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, and Q ∈ R^{n x n} and L ∈ R^{n x n} be symmetric matrices. 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

Non-Strict Finsler Lemma

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]

Extra equivalences when L is positive semi-definite

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: ^[4]

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

Matrix Finsler's lemma

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]}

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

under the assumption that

${\displaystyle Q={\begin{bmatrix}Q_{11}&Q_{12}\\Q_{12}^{T}&Q_{22}\end{bmatrix}}}$ and ${\displaystyle L={\begin{bmatrix}L_{11}&L_{12}\\L_{12}^{T}&L_{22}\end{bmatrix}}}$

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 + G^TQ₂₂G > 0 and L₂₂G = L₁₂^T_.

Generalizations

Projection lemma

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]

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

This 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]

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. ^[9]

Applications

Data-driven control

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

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

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]

See also

• Linear matrix inequality
• S-procedure
• 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. S2CID   121751764.
3. Meijer, T. J.; Scheres, K. J. A.; van den Eijnden, S.; Holicki, T.; Scherer, C. W.; Heemels, W. P. M. H. (2024). "A General Non-Strict Finsler's Lemma". arXiv: 2403.10306 [math.OC].
4. 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.
5. 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) (PDF). Austin, TX, USA: IEEE. pp. 5777–5782. doi:10.1109/CDC45484.2021.9683285. ISBN   978-1-6654-3659-5. S2CID   246479914.
6. van Waarde, Henk J.; Camlibel, M. Kanat; Eising, Jaap; Trentelman, Harry L. (2023-08-31). "Quadratic Matrix Inequalities with Applications to Data-Based Control". SIAM Journal on Control and Optimization. 61 (4): 2251–2281. arXiv: 2203.12959 . doi:10.1137/22M1486807. ISSN   0363-0129. S2CID   247627787.
7. 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. S2CID   27307648.
8. Meijer, T. J.; Holicki, T.; Eijnden, S. J. A. M. van den; Scherer, C. W.; Heemels, W. P. M. H. (2024). "The Non-Strict Projection Lemma". IEEE Transactions on Automatic Control: 1–8. arXiv: 2305.08735 . doi:10.1109/TAC.2024.3371374. ISSN   0018-9286.
9. 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. S2CID   20563439.
10. Nguyen, Hoang Hai; Friedel, Maurice; Findeisen, Rolf (2023-03-08). "LMI-based Data-Driven Robust Model Predictive Control". arXiv: 2303.04777 [eess.SY].
11. 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. S2CID   23352506.
12. 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.
13. 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. S2CID   34083031.
14. 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. Vol. 4. pp. 2585–2590 vol.4. doi:10.1109/sice.2002.1195827. ISBN   978-0-7803-7631-1. S2CID   125985256.
15. 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. S2CID   5100489.