# LMIs in Control/pages/H2-OSE

LMIs in Control/pages/H2-OSE

The H2 norm of a stable system H is the root-mean-square of the impulse response of the system. The H2 norm measures the steady-state covariance (or power) of the output response to unit noise input. In this module, the goal of H_{2} optimal state estimation is to design an observer that minimizes the H_{2} norm of the closed loop transfer matrix

**The System**[edit | edit source]

Consider the continuous-time generalized plant * P* with state-space realization

where it is assumed that (A,C_{2}) is detectable. An observer of the form

**The Data**[edit | edit source]

- ∈
, ∈^{n}, ∈^{l}are respectively the state vector, the measured^{m}

output vector, and the output vector of interests

- ∈
and ∈^{p}are the disturbance vector and the control vector,^{r}

respectively

**A, B**are the system coefficient matrices of_{1}, B_{2}, C_{1}, C_{2}, D_{1}, and D_{2}

appropriate dimensions

**The Optimization Problem**[edit | edit source]

Given the system and a positive scalar we have to find the matrix L such that

||||_{2} <

An observer of the form

is to be designed, where L is the observer gain.

Defining the error state as

**
**

The break dynamics are found to be

For the system we introduce a full state observer in the following form:

are the observation vector and the observer gain.

The transfer function for this case is

and thus the problem of state observer design is to find L such that

||

** The LMI: LMI for H**_{2} Observer estimation[edit | edit source]

_{2}Observer estimation

The H_{2} state observer problem has a solution if and only if there exists a matrix **, a symmetric matrix **** and a symmetric matrix **** such that
**

and from the solution of the above LMIs we can obtain the observer matrix as

**Conclusion**[edit | edit source]

Thus by formulation, we have converted the problem of H_{2} state observer design into an LMI feasibility problem by optimizing the above LMIs. In application we are often concerned with the problem of finding the minimal attenuation level **
**

On implementation and optimization of the above LMI using YALMIP and MOSEK (or) SeDuMi we get 3 matrices as output, and also which is used to calculate which is the H_{2} norm of the system.

There exists another set of LMIs which holds true for the same optimization problem as above.

**
**

When a minimal ** is obtained, the minimal attenuation level is
**

**Implementation**[edit | edit source]

A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/ygovada

** Related LMIs**[edit | edit source]

H_{$\infty$} State Observer Design

Discrete time H_{2} State Observer Design

** External Links**[edit | edit source]

- [1] - LMI in Control Systems Analysis, Design and Applications
- 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.