Deterministic and Stochastic Time-Delay Systems

The purpose of this chapter is to define the class of dynamical systems with time delay that we are treating in this book. This class of systems can be defined as a group of systems in which there exists a time delay between the instant the input or the control to the system is applied and the instant or moment effect is observed. In most cases, the time delay cannot be neglected; it should be taken into account during the design phase of an experiment if we want to assure good performance.

It is clear from the examples given previously that the the investigation of the class of dynamical time delay systems is of great importance in dealing with the world economy. It is why much attention has been paid to this class of systems by many researchers from different communities. During recent decades, the control community has contributed to many problems of this class of systems. Part I focuses on this class of systems and gives a summary of what has been done in the area. It addresses mainly the problems of stability and stabilizability of the class of linear systems with time delay. It also deals with the robustness of stability and stabilizability when uncertainties are assumed to be of the norm-bounded type. Memoryless and memory state feedback controllers and output feedback controllers are used to stabilize dynamical time delay systems and to ensure the desired performance.

This chapter is devoted to the study of the stability and the stabilizability of linear time invariant systems with time delay using the Lyapunov method and linear matrix inequality (see Appendix A). For this purpose, let us consider the following linear continuous-time system with time delay:

Most real systems cannot be represented by linear dynamics, but sometimes, under some assumptions, it is possible to model the dynamical behavior of practical systems with a linear model having some uncertainties. The presence of these uncertainties in the dynamics requires the establishment of robust conditions that can guarantee the stability and/or the stabilizability of the practical system under study. This topic has in fact dominated the research effort of the control community during the last two decades. Among the contribution on this area of research we quote the work of [77, 105, 114] on robust stability and the work of [49, 108, 110, 118, 128, 155, 189, 190, 206, 207, 215] on robust stabilizability. This chapter will deal with the robust stability and robust stabilizability of the class of uncertain continuous-time linear time delay systems. Our results are mainly based on the Lyapunov second method. In some sense, given an uncertain dynamical system with time delay, we answer the following questions: How can we check whether the unforced nominal system with time delay is stable or not?How can we check if the unforced uncertain dynamical system with time delay is robust stable for all admissible uncertainties or not?When the unforced uncertain dynamical system with time delay is unstable, how can we design a memoryless state feedback, or memory state feedback or output feedback controller to stabilize the system and guarantee that it will remain stable for all admissible uncertainties?

Almost all practical systems are subject to external disturbances that can in some situations degrade system performance if their effects are not considered during the design phase. In the current literature there are many ways to eliminate the effects of the external disturbances. One of them is the -1_oocontrol technique. It consists of designing a suboptimal control that minimizes the effects of the external disturbance on the output. In other words, the problem can be explained as follows: Given a dynamical time delay system with exogenous input that belongs to G₂[0, oo], design a controller that minimizes the 7-1_ß-norm of the transfer function between the controlled output and the external disturbance, or at least guarantees that the W_oo-norm will not exceed a given level γ > 0.

In the previous chapter, we developed algorithms that can be used to design H_∞controllers and ?H_∞filters for dynamical linear systems with time delay. Since these algorithms are based on nominal systems, there is no guarantee that the robustness of system performance will be assured in the presence of uncertainties. To overcome this and avoid any trouble we may have, we should take into account system uncertainties during the analysis and the design phase. Therefore, the problems we studied in the previous chapter should be extended to cope with system uncertainties. Here we will consider norm-bounded uncertainties and deal with the robust H_∞control problem and the robust H_∞-filtering problem.

During the past decades, we have seen more and more examples showing the importance of dynamical systems subject to abrupt variations in their structures. This is partly due to the fact that very often dynamical systems are inherently vulnerable to component failures or repairs, sudden environmental disturbances, changing subsystem interconnections, abrupt variations in the operating point of a nonlinear plant, and so on. The class of Markov jump linear systems (MJLS) is an example of this class of dynamical systems, and it represents an important class of stochastic dynamical systems which, in several situations, is most appropriate for modeling the above phenomenon.

This chapter deals with stability and stabilizability problems of the class of linear systems with Markov jumps and time delay. For this purpose, consider a hybrid system withNmodes, that isS ={1, 2, • • •, N}. Mode switching is assumed to be governed by a continuous-time Markov process {r_tt > Of taking values in the state spaceSand having the following infinitesimal generator

In chapter 8, we dealt with the class of time delay systems with Markov jumps, and we developed results for stability and the stabilizability problems. The dynamic was assumed to be free of uncertainties. But since in practice uncertainties cannot be neglected, this chapter deals with the robust stability and stabilizability of the class of linear uncertain system with Markov jumps and time delay. For this purpose, consider a hybrid system withNmodes, that isS ={1, 2, • • •, N}. Mode switching is governed by a continuous-time Markov process {r_tt> 0} as defined previously.

Practical systems are always subject to exogenous disturbance input which can cause performances degradations. The tools we developed are unfortunately inefficient for eliminating the effect of the disturbance. One alternative is to use the R_oocontrol, which entails searching for a controller that stabilizes the system, and at the same time assures disturbance rejection at a given level. The7-l _oo control problem has received considerable attention and thus numerous achievements have been reported to the literature. In the linear time-invariant context, early7-1control theory emerged as a frequency domain design technique [83].

The linear Markov jump model that is usually used in the analysis and design phases is an approximation of a real nonlinear system with Markov jumps in the neighborhood of the operating point. For many reasons well known in the control community, the systems parameters and the operating point change with time and therefore, the fixed linear Markov jump model is not adequate to guarantee robustness of system performance. Besides this, the system can be affected by exogenous disturbances, which will make the degradation worse. To overcome this surprise, the control engineer should take care of these uncertainties and exogenous disturbances during the analysis and design phases to guarantee the required stability and other system performance, despite the presence of uncertainties in the system. The results presented so far for the class of linear systems with Markov jumps and time delay are not adequate to guarantee the robustness of the desired performance. The robust H∞ control problem was developed to maintain robustness of stability and performance when it known algorithms lack robustness, that is, the system parameters have uncertainties. This chapter deals with the robust H∞ control and the guaranteed cost control problems for jump linear systems with norm-bounded uncertainties and time delay. The rest of the chapter is organized as follows.we deal with the robust H∞ control problem. considers he guaranteed cost control problem. we cover the output feedback guaranteed cost control problem.

In the previous chapters we dealt with linear deterministic and stochastic time delay systems, and we solved many problems like stability, stabilizability (using different types of controllers), 7-t_ßcontrol, filtering, and robustness of the techniques therein. The tools presented in these chapters will be efficient for dynamical time delay linear systems, but will fail for nonlinear ones. Therefore an alternative for solving stability and stabilizability problems for the class of nonlinear time delay dynamical systems is needed.