On Event Triggered Tracking for Nonlinear Systems

Abstract: 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.

• The paper presents an event-triggered control design for nonlinear systems that ensures uniform ultimate boundedness of tracking errors while preventing accumulation of inter-execution times.
• For systems with piecewise continuous exogenous input derivatives, the work establishes a framework where tracking errors remain within a known bounded region.
• The research facilitates the development of computationally efficient control systems suitable for resource-constrained applications like embedded or decentralized systems.

An Examination of Event Triggered Tracking for Nonlinear Systems

This paper by Pavankumar Tallapragada and Nikhil Chopra presents an analysis of an event-based control mechanism designed for trajectory tracking in nonlinear systems. This work stands out due to its focus on designing event-triggered controllers that accommodate exogenous inputs, specifically emphasizing conditions under which these controllers can ensure bounded tracking error and non-accumulative inter-execution times.

Key Innovations and Results

In the field of control systems, traditional periodic control paradigms offer theorized stability but can be inefficient due to their reliance on worst-case analysis-derived sampling rates. Event-triggered control, contrastingly, adapts the timing of control execution based on system states, offering computational economy and an adaptively reactive approach to tracking dynamic trajectories.

The paper makes the following contributions:

1. Event-Triggered Control Design: The proposed control algorithm guarantees uniform ultimate boundedness of tracking errors even as it ensures the non-accumulation of inter-execution times. By leveraging uniformly bounded reference trajectory and exogenous inputs, the method presents a systematic approach adaptable to a broad class of nonlinear systems.
2. Ultimate Boundedness under Differentiable Conditions: For systems where the derivative of exogenous inputs is piecewise continuous, albeit non-differentiable, the paper delineates a framework where the tracking error remains constrained within a known region of attraction. This nuanced analysis extends the applicability of event-triggered controllers beyond typical state feedback scenarios.
3. Theoretical Foundations and Simulations: Through simulations of a second-order nonlinear system, the authors exemplify their theoretical formulations, substantiating the algorithm's ability to maintain the tracking error within a predefined boundary while ensuring event-triggered updates occur at non-accumulating intervals.

Implications and Future Applications

The paper's insights into event-based control provide several theoretical and practical implications:

• Practical Implications: The results facilitate the development of efficient control systems with reduced computational load, especially relevant in scenarios where resources are constrained, such as embedded systems or large-scale decentralized control systems.
• Theoretical Insights: The relaxation of certain continuity assumptions, such as differentiability of exogenous input derivatives, demonstrates robustness in the algorithm, potentially stimulating further research into more generalized exogenous input handling and stability under broader conditions.

Future Directions

Looking ahead, this research opens multiple avenues for further investigation:

• Extensions to Output Feedback Systems: Developing similar event-triggered techniques for output feedback systems could extend practical applicability, especially in complex systems where full state feedback is not feasible.
• Refining Lower Bound Estimates on Inter-Execution Times: Current theoretical estimates on minimum inter-execution times are notably conservative, suggesting opportunities for refinement using more precise analytical techniques that better capture system dynamics and improve resource allocation efficiency.
• Integration with Machine Learning Approaches: Coupling event-triggered control strategies with adaptive machine learning models might enhance predictive capabilities, allowing control systems to better anticipate trajectory deviations and adaptively adjust trigger conditions in real-time.

Conclusion

Tallapragada and Chopra's work on event-triggered control for nonlinear systems underlines significant strides in achieving computationally efficient control while ensuring performance reliability. This paper lays the groundwork for continued exploration into adaptive control strategies that align theoretical rigor with practical applicability, aiming to tackle the complexities inherent in nonlinear control systems within increasingly dynamic environments.