Controller Design for Structured State-space Models via Contraction Theory

Muhammad Zakwan; Vaibhav Gupta; Alireza Karimi; Efe C. Balta; Giancarlo Ferrari-Trecate

eess.SYApr 8, 2026

Controller Design for Structured State-space Models via Contraction Theory

Authors: Muhammad Zakwan, Vaibhav Gupta, Alireza Karimi, Efe C. Balta, Giancarlo Ferrari-Trecate

Organizations: Control & Automation Group, Inspire AG, 8005 Zürich, Switzerland · Automatic Control Laboratory (IfA), ETH Zürich, 8092 Zürich, Switzerland. · Laboratoire d’Automatique, EPFL, 1015 Lausanne, Switzerland.

Abstract

This paper presents an indirect data-driven output feedback controller synthesis for nonlinear systems, leveraging Structured State-space Models (SSMs) as surrogate models. SSMs have emerged as a compelling alternative in modelling time-series data and dynamical systems. They can capture long-term dependencies while maintaining linear computational complexity with respect to the sequence length, in comparison to the quadratic complexity of Transformer-based architectures. The contributions of this work are threefold. We provide the first analysis of controllability and observability of SSMs, which leads to scalable control design via Linear Matrix Inequalities (LMIs) that leverage contraction theory. Moreover, a separation principle for SSMs is established, enabling the independent design of observers and state-feedback controllers while preserving the exponential stability of the closed-loop system. The effectiveness of the proposed framework is demonstrated through a numerical example, showcasing nonlinear system identification and the synthesis of an output feedback controller.

Figures & tables

Panel of Figure 1 — Fig. 1 : Typical SSM layer

Parameter	Symbol	Value
Armature resistance	$R_{a}$	$3.3$	$\mathrm{\SIUnitSymbolOhm}$
Armature inductance	$L_{a}$	$2.75$	$\mathrm{mH}$
Motor constant	$K_{m}$	$3.24\text{\times}{10}^{-2}$	$\mathrm{N}\text{\,}\mathrm{m}\text{\,}{\mathrm{A}}^{-1}$
Motor inertia	$J_{m}$	$1.16\text{\times}{10}^{-4}$	$\mathrm{kg}\text{\,}{\mathrm{m}}^{2}$
Load inertia	$J_{L}$	$4.0\text{\times}{10}^{-4}$	$\mathrm{kg}\text{\,}{\mathrm{m}}^{2}$
Shaft stiffness	$k_{s}$	$1.35$	$\mathrm{N}\text{\,}\mathrm{m}\text{\,}{\mathrm{rad}}^{-1}$

TABLE I : DC Motor Parameters

Fig. 2 : A sample validation trajectory after the training

Fig. 3 : Controller response on the nonlinear mathematical model for different initial conditions

Contraction Theory

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Regularity and Stability Properties of Selective SSMs with Discontinuous Gating
May 16, 2025Nikola Zubić, Davide ScaramuzzaDynamical Systems Selective SSMs
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Okinawa Institute of Science and Technology · Waseda University
A Nonlinear Separation Principle via Contraction Theory: Applications to Neural Networks, Control, and Learning
Apr 16, 2026Anand Gokhale, Anton V. Proskurnikov, Yu Kawano +1Dynamical Systems Contraction Theory
Center for Control, Dynamical Systems, and Computation, UC Santa Barbara, Santa Barbara, CA 93106 USA · Department of Electronics and Telecommunications, Politecnico di Torino, Italy, 10129 · Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan

May 16, 2025J/K move · Enter open · S save

Regularity and Stability Properties of Selective SSMs with Discontinuous Gating

Nikola Zubić, Davide Scaramuzza

Robotics and Perception Group University of Zurich

Dynamical Systems Selective SSMs

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Selective State-Space Models (SSMs) such as Mamba have become central to long-sequence modeling. Still, their stability is poorly understood: their state-space coefficients are modulated online by a token-dependent gating signal, making the recurrence neither linear time-invariant nor classically nonlinear. We study continuous-time selective SSMs through passivity, dissipativity, and Input-to-State Stability (ISS), explicitly separating the selection signal

x(\cdot)

from the driving input

u(\cdot)

. We obtain four results: exponential forgetting under strict dissipativity; a canonical

\mathrm{AUC}_{\mathrm{loc}}

quadratic storage for the frozen-selection subsystem that accommodates discontinuous gating; a parametric LMI together with universal kernel constraints and "irreversible forgetting" under universal quadratic storage; and sufficient conditions for global ISS uniformly over admissible selection schedules. We then bridge to practice by deriving a sampled block LMI for the Mamba selective-scan core, which is used as a differentiable training-time regularizer. Across seven standard time-series datasets and four prediction horizons, the regularizer reduces sampled Mamba-core LMI violations by roughly

92\%

28/28

pairs at a clean-MSE cost of less than

0.018\%

. It improves internal Mamba passivity and state-norm diagnostics under injected perturbations. Our results turn classical control-theoretic tools into verifiable structural and training criteria for selective SSMs, while honestly scoping which guarantees transfer to a deep selective-scan architecture.

Abstract

Contraction Theory

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