This paper explores the existence of heteroclinic cycles and corresponding chaotic dynamics in a class of 3-dimensional two-zone piecewise affine systems. Moreover, the heteroclinic cycles connect two saddle foci and ...This paper explores the existence of heteroclinic cycles and corresponding chaotic dynamics in a class of 3-dimensional two-zone piecewise affine systems. Moreover, the heteroclinic cycles connect two saddle foci and intersect the switching manifold at two points and the switching manifold is composed of two perpendicular planes.展开更多
The main contribution of this paper is to present stability synthesis results for discrete-time piecewise affine （PWA） systems with polytopic time-varying uncertainties and for discrete-time PWA systems with norm-bo...The main contribution of this paper is to present stability synthesis results for discrete-time piecewise affine （PWA） systems with polytopic time-varying uncertainties and for discrete-time PWA systems with norm-bounded uncertainties respectively.The basic idea of the proposed approaches is to construct piecewise-quadratic （PWQ） Lyapunov functions to guarantee the stability of the closed-loop systems.The partition information of the PWA systems is taken into account and each polytopic operating region is outer approximated by an ellipsoid,then sufficient conditions for the robust stabilization are derived and expressed as a set of linear matrix inequalities （LMIs）.Two examples are given to illustrate the proposed theoretical results.展开更多
In this paper, global input-to-state stability （ISS） for discrete-time piecewise affine systems with time-delay are considered Piecewise quadratic ISS-Lyapunov functions are adopted. Both Lyapunov-Razumikhiu and Lya...In this paper, global input-to-state stability （ISS） for discrete-time piecewise affine systems with time-delay are considered Piecewise quadratic ISS-Lyapunov functions are adopted. Both Lyapunov-Razumikhiu and Lyapunov-Krasovskii methods are used The theorems of Lyapunov-Razumikhin type and Lyapunov-Krasovskii type for piecewise affine systems with time-delay are shown respectively.展开更多
This paper investigates the problem of robust H-infinity state estimation for a class of uncertain discretetime piecewise affine systems where state space instead of measurable output space partitions are assumed so t...This paper investigates the problem of robust H-infinity state estimation for a class of uncertain discretetime piecewise affine systems where state space instead of measurable output space partitions are assumed so that the filter implementation may not be synchronized with plant state trajectory transitions. Based on a piecewise quadratic Lyapunov function combined with S-procedure and some matrix inequality convexifying techniques, two different approaches are developed to the robust filtering design for the underlying piecewise affine systems. It is shown that the filter gains can be obtained by solving a set of linear matrix inequalities （LMIs）. Finally, a simulation example is provided to illustrate the effectiveness of the proposed approaches.展开更多
The problem of how to identify the piecewise affine system is studied in this paper, where this considered piecewise affine system is a special nonlinear system. The reason why it is not easy to identify this piecewis...The problem of how to identify the piecewise affine system is studied in this paper, where this considered piecewise affine system is a special nonlinear system. The reason why it is not easy to identify this piecewise affine system is that each separated region and each unknown parameter vector are all needed to be determined simultaneously. Then, firstly, in order to achieve the identification goal, a multi-class classification process is proposed to determine each separated region. As the proposed multi-class classification process is the same with the classical data clustering strategy, the multi-class classification process can combine the first order algorithm of convex optimization, while achieving the goal of the classification process. Secondly, a zonotope parameter identification algorithm is used to construct a set, which contains the unknown parameter vector. In this zonotope parameter identification algorithm, the strict probabilistic description about the external noise is relaxed, and each unknown parameter vector is also identified. Furthermore, this constructed set is consistent with the measured output and the given bound corresponding to the noise. Thirdly, a sufficient condition about guaranteeing our derived zonotope not growing unbounded with iterations is formulated as an explicit linear matrix inequality. Finally, the effectiveness of this zonotope parameter identification algorithm is proven through a simulation example.展开更多
It is a huge challenge to give an existence theorem for heteroclinic cycles in the high-dimensional discontinuous piecewise systems（DPSs）. This paper first provides a new class of four-dimensional（4 D） two-zone di...It is a huge challenge to give an existence theorem for heteroclinic cycles in the high-dimensional discontinuous piecewise systems（DPSs）. This paper first provides a new class of four-dimensional（4 D） two-zone discontinuous piecewise affine systems（DPASs）, and then gives a useful criterion to ensure the existence of heteroclinic cycles in the systems by rigorous mathematical analysis. To illustrate the feasibility and efficiency of the theory, two numerical examples, exhibiting chaotic behaviors in a small neighborhood of heteroclinic cycles, are discussed.展开更多
The main contribution of this paper is to present a novel robust observer-based controller design method for discrete-time piecewise affine systems with norm-bounded uncertainties. The key ideas are to construct a pie...The main contribution of this paper is to present a novel robust observer-based controller design method for discrete-time piecewise affine systems with norm-bounded uncertainties. The key ideas are to construct a piecewise quadratic Lyapunov function to guarantee the stability of the closed-loop systems, approximate polytopic operating regions by ellipsoids, and use the singular value decomposition technique to treat the constraint of matrix equality. It is shown that the suggested control method can be formulated as linear matrix inequalities that are numerically feasible with commer- cially available software. A numerical example is also given to verify the proposed approach.展开更多
In this paper, global input-to-state stabilization with quantized feedback for discrete-time piecewise affine systems （PWA） with time delays are considered. Both feedback with time delays and feedback without time d...In this paper, global input-to-state stabilization with quantized feedback for discrete-time piecewise affine systems （PWA） with time delays are considered. Both feedback with time delays and feedback without time delays are considered. Piecewise quadratic ISS-Lyapunov functions are adopted. Both Lyapunov-Razumikhin and Lyapunov-Krasovskii methods are adopted. The theorems for global input-to-state stabilization with quantized feedback of discrete PWA systems with time delays are展开更多
This paper is concerned with the problem of designing robust H∞and H2static output feedback controllers for a class of discrete-time piecewise-affine singular systems with norm-bounded time-varying parameters uncerta...This paper is concerned with the problem of designing robust H∞and H2static output feedback controllers for a class of discrete-time piecewise-affine singular systems with norm-bounded time-varying parameters uncertainties. Based on a piecewise singular Lyapunov function combined with S-procedure,Projection lemma and some matrix inequality convexifying techniques,sufficient conditions in terms of linear matrix inequalities are given for the existence of an output-feedback controller for the discrete-time piecewiseaffine singular systems with a prescribed H∞disturbance attenuation level,and the H2norm is smaller than a given positive number. It is shown that the controller gains can be obtained by solving a family of LMIs parameterized by one or two scalar variables. The numerical examples are given to illustrate the effectiveness of the proposed design methods.展开更多
In the context of continuous piecewise affine dynamical systems and affine complementarity systems with inputs, we study the existence of Zeno behavior, i.e., infinite number of mode transitions in a finite-length tim...In the context of continuous piecewise affine dynamical systems and affine complementarity systems with inputs, we study the existence of Zeno behavior, i.e., infinite number of mode transitions in a finite-length time interval, in this paper. The main result reveals that continuous piecewise affine dynamical systems with piecewise real-analytic inputs do not exhibit Zeno behavior. Applied the achieved result to affine complementarity systems with inputs, we also obtained a similar conclusion. A direct benefit of the main result is that one can apply smooth ordinary differential equations theory in a local manner for the analysis of continuous piecewise affine dynamical systems with inputs.展开更多
In this paper,l1 regularisation-based robust fault detection technique for PWARX(PieceWise affine Auto Regressive eXogenous)systems is discussed.The problem is formulated in convex optimisation form using l1 regularis...In this paper,l1 regularisation-based robust fault detection technique for PWARX(PieceWise affine Auto Regressive eXogenous)systems is discussed.The problem is formulated in convex optimisation form using l1 regularisation hence,global solution can be found.The regularisation constant plays a key role for finding the number of submodels.The MSE(mean square error)value between actual system and its model under no fault is used as a threshold for tracking dynamics change and detecting fault.The simplicity of turning only one parameter(regularisation constant)outstands this approach from others.The proposed algorithm also shows the capability of separating the dynamics change from additive fault under noise.One illustration concludes this proposal by demonstrating its effectiveness.展开更多
Purpose-The purpose of this paper is to probe the recursive identification of piecewise affine Hammerstein models directly by using input-output data.To explain the identification process of a parametric piecewise aff...Purpose-The purpose of this paper is to probe the recursive identification of piecewise affine Hammerstein models directly by using input-output data.To explain the identification process of a parametric piecewise affine nonlinear function,the authors prove that the inverse function corresponding to the given piecewise affine nonlinear function is also an equivalent piecewise affine form.Based on this equivalent property,during the detailed identification process with respect to piecewise affine function and linear dynamical system,three recursive least squares methods are proposed to identify those unknown parameters under the probabilistic description or bounded property of noise.Design/methodology/approach-First,the basic recursive least squares method is used to identify those unknown parameters under the probabilistic description of noise.Second,multi-innovation recursive least squares method is proposed to improve the efficiency lacked in basic recursive least squares method.Third,to relax the strict probabilistic description on noise,the authors provide a projection algorithm with a dead zone in the presence of bounded noise and analyze its two properties.Findings-Based on complex mathematical derivation,the inverse function of a given piecewise affine nonlinear function is also an equivalent piecewise affine form.As the least squares method is suited under one condition that the considered noise may be a zero mean random signal,a projection algorithm with a dead zone in the presence of bounded noise can enhance the robustness in the parameter update equation.Originality/value-To the best knowledge of the authors,this is the first attempt at identifying piecewise affine Hammerstein models,which combine a piecewise affine function and a linear dynamical system.In the presence of bounded noise,the modified recursive least squares methods are efficient in identifying two kinds of unknown parameters,so that the common set membership method can be replaced by the proposed methods.展开更多
One of the basic issues in the study of hybrid systems is the well-posedness(existence and uniqueness of solutions)problem of discontinuous dynamical systems.This paper addresses this problem for a class of piecewise ...One of the basic issues in the study of hybrid systems is the well-posedness(existence and uniqueness of solutions)problem of discontinuous dynamical systems.This paper addresses this problem for a class of piecewise affine discontinuous systems with affine inequalities such as systems with pulse-width modulator under the definition of Carathéodory solutions in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions.Furthermore,it is clear that when carrier signal h(t)=0,closed-loop pulse-width modulation(PWM)DC–DC converters are not well posed,and when some condition is satisfied,the closed-loop PWM DC–DC converters with a P controller are well posed.展开更多
This paper addresses the problem of approximating parameter dependent nonlinear systems in a unified framework. This modeling has been presented for the first time in the form of parameter dependent piecewise affine s...This paper addresses the problem of approximating parameter dependent nonlinear systems in a unified framework. This modeling has been presented for the first time in the form of parameter dependent piecewise affine systems. In this model, the matrices and vectors defining piecewise affine systems are affine functions of parameters. Modeling of the system is done based on distinct spaces of state and parameter, and the operating regions are partitioned into the sections that we call 'multiplied simplices'. It is proven that this method of partitioning leads to less complexity of the approximated model compared with the few existing methods for modeling of parameter dependent nonlinear systems. It is also proven that the approximation is continuous for continuous functions and can be arbitrarily close to the original one. Next, the approximation error is calculated for a special class of parameter dependent nonlinear systems. For this class of systems, by solving an optimization problem, the operating regions can be partitioned into the minimum number of hyper-rectangles such that the modeling error does not exceed a specified value. This modeling method can be the first step towards analyzing the parameter dependent nonlinear systems with a uniform method.展开更多
文摘This paper explores the existence of heteroclinic cycles and corresponding chaotic dynamics in a class of 3-dimensional two-zone piecewise affine systems. Moreover, the heteroclinic cycles connect two saddle foci and intersect the switching manifold at two points and the switching manifold is composed of two perpendicular planes.
基金supported by the National Science Fund of China for Distinguished Young Scholars(No.60725311)
文摘The main contribution of this paper is to present stability synthesis results for discrete-time piecewise affine （PWA） systems with polytopic time-varying uncertainties and for discrete-time PWA systems with norm-bounded uncertainties respectively.The basic idea of the proposed approaches is to construct piecewise-quadratic （PWQ） Lyapunov functions to guarantee the stability of the closed-loop systems.The partition information of the PWA systems is taken into account and each polytopic operating region is outer approximated by an ellipsoid,then sufficient conditions for the robust stabilization are derived and expressed as a set of linear matrix inequalities （LMIs）.Two examples are given to illustrate the proposed theoretical results.
基金supported by National Natural Science Foundation of China (No. 60874006)Natural Science Foundation of Hei-longjiang Province for Youth (No. QC2009C99)
文摘In this paper, global input-to-state stability （ISS） for discrete-time piecewise affine systems with time-delay are considered Piecewise quadratic ISS-Lyapunov functions are adopted. Both Lyapunov-Razumikhiu and Lyapunov-Krasovskii methods are used The theorems of Lyapunov-Razumikhin type and Lyapunov-Krasovskii type for piecewise affine systems with time-delay are shown respectively.
基金supported by the Research Grants Council of the Hong Kong Special Administrative Region of China under the Project CityU/113708partly by the National Natural Science Foundation of China (No.60825303, 60834003)+2 种基金partly by the 973 Project (No.2009CB320600)partly by the Postdoctoral Science Foundation of China (No.20100471059)partly by the Overseas Talents Foundation of the Harbin Institute of Technology
文摘This paper investigates the problem of robust H-infinity state estimation for a class of uncertain discretetime piecewise affine systems where state space instead of measurable output space partitions are assumed so that the filter implementation may not be synchronized with plant state trajectory transitions. Based on a piecewise quadratic Lyapunov function combined with S-procedure and some matrix inequality convexifying techniques, two different approaches are developed to the robust filtering design for the underlying piecewise affine systems. It is shown that the filter gains can be obtained by solving a set of linear matrix inequalities （LMIs）. Finally, a simulation example is provided to illustrate the effectiveness of the proposed approaches.
文摘The problem of how to identify the piecewise affine system is studied in this paper, where this considered piecewise affine system is a special nonlinear system. The reason why it is not easy to identify this piecewise affine system is that each separated region and each unknown parameter vector are all needed to be determined simultaneously. Then, firstly, in order to achieve the identification goal, a multi-class classification process is proposed to determine each separated region. As the proposed multi-class classification process is the same with the classical data clustering strategy, the multi-class classification process can combine the first order algorithm of convex optimization, while achieving the goal of the classification process. Secondly, a zonotope parameter identification algorithm is used to construct a set, which contains the unknown parameter vector. In this zonotope parameter identification algorithm, the strict probabilistic description about the external noise is relaxed, and each unknown parameter vector is also identified. Furthermore, this constructed set is consistent with the measured output and the given bound corresponding to the noise. Thirdly, a sufficient condition about guaranteeing our derived zonotope not growing unbounded with iterations is formulated as an explicit linear matrix inequality. Finally, the effectiveness of this zonotope parameter identification algorithm is proven through a simulation example.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11472212 and 11532011)
文摘It is a huge challenge to give an existence theorem for heteroclinic cycles in the high-dimensional discontinuous piecewise systems（DPSs）. This paper first provides a new class of four-dimensional（4 D） two-zone discontinuous piecewise affine systems（DPASs）, and then gives a useful criterion to ensure the existence of heteroclinic cycles in the systems by rigorous mathematical analysis. To illustrate the feasibility and efficiency of the theory, two numerical examples, exhibiting chaotic behaviors in a small neighborhood of heteroclinic cycles, are discussed.
文摘The main contribution of this paper is to present a novel robust observer-based controller design method for discrete-time piecewise affine systems with norm-bounded uncertainties. The key ideas are to construct a piecewise quadratic Lyapunov function to guarantee the stability of the closed-loop systems, approximate polytopic operating regions by ellipsoids, and use the singular value decomposition technique to treat the constraint of matrix equality. It is shown that the suggested control method can be formulated as linear matrix inequalities that are numerically feasible with commer- cially available software. A numerical example is also given to verify the proposed approach.
基金supported by the National Natural Science Foundation of China under Grant No.60874006Natural Science Foundation of Heilong jiang Province for Youth under Grant No.QC2009C99
文摘In this paper, global input-to-state stabilization with quantized feedback for discrete-time piecewise affine systems （PWA） with time delays are considered. Both feedback with time delays and feedback without time delays are considered. Piecewise quadratic ISS-Lyapunov functions are adopted. Both Lyapunov-Razumikhin and Lyapunov-Krasovskii methods are adopted. The theorems for global input-to-state stabilization with quantized feedback of discrete PWA systems with time delays are
基金Sponsored by the National Natural Science Foundation of China(Grant No.61004038)
文摘This paper is concerned with the problem of designing robust H∞and H2static output feedback controllers for a class of discrete-time piecewise-affine singular systems with norm-bounded time-varying parameters uncertainties. Based on a piecewise singular Lyapunov function combined with S-procedure,Projection lemma and some matrix inequality convexifying techniques,sufficient conditions in terms of linear matrix inequalities are given for the existence of an output-feedback controller for the discrete-time piecewiseaffine singular systems with a prescribed H∞disturbance attenuation level,and the H2norm is smaller than a given positive number. It is shown that the controller gains can be obtained by solving a family of LMIs parameterized by one or two scalar variables. The numerical examples are given to illustrate the effectiveness of the proposed design methods.
文摘In the context of continuous piecewise affine dynamical systems and affine complementarity systems with inputs, we study the existence of Zeno behavior, i.e., infinite number of mode transitions in a finite-length time interval, in this paper. The main result reveals that continuous piecewise affine dynamical systems with piecewise real-analytic inputs do not exhibit Zeno behavior. Applied the achieved result to affine complementarity systems with inputs, we also obtained a similar conclusion. A direct benefit of the main result is that one can apply smooth ordinary differential equations theory in a local manner for the analysis of continuous piecewise affine dynamical systems with inputs.
文摘In this paper,l1 regularisation-based robust fault detection technique for PWARX(PieceWise affine Auto Regressive eXogenous)systems is discussed.The problem is formulated in convex optimisation form using l1 regularisation hence,global solution can be found.The regularisation constant plays a key role for finding the number of submodels.The MSE(mean square error)value between actual system and its model under no fault is used as a threshold for tracking dynamics change and detecting fault.The simplicity of turning only one parameter(regularisation constant)outstands this approach from others.The proposed algorithm also shows the capability of separating the dynamics change from additive fault under noise.One illustration concludes this proposal by demonstrating its effectiveness.
文摘Purpose-The purpose of this paper is to probe the recursive identification of piecewise affine Hammerstein models directly by using input-output data.To explain the identification process of a parametric piecewise affine nonlinear function,the authors prove that the inverse function corresponding to the given piecewise affine nonlinear function is also an equivalent piecewise affine form.Based on this equivalent property,during the detailed identification process with respect to piecewise affine function and linear dynamical system,three recursive least squares methods are proposed to identify those unknown parameters under the probabilistic description or bounded property of noise.Design/methodology/approach-First,the basic recursive least squares method is used to identify those unknown parameters under the probabilistic description of noise.Second,multi-innovation recursive least squares method is proposed to improve the efficiency lacked in basic recursive least squares method.Third,to relax the strict probabilistic description on noise,the authors provide a projection algorithm with a dead zone in the presence of bounded noise and analyze its two properties.Findings-Based on complex mathematical derivation,the inverse function of a given piecewise affine nonlinear function is also an equivalent piecewise affine form.As the least squares method is suited under one condition that the considered noise may be a zero mean random signal,a projection algorithm with a dead zone in the presence of bounded noise can enhance the robustness in the parameter update equation.Originality/value-To the best knowledge of the authors,this is the first attempt at identifying piecewise affine Hammerstein models,which combine a piecewise affine function and a linear dynamical system.In the presence of bounded noise,the modified recursive least squares methods are efficient in identifying two kinds of unknown parameters,so that the common set membership method can be replaced by the proposed methods.
基金supported by the Postdoctoral Foundation of China (17th).
文摘One of the basic issues in the study of hybrid systems is the well-posedness(existence and uniqueness of solutions)problem of discontinuous dynamical systems.This paper addresses this problem for a class of piecewise affine discontinuous systems with affine inequalities such as systems with pulse-width modulator under the definition of Carathéodory solutions in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions.Furthermore,it is clear that when carrier signal h(t)=0,closed-loop pulse-width modulation(PWM)DC–DC converters are not well posed,and when some condition is satisfied,the closed-loop PWM DC–DC converters with a P controller are well posed.
文摘This paper addresses the problem of approximating parameter dependent nonlinear systems in a unified framework. This modeling has been presented for the first time in the form of parameter dependent piecewise affine systems. In this model, the matrices and vectors defining piecewise affine systems are affine functions of parameters. Modeling of the system is done based on distinct spaces of state and parameter, and the operating regions are partitioned into the sections that we call 'multiplied simplices'. It is proven that this method of partitioning leads to less complexity of the approximated model compared with the few existing methods for modeling of parameter dependent nonlinear systems. It is also proven that the approximation is continuous for continuous functions and can be arbitrarily close to the original one. Next, the approximation error is calculated for a special class of parameter dependent nonlinear systems. For this class of systems, by solving an optimization problem, the operating regions can be partitioned into the minimum number of hyper-rectangles such that the modeling error does not exceed a specified value. This modeling method can be the first step towards analyzing the parameter dependent nonlinear systems with a uniform method.
