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Data-Driven Control of Distributed Event-Triggered Network Systems

Xin Wang, Jian Sun, Gang Wang, Frank Allgöwer, Jie Chen

arXiv:2208.10303v1eess.SY

TL;DR

Unknown discrete-time network systems require control under limited communication resources and may lack usable explicit models. The paper combines periodic distributed event-triggering, looped-functional stability analysis, and data-driven system representation to obtain LMI-based co-design from offline data. Numerical results support reduced data transmissions and the effectiveness of the co-design procedure.

  • Problem

    The paper studies data-driven event-triggered control for unknown discrete-time interconnected systems under limited communication resources and unavailable explicit system models.

  • Method

    The paper combines a periodic distributed event-triggering strategy, discrete-time looped-functional analysis, and data-driven system representation to derive LMI-based stability and co-design criteria.

  • Results

    Numerical results support the proposed distributed data-driven event-triggering schemes in reducing transmissions while achieving distributed control.

  • Takeaways & Limitations

    The framework jointly designs event-triggering coefficients and feedback control gains using offline collected state-input data.

Abstract

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The present paper deals with data-driven event-triggered control of a class of unknown discrete-time interconnected systems (a.k.a. network systems). To this end, we start by putting forth a novel distributed event-triggering transmission strategy based on periodic sampling, under which a model-based stability criterion for the closed-loop network system is derived, by leveraging a discrete-time looped-functional approach. Marrying the model-based criterion with a data-driven system representation recently developed in the literature, a purely data-driven stability criterion expressed in the form of linear matrix inequalities (LMIs) is established. Meanwhile, the data-driven stability criterion suggests a means for co-designing the event-triggering coefficient matrix and the feedback control gain matrix using only some offline collected state-input data. Finally, numerical results corroborate the efficacy of the proposed distributed data-driven ETS in cutting off data transmissions and the co-design procedure.

I. INTRODUCTION

The paper targets unknown discrete-time network systems whose communication constraints make redundant sampled-data transmissions costly. It develops a distributed periodic event-triggering strategy and data-driven stability and co-design methods using offline state-input data.

  • I. INTRODUCTION: The proposed dynamic distributed event-triggering strategy uses periodic sampling and local subsystem and neighbor state information at sampling instants.It generalizes decentralized and distributed event-triggering strategies while incorporating both local and neighboring information.
  • I. INTRODUCTION: A discrete-time looped-functional yields a model-based stability criterion for the network system under the proposed distributed event-triggering strategy.The criterion supports analysis of the closed-loop network system under event-triggered transmissions.
  • I. INTRODUCTION: Combining the stability criterion with a data-driven system representation produces an LMI-based criterion that co-designs feedback gains and event-triggering coefficients from offline state-input data.The approach avoids requiring explicit knowledge of the system matrices for controller and event-triggering design.
  • I. INTRODUCTION: The network model comprises coupled discrete-time linear subsystems connected through communication channels and controlled by local controllers using neighboring information.The system matrices describing subsystem dynamics, inputs, and interconnections are assumed unknown in the paper’s setting.
  • I. INTRODUCTION: The paper addresses unknown discrete-time network systems where limited bandwidth and energy make redundant transmissions costly.Existing event-triggered strategies are often model-based, while accurate system models can be expensive or impossible to obtain.

B. Data-driven system representation for noisy data

The paper represents unknown network-system dynamics using locally collected state-input data corrupted by bounded disturbances. This yields a data-consistent system set that supports stability analysis without knowing the system matrices.

  • Data collection: The unknown system matrices are characterized from locally collected state, successor-state, and input data affected by additive disturbances.The disturbance can represent noise or unmodeled dynamics, with known disturbance-channel structure.
  • Data collection: The local data are stacked into global matrices, while the corresponding disturbance matrix remains unknown.The global representation summarizes the interconnected subsystems and their measured transitions.
  • Noise model: A global bounded-noise assumption models all local disturbances using known matrices and a bounded noise set.For a single subsystem, this assumption reduces to the centralized bounded-noise formulation used in prior work.
  • Data-driven representation: The consistent system set ΣAB contains all system matrices compatible with the measurements and bounded-noise description.It is defined through a quadratic matrix inequality derived from the data and noise constraint.
  • Data-driven representation: The resulting parametrization uses only offline data X, X+, and U, enabling stability guarantees for every system in ΣAB.The offline open-loop data are separate from the sampling times generated during closed-loop operation.

III. MAIN RESULTS

The main results introduce a dynamic distributed event-triggering scheme based on periodic sampling and develop its stability and design foundations. The scheme uses sampled local and neighboring information while updating its dynamic trigger variable only at sampling instants.

  • A. Distributed dynamic ETS: The proposed event generator decides whether periodically sampled subsystem data are transmitted, avoiding state monitoring at every discrete time.The sampling interval can exceed one system step, with lower and upper bounds imposed on the interval.
  • A. Distributed dynamic ETS: The triggering condition uses local sampling errors, neighboring relative-state information, designed coefficients, and a dynamic variable.Neighbor information can enlarge triggering intervals relative to schemes that ignore relative states.
  • A. Distributed dynamic ETS: The dynamic variable evolves according to a discrete-time difference equation and avoids the continuous computation required by continuous-time dynamic triggers.Prior results cited by the paper indicate that the dynamic scheme can reduce triggering frequency relative to a static strategy.
  • A. Distributed dynamic ETS: A simulation example reports fewer transmissions than the decentralized ETS while maintaining similar performance.The comparison is made using the proposed ETS with the neighbor-relative-state contribution removed.
  • A. Distributed dynamic ETS: The proposed ETS generalizes several dynamic, static decentralized, and classic distributed triggering schemes as special parameter cases.These include discrete-time dynamic, static decentralized periodic, static decentralized, and classic distributed ETSs.

B. Model-based stability analysis

The model-based analysis establishes stability of the event-triggered network system through a discrete-time looped functional and vertex-checkable LMIs. The resulting criterion also ensures convergence of the auxiliary triggering variables.

  • B. Model-based stability analysis: Under the stated parameter conditions, the system state and each auxiliary trigger variable converge to the origin, establishing asymptotic stability.The result applies for nonnegative initial trigger-variable values and admissible sampling intervals.
  • B. Model-based stability analysis: The stability criterion is expressed through LMIs involving positive-definite matrices and must be checked at the lower and upper sampling-interval vertices.Affineness in the sampling interval makes endpoint checks sufficient over the prescribed interval range.
  • B. Model-based stability analysis: The discrete-time looped functional relaxes the requirement that a common Lyapunov functional decrease at every discrete time.It instead enforces descent at designed sampling points, which can yield less conservative conditions.
  • B. Model-based stability analysis: The looped-functional construction also produces sampling-dependent conditions that can be used to search for allowable sampling intervals.This sampling dependence is incorporated into the LMI conditions.

C. Model-based controller design

The model-based controller-design result removes the direct gain coupling in the analysis by applying a nonsingular coordinate transformation. It then formulates controller and triggering-matrix co-design through LMIs while preserving system stability.

  • C. Model-based controller design: A nonsingular transformation x(t)=Gz(t) decouples the controller-gain design from the free matrix coupling present in the stability-analysis LMIs.The transformed system has the same stability behavior as the original system.
  • C. Model-based controller design: Theorem 2 provides an LMI-based condition for jointly designing a block controller gain and triggering matrices while achieving asymptotic stability.The decision variables include transformed controller quantities, triggering matrices, Lyapunov variables, and auxiliary matrices.
  • C. Model-based controller design: The original controller and triggering matrices are recovered from the transformed design variables using G and its inverse.The paper gives K=KcG^-1 and the corresponding transformation for the triggering matrix.
  • C. Model-based controller design: The transformed looped-functional proof shows that the resulting LMIs ensure asymptotic stability of the transformed and original systems.The equivalence follows because the coordinate transformation is algebraically invertible.

D. Data-driven controller design

The paper develops a data-driven LMI criterion for co-designing controller and distributed triggering matrices for unknown network systems. The criterion guarantees asymptotic stability for all system matrices consistent with the collected data, while its conservatism and implementation scope are explicitly characterized.

  • D. Data-driven controller design: Theorem 3 co-designs the block controller gain and triggering matrices through LMIs using a data-driven representation of the unknown system.The design variables include positive-definite matrices, auxiliary matrices, and a scalar ε satisfying endpoint LMIs in the sampling interval.
  • D. Data-driven controller design: The resulting condition guarantees asymptotic stability for every system [A B] in the data-consistent set ΣAB, with the triggering error converging to the origin.The guarantee holds under the stated parameter conditions and for admissible initial triggering-error values.
  • D. Data-driven controller design: The design recovers the controller gain and triggering matrices from auxiliary decision variables as K = KcG−1 and Ωa = G−1⊤Ωz.These transformations connect the LMI variables to the desired feedback and triggering parameters.
  • D. Data-driven controller design: The data-driven condition is more conservative than the model-based condition because it must certify all systems in ΣAB rather than one known system.The paper relates this conservatism to less room for optimizing transmission frequency and system performance, and compares the methods numerically in Section IV.
  • D. Data-driven controller design: The LMI dimensions are independent of the amount of collected state-input data, although increasing data length can improve system-description accuracy while increasing computational complexity.The method is described as scalable to any data length, but large data sets can challenge large-scale network systems.
  • D. Data-driven controller design: The control and triggering strategy uses local sampled-state information, but the data-driven protocols still require global network-graph information when constructing the system representation.Thus, the distributed transmission and control structure does not make the complete data-driven design fully distributed.

IV. EXAMPLES AND SIMULATION

The simulation study uses a network of three identical coupled inverted pendulums to evaluate the proposed data-driven and model-based distributed event-triggered control methods. The pendulum network is discretized into a linear discrete-time system and controlled with distributed state feedback.

  • IV. EXAMPLES AND SIMULATION: The example consists of three identical coupled inverted pendulums whose communication graph links neighboring subsystems.The pendulum parameters include g = 10m/s2, m = 1kg, d = 2m, and f = 5N/m.
  • IV. EXAMPLES AND SIMULATION: After discretization, the pendulum network is represented as a discrete-time linear system with subsystem states containing angular position and angular velocity.The discretization interval is denoted by T_k, with A(T_k) and B(T_k) obtained from the continuous-time model.
  • IV. EXAMPLES AND SIMULATION: A distributed linear state-feedback controller is applied to the discretized network, and the proposed data-driven event-triggering strategy is tested on the resulting system.The numerical computations use Matlab and the SeDuMi toolbox.

A. Testing data-based method

The data-based simulation evaluates controller and triggering matrices designed from offline measurements on the coupled-pendulum network. The trajectories converge while event-triggered transmission substantially reduces communication relative to periodic sampling and alternative triggering settings.

  • A. Testing data-based method: The data-driven controller design assumes unknown A and B, uses T_k = 0.01, and collects ρ = 200 state-input measurements before solving the data-based LMIs.The simulation also specifies sampling, disturbance, and triggering parameters for the numerical test.
  • A. Testing data-based method: 34, 44, and 31 measurements were transmitted for subsystems 1, 2, and 3, respectively, out of 150 sampled data per subsystem, while states and dynamic variables converged to zero.These results support both the stability of the data-driven scheme and its communication-saving behavior.
  • A. Testing data-based method: As θ_i increased from 2 to 10^4, transmissions increased from 109 to a fixed value of 154.The paper explains that the triggering scheme approaches the static decentralized ETS as θ_i tends toward infinity.

B. Testing the model-based method

The model-based simulation applies the same triggering setup while assuming the system matrices are known. It achieves convergence with fewer transmissions than the data-driven design, but with a substantially longer settling time.

  • B. Testing the model-based method: Both the subsystem trajectories and dynamic variables converge to zero under the model-based event-triggered controller.The result is reported for the same initial states used in the data-driven simulation.
  • B. Testing the model-based method: 42 sampled data were transmitted under the model-based method, compared with 109 under the data-driven method using the same triggering parameters.The model-based transmission count was 13, 13, and 16 for subsystems 1, 2, and 3.

C. Comparing with centralized and distributed ETSs

The data-driven distributed ETS is compared with decentralized triggering schemes using co-designed controllers and triggering matrices. It reduces transmissions while preserving comparable steady-state behavior.

  • C. Comparing with centralized and distributed ETSs: The distributed ETS incorporates neighbors’ information through σi_1 = 0, whereas the decentralized comparison uses σij_2 = 0.The two triggering configurations were evaluated using the same collected state-input data and triggering parameters apart from the specified coefficients.
  • C. Comparing with centralized and distributed ETSs: The compared trajectories and triggered events were evaluated under the data-driven ETS and alternative periodic or decentralized transmission schemes.Figures 6–8 provide the trajectory comparisons for the triggering and controller configurations.
  • C. Comparing with centralized and distributed ETSs: 109 transmissions were required by the data-driven ETS, versus 148 and 130 for the compared triggering schemes, with settling time around t = 150.The schemes maintained the same steady-state level while differing in communication frequency.

D. Comparison between centralized and distributed controllers

The study compares decentralized and distributed controllers under the same periodic transmission scheme. The distributed controller achieves faster convergence.

  • D. Comparison between centralized and distributed controllers: The decentralized and distributed controller gains were designed from the same state-input data using Theorem 3.The decentralized case sets Kij = 0, while the distributed controller uses neighbor information.
  • D. Comparison between centralized and distributed controllers: The distributed controller reached settling time around t = 50, compared with t = 150 for the decentralized controller.Both controllers were evaluated with the same initial points and periodic transmission scheme.

V. CONCLUDING REMARKS

The paper develops a data-driven co-design approach for distributed event-triggered control of unknown interconnected discrete-time systems. Numerical comparisons support reduced data transmissions while maintaining desired system performance.

  • V. CONCLUDING REMARKS: The proposed approach co-designs the triggering matrix and controller gain using a data-based system representation and analyzes closed-loop stability with a looped-functional.The stability and co-design procedures target unknown interconnected discrete-time systems.
  • V. CONCLUDING REMARKS: Numerical comparisons support the proposed data-driven distributed strategy’s ability to reduce data transmissions while maintaining desired system performance.The conclusion reports this outcome relative to existing decentralized and distributed strategies.
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