Abstract
This paper mainly studies the problem of the robust stability analysis for sampled-data system with long time delay. By constructing an improved Lyapunov-Krasovskii functional and employing some free weighting matrices, some new robust stability criteria can be established in terms of linear matrix inequalities. Furthermore, the proposed equivalent criterion eliminates the effect of free weighing matrices such that numbers of decision variables and computational burden are less than some existing results. A numerical example is also presented and compared with previously proposed algorithm to illustrate the feasibility and effectiveness of the developed results.
1. Introduction
In the last few decades, there has been much interest in long time delay system. This is due to its key role in theory research and practical application, such as welding industry, communication networks, and electrical power system. Lots of relevant research results to long time delay systems have been reported in the literature. To mention a few, modeling of long time delay system was considered in [1]; analysis and synthesis results of such system were reported in [2–8]; the fault detection and filtering problem were solved in [9, 10]. However, relationship between sampling period and time delay in the plant is not fully considered [11, 12], which has inevitably limited the applicability of the aforementioned resulted.
In the recent years, increasing attention has been devoted to the problem of sampled-data system with long time delay, in which time delay of the plant is usually longer than a sampling period. When such problem is researched in the discrete-time framework, concentrated augmentation approach and direct distribution approach can be chosen. However, in concentrated augmentation approach, time delay is not taken into consideration in process of deriving stability criterion and designing desired controller [5, 6, 13]. Therefore, this approach is usually regarded as being more numbers of decision variable and computational burden than direct distribution approach.
In this paper, we make an attempt to solve the robust stability problem of sampled-data systems with long time delay. Some free weighting matrices and an equivalent criterion are introduced, in order to reduce numbers of decide variable and computation burden. Numerical examples are also presented to illustrate the feasibility and effectiveness of the developed results.
2. Problem Formulation
Consider a continuous plant of sampled-data system with long time delay: where is the state vector and and are the constant matrices of appropriate dimensions. is the initial state vector.
Assumption 1. Sampler is time-driven with a constant sampling period .
Assumption 2. The time delay is uncertain but is evaluated between two adjacent sampling periods; namely, , where is a known constant.
Lemma 3 (see [14]). For given matrices , , and , with appropriate dimensions, holds for all satisfying if and only if there exists :
3. Main Results
Discretizing system (1) in one period, we can obtain the discrete state equation of sampled-data system: where Let Then In the same way, we can get
Choosing appropriate variables , so that they satisfy , then
Remark 4. Modeling of sampled-data system with long time delay has been reported in [13].
3.1. Stability Condition
The stability condition presented in this section is based on system (4); we will proceed with the following theorem.
Theorem 5. System (4) is asymptotically stable if there exist matrices , , , , , , , , , and , satisfying where
Proof. Letting
then
Choosing a Lyapunov function candidate as
where , , , and , , are undetermined.
Defining , and then along the solution of (4), we have
Defining , then we have
Since , , the last three terms are all nonpositive. By the Schur complement [15], inequality (10) guarantees . From (17), we can attain , for a sufficiently small , and , and the asymptotic stability is established.
3.2. Robust Stability Condition
There exist uncertainties in the stability condition of Theorem 5. In this section, we will cope with uncertainties of stability condition (10), such that robust stability condition is derived.
Theorem 6. System (4) is asymptotically stable if there exist matrices , , , , , , , , , and , satisfying where
Proof. With Theorem 5, the system in (4) is asymptotically stable, if there exist matrices , , , and , , satisfying
where
Substitution of the matrices and defined in (7)~(8) leads to
where
By Lemma 3, we readily obtain
where
A congruence transformation is applied to (24), by , we can attain LMI (18).
Fact 1. Definiteness of a matrix is invariant under congruent transformation by a full rank matrix. For instance, if , and is of full rank, then .
Theorem 7. System (4) is asymptotically stable if there exist real symmetric positive definite matrices , , , , , and matrices , , and , such that the following LMI holds: where and , to , are block entry matrices of appropriate dimensions; namely, , , and .
Proof. Consider a nonsingular transformation matrix . Congruent transformation of the LMI given in (10) with yields the following equivalent criterion:
Using Schur complement, we can express the right-side LMI of (28) as follows:
Since holds, we can write the equivalent stability criterion as , if we let the slack variable matrices , , and be as follows:
Hence, is equivalent to (10). In other words, Theorem 6 is equivalent to Theorem 5.
Remark 8. Theorem 7 presents an equivalent condition for the solvability of the stability condition problem for sampled-data system with long time delay. We note that and include norm-bounded uncertain parameters, such that (26) cannot be solved directly. In order to solve this problem, Theorem 9 is derived as follows.
Theorem 9. System (4) is asymptotically stable if there exist real symmetric positive definite matrices , , , , , such that the following LMI holds: where
Proof. The process of proof is similar to Theorem 6, so it is omitted.
Remark 10. The equivalent stability criterion presented in Theorems 7 and 9 has no free weighting matrices, and they have only the matrix variables that are associated in the Lyapunov-Krasovskii function. Therefore, as the total number of matrix variables involved with the proposed equivalent criterion in minimum, it has less offline computational burden compared to Theorem 5. To highlight this aspect, we compare the number of decision variables involved in both the stability criterion in Table 1.
4. Numerical Example
Considering the system in (1) Letting , , and discretizing system (1), new parameter of discrete-time system is attained: Let
Remark 11. The problem of relevant norm-bounded uncertain parameter is further solved in Theorem 9. Compared with [16], the number of decision variables is , which is less than . Moreover, state response curve of Theorem 9’s method has smaller overshoot and quicker convergence speed, which is shown in Figure 1.
| (a) |
| (b) |
5. Conclusions
In this correspondence paper, we have presented an equivalent stability criterion with less number of LMI variables for a robust stability criterion reported in Theorem 6. By employing congruent transformation with nonsingular matrix, the free weighting matrices in the existing criterion are eliminated without sacrificing the conservation of stability criterion; this, in turn, yields an equivalent criterion that is mathematically less complex and computationally less expensive.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported in part by the National Natural Science Funds for Young Scholar (no. 51307045), and the authors are indebted to the editor and reviewers for their valuable comments and suggestions.