LMIs in Control/Stability Analysis/Discrete Time/Output Energy Bound for Non-Autonomous LTI Systems

Introduction
Wile in the case of autonomous systems, they only need initial input constraints and cannot be changed using external inputs; the non-autonomous systems on the other hand can have changes happen to its behavior using a controller.

The System
Consider the discrete-time LTI system with state space realization $$\bold{x}_{k+1} = \bold{A}_d \bold{x}_k + \bold{B}_d \bold{u}_k$$ $$\bold{y}_k = \bold{C}_d \bold{x}_k + \bold{D}_d \bold{u}_k$$

The Data
$$\bold A_d \in\mathbb R^{n \times n}$$, $$\bold B_d \in\mathbb R^{n \times m}$$, $$\bold C_d \in\mathbb R^{p \times n}$$, $$\bold D_d \in\mathbb R^{p \times m}$$

Determining the bound
The output of this system must satisfy $$ \sum_{i=0}^{k} \bold{y}_i^T\bold{y}_i=\left\vert\left\vert \bold{y} \right\vert\right\vert_{2k}^2 \leq \gamma^2 (\left\vert\left\vert \bold{x}_0 \right\vert\right\vert_2^2 + \left\vert\left\vert \bold{u} \right\vert\right\vert_{2k}^2 ,\forall k \in \Z_{\geq0}$$ if there exists some matrix $$ \bold{P} \in \S^p$$and scalar $$ \gamma \in \R_{>0}$$, where $$ \bold{P} > 0$$, such that $$ \bold{P} - \gamma\bold{I} \leq0$$,

$$ \begin{bmatrix} \bold{PA}+\bold{A^TP} & \bold{PB} & \bold{C^T} \\ \end{bmatrix} \leq0$$.
 * & -\gamma\bold{I} & \bold D^T \\
 * & * & -\gamma\bold{I}

Implementation
This can be implemented in any LMI solver such as YALMIP, using an algorithmic solver like Gurobi.

Conclusion
Given a non-autonomous system with initial operating conditions and a controller, the parameter $$ \gamma$$ can be used to determine the feasible bound on the output of that system.

Related LMIs

 * LMIs in Control/Stability Analysis/Discrete Time/Output Energy Bound for Autonomous LTI Systems
 * LMIs in Control/Stability Analysis/Continuous Time/Output Energy Bound for Non-Autonomous LTI Systems