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

Output Energy Bound for Discrete-Time Autonomous LTI Systems

Autonomous systems are initialized under a given set of initial conditions, and then run without any additional inputs. It is useful to know ahead of time the bounds such a system will operate within. This analysis can be used to determine the upper bound on the output of a given autonomous LTI system operating in discrete time.

The System
Consider the discrete-time, LTI autonomous system with state space representation

$$ \bold{x}_{k+1} = \bold{A}_d\bold{x}_k$$,

$$ \bold{y}_k=\bold{C}_d\bold{x}_k$$,

where $$ \bold{A}_d \in \R^{n\times n}$$ and $$ \bold{C}_d \in \R^{p\times n}.$$

Determining an Upper Bound
The output of this system will satisfy

$$ \left\vert\left\vert \bold{y} \right\vert\right\vert_{2k} \leq \gamma \left\vert\left\vert \bold{x}_0 \right\vert\right\vert_2 ,\forall k \in \Z_{\geq0}$$

if there exists some matrix $$ \bold{P} \in \S^n$$and scalar $$ \gamma \in \R_{>0}$$ such that

$$ \bold{P} > 0$$,

$$ \bold{P} - \gamma\bold{I} \leq0$$,

$$ \begin{bmatrix} \bold{A_d^T P A_d}-\bold{P} & \bold{C_d^T} \\ \end{bmatrix} \leq0$$.
 * & -\gamma\bold{I}

Conclusion
Given an autonomous system operating in discrete-time conditions, the parameter $$ \gamma$$ can be used to determine the largest feasible bound on the output of that system.