InnerControl: Frameworks & Applications

• InnerControl is a framework that uses internal states, models, and feedback to achieve precise and self-regulated control.
• It integrates model-based techniques like IADRC with data-driven methods such as convolutional probes in diffusion models for effective disturbance rejection and alignment.
• Applications span from high-precision motion and switched systems to reinforcement learning, enhancing autonomy and interpretability in complex environments.

InnerControl encompasses a variety of frameworks and methodologies in control theory, reinforcement learning, neural modeling, and deep generative modeling, where the goal is to achieve precise, robust, and self-regulated behavior by leveraging internal mechanisms, models, or feedback signals. Specific realizations of InnerControl target tasks such as disturbance rejection, invariance set synthesis, unsupervised option discovery, homeostasis control, precise model conditioning, and structured internal feedback for generation in deep diffusion models.

1. Principles and Conceptual Frameworks

The conceptual underpinnings of InnerControl are rooted in the design of systems that achieve control goals not purely through external supervision or terminal constraints but by leveraging internal states, models, structural knowledge, or intrinsic feedback. Depending on the domain, this may involve:

• Separation of disturbances into components that can be directly modeled and those that must be robustly rejected using observers, as in Internal Model Based Active Disturbance Rejection Control (IADRC) (1603.03734).
• Characterization and synthesis of invariant sets, particularly in switched systems or impulsively controlled dynamics, to ensure that trajectories remain safely within functional regions under complex or time-varying behaviors (1608.08683, 2103.13831).
• Internalization of behavioral objectives or constraints, either through the construction of internal belief models (e.g., Bayesian agents in neuroscience (2009.12576)) or by using intermediate feedback to shape ongoing computation (e.g., convolutional probes for spatial alignment in diffusion models (2507.02321)).
• Intrinsic motivation and control objectives, where a system actively seeks to maximize empowerment or minimize uncertainty in its internal representations using information-theoretic objectives (1611.07507, 2112.03899).

2. Model-Based and Data-Driven Internal Control

InnerControl methods span both model-based and data-driven settings:

• Model-Based Approaches: Techniques such as IADRC (1603.03734) embed known dynamics of disturbances into observer-controller structures. Mathematical formulations involve solving Sylvester equations to link internal model observers with plant dynamics, allowing for perfect or exponentially fast disturbance estimation.
• Data-Driven/internal Feedback: In advanced generative models, such as text-to-image diffusion, InnerControl can denote the enforcement of alignment between control signals (e.g., edges, depth) and internal UNet representations throughout the generation trajectory. Lightweight convolutional probes extract pseudo ground truth signals from intermediate activations, enabling a new alignment loss applied at every diffusion step for robust spatial control (2507.02321).
• Behavioral System Theory: Internal Behavior Control (IBC) recasts Internal Model Control (IMC) using behavioral system theory and Willems’ fundamental lemma, constructing controllers directly from trajectory data without a parametric model, thereby internalizing robust control via data-driven predictions (2311.12696).

3. Mathematical Formulations and Algorithmic Strategies

Mathematical formalism is central to InnerControl methodologies:

• Observer Synthesis: The IADRC observer estimates disturbance components through dynamic systems with known or adaptively estimated parameters, involving solutions to equations such as

$Q S = (A - l c) Q + b h^{\top}$

and adaptive laws for unknown dynamics (1603.03734).

• Invariant Set Computation: Algorithms for invariance control use interval arithmetic, branch-and-prune (SIVIA-type) set approximations, and Pontryagin difference operations to construct maximal or robustly invariant sets for nonlinear, switched, or impulsive systems (1608.08683, 2103.13831). The computed set naturally partitions the state space, with controllers mapping state intervals to admissible modes.
• Alignment Losses in Diffusion Models: InnerControl in diffusion models defines an alignment loss at each denoising step:

$$\mathcal{L}_{\text{alignment}} = \mathcal{L}(c_{\text{spatial}}, H[\text{UNet}(c_{\text{spatial}}, c_{\text{txt}}, x_T, t)], t)$$

where $H(·, t)$ is a convolutional probe reconstructing control targets from intermediate features (2507.02321).

• Information-Theoretic Objectives: Variational Intrinsic Control maximizes mutual information $I(\Omega, s_f | s_0)$ between options and final states to discover controllable behaviors, and Information Capture methods use entropy minimization of the latent state visitation distribution as a control goal (1611.07507, 2112.03899).
• Dual Control and Separation Principles: For constrained linear systems with output feedback, the intrinsic separation principle splits the control objective into meta-learning (intrinsic information minimization) and robust control (extrinsic risk), using equivalence classes of information sets (2307.04146).

4. Applications Across Domains

InnerControl frameworks have been tailored for a range of control and learning tasks:

Domain	InnerControl Focus	Notable Techniques
High-precision motion	Decoupling multi-axis contouring with position-domain internal model control	TV-IMCC, signal conversion, LMIs (2203.12232)
Switched/nonlinear systems	Invariant set synthesis, partition-based controller design	Interval analysis, set-iterative algorithms (1608.08683)
Lattice/statistical physics	Control of energy transport via embedded kernel design	Combinatorial/geometric parameterizations (2407.03710)
Reinforcement learning	Discovery and exploitation of intrinsic control options	Mutual information maximization, empowerment (1611.07507)
LLM behavior alignment	Real-time inference-time behavior control by latent prefix adjustments	Suffix gradient compression, prefix module (2406.02721)
Biomedical/robotic systems	Biologically inspired multisensory integration and learning	Neural assembly adaptation, homeostasis control (2206.04400, 2311.14840)

5. Performance, Evaluation, and Constraints

The effectiveness of InnerControl is demonstrated via both quantitative and structural metrics, including:

• Disturbance rejection (IADRC): Reduced phase lag, faster state convergence, perfect disturbance compensation under expressible dynamics (1603.03734).
• Invariant set accuracy, precision, and computational metrics: Finer interval approximations yield tighter controlled invariance with lower computational burden than abstraction-based models (1608.08683).
• Alignment and generation fidelity: InnerControl achieves lower RMSE and higher SSIM in vision control tasks, preserving input feature alignment over all denoising steps, with no observed degradation in FID/CLIP scores compared to prior methods (2507.02321).
• Unsupervised agent skills: Intrinsic control agents learn more reliably distinct options and reach higher empowerment, scaling to complex structured and visual domains (1611.07507, 2112.03899).
• Constraints and limitations: Many approaches require certain system properties or structure—an expressible disturbance model, the existence of robustly invariant sets, or rich enough data for behavioral system realization. Adaptive strategies may only converge under persistent excitation or structural identifiability.

6. Broader Implications, Extensions, and Future Directions

InnerControl approaches have broad implications for advancing modular, interpretable, and robust autonomy:

• Interplay of internal and external control: Internal modeling and feedback can dramatically boost robustness compared to methods which solely depend on reactive, external corrections, and serve as a foundation for interpretable and composable controllers.
• Rich information structures: Exploiting information-theoretic, behavioral, and structural representations allows for finer partitioning of risk (intrinsic vs. extrinsic) and may improve controller efficacy and tractability for constrained or high-dimensional systems.
• Unified treatment of adaptation and self-regulation: Biologically inspired homeostasis and invariant equalization control formalizations (via speed gradient and Lyapunov techniques) provide adaptive inner regulation strategies broadly applicable to distributed, multi-agent, and networked systems (2311.14840, 2206.04400).
• Extensions to nonlinear, uncertain, and hybrid domains: Open research directions include extending interval and spectrahedral invariance methods to nonlinear and hybrid impulsive systems, robustifying data-driven controllers against noise and actuation limits, and optimizing the interface between internal model learning and real-time policy updates.

InnerControl thus constitutes a diverse, rigorously founded set of methods for achieving high-fidelity, interpretable, and internally robust adaptive behavior in both engineered and biological systems.