User:Margav06/sandbox/Click here to continue/Time-Delay Systems/Bounded Real Lemma under Slowly-Varying Delay

This page describes a bounded real lemma for a continuous-time system with a time-varying delay. In particular, a condition is provided to obtain a bound on the $$ L_2 $$-gain of a retarded differential system through feasibility of an LMI. The system under consideration pertains a single discrete delay, with the extent of the delay at any time bounded by some known value. This delay is only present in the state, with no direct delay in the effects of exogenous inputs on the state. In addition, the delay is assumed to vary only slowly in time, with a temporal derivative bounded by a value less than one, although results can also be attained if no bound is known. Solving the LMI for a particular value of the bound, while minimizing a scalar variable, an upper limit on the $$ L_2 $$-gain of the system can be shown for any time-delay satisfying this bound.

The System
The system under consideration is one of the form:



\begin{align}

\dot{x}(t) &= Ax(t) + A_1x(t-\tau(t)) + B_0 w(t) & t&\geq t_0, & 0&\leq\tau(t) \leq h, & \dot{\tau}(t)&\leq d<1 \\ z(t) &= C_0x(t) + C_{1}x(t-\tau(t))

\end{align} $$

In this description, $$ A $$ and $$ A_1 $$ are constant matrices in $$ \mathbb{R}^{n\times n} $$. In addition, $$ B_0 $$ is a constant matrix in $$ \mathbb{R}^{n\times n_w}$$, and $$ C_0,C_{1} $$ are constant matrices in $$ \mathbb{R}^{n_z\times n}$$ where $$ n_w,n_z\in\mathbb{N} $$ denote the number of exogenous inputs and regulated outputs respectively. The variable $$ \tau(t) $$ denotes a delay in the state at time $$ t\geq t_0 $$, assuming a value no greater than some $$ h\in\mathbb{R}_+ $$. Moreover, we assume that the function $$ \tau(t) $$ is differentiable at any time, with the derivative bounded by some value $$ d<1 $$, assuring the delay to be slowly-varying in time.

The Data
To obtain a bound on the $$ L_2 $$-gain of the system, the following parameters must be known:

$$ \begin{align} A,A_1&\in\mathbb{R}^{n\times n} \\ B_0&\in\mathbb{R}^{n\times n_w} \\ C_0,C_1&\in\mathbb{R}^{n_z\times n} \\ h&\in \mathbb{R}_+ \\ d&\in [0,1) \end{align} $$

The Optimization Problem
Based on the provided data, we can obtain a bound on the $$ L_2$$-gain of the system by testing feasibility of an LMI. In particular, the bounded real lemma states that if the LMI presented below is feasible for some $$ \gamma>0 $$, the $$ L_2 $$-gain of the system is less than or equal to this $$ \gamma $$. To attain a bound that is as small as possible, we minimize the value of $$ \gamma $$ while solving the LMI:

The LMI: L2-gain for TDS with Slowly-Varying Delay


\begin{align} &\text{Solve}:\\ &\qquad\min \gamma \\ &\text{such that there exist}:\\ &\qquad P,P_2,P_3,R,S,S_{12},Q\in\mathbb{R}^{n\times n}\\ &\text{for which}:\\ &\qquad P>0,\quad R>0,\quad S>0,\quad Q>0  \\ &\qquad \begin{bmatrix} \begin{array}{c c c c | c c} \Phi_{11} & \Phi_{12} & S_{12} & R-S_{12}+P_2^T A_1 & P_2^T B_0 & C_0^T \\ \end{array} \end{bmatrix}<0 \\ &\text{where}:\\ &\qquad \Phi_{11} = A^T P_2 + P_2^T A + S + Q -R\\ &\qquad \Phi_{12} = P-P_2^T+A^T P_3 \\ &\qquad \Phi_{22} = -P_3-P_3^T +h^2 R \end{align}
 * & \Phi_{22} & 0 & P_3^T A_1 & P_3^T B_0 & 0\\
 * & * & -S-R & R-S_{12}^T & 0 & 0 \\
 * & * & * & -2R - S_{12} - S_{12}^T - (1-d)Q & 0 & C_1^T \\\hline
 * & * & * & * & -\gamma^2 I & 0 \\
 * & * & * & * & * & -I

$$

In this notation, the symbols $$ * $$ are used to indicate appropriate matrices to assure the overall matrix is symmetric.

Conclusion:
If the presented LMI is feasible for some $$ \gamma $$, the system is internally stable, and will have an $$ L_2$$-gain less than $$ \gamma $$. That is, independent of the values of the delays $$ \tau(t) $$:


 * $$ \|z\|_{L_2} < \gamma \|w\|_{L_2} $$

It should be noted that this result is conservative. That is, even when minimizing the value of $$ \gamma $$, there is no guarantee that the bound obtained on the $$ L_2 $$-gain is sharp.

In a scenario where no bound $$ d $$ on the change in the delay is known, the above LMI can still be used to obtain a bound on the $$ L_2 $$-gain. In particular, setting $$ Q=0 $$ in the above LMI, a bound can be attained independent of the value of the derivative of the delay.

Implementation
An example of the implementation of this LMI in Matlab is provided on the following site:


 * https://github.com/djagt/LMI_Codes/blob/main/L2gain_cTDS.m

Note that this implementation requires packages for YALMIP with solver mosek, though a different solver can also be implemented.

Related LMIs

 * - Bounded real lemma for continuous-time system without delay


 * - Bounded real lemma for discrete-time system without delay


 * - Stability LMI for continuous-time RDE with slowly-varying delay