L2-gain analysis for a class of hybrid systems with applications to reset and event-triggered control: A lifting approach

In this paper we study the stability and L2-gain properties of a class of hybrid systems that exhibit linear flow dynamics, periodic time-triggered jumps and arbitrary nonlinear jump maps. This class of hybrid systems is relevant for a broad range of applications including periodic event-triggered control, sampled-data reset control, sampled-data saturated control, and certain networked control systems with scheduling protocols. For this class of continuous-time hybrid systems we provide new stability and L2-gain analysis methods. Inspired by ideas from lifting we show that the stability and the contractivity in L2-sense (meaning that the L2-gain is smaller than 1) of the continuous-time hybrid system is equivalent to the stability and the contractivity in l2-sense (meaning that the l2-gain is smaller than 1) of an appropriate discrete-time nonlinear system. These new characterizations generalize earlier (more conservative) conditions provided in the literature.We show via a reset control example and an event- triggered control application, for which stability and contractivity in L2-sense is the same as stability and contractivity in l2-sense of a discrete-time piecewise linear system, that the new conditions are significantly less conservative than the existing ones in the literature. Moreover, we show that the existing conditions can be reinterpreted as a conservative l2-gain analysis of a discretetime piecewise linear system based on common quadratic storage/ Lyapunov functions. These new insights are obtained by the adopted lifting-based perspective on this problem, which leads to computable l2-gain (and thus L2-gain) conditions, despite the fact that the linearity assumption, which is usually needed in the lifting literature, is not satisfied

L2-gain analysis of periodic event-Triggered systems with varying delays using lifting techniques

In this paper we study the stability and L2-gain properties of periodic event-Triggered control (PETC) systems including time-varying delays. We introduce a general framework that captures these PETC systems and encompasses a class of hybrid systems that exhibit linear flow, aperiodic time-Triggered jumps (possibly with different deadlines) and arbitrary nonlinear time-varying jump maps. New notions on the stability and contractivity (L2-gain strictly smaller than 1) from the beginning of the flow and from the end of the flow are introduced and formal relationships are deduced between these notions, revealing that some are stronger than others. Inspired by ideas from lifting, it is shown that the internal stability and contractivity in L2-sense of a continuous-Time hybrid system in the framework is equivalent to the stability and contractivity in l2-sense of an appropriate time-varying discrete-Time nonlinear system. These results recover existing works in the literature as special cases and indicate that analysing different discrete-Time nonlinear systems (of the same level of complexity) than in existing works yield stronger conclusions on the L2-gain. At the end of the paper we indicate several extensions of the framework, which even include the possibility of the interjump times depending on the state, such that, for instance, self-Triggered control systems can also be included allowing their stability and contractivity analysis. A numerical example is presented showing how stability and contractivity analyses are carried out for PETC systems with delays

Time-regularized and periodic event-triggered control for linear systems

In this chapter, we provide an overview of our recent results for the analysis and design of Event-Triggered controllers that are tailored to linear systems as provided in Heemels et al., IEEE Trans Autom Control 58(4):847–861, 2013, Heemels et al., IEEE Trans Autom Control 61(10):2766–2781, 2016, Borgers et al., IEEE Trans Autom Control, 2018. In particular, we discuss two different frameworks for the stability and contractivity analysis and design of (static) periodic Event-Triggered control (PETC) and time-regularized continuous Event-Triggered control (CETC) systems: the lifting-based framework of Heemels et al., IEEE Trans Autom Control 61(10):2766–2781, 2016, which applies to PETC systems, and the Riccati-based framework of Heemels et al., IEEE Trans Autom Control 58(4):847–861, 2013, Borgers et al., IEEE Trans Autom Control (2018), which applies to both PETC systems and time-regularized CETC systems. Moreover, we identify the connections and differences between the two frameworks. Finally, for PETC and time-regularized CETC systems, we show how the Riccati-based analysis leads to new designs for dynamic Event-Triggered controllers, which (for identical stability and contractivity guarantees) lead to a significantly reduced consumption of communication and energy resources compared to their static counterparts

Cyber attack mitigation for cyber-physical systems: hybrid system approach to controller design

This study considers robust controller design for cyber-physical systems (CPSs) subject to cyber attacks. While previous studies have investigated secure control by assuming specific attack strategies, in this study the authors propose a robust hybrid control scheme containing multiple sub-controllers, each matched to a different type of cyber attack. A system using this control scheme is able to adapt its behaviour to various cyber attacks (including those which have not been specifically addressed in the sub-controller designs) by switching sub-controllers to achieve the best performance. They propose a method for designing the secure switching logic to counter possible cyber attacks and to mathematically verify the system's performance and stability as well. The performance of the proposed control scheme is demonstrated by the hybrid H-2-H-infinity controller applied to a CPS subject to cyber attacks

Robust Gain Scheduling for Smart-Structures in Parallel Robots

Smart-structures offer the potential to increase the productivity of parallel robots by reducing disturbing vibrations caused by high dynamic loads. In parallel robots the vibration behavior of the structure is position dependent. A single robust controller is not able to gain satisfying control performance within the entire workspace. Hence, vibration behavior is linearized at several operating points and robust controllers are designed. Controllers can be smoothly switched by gain-scheduling. A stability proof for fast varying scheduling parameters based on the Small-Gain Theorem is developed. Experimental data from Triglide, a four degree of freedom (DOF) parallel robot of the Collaborative Research Center 562, validate the presented concepts