Wei Wang

Input-to-State Stability and Averaging of Linear Fast Switching Systems

We consider the averaging method for stability of rapidly switching linear systems with disturbances. We show that the notions of strong and weak averages proposed, with partial strong average defined in this note, play an important role in the context of switched systems. Using these notions of average, we show that exponential input-to-state stability (ISS) of the strong and the partial strong average system with linear gain imply exponential ISS with linear gain of the actual system. Similarly, exponential ISS of the weak average guarantees an appropriate exponential derivative ISS (DISS) property for the actual system. Moreover, using the Lyapunov method, we show that linear ISS gains of the actual system and its average converge to each other as the switching rate is increased.

Analysis for a class of singularly perturbed hybrid systems via averaging

A class of singularly perturbed hybrid dynamical systems is analyzed. The fast states are restricted to a compact set a priori. The continuous-time boundary layer dynamics produce solutions that are assumed to generate a well-defined average vector field for the slow dynamics. This average, the projection of the jump map in the direction of the slow states, and flow and jump sets from the original dynamics define the reduced, or average, hybrid dynamical system. Assumptions about the average system lead to conclusions about the original, higher-dimensional system. For example, forward pre-completeness for the average system leads to a result on closeness of solutions between the original and average system on compact time domains. In addition, global asymptotic stability for the average system implies semiglobal, practical asymptotic stability for the original system. We give examples to illustrate the averaging concept and to relate it to classical singular perturbation results as well as to other singular perturbation results that have appeared recently for hybrid systems. We also use an example to show that our results can be used as an analysis tool to design hybrid feedbacks for continuous-time plants implemented by fast but continuous actuators.

Path Following for Nonlinear Systems With Unstable Zero Dynamics: An Averaging Solution

We consider a path-following problem in which the goal is to ensure that the error between the system output and the geometric path is asymptotically less than a prespecified constant, while guaranteeing a forward motion along the path and boundedness of all states. Comparing with the results on this problem, we exploit averaging techniques to develop an alternative simpler solution for a class of nonlinear systems and for paths satisfying a certain geometric condition.

Input-to-state stability for a class of hybrid dynamical systems via averaging

Input-to-state stability (ISS) properties for a class of time-varying hybrid dynamical systems via averaging method are considered. Two definitions of averages, strong average and weak average, are used to approximate the time-varying hybrid systems with time-invariant hybrid systems. Closeness of solutions between the time-varying system and solutions of its weak or strong average on compact time domains is given under the assumption of forward completeness for the average system. We also show that ISS of the strong average implies semi-global practical (SGP)-ISS of the actual system. In a similar fashion, ISS of the weak average implies semi-global practical derivative ISS (SGP-DISS) of the actual system. Through a power converter example, we show that the main results can be used in a framework for a systematic design of hybrid feedbacks for pulse-width modulated control systems.

Emulation-based stabilization of networked control systems implemented on FlexRay

We investigate the emulation controller design approach for nonlinear networked control systems (NCS) with FlexRay. FlexRay is a deterministic communication protocol which is increasingly used in the automotive industry as it provides a high bandwidth and allows for safety critical applications. It is characterized by pre-set communication cycles that are subdivided into static and dynamic segments; the data transmissions are scheduled by different rules depending on the segment. We propose for the first time a hybrid model of NCS with FlexRay for this purpose. We show, under reasonable assumptions, that the asymptotic stability property ensured by the controller in the absence of communication constraints is preserved when the latter is implemented over FlexRay with sufficiently frequent data transmission. In particular, we assume that on each communication segment, the data transmissions are governed by uniformly globally exponentially stable protocols. This covers the case when the round-robin protocol is implemented on the static segment and the try-once-discard protocol is implemented on the dynamic segment. We provide explicit maximum allowable transmission interval bounds that guarantee stability.

Stabilization of nonlinear systems using state-feedback periodic event-triggered controllers

We investigate the scenario where a controller communicates with a plant at discrete time instants generated by an event-triggering mechanism. In particular, the latter collects sampled data from the plant and the controller at each sampling instant, and then decides whether the control input needs to be updated, leading to periodic event-triggered control (PETC). In this paper, we propose a systematic design procedure for PETC that stabilize general nonlinear systems. The design is based on the existence of a continuous-time state-feedback controller, which stabilizes the system in the absence of communication constraints. We then take into account the sampling and we design an event-triggering condition, which is only updated at some of the sampling instants, to preserve stability. An explicit bound on the maximum sampling period with which the triggering rule is evaluated is provided. We show that there exists a trade-off between the latter and a parameter used to define the triggering condition. The results are applied to a van de Pol oscillator as an illustration.

Averaging in singularly perturbed hybrid systems with hybrid boundary layer systems

We analyze a class of singularly perturbed hybrid systems based on two auxiliary hybrid systems: the averaged system, which approximates the slow dynamics, and the boundary layer system, which approximates the fast dynamics. The average system is generated by averaging the solutions of the boundary layer system. The novelty of this work is that the boundary layer system is a hybrid system rather than a continuous-time system. This extends available results to cover new classes of hybrid systems. We illustrate how to apply our results through an example that is a power converter system under hybrid feedbacks implemented by pulse-width modulation (PWM).

Observer design for networked control systems with FlexRay

Abstract We design state observers for nonlinear networked control systems (NCS) implemented over FlexRay. FlexRay is a communication protocol used in the automotive industry, which has the feature to switch between two scheduling rules during its communication cycles. These switches induce technical difficulties when modeling, designing and analyzing observers for such systems compared to standard NCS. We present a solution based on the emulation approach. Given an observer in the absence of communication constraints, we implement it over the network and we provide sufficient conditions on the latter, to preserve the stability property of the observer. In particular, we provide explicit bounds on the maximal allowable transmission intervals, which adapt to the lengths of the segment associated to each scheduling rule. We assume that the plant dynamics and measurements are affected by noise and we guarantee an input-to-state stability property for the corresponding estimation error system. The overall system is modeled as a hybrid system and the analysis relies on the use of a novel hybrid Lyapunov function.

Design of observers implemented over FlexRay networks

We investigate the observer design for nonlinear systems whose measurements are sent over a network governed by FlexRay. FlexRay is a communication protocol used in the automotive industry which has the feature to switch between two scheduling rules associated with the two segments of its communication cycles. The objective of this paper is to generalize existing works on emulated observers for networked control systems (NCS) to be applicable to NCS with FlexRay. We propose for that purpose a novel hybrid model and guarantee the observer convergence provided that, for each segment, the scheduling rules are uniformly globally exponentially stable and the maximal allowable transmission intervals satisfy given explicit bounds. The analysis relies on the use of an hybrid Lyapunov function we recently constructed to investigate the stabilization of NCS with FlexRay. We finally apply the approach to a class of globally Lipschitz systems, which includes linear time-invariant systems as a particular case.

Novel results in averaging analysis of singularly perturbed hybrid systems

We investigate stability of a class of singularly perturbed systems whose slow system is a set-valued average defined via an appropriate averaging procedure of the solutions of the continuous-time boundary layer system. An approximate hybrid system consisting of this average, the projection of the jump map in the direction of the slow states and flow and jump sets from the original dynamics is shown to approximate the actual singularly perturbed hybrid system. In particular, using forward pre-completeness of the average system we show that solutions of the actual and approximate systems are close in an appropriate sense on compact time intervals. It is also shown that global asymptotic stability of the average system implies semi-global practical asymptotic stability of the actual system. Several examples are presented to illustrate our results and relate them to previously published results in the literature.