Latent Control Signal in Model-Based Control

• Latent control signals are low-dimensional, non-directly observed variables that encode the essential dynamics of high-dimensional systems.
• They are extracted using encoder training and dynamic identification methods such as locally linear models, sparse recovery, and graphical models.
• Applications in robotics, neuroscience, visual control, and adaptive filtering demonstrate their capability to enable efficient, robust, and transferable control.

A latent control signal is a low-dimensional, non-directly observed variable or set of variables that mediates, encodes, or parameterizes how a dynamical system is controlled, typically within a learned or otherwise abstracted state space. This concept is foundational across modern model-based control, reinforcement learning, system identification from high-dimensional observations, adaptive filtering, and interpretable generative models. Latent control signals serve as compressed interfaces between high-dimensional sensor data (such as images, neural recordings, or wireless environment features) and control inputs, facilitating efficient, robust, and transferable control by operating in a reduced representation that preserves the essential dynamics and actuation structure.

1. Mathematical Foundations and Model Classes

Latent control signals arise within various mathematical frameworks that embed control tasks in latent representations:

• Locally Linear Latent State-Space Models: Embed high-dimensional sensory measurements $x_t$ into a latent state $z_t = E(x_t)$, yielding models of the form

$z_{t+1} = A_t z_t + B_t u_t + o_t + \omega_t,$

where $A_t, B_t, o_t$ are functions of $z_t$ and $\omega_t$ models process noise. Control input $u_t$ acts directly in the latent domain, allowing classic optimal control tools (e.g., LQG, iLQR) to operate efficiently on the compressed dynamics (Watter et al., 2015).

• Globally Linear with Sparse Control: In neuroscience, the "latent control signal" is an internal, often sparse, vector $u_t$ actuating linear dynamics

$x_{t+1} = A x_t + B u_t,$

with $x_t$ denoting neural state vectors and $B u_t$ encoding stereotyped behavioral transitions. The latent $u_t$ is recovered unsupervised via sparsity-promoting identification (Fieseler et al., 2020).

• KL-Control in Graphical Models: Control objectives are transcribed into discrete latent spaces via hidden Markov models (HMMs or FHMMs), where policy computation (principal eigenvector of a controlled transition matrix) becomes efficient (Matsubara et al., 2014).
• Latent Force Models (LFMs): The unknown control input $u(t)$ is itself a latent variable, modeled via a stochastic process (typically a GP), and inferred along with physical state using state-space methods (Särkkä et al., 2017).
• Latent Policy Transfer: Latent-to-latent policy architectures employ encoders $f_e$ and decoders $f_d$ with policies $\pi_\theta$ acting entirely in a latent space, enabling efficient adaptation across morphologically diverse agents (Zheng et al., 22 Mar 2025).
• Compressed Adaptive Filtering: Low-dimensional latent variables $z$ parameterize high-DOF controllers, with the time-evolution of $z$ serving as the latent control signal (e.g., in active noise control) (Sarkar et al., 5 Jul 2025).

2. Learning and Identification in Latent Spaces

Given observations $y_t$ or $x_t$ and actions $u_t$, latent control signals are extracted by jointly learning an embedding and identifying dynamics amenable to control:

• Encoder Training: Encoders $E$ (CNNs, VAEs, MLPs) map observations to latent states $z_t$, regularized to ensure information relevant to prediction and controllability is retained (Watter et al., 2015, Shu et al., 2020, Stölzle et al., 2024).
• Sparse Control Discovery: In systems like neural population recordings, optimization problems of the form

$$\min_{A,B,U} \|X' - A X - B U\|_F^2 + \lambda\|U\|_0$$

yield $u_t$ as interpretable, temporally sparse latent control signals (Fieseler et al., 2020).

• Dynamics Consistency and Predictive Coding: Information-theoretic losses (InfoNCE bounds, consistency KL) train the embedding and dynamics to maximize predictability and linearizability in the latent space (Shu et al., 2020).
• Structured Latent Dynamics: Structures such as input-affine or Lagrangian (CON) dynamics in the latent space underpin robust control design, identifiability up to diffeomorphism, and the transfer of stability guarantees (Zhang et al., 2024, Stölzle et al., 2024).

3. Control Synthesis and Policy Optimization

After identification, latent control signals support a range of policy and control synthesis procedures:

• Optimal Control in Latent Space: Policy computation—including LQR, iLQR, KL-control, and barrier/Lyapunov approaches—is performed efficiently on the low-dimensional latent system, with the control input $u$ or its latent counterpart $z$ serving as the actionable variable (Matsubara et al., 2014, Watter et al., 2015, Shu et al., 2020).
• End-to-End Reinforcement Learning: Pretrain-and-finetune pipelines leverage latent-to-latent policies, optimizing PPO or similar objectives directly on latent trajectories while primarily updating encoders and decoders during adaptation (Zheng et al., 22 Mar 2025).
• Latent-Space Planning (MCTS, Rollout): Model-based RL architectures (e.g., Dream and Search to Control) perform lookahead or tree-based planning by simulating and optimizing over latent dynamics and policy rollouts, deriving action sequences as functions of inferred or maintained latent states (Koul et al., 2020, Chaaya et al., 19 Jun 2025).
• Gradient-Based Adaptive Filtering: Updates in latent codes $z[n+1] = z[n] - \mu_z \nabla_z J(z[n])$, where $J$ is instantaneous cost, replace high-dimensional parametric adaptation, providing both computational and statistical gains (Sarkar et al., 5 Jul 2025).

4. Theoretical Guarantees and Structural Properties

Latent control signals carry meaningful theoretical properties when the embedding and dynamics satisfy structural and identifiability conditions:

• Stability and Safety Transfer: If the latent system admits explicit Lyapunov or barrier certificates, and approximate conjugacy conditions are satisfied, stability and invariance guarantees can be transferred to the full observed system [see e.g., (Lutkus et al., 29 May 2025) abstract].
• Input-to-State Stability (ISS): Coupled oscillator latent models (CONs) admit analytic Lyapunov functions, guaranteeing ISS under forced dynamics—including robust control laws with saturated integral feedback—while learned decoders invertibly map latent forces back to real actuators (Stölzle et al., 2024).
• System-Theoretic Properties: Observability and controllability analysis of joint physical–latent models (e.g., LFMs) supports the design of output-controllable controllers, even when latent forces are not directly actuated (Särkkä et al., 2017).
• Parameterization and Identifiability: Assumptions such as injectivity of the observation map and positivity of latent actuation ensure that the learned representation recovers the true controllable coordinates up to diffeomorphism, supporting global controller design (Zhang et al., 2024).

5. Applications and Empirical Performance

The latent control signal paradigm has enabled advances across multiple domains:

Domain	Latent Signal Model	Impact/Role
Neuroscience	Sparse internal $u_t$ pulses	Predicting behavior transitions from distributed activity (Fieseler et al., 2020)
Robotics/Locomotion	Embedding-based latent policy	Zero-shot adaptation to new bodies and tasks, data efficiency (Zheng et al., 22 Mar 2025)
Visual Control	VAE or predictive-coding latent	Planning and feedback from raw images, improved control success rates (Watter et al., 2015, Shu et al., 2020)
Wireless/ANC	Low-dim $z$ controlling $w$	Faster adaptation, energy and channel resource minimization (Sarkar et al., 5 Jul 2025, Chaaya et al., 19 Jun 2025)
Image Synthesis	PCA-projected latent vector	Efficient, interpretable GAN manipulations, cross-model edits (Odendaal et al., 26 Sep 2025)
Physical Systems	GP latent force/CON	Output-controllable stochastic systems, Lagrangian latent models (Särkkä et al., 2017, Stölzle et al., 2024)

In empirical benchmarks, latent-space controllers typically achieve comparable or superior control success, faster convergence, and greatly reduced resource consumption compared to full-state or direct observation approaches. Notably, real-world applicability extends to control from visual observations in robotics, energy-saving adaptive filtering, interpretable brain dynamics, and high-throughput wireless systems.

6. Challenges, Limitations, and Outlook

Despite their broad applicability, latent control signal frameworks face several persistent challenges:

• Dependence on Representation Quality: Performance is sensitive to encoder capacity, latent dimensionality, and coverage of exploration data (e.g., in KL-control/FHMM approaches, (Matsubara et al., 2014)).
• Approximation Artifacts: Structured latent models (e.g., factorial HMMs, locally linear flows) trade off scalability for approximation: factorized variational methods may get stuck in local minima, and learned models may "bleed" critical features unless regularized (diffusion recovery modules in (Zheng et al., 22 Mar 2025)).
• Robustness Across Domains: Adapting to new exogenous/hidden dynamics (e.g., environment-influenced latents) can require multi-environment training or careful disentangling (as formalized in identifiability extensions (Zhang et al., 2024)).
• Invertibility and Realizability: Ensuring the invertibility of latent-to-physical actuation (decoder regularization), or finding interpretable mappings, remains a topic of active research (Stölzle et al., 2024).
• Theoretical Certification: While some settings admit transferred guarantees via Lyapunov functions or ISS, full certification under high-dimensional nonlinearities or adversarial perturbations is an open area [(Lutkus et al., 29 May 2025) abstract].

A plausible implication is that advances in joint constrained representation learning, interpretable latent geometry, and certified transfer of control-theoretic properties will further deepen the reach and reliability of latent control signals across increasingly complex domains.