# Region stability of linear stochastic discrete systems with time-delays

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## Abstract

In this paper, our aim is to investigate the region stability of stochastic discrete time-delay systems. Firstly, region stabilities of systems are defined by a spectrum operator. Secondly, with the aid of linear matrix inequality, some equivalent conditions and related results for region stabilities of systems are obtained. Finally, the relation between region stabilities and the convergence speed of the trajectories of systems is given.

## Keywords

Region-stability Time-delay Discrete stochastic system## 1 Introduction

Stability is one of essential concepts in dynamical system theory, which has been considered by the researchers in many fields, such as [1, 2, 3, 4, 5, 6, 7, 8], and [9, 10, 11, 12]. For a linear system without delays, as is well known, the stability is related to the system matrix root-clustering in subregions of the complex plane. That is, the stability of systems has a close relation with the spectrum placement of the system matrix. For example, the further left the spectrum set of the considered system is located, the faster the system response rate is.

Mean square stability has become a hot issue, which has many important applications in system analysis and designs. Some equivalent conditions for mean square stability of linear stochastic systems have been obtained in terms of generalized algebraic Riccati equation (see [13, 14, 15, 16, 17]), or linear matrix inequality (LMI) (see [18, 19, 20, 21] and [22, 23, 24, 25]), or spectra of some operators (see [26, 27, 28]). The spectrum technique is very powerful in system analysis and design, which began with [26] for a stochastic system.

Region stability considered in this paper is related with the mean square stability. Region stability problem is to check whether the spectrum set of deterministic or stochastic systems lies in a given region of the complex plane, which has become a popular research focus in the past decades (see [29, 30, 31, 32, 33], and the references therein). Compared with deterministic systems, stochastic systems can describe the complexity of practical problems more accurately. For stochastic systems, a special case of region-stability and region-stabilization, called respectively interval stability and interval stabilization, was studied, and a sufficient condition was given for the interval stabilization of general linear stochastic systems with state and control dependent noises in [29]. In [30], the authors further defined and discussed a class of relative stability and stabilization, which is a more special case of region-stability and region-stabilization of linear stochastic systems. The objective of the present work is to make a further investigation in the region stability of linear stochastic discrete systems. New results concerning the region stability of linear stochastic discrete systems are developed. By means of LMI, some necessary and sufficient conditions are presented for region-stability with generalized LMI regions. To the best of our knowledge, there were no similar results reported for such dynamical systems up to now.

This paper is organized as follows: In Sect. 2, we introduce some LMI regions and generalized LMI regions. In Sect. 3, we give some definitions of region stability and obtain some equivalent conditions and related results for the region stability of stochastic discrete systems with delays.

For convenience, we adopt the following traditional notations: \(\mathbb{S}^{n}\) is the set of all symmetric matrices, whose components may be complex; \(\mathbb{R}^{n\times m}\) is the set of all \(n\times m\) matrices, whose components are real; \(A'\) (\(\ker (A)\)) denotes the transpose (kernel space) of the matrix *A*; \(A\geq 0\) (\(A>0\)) is a positive semidefinite (positive definite) symmetric matrix; *I* is the identity matrix; \(\sigma (L)\) is the spectral set of the operator or matrix *L*; \(\mathbb{N}_{0}=\{0,1,2,\ldots \}\). Here ⊗ denotes Kronecker product operation of matrices (see [34] for details); \(\mathbb{R}^{n}\) is the set of all *n*-dimensional vectors, whose components are real. For \(x\in \mathbb{R}^{n}\), \(\Vert x \Vert =\sqrt[2]{x'x}\).

## 2 LMI regions

In this paper, we shall discuss the stability of time-delay systems in generalized LMI regions. First of all, we introduce some definitions and related examples in LMI regions and generalized LMI regions.

### Definition 2.1

([35])

*D*in the complex plane is called an LMI region if there exist symmetric matrices \(H_{1}\) and \(H_{2}\) such that

Here we give two examples of LMI region.

### Example 2.1

In the complex plane, the circle region \(D(q,r)\), which has center \({(q,0)}\) and radius *r*, is an LMI region, where *q* and *r* satisfy \(\vert q \vert +r<1\).

### Example 2.2

Different from an LMI region, we introduce another region, which is defined as follows.

### Definition 2.2

([33])

### Example 2.3

## 3 Criteria of region-stability

Now we give some definitions of region-stability of stochastic time-delay systems by the spectrum operator \(\sigma (L_{F,G})\).

### Definition 3.1

System (5) is \(D_{R}\)-stable, if \({\sigma (L_{F,G})}\subset {D_{R}}\subset {D(0,1)}\).

### Definition 3.2

System (5) is \(D(q,r)\)-stable, if \({\sigma (L_{F,G})}\subset {D(q,r)}\subset {D(0,1)}\).

### Definition 3.3

System (5) is \(D(r,\theta )\)-stable, if \({\sigma (L_{F,G})}\subset {D(r,\theta )}\subset {D(0,1)}\).

### Definition 3.4

System (5) is \(D(0,\alpha ,\beta )\)-stable, if \({\sigma (L_{F,G})}\subset {D(0,\alpha ,\beta )}\subset {D(0,1)}\).

Now we give some equivalent conditions of these region stabilities.

### Theorem 3.1

*The time*-

*delay system*(5)

*is*\(D_{R}\)-

*stable if and only if there exists a positive matrix*\(P>0\)

*such that*

*where*\(\varTheta (H(n,m),F,G) =[H(n,m)'H(n,m)]^{-1}H(n,m)'[{F}\otimes {F}+{G}\otimes {G}]H(n,m)\).

### Proof

*H*representation method in [36], there exists a full column rank matrix \(H(n,m)\in \mathbb{R}^{n^{2}(m+1)^{2}\times \frac{n(m+1)[n(m+1)+1]}{2}}\) such that \(\overrightarrow{X}=H(n,m) \widetilde{X}\), \(\widetilde{X}\in \mathbb{R}^{ \frac{n(m+1)[n(m+1)+1]}{2}}\), so system (9) is equivalent to the following discrete system:

From Eqs. (8) and (9) in Theorem 3.1 and Theorem 1 in [27], the relation between the spectrum of operator \(L_{F,G}\) and matrix \(\varTheta (H(n,m),F,G)\) is obtained as follows.

### Corollary 3.1

\(\sigma (L_{F,G})=\sigma (\varTheta (H(n,m),F,G))\).

### Remark 3.1

From Theorem 3.1, we know that the time-delay system (5) is \(D_{R}\)-stable if and only if a certain LMI is feasible.

Similar to Theorem 3.1, it is easy to draw some results as follows.

### Theorem 3.2

*System*(5)

*is*\(D(q,r)\)-

*stable if and only if there exists a positive matrix*\(P>0\)

*such that the following inequality holds*:

### Theorem 3.3

*System*(5)

*is*\(D(r,\theta )\)-

*stable if and only if there exists a positive matrix*\(P>0\)

*satisfying the inequality in Theorem*3.2

*and the following inequality*:

*where*

### Theorem 3.4

*System*(5)

*is*\(D(0,\alpha ,\beta )\)-

*stable if and only if there exists a positive matrix*\(P>0\)

*satisfying*

Using MATLAB, it is quick to check the above criterion of \(D_{R}\)-stability with LMI. Now we give an example as follows.

### Example 3.1

We use the LMI toolbox of Matlab and commands for obtaining *P* as follows:

H=[1 0 0;0 1 0;0 1 0;0 0 1];

F=[-1/4 1/8;1 0];

G=[-1/16 1/32;0 0];

T1=H.'*H;

T3=[kron(F,F)+kron(G,G)]*H;

T=inv(T1)*H.'*T3;

E=[1 0 0;0 1 0;0 0 1];

q=1/4;

A2=-q*E+T;

r=1/2;

B1=-r*E;

setlmis([])

P=lmivar(1,[3,1])

lmiterm([1 1 1 P],B1,1)

lmiterm([1 1 2 P],1,A2)

lmiterm([1 2 2 P],B1,1)

lmiterm([-2 1 1 P],1,1)

lmisys=getlmis;

[tmin,xfeas]=feasp(lmisys)

P=dec2mat(lmisys,xfeas,P)

eig(P)

### Remark 3.2

Theorems 3.1–3.4 give a kind of characterization for region stability. But the relation between region stabilities and the convergence speed of the trajectories of a system need to be further investigated, which is an interesting and practical problem.

### Theorem 3.5

*System*(5)

*is*\(D(0,\alpha ,\beta )\)-

*stable if and only if system*(1)

*is such for arbitrary sufficiently small real number*\(\varepsilon >0\),

*the convergence speed of the system*\([\sum_{j=0}^{m}F_{j},\sum_{j=0}^{m}G_{j}]\)

*is slower than*\(O((\alpha +\varepsilon )^{t})\),

*but is faster than*\(O((\beta -\varepsilon )^{t})\).

*That is*,

*there exist two constants*\(C_{1}>0\), \(C_{2}>0\)

*such that*

### Proof

*(Necessity)*Assume that system (5) is \(D(0,\alpha ,\beta )\)-stable. For \({\lambda _{i}}\in {\sigma (L_{F,G})}\), \(i=1,2,\ldots, \frac{n(m+1)[n(m+1)+1]}{2}\), without loss of generality, let

*H*representation method in [36], the initial problem of system (10) is equivalent to the initial problem of the following discrete system:

*(Sufficiency)*Suppose that system (5) is \((\alpha ,\beta )\) region stable. For arbitrary real number \(\varepsilon >0\), the convergence speed of the system \([\sum_{j=0}^{m}F_{j},\sum_{j=0}^{m}G _{j}]\) is slower than \(O((\alpha +\varepsilon )^{t})\), but is faster than \(O((\beta -\varepsilon )^{t})\). That is, there exist two constants \(C_{1}>0\), \(C_{2}>0\) such that

### Corollary 3.2

*If system* (5) *is*\(D_{R}\)-*stable then there exists a constant*\(C>0\)*such that*\(\lim_{t \rightarrow +\infty }E { \Vert x(t) \Vert }^{2}=0\).

### Corollary 3.3

*If system* (5) *is*\(D(q,r)\)-*stable then there exist two constants*\(C_{1}>0\), \(C_{2}>0\)*such that*\(C_{2}{ \Vert \overline{x} _{0} \Vert }^{2}( \Vert q \Vert +\varepsilon )^{t} \leq E{ \Vert x(t) \Vert }^{2}\leq {C_{1} { \Vert \overline{x}_{0} \Vert }^{2} (( \Vert q \Vert +r)-\varepsilon )^{t}}\).

### Corollary 3.4

*If system* (5) *is*\(D(r,\theta )\)-*stable then there exists a constant*\(C>0\)*such that*\(E{ \Vert x(t) \Vert }^{2}\leq {C{ \Vert \overline{x}_{0} \Vert }^{2} (r-\varepsilon )^{t}}\).

## 4 Conclusion

In this paper, we investigated the region stability of discrete stochastic time-delay systems. First of all, we defined the notions of region-stability of discrete stochastic time-delay systems. We then gave some equivalent conditions for the region stabilization of linear stochastic discrete time-delay systems. We only considered the stability of discrete stochastic time-delay systems on some LMI regions. But the stability of discrete stochastic time-delay systems on general regions is still a challenging problem.

## Notes

### Acknowledgements

The authors wish to thank editor and anonymous reviewer for his/her valuable suggestions to this paper.

### Availability of data and materials

We don’t apply new software, databases and relevant raw data.

### Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final paper.

### Funding

This work was supported by the National Natural Science Foundation of China under Grants 61503224, National Science Foundation of Shandong Province ZR2017MF054, Qingdao Postdoctoral Applied Research Project No. 2015188 and SDUST Research Fund No. 2015TDJH105.

### Competing interests

All the authors declare that there is no conflict of interest regarding the publication of this paper.

## References

- 1.Mao, X.: Stochastic Differential Equations and Their Applications. Horwood, Chichester (1997) zbMATHGoogle Scholar
- 2.Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955) zbMATHGoogle Scholar
- 3.Has’minskii, R.Z.: Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Alphen (1980) CrossRefGoogle Scholar
- 4.Kushner, H.J.: Stochastic Stability and Control. Academic Press, New York (1967) zbMATHGoogle Scholar
- 5.Chen, M., Xia, D., Wang, D., Han, J., Liu, Z.: An analytical method for reducing metal artifacts in X-ray CT images. Math. Probl. Eng.
**2019**, Article ID 2351878 (2019) MathSciNetGoogle Scholar - 6.Li, G., Chen, M.: Intertwined phenomenon of a kind of dynamical system. Adv. Differ. Equ.
**2013**(1), Article ID 265 (2013) MathSciNetCrossRefGoogle Scholar - 7.Li, G., Ding, C., Chen, M.: Intertwined basins of attraction of dynamical systems. Appl. Math. Comput.
**213**(1), 272–274 (2009) MathSciNetzbMATHGoogle Scholar - 8.Zhang, T., Meng, X., Zhang, T.: Global analysis for a delayed SIV model with direct and environmental transmissions. J. Appl. Anal. Comput.
**6**(2), 479–491 (2016) MathSciNetGoogle Scholar - 9.Baleanu, D., Wu, G., Bai, Y., Chen, F.: Stability analysis of Caputo-like discrete fractional systems. Commun. Nonlinear Sci. Numer. Simul.
**48**, 520–530 (2017) MathSciNetCrossRefGoogle Scholar - 10.Wu, G., Baleanu, D., Zeng, S.: Finite-time stability of discrete fractional delay systems: Gronwall inequality and stability criterion. Commun. Nonlinear Sci. Numer. Simul.
**57**, 299–308 (2018) MathSciNetCrossRefGoogle Scholar - 11.Wu, G., Baleanu, D., Zeng, S.: Several fractional differences and their applications to discrete maps. J. Appl. Nonlinear Dyn.
**4**, 339–348 (2015) CrossRefGoogle Scholar - 12.Li, G., Gao, Y., Chen, M.: Attractors of dynamical systems in locally compact spaces. Open Math.
**17**, 465–471 (2019) MathSciNetCrossRefGoogle Scholar - 13.Li, G., Chen, M.: On uniqueness of strong solution of stochastic systems. Abstr. Appl. Anal.
**2014**, Article ID 890925 (2014) MathSciNetzbMATHGoogle Scholar - 14.Gao, M., Sheng, L., Zhang, W.: Stochastic \(H_{2}/H_{\infty }\) control of nonlinear systems with time-delay and state-dependent noise. Appl. Math. Comput.
**266**, 429–440 (2015) MathSciNetzbMATHGoogle Scholar - 15.Liu, X., Li, Y., Zhang, W.: Stochastic linear quadratic optimal control with constraint for discrete-time systems. Appl. Math. Comput.
**228**, 264–270 (2014) MathSciNetzbMATHGoogle Scholar - 16.Zhang, W., Zhang, H., Chen, B.S.: Generalized Lyapunov equation approach to state-dependent stochastic stabilization/detectability criterion. IEEE Trans. Autom. Control
**53**, 1630–1642 (2008) MathSciNetCrossRefGoogle Scholar - 17.Li, G., Zhang, W., Chen, M.: Robust \(H_{2}/H_{\infty }\) control for periodic stochastic difference systems with multiplicative noise. IET Control Theory Appl.
**9**(16), 2451–2457 (2015) MathSciNetCrossRefGoogle Scholar - 18.Yan, Z., Zhang, G., Wang, J., Zhang, W.: State and output feedback finite-time guaranteed cost control of linear ito stochastic systems. J. Syst. Sci. Complex.
**28**, 813–829 (2015) MathSciNetCrossRefGoogle Scholar - 19.Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994) CrossRefGoogle Scholar
- 20.Ait Rami, M., Zhou, X.Y.: Linear matrix inequalities, Riccati equations and indefinite stochastic linear quadratic control. IEEE Trans. Autom. Control
**45**, 1131–1142 (2000) MathSciNetCrossRefGoogle Scholar - 21.Boyd, S., Ghaoui, E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994) CrossRefGoogle Scholar
- 22.Mobayen, S., Baleanu, D.: Linear matrix inequalities design approach for robust stabilization of uncertain nonlinear systems with perturbation based on optimally-tuned global sliding mode control. J. Vib. Control
**23**, 1285–1295 (2017) MathSciNetCrossRefGoogle Scholar - 23.Faieghi, M.R., Kuntanapreeda, S., Delavari, H., Baleanu, D.: Robust stabilization of fractional-order chaotic systems with linear controllers: LMI-based sufficient conditions. J. Vib. Control
**20**, 1042–1051 (2014) MathSciNetCrossRefGoogle Scholar - 24.Mobayen, S., Baleanu, D., Tchier, F.: Second-order fast terminal sliding mode control design based on LMI for a class of non-linear uncertain systems and its application to chaotic systems. J. Vib. Control
**23**, 2912–2925 (2016) MathSciNetCrossRefGoogle Scholar - 25.Li, G., Chen, M.: Infinite horizon linear quadratic optimal control for stochastic difference time-delay systems. Adv. Differ. Equ.
**2015**, Article ID 14 (2015) MathSciNetCrossRefGoogle Scholar - 26.Zhang, W., Chen, B.S.: On stabilizability and exact observability of stochastic systems with their applications. Automatica
**40**, 87–94 (2004) MathSciNetCrossRefGoogle Scholar - 27.Li, G., Chen, M.: The stability and stabilization of stochastic delay-time systems. Math. Probl. Eng.
**2014**, Article ID 272745 (2014) MathSciNetzbMATHGoogle Scholar - 28.Hou, T., Ma, H., Zhang, W.: Spectral tests for observability and detectability of periodic Markov jump systems with nonhomogeneous Markov chain. Automatica
**63**, 175–181 (2016) MathSciNetCrossRefGoogle Scholar - 29.Zhang, W., Xie, L.: Interval stability and stabilization of linear stochastic systems. IEEE Trans. Autom. Control
**54**, 810–815 (2009) MathSciNetCrossRefGoogle Scholar - 30.Zhang, W.: General D-stability and D-stabilization for linear stochastic systems: continuous-time case. In: 2010 8th IEEE International Conference on Control and Automation (2010) Google Scholar
- 31.Fragoso, M.D., Costa, O.L.V., de Souza, C.E.: A new approach to linearly perturbed Riccati equations arising in stochastic control. Appl. Math. Optim.
**37**, 99–126 (1998) MathSciNetCrossRefGoogle Scholar - 32.El Harraki, I., El Alami, A., Boutoulout, A., Serhani, M.: Regional stabilization of semi-linear parabolic systems. IMA J. Math. Control Inf.
**34**, 961–971 (2017) MathSciNetzbMATHGoogle Scholar - 33.Peaucelle, D., Arzelier, D., Bachelier, O., Bernussou, J.R.K.: A new robust D-stability condition for real convex polytypic uncertainty. Syst. Control Lett.
**40**, 21–30 (2000) CrossRefGoogle Scholar - 34.Ortega, J.M.: Matrix Theory. Plenum, New York (1987) CrossRefGoogle Scholar
- 35.Chilali, M., Gahinet, P.: \(H_{\infty }\) design with pole placement constraints: an LMI approach. IEEE Trans. Autom. Control
**41**, 358–367 (1996) MathSciNetCrossRefGoogle Scholar - 36.Zhang, W., Chen, B.S.: H-representation and applications to generalized Lyapunov equations and linear stochastic systems. IEEE Trans. Autom. Control
**57**, 3009–3022 (2012) MathSciNetCrossRefGoogle Scholar

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