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The main contribution of this work is to propose a design technique for state-feedback control of continuous-time piecewise-affine (PWA) systems that is robust not only to the system uncertainties, but also to variations of the controller gains. More specifically, this study presents sufficient conditions to synthesise a robust non-fragile PWA controller that exponentially stabilises the closed-loop equilibrium point. Furthermore, these conditions are cast as an optimisation problem subject to a set of linear matrix inequalities (LMIs), which can then be solved efficiently. In addition, a set of LMI conditions are derived to ensure that the control input will always meet a pre-assigned upper bound. Simulation results demonstrate the effectiveness of the proposed approach.

This paper deals with state feedback control synthesis for a boiler-turbine system using a set invariance property and a piecewise affine modelling procedure. The nonlinear model of the considered system is linearized around different equilibrium points. After that, the obtained piecewise affine model is exploited to design feedback gains that guarantee compliance with state and control constraints, as well as asymptotic stability of the closed-loop system. The feedback gains are computed by solving a linear programming problem, which can be achieved by numerous effective solvers. Finally, the validity and efficacy of the proposed approach are demonstrated through a numerical example.

Sliding dynamics is a peculiar phenomenon to discontinuous dynamical systems, while homoclinic orbits play a role in studying the global dynamics of dynamical systems. This paper provides a method to ensure the existence of sliding homoclinic orbits of three-dimensional piecewise affine systems. In addition, sliding cycles are obtained by bifurcations of the systems with sliding homoclinic orbits to saddles. Two examples with simulations of sliding homoclinic orbits and sliding cycles are provided to illustrate the effectiveness of the results.

The traditional model predictive control (tMPC) algorithms have a large amount of online calculation, which makes it difficult to apply them directly to turboshaft engine–rotor systems because of real time requirements. Therefore, based on the theory of the perturbed piecewise affine system (PWA) and multi-parameter quadratic programming explicit model predictive control (mpQP-eMPC) algorithm, we develop a controller design method for turboshaft engine–rotor systems, which can be used for engine steady-state, transient state and limit protection control. This method consists of two steps: controller offline design and online implementation. Firstly, the parameter space of the PWA system is divided into several partitions offline based on the disturbance and performance constraints. Each partition has its own control law, which is in the form of piecewise affine linear function between the controller and the parameters. The control laws for those partitions are also obtained in this offline step. After which, for the online control implementation step, the corresponding control law can be obtained by a real-time query of a corresponding partition, which the current engine state falls into. This greatly reduces the amount of online calculation and thus improves the real-time performance of the MPC controller. The effectiveness of the proposed method is verified by simulating the steady-state and transient process of a turboshaft engine–rotor system with a limit protection requirement. Compared with tMPC, an mpQP-eMPC based controller can not only guarantee good steady-state, dynamic control performance and limit protection, but can also significantly improve the real-time performance of the control system.

This paper considers design of chaos generator with a class of three-dimensional two-zone piecewise affine systems. A criterion is established to guarantee the existence of a heteroclinic cycle connecting two saddle-focus equilibrium points located in two zones, respectively, by which the existence of chaotic attractors in a neighborhood of the heteroclinic cycle can be easily obtained by Shil’nikov type theory. Based on the mathematical criteria proposed, we present a methodology for design of chaos generator.

It is a challenging task to prove mathematically the existence of homoclinic orbits in a high-dimensional dynamical system. Here we first introduce a new four-dimensional (4D) piecewise affine system and then establishes useful yet general conditions on the existence of an orbit homoclinic to a spiral saddle-foci and chaos in the system. Rigorously mathematical analysis is also provided. A 4D example with both a homoclinic orbit and an invariant chaotic set is used to depict the effectiveness of analysis and prediction.

In this paper, three chaos synchronization approaches using Linear Matrix Inequality (LMI) tools are evaluated and compared. The comparative analysis is supported by four examples of Piecewise affine (PWA) chaotic systems: The Chua’s original circuit, the Chua’s modified system, the Lur’e like circuit and the five-scroll attractor system. To evaluate the performances of each synchronization approach, we examine first, the practical implementation of the LMIs. We analyze then, by simulation results, the feasibility of each approach for each PWA chaotic system. The elapsed time for solving the predefined LMIs and the influence of their tuning parameters’ domain belonging on the feasibility and the performances of each approach are finally the considered comparative criteria.

The problem of event-triggered control is investigated for discrete-time piecewise affine systems subject to input saturation. Linear matrix inequality (LMI) based on local stability criterion is formed by introducing a nonlinear dead-zone function which represent saturation characteristics. Unique state feedback control gains can be obtained by solving the LMI problem under the event-triggering strategy. The domain of attraction is estimated and maximized with the controller synthesis via ellipsoidal approximations. The effectiveness of controller is illustrated by simulations of numerical examples.

The existence of homoclinic orbits or heteroclinic cycle plays a crucial role in chaos research. This paper investigates the existence of the homoclinic orbits to a saddle-focus equilibrium point in several classes of three-dimensional piecewise affine systems with two switching planes regardless of the symmetry. An analytic proof is provided using the concrete expression forms of the analytic solution, stable manifold, and unstable manifold. Meanwhile, a sufficient condition for the existence of two homoclinic orbits is also obtained. Furthermore, two concrete piecewise affine asymmetric systems with two homoclinic orbits have been constructed successfully, demonstrating the method’s effectiveness.

Since complicated dynamical behavior can occur easily near homoclinic trajectory or heteroclinic cycle in dynamical systems with dimension not less than three, this paper investigates the existence of heteroclinic cycles in some class of 3-dimensional three-zone piecewise affine systems with two switching planes. Based on the exact determination of the stable manifold, unstable manifold and analytic solution, a rigorous analytic methodology of designing chaos generators is proposed, which may be of potential applications to chaos secure communication. Furthermore, we obtain three sufficient conditions for the existence of a single or two heteroclinic cycles in three different cases. Finally, some examples are given to illustrate our theoretical results.