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A linear matrix inequality (LMI) is any constraint of the form
where
x = (x_{1}, . . . , x_{N}) is a vector of unknown scalars (the decision or optimization variables)
A_{0}, . . . , A_{N} 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 $$A\left(\frac{y+z}{2}\right)<0$$. 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
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.