A Novel Finite-Time Stability Criterion for Linear Discrete-Time Stochastic System with Applications to Consensus of Multi-Agent System

Abstract

This paper is concerned with the finite-time stability of linear discrete-time stochastic system with time-varying delays and its applications to the consensus problem of multi-agent system. A novel finite-time stability criterion is presented to guarantee that the state of the system does not exceed a prescribed bound during a fixed time interval using the piecewise-like delay method. Then, a corollary is derived for the case without stochastic perturbations. Numerical examples are provided to show the less conservatism and effectiveness of the proposed linear matrix inequality conditions. Finally, the stability results are directly applied to develop the finite-time consensus conditions for the linear multi-agent system with time-varying communication delays. An illustrative example is given to validate the effectiveness of the theoretical results.

Appendices

Appendices

1.1 Appendix 1: The notations for the conditions in Theorem 1

$$\begin{aligned} \varOmega _1&= \Omega -P,~~~\varOmega =P+\delta _1^2R_1+\delta _2^2R_2+d_m^2R_3,\\ \bar{P}&= R^{-\frac{1}{2}}PR^{-\frac{1}{2}},~~~ \bar{Q}_1=R^{-\frac{1}{2}}Q_1R^{-\frac{1}{2}}, ~~~\bar{Q}_2=R^{-\frac{1}{2}}Q_2R^{-\frac{1}{2}}, \\ \bar{Q}_3&= R^{-\frac{1}{2}}Q_3R^{-\frac{1}{2}},~~~ \bar{Q}_4=R^{-\frac{1}{2}}Q_4R^{-\frac{1}{2}}, \\ \eta ^T(k)&= \Big [x^T(k), x^T(k-d_m), x^T(k-d_0), x^T(k-d(k)),x^T(k-d_M)\Big ],\\ \bar{\Sigma }&= \gamma ^Kc_1\Big [\lambda _2+\gamma ^{d_0-1}d_0\lambda _3+\gamma ^{d_M-1}d_M\lambda _4 +\gamma ^{d_m-1}d_m\lambda _5+\gamma ^{d_M-1}d_M\lambda _6\\&+\,\gamma ^{d_M-2}\frac{d_M(d_M-1)-d_m(d_m-1)}{2}\lambda _6\Big ]-\lambda _1c_2\\&+\,\gamma ^K\delta \Big [\delta _1\gamma ^{d_0-1}\lambda _7\frac{(d_0-d_m)(d_m+d_0+1)}{2}\\&+\,\delta _2\gamma ^{d_M-1}\lambda _8\frac{(d_M-d_0)(d_M+d_0+1)}{2}+d_m\gamma ^{d_m-1}\lambda _9\frac{d_m(d_m+1)}{2}\Big ],\\ \Xi&= \begin{bmatrix}\Xi _{11}&\quad R_3&\quad 0&\quad \Xi _{14}&\quad 0 \\ *&\quad \Xi _{22}&\quad 0&\quad 0&\quad 0\\ *&\quad *&\quad -\gamma ^{d_0}Q_1&\quad 0&\quad 0\\ *&\quad *&\quad *&\quad \Xi _{44}&\quad 0\\ *&\quad *&\quad *&\quad *&\quad -\gamma ^{d_M}Q_2\end{bmatrix},\\ \Xi _{11}&= A^TPA-\gamma P+Q_1+Q_2+Q_3+(d_M-d_m+1)Q_4\\&+\,(A-I)^T\varOmega _1(A-I)+\lambda \rho _1I-R_3,\\ \Xi _{14}&= A^TPB+(A-I)^T\varOmega _1B, \; \Xi _{22}=-\gamma ^{d_m}Q_3-R_3,\\ \Xi _{44}&= B^TPB-\gamma ^{d_m}Q_4+B^T\varOmega _1B+\lambda \rho _2I,\\ \beta _1&= [0,1,0,-1,0], \;\beta _2=[0,0,-1,1,0], \; \beta _3=[0,0,-1,0,1],\\ \beta _4&= [0,0,0,1,-1], \;\beta _5=[0,-1,1,0,0],\\ \Gamma _1&= \delta _1\gamma ^{d_m}(M\beta _1+\beta _1^TM^T+F\beta _2+\beta _2^TF^T)-\gamma ^{d_0}\beta _3^TR_2\beta _3, \\ \Gamma _2&= \delta _2\gamma ^{d_0}(F\beta _2+\beta _2^TF^T+H\beta _4+\beta _4^TH^T)-\gamma ^{d_m}\beta _5^TR_1\beta _5,\\ \Gamma _{11}&= \delta _1\gamma ^{d_m}M, \;\Gamma _{12}=\delta _1\gamma ^{d_m}F,\;\Gamma _{21}=\delta _2\gamma ^{d_0}F,\;\Gamma _{22}=\delta _2\gamma ^{d_0}H,\\ \tilde{R}_1&= \gamma ^{d_m}R_1,\; \tilde{R}_2=\gamma ^{d_0}R_2,\\ M^T&= [0, M_1^T, 0, M_2^T, 0],\;F^T=[0, 0, F_1^T, F_2^T, 0],\;H^T=[0,0, 0, H_1^T, H_2^T]. \end{aligned}$$

1.2 Appendix 2: The bounded estimation of $V(0)$

$$\begin{aligned} V(0)&= x^T(0)Px(0)+\sum _{s=-d_0}^{-1}\gamma ^{-1-s}x^T(s)Q_1x(s)\\&+\sum _{s=-d_M}^{-1}\gamma ^{-1-s}x^T(s)Q_2x(s)+\sum _{s=-d_m}^{-1}\gamma ^{-1-s}x^T(s)Q_3x(s) \end{aligned}$$

$$\begin{aligned}&\sum _{s=-d(0)}^{-1}\gamma ^{-1-s}x^T(s)Q_4x(s)+\sum _{s=-d_M+1}^{-d_m}\sum _{v=s}^{-1}\gamma ^{-1-v}x^T(v)Q_4x(v)\\&\qquad +\,\delta _1\sum _{s=-d_0}^{-d_m-1}\sum _{v=s}^{-1}\gamma ^{-1-v}y^T(v)R_1y(v)\\&\qquad +\,\delta _2\sum _{s=-d_M}^{-d_0-1}\sum _{v=s}^{-1}\gamma ^{-1-v}y^T(v)R_2y(v)\\&\qquad +\,d_m\sum _{s=-d_m}^{-1}\sum _{v=s}^{-1}\gamma ^{-1-v}y^T(v)R_3y(v)\\&\quad \le \lambda _{\max }(\bar{P})x^T(0)Rx(0)+\gamma ^{d_0-1}\lambda _{\max }(\bar{Q}_1)\sum _{s=-d_0}^{-1}x^T(s)Rx(s)\\&\qquad +\,\gamma ^{d_M-1}\lambda _{\max }(\bar{Q}_2)\sum _{s=-d_M}^{-1}x^T(s)Rx(s)\\&\qquad +\,\gamma ^{d_m-1}\lambda _{\max }(\bar{Q}_3)\sum _{s=-d_m}^{-1}x^T(s)Rx(s)\\&\qquad +\,\gamma ^{d_M-1}\lambda _{\max }(\bar{Q}_4)\sum _{s=-d_M}^{-1}x^T(s)Rx(s)\\&\qquad +\,\gamma ^{d_M-2}\lambda _{\max }(\bar{Q}_4)\sum _{s=-d_M+1}^{-d_m}\sum _{v=s}^{-1}x^T(v)Rx(v)\\&\qquad +\,\delta _1\gamma ^{d_0-1}\lambda _{\max }(R_1)\sum _{s=-d_0}^{-d_m-1}\sum _{v=s}^{-1}y^T(v)y(v)\\&\qquad +\,\delta _2\gamma ^{d_M-1}\lambda _{\max }(R_2)\sum _{s=-d_M}^{-d_0-1}\sum _{v=s}^{-1}y^T(v)y(v)\\&\qquad +\,d_m\gamma ^{d_m-1}\lambda _{\max }(R_3)\sum _{s=-d_m}^{-1}\sum _{v=s}^{-1}y^T(v)y(v)\\&\quad \le \Sigma \end{aligned}$$

with

$$\begin{aligned} \Sigma&= \left[ \lambda _{\max }(\bar{P})+\gamma ^{d_0-1}d_0\lambda _{\max }(\bar{Q}_1)+\gamma ^{d_M-1}d_M\lambda _{\max }(\bar{Q}_2)\right. \\&+\,\gamma ^{d_m-1}d_m\lambda _{\max }(\bar{Q}_3)+\gamma ^{d_M-1}d_M\lambda _{\max }(\bar{Q}_4)\\&\left. +\,\gamma ^{d_M-2}\frac{d_M(d_M-1)-d_m(d_m-1)}{2}\lambda _{\max }(\bar{Q}_4)\right] c_1\\&+\,\left[ \delta _1\gamma ^{d_0-1}\lambda _{\max }(R_1)\frac{(d_0-d_m)(d_m+d_0+1)}{2}\right. \\&+\,\delta _2\gamma ^{d_M-1}\lambda _{\max }(R_2)\frac{(d_M-d_0)(d_M+d_0+1)}{2}\\&\left. +\,d_m\gamma ^{d_m-1}\lambda _{\max }(R_3)\frac{d_m(d_m+1)}{2}\right] \delta . \end{aligned}$$