Observer-based Control for Time–varying Delay Systems with Delay-dependence

This paper is concerned with the problem of observer control of a class of time-varying delay systems with delayed measurements. By using information on delay derivative, improved asymptotic stability conditions for time-delay systems are presented. Unlike previous methods, upper bound of delay derivative is taken into consideration, even if this upper bound is larger than or equal to 1. We develop delay-dependent method for designing linear observer controllers, thus ensuring global uniform asymptotic stabilization for any time delay no larger than a given bound. Numerical examples are given to illustrate effectiveness and less conservatism of obtained stability conditions.


Introduction
Time-delay systems constitute a special class of dynamical systems that are commonly found in chemical and biochemical engineering systems, as well in other fields of engineering. Time-delay usually lead to unsatisfactory performance and is frequently a source of instability. Stability properties of time-delay systems have been extensively studied in literature, mainly in applied mathematics and control theory related journals [1-8, 10-13, 15-18, 20-24]. It can be claimed that any real-life system has time delay associated with it. The delay in a system may be due to one or more of the following causes: (i) measurement of system variables, (ii) physical properties of the equipment used in the system, and (iii) signal transmission (transport delay) [20].
For systems with time-varying delays, the above mentioned literature usually demand that the upper bound of the derivative of delays must be smaller than 1. If the upper bound of the derivative of delays is larger than 1, the results in [8,10,11,22] are not applicable.
It is well known that a static (or dynamic) output controller, which uses only the output for feedback, is more practical to deal with time-varying delay systems [7,14]. In particular, the observer-based output feedback controller, which is a dynamic output feedback controller, can on-line estimate the system states. The control of such systems has been thoroughly studied. One of the major difficulties in implementing a control law is that all of the state variables of the system are required for the controller synthesis. However, in the most practical situations, this condition is rarely satisfied and an observer has to be building up in order to estimate the state variables from the output and the input measurements. Observer analysis of such kind of systems has been the subject of numerous papers and monographs [2, 4, 5, 9, 18, 21, 23; 24] and the references therein. Referring to the extensive results on stabilization and observation of time-delay systems, the conditions under which stabilizing feedbacks exist are classified into two possible categories: delay independent conditions and delay-dependent ones [17]. Even, delay-independent conditions do not take into account the size of the delay and may lead to robust control design, the delay-dependent results reveal less conservative than delay independent conditions. Nevertheless, in most cases, a small delay is tolerable to maintain stability by output feedbacks. Over past years, the stability analysis and control for dynamic time-delay systems attracted a large number of researchers. Recently, using Lyapunov Krasovskii functions, a lot of control methods have been developed to treat the various non-linear systems involving time delays, such as linear matrix inequality method [11], free weighting matrix method [6], an integral inequality approach method [16] and so on. However, free-weighting matrix method is too complicated and it needs heavy computational burden. In order to overcome this limitation, recently, new convex combination conditions for the stability of the systems with time delays are established without free-weighting matrices. We apply an observer controller to reconstruct an approximation of the unavailable state from the available input and output of the system. However, in the control of time-delay systems, it is usually desirable to design a controller which not only robustly stabilizes the system but also estimate the bound of delay time h to keep the stabililization of system.

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Observer-based Control for Time-varying Delay Systems with Delay-dependence The present paper addresses problem of stabilization for linear time-varying delay systems under observer controller. A delay dependent robust stabilization condition is derived based on linear matrix inequalities (LMIs). The stabilization analysis is derived in three steps: (i) applying a state-observer controller to reconstruct an approximation of the unavailable state from the available input and output of the systems, (ii) the choosing of Lyapunov-Krasovskii functional with linear matrix inequality (LMI) techniques and integral inequality approach (IIA) in the designed observer controller and (iii) finally, a maximum allowable delay bound which ensures that the class of time-varying delay system with observer controller considered in this paper is stabilizable for any h is determined by solving an optimization problem.

Stability Description and Preliminaries
Let us consider the system described by the following linear differential-difference equation is the state vector of the system; where h and d h are given positive constants.
In this control system, we assume that ( 0 A ,B) is controllable, i.e. the process state x(t) can be determined on the basis of control input ( ) u s for . s t ≤ For system (1) subject to delayed measurement, if all the states are not measurable, design an observer controller as follows: such that the closed-loop system is stable, where K is the controller gain matrix and L is the observer gain matrix. Introducing the observer error Combining (1), (2) and (3), we can get Before embarking on main results, the following lemmas are introduced, which play important roles in the proof of the main results. First, Lemma 1 induces the integral inequality approach (IIA). Lemma 1 [16]. For any positive semi-definite matrices: 11 12 13 12 22 23 the following integral inequality holds: 33 ( ) ( ) 11 12 13 Secondary, we introduce the following Schur complement which is essential in the proofs of our results. Lemma 2 [1]. The following matrix inequality: x is equivalent to Substituting (11) and   13  13  11  13  23  1  12  1   23  13  1  12  1  23  23  22  1  1   .  (1  )   T  T  T  T  T  e  e  e  e  e  e  e   T  T  T  T  T  T  e  e  e  e  e  d   P P  Q  h (1)

Observer Control Design
In this section, we seek a design method of the observer control for a time-varying delay system. Unfortunately, Theorem 1 does not give a feasible LMI condition for obtaining a state feedback control gains matrices K and . L Hence, we look for another stabilization condition. To this end, we make a congruence transformation and obtain a feasible LMI stability condition. Based on it, we give a design method of an observer controller. The following Theorem 2 gives an LMI-based computational procedure to determine dynamic controls. Then we have the following result.  T  T   T  T  T   T  T  T  T  T   T  T  T   T  aa  aa   T  T  ab  bb   h  h  W  A  T  T  T  T  T  T  h  h  T  T  T  T  T   (1 ) , .
In this case, observer control gains in (3) is given by In view of Theorem 1, to prove the asymptotic stability of the closed-loop system with control it suffices to show that there exist symmetric, positive-definite matrices 0, P > 0 Q > and 0 R > such that (8) leads to (11). Thus, if 1 2 , , , , W U Z Y Y and ijab T are a set of feasible solution to LMI (13), then (13). This ends the proof.

Illustrative Examples
Example 1: Consider the following time delay dynamic system as follows: We assumed that the states of the system are not available. How do we find the maximum allowable delay bound (MADB) h with an observer-based controller to guarantee the system (15) to be asymptotically stable.    (15) is asymptotically stable under observer controller (3) with controller gain matrix K and the observer gain matrix .
L The simulation of the above closed system (15) for h =2.17 is depicted in Fig.1 and 2.
Now, our problem is to calculate the maximum allowable delay bound (MADB) h for which an observer control law, controller gain matrix K and the observer gain matrix , L exists to stabilize (16). Solution: To begin with, for 0, d h = equation (16) reduces to the system discussed in [13]. Solving the quasi-convex optimization problem (14), according to the Theorem 2, using the soft-ware package LMI Toolbox, we obtain the corresponding controller gain matrix K and the observer gain matrix L are -0.  Fig.3 and 4. By the criteria in [13], the system (16) is asymptotically stable for <0.2650. h Hence, for this example, the criteria proposed here significantly improve the estimate of the stability limit compared for the result of [13].

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Observer-based Control for Time-varying Delay Systems with Delay-dependence   [13] cannot be applied to check stabilization of system (16). It is clear that the results obtained in this paper are better than those in [13].

Conclusion
The state feedback and observer control problem of a class of time-varying delay systems subject to delayed measurements has been investigated. This problem has been cast into a framework of convex optimization. By constructing a Lyapunov-Krasovskii functional and using a recently developed integral inequality approach, existence conditions for the state feedback and the observer controller have been established. The stability conditions obtained are dependent of the delay values, and are generally less restrictive than those previously presented in the literature. Numerical simulations are given and the results show that the designed observers and controllers are feasible and valid.