Improvements on Robust Stability of Sampled-Data System with Long Time

This paper mainly studies the problem of the robust stability analysis for sampled-data systemwith 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.


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][3][4][5][6][7][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.

Problem Formulation
Consider a continuous plant of sampled-data system with long time delay: where () ∈ R  is the state vector and  0 and  1 are the constant matrices of appropriate dimensions. 0 is the initial state vector.

Mathematical Problems in Engineering
Assumption 2. The time delay  is uncertain but is evaluated between two adjacent sampling periods; namely, ( − 1) ℎ ≤  ≤ ℎ, where  > 1 is a known constant.
Lemma 3 (see [14]).For given matrices Q = Q T , H, and E, with appropriate dimensions, holds for all F() satisfying F T ()F() ≤  if and only if there exists  > 0:

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  1 ,  2 , . . .,   , so that they satisfy    (ℎ−−  ) ≤ 1, then Remark 4. Modeling of sampled-data system with long time delay has been reported in [13].

Stability Condition.
The stability condition presented in this section is based on system (4); we will proceed with the following theorem.
Fact 1. Definiteness of a matrix is invariant under congruent transformation by a full rank matrix.For instance, if  =  T ∈ R × , and  ∈ R × is of full rank, then  > 0 →  T > 0.
Congruent transformation of the LMI given in (10) with Λ yields the following equivalent criterion: Using Schur complement, we can express the right-side LMI of (28) as follows: Since Γ 3 < 0 holds, we can write the equivalent stability criterion as Π 1 + Γ 1 + Π 3 < 0, if we let the slack variable matrices , , and  be as follows: Hence, Π 1 + Γ 1 + Π 3 < 0 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 sampleddata system with long time delay.We note that Π 1 and Π 3 include norm-bounded uncertain parameters, such that (26) cannot be solved directly.In order to solve this problem, Theorem 9 is derived as follows.
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.

Numerical Example
Considering the system S in (1)    [16], the number of decision variables is 3.5 2 + 3.5, which is less than 20.5 2 + 4.5.Moreover, state response curve of Theorem 9's method has smaller overshoot and quicker convergence speed, which is shown in Figure 1.

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.

Table 1 :
Comparison of decision variables.Remark 11.The problem of relevant norm-bounded uncertain parameter is further solved in Theorem 9. Compared with