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On Event Triggered Tracking for Nonlinear Systems
Pavankumar Tallapragada, Nikhil Chopra
TL;DR
The paper addresses event-based trajectory tracking for nonlinear systems with reference trajectories generated by exogenous inputs. It designs a Lyapunov-based triggering rule from a continuous-time asymptotically tracking controller, obtaining bounded tracking error and inter-execution times bounded away from zero. Under a uniformly bounded exogenous-input derivative, the ultimate bound can be arbitrarily small; with piecewise-continuous inputs, the bound is constrained and the result is local.
Problem
Event-triggered tracking has mostly been studied without exogenous inputs, motivating trajectory tracking through a reference system with such inputs.
Method
The paper converts a continuously updated asymptotically tracking controller into an event-based controller using a Lyapunov-derived triggering condition with design parameter r.
Results
The event-based controller guarantees uniformly ultimately bounded tracking error and inter-execution times bounded away from zero; derivative-bounded inputs allow an arbitrarily small ultimate bound.
Takeaways & Limitations
Event-based trajectory tracking can retain the continuous controller’s tracking objective while reducing dependence on periodic, worst-case execution timing.
Takeaways & Limitations
With piecewise-continuous, nondifferentiable exogenous inputs, the achievable ultimate bound is constrained and the result is local with a known region of attraction.
Abstract
from arXiv · showhide
In this paper we study an event based control algorithm for trajectory tracking in nonlinear systems. The desired trajectory is modelled as the solution of a reference system with an exogenous input and it is assumed that the desired trajectory and the exogenous input to the reference system are uniformly bounded. Given a continuous-time control law that guarantees global uniform asymptotic tracking of the desired trajectory, our algorithm provides an event based controller that not only guarantees uniform ultimate boundedness of the tracking error, but also ensures non-accumulation of inter-execution times. In the case that the derivative of the exogenous input to the reference system is also uniformly bounded, an arbitrarily small ultimate bound can be designed. If the exogenous input to the reference system is piecewise continuous and not differentiable everywhere then the achievable ultimate bound is constrained and the result is local, though with a known region of attraction. The main ideas in the paper are illustrated through simulations of trajectory tracking by a nonlinear system.
I. INTRODUCTION
The paper motivates event-based control for trajectory tracking in nonlinear systems, addressing the limitations of periodic execution and prior event-triggered work with exogenous inputs. It proposes a controller that preserves bounded tracking and nonzero inter-execution times under stated boundedness assumptions.
- Motivation: Periodic sampling fixes control execution to a worst-case rate independent of system state, which can waste computational resources.Event-based control instead determines execution timing dynamically and can make better use of computational and communication resources.
- Related work: Systematic event-based controller design for nonlinear stabilization has emerged only recently, with prior work establishing stability and lower-bounded inter-execution times under an ISS condition.The paper’s controller is significantly influenced by the cited work in.
- Problem scope: Prior event-triggered tracking studies mostly assumed state feedback without exogenous inputs, whereas this paper considers available exogenous inputs through a reference system.The desired trajectory is generated by the reference system.
- Contribution: Given a continuous-time controller with global uniform asymptotic tracking, the proposed event-based controller guarantees uniform ultimate boundedness and inter-execution times bounded away from zero.If the reference-system input derivative is uniformly bounded, the ultimate tracking bound can be designed arbitrarily small.
- Organization: The paper develops the problem formulation, triggering condition, analytical results, and second-order nonlinear-system simulations in successive sections.The results are summarized in the final section.
II. PROBLEM STATEMENT AND NOTATION
The paper formulates trajectory tracking around a reference system driven by an exogenous signal and represents intermittent control updates as measurement error in a perturbed tracking-error system. Event-triggered update times are dynamically determined rather than uniformly spaced.
- Reference system: The reference trajectory is determined by the exogenous signal v and the initial condition of xd.A tracking controller generally depends on both the tracking error and the reference trajectory.
- Control law: The control signal is modeled as a function of the tracking error, reference trajectory, and exogenous input.The closed-loop dynamics are then expressed in terms of the tracking error.
- Intermittent control: Intermittent updates occur at times ti, and the tracking-error dynamics evolve between successive update instants.The control is computed and updated only at these event times.
- Measurement-error interpretation: The intermittent controller can be viewed as continuous control subject to measurement error in the state and exogenous input.The measurement error is reset at update instants and is discontinuous there.
- Timing model: Periodic control uses ti+1 − ti = Ts, whereas event-triggered control determines nonuniform update times dynamically through a triggering condition.This distinction makes execution timing dependent on system evolution rather than fixed sampling.
- Problem formulation: The objective is to track a reference trajectory asymptotically under continuous control and, subsequently, within a desired ultimate bound using event-based control.The tracking error is defined as ˜x = x − xd.
III. EVENT-TRIGGERING CONDITION FOR EMULATION BASED TRAJECTORY TRACKING CONTROL
The triggering design targets uniform ultimate boundedness of tracking error and non-accumulating updates using a Lyapunov-based condition. Its threshold parameter r directly sets the desired ultimate error bound, under boundedness and regularity assumptions.
- Requirements: The event-based controller must ensure both uniformly ultimately bounded tracking error and no accumulation of execution times.These requirements motivate the assumptions and triggering condition developed in the section.
- Assumptions: The analysis assumes a Lyapunov function for the continuously controlled tracking-error system, locally Lipschitz system and controller functions, and bounded reference signals.The exogenous input is piecewise continuous; a separate assumption also considers a uniformly bounded derivative.
- Lyapunov analysis: The analysis uses compact sets containing admissible reference signals and bounds control sensitivity to measurement error on those sets.The sensitivity bound is expressed through L(R) and the norm of the measurement error.
- Trigger construction: The triggering condition is derived from the Lyapunov derivative and updates control according to the resulting error-dependent condition.The derivation interprets the event-triggered dynamics as a perturbed version of the continuous-time tracking-error system.
- Trigger parameters: The parameter r determines the ultimate tracking-error bound, while 0 < σ < 1 is another triggering-condition design parameter.Updates are prohibited when ∥˜x∥ < r to avoid accumulation, and the initial update time may therefore exceed zero.
IV. UNIFORM ULTIMATE BOUNDEDNESS OF THE TRAJECTORY TRACKING ERROR
The event-triggering condition establishes uniform ultimate boundedness of the tracking error while preventing accumulation of inter-execution times. The guarantees are global under stronger reference-signal assumptions and local with constrained parameters when those assumptions are relaxed.
- The triggering condition keeps the tracking error ultimately bounded when control execution times do not exhibit Zeno behavior.The resulting bound is a ball of radius r1 = α^-1(α2(r)).
- The Lyapunov derivative is negative whenever the tracking error exceeds r, and the corresponding sub-level sets remain positively invariant.For ∥˜x∥ ≥ r, ˙V ≤ −(1 − σ)α3(∥˜x∥) < 0.
- The analysis identifies α3 and β as determining event frequency and the lower bound on inter-execution times.A more useful relaxed-reference result assumes separated jumps and Lipschitz behavior between jumps, while Theorem 2 uses only a uniform bound on ∥v∥.
- Under assumptions (A1)–(A4), the tracking error is uniformly ultimately bounded and inter-execution times have a positive uniform lower bound.The lower bound depends on the initial tracking-error bound.
- The lower bound on inter-execution times follows because measurement error must grow from zero to a positive triggering threshold.The proof bounds the growth using ∥L0∥ and shows the resulting time T is greater than zero.
- Without assumption (A4), bounded exogenous input yields only local uniform ultimate boundedness and constrains the admissible triggering radius.The result applies when ∥˜x(0)∥ ≤ R0 and requires ∆µ0 r − 2dv∥M(R0)∥ > 0.
V. EXAMPLES AND SIMULATION RESULTS
Simulations apply the event-triggered tracking design to a second-order nonlinear system with smooth and piecewise-continuous reference inputs. In both cases, tracking is ultimately bounded with observed minimum execution times near 0.005 s, while theoretical lower bounds are much more conservative.
- Simulation setup: The simulated plant is a second-order nonlinear system tracking a reference generated by [ẋd,1; ẋd,2] = [xd,2; v].The exogenous input v and the reference initial conditions determine the desired trajectory.
- Simulation setup: The controller uses γ(ξ) = Kx̃ + v + (x̃1 + xd,1)^3 + xd,2 with K selected so à = A + BK is Hurwitz.The event-triggered closed-loop error dynamics are analyzed through the measurement-error interpretation.
- Case I: smooth exogenous input: Case I tracking error remained ultimately bounded and well below the desired bound, with the zoomed plot showing that the error is not driven to zero.The triggering condition updates control when the weighted measurement error may exceed the tracking-error norm while the error is outside the radius-r ball.
- Case I: smooth exogenous input: 301 control executions and a 0.005s minimum inter-execution time were observed in Case I, with average update frequencies of around 30Hz overall and 46Hz before entering the radius-r ball.The theoretical minimum inter-execution time was around 6 × 10^-8s, orders of magnitude below the observed value.
- Case II: piecewise-continuous input: 304 control updates and a 0.005s minimum execution time were observed in Case II, while the theoretical minimum was around 3 × 10^-8s and again very conservative.Case II uses a piecewise constant exogenous input, and its observed average frequencies were around 30Hz overall and 46Hz before entering the radius-r ball.
- Case II: piecewise-continuous input: In Case II, the triggering parameter must satisfy r > 0.0075 because the theorem requires r to exceed Jv = 0.1.The simulation nevertheless used r = 0.0154, with other parameters matching Case I.
VI. CONCLUSIONS
The paper develops event-based trajectory tracking for nonlinear systems from a continuously updated controller with uniform asymptotic tracking. Its results provide ultimate boundedness and non-accumulating executions, while exposing conservatism in timing estimates and limits for nonsmooth inputs.
- Main conclusions: The event-based controller preserves uniform ultimate boundedness of tracking error and keeps inter-execution times uniformly bounded away from zero.These guarantees are obtained from a continuous-time controller that ensures uniform asymptotic tracking.
- Main conclusions: When the reference trajectory, exogenous input, and its derivative are uniformly bounded, an arbitrary ultimate tracking-error bound can be achieved.The guaranteed minimum inter-execution time decreases as the selected ultimate bound decreases.
- Scope limits: Without the derivative bound, the ultimate error bound may not be reducible below a threshold, and the analytical result is generally only local.The paper identifies this as the scope of its second and third results.
- Simulation evidence and future work: Simulations of a second-order nonlinear system support the theory, but theoretical lower bounds on inter-update times are very conservative.The authors attribute this partly to estimating the rate of change of ∥e∥ in the presence of exogenous signals; extending the results to output feedback remains future work.