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This paper studies sliding homoclinic bifurcations in a class of symmetric three-zone three-dimensional piecewise affine systems. The systems have one parameter and the unperturbed systems have a pair of sliding homoclinic orbits to a saddle. Based on the analysis of the one-dimensional Poincaré maps, two types of sliding cycles are obtained from the sliding homoclinic bifurcations of the systems. In addition, two examples of sliding homoclinic orbits and sliding cycles are provided with simulations to illustrate the effectiveness of the theorems. [ABSTRACT FROM AUTHOR]

Singular cycles (homoclinic orbits and heteroclinic cycles) play an important role in the study of chaotic dynamics of dynamical systems. This paper provides the coexistence of singular cycles that intersect the switching manifold transversely at two points in a class of three-dimensional two-zone piecewise affine systems. Moreover, the switching manifold of the systems is constructed by two perpendicular planes. Different to the three-dimensional piecewise affine systems with a switching plane, the system can ensure the coexistence of two homoclinic orbits to the same one equilibrium point and two heteroclinic cycles constructing by three heteroclinic orbits. In addition, three examples with simulations of the singular cycles are provided to illustrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]

Many physical and engineering systems have certain symmetric properties. Homoclinic orbits play an important role in studying the global dynamics of dynamical systems. This paper focuses on the existence and bifurcations of homoclinic orbits to a saddle in a class of three-dimensional one-parameter three-zone symmetric piecewise affine systems. Based on the analysis of the Poincaré maps, the systems have two types of limit cycles and do not have chaotic invariant sets near the homoclinic orbits. In addition, the paper provides a constant D to study the homoclinic bifurcations to limit cycles for the case | λ 1 | = λ 3 . Two examples with simulations of the homoclinic orbits and the limit cycles are given to illustrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper is concerned with model predictive control of piecewise affine systems with dead zone constraints based on the mixed logical dynamical modeling approach. A predictive control strategy based on a mixed logic dynamical model is proposed. The strategy contains two steps: the first step is to transform the piecewise affine systems into mixed logic dynamical models; the second step is to apply a predictive control scheme based on this model. Then, the optimal control sequence can be obtained by solving the constructed mixed‐integer quadratic/linear programming problem. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed approaches. [ABSTRACT FROM AUTHOR]

Detecting an isolated homoclinic or heteroclinic cycle is a great challenge in a concrete system, letting alone the case of coexisting scenarios and more complicated chaotic behaviors. This paper systematically investigates the dynamics for a class of three-dimensional (3D) three-zone piecewise affine systems (PWASs) consisting of three sub-systems. Interestingly, under different conditions the considered system can display three types of coexisting singular cycles including: homoclinic and homoclinic cycles, heteroclinic and heteroclinic cycles, homoclinic and heteroclinic cycles. Furthermore, it establishes sufficient conditions for the presence of chaotic invariant sets emerged from such coexisting cycles. Finally, three numerical examples are provided to verify the proposed theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates self-oscillation behaviors occurring in a class of piecewise affine (PWA) systems. We address a particular case of PWA systems with two subsystems and a linear switching surface. Conditions are given for characterizing forward invariant sets containing self-oscillating solutions and domains of attraction Rather than relying on purely computational tools, we provide conditions taking into account the structure of the system. [ABSTRACT FROM AUTHOR]

This paper studies one identification problem for a piecewise affine system which is a special nonlinear system. As the difficulty in identifying the piecewise affine system is to determine each separated region and each unknown parameter vector simultaneously, here we propose a multi-class classification process to determine each separated region. This multi-class classification process is similar to the classical data clustering process, and the merit of our strategy is that the first-order algorithm of convex optimization can be applied to achieve this classification process. Furthermore, to relax the strict probabilistic description on external noise and identify each unknown parameter vector, a zonotope parameter identification algorithm is proposed to compute a set that contains the parameter vector, consistent with the measured output and the given bound of the noise. To guarantee our derived zonotope not growing unbounded with iterations, a sufficient condition for this requirement to hold may be formulated as one linear matrix inequality. Finally, a numerical example confirms our theoretical results.

This paper studies one identification problem for a piecewise affine system which is a special nonlinear system. As the difficulty in identifying the piecewise affine system is to determine each separated region and each unknown parameter vector simultaneously, here we propose a multi-class classification process to determine each separated region. This multi-class classification process is similar to the classical data clustering process, and the merit of our strategy is that the first-order algorithm of convex optimization can be applied to achieve this classification process. Furthermore, to relax the strict probabilistic description on external noise and identify each unknown parameter vector, a zonotope parameter identification algorithm is proposed to compute a set that contains the parameter vector, consistent with the measured output and the given bound of the noise. To guarantee our derived zonotope not growing unbounded with iterations, a sufficient condition for this requirement to hold may be formulated as one linear matrix inequality. Finally, a numerical example confirms our theoretical results. [ABSTRACT FROM AUTHOR]

It is a significant and challenging task to detect both the coexistence of singular cycles, mainly homoclinic and heteroclinic cycles, and chaos induced by the coexistence in nonsmooth systems. By analyzing the dynamical behaviors on manifolds, this paper proposes some criteria to accurately locate the coexistence of homoclinic cycles and of heteroclinic cycles in a class of three-dimensional (3D) piecewise affine systems (PASs), respectively. It further establishes the existence conditions of chaos arising from such coexistence, and presents a mathematical proof by analyzing the constructed Poincaré map. Finally, the simulations for two numerical examples are provided to validate the established results. [ABSTRACT FROM AUTHOR]

Little seems to be known about heteroclinic cycles and chaos in three-dimensional piecewise smooth dynamical systems with two or more discontinuous boundaries. This article presents a new class of three-dimensional three-zone piecewise affine systems with two discontinuous boundaries and provides some criteria for the existence of heteroclinic cycles in the following cases: (i) one saddle point and two focus points, (ii) two saddle points and one focus point, and (iii) three saddle points. Moreover, sufficient conditions for the existence of chaos are established. Finally, two numerical examples are provided to show the feasibility of our theoretical approach. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with uncertain linear discrete-time systems controlled by a piecewise affine feedback law defined on a polyhedral domain. Assuming the initial state is contained in a given set, a novel algorithm is proposed to identify regions of the polyhedral partition which are guaranteed not to be visited during real-time operation. By dispensing with the need to store the aforementioned regions and associated control functions, the proposed method enables a reduction in both storage and on-line computational demands. [ABSTRACT FROM AUTHOR]

Abstract In the face of the frequent occurrence of urban traffic congestion, only part traffic sensors can be utilized in the urban freeway, in order to more accurately estimate traffic densities and timely identify the congested road segments, a novel estimated algorithm of proportional integral state observer is proposed. Firstly, by embedding the Cell Transmission Model (CTM) into the Dynamic Graph Hybrid Automata (DGHA) framework, we model traffic flow over the urban freeway network. In modeling procedure, an urban freeway network is partitioned in term of three hierarchical structures which are freeway networks, links and cells, respectively. On the one hand, both the dual digraph of road segment structure and hybrid automata respectively are adopted to describe the freeway topology and the multi-modes of dynamic densities in road segments. On the other hand, a deterministic and reasonable constraint is used to reduce the number of combination modes, and thus the modeling procedure is modularized, rule-based and easily-extensible with the help of a combination algorithm. Therefore, the Piecewise Affine System (PWAS) model of the freeway network can be obtained. Secondly, In order to estimate the traffic densities of the urban freeway network, based on the dynamic model, we design a switched proportional integral state observer, and the observer design problem is transferred into the stability problem of the switching system. So the proportional gains and integral gains can be computed via introducing the Lyapunov function and using LMI technique, and thus the traffic densities can be estimated by the designed observer. Especially, the traffic congestion can be identified by checking whether the estimated densities are greater than the critical ones. Finally, a section of Beijing south third ring freeway as an example to verify the proposed approach, and the effectiveness and the feasibility are shown in the experiment results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The recoverability reflects the capability of the system to tolerate the worst faults over a prescribed set under admissible energy constrains. In this paper, a piecewise affine (PWA) controller is designed for a PWA approximation of the nonlinear dynamics, which can stabilize the original nonlinear system under some conditions. By analyzing the upper bound of the piecewise linear quadratic control performance of the PWA system, a novel fault recoverability evaluation scheme is further proposed for the nonlinear system with respect to the given limit of admissible control performance. A longitudinal flight control example is taken to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Type 1 diabetic patients require adjustments of their basal therapy due to insulin requirement changes. A very promising automatic approach is based on the so-called Run-to-Run (R2R) strategy, which adjusts the insulin therapy based on the performance measured during the previous run, usually of 1 day. Previous R2R approaches were based on a few blood glucose measurements. In this paper, the use of a subcutaneous continuous glucose monitoring is exploited in order to obtain more relevant clinical performance indices, such as the percentage of time spent below 70 mg/dl, above 180 mg/dl, and the average glucose. Different priority is given to hypo- and hyperglycemia control and, in particular, the primary goal is to reduce the time below 70 mg/dl, whereas the secondary is to reduce the time above 180 mg/dl and to reach a desirable glucose concentration. In doing so, a switching control law is achieved. A procedure for the convergence analysis of the proposed R2R approach is introduced by resorting to the Lyapunov theory for piecewise affine systems. Performance is studied for the algorithm by means of an extensive in silico trial performed on 100 adult patients of the UVA/Padova simulator. Five scenarios characterized by over- and underestimation of the initial basal therapy, random variations of the patient insulin sensitivity, meal variations, and intra-day circadian variability are considered. After a week the time in range [70–180] mg/dl is significantly ( p -value < 0.001) increased with a reduction of the time spent below 70 mg/dl by using the proposed algorithm. [ABSTRACT FROM AUTHOR]

Discrete Event Systems (DES) theory and engineering are mainly driven by needs that arise in many different human-made systems (manufacturing, communications, logistics, workflow management, traffic, etc.). With the accelerated increase in the complexity and size of new technological constructions, the state explosion problem in DES analysis and synthesis becomes more and more acute. Two traditional conceptual and complementary ways of dealing with the computational complexities in the Petri nets (PN) framework are structure theory (that investigates the relationship between the behavior of a net system and its structure) and fluid relaxations , here leading to particular classes of hybrid systems. In the second case, the expected computational gains for analysis and synthesis problems are usually achieved at the expense of the fidelity or accuracy of the relaxed model. This invited overview will mainly focus on the second strategy, nevertheless always interspersed with basic structural concepts and methods. Using an example-driven approach, starting with a DES “view of the system”, the legitimization and improvement of fluidization process, the aggregation of local states by symmetries and the decolorization of models will be briefly addressed, together with reflections about the analysis of the new models obtained. As the linearization of a continuous dynamical system, the fluidization of a DES is a relaxation that has to be used with care, depending on the problem at hand. This abstraction is here considered from two complementary perspectives: at logical and at performance levels, both for untimed and timed PNs. On the one hand, the expressive power of timed fluid PNs under infinite server semantics is such that the simulation of Turing machines is possible. From a complementary perspective, the expression of modeling capabilities such as non-monotonicities and bifurcations may also be revealed for steady-state behaviors. Symmetries (more generally, lumping ) seek to group together “equivalent” behaviors and decolorization seeks to abstract identities, in order to create new collectivities of processes and resources. The synergy between symmetry-decolorization state-aggregation approaches and fluid relaxations is highlighted. In fact, the first approaches not only reduce the state space, but also “produce” populations, thus proceed upgrading the applicability of fluidization. Opening the window, related issues such as control, optimization, observation or diagnosis are briefly pointed out. For conciseness, this work is limited to fully fluid (or continuous) PN models and their relationships with the corresponding discrete systems. [ABSTRACT FROM AUTHOR]

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In this paper, the control signal of the series-resonant converter is considered as a slope of a switching surface. The output characteristic of the converter, for an ideal case and above the resonant frequency, is achieved based on the slope of the switching surface. This output characteristic leads us to find the related slope for a specific converter gain. Nonlinear state equations of the series-resonant converter with new control input are represented in widely used class of hybrid systems that have been called piecewise affine systems. Considering the effects of the slope variations in the switching surface, a simple and efficient control law is achieved. The main advantage of the proposed method is the absence of microcontroller and frequency modulator integrated circuits in hardware implementation. This property makes the proposed method more beneficial in high-frequency applications. Experimental results confirm our theoretical investigations. [ABSTRACT FROM AUTHOR]

SUMMARY An active fault tolerant control (AFTC) method is proposed for discrete-time piecewise affine (PWA) systems. Only actuator faults are considered. The AFTC framework contains a supervisory scheme, which selects a suitable controller in a set of controllers such that the stability and an acceptable performance of the faulty system are held. The design of the supervisory scheme is not considered here. The set of controllers is composed of a normal controller for the fault-free case, an active fault detection and isolation controller for isolation and identification of the faults, and a set of passive fault tolerant controllers (PFTCs) modules designed to be robust against a set of actuator faults. In this research, the piecewise nonlinear model is approximated by a PWA system. The PFTCs are state feedback laws. Each one is robust against a fixed set of actuator faults and is able to track the reference signal while the control inputs are bounded. The PFTC problem is transformed into a feasibility problem of a set of LMIs. The method is applied on a large-scale live-stock ventilation model. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this article, a switching Model Predictive Controller (sMPC) for a pneumatic artificial muscle (PAM) is presented. The control scheme is based on a switching PieceWise Affine (PWA) system model approximation that is able to capture the high nonlinearities of the PAM, while improving the overall model accuracy, and is composed of: (a) a feed-forward term regulating control input at specific reference set-points, and (b) a switching Model Predictive Controller handling any deviations from the system's equilibrium points. Extended experimental studies are being presented that prove the overall scheme's efficiency. [Copyright &y& Elsevier]

In this article, a switching Model Predictive Controller (sMPC) scheme for the position control of a Pneumatic Artificial Muscle (PAM) is being presented. The control scheme is based on a constrained linear and PieceWise Affine (PWA) system model approximation that is able to capture the high nonlinearities of the PAM and improve the overall model accuracy, and is composed of: a) a feed-forward term regulating control input at specific reference set-points, and b) a switching Model Predictive Controller handling any deviations from the system’s equilibrium points. Extended simulation studies were utilized in order to investigate and evaluate the efficacy of the suggested controller in the positioning problem of a PAM. [ABSTRACT FROM AUTHOR]

This paper deals with the analysis of the nonlinear glucose regulation system from its Piecewise Affine (PWA) equivalent Piecewise Affine (PWA) equivalent model. The use of PWA model was motivated by the fact that it has developed as an attractive tool for nonlinear system analysis in recent years, since it allows the use of well developed concepts in the field of linear systems and control theory. The analysis mainly involves the behavioral study of the system at steady state for different types of input and different sets of physiological parameters. The number of affine systems required to represent a nonlinear system is an important consideration in PWA modeling. In this paper, the optimal number of piecewise affine systems is obtained by a technique based on a fuzzy clustering. From the PWA representation, it is possible to determine the equilibrium point (basal condition) of the glucose regulation system quite easily without solving a complex transcendental equation. The presence of a compensating mechanism between insulin sensitivity and insulin secretion is utilized by the PWA model to determine a particular profile (characteristics) for insulin sensitivity for a modified insulin secretion characteristics that would maintain a normal basal glucose level. A condition for the existence of a stable limit cycle of plasma glucose concentration as a function of delay in hepatic glucose production is determined from the model. This allowed determining the point of hopf bifurcation without carrying out extensive simulations, as is done with the existing models. The response obtained from the numerical simulation of the model, is in line with the experimental results. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a novel observer-based control scheme for discrete-time piecewise affine systems based on a piecewise-quadratic Lyapunov function. The key issue addressed in this paper is that the currently active region of the system is unknown, and can not be inferred from the measured outputs. By approximating polytopic operating regions by ellipsoids and using the singular value decomposition technique to treat the constraint of matrix equality, the suggested control method can be formulated as linear matrix inequalities (LMIs), and solved much more efficiently than existing methods which could be only cast as bilinear matrix inequalities. A numerical example is also given to verify the proposed approach. [Copyright &y& Elsevier]

Abstract: This paper focuses on the -hard problem of reducing the complexity of piecewise polyhedral systems (e.g. polyhedral piecewise affine (PWA) systems). The results are fourfold. Firstly, the paper presents two computationally attractive algorithms for optimal complexity reduction that, under the assumption that the system is defined over the cells of a hyperplane arrangement, derive an equivalent polyhedral piecewise system that is minimal in the number of polyhedra. The algorithms are based on the cells and the markings of the hyperplane arrangement. In particular, the first algorithm yields a set of disjoint (non overlapping) merged polyhedra by executing a branch and bound search on the markings of the cells. The second approach leads to non-disjoint (overlapping) polyhedra by formulating and solving an equivalent (and well-studied) logic minimization problem. Secondly, the results are extended to systems defined on general polyhedral partitions (and not on cells of hyperplane arrangements). Thirdly, the paper proposes a technique to further reduce the complexity of piecewise polyhedral systems if the introduction of an adjustable degree of error is acceptable. Fourthly, the paper shows that based on the notion of the hyperplane arrangement PWA state feedback control laws can be implemented efficiently. Three examples, including a challenging industrial problem, illustrate the algorithms and show their computational effectiveness in reducing the complexity by up to one order of magnitude. [Copyright &y& Elsevier]

This paper addresses system identification of a switched system in which each subsystem is expressed by a piecewise affine system. The system considered is expressed by a mixed logical dynamical system for which some design methods are already known. For identification, in most cases, a number of subsystems, that is, switching points, are given in advance. However, it is not easy to find switching points for multivariable systems. Therefore, in this paper, we propose a simultaneous identification technique that takes into account the number of subsystems and their parameters. The proposed method is applied to several examples to show its effectiveness. © 2008 Wiley Periodicals, Inc. Electron Comm Jpn, 91(5): 1–10, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecj.10109 [ABSTRACT FROM AUTHOR]

Abstract: Piecewise affine (PWA) systems are powerful models for describing both non-linear and hybrid systems. One of the key problems in controlling these systems is the inherent computational complexity of controller synthesis and analysis, especially if constraints on states and inputs are present. In addition, few results are available which address the issue of computing stabilizing controllers for PWA systems without placing constraints on the location of the origin. This paper first introduces a method to obtain stability guarantees for receding horizon control of discrete-time PWA systems. Based on this result, two algorithms which provide low complexity state feedback controllers are introduced. Specifically, we demonstrate how multi-parametric programming can be used to obtain minimum-time controllers, i.e., controllers which drive the state into a pre-specified target set in minimum time. In a second segment, we show how controllers of even lower complexity can be obtained by separately dealing with constraint satisfaction and stability properties. To this end, we introduce a method to compute PWA Lyapunov functions for discrete-time PWA systems via linear programming. Finally, we report results of an extensive case study which justify our claims of complexity reduction. [Copyright &y& Elsevier]

• A complete model of vehicle longitudinal dynamics is built based on the PWA method. • A computational MLD framework is derived based on the developed system PWA model. • The HMPC control strategy for the MLD model of the intelligent vehicle is designed. In this paper, a novel piecewise affine (PWA) modeling framework and a hybrid optimal control scheme based on the developed PWA model for the intelligent vehicle longitudinal dynamics are proposed. The proposed methodology approximates the nonlinear characteristics of the system major components, i.e. the engine, the automatic transmission and the tire, using the PWA representation method firstly. Due to the affine submodel switching behaviors of the PWA model and the fact that the intelligent vehicle must be worked in two discrete modes (driving and braking) for regulating the longitudinal velocity automatically, the control procedure of the intelligent vehicle longitudinal dynamics actually shows a hybrid nature with both continuous variables and discrete events. Thus, to further solve the system hybrid control problem, the intelligent vehicle longitudinal dynamics whose major components are modelled in the PWA form is further transformed into a computational mixed logical dynamical (MLD) model in this work, based on which a hybrid model predictive control (HMPC) strategy, which can simultaneously calculate the switching sequences of the intelligent vehicle longitudinal working modes (binary control inputs) and the throttle angle and braking pressure (continuous control inputs) during autonomous velocity regulation, is designed by solving a mixed-integer quadratic programming (MIQP) problem. Simulation and experimental results are finally provided to verify the superior control performance of the designed hybrid controller in longitudinal velocity regulation under typical driving conditions. [ABSTRACT FROM AUTHOR]