Muhammad Zakwan, Vaibhav Gupta, Alireza Karimi, Efe C. Balta, Giancarlo Ferrari-Trecate
8 min
This paper bridges machine learning and control theory by showing how Structured State-space Models (SSMs)—popular in sequence modeling like Mamba—can serve as surrogate models for nonlinear dynamical systems. Unlike black-box neural networks, SSMs enable rigorous controller design thanks to their structured form and newly proven control-theoretic properties. The authors provide the first controllability/observability analysis for SSMs, derive scalable synthesis methods via contraction theory, and prove a separation principle for stable output feedback. This unlocks data-driven control for nonlinear systems without sacrificing guarantees like exponential stability.
SSMs extend linear recurrent units (LRUs) with nonlinear 'scaffolding' functions: and . Here, are trainable matrices, and are bi-Lipschitz neural networks mapping inputs/outputs to higher dimensions. This structure captures long-term dependencies in time series with linear computational complexity (vs. Transformer's quadratic), making SSMs ideal surrogates for nonlinear system identification (SysId). Trained on data, they approximate unknown nonlinear dynamics for indirect controller synthesis—fit SSM to input-output data, then design controllers as if the SSM is the true plant.
The paper's breakthrough: SSMs inherit controllability/observability from their linear core if scaffoldings are bi-Lipschitz (locally invertible with bounded condition number). For local controllability, the pair must be controllable, ensuring the controllability Gramian is invertible despite nonlinear perturbations. Similarly for observability via . Intuition: Bi-Lipschitz acts like a 'nice' coordinate change, preserving structural properties. Verified via differential dynamics .
Stability via discrete-time Control Contraction Metrics (DCCMs): Seek metric such that with , ensuring exponential convergence of trajectories. For state-feedback, solve convex LMIs: find and satisfying contraction conditions on closed-loop . Observers use dual LMIs for . Scalable: LMIs are SDPs solvable in polynomial time, unlike nonconvex Riccati equations for nonlinear H∞ control.
Proves controllers and observers can be designed independently while guaranteeing closed-loop exponential stability—analogous to linear Kalman decomposition. Combine stable state-feedback with observer . Error dynamics decouple, converging at rate . Enables indirect output-feedback: identify SSM from data, design separately, deploy on true nonlinear plant (input-to-state stable if SSM approximates well).
Demonstrated on numerical example: SSM identifies nonlinear plant, LMIs synthesize controller outperforming baselines. Opens data-driven control co-design—joint SysId + control without black-box headaches. Limitations: assumes bi-Lipschitz scaffoldings (enforceable via regularization); local properties require persistent excitation. Future: real-world benchmarks, extensions to continuous-time/partial feedback.
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.
Alex: So that's why they can guarantee stability. But how do they actually build the controller from there?
Sam: They use contraction theory, a way to ensure paths in the system pull together exponentially, like rubber bands snapping trajectories toward a target regardless of starting point. This leads to math conditions called Linear Matrix Inequalities—solvable efficiently like finding the safest path in a grid—to design gains for a state-feedback controller and an observer. A separation principle lets you design them independently, just like in linear systems, while proving the combined loop is exponentially stable—trajectories converge reliably.
Alex: And they test this on that motor example to show it works in practice?
Sam: Yes, the numerical example on the DC motor demonstrates nonlinear identification and stable output feedback control. It bridges data-driven modeling with certifiable guarantees, a meaningful step for scalable nonlinear control.
Alex: In that DC motor test, they use the SSM to identify the nonlinear behavior from data first, then build the controller on top?
Sam: Yes. Nonlinear system identification here means fitting an SSM to real data as a stand-in model for the motor's tricky friction and dead zones. From there, they design controllers indirectly—without needing the true internal states—by combining a state-feedback controller with an observer that estimates those states from outputs alone.
Alex: A state-feedback controller... that's like telling the system what to do based on its full current position?
Sam: Close. Imagine the controller looks at the exact hidden state—like knowing a ball's position and speed inside a box—and picks inputs to steer it to the goal. They solve for the gain using those math conditions, ensuring paths contract exponentially no matter the bi-Lipschitz input nonlinearity.
Alex: Okay, that tunes the controller. But for output feedback, since we only see the motor's position, not the full state—how does the observer fit in?
Sam: The observer estimates the hidden state from visible outputs, much like piecing together a puzzle from clues. Its gain comes from similar math conditions ensuring it tracks accurately. A separation principle lets them design this observer and the controller separately, then prove the full loop—real system plus estimates—is exponentially stable.
Alex: Huh. So even with data-driven SSMs approximating the real nonlinear motor, these guarantees hold because of the contraction setup.
Sam: Precisely. The core is the SSM's linear recurrent unit—a simple state update. Wrapped in bi-Lipschitz layers, its tiny variations around a path stay controllable and observable, bounding errors for robust designs. This all scales because SSMs handle long data linearly, unlike slower models.
Alex: Wait—so this separation principle lets you design the controller and observer separately, even for these nonlinear SSMs? How does that actually hold up without everything falling apart when combined?
Sam: Yes. The key is showing contraction holds even with disturbances—like small errors from the observer fed back as noise. Paths still pull together exponentially, bounding the distance to the target with terms that decay over time.
Alex: So the controller treats observer errors like external noise it can handle?
Sam: Exactly. They derive the controller gain from math conditions that account for uncertainty in the input nonlinearity's slope—ensuring paths converge no matter how the rubber band deforms within limits. The observer mirrors that for estimating states from outputs.
Alex: And in the DC motor example, this output feedback stabilizes tracking despite friction and dead zones?
Sam: Yes—the identified SSM enables solved gains for closed-loop simulation, showing reliable convergence. Simulations showed the controller and observer robustly converging the load velocity to the reference, regardless of starting conditions or disturbances like external torque. Figure 3 captures this: paths pull together reliably, validating the indirect data-driven approach on a system with dead-zone inputs and Coulomb friction that trips up black-box methods.
Alex: That's a solid check on paper. But what are the limits here—does it generalize beyond this motor?
Sam: Fair point. The bi-Lipschitz requirement on the scaffolding layers is restrictive—it holds for this constrained neural net but not arbitrary deep networks, where slopes can vary wildly. They stick to single-layer SSMs, and validation is just this DC motor benchmark, so scaling to multi-layer or other dynamics needs more work.
Alex: So practical next steps—where could this fit in real systems?
Sam: The paper points to real-time adaptive control for setups like industrial robots handling variable loads, HVAC systems balancing flows with sensor data, or power grids stabilizing amid fluctuations—all using SSMs trained on data streams, with math conditions giving stability proofs that black-box ML lacks.
Alex: Huh. Pulling it together, this bridges efficient sequence models with certifiable nonlinear control—a meaningful step for data-driven stability without the usual risks.
Sam: Precisely. By establishing math-based synthesis via contraction for SSMs, complete with separation, it offers scalable guarantees where classical tools falter. That's the core contribution here.
Alex: Thanks, Sam—clear breakdown of a precise advance. Listeners, that's our look at controller design for structured state-space models. Thanks for listening to ResearchPod.