An Improved Finite-Time Stability and Stabilization of Linear System with Constant Delay

Practical systems in engineering fields often require that values of state variables, during the finite-time interval, must not exceed a certain value when the initial values of state are given. This leads us to investigate the finite-time stability and stabilization of a linear system with a constant time-delay. Sufficient conditions to guarantee the finite-time stability and stabilization are derived by using a new form of Lyapunov-Krasovskii functional and a desired state-feedback controller. These conditions are in the form of LMIs and inequalities. Two numerical examples are given to show the effectiveness of the proposed criteria. Results show that our proposed criteria are less conservative than previous works in terms of versatility of minimum bounds and larger bounds of time-delay.


Introduction
In the past decades, researchers have paid much attention to asymptotic stability which concerns behaviors of state variables over an infinite time interval.One disadvantage of the asymptotic stability behavior is that large values of state variables may present during transient period.In practical system, the presence of large values should not exceed its limit, for example, the presence of saturations or the excitation of nonlinear dynamics [1,2].This leads us to a concept called finite-time stability, introduced back in 1960s.This concept is focusing on the behavior of state variables during the transient period which must not exceed a certain value when the upper bound of initial condition is given (see [1,[3][4][5]).Researchers have studied finite-time stability on various systems such as linear system, impulsive system, neural networks, and switched systems (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein) and proposed sufficient conditions to guarantee finite-time stability in the forms of linear matrix inequality (LMI), Lyapunov differential matrix equation, or algebraic inequality, and so forth.
Time-delay often occurs in practical systems such as biological systems, chemical systems, electrical networks, and engineering fields.It is known that the small change of timedelay can cause instability and poor performance of such systems.Considering the broad applications related to the time-delay and the suitable values of state variables during the transient, it is important to investigate the finite-time stability and stabilization of systems with time-delay.From literature, the studies of finite-time stability and finite-time stabilization on time-delay system are not many [9,[12][13][14][15][16][18][19][20][21] and only few studies are on linear system with time-delay [9,11,16].
In this paper, we propose new sufficient conditions on finite-time stability and stabilization for a linear system with constant time-delay in the form of LMIs.The proposed conditions are formulated using a new form Lyapunov-Krasovskii functional.To illustrate the efficiency of the proposed conditions, two numerical examples are presented at the end.

Preliminaries
The following notations will be used in this paper.  denotes the -dimensional space with the scalar product    and the vector norm ‖ ⋅ ‖;  × denotes an  ×  matrix with real value elements;   denotes the transpose of the matrix ; () 2 Mathematical Problems in Engineering denotes eigenvalues of ;  max () and  min () represent the maximum and minimum of real part of (), respectively.  := {(+) :  ∈ [−, 0]}; ‖  ‖ := sup ∈[−,0] {‖(+)‖, ‖ ẋ (+ )‖};  > 0 ( < 0) means  is positive (negative) definite;  >  is equivalent to  −  > 0. Entries * in a matrix represent the symmetric elements of the symmetric matrix.Consider a linear system with constant time-delay where  > 0 and () ∈   is the state vector of the system.() ∈   is the control input. 0 ,  1 ∈  × and  ∈  × are known constant matrices.The delay  is a positive constant.

Main Results
In this section, we first formulate the finite-time stability of the linear system (1) without controller, that is, () = 0.
We then provide proof of the finite-time stabilization of the linear system (1) with a feedback controller defined in (2).The formulations are as follows.
Next consider the linear system (1) with a feedback controller of the form () = −0.5 () as defined in (2).This equation can be rewritten as where Â0 =  0 − 0.5  .
Remark 5.One can notice that the finite-time stabilization of (1) with the feedback controller above is equivalent to finitetime stability of (19).Thus we can formulate the finite-time stabilization as in the following theorem. where Proof.Replacing  0 in the LMI (8) with Â0 =  0 − 0.5  , we obtain where
With the setting relations above, inequalities (17) in Theorem 4 can be bounded by The proof is complete.

Numerical Examples
In this section, we give two numerical examples to show the effectiveness of our main results by investigating the linear system of the form where Remark 7. The linear system (28) with () = 0 is not asymptotically stable with initial condition   () = [0.7,0, 0] as shown in Figure 1.The figure reveals that the state variables   () → ∞,  = 1, 2, 3, as  → ∞.In the next example we will show that this system is finite-time stable.Example 1.Consider the finite-time stability of linear system (28) with respect to ( 1 ,  2 , ) = (0.55, 100, 2) and   () = [0.7,0, 0] without the controller; that is, () = 0. Note that   ()() = 0.49 < 0.55 =  1 .For fixed  = 1.95, we solve inequalities (6) and LMIs ( 7) and ( 8 We further investigate the finite-time stability of the linear system by comparing the smallest eligible value of  2 and the largest eligible value of delay  when  varies between condition given in [16]  In addition, we compare the smallest eligible value of  2 for two different time-delays ( = 0.2, 0.5) with  = 2.1.bounds of  2 are smaller than those given in [16] by 17.1% and 6.8% for  = 0.2, 0.5, respectively.In other words, our proposed condition for finite-time stability of the linear system (28) would be more tolerant for smaller values of  2 than conditions given in [4,12,16].Note that results obtained in [4,12]    (31)
Remark 8.The derivations of the main theorem above are based upon the Lyapunov-Krasovskii approach.Thus, these conditions are not only finite-time stable, but also asymptotically stable.This behavior can be seen when the time domain in Figure 3 is extended.

Conclusion
In this paper, the finite-time stability and stabilization conditions of the linear system with constant delay are obtained.The sufficient conditions are formulated using a new form of Lyapunov-Krasovskii functional.Results from both examples illustrate that our proposed criteria are less conservative than other existing works.

Figure 2 :
Figure 2: Comparing the values of  2 (a) and the time-delay  (b) when  is varied guaranteeing FTS of the linear system (28).
and our condition in Theorem 4. Comparing results are plotted in Figure 2. From this figure, one can clearly see that our condition allows smaller value of  2 (Figure 2(a)) and larger value of delay  (Figure 2(b)) for all values of .Moreover, we observe that the optimal values of  2 and  are obtained when  ∈ [2.05, 2.1].For  = 2.05 is fixed, Theorem 4 allows smaller value of  2 by 17.1% and larger value of  by 11.2%.

Table 1 ,
reveal that Theorem 4 allows smallest value of  2 compared with the others.Our lower
do not use LMI technique.