LMIs in Control/Matrix and LMI Properties and Tools/Change of Subject

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LMIs in Control/Matrix and LMI Properties and Tools/Change of Subject

A Bilinear Matrix Inequality (BMI) can sometimes be converted into a Linear Matrix Inequality (LMI) using a change of variables. This is a basic mathematical technique of changing the position of variables with respect to equal signs and the inequality operators. The change of subject will be demonstrated by the example below.


Example[edit | edit source]

Consider , and , where .

The matrix inequality given by:

is bilinear in the variables and .

Defining a change of variable as to obtain

,

which is an LMI in the variables and .

Once this LMI is solved, the original variable can be recovered by .

Conclusion[edit | edit source]

It is important that a change of variables is chosen to be a one-to-one mapping in order for the new matrix inequality to be equivalent to the original matrix inequality. The change of variable from the above example is a one-to-one mapping since is invertible, which gives a unique solution for the reverse change of variable .


External Links[edit | edit source]

A list of references documenting and validating the LMI.


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