Finsler's lemma

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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

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

  • [math]\displaystyle{ \displaystyle x^{T}Lx=0 \text{ and } x \ne 0 \text{ implies } x^T Q x \lt 0. }[/math]
  • [math]\displaystyle{ \exists \mu \in \mathbb{R} : Q - \mu L \prec 0. }[/math]

Variants

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:[3]

  • [math]\displaystyle{ x^T Q x \lt 0 \text{ for all } x \in \ker(B)\smallsetminus\{0\} }[/math]
  • [math]\displaystyle{ B^{\perp^T} Q B^\perp \prec 0 }[/math]
  • [math]\displaystyle{ \exists \mu \in \mathbb{R} : Q - \mu B^T B \prec 0 }[/math]
  • [math]\displaystyle{ \exists X \in \mathbb{R}^{n \times m} : Q + XB + B^T X^T \prec 0 }[/math]

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):[4][5]

  • [math]\displaystyle{ \left[\begin{array}{l} I \\ Z \end{array}\right]^{\top} Q\left[\begin{array}{l} I \\ Z \end{array}\right] \succeq 0 \text{ for all } Z \in \mathbb{R}^{l \times k} \text { such that }\left[\begin{array}{l} I \\ Z \end{array}\right]^{\top} L\left[\begin{array}{l} I \\ Z \end{array}\right]=0 }[/math]
  • [math]\displaystyle{ \exists \mu \in \mathbb{R} : Q - \mu L \succeq 0 }[/math]

under the assumption that

[math]\displaystyle{ Q = \begin{bmatrix} Q_{11} & Q_{12} \\ Q_{12}^T & Q_{22} \end{bmatrix} }[/math] and [math]\displaystyle{ L = \begin{bmatrix} L_{11} & L_{12} \\ L_{12}^T & L_{22} \end{bmatrix} }[/math]

satisfy the following assumptions:

  1. Q12 = 0 and Q22 < 0,
  2. L22 < 0, and L11 - L12L22+L12 = 0, and
  3. there exists a matrix G such that Q11 + GTQ22G > 0 and L22G = L12T.

Generalizations

Projection lemma

The following statement, known as Projection Lemma (or also as Elimination Lemma), is common on the literature of linear matrix inequalities:[6]

  • [math]\displaystyle{ B^{\perp^T} Q B^\perp \prec 0 \text{ and } C^{T \perp T} Q C^{T \perp} \prec 0 }[/math]
  • [math]\displaystyle{ \exists X \in \mathbb{R}^{n \times m} : Q + CXB + B^T X^T C^T \prec 0 }[/math]

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. Furthermore, there exists a version of the projection lemma that utilizes non-strict inequalities.[7]

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 [math]\displaystyle{ Q(s)-\mu B^{T}(s)B(s) \prec 0 }[/math] for all [math]\displaystyle{ s\in S }[/math] (respectively, [math]\displaystyle{ Q(s) + X(s)B(s)+B^T(s)X^T(s) \prec 0 }[/math]). 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.[8]

Applications

Data-driven control

The matrix variant of Finsler lemma has been applied to the data-driven control of Lur'e systems[4] and in a data-driven robust linear matrix inequality-based model predictive control scheme.[9]

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.[10] This approach has recently been called as S-variable approach[11][12] and the LMIs stemming from this approach are known as SV-LMIs (also known as dilated LMIs[13]).

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.[14]

See also

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. ↑ 3.0 3.1 de Oliveira, Maurício C.; Skelton, Robert E. (2001). "Stability tests for constrained linear systems". in Moheimani, S. O. Reza. Perspectives in robust control. London: Springer-Verlag. pp. 241–257. ISBN 978-1-84628-576-9. https://archive.org/details/perspectivesrobu00mohe. 
  4. ↑ 4.0 4.1 van Waarde, Henk J.; Kanat Camlibel, M. (2021-12-14). "A Matrix Finsler's Lemma with Applications to Data-Driven Control". 2021 60th IEEE Conference on Decision and Control (CDC). Austin, TX, USA: IEEE. pp. 5777–5782. doi:10.1109/CDC45484.2021.9683285. ISBN 978-1-6654-3659-5. https://ieeexplore.ieee.org/document/9683285. 
  5. ↑ van Waarde, Henk J.; Camlibel, M. Kanat; Eising, Jaap; Trentelman, Harry L. (2023-08-31). "Quadratic Matrix Inequalities with Applications to Data-Based Control" (in en). SIAM Journal on Control and Optimization 61 (4): 2251–2281. doi:10.1137/22M1486807. ISSN 0363-0129. https://epubs.siam.org/doi/10.1137/22M1486807. 
  6. ↑ 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. 
  7. ↑ Meijer, Tomas; Holicki, Tobias; van den Eijnden, Sebastiaan; Scherer, Carsten W.; Heemels, Maurice (2023). "The Non-Strict Projection Lemma". arXiv:2305.08735 [math.OC].
  8. ↑ 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. doi:10.1109/tac.2017.2682221. ISSN 0018-9286. 
  9. ↑ Nguyen, Hoang Hai; Friedel, Maurice; Findeisen, Rolf (2023-03-08). "LMI-based Data-Driven Robust Model Predictive Control". arXiv:2303.04777 [eess.SY].
  10. ↑ 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. 
  11. ↑ Ebihara, Yoshio; Peaucelle, Dimitri; Arzelier, Denis (2015) (in en-gb). 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. 
  12. ↑ 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. 
  13. ↑ 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. 
  14. ↑ 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. doi:10.1109/tac.2017.2668380. ISSN 0018-9286. 



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