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Data-Driven Resilient Predictive Control under Denial-of-Service

Wenjie Liu, Jian Sun, Gang Wang, Francesco Bullo, Jie Chen

arXiv:2110.12766v1eess.SY

TL;DR

The paper tackles resilient stabilization of unknown stochastic LTI systems under DoS attacks without explicit models or prior system identification. It develops a data-driven resilient MPC scheme based on pre-collected input-output trajectories and data-dependent convex conditions. The method achieves the same resilience level as model-based control, including local ISS under mild noise and attack assumptions, while global ISS requires either reduced resilience or greater computational complexity.

  • Problem

    Prior predictor-based resilient controllers require explicit system models or prior system identification, motivating stabilization of unknown LTI systems under DoS attacks using only input-output trajectories.

  • Method

    The paper develops a resilient data-driven MPC that uses pre-collected input-output data and convex, data-dependent optimization to generate predictions and control inputs during DoS attacks.

  • Results

    The controller establishes local ISS under stated DoS and noise conditions, with the same attack-resilience condition as existing model-based control; global ISS requires one of two proposed modifications.

  • Takeaways & Limitations

    Data-driven control can provide resilient stabilization of unknown stochastic LTI systems under DoS attacks without system-model knowledge or a priori identification.

Abstract

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The study of resilient control of linear time-invariant (LTI) systems against denial-of-service (DoS) attacks is gaining popularity in emerging cyber-physical applications. In previous works, explicit system models are required to design a predictor-based resilient controller. These models can be either given a priori or obtained through a prior system identification step. Recent research efforts have focused on data-driven control based on pre-collected input-output trajectories (i.e., without explicit system models). In this paper, we take an initial step toward data-driven stabilization of stochastic LTI systems under DoS attacks, and develop a resilient model predictive control (MPC) scheme driven purely by data-dependent conditions. The proposed data-driven control method achieves the same level of resilience as the model-based control method. For example, local input-to-state stability (ISS) is achieved under mild assumptions on the noise and the DoS attacks. To recover global ISS, two modifications are further suggested at the price of reduced resilience against DoS attacks or increased computational complexity. Finally, a numerical example is given to validate the effectiveness of the proposed control method.

I. INTRODUCTION

The paper addresses resilient stabilization of unknown LTI systems under DoS attacks without explicit models, using pre-collected input-output data. It develops a data-driven MPC controller that accounts for noise and achieves local ISS under attack conditions, with modifications for global ISS.

  • Motivation: Existing resilient control strategies require explicit system models or prior system identification, which can be difficult or costly for real-world and large-scale systems.Data-driven methods instead use input-output trajectories to control unknown dynamics.
  • Method: The proposed resilient data-driven MPC computes control inputs and predicted outputs through a convex optimization problem using purely data-dependent constraints.Optimization is performed when measurements arrive, while stored predictions or zero input are used during sufficiently long DoS intervals.
  • Method: The method models process and measurement noise while retaining computationally appealing convexity, unlike a related formulation that is generally non-convex and NP-hard.The paper also distinguishes its approach from prior convex data-driven MPC without stability and robustness guarantees.
  • Contributions: The main contribution is a three-part data-driven resilient MPC framework for unknown systems, including convex prediction, DoS accommodation, and ISS analysis.The introduction summarizes the work as a threefold contribution.
  • Results: The paper establishes local input-to-state stability for the closed loop under standard conditions on DoS attacks and noise.The controller is designed to accommodate DoS attacks while preserving the resilience level associated with model-based predictor control.

A. Networked control systems

The paper considers a stochastic discrete-time LTI system with unknown matrices, noisy process and output measurements, and a communication channel subject to DoS attacks. It uses ISS and UIOSS concepts to analyze stability when transmitted measurements are intermittently unavailable.

  • System model: The plant is a stochastic discrete-time LTI system whose state, input, output, and process noise are represented by x_t, u_t, y_t, and w_t.The initial state is arbitrary.
  • Assumptions: The system is assumed stabilizable and observable, while its matrices are unknown and only input-output data from offline experiments are available.These assumptions support data-driven control without explicit system identification in the controller design.
  • Network and noise: At each time instant, a packet containing the past η output measurements is transmitted to the remote controller, but DoS attacks can prevent packet reception.Received outputs also contain additive network-induced noise, while the controller-to-plant input channel is ideal.
  • Network and noise: Process and network-induced measurement noises are assumed bounded by a known constant v̄.This bound is used in the stability analysis.
  • Stability concepts: Because noise prevents relying on asymptotic stability directly, the analysis uses input-to-state stability and related UIOSS properties.UIOSS supplies trajectory and Lyapunov-function bounds under bounded inputs and noise.

B. Denial-of-Service attack

DoS attacks block communication and can destabilize open-loop unstable systems. The paper models attacks through frequency and duration, imposing conditions that ensure successful transmissions and characterize maximum resilience.

  • DoS attacks block communication channels, causing packet losses that can degrade performance and induce instability in open-loop unstable plants.
  • DoS attacks are characterized using an indicator that records whether an attack occurs and associated duration and frequency measures.
  • Assumptions constrain DoS frequency through a chatter bound and average dwell-time, and DoS duration through a chatter bound and average duration ratio.
  • These assumptions upper bound the number of time steps between consecutive successful transmissions.
  • 1/νf + 1/νd < 1 is required for maximum resilience; otherwise, an attack sequence can prevent stabilization of an open-loop unstable system.

C. Fundamental Lemma

The Fundamental Lemma connects persistently exciting input-output data with trajectories of an LTI system. This connection supplies a data-dependent representation needed for control without an explicit system model.

  • The paper targets resilient stabilization of unknown LTI systems under DoS attacks using only measured input-output data.
  • Persistency of excitation requires rank(HL(uN)) = nuL for an input sequence of order L.
  • The Fundamental Lemma states that input-output trajectories can be expressed as linear combinations of pre-collected persistently exciting input-output data.
  • A trajectory of the noise-free LTI system can be represented using pre-collected data generated by a persistently exciting input sequence of order L + nx.
  • The constructed trajectory length depends on the persistently exciting order, and data can be collected from one long trajectory or multiple shorter trajectories.

D. Model-based observer-based controller

The model-based controller uses an observer and predictor to maintain state estimates through DoS interruptions. Successful transmissions reset prediction errors, while the attack condition guarantees such resets recur regularly.

  • The model-based strategy combines an observer at the sensor with predictor-based state feedback using the estimated state.
  • The predictor updates from a new estimated state when transmission succeeds and performs a model-based prediction step during DoS attacks.
  • The observer gain satisfies (A − LC)^η = 0, where η is the system observability index, while K is a stabilizing feedback gain.
  • For attacks satisfying the resilience condition, the prediction error is reset to a bounded value at least once every T time instants.
  • The resulting predict-then-reset cycle repeatedly keeps the estimated state close to the actual state under bounded noise.

III. DATA-DRIVEN RESILIENT CONTROL UNDER DOS

The proposed controller redesigns the robust predictor-based approach using the Fundamental Lemma. It constructs a data-dependent controller and embeds it in a DoS-resilient MPC scheme whose stability is analyzed.

  • The paper addresses stabilization of unknown LTI systems under DoS attacks by redesigning a robust predictor-based controller using the Fundamental Lemma.
  • The resulting data-dependent controller follows a predict-then-reset cycle and supports a DoS-resilient MPC algorithm.
  • The proposed control scheme is corroborated by a stability analysis.

A. Data-driven resilient predictive control

The paper develops a data-driven MPC scheme for unknown LTI systems using offline input-output trajectories, convex data-dependent optimization, and buffered inputs during DoS attacks. Its design relies on persistent excitation, a sufficiently long prediction horizon, noise-aware regularization, and repeated replanning after successful transmissions.

  • Data-driven MPC formulation: The MPC predicts an L-step future trajectory from offline input-output data and applies only the first computed input at each successful transmission.The prediction horizon must satisfy L ≥ η + n_x, and the offline input sequence must be persistently exciting of order L + n_x + η.
  • Data-driven MPC formulation: The optimization uses slack variables to compensate for network-induced noise and constraints that preserve trajectory continuity and predicted-trajectory convergence.The objective penalizes constraint violation and process-noise effects through weighted terms involving h(t) and g(t).
  • Data-driven MPC formulation: All constraints and the objective are convex, allowing Problem (20) to be solved efficiently with off-the-shelf convex programming solvers.Noise-dependent regularization coefficients balance cost minimization against slack and process-noise penalties.
  • Comparison and scope: Compared with prior data-driven MPC, the formulation accounts for process noise and avoids the earlier non-convex stability constraint, while assuming noise-free pre-collected outputs for exposition.The authors state that stability analysis with noisy pre-collected outputs is possible but involves more complicated noise terms.
  • Resilient operation under DoS: During a DoS attack, the controller sequentially applies inputs from the most recent MPC solution for up to L steps, then applies zero input until transmission resumes.At each successful transmission, new measurements are used to solve Problem (20) and generate a refreshed input sequence.
  • Resilient operation under DoS: The resilient scheme combines data-driven MPC with a predictor-based response to missing packets and requires no system matrices, only offline input-output trajectories.The paper summarizes this procedure in Algorithm 1, including horizon, cost, noise-bound, and data-excitation inputs.

B. Stability analysis

The stability analysis first bounds prediction error, then uses a Lyapunov function to establish recursive feasibility and ISS under bounded noise and constrained DoS attacks. The baseline controller achieves local ISS, while global ISS requires modifications that trade resilience for computation.

  • Stability analysis: The analysis bounds the difference between predicted and actual outputs before establishing closed-loop stability with a Lyapunov function.This two-step strategy follows the model-based resilient-control analysis while incorporating data-driven predictions.
  • Noise–DoS trade-off: Prediction error increases between successful transmissions and resets to a noise-dependent constant when a transmission succeeds.Stronger DoS attacks can lengthen the interval between successful transmissions, increasing prediction error.
  • Recursive feasibility: Under the stated assumptions, the optimization problem remains feasible at every successful transmission instant.The Lyapunov function decreases at each successful transmission, supporting recursive feasibility.
  • ISS result: System (3) is input-to-state stable under Algorithm 1 when the DoS conditions, initial feasibility, Lyapunov bound, and noise threshold assumptions hold.The theorem applies for sufficiently small bounded noise and initial conditions within a specified neighborhood.
  • Scope and extensions: The proposed data-driven controller achieves only local ISS, while global ISS requires either solving the problem less frequently or accepting increased computational complexity.The paper identifies reduced DoS resilience or increased computation as the corresponding costs of the two global-ISS modifications.

IV. NUMERICAL EXAMPLE

The numerical example evaluates the proposed resilient controller on an unstable batch reactor and compares data-driven with model-based control under varying noise and DoS attacks. It also examines how prediction horizon L and dataset length N affect performance.

  • Setup: The experiment uses a discrete-time unstable batch reactor with 0.1s sampling and pre-collected input-output trajectories of length N = 100.The prediction horizon is set to L = 10, with L ≥ nx + 2η = 8.
  • Comparison: The data-driven and model-based methods are compared under different noise and DoS attack levels over a 200-step simulation.The identification step uses the same data as the data-driven method.
  • Comparison: The comparison reveals a trade-off between resilience against DoS attacks and robustness against noise.This trade-off is observed in the output trajectories under varying gray-shaded attack and noise conditions.
  • Data length and horizon: When 33 < N ≤60, system performance improves rapidly as N increases, then remains almost steady for larger datasets.The stated lower bound is N ≥33.
  • Data length and horizon: For N = 40, the admissible prediction-horizon range is 8 ≤L ≤12, and sufficiently large L and N have little effect on performance.With small datasets, the data-driven method performs better than the model-based method.

V. CONCLUSIONS

The paper develops data-driven resilient MPC for unknown stochastic LTI systems under DoS attacks using only pre-collected input-output data. It achieves local ISS under conditions on attacks and noise, while global stability requires either reduced resilience or greater computational complexity.

  • Contribution: The proposed data-driven MPC computes input sequences and predicted outputs from pre-collected input-output data by solving convex programs.No explicit system model is required for this computation.
  • Stability: The resilient controller achieves local input-to-state stability under conditions on DoS attacks and noise without a system model or prior identification.The controller is built on the data-driven MPC scheme.
  • Resilience: The required DoS condition matches the maximum resilience achieved by existing model-based controllers.This comparison is stated for the proposed local-stability result.
  • Stability: Global stability can be obtained through two modifications, at the cost of reduced DoS resilience or increased computational complexity.The paper presents these as alternatives for recovering global stability.
  • Validation: A numerical example demonstrates the effectiveness and practical merits of the proposed method and theory.The example evaluates the controller on an unstable batch reactor.
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