# LMIs in Control/pages/Discrete-Time Mixed H2 HInf Optimal Observer

LMIs in Control/pages/Discrete-Time Mixed H2 HInf Optimal Observer

In many applications, perhaps even most, the state of the system cannot be directly known. In this case, you will need to strategically to measure key system outputs that will make the system states indirectly observable. Observers need to converge much faster than the system dynamics in order for their estimations to be accurate. Optimal observer synthesis is therefore advantageous. In this LMI, we seek to optimize both H2 and Hinf norms, to minimize both the average and the maximum error of the observer.

**The System**Edit

where and is the state vector, and is the state matrix, and is the input matrix, and is the exogenous input, and is the output matrix, and is the feedthrough matrix, and is the output, and it is assumed that is detectable.

```
```

**The Data**Edit

The matrices .

**The Optimization Problem**Edit

An observer of the form:

is to be designed, where is the observer gain.

Defining the error state , the error dynamics are found to be

,

and the performance output is defined as

.

The observer gain is to be designed to minimize the norm of the closed loop transfer matrix from the exogenous input to the performance output is less than , where

**The LMI:** Discrete-Time Mixed H2-Hinf-Optimal ObserverEdit

The discrete-time mixed- -optimal observer gain is synthesized by solving for , , , and that minimize *J* subject to ,

where refers to the trace of a matrix.

**Conclusion:**Edit

The mixed- -optimal observer gain is recovered by , the norm of is less than , and the norm of is less than . This result gives us a matrix of observer gains that allow us to optimally observe the states of the system indirectly as:

**Implementation**Edit

This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/mixedh2hinfobsdiscretetime.m

**Related LMIs**Edit

## External LinksEdit

This LMI comes from Ryan Caverly's text on LMI's (Section 5.3.2):

- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.

Other resources:

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.