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

A(x) := A_{0} + x_{1}A_{1} + ... + x < 0_{N}A_{N} | (3-1) |

where

*x*= (*x*_{1}, . . . ,*x*) is a vector of unknown scalars (the_{N}*decision*or*optimization*variables)*A*_{0}, . . . ,*A*are given_{N}*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 $$A\left(\frac{y+z}{2}\right)<0$$. As a result,

Its solution set, called the

*feasible set*, is a convex subset of*R*^{N}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

$$\{\begin{array}{c}{A}_{1}\left(x\right)<0\\ \vdots \\ {A}_{K}\left(x\right)<0\end{array}$$

is equivalent to

$$A\left(x\right):=\text{diag}\left({\text{A}}_{\text{1}}\left(x\right),\dots ,{\text{A}}_{\text{K}}\left(x\right)\right)<0$$

where diag (*A*_{1}(*x*), . . . ,
*A*_{K}(*x*)) denotes the
block-diagonal matrix with

*A*_{1}(*x*), . . .
, *A*_{K}(*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*(*X*_{1}, . . . ,
*X*_{n}) <
*R*(*X*_{1}, . . . ,
*X*_{n})

where *L*(.) and *R*(.) are affine functions of some
structured *matrix* variables *X*_{1},
. . . , *X*_{n}. *A* simple example is
the Lyapunov inequality

A + ^{T}XXA < 0 | (3-2) |

where the unknown *X* is a symmetric matrix. Defining
*x*_{1}, . . . ,
*x*_{N} 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.

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