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

The System

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

$x_{k+1}=A_{d}x_{k}+B_{d}u_{k},$

where $A_{d}\in \mathbb {R} ^{nxn}$, and $B_{d}\in \mathbb {R} ^{nxm}$,

the transient bound can be analyzed with the LMI below.

The Data

$A_{d}\in \mathbb {R} ^{nxn}$, and $B_{d}\in \mathbb {R} ^{nxm}$

The LMI:

The Euclidean norm of the state satisfies:

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

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

• $P-\gamma 1\leq 0,$
• ${\begin{bmatrix}P&1\\*&\gamma 1\end{bmatrix}}\geq 0,$
• ${\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}\\*&B_{d}^{T}B_{d}-\gamma 1\end{bmatrix}}\leq 0.$

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

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 Discrete-Time 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.