Discrete-Time Approximation of Multivariable Continuous-Time Delay Systems

Discrete-Time Approximation of Multivariable Continuous-Time Delay Systems

Bemri H'mida (National Engineers School of Tunis BP 37, Tunisia), Mezlini Sahbi (National Engineers School of Tunis BP 37, Tunisia) and Soudani Dhaou (National Engineers School of Tunis BP 37, Tunisia)
DOI: 10.4018/978-1-4666-7248-2.ch019
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Many works are related to the analysis and control of either continuous or else discrete time-delay systems. In general, discrete-time controls have become more and more preferable in engineering because of their easy implementation and simple computation. However, the available discretization approaches for the systems having time delays increase the system dimensions and have a high computational cost. The case studies in this chapter support the efficiency of the two methods. However, the discretization of continuous time-delay systems has not been sufficiently/extensively studied in many works. In this work, the authors present two methods of the effective discretization approach for the continuous-time systems with an input and output delays. Sampled-data time-delay systems with internal and external point delays are described by approximate discrete time-delay systems in the discrete domain. These approximate discrete systems allow the hybrid control of time-delay systems.
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1. Introduction

Systems with delays abound in the world. One reason is that nature is full of transparent delays. Another reason is that time-delay systems are often used to model a large class of systems with delays that frequently appearing in engineering. Typical examples of time-delay systems are communication networks, chemical processes, teleoperation systems, biosystems, and underwater vehicles and so on. The presence of delays makes system analysis and control design much more complicated. The emphasis is on systems with a single input/output delay although the delay-free part of the plant can be multi-input, multi-output (MIMO) when the delays in different channels are the same. The major tools used in this chapter are State-transition method and a Method based on the trapezoidal rule for integration.

With the pace of development in the field of microelectronics, analog controllers yield more places for their digital computers. Indeed, and given the importance of these control systems, it uses methods and numerical models to analyze and/or control industrial processes. To implement such a control structure and ensure the desired objectives, modeling in discrete-time analog systems is required.

Two types of representation are available to model continuous or discrete systems namely the external representation that uses input-output (or transfer function matrix) relations and the internal representation of dynamic system which is based on the concept of state.

Digital control of physical systems usually requires developing discrete models. Several modeling strategies, developed in the literature reflecting a meaningful description of dynamical systems to study, led to mathematical tools generally resulting in linear or nonlinear systems with or without delays whose behavior is close to the real system models. These models, which are described by relations between input and output variables, can be modified by inputs considered secondary (disturbances) that still exist in practice. Knowing the benefits given by the state description of dynamical systems, opt for following such descriptions.

A discrete time delay system is defined as an operator between two discrete-time signals that involve a time called delays. A physical system for which is the general term of the sequence input and is the general term of the sequence and state of delayed.

Initial modeling of a system to discrete time delays often leads to writing a recurrent equation between different terms of the input and output sequences . This formulation of the recursive equation is well suited for numerical calculation. This is the form in which these algorithms are control methods. The system is fully defined and recurrent equation can be solved if the initial conditions are specified. These forms can be simplified mathematically using the lag operator and allow formalizing recurrent equations as follows:

(1) where and are matrices of suitable dimensions, and is assumed to be constant delay.

Time delay is the property of a physical system by which the response to an applied force (action) is delayed in its effect (Chen & Latchman,1994;Shinskey, 1967). Whenever material, information or energy is physically transmitted from one place to another, there is a delay associated with the transmission. The value of the delay is determined by the distance and the transmission speed. Some delays are short, others are very long. The presence of long delays makes system analysis and control design much more complex. What is worse is that some delays are too long to perceive and the system is misperceived as one without delays.

Key Terms in this Chapter

Multivariable Control Systems: The control of systems characterized by multiple inputs, which are usually referred to as the controls; or by multiple outputs, which are often the measured variables to be controlled; or by both multiple inputs and outputs. Automobiles, chemical processing plants, aerospace vehicles, biological systems, and the national economy are all examples of multivariable systems which require and receive some form of regulation or control, be it mathematically contrived or not.

Linear Systems: A linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case. As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications.

Linear Matrix Inequalities: There are efficient numerical methods to determine whether an LMI is feasible (e.g., whether there exists a vector y such that LMI(y) = 0), or to solve a convex optimization problem with LMI constraints. Many optimization problems in control theory, system identification and signal processing can be formulated using LMIs.

Time-Delay Systems: A system can be built with an inherent delay. Delays are units that cause a time-shift in the input signal, but that don't affect the signal characteristics. An ideal delay is a delay system that doesn't affect the signal characteristics at all, and that delays the signal for an exact amount of time. Some delays, like processing delays or transmission delays, are unintentional. Other delays however, such as synchronization delays, are an integral part of a system.

Asymptotic stability: A time-invariant system is asymptotically stable if all the eigenvalue of the system matrix A have negative real parts. If a system is asymptotically stable, it is also BIBO stable. However the inverse is not true: A system that is BIBO stable might not be asymptotically stable.

Exponential Stability: In control theory, a continuous linear time-invariant system is exponentially stable if and only if the system has eigenvalue with strictly negative real parts. A discrete-time input-to-output LTI system is exponentially stable if and only if the poles of its transfer function lie strictly within the unit circle centered on the origin of the complex plane. Exponential stability is a form of asymptotic stability. Systems that are not LTI are exponentially stable if their convergence is bounded by exponential decay.

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