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

The System

Consider a continuous-time LTI system with a state-space representation of:

${\dot {x}}=Ax+Bu,$

$y=Cx,$

where $A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$ and $x(0)=x_{0}$,

the transient bound on the output can be evaluated with the following LMI.

The Data

$A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$ and $x(0)=x_{0}$.

The LMI:

The Euclidean norm of the output satisfies

$\lVert y(T)\rVert _{2}^{2}\leq \gamma ^{2}(\lVert x_{0}\rVert _{2}^{2}+\lVert u\rVert _{2}T^{2}),\forall T\in \mathbb {R} _{+},$

if there exists $P\in \mathbb {S} ^{p}$ and $\gamma \in \mathbb {R} _{++}$, where $P>0$, such that

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

Conclusion

By using these LMIs, the transient output bound can be found for a given non-autonomous LTI system such that if $x_{0}=0$ and $u$ is a unit-energy input (i.e., $\lVert u\rVert _{2}T^{2}\leq 1,\forall T\in \mathbb {R} _{+}$), then the preceding conditions ensure that $\lVert y(T)\rVert _{2}\leq \gamma ,\forall T\in \mathbb {R} _{+}$.

Implementation

The LMI given above can be implemented and solved using a parser such as YALMIP, along with an LMI solver such as SeDuMi, or MOSEK. (For a detailed list of solvers, please visit this page on Relevant Solvers).

References

Caverly, Ryan; Forbes, James (2021). LMI Properties and Applications in Systems, Stability, and Control Theory.