# LMIs in Control/pages/Discrete-Time Lyapunov Stability

Discrete-Time Lyapunov Stability

## The System

Discrete-Time System

{\displaystyle {\begin{aligned}x(t)_{k+1}&=A_{d}x(t)_{k},&A_{d}\in {\bf {{R^{n*n}}\;}}\\\end{aligned}}}

## The Data

The matrices ${\displaystyle SystemA_{d},P}$ .

## The Optimization Problem

The following feasibility problem should be solved:

Find P obeying the LMI constraints.

## The LMI:

Discrete-Time Lyapunov Stability

The LMI formulation

{\displaystyle {\begin{aligned}P\in {\bf {S^{n}}}\\Find\;&P>0,\\{\begin{bmatrix}A_{d}^{T}PA_{d}-P\end{bmatrix}}&\leq 0\end{aligned}}}

## Conclusion:

If there exists a ${\displaystyle P\in {\bf {S^{n}}}}$  satisfying the LMI then, ${\displaystyle |\lambda _{i}(A_{d})|\leq 1,\forall i=1,2,...,n;}$  and the equilibrium point ${\displaystyle {\bar {x}}=0}$  of the system is Lyapunov stable.

## Implementation

• [] - MATLAB implementation of the LMI.

## Related LMIs

Continuous_Time_Lyapunov_Inequality - Lyapunov_Inequality