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

The System

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

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

where $A\in \mathbb {R} ^{nxn}$, $B\in \mathbb {R} ^{nxm}$ and x(0) = x₀,

the transient bound can be evaluated with the following LMI.

The Data

$A\in \mathbb {R} ^{nxn}$, $B\in \mathbb {R} ^{nxm}$ and x(0) = x₀.

The LMI:

The Euclidean norm of the state satisfies:

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

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

• $P-\gamma 1\leq 0,$
• ${\begin{bmatrix}P&1\\*&\gamma 1\end{bmatrix}}\geq 0,$
• ${\begin{bmatrix}PA+A^{T}P&PB\\*&-\gamma 1\end{bmatrix}}\leq 0.$

if x₀ = 0 and u is a unit-energy input ($\lVert u\rVert _{2T}\leq 1,\forall T\in \mathbb {R} _{\geq 0}$), then the above LMIs ensure that $\lVert x(T)\rVert _{2}\leq \gamma ,\forall T\in \mathbb {R} _{\geq 0}$

Conclusion

By using this LMI the transient state bound can be analyzed for a given non-autonomous LTI system.

Implementation

The LMI given above can be implemented and solved using a tool such as YALMIP, along with an LMI solver such as MOSEK.

Related LMIs

• Output Energy Bound for Autonomous LTI Systems

External Links

• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.