[Paper Review] Observer-Based Controllers For Incrementally Quadratic Nonlinear Systems with Disturbances: Continuous-time and Event-triggered Cases

Xiangru Xu, Behçet Açıkmeşe+1 more|arXiv (Cornell University)|Feb 24, 2018

Stability and Control of Uncertain Systems60 references3 citations

TL;DR

This paper proposes observer-based output feedback controllers for incrementally quadratic nonlinear systems subject to bounded disturbances, using linear matrix inequality (LMI) techniques to design both continuous-time and event-triggered controllers. The approach ensures input-to-state stability (ISS) and practical stability under Zeno-free event-triggering, demonstrated via a single-link robot arm simulation.

ABSTRACT

This paper investigates observer-based output feedback control design for incrementally quadratic nonlinear systems with bounded external disturbances. The nonlinearities considered satisfy incremental quadratic constraints, which are characterized by incremental multiplier matrices and include many common nonlinearities as special cases. Linear matrix inequality (LMI) conditions are presented to separately construct the continuous-time observer and the feedback control law for two parameterizations of the incremental multiplier matrices. The observer and the controller gains are obtained from the LMI solutions and proved to render the closed-loop system input-to-state stable with respect to external disturbances. Based on the continuous time observer-based controller designed, event-triggered controllers with Zeno-freeness are constructed for two triggering configurations where the first one implements the event-triggering mechanisms (ETMs) in the controller-to-actuator channel and the second one implements ETMs in both the controller-to-actuator channel and the sensor-to-observer channel asynchronously. For each configuration the closed-loop system is proved to be input-to-state practically stable. The theoretical results are illustrated via simulations of a single-link robot arm example.

Motivation & Objective

• Address the challenge of output feedback control for nonlinear systems with bounded external disturbances.
• Develop a systematic design method for observer-based controllers that ensures stability despite incremental quadratic nonlinearities.
• Extend the framework to event-triggered control with Zeno-freeness in both controller-to-actuator and sensor-to-observer channels.
• Ensure robustness against disturbances through input-to-state practical stability (ISSp) guarantees.
• Demonstrate the effectiveness of the proposed approach through a realistic single-link robot arm case study.

Proposed method

• Model nonlinear systems using incremental quadratic constraints parameterized by multiplier matrices, enabling representation of diverse nonlinearities.
• Formulate LMI conditions to separately design the observer and feedback control law for two parameterizations of the incremental multiplier matrices.
• Construct continuous-time observer-based controllers that achieve input-to-state stability (ISS) with respect to external disturbances.
• Design event-triggered controllers with two configurations: one triggering in the controller-to-actuator channel, and another with asynchronous triggering in both controller-to-actuator and sensor-to-observer channels.
• Prove Zeno-freeness for both event-triggering mechanisms by ensuring positive inter-event times.
• Establish input-to-state practical stability (ISSp) for the closed-loop systems under both event-triggering configurations.

Experimental results

Research questions

• RQ1 How can observer-based output feedback control be designed for incrementally quadratic nonlinear systems with bounded disturbances?
• RQ2 What LMI-based conditions ensure input-to-state stability (ISS) in the continuous-time observer-controller design?
• RQ3 How can event-triggered control be implemented in both controller-to-actuator and sensor-to-observer channels while avoiding Zeno behavior?
• RQ4 What stability guarantees (e.g., ISSp) can be achieved under the proposed event-triggering mechanisms?
• RQ5 How does the proposed framework perform in a realistic mechanical system, such as a single-link robot arm?

Key findings

• The proposed LMI-based design successfully constructs observer and controller gains that ensure input-to-state stability (ISS) for the continuous-time closed-loop system under bounded disturbances.
• Event-triggered controllers with Zeno-freeness are achieved in both configurations, ensuring that the inter-event time remains bounded away from zero.
• The system remains input-to-state practically stable (ISSp) under both event-triggering mechanisms, with stability proven via Lyapunov-based analysis.
• The simulation results on a single-link robot arm demonstrate the effectiveness of the proposed controller in maintaining system stability despite nonlinearities and disturbances.
• The method supports two distinct parameterizations of incremental multiplier matrices, enabling broader applicability to various nonlinear systems.
• The separation of observer and controller design via LMI conditions allows for systematic and numerically tractable synthesis of the control laws.