Control Of Time-delay Systems Research Articles

A Message From the Special Issue Editor

Time delay in dynamic systems has long been considered a complex phenomenon. One can see increased research activity on this topic since the early 1980s, and primarily from the mathematics research community. Most investigations are directed toward the issue of stability, which is considerably different from those of delay free dynamics. Stability research is still alive and progressing at an increased pace after four or five decades of brilliant work by prominent researchers. In the past decade, we observe increased participation by the engineering and applied science researchers. This is the primary reason for this Special Issue on Time Delayed Systems (TDS). We have collected over 65 papers around the main theme, and selected 20 after a careful review process involving active researchers in the area. The papers in this issue present a variety of interesting research results in the Time Delayed Systems (TDS) area. The reader will find three categories among these papers dealing with: a) Stability and control of TDS, b) Practical application of TDS, and c) Modeling and identification of system dynamics with time delays. Stability is, without dispute, the hottest area in TDS research today. It spans from Linear Time Invariant (LTI) single delay TDS to multiple and unrelated time delay cases, and further to nonlinear and uncertain time varying time delayed structures. A common objective is to find the stability interval(s) of time delay for a given dynamics, another is to find the ranges of some structural parameters which render stability while the time delay remains fixed. The capabilities of most current methods are often limited due to some inherent assumptions for the methods to be viable. And they offer many valuable directions for future research. It is obvious that our community is still looking for more precise, universal and practicable procedures in order for the TDS to be a part of the every-day arsenal. In many practical applications, time delay may appear due to two main reasons: either the dynamics is inherently time delayed (such as the machine tool chatter problem), or the feedback control structure introduces the delay. Although the source of the delay may be different, these two groups render mathematically identical representations and ultimately the same tools for analysis apply to both. In modeling, if there is a time delay, then its proper representation in a dynamic model is a very critical step. Many interesting problems are encountered in this class especially as computerized control systems improve rapidly yielding better and faster system identification. We are at a very exciting period. Both the reviewers and I were mindful, in selecting the final papers for this Special Issue, that it provide a source for inspiration. I hope our readers will be pleased with the content. As I complete the editorial work, I would like to acknowledge the support of my graduate students in my lab at UCONN (ALARM Lab), without which the task of the Guest Editor would have been an impossible mission. I extend my deep appreciation to the reviewers of the papers. The value of the assistance I received from the Journal Office (G. Ulsoy and T. Marion) is not measurable. I also thank wholeheartedly, all the authors who participated in this activity, including those who did not make the final selection. I hope to handle their future work through regular issues of our Journal. Best wishes to you all.

State-Feedback Stabilization and Control of Systems with Time-Delay and Saturating Actuators

A parameter dependent approach is introduced for the design of a constant state-feedback controller that locally stabilizes linear systems with state time varying delays and saturating actuators and achieves a prescribed performance level for all disturbances with uniformly bounded magnitudes. A polytopic representation is used to describe the saturation behavior. Delay-dependent sufficient conditions in terms of LMIs are obtained for the existence of such a controller. An estimate is made of the domain of attraction for the disturbance-free system. The conditions for the stabilizability and H∞ performance of the system apply the Lyapunov-Krasovskii functional and the recent descriptor approach to the control of time-delay systems, whereas the conditions for finding an ellipsoid that bounds the set of the states (in the Euclidean space) that are reachable from the origin in finite time are obtained via the Razumikhin approach.

Robust sliding-mode control for uncertain time-delay systems: an lmi approach

This note is devoted to robust sliding-mode control for time-delay systems with mismatched parametric uncertainties. A delay-independent sufficient condition for the existence of linear sliding surfaces is given in terms of linear matrix inequalities, based on which the corresponding reaching motion controller is also developed. The results are illustrated by an example.

An adaptive Smith-controller for time-delay systems with relative degree [formula omitted

This paper addresses the control of time-delay systems whose relative degree does not exceed two. An adaptive Smith controller together with an adaptive law similar to the delay-free case is proposed. By using Lyapunov–Krasovskii functionals for an appropriate model transformation of the original system, semiglobal stability of the closed-loop system and asymptotic convergence of the output error is established. Strict positive realness together with the low relative degree of the plant is exploited to establish the stability properties. Robustness properties of the adaptive controller are briefly discussed.

Robust stability, stabilization and ℋ∞ control of time‐delay systems with Markovian jump parameters

AbstractIn this paper, the problems of stochastic stability and stabilization for a class of uncertain time‐delay systems with Markovian jump parameters are investigated. The jumping parameters are modelled as a continuous‐time, discrete‐state Markov process. The parametric uncertainties are assumed to be real, time‐varying and norm‐bounded that appear in the state, input and delayed‐state matrices. The time‐delay factor is constant and unknown with a known bound. Complete results for both delay‐independent and delay‐dependent stochastic stability criteria for the nominal and uncertain time‐delay jumping systems are developed. The control objective is to design a state feedback controller such that stochastic stability and a prescribed ℋ∞‐performance are guaranteed. We establish that the control problem for the time‐delay Markovian jump systems with and without uncertain parameters can be essentially solved in terms of the solutions of a finite set of coupled algebraic Riccati inequalities or linear matrix inequalities. Extension of the developed results to the case of uncertain jumping rates is also provided. Copyright © 2003 John Wiley & Sons, Ltd.

Reachable Set Analysis of Linear Time-Delay Systems and Application to Constrained H.INF. Controller Synthesis.

In this paper, a new notion of reachable set for linear time-delay systems is introduced and a method to evaluate the reachable set is proposed. The evaluation method is given in linear matrix inequality form. An important application of reachable set analisys is constrained controller synthesis. We also propose synthesis method of constrained H∞ controllers for time-delay systems.

Output Time Delayed Control System Design Subject to the Mixed Sensitivity Problem.

This paper presents a design methodology for MIMO output time delayed systems satisfying the mixed sensitivity specification. Nevertheless the procedure is on the basis of the state space approach, it provides a controller with tuneable parameters, which makes the technique useful for practical applications. The key features of the proposed method are its capability to handle multiple time delays and to allow the calculations to be performed as in the time delay free system case. A solution for output time delay unstable system is illustrated with a numerical example.

Strict positive realness for linear time-delay systems

The strict positive realness (SPR) of a linear time-delay system via a linear matrix inequality (LMI) technique is studied. Presented are sufficient conditions via LMIs such that the linear delayed system is strictly positive real. More generally, we present a memoryless state feedback controller via LMIs such that the resulting closed-loop system is SPR with an α -asymptotic stability constraint (α -SPR) for a class of linear time-delay control systems. Furthermore, an LMI approach to the optimization problem of computation of the maximal allowable bound on the time delays is given such that the closed-loop system is α -SPR. Finally, a numerical example illustrates the effectiveness of the results.

A simple receding horizon control for state delayed systems and its stability criterion

Abstract In this paper, a simple receding horizon (or model predictive) control for state delayed systems is presented and its solution is given in a closed form by a reduction method. While the control for a time-delay system is usually complex, the proposed controller is simple to construct and therefore can be simply implemented in real applications. To check the closed-loop stability of the proposed controller, a sufficient condition is provided by linear matrix inequalities. In addition, a numerical algorithm is presented for computing the eigenvalues of systems with distributed time delays, which can be used as a necessary and sufficient condition to check closed-loop stability. It is shown by simulation that this simple control can be a stabilizing control for time-delay systems.

On the reachable robustness limits for time delay control systems

The paper presents a new decomposition method to handle optimal control design for two‐degree of freedom time delay control systems. In this approach exact relationships between the actuator, process and design parameters and also robustness measures are developed. The ultimate robustness limit of any control can be explicitly calculated by this approach.

Melnikov's analysis of time-delayed feedback control in chaotic dynamics

The problem of suppressing chaos in the time-delayed feedback control (TDFC) system is considered. The validity of Melnikov's technique in the planar Hamiltonian system perturbed by weak periodic terms and time-delay terms is discussed. Melnikov's technique is used to analyze a TDFC chaotic system that is nonautonomous. It is revealed with analytical results that the delayed time can be independent of the period of the inherent unstable orbit in the chaotic attractor. Numerical simulations demonstrate the effectiveness of Melnikov's analysis.

A note on the two disk mixed sensitivity design for control system with multiple output time delays

This note presents a tuning design method for output time delayed control systems that satisfies the two disk mixed sensitivity control design specification. The key features of this method are that it focuses on the two disk mixed sensitivity problem; it handles multiple time delays and allows the majority of the calculations to be carried out as in the time delay free case. A numerical example is given to show how the procedure works.

Sampled-data control for time-delay systems

The sampled-data systems are hybrid ones involving continuous time and discrete time signals, which makes the traditional analysis and synthesis methodologies of time-delay systems unable to be directly used in the cases of hybrid systems with time-delay. The primary disadvantages of current design techniques of sampled-data control are their inabilities to deal effectively with time-delay and the model uncertainty. In this paper, we generalized the analysis methodology of time-delay systems to that of the hybrid systems with time-delay and uncertainty, which developed a design procedure of sampled-data control for time-delay systems. Asymptotic stability of the time-delay hybrid systems was developed. The time-delay dependent robust sampled-data control for the time-varying delay of an uncertain linear system was then discussed. The results were described as linear matrix inequalities, which can be solved using newly released LMITool.

ON THE REACHABLE PERFORMANCE/ROBUSTNESS LIMITS FOR LINEAR CONTROL SYSTEMS

A new decomposition method is presented to handle optimal control design for two-degree of freedom time delay control systems. In this approach exact relationships between the actuator, process and design parameters furthermore the Nyquist stability margin are developed for a first order time delay process. The ultimate robustness limit of any control can be explicitly calculated using this approach.

DISCRETIZATION OF CONTINUOUS TIME-DELAY SYSTEMS

Many works are related to the analysis and control of either continuous or else discrete time-delay systems. However, the discretization of continuous time-delay systems has not been extensively studied. In this work, sampled-data time-delay systems with internal and external point delays are described by approximate discrete time-delay systems in the discrete domain. Those approximate discrete systems allow the hybrid control of time-delay systems. Several examples complete the paper, showing the correctness of the discretization process.

Predictive fuzzy PID control: theory, design and simulation

Predictive fuzzy PID control theory is developed in this paper, which offers a new approach for robust control of time-delay systems. The paper describes the functional structure, design principle, and stability analysis of a new predictive fuzzy PID controller. Sufficient computer simulations are provided for illustration and verification. First, the structure of the controller is derived from both the fuzzy PID control and the generalized predictive control concepts. Then, on-line model identification, optimal cost index, fuzzification , rule base, and defuzzification of the representative predictive fuzzy PD+I controller are discussed in detail. Lyapunov asymptotic stability analysis is conducted. Then, many computer simulations are performed to compare with several closely related controllers such as the fuzzy PD+I controller and the Smith-type predictive fuzzy PD+I controller. In the simulations, second-order linear systems with/without time delays, nonlinear systems with/without time delays, uncertain linear systems with time delays, and uncertain nonlinear systems with time delays are used to confirm the advantages of the new predictive fuzzy PD+I controller. Finally, this method is applied to control some chaotic systems with success. This predictive fuzzy control method provides a new way for controlling uncertainty and complex linear and nonlinear systems, even with significant time delay.

Continuous time self-tuning control of time delay system

Much of the work reported on continuous time self tuning control addresses the class of systems with known time delay. In this paper, the system parameters including the dead time are estimated by the proposed extended polynomial identification method. The original polynomial identification is further modified in this paper to identified both poles, zeros and unknown time delay. An explicit self tuner is then designed based on the estimated parameters. Experimental studies are used to evaluate the performance of this algorithm.

Anisochronic internal model control of time-delay systems

This paper is concerned with the robust control of time-delay systems with both state and control delays. The Smith predictor is not a suitable control scheme for this class of systems and special approaches developed hitherto are not practically applicable due to their poor robustness and complicated design and structure. However, a suitable controller design method that is both robust and practicably applicable can be based on the extension of the internal model control scheme. This paper describes the design of internal model controllers for this class of time-delay systems in both one degree-of-freedom and two degrees-of-freedom arrangement. A common feature of the proposed controllers is the application of both delayors and integrators in their final implementation, resulting in a better compensation of infinite order process dynamics. The practical applicability of this approach is demonstrated on a small scale heating plant with several internal delays and nonlinear steady state behaviour.

Numerical solution of time-delayed functional differential equation control systems

We develop a pseudospectral approximation scheme for solving the class of time-delayed functional differential equation control systems. The problem is first formulated as a delay free optimal control problem governed by a system of partial differential equations with nonlocal boundary conditions. Next, a Chebyshev spectral method together with the cell-averaging Chebyshev integration technique are used to discretize the delay free optimal control problem. The optimal control problem is thereby transformed into a nonlinear programming problem which can be solved by well-developed nonlinear programming algorithms. Due to its dynamic nature, the proposed method avoids many of the numerical difficulties typically encountered in solving standard time-delayed optimal control problems. Moreover, a comparison is made with optimal solutions obtained by closed-form analysis and/or other numerical methods in the literature.

A new modified Smith predictor: the concept, design and tuning

The paper presents a new modified Smith predictor (MSP) for processes with a long time delay. The MSP appears as an extension of the double controller-scheme (DCS) proposed by Tian and Gao. The important feature of the MSP is that the trade-off between disturbance rejection and robustness to variations in process parameters can be adjusted by means of a single free parameter. The main contribution of the paper concerns tuning of the MSP, which relies on a combination of magnitude optimum criterion with process parameterisation based on multiple integrals of the open-loop step response. In a simulation study the performance of the MSP is compared with that of two known controllers for time delay systems, i.e. DCS of Tian and Gao and Hägglund's predictive PI controller. The results show the advantage of the MSP compared to the two other controllers.