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To sample or not to sample: Self-triggered control for nonlinear systems
Adolfo Anta, Paulo Tabuada
TL;DR
Periodic digital control can use resources inefficiently because fixed execution rates ignore the plant’s current state, while nonlinear systems still have open implementation questions. The paper develops self-triggered control, using current state measurements to choose future executions for state-dependent homogeneous and polynomial systems. Its theory establishes a resource–performance tradeoff and is illustrated on a jet engine compressor and a rigid body.
Problem
Periodic execution can use processor and other resources inefficiently, while implementation effects and sampling questions remain open for nonlinear control systems.
Method
The paper uses current state measurements to compute self-trigger conditions for state-dependent homogeneous and polynomial systems.
Results
The developed self-trigger conditions characterize a tradeoff between controller resource allocation and achieved performance, with examples including a jet engine compressor and a rigid body.
Takeaways & Limitations
Self-triggered control provides an aperiodic implementation that uses plant-state information to decide when the controller should execute again.
Takeaways & Limitations
For polynomial systems, the method computes inter-execution estimates in a higher-dimensional auxiliary space and cannot guarantee longer intervals than periodic control.
Abstract
from arXiv · showhide
Feedback control laws have been traditionally implemented in a periodic fashion on digital hardware. Although periodicity simplifies the analysis of the mismatch between the control design and its digital implementation, it also leads to conservative usage of resources such as CPU utilization in the case of embedded control. We present a novel technique that abandons the periodicity assumption by using the current state of the plant to decide the next time instant in which the state should be measured, the control law computed, and the actuators updated. This technique, termed self-triggered control, is developed for two classes of nonlinear control systems, namely, state-dependent homogeneous systems and polynomial systems. The wide applicability of the proposed results is illustrated in two well known physical examples: a jet engine compressor and the rigid body.
1 Introduction
The paper addresses resource inefficiency and unresolved implementation questions caused by periodic control, especially for nonlinear systems. It proposes self-triggered control for two nonlinear system classes and develops theory connecting execution timing, resources, and performance.
- Motivation: Periodic control executes at a fixed rate based on worst-case operation, causing inefficient use of processor time, communication bandwidth, and energy.Digital implementations also face difficulties choosing a sampling period that preserves desired performance.
- Motivation: Existing nonlinear-system analyses remain limited because nonlinear dynamics generally cannot be discretized exactly in closed form.Approximate discrete-time models are commonly used instead.
- Prior approach: Event-triggered control reduces resource usage by triggering execution from the plant state, but permanent monitoring may require unavailable dedicated hardware.This limitation motivates a software-compatible alternative.
- Contribution: Self-triggered control uses the current state measurement to decide the controller’s next execution time, avoiding extra monitoring hardware.The controller therefore decides when it will execute again rather than following a fixed periodic schedule.
- Contribution: The paper develops online self-trigger conditions for state-dependent homogeneous and polynomial systems.The conditions depend on system dynamics, desired performance, and the current state measurement.
- Scope: The proposed conditions expose a tradeoff between controller resource allocation and achieved performance, with applications to a jet engine compressor and a rigid body.The results are presented as contributions to control under information constraints.
2 Notation and input-to-state stability
This section establishes notation for smooth dynamical systems, vector fields, mappings, and trajectories, then characterizes input-to-state stability through ISS Lyapunov functions.
- 2.1 Notation: The paper uses |x| for the Euclidean norm of a vector x ∈ R^n.It also introduces class K and K∞ functions for later stability statements.
- 2.1 Notation: Class K functions are strictly increasing and vanish at zero, while class K∞ functions are additionally defined on [0,∞[ and unbounded.The K∞ condition is α(r) → ∞ as r → ∞.
- 2.1 Notation: The paper assumes objects are smooth unless stated otherwise and defines vector-field Lie products, differentials, and related vector fields.Mappings with smooth inverses also support pullbacks of vector fields.
- 2.2 Input-to-state stability: The state is denoted x, its trajectory x(t), and the input trajectory u(t), with x(t,x0) emphasizing the initial condition.These conventions support the subsequent control-system and stability definitions.
- 2.2 Input-to-state stability: A system is input-to-state stable exactly when it has an ISS Lyapunov function satisfying the stated class-K∞ conditions.The paper uses this characterization instead of the standard ISS definition.
3 Event-triggered stabilization of nonlinear control systems
This section models sampled-data implementation through measurement error and replaces periodic execution with event- and self-triggered rules. Stability is maintained by restricting the error using an ISS-based Lyapunov condition.
- Implementation model: Digital implementation samples the state, computes u(t_i)=k(x(t_i)), and updates actuators at execution times that are traditionally separated by a fixed period T.The paper drops this periodicity assumption in favor of self-triggered executions.
- Measurement error: The measurement error e is the difference between the last measured state and the current state.At an execution time t_i, the error becomes zero because the state is resampled.
- Measurement error: Under sample-and-hold implementation, the closed-loop dynamics use the current state and the control computed from the last sampled state.This produces the implementation model analyzed for triggering and stability.
- Stability condition: If the feedback law renders the system ISS with respect to measurement error, an ISS Lyapunov function provides a condition for stability.The Lyapunov derivative is bounded using class-K∞ functions of the state and error.
- Stability condition: When the relevant functions are Lipschitz on compact sets, a simpler inequality can enforce the error restriction and guarantee decreasing V when σ<1.The simplified condition uses constants selected from the corresponding Lipschitz bounds.
- Self-triggering: Self-triggering uses the current state measurement to determine the next execution time instead of repeatedly testing the event condition.The resulting inter-execution time is denoted τ(x(t_i)) because it depends on the last sampled state.
4 Homogeneous control systems
The paper develops scaling laws for homogeneous and state-dependent homogeneous systems, then uses them to derive self-trigger conditions from the current state. These laws relate inter-execution times across homogeneous rays and support online computation with stability guarantees.
- Homogeneous systems: For standard homogeneous systems, the inter-execution time scales according to the degree of homogeneity and the spatial dilation.The closed-loop system is assumed homogeneous with respect to the standard dilation, and the execution rule is |e| = c|x| with c > 0.
- Homogeneous systems: Homogeneous vector fields provide symmetry-based trajectory relations that allow inter-execution times at one state to be extended along its homogeneous ray.For a known point x, the time for any scaled state λx can be inferred for λ > 0.
- State-dependent homogeneity: The resulting framework defines homogeneous rays through a dilation vector field and relates the flows of the system and dilation field through space-time scaling.The trajectories of the dilation field are the homogeneous rays, while the associated flow determines how time changes along them.
- Self-triggered implementation: The self-trigger condition uses these scaling properties to compute the next execution time from the current state without continuously monitoring the plant.The approach is intended for online computation and uses a precomputed stabilizing time estimate to obtain a lower bound on inter-execution times.
- State-dependent homogeneity: State-dependent homogeneity generalizes constant-degree homogeneity by allowing the degree function, and therefore time scaling, to depend on the state and homogeneous ray.The dilation flow acts as a spatial operator whose application induces a scaling in the system trajectory's time.
- Self-triggered implementation: In the jet-engine compressor example, the linear-model estimate is τ* = 7.63ms, and inter-execution times enlarge as the system approaches equilibrium while preserving stability.The self-trigger strategy achieves a similar decay rate to periodic control with fewer executions, including under sensor noise and actuator disturbance.
5 Polynomial control systems
The paper extends self-triggered control from homogeneous systems to polynomial systems by embedding them in a higher-dimensional homogeneous system. The resulting construction preserves inter-execution times and yields a computable self-trigger condition, while introducing scope limitations compared with the homogeneous case.
- Inter-execution-time equivalence: A commuting-map construction transfers the evolution of the original error-to-state ratio to an auxiliary vector field.The auxiliary system uses an output map based on the ratio of projected error and state norms, allowing its inter-execution times to coincide with those of the original system.
- Homogenization: Polynomial systems are handled by adding an auxiliary state so the extended vector field becomes homogeneous.Lower-degree monomials are multiplied by powers of the auxiliary variable, and a dummy error variable is added so the extended system has matching state and error dimensions.
- Inter-execution-time equivalence: The polynomial-system corollary computes original inter-execution times through the higher-dimensional auxiliary system rather than directly in the original state space.The same execution rule is applied to both systems, with the relation between their times determined by the homogenization degree.
- Computing the self-trigger condition: Embedding the system in a higher-dimensional space enables invariant-set construction needed to compute a lower bound for inter-execution times.Invariant sets of the original system generate invariant sets for the extended system, where the bound τ* can be computed and then extended through the corollary.
- Rigid-body example: The rigid-body example applies the polynomial-system method to derive a self-trigger condition for a globally asymptotically stabilizing nonlinear feedback law.The example uses an auxiliary variable to obtain a degree-2 homogeneous extended system and computes the condition through the extended invariant set.
- Rigid-body example: For the rigid-body example, the derived condition is obtained by applying the polynomial-system corollary after constructing an invariant set covering the operation region.The operation region is a radius-15 ball, with σ = 0.01 and a computed stabilizing period τ* = 4.5 · 10^-5s.
6 Appendix
The appendix establishes how homogeneity, flow relationships, and auxiliary systems determine self-triggered inter-execution times and invariant sets.
- Homogeneous systems: Theorem 4.11 states that inter-execution times scale for homogeneous closed-loop systems under the execution rule |e| = c|x|.The result applies to feedback laws rendering the closed loop homogeneous with a degree function.
- Homogeneous systems: A dilation e^rs in the initial condition produces a corresponding scaling e^ρ(s) in the inter-execution times.The scaling follows from the faster evolution of the error trajectory along homogeneous rays.
- Related systems: Theorem 5.1 shows that related vector fields with commuting maps generate identical inter-execution times under corresponding output conditions.The equality follows because the relevant output trajectories are identical for every point.
- Auxiliary systems: For the original and auxiliary systems, inter-execution times coincide under the same execution rule, while homogeneity determines their scaling.The auxiliary system is related to the original through a map and is homogeneous of degree l −1.
- Auxiliary systems: Every invariant set of the original system induces a family of invariant sets for the extended auxiliary system indexed by λ > 0.The induced sets have the form ˜Σλ = {(λx, λ, λe, 0) : (x, e) ∈Σ}.