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Stabilization of nonlinear systems using event-triggered output feedback controllers

Mahmoud Abdelrahim, Romain Postoyan, Jamal Daafouz, Dragan Nešić

arXiv:1408.5738v1eess.SY

TL;DR

The paper addresses output-feedback event-triggered stabilization of nonlinear systems while ensuring a strictly positive minimum time between transmissions. It combines event-triggered and time-triggered control by activating event evaluation only after a fixed interval, obtaining asymptotic stability and applicability to stabilizable and detectable LTI systems.

  • Problem

    Output-feedback event-triggered controllers must stabilize nonlinear systems while respecting hardware-imposed minimum times between transmissions.

  • Method

    The strategy evaluates an output-based event-triggering condition only after T time units, with T obtained from time-driven sampled-data stabilization results.

  • Results

    The proposed mechanism guarantees asymptotic stability while lower-bounding inter-transmission times by a strictly positive constant.

  • Takeaways & Limitations

    The conditions are satisfied by any stabilizable and detectable LTI system, extending the strategy beyond the nonlinear setting.

Abstract

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The objective is to design output feedback event-triggered controllers to stabilize a class of nonlinear systems. One of the main difficulties of the problem is to ensure the existence of a minimum amount of time between two consecutive transmissions, which is essential in practice. We solve this issue by combining techniques from event-triggered and time-triggered control. The idea is to turn on the event-triggering mechanism only after a fixed amount of time has elapsed since the last transmission. This time is computed based on results on the stabilization of time-driven sampled-data systems. The overall strategy ensures an asymptotic stability property for the closed-loop system. The results are proved to be applicable to linear time-invariant (LTI) systems as a particular case.

I. INTRODUCTION

Networked control systems require communication-aware strategies because shared digital channels have limited bandwidth. The paper addresses output-feedback event-triggered stabilization while enforcing a strictly positive minimum inter-transmission time.

  • Event-triggered control adapts transmissions to the plant state so communication occurs only when needed for control objectives.
  • Output-feedback event-triggered stabilization is difficult because hardware requires a minimum time between transmissions, especially when only outputs are available.
  • The proposed strategy evaluates an output-based event condition only after T time units, combining event-triggered control with sampled-data stabilization results.T is selected as the maximum allowable sampling period from the sampled-data analysis.
  • The results use assumptions enabling local and global analyses and apply to stabilizable and detectable LTI systems through an LMI reformulation.
  • Unlike observer-based approaches, the output-feedback law need not use an observer, and the triggering mechanism requires only plant-output access rather than controller-variable access.

II. PRELIMINARIES

The preliminaries define the mathematical classes, norms, derivatives, and hybrid-system framework used to analyze the controller and its event-triggered dynamics.

  • Class K, class K∞, and class KL functions provide standard comparison-function notation for stability statements.
  • The preliminaries specify Euclidean and matrix norms, eigenvalue notation, transposes, identity matrices, and vector concatenation.
  • Clarke’s generalized directional derivative is used because the Lyapunov functions considered are locally Lipschitz and need not be differentiable everywhere.
  • For continuously differentiable functions, Clarke’s generalized derivative reduces to the standard gradient-based directional derivative.
  • The maximum of two functions has a generalized derivative determined by the active function, with an upper bound by both derivatives on the equality set.
  • The model uses hybrid systems with continuous flows on C and discrete jumps on D, defined over hybrid time domains.

III. PROBLEM STATEMENT

The paper models a nonlinear plant with a dynamic output-feedback controller communicating through zero-order holds. It formulates triggering as flow and jump sets for a hybrid system and seeks asymptotic stability.

  • The controller uses measured plant output y and has controller state xc, while static output feedback is included as a special case.
  • The emulation approach assumes the controller globally asymptotically stabilizes the plant without network constraints before synthesizing the triggering condition.
  • Zero-order holds keep the last transmitted output and control-input values constant between transmission instants.
  • The network-induced error e combines output and input errors and is reset to zero at each transmission.
  • The closed loop is represented as a hybrid system in which flows continue while the triggering condition holds and jumps represent transmissions.
  • The design objective is to choose the flow and jump sets so the hybrid system has a global asymptotic stability property.

IV. MAIN RESULTS

The paper combines output-based event triggering with time-driven sampling to stabilize nonlinear systems while enforcing a strictly positive lower bound on inter-transmission times. Under local or global assumptions, the resulting hybrid closed loop has corresponding asymptotic stability guarantees.

  • Stability conditions: Assumption 1 provides conditions for deriving local or global stability results and implies L2-gain stability and input-to-state stability for the error-driven system.The global version requires conditions (7) and (8) to hold for almost all states and errors.
  • Triggering strategy: The proposed trigger evaluates the event condition only after T time units, preventing Zeno behavior while preserving asymptotic stability.T is chosen below the maximum allowable sampling period T(γ, L), and inter-jump times are uniformly lower bounded by T.
  • Main theorem: For T ∈ (0, T(γ, L)), there exist Δ > 0 and β ∈ KL such that sufficiently small initial states satisfy the theorem’s asymptotic bound.The bound is |φx(t, j)| ≤ β(|(φx(0, 0), φe(0, 0))|, t+j) for all points in the solution domain.
  • Main theorem: Maximal solutions are complete, and the stability estimate becomes global when Assumption 1 holds globally.Completeness rules out finite-time termination of maximal solutions under the stated hybrid-system construction.
  • Example: For the controlled Lorenz equations, choosing p1 = 2 and p2 = 3a with a = 10, b = 28, and c = 8/3 yields T = 0.01.The example uses static output feedback and a network-induced measurement error with W(e) = |e|.

V. LINEAR SYSTEMS

For LTI systems, the paper converts the nonlinear assumptions into an LMI-based sufficient condition and shows that stabilizable and detectable systems satisfy it. An example reports a guaranteed minimum inter-transmission time and substantially fewer transmissions than a prior method.

  • LTI formulation: The LTI construction uses a dynamic controller and incorporates communication constraints into the hybrid model used for triggering.The controller stabilizes the plant without sampling before the network constraints are added.
  • LMI condition: Proposition 1 gives an LMI sufficient condition for verifying Assumption 1 and applying the main nonlinear-system results to LTI systems.The condition can always be satisfied when the LTI system is stabilizable and detectable, by selecting a controller that makes A1 Hurwitz.
  • Numerical example: For 100 random initial conditions, the example obtains L = 4, ε1 = 1.5839, ε2 = 13.9969, and γ = 89.9666 from the LMI.The parameters are computed using the SEDUMI solver with the YALMIP interface.
  • Numerical example: The guaranteed minimum inter-transmission time is T = 0.017, while the average transmission count is approximately 100 times lower than under.The comparison uses the minimum and average inter-transmission times reported for the simulation conditions in Table I.
  • Comparison: The proposed method achieves global asymptotic stability in the example, whereas provides only practical stability.The paper also states that the comparison with is not relevant because its triggering mechanism and observer-based dynamic controller differ.

VI. STATE FEEDBACK CONTROLLERS

The technique also applies to state feedback, where T tunes the minimum inter-transmission time. Table II reports minimum and average inter-execution times for 200 initial conditions.

  • State feedback sets y = x, allowing T to directly tune the minimum inter-transmission time up to the bound in (11).
  • Table II evaluates minimum and average inter-execution times over 200 initial conditions with |(x(0, 0), e(0, 0))| ≤100 and τ(0, 0) = 0.
  • Figure 2 shows inter-transmission times for T = 0.075.
  • Figure 3 shows inter-transmission times under the comparison condition identified as [3].

VII. CONCLUSION

The paper develops output-based event-triggered controllers for nonlinear-system stabilization. The technique ensures asymptotic stability and a positive minimum time between transmissions, with conditions also covering stabilizable and detectable LTI systems.

  • The proposed output-based event-triggered controllers stabilize nonlinear systems while enforcing a minimum time between consecutive transmission instants.
  • The required conditions are satisfied by any stabilizable and detectable LTI system.
  • The results provide a starting point for co-designing an output feedback law jointly with the triggering condition.

APPENDIX

The appendix establishes stability through a timer-based Lyapunov analysis. It selects parameters so the timer enforces a positive inter-jump period, then proves decay, completeness, and LTI verification under the stated assumptions.

  • The parameter choice ensures T < eT(θ, η, γ, L), so ζ(τ) < 0 implies τ > T.
  • The proof introduces ζ as the solution of a scalar timer equation and defines a composite function R combining V(x) with an error term weighted by ζ.
  • At jumps, the error and timer reset to zero, and R does not increase.
  • During flows, the generalized derivative of R is bounded by a negative definite function, establishing decay of the composite Lyapunov measure.
  • The stability estimate has the form R(φ(t, j)) ≤ β̃(R(φ(0, 0)), 0.5t + 0.5Tj), explicitly incorporating flow time and jumps.
  • The proof also establishes complete maximal solutions because the error cannot blow up in finite time and jumps map states back into the flow set.
  • For locally valid assumptions, the resulting stability estimate holds locally on an invariant neighborhood.
  • For LTI systems, the assumptions can be verified using a quadratic Lyapunov function and an LMI.
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