Linear Matrix Inequalities

LMI solvers, control system analysis and design with LMIs

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.

Tools

LMI EditorSpecify or display systems of LMIs as MATLAB expressions

Functions

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setlmisInitialize description of LMI system
lmivarSpecify matrix variables in LMI problem
lmitermSpecify term content of LMIs
newlmiAttach identifying tag to LMIs
getlmisGet internal 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
hinfgsSynthesis of gain-scheduled H∞ controllers
hinfsynCompute H-infinity optimal controller
h2hinfsynMixed H2/H∞ synthesis with regional pole placement constraints
msfsynMulti-model/multi-objective state-feedback synthesis

Topics