Representation-Adaptive Control Theory (RACT): A Unified Framework Parts I–V

1 Introduction

Every feedback controller relies on some internal representation of the control problem. Depending on the control architecture, this representation may consist of a state estimate, a mathematical model, a collection of controller gains, or a set of estimated parameters. In classical control, this representation is fixed during the design process and remains unchanged throughout operation. In adaptive control, the representation evolves online, but only within a narrowly prescribed mathematical structure, most commonly a vector of unknown parameters that enter the system dynamics linearly or affinely. This structural restriction is fundamental to the classical Lyapunov framework because it permits the construction of parameter-update laws that exactly cancel the coupling terms between the tracking-error dynamics and the parameter-estimation dynamics.

This paper begins from a different premise. Rather than viewing adaptation solely as the adjustment of values within a predetermined representation, we consider the representation itself to be an adaptive mathematical object. The central idea is that the internal structure through which a controller interprets the plant can evolve dynamically while remaining subject to rigorous stability analysis. We refer to this general framework as Representation-Adaptive Control Theory (RACT).

In conventional state-feedback control the control law is written as (u=\pi(x)). Observer-based control extends this to (u=\pi(\hat{x})), while adaptive control introduces parameter estimates through (u=\pi(\hat{x},\hat{\theta})). RACT instead considers

[
u=\pi(x,\sigma), \qquad \sigma\in\mathcal{R},
]

where (\mathcal{R}) denotes a representation space whose elements need not be limited to Euclidean parameter vectors. Depending on the application, (\sigma) may represent estimated physical parameters, confidence measures, geometric descriptors, temporal prediction structures, executive coordination graphs, memory states, or other mathematical representations relevant to the control task. The representation evolves according to its own dynamics,

[
\dot{\sigma}=F(\sigma,x,u,t),
]

and is coupled to the plant through both the system state and the control input. The resulting closed-loop system therefore consists of two interacting dynamical subsystems,

[
\dot{x}=f(x,\pi(x,\sigma),d),
]

[
\dot{\sigma}=F(\sigma,x,\pi(x,\sigma),t),
]

where (d) represents disturbances, modeling uncertainty, and exogenous inputs.

The ambition of Representation-Adaptive Control Theory extends well beyond the scope of a single paper. The long-term research program encompasses the development of a general representation algebra, multiple classes of representation modules, software architectures, benchmarking methodologies, and successive theoretical extensions spanning RACT-I through RACT-VI. It would therefore be misleading to present this work as a complete theory. Instead, this paper establishes only the minimal mathematical foundations required for the framework. The focus is intentionally restricted to the definition of representation spaces, representation dynamics, stability theory for coupled plant-representation systems, and the composite Lyapunov framework required to analyze them. To demonstrate that these abstractions possess practical meaning rather than serving as purely conceptual constructs, the paper also develops one complete instantiation and validates it numerically. More sophisticated representation classes, algebraic composition rules, hardware implementations, and extensive benchmarking are deferred to later parts of the research program.

1.1 Contributions

The primary objective of this first part of the RACT program is to establish a mathematically rigorous foundation for treating internal representations as dynamical objects within feedback control. We begin by introducing formal definitions of representation spaces, representation dynamics, and representation controllers together with the regularity assumptions required to guarantee existence, uniqueness, and well-posedness of the resulting closed-loop systems.

Building on these definitions, we extend Lyapunov stability concepts to systems whose dynamics include both physical and representational states. This leads to notions of representation stability, representation convergence, practical representation stability, and joint stability of coupled plant-representation systems defined with respect to a general metric on the representation space.

The theoretical core of the paper consists of three composite Lyapunov stability theorems. The first generalizes the classical exact-cancellation argument familiar from adaptive control. The second establishes stability under bounded coupling using Young’s inequality and small-gain reasoning. The third formulates an ISS-based small-gain composition theorem that provides sufficient conditions for stability without requiring the representation space to possess Euclidean structure.

To illustrate how the abstract framework can be instantiated in practice, we develop an Information-Weighted Robust Controller (IWRC) in which the representation state combines a conventional parameter estimate with a dynamically evolving confidence variable. The confidence state continuously adjusts the magnitude of a robustifying control term, allowing the controller to vary its conservatism according to the estimated quality of its internal representation. For this specific instantiation, a uniform ultimate boundedness result is established together with an explicit bound on the closed-loop trajectories.

Finally, the proposed controller is evaluated through executed numerical simulations using a fourth-order Runge-Kutta implementation. The experimental study compares the proposed approach with a fixed worst-case baseline and reports quantitative measures including tracking accuracy, parameter convergence, and control effort.

Throughout the paper we carefully distinguish between theoretical results established in full generality and conclusions that apply only to the specific IWRC instantiation. General mathematical statements are presented as proved theorems, whereas properties supported solely by numerical evaluation are explicitly identified as instantiation-specific results. Broader objectives that lie beyond the scope of the present work are stated separately as research goals for subsequent parts of the RACT program.

2 Motivation: Why a Representation-Adaptive Framework?

Consider three controllers designed for the same uncertain nonlinear plant. A robust controller assumes uncertainty belongs to a prescribed set and synthesizes a single fixed control law capable of tolerating every admissible realization. This approach provides strong worst-case guarantees but necessarily sacrifices performance because the controller remains permanently tuned for the most adverse conditions, even after repeated observations suggest that the uncertainty is significantly smaller.

Adaptive control addresses this conservatism by estimating unknown parameters online and modifying the control law accordingly. When its structural assumptions are satisfied, adaptation can recover much of the lost performance. However, the classical Lyapunov framework depends critically on a particular algebraic structure, typically linear parameterizations with matched uncertainty. Moreover, while adaptive control updates parameter values, it offers no principled mechanism for representing the controller’s confidence in those estimates or for adapting the representation itself when the nature of the uncertainty changes.

Gain-scheduled and switched controllers accommodate different operating regimes by selecting among a finite collection of pre-designed control laws. Although effective in many engineering applications, both the available controllers and the switching logic are determined offline. The representation used to characterize operating conditions is therefore fixed before deployment rather than evolving in response to experience.

Representation-Adaptive Control Theory is motivated by the observation that each of these approaches implicitly maintains an internal representation that mediates between measured system behavior and control action. Existing methods differ primarily in how that representation is constructed, updated, and analyzed. RACT proposes elevating this representation to the status of an explicit dynamical state whose evolution is governed by its own mathematical laws and whose interaction with the physical system is analyzed directly.

Viewing representations as first-class dynamical objects provides several conceptual advantages. It permits a common stability framework to encompass representations that extend well beyond parameter vectors, including confidence measures, geometric structures, predictive temporal models, memory systems, and graph-based executive representations. It separates the abstract conditions required for stability from the specific representation employed, allowing diverse representation mechanisms to be verified against a common set of mathematical principles rather than requiring entirely new analyses for each design. It also enables the controller’s level of conservatism or aggressiveness to become an explicit, dynamically evolving variable that is itself incorporated into the stability analysis instead of being introduced as an external heuristic.

The claims made in this paper are deliberately limited. We do not argue that robust control, adaptive control, or gain scheduling require Representation-Adaptive Control Theory in order to function. Each remains a valid and successful methodology within its intended domain. Our claim is instead that these seemingly distinct approaches can be interpreted as particular instances of a broader mathematical perspective in which the internal representation becomes an adaptive dynamical system in its own right. By formalizing that perspective, RACT provides a common language for analyzing and extending a wide family of controllers whose representational mechanisms have traditionally been developed independently and justified through separate, problem-specific arguments.