Source-linked AI summary
A trajectory-based framework for data-driven system analysis and control
Julian Berberich, Frank Allgöwer
TL;DR
The paper addresses how measured data can support rigorous system analysis and control without explicitly identifying a model. It translates single-trajectory representation into classical state-space control, extends it to certain nonlinear systems, and applies the extension to data-driven simulation with kernel methods. The resulting framework constructs unknown-system trajectories from measured data and reports good noisy-output estimation in the kernel-based simulation example.
Problem
Data-driven control needs rigorous methods that avoid relying exclusively on identified models while addressing limitations of some learning-based approaches, including data demands, reproducibility, and guarantees.
Method
The paper characterizes system trajectories from a single measured input-output trajectory, extends the construction to systems linear in suitable known nonlinear coordinates, and uses kernel methods for simulation.
Results
All trajectories of an unknown LTI system can be constructed from one persistently exciting measured trajectory, and the nonlinear extension supports data-driven simulation with good output estimates under noise.
Takeaways & Limitations
A measured trajectory can contain the information needed for system analysis, controller design, and certain data-driven simulation tasks without explicit model identification.
Abstract
from arXiv · showhide
The vector space of all input-output trajectories of a discrete-time linear time-invariant (LTI) system is spanned by time-shifts of a single measured trajectory, given that the respective input signal is persistently exciting. This fact, which was proven in the behavioral control framework, shows that a single measured trajectory can capture the full behavior of an LTI system and might therefore be used directly for system analysis and controller design, without explicitly identifying a model. In this paper, we translate the result from the behavioral context to the classical state-space control framework and we extend it to certain classes of nonlinear systems, which are linear in suitable input-output coordinates. Moreover, we show how this extension can be applied to the data-driven simulation problem, where we introduce kernel-methods to obtain a rich set of basis functions.
I. INTRODUCTION
The paper develops a data-driven control framework that uses measured trajectories directly rather than explicitly identified models. It translates behavioral-system results into classical control and extends them toward nonlinear systems and further applications.
- Data-driven control seeks rigorous alternatives to model-based methods and to learning approaches that often require much data and lack reproducibility or closed-loop guarantees.
- The framework characterizes all trajectories of an unknown system using a single measured input-output trajectory.
- Behavioral systems theory is especially suited to purely data-driven analysis because it defines systems through their spaces of trajectories.
- Prior work applies the trajectory-based result to data-driven MPC, stochastic analysis, power systems, stability, robustness, and closed-loop parametrization.
- The paper translates the main result to classical state-space control, improves persistence-of-excitation requirements by weaving trajectories, extends it to nonlinear systems, and addresses data-driven simulation.
II. SETTING
The setting defines finite input-output trajectories of discrete-time LTI systems and introduces persistent excitation as the central data condition. The framework assumes measured data are available while the unknown system order is known only through a potentially rough upper bound.
- Persistent excitation: Persistent excitation of order L requires the associated Hankel matrix to have full rank, rank(H_L(x)) = nL.
- Persistent excitation: Persistent excitation requires sufficient signal length, not merely linearly independent time-shifts.
- Data assumptions: The measured input-output trajectory may come from simulation or experiment, while the unknown system order is assumed known only through a potentially rough upper bound.
- Trajectory and system setting: An input-output sequence is a trajectory when it is generated by an LTI realization from an initial condition and state sequence.
- Trajectory and system setting: LTI trajectory sets form vector spaces, whose bases can be formed by time-shifts of one measured trajectory under persistent excitation.
III. TRAJECTORY-BASED REPRESENTATION OF LINEAR
The linear result represents every trajectory of an unknown LTI system using one persistently exciting measured trajectory, equivalently through a data-dependent Hankel matrix. Multiple trajectories can be woven together to extend the usable horizon, but persistence of excitation imposes a data-length limit.
- Classical reformulation: The paper reformulates the behavioral trajectory-characterization theorem in the classical state-space control framework.
- Single-trajectory representation: Every length-L trajectory of an unknown LTI system can be constructed from a single measured trajectory when the input is persistently exciting of order L+n.
- Single-trajectory representation: The trajectory space equals the range of a data-dependent Hankel matrix, so the measured trajectory acts as a system representation without explicit model identification.
- Persistence-of-excitation limitation: Theorem 3 requires N ≥ (m+1)(L+n)−1 to span trajectories of length L, limiting the horizon supported by a finite dataset.
- Trajectory weaving: Multiple trajectories can be woven into an arbitrarily long trajectory when their internal states align over at least n steps at each intersection.
IV. TRAJECTORY-BASED REPRESENTATION OF NONLINEAR
The paper extends trajectory-based representation beyond LTI systems to Hammerstein, Wiener, and more general systems that become linear in known nonlinear input-output coordinates. These extensions provide an alternative to identifying such nonlinear systems from data.
- The nonlinear extension covers Hammerstein, Wiener, and other systems that are linear in suitably chosen and known input-output coordinates.
- For these systems, a single measured trajectory can represent the system as an alternative to identification.
A. Hammerstein systems
The paper extends trajectory-based characterization from LTI systems to Hammerstein systems by transforming the static nonlinearity into known basis-function coordinates. Under persistence of excitation, measured data can represent system trajectories without identifying the nonlinear system.
- A. Hammerstein systems: A Hammerstein system combines a static nonlinearity with a subsequent LTI system.
- A. Hammerstein systems: The nonlinear input transformation ψ is represented using r known basis functions, producing an auxiliary input trajectory v.
- A. Hammerstein systems: The transformed system can be treated as an LTI map from the auxiliary input v to the output y.
- A. Hammerstein systems: If v is persistently exciting of order L + n, a length-L trajectory is characterized by a linear combination of time-shifts of the measured trajectory.
- A. Hammerstein systems: The known basis-function representation requires N ≥ (r + 1)(L + n) − 1, limiting how many basis functions can be used explicitly.
B. Wiener systems
The paper applies the trajectory-based approach to Wiener systems by transforming outputs through an inverse static nonlinearity. This yields a corresponding trajectory representation, but with a weaker converse and more difficult basis-function selection than in the Hammerstein case.
- B. Wiener systems: A Wiener system consists of an LTI system followed by a static nonlinearity.
- B. Wiener systems: The inverse output nonlinearity φ^-1 is assumed to admit a decomposition using q known basis functions, which define an auxiliary output trajectory z.
- B. Wiener systems: The transformed output z serves as the output of an equivalent LTI system for applying the trajectory-based argument.
- B. Wiener systems: Under persistence of excitation of u of order L + n, the result gives a trajectory condition expressed through a coefficient vector α.
- B. Wiener systems: Unlike the Hammerstein case, the Wiener result has no limit on the number of basis functions, but selecting basis functions for φ^-1 is more difficult.
- B. Wiener systems: The “only if” direction generally fails because the map from u to z is not necessarily linear.
V. DATA-DRIVEN SIMULATION
The data-driven simulation problem computes an unknown system’s output from a prescribed input using only a measured input-output trajectory. The paper extends this trajectory-based approach to nonlinear systems through lifted coordinates and kernel methods, with regularization improving noisy simulation accuracy.
- Trajectory-based simulation: Data-driven simulation predicts an unknown system’s output for a given input using only a previously measured input-output trajectory.The approach avoids requiring an explicit model.
- Trajectory-based simulation: A persistently exciting input trajectory spans arbitrary LTI trajectories, allowing coefficients solved from the prescribed input to generate the predicted output.The remaining output is computed as ¯y = HL(y)α.
- Kernel reformulation: Kernel methods implicitly provide rich basis functions through inner products, avoiding explicit construction of the nonlinear coordinates.A selected kernel can imply the basis functions used for the nonlinearities.
- Nonlinear extension: The method extends simulation to Hammerstein-Wiener systems by representing trajectories in lifted input-output coordinates.For these systems, trajectory spaces are spanned by Hankel matrices of transformed inputs and outputs.
- Numerical example: With 5% multiplicative output noise, a squared exponential kernel and λ = 10 produced a good estimate for a length-50 random input, whereas omitting regularization or using fixed polynomial bases significantly worsened accuracy.The experiment used N = 1000 measured samples and zero initial conditions.
VI. CONCLUSION
The paper presents a purely data-driven framework in which a single measured trajectory supports system analysis and controller design without explicit model identification. It extends the framework to certain nonlinear systems and applies the extension to kernel-based data-driven simulation.
- Conclusion: All trajectories of an unknown system can be constructed from a single measured trajectory, which supplies the information needed for analysis and controller design without explicit model identification.The framework is first described in the classical control setting.
- Conclusion: The nonlinear extension applies to certain systems that are linear in suitable input-output coordinates and supports kernel-based data-driven simulation.The paper identifies further applications and connections to kernel methods as future research directions.