Finsler's lemma

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

Let x ∈ Rⁿ, Q ∈ R^{n x n} and L ∈ R^{n x n} . The following statements are equivalent:[2]

• ${\displaystyle \displaystyle x^{T}Lx=0{\text{ and }}x\neq 0{\text{ implies }}x^{T}Qx<0.}$
• ${\displaystyle \exists \mu \in \mathbb {R} :Q-\mu L\prec 0.}$

Variants

In the particular case that L is positive semi-definite, it is possible to decompose it as L = B^TB. The following statements, which are also referred as Finsler's lemma in the literature, are equivalent:[3]

• ${\displaystyle x^{T}Qx<0{\text{ for all }}x\in \ker(B)\smallsetminus \{0\}}$
• ${\displaystyle B^{\perp ^{T}}QB^{\perp }\prec 0}$
• ${\displaystyle \exists \mu \in \mathbb {R} :Q-\mu B^{T}B\prec 0}$
• ${\displaystyle \exists X\in \mathbb {R} ^{n\times m}:Q+XB+B^{T}X^{T}\prec 0}$

Generalizations

Projection lemma

The following statement, known as Projection Lemma (or also as Elimination Lemma), is common on the literature of linear matrix inequalities:[4]

• ${\displaystyle B^{\perp ^{T}}QB^{\perp }\prec 0{\text{ and }}C^{T\perp T}QC^{T\perp }\prec 0}$
• ${\displaystyle \exists X\in \mathbb {R} ^{n\times m}:Q+CXB+B^{T}X^{T}C^{T}\prec 0}$

This can be seen as a generalization of one of Finsler's lemma variants with the inclusion of an extra matrix and an extra constraint.

Robust version

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 ${\displaystyle Q(s)-\mu B^{T}(s)B(s)\prec 0}$ for all ${\displaystyle s\in S}$ (respectively, ${\displaystyle Q(s)+X(s)B(s)+B^{T}(s)X^{T}(s)\prec 0}$). 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.[5]

Applications

S-Variable approach to robust control of linear dynamical systems

Finsler's lemma can be used to give novel linear matrix inequality (LMI) characterizations to stability and control problems.[3] 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.[6] This approach has recently been called as S-variable approach[7][8] and the LMIs stemming from this approach are known as SV-LMIs (also known as dilated LMIs[9]).

Sufficient condition for universal stabilizability of non-linear systems

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.[10]

See also

• Linear matrix inequality
• S-lemma
• Elimination lemma

References

1. Zi-Zong, Yan; Jin-Hai, Guo (2010). "Some Equivalent Results with Yakubovich's S-Lemma". SIAM Journal on Control and Optimization. 48 (7): 4474–4480. doi:10.1137/080744219.
2. Finsler, Paul (1936). "Über das Vorkommen definiter und semidefiniter Formen in Scharen quadratischer Formen". Commentarii Mathematici Helvetici. 9 (1): 188–192. doi:10.1007/BF01258188.
3. de Oliveira, Maurício C.; Skelton, Robert E. (2001). "Stability tests for constrained linear systems". In Moheimani, S. O. Reza (ed.). Perspectives in robust control. London: Springer-Verlag. pp. 241–257. ISBN 978-1-84628-576-9.
4. Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V. (1994-01-01). Linear Matrix Inequalities in System and Control Theory. Studies in Applied and Numerical Mathematics. Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611970777. ISBN 9780898714852.
5. Ishihara, J. Y.; Kussaba, H. T. M.; Borges, R. A. (August 2017). "Existence of Continuous or Constant Finsler's Variables for Parameter-Dependent Systems". IEEE Transactions on Automatic Control. 62 (8): 4187–4193. arXiv:1711.04570. doi:10.1109/tac.2017.2682221. ISSN 0018-9286.
6. Oliveira, R. C. L. F.; Peres, P. L. D. (July 2007). "Parameter-Dependent LMIs in Robust Analysis: Characterization of Homogeneous Polynomially Parameter-Dependent Solutions Via LMI Relaxations". IEEE Transactions on Automatic Control. 52 (7): 1334–1340. doi:10.1109/tac.2007.900848. ISSN 0018-9286.
7. Ebihara, Yoshio; Peaucelle, Dimitri; Arzelier, Denis (2015). S-Variable Approach to LMI-Based Robust Control | SpringerLink. Communications and Control Engineering. doi:10.1007/978-1-4471-6606-1. ISBN 978-1-4471-6605-4.
8. Hosoe, Y.; Peaucelle, D. (June 2016). S-variable approach to robust stabilization state feedback synthesis for systems characterized by random polytopes. 2016 European Control Conference (ECC). pp. 2023–2028. doi:10.1109/ecc.2016.7810589. ISBN 978-1-5090-2591-6.
9. Ebihara, Y.; Hagiwara, T. (August 2002). A dilated LMI approach to robust performance analysis of linear time-invariant uncertain systems. Proceedings of the 41st SICE Annual Conference. SICE 2002. 4. pp. 2585–2590 vol.4. doi:10.1109/sice.2002.1195827. ISBN 978-0-7803-7631-1.
10. Manchester, I. R.; Slotine, J. J. E. (June 2017). "Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design". IEEE Transactions on Automatic Control. 62 (6): 3046–3053. arXiv:1503.03144. doi:10.1109/tac.2017.2668380. ISSN 0018-9286.