# LMIs in Control/pages/Deduced LMI Conditions for H-infinity Index

**H-infinity Index Deduced LMI**

Although the KYP Lemma, also known as the Bounded Real Lemma, is a basic condition to evaluate an upper bound on the H_{∞}, the verification of the bound on the H_{∞}-gain of the system can be done via the deduced condition.

**The System**[edit]

A state-space representation of a linear system as given below:

where , and are the system state, output, and the disturbance vector respectively. The transfer function of such a system can be evaluated as:

**The Data**[edit]

Number of states *n*, number of outputs *m* and number of external noise channels *r* need to be known. Moreover, the system matrices *A,B,C,D* are also required to be known.

**The Feasibility LMI**[edit]

For an arbitrary , the transfer function *G(s)* satisfies

if and only if there exists a symmetric matrix *X > 0* and a matrix such that:

where:

The above LMI can be combined with the bisection method to find minimum to find the minimum upper bound on the H_{∞} gain of .

**Conclusion:**[edit]

If there is a feasible solution to the aforementioned LMI, then the upper bounds the infinity norm of the system *G(s)*.

**Implementation**[edit]

To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:

https://github.com/smhassaan/LMI-Examples/blob/master/Deduced_hinf_example.m

**Related LMIs**[edit]

## External Links[edit]

A list of references documenting and validating the LMI.

- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 5.2.2 pp. 153–156.