LMIs in Control/pages/Discrete Time Lyapunov Stability

Discrete-Time Lyapunov Stability

A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.

Stability of DT LTI systems can be determined by solving Lyapunov Inequality.

The System

Discrete-Time System

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

The Data

The matrices: System $A_{d},P$ .

The Optimization Problem

The following feasibility problem should be optimized:

Find P obeying the LMI constraints.

The LMI:

Discrete-Time Bounded Real Lemma

The LMI formulation

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

Conclusion:

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