# Linear Matrix Inequalities

Linear Matrix Inequalities (LMIs) and LMI techniques have emerged as powerful design tools in areas ranging from control engineering to system identification and structural design. Three factors make LMI techniques appealing:

A variety of design specifications and constraints can be expressed as LMIs.

Once formulated in terms of LMIs, a problem can be solved

*exactly*by efficient convex optimization algorithms (see LMI Solvers).While most problems with multiple constraints or objectives lack analytical solutions in terms of matrix equations, they often remain tractable in the LMI framework. This makes LMI-based design a valuable alternative to classical “analytical” methods.

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**Linear Matrix Inequalities**Linear Matrix Inequalities (LMIs) and LMI techniques are powerful design tools in areas ranging from control engineering to system identification and structural design.

**LMI Applications**Applications of LMIs include robust stability, optimal LQG control, estimation, and many others.

**Tools for Specifying and Solving LMIs**The LMI Lab blends tools for the specification and manipulation of LMIs with powerful LMI solvers for three generic LMI problems.

**Specifying a System of LMIs**

To specify a system of LMIs, declare the dimensions and structure of each matrix variable, and then describe the terms of each LMI.

**LMI Solvers**There is a solver for each of the three generic optimization problems.

**Minimize Linear Objectives Under LMI Constraints**Solve an optimization problem using the

`mincx`

solver.**Conversion Between Decision and Matrix Variables**LMI solvers optimize a vector of the free scalar entries of the matrix variables. These entries are called the decision variables.

**Validating Results**Use

`evallmi`

and`showlmi`

to analyze and validate the results of an LMI optimization.**Advanced LMI Techniques**LMI Lab supports structured matrix variables, complex-valued LMIs, custom objectives.