IFAC WC 2026 Open Invited Track

"LMIs and S-variable Approach in Control"
code: df36u

Organizers: Dimitri Peaucelle$^{1}$ and Yoshio Ebihara$^2$
$^1$ LAAS-CNRS, Univ. Toulouse, CNRS, Toulouse, France,
$^2$ Kyushu University, Japan

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Abstract

This Open Invited Track proposal aims at gathering contributions where LMI results for control problems make the usage of the ''S-variable approach", which has many other names in the literature among which: ''dilated LMIs", ''extended LMIs", ''Finsler lemma based", ''Descriptor approach" are some of these. This approach has had a significant impact on the Robust control field for 25 years and is still widely used for many control problems such as the analysis of systems with uncertainties, TS-fuzzy systems, linear systems with non-linearities, time-varying systems, parameter-varying systems and well as for control design ranging from state-feedback, observer design and outputdesign. The Open Invited Track will give the opportunity to gather in joint session(s) various viewpoints on the usage of this technique, its advantages but also the drawbacks and how to circumvent these. Contributions aiming at relating the S-variable approach with closely related techniques such as the Integral Quadratic Constraints framework, the KYP-lemma, Lagrange relaxations, Lyapunov theory, Sum-of-Squares and others, are of high interest. Both theoretical and application papers are most welcome.

Keywords: S-variable approach, Finsler's lemma, Dilated LMIs, Extended LMIs, Descriptor approach, Polytopic uncertainties, TS-fuzzy systems, Robustness, LPV control, Time-varying systems, Time-delay systems, Descriptor systems, Analysis, State-feedback, Observers, Output-feedback

Detailed Description

The robust control field has benefitted widely in the past 35 years of the fantastic flexibility and efficiency of Linear Matrix Inequality (LMI) based formulations [1, 6]. Among the several techniques employed to derive results in the form of LMIs one can mention the Lyapunov theory which naturally leads to inequality conditions that have a matrix formulations, as well the Integral Quadratic Constraints (IQCs) framework which combined to the Kalman-Yakubovich-Popov lemma offers wide declinations for systems represented as a feedback loop of a linear plant with some operator. A complementary approach has emerged after the publication [4] which provided new kind of LMIs for a specific problem. Since then, many authors have exploited similar methodologies which in all cases amount to increase the dimensions of the LMIs (both in the number of variables and the size of the constraints) with the benefit of bringing useful degrees of freedom to the original conservative formulation. As soon as [17] and in [20] it was shown that the technique is related to Finsler's lemma that states the equivalence of the two following matrix inequalities

$ M^{\perp T}QM^\perp\prec 0 \quad \Leftrightarrow \quad \exists S \,:\, Q\prec SM+M^T S^T. $

While that lemma was already adopted for building LMIs, it was, until then, used as a projection lemma allowing to reduce the dimensions of the constraints: remove the $S$ variable by projection onto the null space $M^\perp$ of $M$. As shown with many examples in [5] the converse creation of the $S$ variable provides many advantages and can been seen as more than a technical ''trick".

Descriptor. Among important implications of this S-variable approach is the fact that it allows dealing with descriptor forms. This is witnessed as soon as [3], [8] and can be summarized by the following formulation of that same result

$ \begin{bmatrix} \dot x \\ x\end{bmatrix}^T Q \begin{bmatrix} \dot x \\ x\end{bmatrix}\le 0 \quad \forall \begin{bmatrix} -E & A \end{bmatrix} \begin{bmatrix} \dot x \\ x\end{bmatrix}=0 \quad \Leftrightarrow \quad \exists S \,:\, Q\prec S\begin{bmatrix} -E & A \end{bmatrix}+\begin{bmatrix} -E & A \end{bmatrix}^TS^T. $

This fact not only enables to consider truly descriptor systems described by differential algebraic equations [13], but allows to manipulate with ease models containing polynomial and rational expressions in the states or the parameters as if being affine functions [3, 18, 29, 2].

Feedback design. Without giving any final answer to the open problem of feedback design, for which there are systematic methods only in special cases, the S-variable approach has proved to bring efficient heuristics [24]. It has led to many recent results such as in [22, 7, 23, 11] and comparisons to other techniques [9, 26].

But these are only some examples of the impact of S-variable approach to the issue of building understandable efficient LMI results. From the recent literature one can see that the approach continues to feed many contributions among which some tackle the fundamentals of the approach itself [25, 12, 15], others employ the technique for problems outside of the strict robust control framework as in [27] for data-driven control, in [14] for model predictive control, in [16, 21] for linear parameter varying control, in [2] for regional stabilization of non-linear systems or [28] for fault-tolerant control. While papers such as [10, 26] provide most interesting examples of concrete impact on applications.

The Open Invited Track aims at gathering all such contributions dealing with this highly present approach among LMI results for control problems. It may be an opportunity to acknowledge its benefits / drawbacks, and clarify the remaining open issues.

References

[1] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics, Philadelphia, 1994.

[2] D. Coutinho, C.E. de Souza, J. M. Gomes da Silva Jr., A.F. Caldeira, and C. Prieur. Regional stabilization of input-delayed uncertain nonlinear polynomial systems. IEEE Transactions on Automatic Control, 65(5):2300-2307, May 2020.

[3] D. Coutinho, A. Trofino, and M. Fu. Guaranteed cost control of uncertain nonlinear systems via polynomial Lyapunov functions. IEEE Trans. Automatic Control, 47(9):1575-1580, 2002.

[4] M.C. de Oliveira, J. Bernussou, and J.C. Geromel. A new discrete-time stability condition. Systems & Control Letters, 37(4):261-265, July 1999.

[5] Y. Ebihara, D. Peaucelle, and D. Arzelier. S-Variable Approach to LMI-Based Robust Control. Elsevier, 2015.

[6] L. El Ghaoui and S.-I. Niculescu, editors. Advances in Linear Matrix Inequality Methods in Control. Advances in Design and Control. SIAM, Philadelphia, 2000.

[7] Alexandre Felipe and Ricardo CLF Oliveira. An LMI-based algorithm to compute robust stabilizing feedback gains directly as optimization variables. IEEE Transactions on Automatic Control, 66(9):4365-4370, 2020.

[8] E. Fridman and U. Shaked. A descriptor system approach to H-infinity control of time-delay systems. IEEE Trans. on Automat. Control, 47:253-270, 2002.

[9] T. Holicki and C.W. Scherer. Revisiting and generalizing the dual iteration for static and robust output-feedback synthesis. International Journal of Robust and Nonlinear Control, 31(11):5427-5459, 2021.

[10] Gustavo Guilherme Koch, Lucas Borin, Caio Os´orio, Mokthar Aly, Margarita Norambuena, Jose Rodriguez, Fernanda Carnieluti, Humberto Pinheiro, Ricardo CLF Oliveira, and Vin´ıcius F Montagner. Improved control of grid-connected converters from strong to very weak conditions integrating more effective lmis and c-hil. Available at SSRN 5352987.

[11] Dario Giuseppe Lui, Alberto Petrillo, and Stefania Santini. Linear matrix inequality-based design of distributed proportional-integral-derivative for the output consensus tracking in heterogeneous high-order multi-agent systems. Asian Journal of Control, 2025.

[12] T.J. Meijer, T. Holicki, S. van den Eijnden, C.W. Scherer, and WPMH Heemels. The non-strict projection lemma. IEEE Transactions on Automatic Control, 69(8):5584-5590, 2024.

[13] Tomas Jesse Meijer. Estimation and control for discrete-time parameter-varying and descriptor systems. 2024.

[14] Marcelo M Morato, Tobias Holicki, and CarstenWScherer. Stabilizing model predictive control synthesis using integral quadratic constraints and full-block multipliers. International Journal of Robust and Nonlinear Control, 33(18):11434-11457, 2023.

[15] H-N Nguyen. Parameterised ellipsoid and its applications to s-lemma and control of uncertain, time-varying linear systems. International Journal of Control, pages 1-15, 2025.

[16] Lucas A.L. Oliveira, Kevin Guelton, Koffi M.D. Motchon, and Valter J.S. Leite. New negative definiteness conditions for quadratic functions with illustration in LPV sampled-data control. Automatica, 173:112077, 2025.

[17] D. Peaucelle, D. Arzelier, O. Bachelier, and J. Bernussou. A new robust D-stability condition for real convex polytopic uncertainty. Systems & Control Letters, 40(1):21-30, May 2000.

[18] D. Peaucelle and M. Sato. LMI tests for positive definite polynomials: Slack variable approach. Technical Report 07452, LAAS-CNRS, Toulouse, August 2007. Extended version of [19].

[19] D. Peaucelle and M. Sato. LMI tests for positive definite polynomials: Slack variable approach. IEEE Trans. on Automat. Control, 54(4):886 - 891, April 2009.

[20] G. Pipeleers, B. Demeulenaere, J. Swevers, and L. Vandenberghe. Extended LMI characterizations for stability and performance of linear systems. Systems & Control Letters, 58(7):510 - 518, 2009.

[21] P. Polcz, T. P´eni, B. Kulcsar, and G. Szederkenyi. Induced L2-gain computation for rational LPV systems using Finsler's lemma and minimal generators. Systems & Control Letters, 142, 2020.

[22] Gabriela L. Reis, Rodrigo F. Ara´ujo, Leonardo A. B. Torres, and Reinaldo M. Palhares. Regional static output feedback stabilization based on polynomial lyapunov functions for a class of nonlinear systems. Journal of Control, Automation and Electrical Systems, 35(4):601-613, 2024.

[23] LA Rodrigues, RCLF Oliveira, and JF Camino. Parameterized LMIs for robust and state feedback control of continuous-time polytopic systems. International Journal of Robust and Nonlinear Control, 28(3):940-952, 2018.

[24] M.S. Sadabadi and D. Peaucelle. From static output feedback to structured robust static output feedback: A survey. Annual Reviews in Control, 42:11-26, 2016.

[25] M Sato and N Sebe. A variant of extended lmi: Performance analysis with static multiplier for discrete-time lifted systems. IFAC-PapersOnLine, 56(2):5825-5830, 2023.

[26] Daam Schoon and Spilios Theodoulis. Review of H-infinity static output feedback controller synthesis methods for fighter aircraft control. In AIAA SCITECH 2025 Forum, page 2241, 2025.

[27] Alexandre Seuret, Carolina Albea, and Francesco Gordillo. Linear matrix inequality relaxations and its application to data-driven control design for switched affine systems. International Journal of Robust and Nonlinear Control, 33(12):6597-6618, 2023.

[28] Hibiki Shiroiwa, Hiroyuki Ichihara, and Masayuki Sato. Fault-tolerant control using recursive gaussian processes and observer-structured robust output feedback controller. IET Control Theory & Applications, 19(1):e70058, 2025.

[29] A. Trofino and T.J.M. Dezuo. LMI stability conditions for uncertain rational nonlinear systems. International Journal of Robust and Nonlinear Control, 2013.