Self-Triggered Control for Linear Systems

• Self-triggered control is a resource-aware paradigm that computes future update instants from the current state to reduce unnecessary sensing, communication, and actuation.
• It uses methods such as Lyapunov-based analysis and model predictive control to ensure exponential stability, constraint satisfaction, and optimized performance.
• Implementation spans centralized, distributed, and data-driven approaches, offering robust guarantees even under quantization and asynchronous measurements.

Self-triggered control for linear systems is a resource-aware control paradigm that replaces periodic or continuously-monitored event-triggered updates with explicit prediction of future update instants. At each control action, the next transmission time (or sequence of times) is computed a priori as a function of the current state, control, and system model. This design paradigm substantially reduces communication, actuation, or sensing frequency in embedded, networked, and distributed control scenarios, while retaining rigorous guarantees on stability, constraint satisfaction, and, when relevant, performance or robustness. The development of self-triggered control for linear systems encompasses both centralized and distributed architectures, continuous- and discrete-time plants, quantized and asynchronous measurements, and extends to infinite-dimensional and networked multi-agent contexts.

1. Core Principles and System Architectures

In self-triggered control of linear systems, the feedback law—usually state or output feedback with zero-order hold—is implemented at non-uniformly spaced instants $\{t_k\}$. The controller, upon acting at time $t_k$, predicts (using a model of the closed-loop and, possibly, state estimation uncertainty) the earliest future instant $t_{k+1}>t_k$ at which the system will require another control update or measurement so as to ensure a prescribed stability, performance, or constraint property.

A representative setup (Tariverdi, 20 Nov 2025):

• Plant: $\dot{x}(t) = Ax(t) + Bu(t)$, $(A,B)$ stabilizable, with or without bounded disturbances.
• Controller: $u(t) = K\hat{x}(t_k)$, $t \in [t_k, t_{k+1})$, where $\hat{x}(t_k)$ is a current, possibly partial, state estimate.
• Measurement: May be synchronous (all sensors at once) or asynchronous (one sensor at a time) (Tariverdi, 20 Nov 2025).
• Triggering law: At time $t_k$, compute the next transmission time $t_{k+1}$ (or a sequence) from triggering conditions that ensure the closed-loop satisfies contraction (e.g., Lyapunov decrease) or other desired properties.

This design contrasts with event-triggered control, which requires continuous or periodic monitoring to detect trigger events, and periodic control, which acts with fixed period regardless of state evolution.

2. Design Methodologies and Triggering Laws

Two dominant approaches structure most contemporary self-triggered designs for linear systems:

a. Lyapunov-Based and Finite-Horizon Optimization

For continuous-state, model-known systems, a quadratic Lyapunov function $V(x) = x^T P x$ is constructed, with $P \succ 0$ satisfying the Lyapunov or Riccati equation for some stabilizing $K$. Self-triggered updates ensure $V(x_{k+1}) \leq e^{-\beta (t_{k+1}-t_k)} V(x_k)$ by enforcing a contraction condition across the entire predicted interval, even accounting for measurement or control errors due to zero-order hold, quantization, or partial observations (Tariverdi, 20 Nov 2025, Tariverdi, 2 Nov 2025, Zobiri et al., 2019, Anta et al., 2010, Wakaiki, 2021).

In more advanced schemes, notably with asynchronous measurements or for enhanced resource optimization, the controller may optimize over sequences of actions. For example, a finite-horizon combinatorial problem is solved to select the sensor reading and holding intervals so as to maximize the average inter-sample interval subject to a Lyapunov decrease constraint (Tariverdi, 20 Nov 2025):

• Variables: Sampling sequence $\sigma = (S^1, ..., S^N)$, $S^j$ is which sensor (or none) to read.
• Objective: Maximize a function of zeros (no sensor read) and interval length.
• Constraint: State contraction over the sequence, $$x_k^T(\Phi_\sigma^T P \Phi_\sigma - e^{-\beta N T} P)x_k \leq 0$$.

b. Model Predictive and Data-Driven Self-Triggering

Self-triggered model predictive control (MPC), as in (Hashimoto et al., 2016), computes (at each update) the optimal control and the largest admissible next interval by solving a family of MPC problems with different discretization/sampling patterns. The longest pattern satisfying value function decay and performance upper bound is selected, offering a formally-guaranteed trade-off between communication and control performance.

Data-driven variants have emerged for systems with unknown $(A,B)$, using offline-collected trajectories and input-state Hankel matrices to predict future evolution via fundamental lemma-based fitting, and to synthesize self-triggering rules by propagation of predicted estimation error and constraint satisfaction (Liu et al., 2022, Wang et al., 2022, Li et al., 2023). These allow model-free operation, provided the offline excitation data is sufficiently rich.

An example structure (Wang et al., 2022, Li et al., 2023):

• Lifting the predicted closed-loop evolution over $s$ steps,
• Checking for the largest $s$ such that robust quadratic matrix inequality constraints (derived from looped or Lyapunov-functional analysis) are satisfied for all admissible system matrices.

3. Implementation Techniques and Offline-Online Trade-offs

Significant work addresses the real-time computational burden of self-triggered laws, especially for high-dimensional and networked systems:

• Online optimization: Solves at each update a small combinatorial or optimization problem, e.g., enumerating all admissible sensor-action sequences, or checking over possible future intervals (Tariverdi, 20 Nov 2025, Hashimoto et al., 2016);
• Offline explicit partitioning: State space partitioned (e.g., into cones, polytopes, or rings), and for each partition region, optimal (or at least feasible) sampling horizons/sequences are precomputed and stored (Tariverdi, 20 Nov 2025, Tariverdi, 2 Nov 2025, Rajan et al., 2022, Hashimoto et al., 2018);
• Set invariance: Maximal contractive or controlled-invariant sets are computed and decomposed, with self-triggering maps built as a function of the region/radial ring containing the current state (Hashimoto et al., 2018), leading to explicit mappings from state to next inter-execution time.

These strategies yield scalable, resource-efficient self-triggered controllers suitable for resource-constrained embedded implementations. For systems with asynchronicity or distributed sensing/actuation, the conic-partitioning method generalizes to generate a map from partial or quantized state information to feasible horizon sequences (Tariverdi, 20 Nov 2025, Hashimoto et al., 2018, Rajan et al., 2022).

4. Extensions: Constraints, Quantization, Disturbance Robustness, and Infinite Dimensions

Self-triggered control for linear systems has been adapted to address a variety of operational constraints and uncertainties:

• Polyhedral state/input constraints: Set-invariance based approaches construct aperiodic self-triggered controllers that guarantee constraint satisfaction and exponential convergence within controlled-invariant polytopes (Hashimoto et al., 2018);
• Quantized and asynchronous measurements: Self-triggered mechanisms combine adaptive quantizer design (with bits per transmission trading off accuracy and interval length) and horizon selection that factors in quantization-induced errors, ensuring exponential convergence where

$$\|K\|_\infty/N \leq \sigma < (1-\gamma)/(\Gamma\|B\|_\infty)$$

prescribes admissible bit-rate and triggering thresholds (Wakaiki, 2021, Liu et al., 2023);

• Networked and distributed systems: Distributed self-triggered DMPC frameworks address both local and global (coupled) constraints, using tube-based tightening for disturbance rejection and distributed optimization (ADMM consensus) for practical implementation. Between updates, open-loop optimal sequences are applied, and the next sampling step is chosen as the minimum over all subsystems to ensure recursive feasibility and ISS (Li, 2020);
• Disturbance and noise robustness: ISS-type guarantees are provided by bounding the cumulative effect of bounded disturbances or noise, both in the contraction constraint and in the ultimate bound on state trajectories (Tariverdi, 20 Nov 2025, Tariverdi, 2 Nov 2025, Li, 2020, Gleizer et al., 2020, Wakaiki et al., 2019);
• Infinite-dimensional and PDE systems: Observer-based self-triggered rules have been constructed for boundary feedback control of coupled hyperbolic PDEs, with Lyapunov-based prediction of future events based on current boundary or observer state functionals, and provably Zeno-free global exponential stability in spatial $L^2$ norms (Wakaiki et al., 2019, Somathilake et al., 2 Apr 2024).

5. Stability, Performance, and Communication Savings Guarantees

The fundamental performance guarantees provided by self-triggered control in linear systems are:

• Exponential stability: Guaranteed if the self-triggering constraint ensures strict Lyapunov decrease at each transmission, typically evidenced by

$$V(x_{k+1}) \leq e^{-\beta \tau_k} V(x_0) \quad \text{with} \quad \beta > 0.$$

• Global uniform ultimate boundedness (GUUB): With bounded disturbances, trajectory norm converges to a computable ellipsoidal bound, by extension of the Lyapunov decrease condition or via robust LMIs (Tariverdi, 20 Nov 2025, Tariverdi, 2 Nov 2025).
• Constraint satisfaction: Enforced by set-invariance, constraint-tightening, or tube-based procedures in the presence of disturbances or state/input quantization (Hashimoto et al., 2018, Li, 2020, Wakaiki, 2021).
• Communication reductions: In practice and simulation, sensor or network communication can be reduced by 59–74% compared to periodic sampling, with exponential stability or GUUB preserved (Tariverdi, 20 Nov 2025, Tariverdi, 2 Nov 2025, Wang et al., 2022, Rajan et al., 2022).

A summary table of selected properties:

Reference	Plant Type	Measurement/Actuation	Guarantee	Communication Savings
(Tariverdi, 20 Nov 2025)	Linear CT	Asynchronous sensors	Exp. stab./GUUB	59–74% (simulated)
(Tariverdi, 2 Nov 2025)	Linear CT	Synchronous sampling	Exp. stab./GUUB	Similar savings
(Li, 2020)	Discrete/dist.	DMPC, tube/ADMM	ISS, constraints	Reduced CPU & comm.
(Hashimoto et al., 2018)	Discrete	State/input constraints	Constraint/stab.	Low comm., explicit maps
(Rajan et al., 2022)	Continuous	Partition (offline)	IET analysis	Predictable event timing

6. Analysis of Inter-Event Evolution and Invariance

The dynamics of inter-event times (IETs) and their convergence characteristics are now rigorously understood for region-based self-triggered schemes (Rajan et al., 2022). By partitioning the state space into conic regions assigned fixed IETs, the evolution of both state and IETs can be analyzed as discrete maps on the unit sphere. Convergence to constant or periodic IETs is characterized via positively invariant subregions (PIS) for the normalized state direction map. Stability of the PIS is necessary and sufficient for local or global convergence of IETs, with explicit spectral criteria (e.g., leading eigenvalue magnitude, defect structure) (Rajan et al., 2022).

A plausible implication is that steady-state communication rates under region-based self-triggered control are explicitly predictable from the dynamics and partition structure.

7. Practical Considerations, Trade-offs, and Extensions

Key practical aspects include:

• Computational complexity: Online combinatorial search scales poorly in the number of sensors and horizon steps; offline region-based lookup tables enable rapid real-time execution (Tariverdi, 20 Nov 2025, Hashimoto et al., 2018).
• Trade-offs: Performance parameters, such as inter-event contraction rate ($\beta$), quantization step ($N$), or allowed cost degradation ($\beta$ in MPC), trade directly against average inter-event interval and remote resource utilization.
• Data-driven and model-free capabilities: Recent frameworks, leveraging offline data, can synthesize self-triggering maps and feedback matrices via sequence prediction, co-design LMIs, and S-lemma tools, eliminating the need for explicit system identification (Liu et al., 2022, Wang et al., 2022, Li et al., 2023).
• Extensions to quantized, output-based, and consensus control: Robust self-triggered laws now account for quantized measurements, communication denial-of-service, and consensus in multi-agent systems, with communication-performance trade-offs computed via explicit LMIs (Wakaiki, 2021, Liu et al., 2023, Li et al., 2023).

Simulation and experimental results consistently confirm that self-triggered control achieves the desired stability and constraint properties with substantial reductions in sensing, actuation, and/or communication frequency relative to classical periodic or event-triggered schemes.

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References

Primary foundational and recent works on self-triggered control for linear systems include (Tariverdi, 20 Nov 2025, Tariverdi, 2 Nov 2025, Zobiri et al., 2019, Rajan et al., 2022, Hashimoto et al., 2016, Li, 2020, Hashimoto et al., 2018, Wang et al., 2022, Wakaiki, 2021, Liu et al., 2023, Gleizer et al., 2020, Wakaiki et al., 2019, Somathilake et al., 2 Apr 2024, Liu et al., 2022), and (Li et al., 2023). These works collectively formalize the theoretical, computational, and practical principles underpinning this field.