Continuous-time Hybrid Systems Research Articles

Hybrid Optimal Tracking Control for Linear Continuous-Time Systems by Simultaneous Optimization Approach of System States and Mode Distributions

In this paper, we study optimal LQG tracking control problems for linear continuous-time hybrid systems. We adopt an approach to prepare some candidates of mode distributions and minimize value functions at each time for the systems and performance indices. By the approach, system states and mode distributions are simultaneously optimized so that the performance indices are minimized for available information of reference signals. We consider optimization problems for averaged systems and averaged performance indices throughout mode distributions. Finally we give numerical examples and verify that we can obtain both optimal tracking performance and optimal candidate of mode distributions by the algorithm presented in this paper.

Hybrid Optimal Tracking Control by Most Probable Trajectory Approach

In this paper, we study optimal tracking control problems for linear continuous-time hybrid systems. We adopt most probable trajectory approach to control system states and mode distributions simultaneously. We consider optimization problems for averaged systems and averaged performance indices throughout mode distributions.

<inline-formula> <tex-math notation="LaTeX">$\mathcal{L}_{2}$</tex-math> </inline-formula>-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 L 2 -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 L 2 -gain analysis methods. Inspired by ideas from lifting we show that the stability and the contractivity in L 2 -sense (meaning that the L 2 -gain is smaller than 1) of the continuous-time hybrid system is equivalent to the stability and the contractivity in L 2 -sense (meaning that the l 2 -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 L 2 -sense is the same as stability and contractivity in l 2 -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 l 2 -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 L 2 -gain (and thus L 2 -gain) conditions, despite the fact that the linearity assumption, which is usually needed in the lifting literature, is not satisfied.

An EM-based identification algorithm for a class of hybrid systems with application to power electronics

In this paper we present an identification algorithm for a class of continuous-time hybrid systems. In such systems, both continuous-time and discrete-time dynamics are involved. We apply the expectation-maximisation algorithm to obtain the maximum likelihood estimate of the parameters of a discrete-time model expressed in incremental form. The main advantage of this approach is that the continuous-time parameters can directly be recovered. The technique is particularly well suited to fast-sampling rates. As an application, we focus on a standard identification problem in power electronics. In this field, our proposed algorithm is of importance since accurate modelling of power converters is required in high- performance applications and for fault diagnosis. As an illustrative example, and to verify the performance of our proposed algorithm, we apply our results to a flying capacitor multicell converter.

Safety verification and reachability analysis for hybrid systems

Safety verification and reachability analysis for hybrid systems is a very active research domain. Many approaches that seem quite different, have been proposed to solve this complex problem. This paper presents an overview of various approaches for autonomous, continuous-time hybrid systems and presents them with respect to basic problems related to verification.

Sampled-data control of hybrid systems with discrete inputs and outputs

We address the control synthesis of hybrid systems with discrete inputs, disturbances and outputs. The control objective is to ensure that the events of the closed-loop system belong to the language of the control requirements. The controller is sampling-based and it is representable by a finite-state machine. We formalize the control problem and provide a theoretically sound solution. The solution is based on solving a discrete-event control problem for a finite-state abstraction of the plant. We propose a specific construction for the finite-state abstraction. This construction is not based on discretizing the state-space, but rather on converting the continuous-time hybrid system to a discrete-time one based on sampling. The construction works only for a specific class of hybrid systems. We describe this class of systems and we provide an example of such a system, inspired by an industrial use-case.

Rigorous model-based safety analysis for nonlinear continuous-time systems

A method is presented for the quantitative, model-based safety analysis of nonlinear continuous-time hybrid systems. This method uses the region-transition-model (RTM) framework of [Huang, H., Adjiman, C. S., & Shah, N. (2002). Quantitative framework for reliable safety analysis. AIChE Journal, 48, 78–96], together with a recently developed technique [Lin, Y., & Stadtherr, M. A. (2007). Validated solutions of initial value problems for parametric ODEs. Applied Numerical Mathematics, 57, 1145–1162] for the rigorous global analysis of nonlinear, continuous-time systems with uncertain initial conditions and/or parameters. Given an operating region described by bounds on possible initial conditions, inputs and model parameters, and a finite time horizon, the method can determine which operating subregions lead to safe operation. Numerical examples are presented that demonstrate the effectiveness of the method. This approach can supplement and complement the more qualitative techniques that are widely used for hazard identification and safety analysis.

OPTIMIZATION OF DYNAMIC OUTPUT FEEDBACK CONTROLLER FOR PIECEWISE AFFINE SYSTEMS: AN LMI APPROACH

This note proposes a systematic method to optimizing dynamic output feedback controller for continuous-time hybrid systems in the Piecewise Affine (PWA) form. In this note, design of the dynamic output feedback controller is shown in the frame of L2 gain synthesis for PWA systems. Matrix inequalities derived from dynamic output feedback controller is not usually linear. Using a congruence transformation, the nonlinear matrix inequalities are mapped into linear matrix inequalities, whose solutions are quite feasible using commercially available software. The proposed techniques satisfy features such as: globally quadratic stability, continuity of control signal, and tracking a reference input. A numerical example is included to demonstrate the applicability and effectiveness of the proposed technique.

Safety verification and reachability analysis for hybrid systems

Safety verification and reachability analysis for hybrid systems is a very active research domain. Many approaches that seem quite different, have been proposed to solve this complex problem. This paper presents an overview of various approaches for autonomous, continuous time hybrid systems and present them with respect to basic problems related to verification.

DYNAMICAL ANALYSIS OF A NETWORKED CONTROL SYSTEM

A new network data transmission strategy was proposed in [Zhang & Chen, 2005], where the resulting nonlinear system was analyzed and the effectiveness of the transmission strategy was demonstrated via simulations. In this paper, we further generalize the results of Zhang and Chen [2005] in the following ways: (1) Construct first-return maps of the nonlinear systems formulated in [Zhang & Chen, 2005] and derive several existence conditions of periodic orbits and study their properties. (2) Formulate the new system as a hybrid system, which will ease the succeeding analysis. (3) Prove that this type of hybrid systems is not structurally stable based on phase transition which can be applied to higher-dimensional cases effortlessly. (4) Simulate a higher-dimensional model with emphasis on their rich dynamics. (5) Study a class of continuous-time hybrid systems as the counterparts of the discrete-time systems discussed above. (6) Propose new controller design methods based on this network data transmission strategy to improve the performance of each individual system and the whole network. We hope that this research and the problems posed here will rouse the interest of researchers in such fields as control, dynamical systems and numerical analysis.

Synthesis of supervisory controllers for hybrid systems based on approximating automata

The paper concerns the synthesis of supervisory controllers for a class of continuous-time hybrid systems with discrete-valued input signals that select differential inclusions for continuous-valued state trajectories and event-valued output signals generated by threshold crossings in the continuous state space, the supervisor is allowed to switch the input signal value when threshold events are observed. The objective is to synthesize a nonblocking supervisor such that the set of possible sequences of control and threshold event pairs for the closed-loop system lies between given upper and lower bounds in the sense of set containment. We show how this problem can be converted into a supervisor synthesis problem for a standard controlled discrete-event system (DES). A finite representation may not exist for the exact DES model of the hybrid system, however. To circumvent this difficulty, we present an algorithm for constructing finite-state Muller automata that accept outer approximations to the exact controlled threshold-event language, and we demonstrate that supervisors that solve the synthesis problem for the approximating automata achieve the control specifications when applied to the original hybrid system.