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LMI Condition For Exponential Stability of Linear Systems With Interval Time-Varying Delays

For systems experiencing time-varying delays where the delays are bounded, the feasibility LMI in this section can be used to determine if the system is $\alpha$-exponentially stable.

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Dx(t-h(t)),&t\in \mathbb {R} ^{+},\\x(t)&=\phi (t),&t\in [-h_{2},0],\end{aligned}}$

where $x(t)\in \mathbb {R} ^{n}$ is the state, $A,D\in \mathbb {R} ^{n\times n}$ are the matrices of delay dynamics, and $\phi (t)\in \mathbb {R} ^{n}$ is the initial function with norm $\|\phi \|=sup_{-{\bar {h}}\leq t\leq 0}\{\|\phi (t)\|,\|{\dot {\phi }}(t)\|\}$ and it is continuously differentiable function on $[-h_{2},0]$. The tyime-varying delay function $h(t)$ satisfies:

$0\leq h_{1}\leq h(t)\leq h_{2},t\in \mathbb {R} ^{+},$

The Data

The matrices $(A,D)$ are known, as well as the bounds $(h_{1},h_{2})$ of the time-varying delay.

The Optimization Problem

For a given $\alpha >0$, the zero solution of the system described above is $\alpha$-exponentially stable if there exists a positive number $N>0$ such that every solution $x(t,\phi )$ satisfies the following condition:

$\|x(t,\phi )\|\leq Ne^{-\alpha t}\|\phi \|,\forall t\in \mathbb {R} ^{+}$

The LMI: $\alpha$-Stability Condition

The following feasibility LMI can be used to check if the system is $\alpha$-exponentially stable or not for a given $\alpha >0$:

${\begin{aligned}&{\text{Find}}\;P,Q,R,U,S_{i},{\text{ where }}i=1,2,...,5:\\&\quad \quad {\begin{bmatrix}M_{11}&M_{12}&M_{13}&M_{14}&M_{15}\\*&M_{22}&0&M_{24}&S_{2}\\*&*&M_{33}&M_{34}&S_{3}\\*&*&*&M_{44}&S_{4}-S_{5}D\\*&*&*&*&M_{55}\end{bmatrix}}<0\\&{\text{where:}}\\&\quad \quad M_{11}=A^{\top }P+PA+2\alpha P-(e^{-2\alpha h_{1}}+e^{-2\alpha h_{2}})R+0.5S_{1}(I-A)+0.5(I-A^{\top })S_{1}^{\top }+2Q,\\&\quad \quad M_{12}=e^{-2\alpha h_{1}}R-S_{2}A,\quad M_{13}=e^{-2\alpha h_{2}}R-S_{3}A,\\&\quad \quad M_{14}=PD-S_{1}D-S_{4}A,\quad M_{15}=S_{1}-S_{5}A,\\&\quad \quad M_{22}=-e^{-2\alpha h_{1}}(Q+R),\quad M_{24}=S_{2}D+e^{-2\alpha h_{2}}U,\\&\quad \quad M_{33}=-e^{-2\alpha h_{1}}(Q+R+U),\quad M_{34}=-S_{3}D+e^{-2\alpha h_{2}}U,\\&\quad \quad M_{44}=0.5(S_{4}D+D^{\top }S_{4}^{\top })-e^{-2\alpha h_{2}}U,\\&\quad \quad M_{55}=S_{5}+S_{5}^{\top }+(h_{1}^{2}+h_{2}^{2})R+(h_{2}-h_{1})^{2}U,\end{aligned}}$

The above LMI can be combined with the bisection method to find $\alpha$.

Conclusion:

For systems with time-varying delays with intervals, the LMI in this section can be used to check if the system is exponentially stable with a certain $\alpha$. The bisection algorithm can be additionally used to compute $\alpha$.

Implementation

To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:

https://github.com/smhassaan/LMI-Examples/blob/master/Intervaled_Delay_Sys_Stability_example.m

External Links

A list of references documenting and validating the LMI.

• LMI approach to exponential stability of linear systems with interval time-varying delays - Original Article by Phat et al.