LMIs in Control/pages/LMI for L2-Optimal State-Feedback Control under Time-Varying Input Delay

This page describes a method for constructing a full-state-feedback controller for a continuous-time system with a time-varying input delay. In particular, a condition is provided to obtain a bound on the $$ L_2 $$-gain of closed-loop system under time-varying delay through feasibility of an LMI. The system under consideration pertains a single discrete delay in the actuator input, with the extent of the delay at any time bounded by some known value. Moreover, the delay is assumed to vary only slowly in time, with a temporal derivative bounded by a value less than one, although results may 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) + B_2u(t-\tau(t)) + B_1 w(t) & t&\geq t_0, & 0&\leq\tau(t) \leq h, & \dot{\tau}(t)&\leq d<1 \\ z(t) &= C_1x(t) + D_{12}u(t-\tau(t))

\end{align} $$

In this description, $$ A $$ and $$ A_1 $$ are constant matrices in $$ \mathbb{R}^{n\times n} $$. In addition, $$ B_1 $$ is a constant matrix in $$ \mathbb{R}^{n\times n_w}$$, and $$ B_2 $$ is a constant matrix in $$ \mathbb{R}^{n\times n_u}$$, where $$ n_w,n_u\in\mathbb{N} $$ denote the number of exogenous and actuator inputs respectively. Finally, $$ C_{1} $$ and $$ D_{12} $$ are constant matrices in $$ \mathbb{R}^{n_z\times n}$$ and $$ \mathbb{R}^{n_z\times n_u}$$ respectively, where $$ n_z\in\mathbb{N} $$ denotes the number of regulated outputs. The variable $$ \tau(t) $$ denotes a delay in the actuator input 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 construct an $$ L_2 $$-optimal controller of the system, the following parameters must be known:

$$ \begin{align} A&\in\mathbb{R}^{n\times n} \\ B_1&\in\mathbb{R}^{n\times n_w} \\ B_2&\in\mathbb{R}^{n\times n_u} \\ C_1&\in\mathbb{R}^{n_z\times n} \\ D_{12}&\in\mathbb{R}^{n_z\times n_u} \\ h&\in \mathbb{R}_+ \\ d&\in [0,1) \end{align} $$

In addition to these parameters, a tuning scalar $$ \epsilon>0 $$ is also implemented in the LMI.

The Optimization Problem
Based on the provided data, we can construct an $$ L_2$$-optimal full-state-feedback controller of the system by testing feasibility of an LMI. In particular, we note that if the LMI presented below is feasible for some $$ \gamma>0 $$ and matrices $$ \bar{P}_2^{-1}>0 $$ and $$ Y $$, implementing the state-feedback $$ u(t)=Kx(t) $$ with $$ K=Y\bar{P}_2^{-1} $$, the $$ L_2 $$-gain of the closed-loop system will be less than or equal to $$ \gamma $$. To attain a bound that is as small as possible, we minimize the value of $$ \gamma $$ while solving the LMI:

The LMI: L2-Optimal Full-State-Feedback for TDS with Slowly-Varying Input Delay


\begin{align} &\text{Solve}:\\ &\qquad\min \gamma \\ &\text{such that there exist}:\\ &\qquad \bar{P},\bar{P}_2,\bar{R},\bar{S},\bar{S}_{12},\bar{Q}\in\mathbb{R}^{n\times n}, \quad Y\in\mathbb{R}^{n_u\times n}\\ &\text{for which}:\\ &\qquad \bar{P}>0,\quad \bar{P}_2>0,\quad \bar{R}>0,\quad \bar{S}>0  \\ &\qquad \begin{bmatrix} \begin{array}{c c c c | c c} \bar{\Phi}_{11} & \bar{\Phi}_{12} & \bar{S}_{12} & B_2Y+\bar{R}-\bar{S}_{12} & B_1 & \bar{P}_2^T C_1^T \\ \end{array} \end{bmatrix}<0 \\ &\text{where}:\\ &\qquad \Phi_{11} = A \bar{P}_2 + \bar{P}_2^T A^T + \bar{S} + \bar{Q} -\bar{R}\\ &\qquad \Phi_{12} = \bar{P}-\bar{P}_2+\epsilon \bar{P}_2^T A^T \\ &\qquad \Phi_{22} = -\epsilon \bar{P}_2-\epsilon\bar{P}_2^T + h^2 R \end{align}
 * & \bar{\Phi}_{22} & 0 & \epsilon B_2Y & \epsilon B_1 & 0\\
 * & * & -\bar{S}-\bar{R} & \bar{R}-\bar{S}_{12}^T & 0 & 0 \\
 * & * & * & -(1-d)\bar{Q}-2\bar{R}+\bar{S}_{12}+\bar{S}_{12}^T & 0 & Y^T D_{12}^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 $$ \gamma,Y,\bar{P}_2 x(t) $$, implementing the full-state-feedback controller $$ u(t)=Kx(t)=Y\bar{P}_2^{-1} $$, the closed-loop system will be asymptotically stable, and will have an $$ L_2$$-gain less than $$ \gamma $$. That is, independent of the values of the delays $$ \tau(t) $$, the system:



\begin{align} \dot{x}(t)&=Ax(t)+B_2Kx(t-\tau(t))+B_1 w(t) \\ z(t) &= C_1x(t)+D_{12}Kx(t-\tau(t)) \end{align} $$

with:


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

\begin{align} K=Y\bar{P}_2^{-1} \end{align} $$

will satisfy:


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

Here we note that $$ \bar{P}_2^{-1}x(t) $$ is guaranteed to exist as $$ P_2 $$ is positive definite, and thus nonsingular.

It should be noted that the obtained 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, meaning that the actual $$ L_2 $$-gain of the closed-loop can be (significantly) smaller than $$ \gamma $$.

In a scenario where no bound $$ d $$ on the change in the delay is known, or this bound is greater than one, the above LMI may still be used to construct a controller. In particular, if the presented LMI is feasible with $$ \bar{Q}=0 $$, the closed-loop system imposing $$ u(t)=Kx(t)=Y\bar{P}_2^{-1} $$ will be internally exponentially stable with an $$ L_2 $$-gain less than $$ \gamma $$ independent of the value of $$ \dot{\tau}(t) $$.

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/L2_OptStateFdbck_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 with slowly-varying delay


 * - LMI for Hinf-optimal full-state-feedback control in a non-delayed continuous-time system


 * - LMI for Hinf-optimal output-feedback control in a non-delayed continuous-time system