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Output Energy Bound for 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.

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

$$ \bold{\dot x} = \bold{Ax}$$,

$$ \bold{y}=\bold{Cx}$$,

where $$ \bold{A} \in \R^{n\times n}, \bold{C} \in \R^{p\times n}$$ and $$ \bold{x}(0)=\bold{x}_0.$$

Determining an Upper Bound
The output of this system will satisfy

$$ \sqrt{\int_{0}^{T} \bold{y^Ty}dt}=\left\vert\left\vert \bold{y} \right\vert\right\vert_{2T} \leq \gamma \left\vert\left\vert \bold{x}_0 \right\vert\right\vert_2 ,\forall T \in \R_{\geq0}$$

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

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

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

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

Conclusion
Given an autonomous system with an initial operating condition, the parameter $$ \gamma$$ can be used to determine the largest feasible bound on the output of that system.