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We will also encounter these problems with nonstrict LMIs, or more generally, with a mixture of strict and nonstrict LMIs. Copyright c 1994 by the Society for Industrial and Applied Mathematics. e. minimize cT x subject to F (x) ≥ 0 Intuition suggests that we could simply solve the strict EVP (by, say, an interior-point method) to obtain the solution of the nonstrict EVP. This is correct in most but not all cases. , the feasible set of the nonstrict LMI is the closure of the feasible set of the strict LMI.

This fact can be used to form a Routh– Hurwitz (Sturm) type test for passivity as a set of polynomial inequalities in the data A, B, C, and D. When M has no pure imaginary eigenvalues we can construct a solution Pare as follows. , V = [v1 · · · vn ] where v1 , . . , vn are a set of independent eigenvectors of M associated with its n eigenvalues with negative real part. Partition V as V = V1 V2 , where V1 and V2 are square; then set Pare = V2 V1−1 . 38), then P ≥ Pare . Much more discussion of this method, including the precise technical conditions, can be found in the References.

There are simple methods for determining whether this is the case, and when it is, of constructing an appropriate P . We will describe this and other results in Chapter 7. Several standard results from linear algebra can also be interpreted as analytic solutions to certain LMI problems of special form. For example, it is well-known that among all symmetric positive-definite matrices with given diagonal elements, the diagonal matrix has maximum determinant. 5). Notes and References LMIs The term “Linear Matrix Inequality” was coined by J.

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