LMIs in Control/pages/Discrete-Time Quadratic Stability

Discrete-Time Quadratic Stability
Stability is an important property, stability analysis is necessary for control theory. For robust control, this criterion is applicable for the uncertain discrete-time linear system. It is based on the Discrete Time Lyapunov Stability.

The System


\begin{align} x_{k+1}&=A_d(\alpha) x_k \\ Where: \\ & A_d(\alpha) = A_{d} + \Delta A_d (\delta(t)) \\ & \Delta A_d (\delta(t)) = \sum_{k = 1}^n \delta_k(t) A_{d;k} \in \mathrm{R}^{n\times n} \\ & \delta(t) = [\delta_1(t), ... \delta_n(t)] - \text{The set of perturbation parameters} \\ & \delta(t) \in \mathrm{R} \;\;\; A_{d;i} \in \mathrm{R}^{n\times n} \end{align} $$

The Data
The matrices $$ A \in R^{n\times n}\; A_{d;i}\in R^{n\times n} $$.

The Optimization Problem
The following feasibility problem should be solved:

\begin{align} \text{Find} \; &P >0:\\ & (A_{d;0} + \Delta A_d (\delta(t)) )^T P (A_{d;0} + \Delta A_d (\delta(t))) - P < 0 \text{ for all } \delta \end{align} $$

Where $$ P \in R^{n,n} $$.

In case of polytopic uncertainty:

\begin{align} \text{Find} \; &P >0:\\ & (A_{d;0} + A_{d;i})^T P (A_{d;0} + A_{d;i}) - P < 0 \text{ for all } i = 1,... n \end{align} $$

Conclusion:
This LMI allows us to investigate stability for the robust control problem in the case of polytopic uncertainty and gives on the controller for this case

Implementation:

 * - Matlab implementation using the YALMIP framework and Mosek solver

Related LMIs:

 * - Discrete Time Stabilizability
 * Polytopic stability for continuous time case
 * Quadratic polytopic stabilization
 * Discrete Time Lyapunov Stability

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