Adaptive Event-triggered Control with Sampled Transmitted Output and Controller Dynamics

The event-triggered control with intermittent output can reduce the communication burden between the controller and plant side over the network. It has been exploited for adaptive output feedback control of uncertain nonlinear systems in the literature, however the controller must partially reside at the plant side where the computation capacity is required. In this paper, all controller components are moved to the controller side and their dynamics use sampled states rather than continuous one with the benefit of directly estimating next triggering instance of some conditions and avoiding constantly checking event condition at the controller side. However, these bring two major challenges. First, the virtual input designed in the dynamic filtering technique for the stabilization is no longer differentiable. Second, the plant output is sampled to transmit at plant side and sampled again at controller side to construct the controller, and the two asynchronous samplings make the analysis more involving. This paper solves these two issues by introducing a new state observer to simplify the adaptive law, a set of continuous companion variables for stability analysis and a new lemma quantifying the error bound between actual output signal and sampled transmitted output. It is theoretically guaranteed that all internal signals in the closed-loop system are semiglobally bounded and the output is practically stabilized to the origin. Finally, the numerical simulation illustrates the effectiveness of proposed scheme.

💡 Research Summary

The paper addresses the problem of reducing communication and computational load in networked control systems by moving the entire controller—including observer, adaptation law, and dynamic filter—to the remote side, while only transmitting sampled output measurements from the plant. Traditional event‑triggered adaptive output‑feedback schemes either require continuous monitoring of plant states or place part of the controller (the dynamic filter) on the plant, demanding computational resources at the plant side. The authors propose a novel architecture with two event detectors: (i) a plant‑side detector (ED1) that triggers transmission of the plant output whenever the deviation from the last transmitted value exceeds a threshold γ_y, and (ii) a controller‑side detector (ED2) that triggers an update of the control law when any of several sampled quantities—transmitted output, observer states ξ and ζ, parameter estimate (\hat\theta), or filter state α_f—change beyond their respective thresholds.

A key technical contribution is the handling of the non‑differentiable “virtual input” that arises when the dynamic filter is driven by sampled signals. To overcome this, the plant state is decomposed as (x = \xi + \theta \zeta). The observer dynamics for ξ and ζ are linear with a Hurwitz matrix (A_c) and are updated only at sampling instants, which makes their evolution piecewise linear and allows the next triggering instant to be calculated analytically (e.g., (t_{\zeta,j+1}=t_j + \gamma_\zeta / |A_c\zeta(t_j)+\psi(\bar y(t_j))|)). This eliminates the need for continuous condition checking at the controller.

The adaptive law is simplified: parameter estimation appears only in the first backstepping step, reducing the complexity of the adaptation dynamics. Continuous companion variables are introduced to facilitate the Lyapunov analysis despite the discontinuities introduced by sampling. Lemma 3.1 establishes that the error between the true output y(t) and the sampled transmitted output (\bar y(t_j)) is bounded by (\tilde\gamma_y = \gamma_y + \gamma_{\bar y}). Using this bound, the authors construct Lyapunov functions for the estimation error ε and for the ζ‑subsystem, obtaining differential inequalities that guarantee that all internal signals remain semiglobally bounded. Consequently, the output y is practically stabilized: (\limsup_{t\to\infty} |y(t)| \le \varepsilon) for a design‑chosen ε.

Simulation on a third‑order chain‑integrator with unknown parameter θ and nonlinear functions ψ_i demonstrates a substantial reduction in transmission events compared with periodic sampling, while the output converges to a small neighborhood of the origin (≈0.05). The plant side performs only a simple zero‑order hold and output sampling, confirming the removal of computational burden.

Overall, the paper contributes a fully remote adaptive event‑triggered output‑feedback scheme that avoids continuous monitoring, reduces network traffic, and provides a rigorous semiglobal practical stability guarantee. Limitations include the practical‑rather‑than asymptotic convergence, the need to tune multiple thresholds, and the reliance on bounded network delays (not explicitly modeled). Future work could extend the approach to multi‑parameter uncertainties, develop automatic threshold adaptation, and incorporate robustness against packet loss and time‑varying delays.