to control systems. The coordinate-free approach is a factorization approach but does not require the coprime factorizations of the plant. We present two criteria for feedback stabilizability for multi-input multi-output (MIMO) systems in which transfer functions belong to the total rings of fractions of commutative rings. Both of them are generalizations of Sule's results in [SIAM J. Control Optim., 32 (1994), pp. 1675--1695]. The first criterion is expressed in terms of modules generated from a causal plant and does not require the plant to be strictly causal. It shows that if the plant is stabilizable, the modules are projective. The other criterion is expressed in terms of ideals called generalized elementary factors. This gives the stabilizability of a causal plant in terms of the coprimeness of the generalized elementary factors. As an example, a discrete finite-time delay system is considered.

Some new stability conditions for two-dimensional difference systems

This paper considers robust stability of two-dimensional (2D) difference systems in the Fornasini-Marchesini (F-M) state space setting. An approach based on the MacL aurine series expansion is presented. First, a new and less conservative sufficient condition for asymptotic stability of nominal 2D discrete systems is derived. Then sufficient conditions for robust stability of 2D discrete systems subjected to both structured and unstructured perturbations are given. Finally, new robust stability conditions for 2D discrete-delay systems are derived. Numerical examples are given to illustrate the results.

Diagonal balanced truncation of discrete delay systems

A new model reduction method is presented for discrete delay systems, where the reduced system is also a discrete delay system of a smaller size in the dimension of the state. Traditional model reduction approaches to discrete delay systems are based on transforming delay systems into non-delay equivalent systems. However, when the state delay is large, they are computationally demanding. The proposed model reduction method can be efficiently computed irrespective of the state delay. To do this, a diagonal balanced realization is proposed based on block diagonal gramian-like matrices, and properties of truncation of the diagonal balanced realization are investigated.

Stability of discrete-delay systems: Boundary criteria

This paper deals with the stability of linear discrete-delay systems. Two stability criteria that are described by evaluating a corresponding harmonic function on the boundary of a torus region are presented. Testing these criteria can be proved to be much easier than those reported in the literature.

The Use of Delay Differential Equations in Chemical Kinetics

The concept of a chemically acceptable model is developed. Chemically acceptable models are causal and maintain the nonnegativity of concentrations. An extension of the law of mass action allowing delayed effects is described and shown to lead to chemically acceptable models. Delayed variable enzyme catalysis and Oregonator models are studied and shown to be equivalent to their classical mass-action counterparts from a number of perspectives. This equivalence is shown to be a consequence of an exact correspondence between certain ordinary and distributed-delay differential equations. For sufficiently small delays, this correspondence extends to discrete-delay systems. However, reversible chemical reaction networks cannot be adequately modeled with discrete-delay equations.

D‐Stability analysis for discrete delay systems

Abstract This paper discusses the D‐pole placement problem of discrete time‐delay systems where the delay duration can be any positive integer. One sufficient condition is proposed to insure all the closed‐loop eigenvalues of discrete delay systems be located inside a specific disk D(α, r). Several criteria are used to retain all the eigenvalues of discrete delay systems with structured or unstructured parametric perturbations inside the same disk. Finally, some illustrative examples are given to demonstrate our results.

Robust stability of singularly perturbed discrete-delay systems

This paper presents new sufficient conditions for the robust stability of singularly perturbed discrete systems subjected to multiple delayed perturbations. For a given /spl epsi//spl isin/ [0, /spl epsi//sub 0/], where /spl epsi//sub 0/ >

Delay-independent exponential stability criteria for time-varying discrete delay systems

Derives delay-independent exponential stability conditions for linear/nonlinear time-varying discrete delay systems. Since these conditions are of delay-independence and easily verifiable, they may provide handy tools for the stability analysis. >

Stabilization of two-dimensional systems

Several new results on stabilization of discrete two-dimensional Systems are presented. If the horizontal (or vertical) part of the system in the Roesser model is controllable, then the stabilizability question is the same as that for a related discrete delay system.

Delay-independent stability criteria for discrete-delay systems

Several sufficient conditions for stability of linear discrete-delay systems are derived. Since these conditions are independent of the delay and possess simple forms, they will provide useful tools to check stability of the systems at the first stage.

A note on the degenerate discrete-delay system

The discrete-delay system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_{k+1}=Ax_{k} + Bx_{k-l}, l \geq 1, k= 0,1,2,3,... \infty</tex> ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> is invertible, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</tex> is the delay) is said to be degenerate with respect to the vector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</tex> if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y_{k}=d^{T}x_{k}\equiv 0, k \geq l_{0}</tex> , for all choices of initial points <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(x_{0},x_{-1},x_{-2},... ,x_{-l})</tex> . In this note we shall show that if the above system is degenerate, then the minmum value of l <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> is ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l+ 1</tex> ). Given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A,d,l</tex> , necessary and sufficient conditions are obtained such that the above system is degenerate at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l_{0}=l+ 1</tex> , and the construction of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</tex> is given in this case. These constructions are used to find delayed state-feedback controls for the linear discrete system to steer the output to zero in a minimum number of steps.