Further Results on Reachable Set Bounding for Discrete-Time System with Time-Varying Delay and Bounded Disturbance Inputs

1School of Information Engineering, Fuyang Normal University, Fuyang 236041, China 2School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China 3School of Mathematics Science, Huaibei Normal University, Huaibei 235000, China 4School of Information Science and Engineering, Chengdu University, Chengdu 610106, China 5School of Science, Hubei University for Nationalities, Enshi, Hubei 445000, China


Introduction
The reachable set of dynamic system is defined as a collection of system state vectors in presence of all admissible input disturbances.As is known to all, the phenomenon of disturbance inputs is usually unavoidable in engineering and practical systems because of data transformation, measurement errors, linearisation approximations, and some unknown disturbances.Therefore, the reachable set bounding for dynamic systems with input disturbances is an important and challenging research topic in robust control theory and practical engineering.Then, applications of reachable set bounding can be found in various fields such as peak-topeak minimization, aircraft collision avoidance, parameter estimation, and constrained control design [1][2][3].On the other hand, time delay often occurs in various real-world systems and the existence of it may lead to instability, oscillation, or bad system performance [4][5][6].Thus, stability and control of time-delay systems plays an important role in the field of engineering.As a result, many scholars have devoted themselves to the study of time-delay systems [7][8][9][10][11][12][13][14][15][16][17][18] and varieties of methods have been introduced to solve the stability analysis and control synthesis of delayed systems, such as delay-partitioning method [10,11], reciprocally convex approach [13], and the free-weighting matrices technique [14].
In many practical applications, we often require to find a bound of a set of all the states that are reachable from a given set.Two definitions on forward and backward reachable sets are introduced in [18][19][20].Those notions have been widely applied to state bounding observers, safety verification, and model checking.Forward reachable set with regard to a given initial set of a dynamic system is the set of all the states starting from the given initial set.It should be pointed out that reachable set defined in [21][22][23][24][25] is a special case of forward reachable set when the given initial set contains only the origin point.Most of existing results about forward reachable set bounding were reported for dynamic systems without time delay.As for the reachable set bounding problem of timedelay systems, it was firstly solved by Fridman and Shaked 2 Complexity in [2] based on Lyapunov-Razumikhin approach.Then, some excellent achievements on reachable set bounding for dynamical systems have been proposed in the literature [21][22][23][24][25][26][27][28][29][30][31].By using an enlarging Lyapunov-Krasovskii functional and delay-partitioning method, an improved condition was provided in [21].Zuo et al. introduced the maximal Lyapunov functional method to deal with the problem of reachable set bounding for linear systems in [22].Subsequently, the maximal Lyapunov functional approach has been adopted to research the reachable set estimation for delayed neural network [23].In [25], the authors studied the problem of reachable set estimation and synthesis for delayed systems by using delay-decomposition technique and reciprocally convex approach.Recently, the problem of reachable set estimation for delayed singular systems was considered in [31] based on Lyapunov method.In [27], new explicit delayindependent conditions are obtained by utilizing a novel method which does not involve the Lyapunov-Krasovskii functional way.In [32], the authors presented a new approach to obtain the smallest box which bounds all reachable sets of a class of nonlinear system.The problem of state bounding for homogeneous positive nonlinear systems was studied in [33].In this paper, we will further study the forward reachable set bounding problem for delayed system.
It is worth noting that all the above-mentioned research has been focused on continuous-time systems.However, with the progress of computer-based computational techniques, discrete-time systems play a considerable role in many practical applications.Therefore, the stability and control of discrete-time delayed systems is an important issue and has been studied by many researchers [34][35][36][37][38][39][40].Up to now, there are few papers considering the reachable set estimation of discrete-time systems with time-varying delay.In [38], an ellipsoidal-like bounding of the reachable set for discrete-time system with time-varying delay is shown by using the delay-decomposition and the free-weighting matrix techniques.In [39], the problem of reachable set bounding for discrete-time polytopic systems with multiple constant delays was studied; delay-dependent conditions were derived in the form of relaxed matrix inequalities.Compared with the results in [38,39], some less conservative conditions are obtained in [40] by employing a novel Lyapunov-Krasovskii functional combining with delay-decomposition technique and reciprocally convex method.In [41], the reachable set estimation of delayed discrete-time T-S fuzzy systems was considered.The reachable set estimation problem for discrete-time Markovian jump neural networks was firstly investigated in [42].However, there is still much room to investigate the problem of forward reachable set bounding for discrete-time system with time-varying delay, which is our motivation.
In this paper, we concentrate on the forward reachable set bounding problem for discrete-time system with timevarying delay.The ellipsoidal reachable set estimation conditions of discrete-time linear systems are obtained by using discrete Wirtinger-based inequality and reciprocally convex approach.The novelty of this paper is three aspects: first, a relaxed Lyapunov-Krasovskii functional, which does not require all the involved symmetric matrices to be positive definite, is employed to solve the addressed problem; second, the initial state vectors are not necessarily zero, which is more general than the existing results [38,39]; third, discrete Wirtinger-based inequality is taken into account in this paper to deal with the problem of reachable set bounding for delayed discrete-time systems.Finally, two numerical examples are provided to validate the effectiveness of the proposed method.
Notations.Throughout this paper,  > 0( ⩾ 0,  < 0,  ⩽ 0) means that the matrix  is positive definite (positive semidefinite, negative definite, and negative semidefinite); R × is the set of  ×  real matrices; the superscripts −1 and  denote the inverse and transpose of a matrix, respectively; * denotes the symmetric block in symmetric matrix;  denotes the identity matrix with compatible dimensions;  denotes the set of natural number; and Z denotes the set of integers.

Problem Statement and Preliminaries
We consider the following discrete-time system with timevarying delay: where () = [ 1 (),  2 (), . . .,   ()]  ∈ R  is the state vector of the system,  1 ,  2 , and  are known constant matrices with appropriate dimensions, () is the timevarying delay and satisfies 0 <   ≤ () ≤   , and () ∈ R  represents a disturbance which satisfies () is the initial condition function satisfying Define that () = ( + 1) − (), which satisfies In this paper, we denote reachable set of system (1) as Our aim is to find an ellipsoid to bound the reachable set R of system (1).
In addition, we give some lemmas which will be used in deriving our results.

Complexity 3
Proof.From ( 6), we can get then, we have then, we can get This completes the proof.
The other vectors are defined as Theorem 4. For given scalar 0 <  < 1, if there exist positive scalars then, the reachable set of system ( 1) is contained in the following ellipsoid: where Proof.We construct the following Lyapunov-Krasovskii functional for system (1): where First of all, the functional () is required to be positive definite.By using Jensen inequality [37], the following can be obtained: Noticing that we have It follows from ( 23)-( 25) that where From condition (15), we have Together with ( 26)-( 28), we can get Then, taking the forward difference of () with respect to (1), we can get Based on Lemma 2, we have When ℎ  < () < ℎ  , by using Lemma 2, we obtain where ] > 0 and using Lemma 3, we have where Thus, we obtain From ( 30)-( 34), we have where For any matrices  1 ,  2 ,  3 with appropriate dimensions, we have  From ( 35)- (36), we can get Noting that Ξ = Ξ 1 + Ξ 2 < 0, we obtain Furthermore, we describe the initial value of Lyapunov functional as By condition (17) and Lemma 1, we have () ≤ 1.Then, together with (15), it is implied that   ()() ≤ 1.Thus, this means that the reachable set of system (1) is contained in the ellipsoid defined by (19).This completes the proof.

Remark 5. The difficulty of this paper is how to handle the term
In this paper, we using discrete Wirtinger-based inequality instead of discrete Jensen inequality [38][39][40] to deal with the term ∑ −1 =−  ∑ −1 =+   () 1 ().Then, by using reciprocally convex approach and discrete Wirtinger-based inequality, the new inequality (33) is introduced to deal with the term ∑ Remark 6.It is worth pointing out that our results are more relaxed and general compared with the results in [38], because the initial condition is not required to be zero in system (1).In the case where the initial set contains only the origin point, that is,  1 =  2 = 0, it is easy to get (0) = 0 without needing (17).Therefore, we can obtain reachable set bounding for discrete-time system with time-varying delay and it is stated in the following theorem.
Remark 8. Traditionally, it was always assumed that each of the involved symmetric matrices must be positive, which ensured that the corresponding Lyapunov-Krasovskii functional would be positive, while, in this paper, we relaxed these conditions.Particularly, we established a novel Lyapunov-Krasovskii functional that the involved matrices such as ,  1 , and  2 may be nonpositive.
Remark 9.The difficulty of this method lies in how to prove that the relevant Lyapunov-Krasovskii functional () must be positive definite.In order to overcome this difficulty, we take full advantage of Jensen's inequality; then, one can obtain the conditions (15), which ensure () ≥   ()() > 0. If the methods in [38] are followed, all the involved symmetric matrices in Lyapunov-Krasovskii functional being positive definite implies that () ≥   ()() > 0. We can easily obtain the following corollaries.
Corollary 11.For given scalars 0 <  < 1, if there exist symmetric matrices then, the reachable set of system ( 1) is contained in the following ellipsoid: where Ξ 1 , Ξ 2 are defined in Theorem 4.
Remark 12.To get the smallest possible ellipsoid with the shortest major axis, we consider the matrix  as a decision variable and satisfying 0 <  ≤ , which is equivalent to where  = 1/.Then the smallest possible ellipsoid can be obtained when  is minimum.

Numerical Examples
In this section, we present two examples to demonstrate the effectiveness of our results.
Example 14.Consider the discrete-time system (1) with the following parameters: and   = 0.
The eigenvalues of  1 are −0.2325 and 0.2127.Obviously, the matrix  1 is not positive definite.Hence, there really exist nonpositive-definite matrices  1 such that the LKF in ( 21) is positive definite.Moreover, by the method in [40], the minimal  = 2.8160 is obtained.Figure 1 gives a more intuitive comparison of the reachable sets by different methods; it is clear that our method in this paper yields much tighter bounds than that of [40].Therefore, it can be seen that our results are less conservative than the ones in [40].Figure 2 shows the reachable set when 2 ≤ () ≤ 8 and 2 ≤ () ≤ 10; from Figure 2, we can see that the reachable set of the system is related to the time delay.
and   = 0.1 and  1 =  2 = 0.The system has been considered in some existing works [38,40].The allowable maximal values of   are listed in Table 1 with different   .Therefore, our conditions proposed in Corollary 11 provide a better range for the time-varying delay as shown in Table 1.

Example 15 .Figure 1 :
Figure 1: The bounding reachable set computed by different methods.

Figure 2 :
Figure 2: The bounding reachable set computed by different timevarying delay.

Table 1 :
Different values of   with different   .