Data-Driven Probabilistic Finite $\mathcal{L}_2$-Gain Stabilization of Stochastic Linear Systems

Abstract: In process operations, it is desirable to manage the sensitivity of the system output against external disturbance in the form of finite $\mathcal{L}_2$-gain stabilization. This matter is, however, nonsensical for stochastic systems because the stochastic uncertainties in the control input almost always lead to an unbounded $\mathcal{L}_2$ gain from the disturbance to the output. To address this issue, this article develops a novel concept that characterizes the $\mathcal{L}_2$ gain of stochastic systems in a probabilistic way. Combined with a large data set, we formulate a data-driven probabilistic finite $\mathcal{L}_2$-gain stabilization design using noisy trajectory measurements and the disturbance forecast that does not necessarily agree with the actual future disturbance. The design approach consists of a data-driven trajectory estimation algorithm, whose resulting estimation error covariance is nicely integrated into the feasibility conditions for controller synthesis, leading to a convex offline design in the form of linear matrix inequalities. The effectiveness of the proposed design, along with the additional insights provided by the approach, is illustrated via a numerical example.

• The paper introduces a novel probabilistic framework that ensures finite L2-gain stabilization for stochastic linear systems using measured data.
• It leverages Hankel matrices and Kalman-like filtering for state estimation to synthesize convex LMI-based controllers guaranteeing performance under uncertainty.
• The approach provides explicit probabilistic bounds on performance, effectively balancing robustness against stochastic input and disturbance uncertainties.

Data-Driven Probabilistic Finite $\mathcal{L}_2$-Gain Stabilization of Stochastic Linear Systems

Overview

The paper "Data-Driven Probabilistic Finite $\mathcal{L}_2$-Gain Stabilization of Stochastic Linear Systems" (2604.13707) introduces a novel probabilistic framework for $\mathcal{L}_2$-gain stabilization in stochastic linear time-invariant (LTI) systems. The approach parametrizes system behavior directly from measured trajectories, contending with stochastic uncertainties in both control inputs and external disturbances. It establishes a data-driven control synthesis methodology leveraging noisy data and disturbance predictions, culminating in a convex offline controller design via linear matrix inequalities (LMIs). The framework delivers explicit probabilistic bounds on the sensitivity of system outputs to disturbances and quantifies performance–robustness tradeoffs.

Problem Formulation and Background

Traditional $\mathcal{L}_2$-gain stabilization methods for deterministic systems aim to bound the worst-case ratio between output and disturbance trajectories. However, in stochastic settings, input uncertainties render such bounds either meaningless or unbounded. The paper leverages behavioral systems theory, particularly the fundamental lemma from Willems et al., to circumvent explicit model identification—a system’s admissible behaviors are parameterized using measured, persistently exciting trajectories.

Within this behavioral context, the system is described as a partitioned manifest variable $(y, u, d)$—with $y$ as controlled output, $u$ as manipulated input (subject to random uncertainty), and $d$ as disturbance (with predicted mean and stochastic deviation). Noisy measurements are modeled for all variables. The control task is recast as probabilistic finite $\mathcal{L}_2$-gain stabilization, defining performance criteria in terms of the probability that the truncated $\mathcal{L}_2$ gain from disturbance to output remains below a prescribed bound.

Algorithmic Framework: Probabilistic $\mathcal{L}_2$0-Gain Control Design

The paper proposes a two-stage design and implementation framework:

Offline Design

1. Data-Driven Parametrization: Noisy measured trajectories are used to construct Hankel matrices approximating the system’s behavior space. Persistently exciting inputs guarantee sufficient expressivity.
2. Estimation of System State: The “parameterizer” ($\mathcal{L}_2$1), which functions as a surrogate observable state, is estimated using a Kalman-like optimal filtering procedure. The resulting estimation error covariance is not only minimized for trajectory prediction but is integrally included in robust controller synthesis.
3. Probabilistic Gain Characterization: The $\mathcal{L}_2$2 gain is probabilistically bounded. Key definitions are:
   • Probabilistic $\mathcal{L}_2$3 Gain: For bound $\mathcal{L}_2$4 and probability threshold $\mathcal{L}_2$5, the probability that the output-to-disturbance gain exceeds $\mathcal{L}_2$6 is less than $\mathcal{L}_2$7 for all time horizons.
   • Almost Sure Stability: The gain is finite almost surely.
4. Convex Controller Synthesis: Controller feasibility is determined via LMIs incorporating the estimation error covariance, disturbance and input covariances, and measurement noise. The closed-form ARE (Algebraic Riccati Equation) solution of the filtering stage is directly linked to controller design. These LMIs ensure probabilistic satisfaction of $\mathcal{L}_2$8-gain bounds for all admissible stochastic trajectories.

Online Implementation

• At each time step, the controller updates the parameterizer estimate using measurements and disturbance predictions, applies the optimal state feedback (computed offline), and iteratively refines the estimation error covariance. The designed input is then applied to the actual system, which experiences a random perturbation due to input uncertainty.

Theoretical Results and Performance Guarantees

Key claims:

• The framework assures almost sure $\mathcal{L}_2$9-gain stability (in the sense of probability one) for arbitrary system initializations and disturbance distributions (subject to known covariances).
• For any fixed gain bound $\mathcal{L}_2$0, the probability of performance violation is explicitly quantified as inversely proportional to $\mathcal{L}_2$1, modulated by tradeoff parameters $\mathcal{L}_2$2, $\mathcal{L}_2$3, and a weighting $\mathcal{L}_2$4 representing relative disturbance mean and variance.
• The design naturally trades off nominal (mean trajectory) performance against robustness to stochastic uncertainties; this can be optimized offline.
• The probabilistic bounds are shown to converge empirically with increasing trajectory horizon length; short-time behavior may be more sensitive to initial errors, but long-horizon guarantees hold uniformly.

Numerical simulation:

• Strong empirical support is provided. Simulations with multiple disturbance realizations under distinct controller configurations confirm stabilization and probabilistic satisfaction of the gain bound. Cases with constant disturbance mean enable tighter performance optimization.

Practical and Theoretical Implications

Practical

• The method eliminates explicit system identification, requiring only measured data, thus enabling rapid controller deployment for complex systems where modeling is challenging.
• It is robust to realistic uncertainties in both manipulation and external disturbances, making it suitable for process industries and any domain with measurement noise and unpredictable actuation.
• The convex LMI-based synthesis is computationally tractable and compatible with established optimization toolchains.

Theoretical

• The framework closes a gap in stochastic system control, where classical worst-case stabilization is not meaningful due to arbitrarily large gain in some stochastic realizations.
• It rigorously links data-driven system parameterization with stochastic robust control theory, integrating estimation and controller design in a unified framework.
• The results establish new probabilistic certification for data-driven controllers, underpinning future research into distributed, interconnected, and large-scale stochastic systems.

Future Directions

• Extension to distributed control architectures, where uncertainty propagation across subsystems can complicate estimation and stability.
• Reduction of conservatism in the probabilistic bound when richer statistical information on disturbances is available.
• Integration with stochastic predictive and economic control, leveraging the probabilistic $\mathcal{L}_2$5-gain bounds as chance-constraints in advanced optimization-based controllers.

Conclusion

This paper advances the theory and practice of data-driven control for stochastic LTI systems by forging a rigorous probabilistic framework for $\mathcal{L}_2$6-gain stabilization. The methodology attaches explicit probabilistic guarantees to trajectory-based controllers synthesized directly from measured data, managing uncertainties in both control and disturbance under realistic noise conditions. The convex LMI design and empirical validation position this approach as a robust alternative to conventional model-based control, with substantial implications for industrial deployment and for research at the interface of data-driven and stochastic robust control.