Class Of Hybrid Dynamical Systems Research Articles

Semiglobal High-Gain Hybrid Observer for a Class of Hybrid Dynamical Systems With Unknown Jump Times

Semiglobal High-Gain Hybrid Observer for a Class of Hybrid Dynamical Systems With Unknown Jump Times

Robust parameter estimation for hybrid dynamical systems with linear parametric uncertainty

We consider the problem of estimating a vector of unknown constant parameters for a class of hybrid dynamical systems – that is, systems whose state variables exhibit both continuous (flow) and discrete (jump) evolution – with dynamics that depend linearly on the unknown parameters. Using a hybrid systems framework, we propose a hybrid estimation algorithm that can operate during both flows and jumps that, under a notion of hybrid persistence of excitation, guarantees convergence of the parameter estimate to the true value. Furthermore, we show that the parameter estimate is input-to-state stable with respect to a class of hybrid disturbances. Simulation results including a spacecraft application show the merits of our proposed approach.

Explicit hybrid MPC for the lateral stabilization of electric vehicle system

This paper presents a hybrid Model Predictive Control (hMPC) approach to improve Electric Vehicle (EV) stability when cornering or in high-risk driving conditions. By using approximations of tire force characteristics, the EV system proposed in this study is considered as a Piecewise Affine (PWA) system which is a class of Hybrid Dynamical Systems (HDSs). First, a hybrid modeling of the studied system is developed within the PWA framework. Then, the synthesis of an explicit hybrid MPC law, based on the integration of the Active Front-wheel Steering control method (AFS) and Direct Yaw moment Control method (DYC), can be established. The purpose of the PWA-MPC is to force the yaw rate and sideslip angle of the EV system to follow the given references to guarantee lateral stability. The calculation of the control laws of this proposed approach is performed offline using multiparametric programming. Finally, this paper investigates the effectiveness of the designed controller by performing a set of simulations to analyze the efficiency and robustness of the proposed approach. A comparative study is addressed in this work to demonstrate the efficiency of the proposed hybrid MPC controller compared to the conventional MPC approach which considers the linear model of the system. Another simulations are carried out to assess the robustness of the proposed control, important disturbances and uncertainties were incorporated such as the change of the additional mass, the change of terrain and the high velocity. The simulation results show that the PWA EV system model with the designed controller is able to ensure lateral stability by keeping the sideslip angle and yaw rate tracking the ideal values.

Diagnosis and fault tolerant control against actuator fault for a class of hybrid dynamic systems

Over the past few decades, there have been increasing research activities in fault diagnosis (FD) and fault-tolerant control (FTC) for switched hybrid systems. This paper addresses the problem of active-fault tolerant control (AFTC) for switched hybrid systems subject to actuator faults to enhance system security and keep system stability. The proposed FTC is designed by adding the state feedback control with integral action to an additive control law which requires accurate fault estimation to compensate for the fault effect. Thus, a data-based projection method (DPM) is extended (EDPM) based on inputs and outputs measures to estimate the fault without using mathematical models. The synthesis of the state feedback control with integral action is proposed for recovering the desired performances. It integrates a set of controllers corresponding to a set of partial models to design a set of switching control laws. Indeed, new linear matrix inequalities (LMIs) using Lyapunov stability analysis are proposed to find the optimal values of the control gains matrices and keeping system stability. A comparative study of the proposed FTC with existing work is given to show the effectiveness of the proposed technique.

Fault-tolerant control and sinusoidal disturbance attenuation for a class of hybrid dynamic systems

The design of an accurate fault-tolerant control fault-tolerant control is still an active researches of switched hybrid system whose switching mechanism is unknown. In this paper, a new approach is proposed to deal with switching mode estimation and control problems for switched system subject to actuator faults and disturbances. First, the switching mode is estimated off-line using the Chiu clustering algorithm. Then, actuator fault estimation is proposed using data-based projection approach based on performance measures that allow fault estimation without relying on mathematical models and fault assumptions. Second, a novel switching control strategy is developed for actuator fault compensation and recovering the desired performances. These strategy integrate a set of controllers, corresponding to a set of partial modes to design a bank of switching control laws, compensating for the actuator fault effect and attenuating the sinusoidal disturbance. A new linear matrix inequality is proposed to compute control gains based on Lyapunov stability. This approach allows finding the optimal control gains matrices and ensure both actuator fault compensation and disturbance attenuation. Simulation results of two numerical examples shows the effectiveness of the proposed estimation and control strategies in spite of the presence of actuator faults and sinusoidal disturbance.

Non-smooth Modeling and Variable Structure Control of a Class of Hybrid Dynamical Systems

In this work, we propose a modeling formulation and controller design for a class of hybrid dynamical systems. In this formulation, a switching dynamical system is modeled as a dynamical system with discontinuous right hand side. More specifically, the system is transformed to a nonlinear system with discontinuous nonlinearities. Then, a synthesis of feedback linearization and sliding mode control is employed for output tracking control problem. Application and implementation of this approach is illustrated via a chemical process example.

Stochastic exponential stabilization of hybrid systems with time‐varying delays via adaptive control

The problem of stochastic stabilization for a class of hybrid dynamical systems with time-varying delays is investigated by adaptive control approach. An adaptive controller is designed to stabilize the hybrid systems, and a sufficient criterion for exponential stabilization is derived based on certain conditions. The time-varying delay observations on the states of noise are made and the least upper bound of time-varying delays is also obtained. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed new techniques.

Smooth Approximations for Hybrid Optimal Control Problems with Application to Robotic Walking

Abstract This paper investigates optimal control problems formulated over a class of hybrid dynamical systems which display event-triggered discrete jumps. Due to the discontinuous nature of the underlying dynamics, previous approaches to solving optimal control problems over this class of systems generally rely on fixing the number and sequence of discrete jumps a priori, or search over possible mode sequences in a combinatorial manner. Employing contributions from the geometric theory of hybrid systems, we instead formulate a family of smooth approximate problems formulated over a family of smooth control systems which faithfully approximate the dynamics of the original hybrid system, in an appropriate metric. Efficient gradient-based methods can be used to solve the smooth approximations, without specifying the sequence of discrete transitions ahead of time. Under appropriate hypothesis, the gradients of the smooth problem are shown to be well-conditioned and closely approximate the gradients of the non-smooth problem (when they exist). Two cases studies demonstrate the utility of the approach, including an in-depth application to generating a stable walking motion for a bipedal robot.

Conley's Fundamental Theorem for a Class of Hybrid Systems

We establish versions of Conley's (i) fundamental theorem and (ii) decomposition theorem for a broad class of hybrid dynamical systems. The hybrid version of (i) asserts that a globally defined hyb...

Finite-time stability and controller design for a class of hybrid dynamical systems with deviating argument

This paper studies the finite-time stability (FTS) for a class of hybrid dynamical systems with deviating argument. An improved hybrid control scheme including sampled-data control as well as impulsive control is presented. Based on the theory of differential equations with piecewise constant argument of generalized type (PCAG) and the method of average impulsive interval (AII), several Lyapunov-based sufficient criteria for FTS are obtained in terms of linear matrix inequalities (LMIs), which can be verified via Matlab. The hybrid controller, in which the sampling instants could be different from the impulse instants, is designed by the established LMIs. The results in present paper are more convenient for application and less conservative than some existing ones. Finally, an example is given to illustrate the effectiveness and advantage of the obtained results.

Identification of a Class of Hybrid Dynamical Systems

This paper presents a novel identification procedure for a class of hybrid dynamical systems. In particular, we consider hybrid dynamical systems which are single flowed and single jumped and whose flow and jump maps linearly depend on two sets of unknown parameters. A systematic way to determine whether the system is flowing or jumping is introduced and used to identify the unknown parameters by employing a linear recursive estimator. Simulations have been performed to prove the validity of the proposed methodology. Results proved the efficiency and accuracy of the developed identification procedure.

Dynamics and Switching Control of a Class of Underactuated Mechanical Systems with Variant Constraints

Locomotion systems with variant constraints are familiar in real world applications, but the dynamics and control issues of variant constraint systems have not been sufficiently discussed to date. From the viewpoint of Lagrange–d’Alembert equations with additional variable constraints, this paper investigates the modeling approaches of a class of hybrid dynamical systems (HDS) with instantaneously variant constraints and the switching control techniques of stabilizing the HDS to given periodic orbits. It is shown that under certain conditions there possibly exist zero impact periodic orbits in the HDS, and the HDS can be stabilized to the period-one orbits by a linear controller with only partial state feedback, even though the HDS are generally underactuated nonholonomic systems. As an example, a one-legged planar hopping robot is employed to demonstrate the main results of modeling and control of a class of HDS.

Stability under events for a class of hybrid dynamical systems with continuous and discrete time variables

This studies the stability under events for a class of hybrid dynamical systems (HDSs). The HDS is modelled via continuous and discrete time (CDT) variables. It is allowed that an HDS has time-varying, unstable, non-linear, different dwell time, and infinite number of subsystems. The stability under events reflects the effects of switching and impulse type events in HDS. By defining the length of CDT variable and extending the notion of hybrid-event-time in the literature, the concepts on global uniform asymptotic stability (GUAS) and event-GUAS are reasonably defined. By employing multiple Lyapunov-like functions, criteria on GUAS and event-GUAS are established. Moreover, by extending average dwell time (ADT) to hybrid ADT, the relations between GUAS and event-GUAS are derived. And the criteria on global uniform exponential stability under events (event-GUES) are obtained. A method via maximal admissible probabilities of unstable subsystems is proposed to test the conditions of event-GUES. As two special cases, the traditional stability criteria of only continuous or discrete variables are also derived. Finally, three examples are provided to verify the main results.

Tracking Fault Tolerant Control for Hybrid System: Two-Link Human Arm Application

This paper proposes a novel fault tolerant control (FTC) scheme for a class of hybrid dynamical system (HDS) subject to sensor faults. The corresponding FTC architecture is designed around a reconfiguration mechanism. It aims to compensate the effects of the sensors degradation and maintain satisfactory performances including continuous stability. Moreover, by using the linear matrix inequalities (LMI) approach, a fault estimation algorithm is fulfilled and the compromise between robustness to disturbances and sensitivity to fault is guaranteed. For the sake of trajectory tracking, a combined robust state feedback and proportional-integral-derivative control system is proposed herein. Finally, extensive simulation results conducted on two-link arm system are included to illustrate the efficiency of the designed FTC scheme.

Modelling, simulation and control of a class of hybrid dynamic systems: using hybrid automaton, APROS and mixed integer quadratic optimisation algorithm

This paper presents modelling, simulation and control of a class of hybrid dynamic systems. Hybrid automaton is used for modelling a transition system with continuous dynamics. The framework consists of a finite set of state and transition for modelling a discrete dynamics called control mode. Each control mode describes continuous dynamics. Using APROS tools, it permits to simulate the behaviour of the hybrid system approaching in the experiment case: actuators, pumps and valves. Mixed logic dynamics formalism allows to describe the dynamics defined by logic rules, continuous dynamics and constraints. These are described by linear dynamic equations subject to linear inequalities involving continuous, discrete and auxiliary variables. This model is used to synthesise a predictive control law under constraints. The controller requires online mixed-integer quadratic programming solution to an optimisation problem. Simulation was performed to illustrate performances and efficiency of these methods and tools.

Modelling, simulation and control of a class of hybrid dynamic systems: using hybrid automaton, APROS and mixed integer quadratic optimisation algorithm

Modelling, simulation and control of a class of hybrid dynamic systems: using hybrid automaton, APROS and mixed integer quadratic optimisation algorithm

Boundness and Linearisation of a Class of Differential Equations with Piecewise Constant Argument

The differential equations with piecewise constant argument (DEPCAs, for short) is a class of hybrid dynamical systems (combining continuous and discrete). In this paper, under the assumption that the nonlinear term is partially unbounded, we study the bounded solution and global topological linearisation of a class of DEPCAs of general type. One of the purpose of this paper is to obtain a new criterion for the existence of a unique bounded solution, which improved the previous results. The other aim of this paper is to establish a generalized Grobman-Hartman-type theorem for the topological conjugacy between a nonlinear perturbation system and its linear system. The method is based on the new obtained criterion for bounded solution. The obtained results generalized and improved some previous papers. Some novel techniques are employed.

Robust Stability of Hybrid Limit Cycles With Multiple Jumps in Hybrid Dynamical Systems

For a broad class of hybrid dynamical systems, we establish results for robust asymptotic stability of hybrid limit cycles with multiple jumps per period. Hybrid systems are given in terms of differential and difference equations with set constraints. Hybrid limit cycles are given by compact sets defined by periodic solutions that flow and jump. Under mild assumptions, we show that asymptotic stability of such hybrid limit cycles is not only equivalent to asymptotic stability of a fixed point of the associated Poincare map but also robust to perturbations. Specifically, robustness to generic perturbations, which capture state noise and unmodeled dynamics, and to inflations of the flow and jump sets are established in terms of ${\mathcal K}{\mathcal L}$ bounds. A two-gene network with binary hysteresis is presented to illustrate the notions and results throughout the paper.

Input-to-state-[formula omitted]-stability and criteria for a class of hybrid dynamical systems

This paper aims to study input-to-state-KL-stability (IS-KL-stability) for a class of hybrid dynamical systems (HDS). The notions of IS-KLK0-property and IS-KL-stability are proposed. The IS-KLK0-property, including input-to-state stability (ISS), IS-KL-stability, and some non-ISS property, is a more general input-to-state property and reflects the specific effects on the dynamic behaviors from the dynamics of subsystems, switching events, and external inputs in HDS. An HDS with IS-KLK0-property is allowed to have non-ISS property. The criteria of IS-KL-stability are established for HDS with IS-KLK0-property subsystems via introducing a hybrid dwell time condition. The hybrid dwell time condition unifies the often used average dwell-time condition and converse average dwell-time condition. Moreover, the criteria on IS-KL-stability including event-ISS are established by the method of multiple Lyapunov-like functions. Finally, two examples are provided to verify the theoretical results.

$\mathcal{KL}_*$ -stability for a class of hybrid dynamical systems

Abstract This article studies $\mathcal{KL}_*$-stability (the stability expressed by $\mathcal{KL}_*$-class function) for a class of hybrid dynamical systems (HDS). The notions of $\mathcal{KL}_{*}\mathcal{K}_{*}$-property and $\mathcal{KL}_{*}$-stability are proposed for HDS with respect to the hybrid-event-time. The $\mathcal{KL}_{*}$-stability, which is based on $\mathcal{K}$ or $\mathcal{L}$ property of the continuous flow, the discrete jump, and the event in an HDS, extends the $\mathcal{KLL}$-stability and the event-stability reported in the literature for HDS. The relationships between $\mathcal{KL}_{*}\mathcal{K}_{*}$-property and $\mathcal{KL}_{*}$-stability are established via introducing the hybrid dwell-time condition (HDT). The HDT generalizes the average dwell-time condition in the literature. For an HDS with $\mathcal{KL}_{*}\mathcal{K}_{*}$-property consisting of stabilizing $\mathcal{L}$-property and destabilizing $\mathcal{K}$-property, it is shown that there exists a common HDT under which the HDS will achieve $\mathcal{KL}_{*}$-stability. Thus HDT may help to derive some easily tested conditions for HDS to achieve uniform asymptotic stability. Moreover, a criterion of $\mathcal{KL}_{*}$-stability is derived by using the multiple Lyapunov-like functions. Examples are given to illustrate the obtained theoretical results.