Source-linked AI summary
Turnpike properties in nonlinear system identification
Julian D. Schiller, Matthias A. Müller
TL;DR
The paper asks how fixed-initial-state SEM affects optimal outputs when learning general nonlinear state-space models, including cases with non-unique solutions. It formulates output turnpike properties and characterizes them through value functions and dissipativity. Under cost reachability, cumulative turnpike behavior is equivalent to value-function coercivity and tailored strict dissipativity, with additional sufficient conditions based on stability, convexity, and optimality.
Problem
Fixed-initial-state SEM is computationally attractive, but its relationship to unconstrained SEM solutions requires analysis, especially with non-unique optimal output sequences.
Method
The paper generalizes cumulative output turnpike theory to non-unique outputs, introduces cardinality turnpike, and analyzes SEM using value functions, dissipativity, stability, convexity, and optimality conditions.
Results
Under cost reachability, cumulative turnpike behavior is equivalent to value-function coercivity and tailored strict dissipativity; cumulative turnpike is strictly stronger than cardinality turnpike.
Takeaways & Limitations
Turnpike behavior supports using tractable fixed-initial-state or truncated SEM while keeping optimal output sequences close to unconstrained optima.
Takeaways & Limitations
The analysis leaves extensions to encoder-augmented SEM formulations and relaxed sufficient conditions for future work.
Abstract
from arXiv · showhide
We analyze the problem of learning general discrete-time nonlinear state-space models using the simulation error minimization (SEM) method. In this setting, model parameters are typically learned by minimizing the mismatch between simulated and measured outputs over a training dataset, or shorter subsequences extracted from it. Specifically, we study the cumulative output turnpike property of the underlying SEM optimization problem, which requires optimal output sequences emanating from a fixed initial state to approach and remain close to an optimal output sequence of the corresponding SEM problem with free initial state. In the presence of non-unique optimal output sequences---as may arise, for instance, in fully black-box system identification using neural networks---the property is formulated with respect to the closest such sequence. Turnpike behavior is generally desirable in practice, as it provides a theoretical justification for employing computationally more tractable SEM formulations with fixed initial states while ensuring that their optimal output sequences remain close to unconstrained optimal ones. Under a mild reachability condition, we establish equivalence between the cumulative turnpike property, coercivity of the value function, and a tailored notion of strict dissipativity. We additionally introduce a cardinality turnpike property and show that it is strictly weaker than the cumulative notion. Finally, we establish sufficient conditions for turnpike behavior based on incremental output stability, convexity of the stage cost, and a suitable optimality condition, and illustrate the theory by means of a numerical example.
1. Introduction
The paper studies simulation error minimization for nonlinear state-space identification, where fixed-initial-state training offers computational advantages but raises questions about solution quality. It develops turnpike theory to analyze when constrained optimal outputs remain close to unconstrained ones.
- Simulation error minimization directly targets multi-step prediction errors needed for finite-horizon analysis, simulation, and control.
- Full SEM is computationally challenging because each optimization iteration requires forward simulation and sensitivity computations over the entire dataset.
- Short subsequences provide a more efficient and parallelizable alternative for SEM training on modern hardware.
- Fixing the initial state reduces optimization complexity and preserves parallelizability, but its effect on the resulting SEM solution requires analysis.
- The paper generalizes cumulative output turnpike theory to non-unique optimal output sequences and relates it to value-function coercivity and strict dissipativity.
2. Problem setting
The paper formulates nonlinear discrete-time state-space identification as an SEM optimization problem over model parameters and initial conditions. It also defines constrained fixed-initial-state variants, regularized output-mismatch costs, and truncated SEM formulations.
- The model uses nonlinear continuous dynamics and output maps parameterized by a vector θ, with states, inputs, and outputs in Euclidean spaces.
- SEM learns parameters by minimizing a weighted output-mismatch cost, optionally augmented with a lower-semicontinuous regularization term.
- The optimization jointly selects the initial state and parameters over compact feasible sets subject to model and knowledge constraints.
- Under the standing feasibility and compactness assumptions, an optimal SEM solution exists, although it may be non-unique.
- A constrained SEM formulation fixes the initial state and optimizes only model parameters, reducing complexity relative to joint optimization.
- Truncated SEM applies the same framework to shorter, possibly overlapping subsequences while coupling them through common parameters.
3. The turnpike phenomenon in system identification
The paper formulates output-based turnpike behavior for SEM with non-unique optima, then characterizes it through coercivity and strict dissipativity and contrasts cumulative with cardinality turnpikes.
- 3.1. Cumulative turnpike: The cumulative turnpike property measures constrained SEM outputs against the closest optimal output sequence of the unconstrained problem.This formulation accommodates non-unique optimal pairs and output sequences.
- 3.1. Cumulative turnpike: Uniformly bounded cumulative discrepancy implies that average output discrepancy vanishes as the dataset size grows.The constrained sequence remains close to the turnpike for most time indices, with uniformly bounded pointwise discrepancy.
- 3. The turnpike phenomenon in system identification: The output-based formulation is natural because different initial conditions and parameters can generate identical output sequences without strong identifiability.This motivates comparing outputs rather than states, controls, or adjoints.
- 3.3. Equivalent characterizations: Under cost reachability, cumulative turnpike, coercivity of the value function, and strict dissipativity are equivalent.Coercivity links the normalized value-function gap to cumulative output discrepancy, while strict dissipativity uses a tailored storage function.
- 3.4. On the cardinality turnpike property: Cumulative turnpike implies cardinality turnpike, but the converse fails generally because cardinality bounds exceptional-index counts without bounding deviation magnitudes.With compact output sets, the two properties become equivalent.
4. Sufficient conditions for turnpike behavior
The paper derives sufficient conditions for turnpike behavior from incremental output stability, stage-cost regularity and convexity, optimality, and uniformly comparable weights. These conditions yield cost reachability and coercivity, which support turnpike conclusions.
- 4.1. Output stability implies cost reachability: Incremental output stability and local Lipschitz continuity of the stage cost provide cost reachability under the stated feasibility and weight conditions.Incremental stability requires outputs generated from different initial states to converge exponentially along the training input sequence.
- 4.2. Optimality implies coercivity: For quadratic costs, the optimality condition controls the first-order cost change along convex combinations of constrained and unconstrained optimal outputs.The negative part of the corresponding first-order quantity must be uniformly bounded, and vanishing negative parts provide an asymptotic sufficient condition.
- 4.2. Optimality implies coercivity: Uniform strict convexity and the paper’s optimality condition imply coercivity of the value function when the weight bounds hold uniformly in N.The result applies to convex stage costs and uses a uniform bound on the value function relative to the average weight.
- 4.2. Optimality implies coercivity: The coercivity theorem requires uniform strict convexity, the optimality condition, and a uniform weight lower-bound condition, and then establishes coercivity of the value function.The theorem’s proof constructs the coercivity modulus from the strict-convexity modulus and the uniform constants.
- 4.2. Optimality implies coercivity: Uniform weight comparability prevents individual weights from becoming negligible or dominant relative to their average.Unweighted and normalized costs satisfy the condition, whereas exponentially discounted costs generally do not satisfy it uniformly in N.
- 4.2. Optimality implies coercivity: Zero-weight indices can be incorporated if the zero-weight set remains uniformly bounded, supporting outlier rejection or a finite burn-in phase.The weight lower bound is imposed only on positive-weight indices in this extension.
5. Numerical example
The numerical example uses a scalar nonlinear state-space model equivalent to a one-unit Elman recurrent neural network and verifies the theoretical assumptions analytically. Its constrained solutions exhibit bounded cumulative output discrepancies and increasingly localized transient differences as the dataset grows.
- Model and setup: The example uses a scalar nonlinear state-space model equivalent to an Elman-type recurrent neural network with one hidden unit.The model has recurrent parameter θ, fixed unit input weight, zero bias, and identity readout.
- Model and setup: The training data use θtrue = −0.4, xtrue_0 = 0.6, a sinusoidal input, and no measurement noise.The noise-free setup isolates the effect of fixing the initial condition and permits direct verification of the optimality condition.
- Verification of theoretical conditions: The model satisfies uniform incremental output stability with C = 1 and λ = 0.9, while the stage cost satisfies the required regularity and convexity conditions.The outputs remain in a compact set, and the burn-in weights satisfy the required comparability extension.
- Non-unique optimal solutions: For N = m = 2, the unconstrained SEM problem has a non-singleton optimal solution set forming a nonlinear curve in the (θ, x0)-plane.Only the terminal output is penalized, allowing multiple parameter–initial-state pairs to attain the zero optimum.
- Non-unique optimal solutions: With fixed initial state x̄ = −0.6, the constrained solution is unique at θ′ = 0.152, while every unconstrained optimum is equally close under the example’s distance measure.This explicitly realizes the non-unique turnpike setting.
- Increasing the dataset size: The cumulative output discrepancy remains uniformly bounded as N increases and approaches approximately 1.685 for the largest considered dataset.The scaled value-function gap also remains bounded and appears to converge to a finite value.
- Increasing the dataset size: The squared output discrepancies are concentrated near the trajectory’s beginning and become progressively smaller later and for larger N.Together with the bounded cumulative discrepancy, this is consistent with the cumulative turnpike property.
6. Conclusion
The paper establishes an equivalence framework for cumulative output turnpikes and identifies conditions that produce them in nonlinear SEM. A recurrent-neural-network example illustrates the theory, while extensions to encoder-augmented formulations and relaxed guarantees remain future work.
- Turnpike behavior supports computationally tractable fixed-initial-state SEM while keeping the resulting output sequence close to an unconstrained optimum.
- The cumulative output turnpike property is generalized to non-unique optimal output sequences and shown equivalent to value-function coercivity and tailored strict dissipativity.
- The cardinality turnpike property is strictly weaker than the cumulative notion, although equivalence can be recovered under additional uniform boundedness assumptions.
- Sufficient conditions for turnpike behavior combine incremental output stability, convexity of the stage cost, and a suitable optimality condition.
- Future work includes more general SEM formulations with additional encoders and relaxed conditions guaranteeing turnpike behavior.
A. Proof of Theorem 1
Theorem 1 is proved through a three-way implication cycle linking coercivity, cumulative turnpike behavior, and strict dissipativity. The proof constructs a storage function for the dissipativity implication and uses value-function bounds for the converse directions.
- Proof strategy: The proof organizes the main equivalence as coercivity implying cumulative turnpike, cumulative turnpike implying strict dissipativity, and strict dissipativity implying coercivity.
- Coercivity implies cumulative turnpike: Coercivity implies the cumulative turnpike property when the cost reachability assumption holds.
- Cumulative turnpike implies dissipativity: Cumulative turnpike behavior implies strict dissipativity through a candidate storage function built from the comparison between constrained and closest unconstrained optimal solutions.
- Strict dissipativity implies coercivity: Strict dissipativity implies coercivity by applying the dissipation inequality together with the storage function’s initial and terminal bounds.
- Proof strategy: The resulting coercivity bound is obtained by applying the inverse comparison function to the accumulated dissipation estimate.
CRediT authorship contribution statement
Julian D. Schiller contributed across conceptualization, formal analysis, methodology, project administration, software, validation, visualization, and original drafting. Matthias A. Müller contributed conceptualization, funding acquisition, project administration, resources, validation, and review and editing.
- Julian D. Schiller handled conceptualization, formal analysis, methodology, project administration, software, validation, visualization, and original drafting.
- Matthias A. Müller handled conceptualization, funding acquisition, project administration, resources, validation, and review and editing.
Declaration of generative AI use in the manuscript preparation process
The authors used ChatGPT for language editing, manuscript organization, and exploratory discussions during preparation. They independently verified the mathematical results and retained responsibility for the article’s content.
- ChatGPT supported language editing, manuscript organization, and exploratory discussions of mathematical arguments, examples, and formulations.
- The authors independently assessed and verified all mathematical results, proofs, interpretations, and conclusions.
- The authors reviewed and edited AI-assisted content and took full responsibility for the article’s content.