IFAC WC 2026 Open Invited Track

"LMIs and S-variable Approach in Control"

Code: df36u
Organizers: Dimitri Peaucelle1 and Yoshio Ebihara2
1 LAAS-CNRS, Univ. Toulouse, CNRS, Toulouse, France
2 Kyushu University, Japan
This web page will be updated in accordance with helpful comments from possible contributors to the OIT. You are invited to contact the organizers of the track for any suggestions.

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", and "Descriptor approach". 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 as well as for control design ranging from state-feedback, observer design and output-design.

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 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:

\[ M^{\perp T} Q M^\perp \prec 0 \quad \Leftrightarrow \quad \exists S \,:\, Q \prec S M + M^T S^T \]

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 be 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:

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

This fact not only enables considering truly descriptor systems described by differential algebraic equations [13], but allows manipulating 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 the 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