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1. Analysis and design of event‐triggered control algorithms using hybrid systems tools

   https://doi.org/10.1002/rnc.5141  

   Chai, Jun; Casau, Pedro; Sanfelice, Ricardo G. (September 2020, International Journal of Robust and Nonlinear Control)

   Summary This article proposes a general framework for analyzing continuous‐time systems controlled by event‐triggered algorithms. Closed‐loop systems resulting from using both static and dynamic output (or state) feedback laws that are implemented via asynchronous event‐triggered techniques are modeled as hybrid systems given in terms of hybrid inclusions. Using recently developed tools for robust stability, properties of the proposed models, including stability of compact sets, robustness, and Zeno behavior of solutions are addressed. The framework and results are illustrated by several event‐triggered strategies available in the literature, and observations about their key dynamical properties are made. 

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2. Dynamic Intermittent Suboptimal Control: Performance Quantification and Comparisons

   https://doi.org/10.23919/ChiCC.2018.8483352  

   Yang, Yongliang; Vamvoudakis, Kyriakos G.; Modares, Hamidreza; Ding, Dawei; Yin, Yixin; Wunsch, Donald C. (July 2018, 2018 37th Chinese Control Conference (CCC))

   This paper presents a novel intermittent suboptimal event-triggered controller design for continuous-time nonlinear systems. The stability of the equilibrium point of the closed-loop system, and the performances are analyzed and quantified theoretically. It is proven that the static and the dynamic event-triggered suboptimal controllers have a known degree of suboptimality compared to the conventional optimal control policy. In order to generate dynamic event-triggering framework, we introduce an internal dynamical system. Moreover, the Zeno behavior is excluded. Finally, a simulation example is conducted to show the effectiveness of the proposed intermittent mechanisms. 

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3. Hybrid Event Shaping to Stabilize Periodic Hybrid Orbits

   Zhu, James; Kong, Nathan J.; Council, George; Johnson, Aaron M. (May 2022, IEEE International Conference on Robotics and Automation)

   Many controllers for legged robotic systems leverage open- or closed-loop control at discrete hybrid events to enhance stability. These controllers appear in several well studied phenomena such as the Raibert stepping controller, paddle juggling, and swing leg retraction. This work introduces hybrid event shaping (HES): a generalized method for analyzing and designing stable hybrid event controllers. HES utilizes the saltation matrix, which gives a closed-form equation for the effect that hybrid events have on stability. We also introduce shape parameters, which are higher order terms that can be tuned completely independently of the system dynamics to promote stability. Optimization methods are used to produce values of these parameters that optimize a stability measure. Hybrid event shaping captures previously developed control methods while also producing new optimally stable trajectories without the need for continuous-domain feedback. 

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4. Model-Free Event-Triggered Containment Control of Multi-Agent Systems

   https://doi.org/10.23919/ACC.2018.8430818  

   Yang, Yongliang; Modares, Hamidreza; Vamvoudakis, Kyriakos G.; Yin, Yixin; Wunsch, Donald C. (June 2018, 2018 Annual American Control Conference (ACC))

   This paper presents a model-free distributed event-triggered containment control scheme for linear multiagent systems. The proposed event-triggered scheme guarantees asymptotic stability of the equilibrium point of the containment error as well the avoidance of the Zeno behavior. To relax the requirement of complete knowledge of the dynamics, we combine an off-policy reinforcement learning algorithm in an actor critic structure with the event-trigger control mechanism to obtain the feedback gain of the distributed containment control protocol. A simulation experiment is conducted to verify the effectiveness of the approach. 

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5. Dynamic Intermittent Q-Learning for Systems with Reduced Bandwidth

   https://doi.org/10.1109/CDC.2018.8619242  

   Yang, Y; Vamvoudakis, K. G; Ferraz, H; Modares, H. (January 2019, 2018 IEEE Conference on Decision and Control (CDC))

   This paper presents a Q-Iearning based dynamic intermittent mechanism to control linear systems evolving in continuous time. In contrast to existing event-triggered mechanisms, where complete knowledge of the system dynamics is required, the proposed dynamic intermittent control obviates this requirement while providing a quantified level of performance. An internal dynamical system will be introduced to generate the triggering condition. Then, a dynamic intermittent Q-Iearning is developed to learn the optimal value function and the hybrid controller. A qualitative performance analysis of the dynamic event-triggered control is given in comparison to the continuous-triggered control law to show the degree of suboptimality. The combined closed-loop system is written as an impulsive system, and it is proved to have an asymptotically stable equilibrium point without any Zeno behavior. A numerical simulation of an unknown unstable system is presented to show the efficacy of the proposed approach. 

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