Statistical Learning of Contractive Dynamical Representations for Composite Adaptive Control

Min Kim; José Leonardo Brenes; Fred Hadaegh; Soon-Jo Chung

eess.SYSep 28, 2026

Statistical Learning of Contractive Dynamical Representations for Composite Adaptive Control

Authors: Min Kim, José Leonardo Brenes, Fred Hadaegh, Soon-Jo Chung

Organizations: California Institute of Technology (Caltech), Pasadena, CA 91125 USA. · Jet Propulsion Laboratory (JPL), Pasadena, CA 91109 USA.

Abstract

We present a representation-learning framework for composite adaptive tracking control under dynamically coupled disturbances. The framework connects classical disturbance-accommodating control (DAC) to recent last-layer adaptive disturbance-rejection methods. Specifically, we introduce a statistically principled hard expectation-maximization (hard-EM) procedure, with a Kalman smoother in the hard E-step, to identify dynamical representations of disturbance whose latent evolution is uniformly contractive. The learned representation evolves a latent disturbance-excitation state from measured plant features and control inputs and decodes that state into the time-varying disturbance acting on the nominal plant, thereby extending prior "fixed-decay" last-layer adaptive methods to a learned, predictive DAC-style formulation. Combined with Bayesian filtering of the learned latent state, this representation yields a composite adaptive tracking controller with predictive capability and provable exponential convergence to a bounded neighborhood. We validate our approach experimentally on a slippery ground vehicle carrying a liquid-sloshing tank and a pendulum load, and we further assess its robustness on a system of coupled Duffing oscillators. Across both settings, the method achieves accurate disturbance prediction and improved overall tracking performance relative to fixed-decay representation-learning ablations, LTI disturbance-accommodating baselines, and model-based PD baselines.

Figures & tables

Fig. 1: Proposed adaptive control architecture and experimental platform. Left: block diagram of the controller. A latent state is estimated online via a Kalman–Bucy-style filter from the disturbance proxy $y$ , using a latent-state evolution model parameterized by $(A,b)$ . The basis functions $(\Phi,\psi)$ map latent-relevant features $\phi$ to an adaptive disturbance-rejection term that augments a nominal tracking controller. The quantities $(A,b,\Phi,\psi)$ are parameterized by neural networks and learned using a statistically principled training procedure. Right: tracked mobile robot used for experimental validation.

Fig. 2: Statistically principled training via hard-EM with exact latent inference. The procedure alternates between a hard E-step and an M-step: given current parameters $(\theta,R_{d},Q_{d})$ , the hard E-step infers the latent trajectory $\mathbf{a}_{0:L-1}$ for each data window, and the M-step updates the noise covariances $(R_{d},Q_{d})$ from residual statistics and updates the network parameters $\theta$ .

Panel of Figure 3 — Fig. 3: (Tracked vehicle) Test-set rollouts for the predicted disturbance $\hat{d}\in\mathbb{R}^{2}$ as functions of time, on two different test-set trajectories. Black: disturbance proxy $y(t)$ ; red: rollout prediction $\hat{d}(t)$ . The proxy $y$ is visibly noisier. The average prediction MSE was $3.53$ ; it was $3.49$ after discarding the initial half second for each trajectory as a burn-in period (vertical dashed line).

Fig. 4: (Tracked vehicle) Tracking runs under four controllers.

Metric	NPD	N4SID v-DAC	FixedDecay	Ours
Ang. rate error RMS [rad/s]	0.449	0.414	0.468	0.407
Vel. error RMS [m/s]	0.151	0.160	0.156	0.149
Pos. error RMS [m]	0.085	0.078	0.081	0.078

TABLE I: Tracked-vehicle RMS tracking errors. NPD: nonlinear PD loop; N4SID v-DAC: LTI-based disturbance-accommodating control; FixedDecay: fixed-decay ablation.

Panel of Figure 6 — Fig. 5: (Coupled nonlinear oscillator) Test-set rollouts from Eq. ( 3 ) for the predicted disturbance as functions of time. Black: disturbance proxy $y(t)$ ; red: rollout prediction $\hat{d}(t)$ . The proxy $y$ is visibly noisier, while $\hat{d}$ is smooth. Mean squared error is $2.54$ overall and $0.88$ after discarding the first second of each trajectory as a burn-in period (vertical dashed line).

Fig. 6: (Coupled nonlinear oscillator) Robustness sweep for tracking control under additional Gaussian noise injected into the sparsely provided disturbance proxy $y$ (averaged over 30 reference trajectories, with each trajectory being 20 seconds long). Mean tracking MSE is plotted versus the added noise level $\sigma$ , where $\sigma=0$ denotes the baseline noise setting. The proposed dynamical-representation controller performs the best. As $\sigma$ increases, the neural adaptive controllers degrade because the disturbance proxy becomes increasingly corrupted. The simpler N4SID v-DAC baseline resulted in a flat graph because the fitted discrete-time $A$ , $B_{1}$ , and $B_{2}$ satisfied $\left\|A\right\|\approx 0.09\ll 1$ , $\left\|B_{1}\right\|\approx 0.08\ll 1$ , $\left\|B_{2}\right\|\approx 0.005\ll 1$ and hence the disturbance compensation mostly came from $\hat{d}\approx D_{1}\phi+D_{2}u$ .

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Jul 29, 2026, cs.ROJ/K move · Enter open · S save

Self-Adaptive Learning and Model Predictive Control for Tracking Unknown Dynamics with No Regret

Atharva Navsalkar, Hongyu Zhou, Vasileios Tzoumas

Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109 USA

Model Predictive Control Optimal Control Regret Adversarial Robustness Evasion

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We propose a self-adaptive online learning for control method for tracking unknown target dynamics. The target dynamics can exhibit switching behavior, particularly, a mixture of structured, random, and/or adversarial motion. Such challenging target tracking scenarios arise in applications of dynamic mapping, traffic control, and pursuit evasion, where robots need to track, pursue, or avoid collision with moving landmarks, objects, humans, etc., whose dynamics are unknown. Our method simultaneously learns multiple predictors from scratch, via self-supervised, one-shot, and computationally efficient learning, and adaptively selects the best one to match the observed target behavior. The method enjoys finite-time near-optimality guarantees in expectation, characterized as a function of the learning error of the target dynamics and the frequency that the target dynamics switch. In the absence of both error and switching, the method asymptotically matches the optimal non-causal control policy that knows a priori the target dynamics, i.e., the method enjoys no regret in expectation. In the presence of learning errors and switching, the method degrades gracefully, \eg when there are errors and no switching, the average regret is proportional to the average learning error and switching times. To prove these guarantees, a novel technical approach is required compared to the existing works that employ RFF-based online learning. We validate our method in Crazyflie simulations and hardware experiments, across target trajectories that vary from structured to random to adversarial, in comparison to non-stochastic, kernel-based, and neural-network-based methods for online learning.

Abstract

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