This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English verison of the page.

Note: This page has been translated by MathWorks. Please click here
To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

LMIs and LMI Problems

A linear matrix inequality (LMI) is any constraint of the form

A(x) := A0 + x1A1 + ... + xNAN < 0(3-1)


  • x = (x1, . . . , xN) is a vector of unknown scalars (the decision or optimization variables)

  • A0, . . . , AN are given symmetric matrices

  • < 0 stands for "negative definite," i.e., the largest eigenvalue of A(x) is negative

Note that the constraints A(x) > 0 and A(x) < B(x) are special cases of Equation 3-1 since they can be rewritten as –A(x) < 0 and A(x) – B(x) < 0, respectively.

The LMI of Equation 3-1 is a convex constraint on x since A(y) < 0 and A(z) < 0 imply that A(y+z2)<0. As a result,

  • Its solution set, called the feasible set, is a convex subset of RN

  • Finding a solution x to Equation 3-1, if any, is a convex optimization problem.

Convexity has an important consequence: even though Equation 3-1 has no analytical solution in general, it can be solved numerically with guarantees of finding a solution when one exists. Note that a system of LMI constraints can be regarded as a single LMI since


is equivalent to


where diag (A1(x), . . . , AK(x)) denotes the block-diagonal matrix with
A1(x), . . . , AK(x) on its diagonal. Hence multiple LMI constraints can be imposed on the vector of decision variables x without destroying convexity.

In most control applications, LMIs do not naturally arise in the canonical form of Equation 3-1 , but rather in the form

L(X1, . . . , Xn) < R(X1, . . . , Xn)

where L(.) and R(.) are affine functions of some structured matrix variables X1, . . . , Xn. A simple example is the Lyapunov inequality

ATX + XA < 0(3-2)

where the unknown X is a symmetric matrix. Defining x1, . . . , xN as the independent scalar entries of X, this LMI could be rewritten in the form of Equation 3-1. Yet it is more convenient and efficient to describe it in its natural form Equation 3-2, which is the approach taken in the LMI Lab.

Related Topics

Was this topic helpful?