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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

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

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

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

Determining an Upper Bound

The output of this system will satisfy

${\sqrt {\int _{0}^{T}{\mathbf {y^{T}y}}dt}}=\left\vert \left\vert {\mathbf {y}}\right\vert \right\vert _{2T}\leq \gamma \left\vert \left\vert {\mathbf {x}}_{0}\right\vert \right\vert _{2},\forall T\in \mathbb {R} _{\geq 0}$

if there exists some matrix ${\mathbf {P}}\in \S ^{p}$and scalar $\gamma \in \mathbb {R} _{>0}$ such that

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

${\mathbf {P}}-\gamma {\mathbf {I}}\leq 0$,

${\begin{bmatrix}{\mathbf {PA}}+{\mathbf {A^{T}P}}&{\mathbf {C^{T}}}\\*&-\gamma {\mathbf {I}}\end{bmatrix}}\leq 0$.

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.

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.