On a generalization of relative degree

We generalize the notion of relative degree for a controlled dynamical system (possibly, discrete). We show that the new notion of leading incomplete relative degree is uniquely determined for a broad class of “square” dynamical systems (i.e., systems with as many inputs as outputs), including controllable systems. If r is the vector of relative degree of a system in the classical sense, then it is so in the generalized sense as well. We indicate a constructive algorithm for computing the leading incomplete relative degree.     

On the expressiveness and decidability of o-minimal hybrid systems

This paper is driven by a general motto: bisimulate a hybrid system by a finite symbolic dynamical system. In the case of o-minimal hybrid systems, the continuous and discrete components can be decoupled, and hence, the problem reduces in building a finite symbolic dynamical system for the continuous dynamics of each location. We show that this can be done for a quite general class of hybrid systems defined on o-minimal structures. In particular, we recover the main result of a paper by G. Lafferriere, G.J. Pappas, and S. Sastry, on o-minimal hybrid systems. We also provide an analysis and extension of results on decidability and complexity of problems and constructions related to o-minimal hybrid systems.     

Some contributions with Petri nets for the modelling, analysis and control of HDS

Petri nets (PN) are useful for the modelling, analysis and control of hybrid dynamical systems (HDS) because PN combine in a comprehensive way discrete events and continuous behaviours. On one hand, PN are suitable for modelling the discrete part of HDS and for providing a discrete abstraction of continuous behaviours. On the other hand, continuous PN are suitable for modelling the continuous part of HDS and for working out a continuous approximation of the discrete part in order to avoid the complexity associated with the exponential growth of discrete states. This paper focuses on the advantages of PN as a modelling tool for HDS. Investigations of such models for diagnosis and control issues are detailed.

Taking inspiration from the discrete event approach, sensor selection for diagnosis is discussed according to the structural analysis of the PN models. Faults are represented with fault transitions and a faulty behaviour occurs when a sequence of transitions is fired that contains at least one fault transition. Minimal sets of observable places are defined for detecting and isolating faulty behaviours.

Taking inspiration from the continuous time approach, flow control of HDS modelled with continuous PN is also investigated. Gradient-based controllers are introduced in order to adapt the firing speeds of some controllable transitions according to a desired trajectory of the marking. The equilibria and stability of the controlled system are studied with Lyapunov functions.     

Hybrid decentralized maximum entropy control for large-scale dynamical systems

In the analysis of complex, large-scale dynamical systems it is often essential to decompose the overall dynamical system into a collection of interacting subsystems. Because of implementation constraints, cost, and reliability considerations, a decentralized controller architecture is often required for controlling large-scale interconnected dynamical systems. In this paper, a novel class of fixed-order, energy-based hybrid decentralized controllers is proposed as a means for achieving enhanced energy dissipation in large-scale lossless and dissipative dynamical systems. These dynamic decentralized controllers combine a logical switching architecture with continuous dynamics to guarantee that the system plant energy is strictly decreasing across switchings. The general framework leads to hybrid closed-loop systems described by impulsive differential equations. In addition, we construct hybrid dynamic controllers that guarantee that each subsystem–subcontroller pair of the hybrid closed-loop system is consistent with basic thermodynamic principles. Special cases of energy-based hybrid controllers involving state-dependent switching are described, and an illustrative combustion control example is given to demonstrate the efficacy of the proposed approach.     

Approximability of nonlinear affine control systems

This paper focuses on a strong approximability property for nonlinear affine control systems. We consider control processes governed by ordinary differential equations (ODEs) and study an initial system and the associated generalized system. Our theoretical approach makes it possible to prove a strong approximability result for the above dynamical systems. The latter can be effectively applied to some classes of variable structure and hybrid control systems. In particular, this paper deals with applications of the strong approximability property obtained to the conventional sliding mode processes and to hybrid control systems with autonomous location transitions. We also take into consideration some optimal control problems for the above class of hybrid systems.     

On the exponentially self-regulating population model

A new type of discrete dynamical systems model for populations, called an exponentially self-regulating (ESR) map, is introduced and analyzed in considerable detail for the case of two competing species. The ESR model exhibits many dynamical features consistent with the observed interactions of populations and subsumes some of the most successful discrete biological models that have been studied in the literature. For example, the well-known Tribolium model is an ESR map. It is shown that in addition to logistic dynamics – ranging from the very simple to manifestly chaotic one-dimensional regimes – the ESR model exhibits, for some parameter values, its own brands of bifurcation and chaos that are essentially two-dimensional in nature. In particular, it is proved that ESR systems have twisted horseshoe with bending tail dynamics associated to an essentially global strange attractor for certain parameter ranges. The existence of a global strange attractor makes the ESR map more plausible as a model for actual populations than several other extant models, including the Lotka–Volterra map.     

Fault diagnosis for a class of nonlinear uncertain hybrid systems

This paper presents a fault diagnosis architecture for a class of hybrid systems with nonlinear uncertain time-driven dynamics, measurement noise, and autonomous and controlled mode transitions. The proposed approach features a hybrid estimator based on a modified hybrid automaton framework. The fault detection scheme employs a filtering approach that attenuates the effect of the measurement noise and allows tighter mode-dependent thresholds for the detection of both discrete and parametric faults while guaranteeing no false alarms due to modeling uncertainty and mode mismatches. Both the hybrid estimator and the fault detection scheme are linked with an autonomous guard events identification (AGEI) scheme that handles the effects of mode mismatches due to autonomous mode transitions and allows effective mode estimation. Finally, the fault isolation scheme anticipates which fault events may have occurred and dynamically employs the appropriate isolation estimators for isolating the fault by calculating suitable thresholds and estimating the parametric fault magnitude through adaptive approximation methods. Simulation results from a five-tank hybrid system illustrate the effectiveness of the proposed approach.     

SimHPN: A MATLAB toolbox for simulation, analysis and design with hybrid Petri nets

This paper presents a MATLAB embedded package for hybrid Petri nets called SimHPN. It offers a collection of tools devoted to simulation, analysis and synthesis of dynamical systems modeled by hybrid Petri nets. The package supports several server semantics for the firing of both, discrete and continuous, types of transitions. Besides providing different simulation options, SimHPN offers the possibility of computing steady state throughput bounds for continuous nets. For such a class of nets, optimal control and observability algorithms are also implemented. The package is fully integrated in MATLAB which allows the creation of powerful algebraic, statistical and graphical instruments that exploit the routines available in MATLAB.     

Hybrid port–Hamiltonian systems: From parameterized incidence matrices to hybrid automata

This paper shows how to formally design a hybrid automaton model for a wide class of dissipative physical systems with sources and switching topology. This method is based on a mathematical representation of the dynamic network graph and of its dual graph, using the hybrid incidence matrix, and on a constructive method for analyzing admissible and constrained configurations. The port–Hamiltonian representation associated with the set of hybrid system configurations, parameterized by the discrete state of the switches, is synthesized to be part of the hybrid automaton of the system. This is a further step towards a generic control synthesis for physical switching systems.     

Differential-geometric structure and spectral properties of nonlinear completely integrable dynamical systems of the Mel'nikov type

One considers V. K. Mel'nikov's new class of nonlinear dynamical systems, which is a generalization of the Korteweg-de Vries dynamical system. One investigates the differential-geometric and spectral properties of dynamical systems of Mel'nikov type, one gives their Hamiltonian form, one establishes the so-called gradient identity. The class of finite-zone potentials of a Sturm-Liouville operator, satisfying the given dynamical systems, is described.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 655–659, May, 1990.     

A dynamic parameter estimator to control chaos with distinct feedback schemes

A dynamic strategy is proposed to estimate parameters of chaotic systems. The dynamic estimator of parameters can be used with diverse control functions; for example, those based on: (i) Lie algebra, (ii) backstepping, or (iii) variable feedback structure (sliding-mode). The proposal has adaptive structure because of interaction between dynamic estimation of parameters and a feedback control function. Without lost of generality, a class of dynamical systems with chaotic behavior is considered as benchmark. The proposed scheme is compared with a previous low-parameterized robust adaptive feedback in terms of execution and performance. The comparison is motivated to ask: What is the suitable adaptive scheme to suppress chaos in an specific implementation? Experimental results of proposed scheme are discussed in terms of control execution and performance and are relevant in specific implementations; for example, in order to induce synchrony in complex networks.     

Zeno phenomenon in hybrid dynamical systems

A way to prolong Zeno solutions to hybrid dynamical system beyond Zeno time is proposed. Additionally, we provide a new definition of asymptotic stability that is equipped with sufficient conditions for its verification. It allows to investigate long-term behavior of the prolonged solutions to hybrid dynamical systems. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the fluidization of Petri nets and marking homothecy

The analysis of Discrete Event Dynamic Systems suffers from the well known state explosion problem. A classical technique to overcome it is to relax the behavior by partially removing the integrality constraints and thus to deal with hybrid or continuous systems. In the Petri nets framework, continuous net systems (technically hybrid systems) are the result of removing the integrality constraint in the firing of transitions. This relaxation may highly reduce the complexity of analysis techniques but may not preserve important properties of the original system. This paper deals with the basic operation of fluidization. More precisely, it aims at establishing conditions that a discrete system must satisfy so that a given property is preserved by the continuous relaxation. These conditions will be mainly based on the marking homothetic behavior of the system. The focus will be on logical properties as boundedness, B-fairness, deadlock-freeness, liveness and reversibility. Furthermore, testing homothetic monotonicity of some properties in the discrete systems is also studied, as well as techniques to improve the quality of the fluid relaxation by removing spurious solutions.     

The Marotto Theorem on planar monotone or competitive maps

In 1978, Marotto generalized Li–Yorke’s results on the criterion for chaos from one-dimensional discrete dynamical systems to n-dimensional discrete dynamical systems, showing that the existence of a non-degenerate snap-back repeller implies chaos in the sense of Li–Yorke. This theorem is very useful in predicting and analyzing discrete chaos in multi-dimensional dynamical systems. Yet, besides it is well known that there exists an error in the conditions of the original Marotto Theorem, and several authors had tried to correct it in different way, Chen, Hsu and Zhou pointed out that the verification of “non-degeneracy” of a snap-back repeller is the most difficult in general and expected, “almost beyond reasonable doubt,” that the existence of only degenerate snap-back repeller still implies chaotic, which was posed as a conjecture by them. In this paper, we shall give necessary and sufficient conditions of chaos in the sense of Li–Yorke for planar monotone or competitive discrete dynamical systems and solve Chen–Hsu–Zhou Conjecture for such kinds of systems.     

Modeling and analysis using hybrid Petri nets

This paper is devoted to the use of hybrid Petri nets (PNs) for modeling and control of hybrid dynamic systems (HDS). Modeling, analysis and control of HDS attract ever more of researchers’ attention and several works have been devoted to these topics. We consider in this paper the extensions of the PN formalism (initially conceived for modeling and analysis of discrete event systems) in the direction of hybrid modeling. We present, first, the continuous PN models. These models are obtained from discrete PNs by the fluidification of the markings. They constitute the first steps in the extension of PNs toward hybrid modeling. Then, we present two hybrid PN models, which differ in the class of HDS they can deal with. The first one is used for deterministic HDS modeling, whereas the second one can deal with HDS with nondeterministic behavior.     

A numerical method for hybrid optimal control based on dynamic programming

This contribution extends a numerical method for solving optimal control problems by dynamic programming to a class of hybrid dynamic systems with autonomous as well as controlled switching. The value function of the hybrid control system is calculated based on a full discretization of the state and input spaces. A bound for the error due to discretization is obtained from modeling the error as perturbation of the continuous dynamics and the cost terms. It is shown that the bound approaches zero and that the value function of the discretized variant converges to the value function of the original problem if the discretization parameters go to zero. The performance of a numerical scheme exploiting the discretized system is illustrated for two different examples treated previously in literature.     

Limit cycles in switched single-server flow networks

The paper addresses the problem of qualitative analysis for a class of hybrid dynamical systems. This class consists of so-called switched flow networks which are used to model various communication, computer, and flexible manufacturing systems. We prove that any hybrid dynamical system from this class has a finite number of asymptotically stable limit cycles and any trajectory of the system converges to one of these cycles.     

Energy dissipating hybrid control for impulsive dynamical systems

A novel class of fixed-order, energy-based hybrid controllers is proposed as a means for achieving enhanced energy dissipation in nonsmooth Euler–Lagrange, hybrid port-controlled Hamiltonian, and lossless impulsive dynamical systems. These dynamic controllers combine a logical switching architecture with hybrid dynamics to guarantee that the system plant energy is strictly decreasing across switchings. The general framework leads to hybrid closed-loop systems described by impulsive differential equations. Special cases of energy-based hybrid controllers involving state-dependent switching are described, and an illustrative numerical example is given to demonstrate the efficacy of the proposed approach.