Building Models That Control: A New Era for System Identification

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Author: Denis Avetisyan

This review explores how modern system identification techniques are moving beyond pure prediction to prioritize control-relevant properties like stability and physical plausibility.

Optimization-based system identification benefits from approaches that explicitly impose control-oriented properties, enabling tailored performance and stability characteristics.
Optimization-based system identification benefits from approaches that explicitly impose control-oriented properties, enabling tailored performance and stability characteristics.

A comprehensive analysis of classical, learning-based, and physics-informed methods for control-oriented system modeling.

While machine learning excels at modeling complex dynamical systems from data, ensuring those models possess properties crucial for robust control remains a significant challenge. This paper, ‘Control-Oriented System Identification: Classical, Learning, and Physics-Informed Approaches’, surveys methods for learning system models that integrate control-relevant characteristics-such as stability, dissipativity, and physical consistency-into the identification process. By categorizing classical, learning-based, and physics-informed techniques, we demonstrate how to formulate system identification as an optimization problem enforcing desired properties through parameterization or constraints. Can a unified framework bridge data-driven learning with the guarantees needed for safe and performant control of increasingly complex systems?

Beyond Prediction: Modeling for Control Integrity

Conventional system identification techniques prioritize accurate prediction of a system’s behavior, often achieving impressive results in forecasting future states. However, this predictive power doesn’t automatically translate to effective control design. While a model might flawlessly predict how a system will respond to an input, it doesn’t inherently guarantee that the resulting closed-loop system will be stable, robust, or even safe. Critical properties for control, such as passivity – the tendency to dissipate energy rather than amplify it – and dissipativity, which ensures bounded inputs lead to bounded outputs, are frequently overlooked in standard identification procedures. Consequently, controllers designed using these purely predictive models may require substantial tuning or even fail altogether, highlighting the need for modeling approaches specifically tailored to the demands of control engineering.

The reliable operation of any control system hinges on guarantees of stability, passivity, and dissipativity – properties that ensure bounded inputs produce bounded outputs, prevent the amplification of disturbances, and manage energy flow, respectively. However, traditional system identification techniques, while proficient at predicting a system’s behavior, do not inherently prioritize or ensure these critical control-oriented characteristics. A model accurately predicting a system can still be dangerously unstable or susceptible to resonance when used within a closed-loop control architecture. Consequently, control engineers often face the arduous task of verifying these properties post hoc, adding complexity and potential for error. The absence of natively guaranteed stability and passivity necessitates specialized modeling approaches that explicitly incorporate these constraints, paving the way for safer, more robust, and ultimately, more effective control designs.

Addressing the limitations of conventional system identification requires a shift towards modeling techniques explicitly designed for control implementation. Standard methods often prioritize predictive accuracy, overlooking critical attributes such as stability, passivity, and dissipativity – properties fundamental to safe and robust control system performance. Consequently, researchers are developing approaches that integrate these control-relevant properties directly into the modeling process itself. This involves formulating models not simply to forecast system behavior, but to guarantee certain performance bounds and safety criteria. By prioritizing these characteristics from the outset, these new methodologies aim to bypass the need for post-hoc validation or potentially risky adjustments, ultimately leading to more reliable and predictable control outcomes, particularly in complex or safety-critical applications where ensuring $L_2$-stability or input-output passivity is paramount.

Deep learning and control-oriented architectures both offer viable approaches to system identification, each with distinct strengths and applications.
Deep learning and control-oriented architectures both offer viable approaches to system identification, each with distinct strengths and applications.

Embedding Prior Knowledge: The Power of Physics-Informed Learning

Physics-Informed Learning (PIL) improves system identification by integrating established physical laws and principles directly into the learning process. Traditional system identification often relies solely on data, which can lead to inaccurate or non-generalizable models, particularly when data is limited or noisy. PIL addresses this by framing the identification problem not just as a data fitting exercise, but as an optimization problem subject to physical constraints. These constraints, derived from governing equations – such as conservation of mass, momentum, or energy – reduce the solution space to physically plausible parameters, resulting in models that exhibit improved accuracy, robustness, and the ability to extrapolate beyond the training data. This is especially beneficial in scenarios where obtaining comprehensive datasets is challenging or expensive, and where adherence to known physical behavior is critical.

Physics-Informed Learning utilizes constraints to shape the learned system model towards physically plausible solutions. ‘Hard’ constraints impose absolute limits on parameter values; for example, specifying that a mass parameter must be positive. ‘Soft’ constraints, implemented through regularization terms in the loss function, penalize deviations from expected behaviors without strictly prohibiting them. Common regularization techniques include L1 and L2 regularization, which minimize parameter magnitude, and terms that enforce specific relationships between parameters, such as ensuring energy conservation. The selection and weighting of these constraints directly influence the model’s behavior and its ability to generalize to unseen data, effectively guiding the learning process towards physically consistent outcomes.

Direct Parameterization and Lagrangian Neural Networks represent a shift in system identification by building physical structure directly into the model architecture. Unlike traditional methods that train a generic model and then apply constraints post-training, these approaches define model parameters in terms of the physical properties of the system, such as mass, damping coefficients, or inertia. For example, a Direct Parameterization might express a system’s state-space matrices, $A$ and $B$, as functions of physically meaningful parameters. Lagrangian Neural Networks, conversely, utilize concepts from Lagrangian mechanics to define the model’s dynamics, ensuring that the resulting model inherently satisfies physical principles like conservation of energy. This proactive integration of physics reduces the need for extensive regularization and improves generalization performance, particularly in data-limited scenarios.

Employing a proactive, physics-informed approach to system identification results in models inherently suited to control tasks, reducing the need for post-hoc adjustments or tuning. Recent research indicates this alignment with control objectives demonstrably improves sample efficiency; specifically, models constructed with embedded physical principles require fewer data points to achieve comparable or superior performance to traditionally identified models. This reduction in data requirements translates to lower experimental costs and faster model development cycles, particularly in scenarios where data acquisition is expensive or time-consuming. The benefit stems from constraining the solution space to physically plausible configurations, thereby mitigating the risk of overfitting and enhancing generalization capabilities, even with limited training data.

Employing a neural Lyapunov function, both hard projection and soft penalty approaches successfully learn stable dynamics, demonstrating viable methods for controlling neural network-based systems.
Employing a neural Lyapunov function, both hard projection and soft penalty approaches successfully learn stable dynamics, demonstrating viable methods for controlling neural network-based systems.

Beyond Representation: Characterizing Behavior Directly

Behavioral System Theory diverges from traditional system identification by focusing on the relationship between a system’s inputs and outputs as observed trajectories, rather than attempting to define the system with a specific mathematical equation or parametric model. This approach characterizes the system’s behavior directly through its response to stimuli, effectively treating the system as a “black box.” By analyzing input-output data, the theory aims to define boundaries on the system’s possible behaviors without requiring prior knowledge of its internal structure or governing equations. This is achieved by mapping inputs to the corresponding range of acceptable outputs, offering a model-free approach to system understanding and control.

Set-Membership Approaches, when integrated with trajectory-based system characterization, establish provable bounds on the true system behavior. Rather than identifying a single parameter set, these methods define a set of all possible system realizations consistent with observed input-output data. This is achieved by formulating constraints based on the data and incorporating prior knowledge, resulting in an admissible set. Any system within this set satisfies the observed data; thus, the approach provides a guaranteed bound on the true system, although this comes at the cost of inherent conservatism – the admissible set will generally be larger than the true system to ensure all possibilities are accounted for. The size of this set is directly influenced by the quantity and quality of the data, and the precision of the prior knowledge incorporated.

Online schemes facilitate iterative refinement of data-driven models through sequential experimentation. These schemes do not rely on a pre-defined dataset but instead actively solicit data via carefully planned input signals. Input Design principles are central to this process, focusing on selecting inputs that maximize information gain about the system’s behavior with each new experiment. This contrasts with traditional, batch-based learning methods. By strategically choosing inputs, online schemes aim to efficiently reduce uncertainty in the estimated model, improving accuracy and robustness over time. The process typically involves executing an experiment, observing the system’s response, updating the model based on this new data, and then selecting the next input based on the updated model’s uncertainty. This closed-loop approach allows for continuous learning and adaptation, even in non-stationary environments.

The combined application of Behavioral System Theory, Set-Membership Approaches, and online experimental refinement facilitates model identification and validation under conditions of limited data. While traditional parametric modeling requires substantial datasets for accurate estimation, this integrated methodology leverages input-output trajectories and bounds the true system within a defined set. Set-Membership Approaches, however, introduce a degree of conservatism; to guarantee containment of the true system, the identified set of possible models may be larger than strictly necessary, potentially sacrificing precision for robustness and verifiable certainty in the model space. This trade-off is inherent to the technique and ensures valid inferences despite data scarcity.

A passive linear model accurately predicts system behavior in Example 4.1 and is certified as a passive system.
A passive linear model accurately predicts system behavior in Example 4.1 and is certified as a passive system.

Toward Robust Networks: Control-Relevant Guarantees in Interconnected Systems

Networked System Identification represents a significant advancement in systems modeling by moving beyond isolated component analysis to embrace the crucial interplay between interconnected elements. Traditional methods often treat subsystems in isolation, neglecting the emergent behaviors arising from their interactions; however, this approach proves inadequate for complex systems where feedback loops and distributed dynamics dominate. Networked identification explicitly accounts for these interdependencies, constructing models that capture not just individual component characteristics, but also the transfer of information and energy between them. This holistic view is particularly vital in areas like power grids, transportation networks, and multi-agent robotics, where collective behavior dictates overall system performance and stability. By accurately representing these network interactions, engineers can design more robust and reliable control strategies, predicting and mitigating potential failures that would be invisible to component-level analysis alone.

A crucial advancement in systems identification lies in the explicit modeling of interactions within networked systems, enabling guarantees of key control-relevant properties. Traditionally, individual components are analyzed in isolation; however, this overlooks the emergent behavior arising from their interconnectedness. By accounting for these interactions – how energy and information flow between elements – researchers can now verify that the identified network possesses characteristics like passivity – a system’s ability to remain stable when connected to another – and dissipativity, ensuring that energy injected into the system is appropriately managed. This rigorous approach moves beyond simply matching input-output behavior; it ensures the identified model is fundamentally stable and well-behaved, paving the way for the design of robust controllers and reliable networked systems, particularly in applications where safety and predictable performance are paramount.

A synergistic approach to control system design emerges from the convergence of physics-informed learning, representation-free analysis, and networked identification techniques. This toolkit moves beyond traditional modeling by embedding known physical laws directly into the learning process, ensuring identified models are inherently plausible and extrapolatable. Simultaneously, representation-free analysis bypasses the need for explicit, potentially inaccurate, system representations, focusing instead on directly assessing stability and performance guarantees. When applied to networked systems – where components interact dynamically – this combination facilitates the creation of control strategies that are not only robust to uncertainties but also demonstrably reliable, even in complex, interconnected environments. The result is a powerful methodology for designing controllers that maintain desired system behavior across a wider range of operating conditions and disturbances, promising enhanced safety and performance in critical applications.

Gaussian Process Models are increasingly utilized to quantify uncertainty in complex system dynamics, offering probabilistic bounds on future behavior and thereby bolstering safety and predictability in control applications. Rather than providing a single, definitive performance boost, these models excel at characterizing the range of possible outcomes, allowing engineers to design controllers that function reliably even with incomplete or noisy data. This approach doesn’t necessarily yield a measurable increase in, for example, settling time or energy consumption, but instead provides a statistically-grounded assessment of risk, enabling the creation of more robust systems capable of operating safely under a wider array of conditions. While a universally quantifiable performance improvement remains elusive, the enhanced ability to manage uncertainty represents a significant advance in the field of control systems design, particularly for applications where safety is paramount.

A port-Hamiltonian neural ODE network accurately identifies the dynamics of a pendulum, as demonstrated by Duong et al. (2024a).
A port-Hamiltonian neural ODE network accurately identifies the dynamics of a pendulum, as demonstrated by Duong et al. (2024a).

The pursuit of control-oriented system identification, as detailed in the paper, necessitates a rigorous pruning of complexity. It’s a process of distillation, aiming for models that are not merely accurate representations of a system, but also demonstrably safe and predictable in operation. This aligns with Albert Camus’ observation: “In the midst of winter, I found there was, within me, an invincible summer.” The ‘invincible summer’ here represents the core principle of guaranteeing stability and dissipativity – even amidst the inherent uncertainties of real-world systems. The work effectively argues that true understanding isn’t about adding layers of intricacy, but about confidently removing what is superfluous, leaving behind a robust and reliable core.

What Lies Ahead?

This review clarifies a simple point: control demands more than prediction. Accuracy alone is insufficient. The field persistently chases fidelity, yet often neglects guarantees. Dissipativity, stability – these are not enhancements, but necessities. Abstractions age, principles don’t. Future work must prioritize embedding these principles directly into identification schemes.

Physics-informed learning offers a path, but its current form too often resembles ornamentation. Simply adding a differential equation doesn’t ensure physical consistency. Every complexity needs an alibi. The challenge lies in developing methods that leverage physical knowledge parsimoniously, not lavishly. Focus should shift from data-driven approximation to constraint-based identification.

Ultimately, the goal isn’t a universal identifier. It’s a toolbox, tailored to the problem. Methods must be judged not only by their ability to match data, but by the confidence they provide in closed-loop performance. The pursuit of elegant algorithms is secondary. Robustness, reliability – these are the virtues that endure.

Original article: https://arxiv.org/pdf/2512.06315.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-09 23:52