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Finsler's lemma is a mathematical result named after Paul Finsler. It states equivalent ways to express the positive definiteness of a quadratic form Q constrained by a linear form L. Since it is equivalent to another lemmas used in optimization and control theory, such as Yakubovich's S-lemma,[1] Finsler's lemma has been given many proofs and has been widely used, particularly in results related to robust optimization and linear matrix inequalities.
Statement of Finsler's lemma[edit]Let x ∈ Rn, and Q ∈ Rn x n and L ∈ Rn x n be symmetric matrices. The following statements are equivalent:[2]
Non-Strict Finsler Lemma[edit]When the matrix L is indefinite, replacing strict inequalities with non-strict ones still maintains the equivalence between the statements of Finsler's lemma. However, if L is not indefinite, additional assumptions are necessary to ensure equivalence between the statements.[3]
In the particular case that L is positive semi-definite, it is possible to decompose it as L = BTB. The following statements, which are also referred as Finsler's lemma in the literature, are equivalent:[4]
Matrix Finsler's lemma[edit]There is also a variant of Finsler's lemma for quadratic matrix inequalities, known as matrix Finsler's lemma, which states that the following statements are equivalent for symmetric matrices Q and L belonging to R(l+k)x(l+k):[5][6]
under the assumption that
Q = [ Q 11 Q 12 Q 12 T Q 22 ] {\displaystyle Q={\begin{bmatrix}Q_{11}&Q_{12}\\Q_{12}^{T}&Q_{22}\end{bmatrix}}} and L = [ L 11 L 12 L 12 T L 22 ] {\displaystyle L={\begin{bmatrix}L_{11}&L_{12}\\L_{12}^{T}&L_{22}\end{bmatrix}}}
satisfy the following assumptions:
The equivalence between the following statements is also common on the literature of linear matrix inequalities, and is known as the Projection Lemma (or also as Elimination Lemma):[7]
This lemma generalizes one of the Finsler's lemma variants by including an extra matrix C and an extra constraint involving this extra matrix.
It is interesting to note that if the strict inequalities are changed to non-strict inequalities, the equivalence does not hold anymore: only the second statement imply the first statement. Nevertheless, it still possible to obtain the equivalence between the statements under extra assumptions.[8]
Finsler's lemma also generalizes for matrices Q and B depending on a parameter s within a set S. In this case, it is natural to ask if the same variable μ (respectively X) can satisfy Q ( s ) − μ B T ( s ) B ( s ) ≺ 0 {\displaystyle Q(s)-\mu B^{T}(s)B(s)\prec 0} for all s ∈ S {\displaystyle s\in S} (respectively, Q ( s ) + X ( s ) B ( s ) + B T ( s ) X T ( s ) ≺ 0 {\displaystyle Q(s)+X(s)B(s)+B^{T}(s)X^{T}(s)\prec 0} ). If Q and B depends continuously on the parameter s, and S is compact, then this is true. If S is not compact, but Q and B are still continuous matrix-valued functions, then μ and X can be guaranteed to be at least continuous functions.[9]
Data-driven control[edit]The matrix variant of Finsler lemma has been applied to the data-driven control of Lur'e systems[5] and in a data-driven robust linear matrix inequality-based model predictive control scheme.[10]
S-Variable approach to robust control of linear dynamical systems[edit]Finsler's lemma can be used to give novel linear matrix inequality (LMI) characterizations to stability and control problems.[4] The set of LMIs stemmed from this procedure yields less conservative results when applied to control problems where the system matrices has dependence on a parameter, such as robust control problems and control of linear-parameter varying systems.[11] This approach has recently been called as S-variable approach[12][13] and the LMIs stemming from this approach are known as SV-LMIs (also known as dilated LMIs[14]).
Sufficient condition for universal stabilizability of non-linear systems[edit]A nonlinear system has the universal stabilizability property if every forward-complete solution of a system can be globally stabilized. By the use of Finsler's lemma, it is possible to derive a sufficient condition for universal stabilizability in terms of a differential linear matrix inequality.[15]
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