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Discrete-Time Bounded Real Lemma

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.

Discrete-Time Bounded Real Lemma or the H∞ norm can be found by solving a LMI.

The System

Discrete-Time LTI System with state space realization $(A_{d},B_{d},C_{d},D_{d})$
${\begin{aligned}&A_{d}\in {\bf {{R^{n*n}},}}&B_{d}\in {\bf {{R^{n*m}},}}&C_{d}\in {\bf {{R^{p*n}},}}&D_{d}\in {\bf {{R^{p*m}}\;}}\\\end{aligned}}$

The Data

The matrices: System $(A_{d},B_{d},C_{d},D_{d}),P$.

The Optimization Problem

The following feasibility problem should be optimized:

$\gamma$ is minimized while obeying the LMI constraints.

The LMI:

Discrete-Time Bounded Real Lemma

The LMI formulation

H∞ norm < $\gamma$

${\begin{aligned}P\in {S^{n}};\gamma \in {R_{>0}}\;\\&P>0,\\{\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}&C_{d}^{T}\\*&B_{d}^{T}PB_{d}-\gamma I&D_{d}^{T}\\*&*&-\gamma I\end{bmatrix}}&<0,\end{aligned}}$

Conclusion:

The H∞ norm is the minimum value of $\gamma \in {R_{>0}}$that satisfies the LMI condition. If $(A_{d},B_{d},C_{d},D_{d})$ is the minimal realization then the inequalities can be non-strict.

Implementation

A link to CodeOcean or other online implementation of the LMI
MATLAB Code

Related LMIs

[1] - Continuous time KYP_Lemma_(Bounded_Real_Lemma)

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.