Submission toISA Transactions
Title: A delay decomposition approach to stability analysis of uncertain systems with time-varying delay
Author: Pin- Lin Liu
Address: Department of Electrical Engineering,ChienkuoTechnologyUniversity,
1 Chien-Shous N. Load, Changhua, Taiwan, R.O.C.
Phone & Fax No. Tel :886-4-7111155 Fax: 886-4-711129
Email: lpl@ ctu.edu.tw
A delay decomposition approach to stability analysis of uncertain systems with time-varying delay
Pin- Lin Liu
Abstract
This paper is concerned with delay-dependent stability for uncertain systems with time-varying delays. The proposed method employs a suitable Lyapunov– Krasovskii’s functional for new augmented system. Then, based on the Lyapunov method, a delay-dependent stabilization criterion is devised by taking the relationship between the terms in the Leibniz-Newton formula into account. By developing a delay 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 (LMIs) which can be easily solved by various optimization algorithms.Numerical examples are included to show that the proposed method is effective and can provide less conservative results.
Keywords:Lyapunov–Krasovskii’s functional, Stability, Time-varying delay, Linear matrix inequalities (LMIs), Maximum admissible upper bound (MAUB)
1. Introduction
Time delay is one of the instability sources for dynamical systems, and is a common phenomenon in many industrial and engineering systems such as those in communication networks, manufacturing, and biology. Since system stability is an essential requirement in many applications, much effort has been made to investigate stability criteria for various time-delay systems during the last two decades. For details, see the works [1-24] and references therein.
Recently, some researchers have paid attention to the issue concerning delay-dependent stability analysis [2-24]. The development of the technologies for delay-dependent stability analysis has been focusing on effective reduction of the conservation of the stability conditions. The main aim is to derive a maximum admissible upper bound (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.
On the other hand, many uncertain factors exist in practical systems. Uncertainty in a control system may be attributed to modeling errors, measurement errors, parameter variations and a linearization approximation [9]. Therefore, study on the robust stability of uncertain systems with time-varying delay becomes significantly important. The results obtained will be very useful to further research for roust stability and control design of uncertain control systems with time-varying delay[4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 23, 24] and references therein.
The development of the technologies for delay-dependent stability analysis has been focusing on effective reduction of the conservation of the stability conditions. Two effective technologies have been widely accepted: the bounding technology [4], and the model transformation technology [2]. The former is used to evaluate the bounds of some cross terms arising from the analysis of the delay-dependent problem; while the latter is employed to transform the original delay models into simpler ones so that the stability analysis becomes easier.
However, it is also known that the bounding technology and the model transformation technique are the main source of conservation. To further improve the performance of delay-dependent stability criteria, there is a need to avoid this source of conservation. Therefore, much effort has been devoted recently to the development of the free weighting matrices method, in which neither the bounding technology nor model transformation is employed [3, 11, 19, 20, 21]. Generally speaking, the free weighting matrices appear in two forms: the one with a null summing term added to theLyapunov functional derivative [19, 20, 21], and the one with free matrices item added to the Lyapunov functional combined with the descriptor model transformation [3, 11]. However, this approach introduced some slack variables apart from the matrix variables appearing in Lyapunov-Krasovskii functionals (LKFs). When the upper bound of delay derivative may be larger than or equal to 1, Zhu and Yang [23, 24] used a delay decomposition approach, and new stability results were derived. Compared with [6], the stability results in [23, 24] are simpler and less conservative.
Motivated by the above discussions, we propose new stability criteria for uncertain systems with time-varying delays. Based on the Lyapunov function method, a novel robust delay-dependent criterion, which is less conservative than delay-independent one when the size of delays is small, is established in terms of LMIs which can be solved efficiently by using the optimization algorithms. In order to derive less conservative results, a new integral inequality approach (IIA) which utilizes free weighting matrices is proposed. Also, the model transformation technique, which leads an additional dynamics, is not used in this work. Finally, three numerical examples are shown to support that our results are less conservative than those of the existing ones.
2. Problem statement
Consider the following uncertain system with a time-varying state delay:
(1a)
(1b)
whereis the state vector, A and B are constant matrices with appropriate dimensions; is a smooth vector-valued initial function.is a time varying delay in the state, and is an upper bound on the delay The uncertainties are assumed to be the form
where and are constant matrices with appropriate dimensions, and is an unknown, real, and possibly time-varying matrix with Lebesgue-measurable elements satisfying
(3)
As in [2], we consider two different cases for time varying delays:
Case I: is a differentiable function, satisfying for all
(4)
Case II: is not differentiable or the upper bound of the derivative of is unknown, and satisfies
(5)
where and are some positive constants.
To do this, two fundamental lemmas are reviewed. First, Lemma 1 induces the integral inequality approach.
Lemma 1 [9]. For any semi-positive definite matrices
(6a)
the following integral inequality holds
Secondary, we introduce the following Schur complement which is essential in the proofs of our results.
Lemma 2[1].The following matrix inequality
(7a)
where depend on affine on is equivalent to
(7b)
(7c)
and
(7d)
Lemma 3 [1]: Given matrices and of appropriate dimensions,
(8)
for all satisfying if and only if there exists some such that
(9)
The purpose of this paper is to find new stability criteria, which are less conservative than the existing results.
Firstly, we consider the nominal from system (1):
(10)
For the nominal system (10), we will give a stability condition by using a delay decomposition approach as follows.
Theorem 1: In Case I, if for given scalars andthe system described by (10) with (4) is asymptotically stable if there exist matrices and semi-positive definite matrices such that
(11)
and
(12)
where
Proof: In Case I, a Lyapunov functional can be constructed as
(13)
where
Taking the time derivative of for along the trajectory of (10), it yields that
(14)
whereand
Now, we estimate the upper bound of the last three terms in inequality (17) as follows:
From integral inequality approach of Lemma 1[9], noticing that and it yields that
Similarly, we obtain
and
The operator for the term is as follows
Combining (13)–(22), it yields
where and with
From nominal system (10) and the Schur complement of Lemma 2, it is easy to see that holds if and
Theorem 2:In Case I, if for given scalars andthe system described by (10) with (4) is asymptotically stable if there exist matrices and semi-positive definite matrices such that
(24)
and
(25)
where
Proof: If it gets
From integral inequality matrix [9], noticing that and it yields that
Combining (13)–(22) and (26)-(29), it yields
where and
From nominal system (10) and the Schur complement, it is easy to see that holds if and
Theorem 3: In Case II, for given scalars , the system described by (10) with (5) is asymptotically stable if there exist matrices and semi-positive definite matrices such that
(31)
and
(32)
where
In Case II, a Lyapunov functional can be chosen as (11) withSimilar to the above analysis, one can get that holds if
Thus, the proof is completed.
Now, extending Theorems 1-3 to uncertain system (1) with time-varying delays yields the following Theorems.
Theorem 4:In Case I, if for given scalars andthe uncertain system described by (1) with (4) is asymptotically stable if there exist matrices and semi- positive definite matrices such that
(33)
and
(34)
where andare defined in (11).
Proof: Replacing and in (11) with and respectively, we apply Lemma 3[1] for system (1) is equivalent to the following condition:
(35)
where
and
By lemma 3 [1], a sufficient condition guaranteeing (11) for system (1) is that there exists a positive number such that
(36)
Applying the Schur complement of Lemma 2 shows that (36) is equivalent to (33). This completes the proof.
Theorem 5: In Case I, if for given scalars andthe uncertain system (1) with (4) is asymptotically stable if there exist matrices and semi-positive definite matrices such that
(37)
and
(38)
where andare defined in (22).
Proof: Replacing and in (24) with and respectively, we apply Lemma 3[1] for system (1) is equivalent to the following condition:
(39)
where
and
By lemma 3 [1], a sufficient condition guaranteeing (11) for system (1) is that there exists a positive number such that
(40)
Applying the Schur complement shows that (40) is equivalent to (37). This completes the proof.
Theorem 6: In Case II, for given scalars , the uncertain system (1) with (5) is asymptotically stable if there exist matrices and semi-positive definite matrices such that
(42)
where andare defined in (31).
Proof: Replacing and in (31) with and respectively, we apply Lemma 3[1] for system (1) is equivalent to the following condition:
(43)
where
and
By lemma 3 [1], a sufficient condition guaranteeing (43) for system (1) is that there exists a positive number such that
(44)
Applying the Schur complement of Lemma 2 shows that (44) is equivalent to (41). This completes the proof.
Remark 1: In the proof of Theorems 1-6, the interval is divided into two subintervals and , the information of delayed state can be taken into account. It is clear that the Lyapunov function defined in Theorems 1-6 are more general than the ones in [6, 20], etc.
Remark 2: In the previous works except [2, 6, 7, 11, 13, 14, 16, 18, 19], the time delay term was usually estimated aswhen estimating the upper bound of some cross term, this may lead to increasing conservatism inevitably. In Theorems 1-6, the value of the upper bound of some cross term is estimated more exactly than the previous methods sinceis confined to the subintervals or So, such decomposition method may lead to reduction of conservatism.
Remark 3:In the stability problem, maximum allowable delay bound (MADB)which ensures that time-varying delay uncertain system (1) is asymptotically stable for any can be determined by solving the following quasi-convex optimization problem when the other bound of time-varying delay is known.
Inequality (45) is a convex optimization problem and can be obtained efficiently using the MATLAB LMI Toolbox.
Five examples will be presented in the following section to highlight the effectiveness of the proposed method.
3. Illustrative examples
In this section, five examples are provided to illustrate the advantages of the proposed stability results.
Example 1: Consider the uncertain system with time varying delay as follows:
(46)
where
Now, our problem is to estimate the bound of delay time to keep the stability of system.
Solution:When using the stability criteria in Parlakci [12], Qian et al.[16] and this paper of Theorem 4, the calculated maximum allowable delay bound (MADB)for the time delay are and and respectively. So the proposed method in this paper yields a less conservative result than that given in Parlakci [12]and Qian et al.[16].
Example 2: Consider the uncertain system with time varying delay as follows:
(47)
where
Solution:For, by Theorem 4, we can obtain the maximum upper bound on the allowable size to be However, applying criteria in [22, 15, 18], the maximum value of for the above system is 0.92, 1.1072 and 1.1075. We also apply Theorems 4-5 to calculate the maximum allowable value for different Table 1 shows the comparison of our results with those in [4, 8, 10, 16, 19, 21]. This example demonstrates that our robust stability condition gives a less conservative result.Hence, it is obvious that the results obtained from our simple method are less conservative than those obtained from the existing methods.
Example 3: Consider the uncertain system with time varying delay as follows:
(48)
where
Solution: For comparison, the Table 2 also lists the maximum allowable delay bound (MADB) obtained from the criteria [2, 6, 7, 11, 13, 14, 16, 18, 19]. It is clear that Theorems 4-5 give much better results than those obtained by [2, 6, 7, 11, 13, 14, 16, 18, 19]. It is illustrated that the proposed robust stability criteria are effective in comparison to earlier and newly published results existing in the literature.
Example 4:Consider the uncertain system with time varying delay as follows:
(49)
where
Solution: When is a constant, by the proposed method of Theorem 6 in this paper and those in [11, 13, 16, 20], for different values ofwe can get the maximum allowable delay bound (MADB) and the results are listed in Table 3, which indicate that our result is less conservative than those in [11, 13, 16, 20].
When Table 4 gives the comparisons of the maximum allowed delay for various It can be seen that the robust stability condition in this paper is less conservative than the one in [6, 23].
Example 5:Consider the system with time varying delay as follows:
(50)
where
Solution: We let as [5] did. From Table 5, we could easily find that the results proposed in this note are better than those of [5, 6, 17]. The conclusion we draw is better than [5, 6, 17]when is small.
4. Conclusions
This paper has studied the problem of robust stability for uncertain systems with time-varying delay. By developing a delay 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 (LMIs) which can be easily solved by various optimization algorithms. Since the delay term is concerned more exactly, less conservative results are presented. Examples have shown that the resulting stability criteria outperform the existing ones in the literature.
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