LMIs in Control/pages/H-2 filtering

LMIs in Control/pages/H-2 filtering

For systems that have disturbances, filtering can be used to reduce the effects of these disturbances. Described on this page is a method of attaining a filter that will reduce the effects of the disturbances as completely as possible. To do this, we look to find a set of new coefficient matrices that describe the filtered system. The process to achieve such a new system is described below. The H2-filter tries to minimize the average magnitude of error.

The SystemEdit

For the application of this LMI, we will look at linear systems that can be represented in state space as

 

where   represent the state vector, the measured output vector, and the output vector of interest, respectively,   is the disturbance vector, and   and   are the system matrices of appropriate dimension. To further define:   is   and is the state vector,   is   and is the state matrix,   is   and is the input matrix,   is   and is the exogenous input,   is   and is the output matrix,   and   are   and are feedthrough matrices, and   and   are   and are the output and the output of interest, respectively.

The DataEdit

The data are   (the disturbance vector), and   and   (the system matrices). Furthermore, the   matrix is assumed to be stable

The Optimization ProblemEdit

We need to design a filter that will eliminate the effects of the disturbances as best we can. For this, we take a filter of the following form:

 

where   is the state vector,   is the estimation vector, and   are the coefficient matrices of appropriate dimensions.

Note that the combined complete system can be represented as

 

where   is the estimation error,

 

is the state vector of the system, and   are the coefficient matrices, defined as:

 

In other words, for the system defined above we need to find   such that

 

where   is a positive constant, and

 

The LMI: H-2 FilteringEdit

For this LMI, the solution exists if one of the following sets of LMIs hold:

Matrices   exist that obey the following LMIs:

 

or

Matrices   exist that obey the following LMIs:

 

Conclusion:Edit

To find the corresponding filter, use the optimized matrices from the first solution to find:

 

Or the second solution to find:

 

These matrices can then be used to produce   to construct the final filter below, that will best eliminate the disturbances of the system.

 

ImplementationEdit

This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/H2_Filtering.m

Related LMIsEdit

H-infinity filtering

External LinksEdit

This LMI comes from

  • [1] - "LMIs in Control Systems: Analysis, Design and Applications" by Guang-Ren Duan and Hai-Hua Yu

Other resources:



ReferencesEdit

Duan, G. (2013). LMIs in control systems: analysis, design and applications. Boca Raton: CRC Press, Taylor & Francis Group.

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