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LMI Solvers

Feasibility, minimization of linear objectives, eigenvalue minimization

Robust Control Toolbox™ LMI functionality serves two purposes:

  • Provide state-of-the-art tools for the LMI-based analysis and design of robust control systems

  • Offer a flexible and user-friendly environment to specify and solve general LMI problems (the LMI Lab)

For users interested in developing their own applications, the LMI Lab provides a general-purpose and fully programmable environment to specify and solve virtually any LMI problem. Note that the scope of this facility is by no means restricted to control-oriented applications.


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lmieditSpecify or display systems of LMIs as MATLAB expressions
setlmisInitialize description of LMI system
lmivarSpecify matrix variables in LMI problem
lmitermSpecify term content of LMIs
newlmiAttach identifying tag to LMIs
getlmisInternal description of LMI system
dellmiRemove LMI from system of LMIs
delmvarRemove one matrix variable from LMI problem
setmvarInstantiate matrix variable and evaluate all LMI terms involving this matrix variable
lmiinfoInformation about variables and term content of LMIs
lminbrReturn number of LMIs in LMI system
matnbrNumber of matrix variables in system of LMIs
decnbrTotal number of decision variables in system of LMIs
dec2matGiven values of decision variables, derive corresponding values of matrix variables
mat2decExtract vector of decision variables from matrix variable values
decinfoDescribe how entries of matrix variable X relate to decision variables
feaspCompute solution to given system of LMIs
mincxMinimize linear objective under LMI constraints
defcxHelp specify cTx objectives for mincx solver
gevpGeneralized eigenvalue minimization under LMI constraints
evallmiGiven particular instance of decision variables, evaluate all variable terms in system of LMIs
showlmiReturn left and right sides of LMI after evaluation of all variable terms


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.