Behavioral Systems Theory Meets Machine Learning: Control-Aware Learning of the Intrinsic Behavior from Big Data

Abstract: The abundance of process operating data in modern industries, along with the rapid advancement of learning techniques, has led to a paradigm shift towards data-centric analysis and control. However, integrating machine learning with control theory for big data-driven control of nonlinear systems remains a challenging open problem. This is because the state-based, model-centric, and causal framework of classical control theory fundamentally contradicts the trajectory-based, set-theoretic, and causality-absent rationale of big data-based learning approaches. Using the behavioral framework, we show that dynamical systems possess an intrinsic state variable that encodes the system behavior in a bijective and causality-free manner, and control design can be carried out entirely within the state space. This approach not only resolves the aforementioned conflict but also complements machine learning techniques well, leading to a neural network architecture that is capable of learning the behavior representation well-suited for control design.

• The paper introduces an intrinsic state variable that enables bijective mapping of trajectory data for unbiased control synthesis.
• It employs the CALIB neural architecture to combine autoencoder-based state learning with control Lyapunov functions for exponential stabilization.
• Empirical validation with drone stabilization demonstrates improved convergence and trajectory smoothness over traditional data-enabled predictive methods.

Behavioral Systems Theory Meets Machine Learning: Control-Aware Learning via Intrinsic State from Big Data

Background and Motivation

The proliferation of industrial process data and advances in machine learning have motivated data-centric control paradigms, particularly where first-principles modeling is infeasible. Classical control theory, with its state-based causal modeling, fundamentally conflicts with trajectory-based, causality-absent learning approaches common in big data-driven methods. While behavioral systems theory provides an alternative viewpoint, focusing on system trajectories rather than explicit input-state-output causality, the synthesis of behavioral systems theory and machine learning presents unresolved technical challenges, especially for nonlinear systems where classical state-space approaches become restrictive or unrepresentative.

Behavioral Systems Theory Foundations

Behavioral systems theory characterizes a dynamical system as a triple $(\mathbb{T},\mathbb{W},B)$, with $\mathbb{T}$ the time axis, $\mathbb{W}$ the signal space, and $B$ the set of all admissible trajectories (the behavior). This framework eschews strict input-output causality, instead emphasizing the structure of the set of all system trajectories. Key concepts include:

• Lag ($L_B$): The minimal interval length required so that trajectory snippets uniquely specify global behavior.
• Kernel Representation: System behavior described by constraints like $r(w_{k-L},\ldots,w_k)=0$.
• State Property: A latent variable $\ell$ possesses the state property if pasting trajectory segments with identical $\ell_k$ yields new valid system trajectories, a foundation for stateful system representations.
• Control as Sub-Behavior Selection: Control is formulated as intersecting the system and controller behaviors, reducing design to the search for feasible sub-behaviors.

This trajectory-centric framework is amenable to analysis and synthesis via data, circumventing explicit causal modeling constraints.

Intrinsic State and Equipotent Behavioral Representation

A core result is the construction of an intrinsic state variable $g$, which is bijectively mapped from length-$(L+1)$ trajectory windows ($\mathbb{T}$0). The dimension of $\mathbb{T}$1 reflects the manifest input dimension and system order: $\mathbb{T}$2, where $\mathbb{T}$3 is the minimal intrinsic complexity.

The intrinsic state $\mathbb{T}$4 offers:

• Bijective Mapping: $\mathbb{T}$5 is invertible, with inverse $\mathbb{T}$6 parameterizing behavior, thus acting as a chart for $\mathbb{T}$7.
• Equipotency: The set of admissible $\mathbb{T}$8-trajectories ($\mathbb{T}$9) is equivalent in structure to $\mathbb{W}$0.
• No Implicit Causality: The representation retains correspondence rather than evolution, congruent with data-driven learning assumptions.

For LTI systems, the map can be explicitly constructed from Hankel matrices in the context of fundamental lemma [Willems:2005]. For nonlinear systems, the existence of such bijective maps is established, and their flexible realization via neural networks is proposed.

Control Design in Intrinsic State Space

Exponential stabilization via intrinsic state proceeds as follows:

• Control Lyapunov Function (CLF) Construction: Via the intrinsic state, stabilizing sub-behaviors are characterized as those permitting monotonic decrease in an appropriate CLF, without a priori controller structure or causality.
• Linear Case: The stabilization problem reduces to LMIs in the intrinsic state representation, with explicit controller realizations.
• Nonlinear Case: Controlled state transitions are formulated as arbitrary functions $\mathbb{W}$1, with CLF-based feasibility constraints ensuring exponential convergence. The existence of such a feasible $\mathbb{W}$2 is treated as a function approximation problem amenable to learning from data.

This methodology eliminates the bias and structural limitations inherent to classical state-space identification, allowing robust control synthesis directly from trajectory data.

CALIB Architecture: Neural Control-Aware Learning

The Control-Aware Learning of Intrinsic Behavior (CALIB) architecture operationalizes the theoretical framework using neural networks:

• Autoencoder Structure: The state map $\mathbb{W}$3 and its inverse $\mathbb{W}$4 are learned as an autoencoder over trajectory windows, with $\mathbb{W}$5 as the latent intrinsic state.
• Controlled Behavior Learning: A neural network $\mathbb{W}$6 implements the controlled state transition, while a CLF network $\mathbb{W}$7 assesses stability.
• Loss Function Design: The loss combines intrinsic behavior reconstruction (bijectivity, chart anchoring, trajectory weaving) and controlled behavior criteria (behavior subset constraints, CLF anchoring and monotonicity), robustly guiding learning to representations suitable for control.

CALIB simultaneously learns representations of the system’s intrinsic complexity and control-relevant features, ensuring aligned behavioral and CLF properties for nonlinear stabilization. The architecture imposes minimal structural or causal constraints, and is flexible across different system classes.

Empirical Validation: Drone Stabilization

The application of CALIB in drone stabilization demonstrates the efficacy of the framework. Large-scale trajectory data are used to learn the intrinsic state representations and control mappings. Results show:

• Smooth convergence to setpoints, with steady-state offsets attributed to small learning errors prevalent near the origin.
• Superior performance compared to DeePC (Data-enabled Predictive Control), particularly in trajectory smoothness and convergence accuracy.
• Robust prediction alignment between realized and CALIB-predicted trajectories.

These results validate the practical integration of behavioral systems theory with deep learning in real-world nonlinear control.

Implications and Future Directions

The introduction of the intrinsic state via behavioral systems theory provides a formal foundation for data-driven control, superseding the limitations of classical modeling and identification. By leveraging bijective behavioral representations and their neural realization, the paper demonstrates integrated learning and control synthesis that is both structurally general and control-aware.

The implications are twofold:

• Theoretical: Redefines state representation for nonlinear systems in big-data settings, enabling intrinsic complexity-preserving and unbiased learning.
• Practical: Provides scalable, flexible neural architectures for complex control tasks with minimal prior structural assumptions.

Future work will address robustness to data noise, extension to tracking and setpoint-independent control, and architecture refinements for error mitigation, potentially expanding CALIB to multi-agent and distributed control environments.

Conclusion

The paper presents a rigorous synthesis of behavioral systems theory and machine learning for control-aware behavior learning from big data. By introducing the intrinsic state and formalizing its neural realization via the CALIB architecture, the approach enables effective, unbiased nonlinear control design directly from trajectory data, with significant theoretical and practical ramifications in data-driven control.