Adaptive Event-Triggered MPC

• Adaptive ETMPC is a control framework that integrates event-triggered mechanisms with model predictive control, enabling online updates to trigger thresholds and model parameters.
• It employs diverse adaptive mechanisms—including dynamic trigger matrices, learned binary policies, and model re-identification—to balance performance, computational load, and communication efficiency.
• Practical implementations in LPV systems, autonomous vehicles, and building thermal control demonstrate reduced communication and tailored control updates, albeit with trade-offs like slower convergence.

Adaptive Event-Triggered Model Predictive Control (ETMPC) denotes a family of MPC schemes in which receding-horizon control is coupled to an event mechanism and at least one element of the closed-loop design is updated online or across repeated executions. In the literature, this adaptation is realized in several non-equivalent ways: dynamic trigger thresholds and trigger-controller co-design for LPV systems with delays, actuator saturation, and disturbances (Zhong et al., 9 Sep 2025); learned binary trigger policies obtained by deep reinforcement learning (Dang et al., 2022); event-triggered re-identification of thermal-envelope models inside economic MPC (Wan et al., 2023); Gaussian-process-uncertainty-dependent trigger thresholds for unknown nonlinear systems under terminal constraints (Onoue et al., 2021); and supervisory performance-monitoring-triggered MPC retuning and re-identification (Mallick et al., 12 Mar 2026). At the same time, several influential ETMPC implementations are explicitly non-adaptive, which is essential for delimiting the subject (Berner et al., 2019).

1. Conceptual scope and boundaries

Adaptive ETMPC is not a single formulation. The most direct interpretation is an MPC law whose event condition contains internal adaptive variables or online-optimized triggering matrices, as in the LPV co-design framework where the event-triggering matrix $\Phi_k$ and control gains $F_k$ are jointly optimized and the internal variable $\beta_k$ evolves dynamically (Zhong et al., 9 Sep 2025). A broader interpretation includes schemes in which the event does not directly change the control-update law, but instead triggers model learning, model replacement, or controller redesign when prediction quality or closed-loop performance deteriorates (Wan et al., 2023).

A recurring misconception is that every event-triggered MPC method is adaptive. The literature gives several counterexamples. In the industrial-hardware implementation of networked MPC, the trigger is the fixed polytope-membership rule $e(x)=1 \iff x\notin P^\star$, and the paper explicitly states that it does not propose an adaptive event-triggering mechanism (Berner et al., 2019). The decentralized robust MPC based on hyper-rectangles around the optimal predicted trajectory recomputes state-dependent trigger sets online, but it does not adapt the model, disturbance description, or trigger template in the usual sense (Kolarijani et al., 2018). The integral-type ETMPC for disturbed nonlinear systems uses a fixed triggering level $\delta$ and fixed design parameters $\beta,T,M,\alpha,\epsilon$, so it is robust and event-triggered rather than adaptive (Sun et al., 2018). Likewise, the shared-server robust ETMPC under hard computation constraints is dynamic and resource-aware, but its priority law $g_i(t)=\|x_i(t)-\hat x_i(1|t-1)\|$ is fixed in form and is described more accurately as state-dependent priority scheduling than as adaptive threshold design (Gräfe et al., 28 Apr 2025).

This boundary matters because the technical questions differ. In non-adaptive ETMPC, the central problem is when a fixed control law may be safely reused. In adaptive ETMPC, the event mechanism also determines when the trigger geometry, the prediction model, or the controller parameterization itself should change. A plausible implication is that “adaptive ETMPC” is best treated as a superfamily whose members are distinguished by what adapts: trigger thresholds, trigger policies, prediction models, controller parameters, or computational-resource allocation.

2. Main adaptive mechanisms

The adaptive mechanisms reported in the literature can be organized by the object that changes at or between events.

Adaptive object	Event criterion	Representative papers
Trigger threshold and trigger matrix	$$\varepsilon(\|e_k\|_{\Phi_{k_t}}^2-\theta\|x_k\|_{\Phi_{k_t}}^2)>\beta_k$$, with dynamic $\beta_k$	(Zhong et al., 9 Sep 2025)
Trigger policy	$a_t\sim\pi_\theta(X_{t_p},\hat x_t)$ with reward $F_k$0	(Dang et al., 2022)
Prediction model parameters	RMSE- or uncertainty-driven re-identification/update of $F_k$1 or GP model	(Wan et al., 2023, Onoue et al., 2021)
Supervisory controller/model adaptation	Mahalanobis-distance trigger $F_k$2	(Mallick et al., 12 Mar 2026)

In the direct adaptive-trigger formulation for LPV systems, the event error is $F_k$3, the next event is the smallest $F_k$4 satisfying $F_k$5, and the internal variable evolves according to $F_k$6, with $F_k$7, $F_k$8, and $F_k$9 (Zhong et al., 9 Sep 2025). Adaptation is therefore twofold: the threshold is dynamic through $\beta_k$0, and the weighting matrix $\beta_k$1 is itself recovered from the online optimization.

In the deep-RL formulation, the adaptive object is not a threshold but a binary trigger policy. The agent observes the current state estimate $\beta_k$2 and previous MPC prediction $\beta_k$3, chooses $\beta_k$4, and learns from the immediate reward $\beta_k$5 (Dang et al., 2022). This is adaptive ETMPC in the sense of state-dependent policy learning rather than analytical trigger synthesis.

In the building-control formulation, the event is attached to model adaptation rather than to the control-update law. The controller keeps solving economic MPC in ordinary receding-horizon fashion, while system identification is triggered only when the recent prediction RMSE over a trigger horizon $\beta_k$6 exceeds a threshold $\beta_k$7; the new parameter estimate then replaces the old one through a binary update law (Wan et al., 2023). The GP-based formulation for unknown nonlinear systems has a similar separation: MPC is nominal with respect to the GP posterior mean, but the event thresholds $\beta_k$8 depend explicitly on GP uncertainty $\beta_k$9, so iterative learning enlarges the admissible inter-event deviations (Onoue et al., 2021). The process-control monitoring framework generalizes this further: the event is not state mismatch but loss of statistical similarity between current closed-loop KPI features and a baseline acceptable dataset, measured by a Mahalanobis statistic $e(x)=1 \iff x\notin P^\star$0 (Mallick et al., 12 Mar 2026).

3. Direct adaptive ETMPC through robust co-design

The most explicit co-design treatment appears in the LPV framework for discrete-time systems with state delays, actuator saturation, and bounded disturbances. The plant is modeled as

$e(x)=1 \iff x\notin P^\star$1

with polytopic parameter dependence, Euclidean-ball disturbance set $e(x)=1 \iff x\notin P^\star$2, and componentwise saturation $e(x)=1 \iff x\notin P^\star$3 (Zhong et al., 9 Sep 2025). The method uses zero-order hold between event times and treats the event-triggering mechanism, saturation handling, and robust MPC design in a unified LMI program.

A distinctive feature is the Lyapunov-Krasovskii-like function

$e(x)=1 \iff x\notin P^\star$4

which embeds the trigger’s internal adaptive variable $e(x)=1 \iff x\notin P^\star$5 directly into the stability and performance analysis (Zhong et al., 9 Sep 2025). Because the $e(x)=1 \iff x\notin P^\star$6-update contributes the term $e(x)=1 \iff x\notin P^\star$7, the event-triggering matrix $e(x)=1 \iff x\notin P^\star$8 becomes a decision variable inside the same dissipation inequalities used to upper-bound the infinite-horizon robust MPC cost. This is the paper’s central co-design mechanism.

The robust MPC objective is the infinite-horizon min-max cost

$e(x)=1 \iff x\notin P^\star$9

which is replaced by minimizing an upper bound $\delta$0 subject to LMIs ensuring dissipation, initial boundedness, saturation admissibility, and invariance (Zhong et al., 9 Sep 2025). Variable substitutions recover the controller gain $\delta$1, the auxiliary saturation matrix $\delta$2, the trigger matrix $\delta$3, and Lyapunov matrices $\delta$4 from the convex decision variables. Saturation is handled through a convex-hull representation $\delta$5, which permits vertex-wise LMI enforcement over LPV uncertainty and saturation modes.

The resulting guarantees are unusually strong for an adaptive ETMPC construction. The feasible LMI solution yields recursive feasibility through invariance of

$\delta$6

and mean-square input-to-state stability under LPV uncertainty, delay, disturbance, saturation, and event-triggered sample-and-hold error (Zhong et al., 9 Sep 2025). In the five-zone industrial electric heating example, adaptive ETMPC achieved a triggering ratio of $\delta$7 versus $\delta$8 for the static ETMPC comparator, with average inter-event interval $\delta$9 steps versus $\beta,T,M,\alpha,\epsilon$0 steps, while the steady-state criterion $\beta,T,M,\alpha,\epsilon$1 increased from $\beta,T,M,\alpha,\epsilon$2 s to $\beta,T,M,\alpha,\epsilon$3 s (Zhong et al., 9 Sep 2025). The reported trade-off is therefore explicit: communication is reduced, but convergence is slower.

4. Learning-, identification-, and monitoring-driven adaptive ETMPC

A second major line of work replaces analytical trigger design with learned or model-quality-driven event decisions. In autonomous driving, the nonlinear MPC controller remains the low-level optimizer, but the event-trigger is a learned policy $\beta,T,M,\alpha,\epsilon$4 deciding whether to solve MPC or to continue executing the buffered optimal sequence computed at the last event (Dang et al., 2022). The trigger state is $\beta,T,M,\alpha,\epsilon$5, the action space is $\beta,T,M,\alpha,\epsilon$6, and the reward $\beta,T,M,\alpha,\epsilon$7 encodes the performance-versus-triggering trade-off. The paper instantiates this with DDQN, PPO, and SAC, augmented by PER and LSTM, and reports that all deep-RL-based eMPC variants outperform both threshold-based eMPC and the previous linear-Q LSTDQ baseline. For $\beta,T,M,\alpha,\epsilon$8, PPO and PPO+LSTM achieved $\beta,T,M,\alpha,\epsilon$9 with $g_i(t)=\|x_i(t)-\hat x_i(1|t-1)\|$0, whereas DDQN+LSTM+PER achieved $g_i(t)=\|x_i(t)-\hat x_i(1|t-1)\|$1 with $g_i(t)=\|x_i(t)-\hat x_i(1|t-1)\|$2; for $g_i(t)=\|x_i(t)-\hat x_i(1|t-1)\|$3, DDQN+LSTM+PER reduced trigger frequency to $g_i(t)=\|x_i(t)-\hat x_i(1|t-1)\|$4 with $g_i(t)=\|x_i(t)-\hat x_i(1|t-1)\|$5 (Dang et al., 2022). The same paper reports average MPC solve time about $g_i(t)=\|x_i(t)-\hat x_i(1|t-1)\|$6 s and average RL policy evaluation time about $g_i(t)=\|x_i(t)-\hat x_i(1|t-1)\|$7 s, making trigger inference negligible relative to optimization cost.

In building thermal control, the adaptive mechanism is parameter-level model refresh inside economic MPC rather than event-triggered control execution. The prediction model is a 1R1C thermal envelope model with parameters $g_i(t)=\|x_i(t)-\hat x_i(1|t-1)\|$8, updated by event-triggered system identification when

$g_i(t)=\|x_i(t)-\hat x_i(1|t-1)\|$9

with $$\varepsilon(\|e_k\|_{\Phi_{k_t}}^2-\theta\|x_k\|_{\Phi_{k_t}}^2)>\beta_k$$0, $$\varepsilon(\|e_k\|_{\Phi_{k_t}}^2-\theta\|x_k\|_{\Phi_{k_t}}^2)>\beta_k$$1 day, and $$\varepsilon(\|e_k\|_{\Phi_{k_t}}^2-\theta\|x_k\|_{\Phi_{k_t}}^2)>\beta_k$$2 days (Wan et al., 2023). The new estimate enters through

$$\varepsilon(\|e_k\|_{\Phi_{k_t}}^2-\theta\|x_k\|_{\Phi_{k_t}}^2)>\beta_k$$3

The framework adds a knowledge-graph layer using BOT, PROPS, SSN/SOSA, Brick, FSO, and SEAS ontologies to automate model setup, data retrieval, and controller configuration. In the reported one-zone office-building study, the event-triggered SI scheme produced a 49% computation saving relative to always-identify operation and the adaptive MPC achieved 12% operating cost reduction versus the rule-based baseline (Wan et al., 2023).

For unknown nonlinear systems, the GP-based ETMPC learns each dynamics component $$\varepsilon(\|e_k\|_{\Phi_{k_t}}^2-\theta\|x_k\|_{\Phi_{k_t}}^2)>\beta_k$$4 using GP regression with deterministic RKHS-based confidence bounds. The MPC uses the GP posterior mean as nominal predictor, terminal constraints are built through symbolic abstraction and a safety game, and the event times are determined by the smallest $$\varepsilon(\|e_k\|_{\Phi_{k_t}}^2-\theta\|x_k\|_{\Phi_{k_t}}^2)>\beta_k$$5 such that the kernel-metric deviation $$\varepsilon(\|e_k\|_{\Phi_{k_t}}^2-\theta\|x_k\|_{\Phi_{k_t}}^2)>\beta_k$$6 exceeds a threshold $$\varepsilon(\|e_k\|_{\Phi_{k_t}}^2-\theta\|x_k\|_{\Phi_{k_t}}^2)>\beta_k$$7 derived from GP uncertainty $$\varepsilon(\|e_k\|_{\Phi_{k_t}}^2-\theta\|x_k\|_{\Phi_{k_t}}^2)>\beta_k$$8 (Onoue et al., 2021). Because $$\varepsilon(\|e_k\|_{\Phi_{k_t}}^2-\theta\|x_k\|_{\Phi_{k_t}}^2)>\beta_k$$9, uncertainty reduction through iterative retraining enlarges the admissible inter-event deviation. In the leader-follower mobile-robot example, the number of triggers over 40-step runs decreased from $\beta_k$0 at iteration 1 to $\beta_k$1 at iteration 5 and $\beta_k$2 at iteration 10; total computation time for OCP plus threshold design decreased from $\beta_k$3 ms to $\beta_k$4 ms, while periodic MPC with the iteration-10 model required $\beta_k$5 ms (Onoue et al., 2021).

A third adaptive pattern is supervisory monitoring-triggered MPC redesign. In process control, closed-loop acceptability is represented by a baseline dataset $\beta_k$6 of feature vectors $\beta_k$7, and the event condition is the Mahalanobis distance

$\beta_k$8

with $\beta_k$9 chosen as a $a_t\sim\pi_\theta(X_{t_p},\hat x_t)$0 confidence bound heuristic (Mallick et al., 12 Mar 2026). The adaptation acts first on controller parameters via MPC-based Q-learning,

$a_t\sim\pi_\theta(X_{t_p},\hat x_t)$1

and, if acceptable performance is not restored, on the prediction model via system identification, followed by resetting $a_t\sim\pi_\theta(X_{t_p},\hat x_t)$2 (Mallick et al., 12 Mar 2026). In the district heating benchmark, the monitoring window was 12 hours, and the reported implementation required three days of persistent threshold violation before adaptation was triggered. This is adaptive ETMPC in a supervisory sense: the event does not skip MPC solves, but it controls when the MPC parameterization is modified.

5. Guarantees, feasibility, and relation to robust ETMPC foundations

The strongest theoretical guarantees in the adaptive literature surveyed here appear in the LMI-based LPV framework and the GP-based learning framework. The LPV method proves recursive feasibility via invariant-set propagation and establishes mean-square ISS for the augmented delayed state under multiple uncertainties (Zhong et al., 9 Sep 2025). The GP-based method proves recursive feasibility of the event-triggered OCP at successive event times and shows finite-time convergence to the terminal set $a_t\sim\pi_\theta(X_{t_p},\hat x_t)$3 within at most $a_t\sim\pi_\theta(X_{t_p},\hat x_t)$4 steps when the model uncertainty is sufficiently small to satisfy the one-step threshold condition $a_t\sim\pi_\theta(X_{t_p},\hat x_t)$5 (Onoue et al., 2021).

By contrast, some practically successful adaptive or adaptive-like schemes are explicitly not accompanied by classical ETMPC guarantees. The deep-RL trigger-learning paper states that it provides no formal theorem proving closed-loop stability, recursive feasibility, convergence, safety, or performance bounds, and identifies stability and convergence of the deep-RL-eMPC framework as future work (Dang et al., 2022). The semantic-assisted building framework states that it does not provide explicit proofs of closed-loop stability, recursive feasibility, boundedness, or convergence of the parameter estimates, and that the trigger threshold $a_t\sim\pi_\theta(X_{t_p},\hat x_t)$6 is empirical (Wan et al., 2023). The monitoring-triggered adaptation architecture is likewise described as comparatively light on formal closed-loop ETMPC theory; its main formal object is the acceptability set $a_t\sim\pi_\theta(X_{t_p},\hat x_t)$7, not a recursive-feasibility or Lyapunov theorem (Mallick et al., 12 Mar 2026).

Non-adaptive robust ETMPC remains a key theoretical foundation for the adaptive variants. The decentralized robust MPC based on hyper-rectangles proves recursive feasibility, hard constraint satisfaction, and asymptotic convergence of states and inputs to target sets, while enabling local trigger evaluation at individual sensors (Kolarijani et al., 2018). The integral-type ETMPC for continuous-time nonlinear systems proves a lower bound $a_t\sim\pi_\theta(X_{t_p},\hat x_t)$8, an upper bound $a_t\sim\pi_\theta(X_{t_p},\hat x_t)$9, recursive feasibility under explicit conditions involving $F_k$00, and practical stability to a disturbance-dependent neighborhood $F_k$01 (Sun et al., 2018). The shared-server robust ETMPC under hard computational limits proves recursive feasibility and generalized ISS for multiple nonlinear systems while capping the number of simultaneous optimizations at $F_k$02 (Gräfe et al., 28 Apr 2025). These results are non-adaptive, but they supply the feasibility, tightening, and ISS machinery on which adaptive extensions can plausibly build.

6. Architectures, computation, applications, and recurring limitations

Implementation architecture is central because many adaptive ETMPC benefits are meaningful only if the per-event computational and communication burdens are controlled. In the industrial-hardware study of networked MPC, a star-shaped network with a central industrial PC and local PLC nodes stores the current affine law and validity polytope locally, while new laws are computed centrally only when the state leaves the current polytope (Berner et al., 2019). Four implementation variants are compared, from transmitting only the active set $F_k$03 to transmitting the full affine law and polytope. The preferred architecture is A1, which transmits only the active set; despite requiring local reconstruction through inversion of $F_k$04, the paper shows for box constraints that the inversion cost satisfies

$F_k$05

and concludes that A1 offers the best communication/computation trade-off on industrial hardware (Berner et al., 2019). This is not adaptive ETMPC by itself, but it is directly relevant whenever adaptation changes event frequency or payload.

A related systems-level issue is hard computational scarcity. In the resource-constrained robust ETMPC for multiple nonlinear systems, only a fixed-size subset of plants may be optimized at each time step, $F_k$06, and priorities are assigned by $F_k$07 (Gräfe et al., 28 Apr 2025). The method is explicitly described as not adaptive ETMPC in the common sense of threshold or parameter adaptation, yet it contributes a resource-aware event-scheduling perspective that becomes important in multi-plant adaptive deployments.

The application range of adaptive ETMPC in the surveyed literature is broad but technically heterogeneous. Direct adaptive co-design has been demonstrated on a five-zone industrial electric heating system (Zhong et al., 9 Sep 2025). Learned trigger policies have been tested on nonlinear autonomous vehicle path following along the reference $F_k$08 (Dang et al., 2022). Event-triggered model re-identification has been studied for a one-zone office building with VAV and radiator heating under occupancy-dependent comfort constraints (Wan et al., 2023). Learning-based uncertainty-adaptive ETMPC has been validated on nonholonomic mobile-robot tracking (Onoue et al., 2021). Monitoring-triggered adaptation has been demonstrated on a high-fidelity district heating benchmark (Mallick et al., 12 Mar 2026). This breadth suggests that “adaptive ETMPC” is better understood as an architectural principle than as a single canonical control law.

The limitations are correspondingly diverse. The direct LPV co-design assumes full state measurement, known fixed delays, and online LMI solving at trigger times, with computational growth from the $F_k$09 saturation vertices (Zhong et al., 9 Sep 2025). The deep-RL trigger framework lacks formal stability and feasibility guarantees and relies on extensive simulation training (Dang et al., 2022). The building framework uses an empirical trigger threshold, lacks anti-chattering results, and remains only semi-automated because some graph construction is manual (Wan et al., 2023). The GP framework relies on deterministic RKHS assumptions, suffers from GP and symbolic-abstraction scalability limits, and is formulated around iterative repeated tasks rather than purely one-shot adaptation (Onoue et al., 2021). The monitoring-based architecture does not provide formal recursive-feasibility or stability guarantees under adaptation and is event-triggered only at the supervisory redesign layer, not at the MPC execution layer (Mallick et al., 12 Mar 2026).

A final misconception is that adaptation in ETMPC must always mean adaptive control retransmission or adaptive skipping of online optimization. The literature surveyed here shows a wider picture. In some formulations, the event adapts the trigger itself; in others, it adapts the prediction model, the controller tuning, or the computational-resource allocation. The unifying feature is not the specific mechanism but the conditional reconfiguration of MPC based on state mismatch, model mismatch, uncertainty, or statistically detected loss of closed-loop acceptability.