Delayed Decomposition Approach to Delay-Dependent Stability for Time-Varying Delay Descriptor Systems

Abstract This paper studies the problem of stability analysis for descriptor systems with time-varying delay. By developing a delayed decomposition approach, information of the delayed plant states can be taken into full consideration, and new delay-dependent sufficient stability criteria are obtained in terms of linear matrix inequalities (LMIs). Then, based on the Lyapunov method, delaydependent stability criteria are devised by taking the relationship between terms in the Leibniz-Newton formula into account. Criteria are derived in terms of LMIs, which can be easily solved by using various convex optimization algorithms. It is proved that the newly proposed criteria may introduce less conservatism than some existing ones. Meanwhile, the computational complexity of the presented stability criteria is reduced greatly since fewer decision variables are involved. Numerical examples are included to show that the proposed method is effective and can provide less conservative results.


Introduction
Time-delays are often encountered in various dynamic systems, such as manufacturing systems, economic systems, biological systems, networked control systems, and so on. The time-delay is frequently a source of instability and performance deterioration. Therefore, stability analysis and controller synthesis for time-delay system have been one of the most challenging issues . The main aim of these studies is to achieve a maximum admissible upper bound (MAUB) such that the system under consideration is globally asymptotically stable for any time delay less than the MAUB. On the other hand, the descriptor system is referred to as a singular system, a generalized state-space system or a semi-state system. It is commonly encountered in many fields such as aircraft attitude control, flexible arm control of robots, large-scale electric network control, chemical engineering systems, lossless transmission lines [2, 4-13, 19-21, 23-28, 30].
Recently, due to the fact that descriptor systems better describe physical systems, many papers have investigated these and some important results have been proposed. To obtain delay-dependent conditions, many efforts have been made in the literature, among which the model transformation technique and bounding technique on cross-product terms are often used [5,6,20,24,25]. As is well known, these two methods will cause some conservatism. Recently, some improved delay dependent stability criteria have been obtained without using model transformation and bounding techniques for cross terms [8, 11, 14-18, 27, 29, 31]. In [8], Jiang et al. proposed free weighting matrices approach to investigate the delay-dependent stability. By employing a Jensen integral inequality technique, [11,27,29,31] further extended the free weighting matrix method. It should be noted that through this methods are effective for the systems with a constant time delay; it can only made a little improvement for the systems with a time-varying delay. Many researchers have realized that many free variables introduced in free weighting matrices method will complicate system analysis, thereby yielding significant increase in computational demand [13]. Hence computational burden involved in solving stability criteria using standard numerical packages becomes minimal.
In [13], an integral inequality was presented for less computational requirement in a alternative stability criterion, and it was intensively used in later literature [14][15][16][17][18]. Later, [14][15][16][17][18] provided integral inequality matrix and delayed decomposition approach, information of the delayed plant states can be taken into full consideration, and new delay-dependent sufficient stability criteria are obtained in terms of linear matrix inequalities (LMIs). Based on the result of [14][15][16][17][18] made some improvements and provided a new integral inequality method to the stability and stabilization analysis of the linear delay-dependent systems, which is proved to be less conservative than the previous Universal Journal of Control and Automation 1(2): 58-67, 2013 59 criteria. Hence computational burden involved in solving stability criteria using standard numerical packages becomes minimal. Furthermore, we not only theoretically prove that our results are less conservative than those in [6,8,24,27,31], but also show that our results have higher computational efficiency than those in [3,6,7,11,21,[23][24][25][26][27] and some existing results in [14][15][16][17][18] are special cases of our results. Yet when estimating the upper bound of the derivative of Lyapunov functional for descriptor systems with timevarying delay, there is room for investigation.
Motivated by the statement above, this paper proposes improved delay-dependent stability of the descriptor time-varying delay systems studied. An improved delaydependent stability criterion is established in terms of linear matrix inequalities (LMIs), which guarantee the descriptor time-delay system to be regular, impulse free and asymptotically stable. A new delay-dependent stability criterion is derived via a linear matrix inequality formulation that can be easily solved by various convex optimization techniques. The main aim is to derive a MAUB of the time-delay such that the time delay system is asymptotically stable for any delay size less than the MAUB. Accordingly, the obtained MAUB becomes a key performance index to measure the conservatism of a delay-dependent stability condition. We will show that the obtained criterion includes some existing results as its special cases and numerical examples are also given to show the less conservativeness of the presented criterion.

Stability Description and Preliminaries
Consider the following descriptor system with a time-varying state delay: Lemma 2 (Liu [13][14][15][16][17][18]). For any positive semi-definite matrices 0, the following integral inequality holds:  [1]). The following matrix inequality: depend on affine on , x is equivalent to Delayed Decomposition Approach to Delay-Dependent Stability for Time-Varying Delay Descriptor Systems
where 1 0 0.5 0 1.1 1 , , our problem is to estimate the MAUB h to keep the stability of system (30).  Table 2. It can be seen that the delay-dependent stability condition in this paper is less conservative than earlier reported ones in the literature [10,21]. Compared with Wang et al. [21] that used 21 LMI variables, we need 11 variables in Theorem 1 the same as [10]. It can be shown that the delay-dependent stability condition in this paper is the best performance.

Solution. Choosing
where 1 0 0.5 0 1 0 , , . 0 0  Table 3 lists the results compared with [7,11,14,[23][24][25][26][27]30]. It can be seen from Table 3 that the maximum admissible upper bound (MAUB) h by using Theorem 3 is the largest with the fewest variables computed. As Fig. 1 shows, the simulation of the above system (31) Table 4 lists the MAUB h as judged by the criteria in [3,6,18,20]. It can be seen that the delay-dependent stability condition in this paper is less conservative than earlier reported ones in the literature [3,6,18,20]. We also know that with fewer matrix variables the stability results obtained in Theorem 3 is less conservative than the one in [3,6,18,20]. Using these data, a simulation program has been written in Matlab. As Fig. 2 shows, the simulation of the above system (32) where 0 ( ) . h t h ≤ ≤

Solution.
When h is a constant. For different methods, the computed MAUB , h which guarantee the stability of system (33) are listed in Table 5. It is clear that the result obtained by Theorems 3 and 4 are less conservative than the ones by the methods in [31], and the result obtained by Theorems 3 and 4 outperforms the existing ones [31].

Conclusion
In this paper, a delayed decomposition approach has been developed to investigate the stability of descriptor systems with a time-varying delay. By developing a delayed decomposition approach, the information of the delayed plant states can be taken into full consideration, and new delay-dependent sufficient stability criteria are obtained in terms of linear matrix inequalities (LMls). The stability condition is expressed in terms of easily computable LMIs. It is proved that the obtained results are less conservative than some existing ones. Meanwhile, the computational complexity of the new stability criteria is reduced greatly since fewer decision variables are involved. An algorithm of seeking appropriate tuning parameter is also presented. Numerical examples have illustrated the effectiveness of the proposed methods.