Continuous Subsystems Research Articles

INTERMITTENT OPERATION OF LINEAR DRIVEN SWITCHED SYSTEMS

The phenomenon of breathing (intermittent operation) is studied in a class of piecewise continuous systems as well as its relation with system parameters. The class of systems under study comprises a continuous time subsystem and a switching rule that induces an oscillatory path by switching alternately between stable and unstable conditions. An interesting feature of the system is that eigenvalues of linear subsystems play an important role in system evolution. It is shown that although regular and chaotic phases evolve irregularly for a given system, their average behavior is surprisingly regular with respect to a bifurcation parameter. It is found that the phenomenon of breathing share some structural characteristics with intermittency; i.e. existence of a critical exponent. However, for switched systems, many critical exponents may be required. Bifurcation maps and other analysis tools allow us to gain insight into the origin of breathing. This work constitutes a first step toward the characterization of intermittent operation in piecewise continuous systems.

The statistical energy analysis of coupled sets of oscillators

The paper concerns the statistical energy analysis (SEA) of two conservatively coupled oscillators, sets of oscillators and continuous subsystems under broadband excitation. The oscillator properties are assumed to be random and ensemble averages found. Account is taken of the correlation between the coupling parameters and the oscillator energies. For coupled sets of oscillators or continuous subsystems, it is assumed that the coupling power between a pair of oscillators is proportional to the difference of either their actual energies or their ‘blocked’ energies, and expressions for the ensemble averages and coupling loss factors (CLFs) are found. Various observations are made, some of which differ from those that are commonly assumed within SEA. The coupling power and CLF are governed by two parameters: the ‘strength of connection’ and the ‘strength of coupling’. The CLF is proportional to damping at low damping and independent of damping in the high damping, weak coupling limit. Equipartition of energy does not occur as damping tends to zero, except for the case of two oscillators that have identical natural frequencies. While attention is focused on spring-coupled oscillators, similar results hold for more general forms of conservative coupling. The examples of two spring-coupled rods and two spring-coupled plates are considered. Conventional SEA and the coupled oscillator results are in good agreement for weak coupling but diverge for strong coupling. For strong coupling and weak connection, the coupled oscillator results agree well with an exact wave analysis and Monte Carlo simulations.

Analysis of Discretely Controlled Continuous Systems by means of Embedded Maps

Discretely controlled continuous systems are characterised by their interacting continuous and discrete dynamics, where the continuous subsystem usually represents the system to be controlled and the discrete part describes the controller that switches the continuous system among different operation modes. This paper analyses systems where a perpetual switching of the operation mode has to occur in order to maintain the system's state in a prescribed operating region. It is shown how the analysis of the overall hybrid system can be simplified by using an embedded map that determines the behaviour at the switching instants.

Modelling and synthesis of discrete–continuous subsystems of machines with damping

Nowadays, discrete–continuous systems are more commonly used in designing models of mechanical systems. Therefore, there has been an increasing interest in such methods of synthesis which could be applied both for systems with converged and continuously distributed parameters. Such synthesis involves a search for a structure of a discrete–continuous system that complies with some specific requirements concerning certain phenomena of dynamics. The continued fraction method of the synthesis of the dynamic characteristics distribution makes it possible to obtain cascade structures of the discrete–continuous system. The investigation into the structures of cascade systems was conducted by means of the software method for distributing the dynamic characteristics to the continued fraction based on graphs and structural numbers.

Computer-aided synthesis of discrete–continuous subsystems of machines with the assumed frequency spectrum represented by graphs

The problem of synthesising discrete physical, mechanical, electrical and electronic systems is widely developed in scientific research. However, it is the problem to find examples of methods of synthesising discrete–continuous mechanical systems. In the paper a foundation of the synthesis such systems has been presented, and its exact formulation and solution is still to deal with.

A theoretical formulation of the dynamical response of a master structure coupled with elastic continuous fuzzy subsystems with discrete attachments

We present the formulation of the dynamic response of a master structure coupled with a locally homogeneous and orthotropic structural fuzzy, with discrete attachment, composed of elastic continuous fuzzy subsystems. As introduced by Soize, the master structure is the part of the coupled system which is accessible by classical modeling, whereas the structural fuzzy represents systems connected to the master structure, whose characteristics are imprecisely known. A deterministic formulation of the boundary impedance of a general continuous structural fuzzy, which models its action on the master structure, is derived: it is shown that the formulation is different from the solution proposed by Soize in the context of the type I fuzzy law, established from the deterministic model of a linear oscillator excited by its support. Finally, the general boundary impedance is applied to the special situation of a structural fuzzy composed of elastic bars whose geometrical parameters are randomly defined, and numerical results are presented.

Intermediate storage in batch/continuous processing systems under stochastic operation conditions

A mathematical model and model-based method is presented to design the intermediate storages aiming to buffer the operational differences between the batch and continuous subsystems in processing systems. The occurrence times of the inputs are assumed to be described by a Poisson process, while the amounts of the material transferred by the batch units allowed changing according to general probability distributions. Based on the stochastic differential equation model of operation, integral equations for determining the overflow and underflow probabilities of a finite storage are formulated for both infinite and finite operation horizons that provide the basis for the rational design of such intermediate storages. Analytical solutions to the integral equations for infinite horizons are derived in the cases of constant and exponentially distributed inputs. For the batch sizes described by general distribution functions, solutions to the integral equations are obtained in the form of approximating functions generated by stochastic simulation. A number of numerical experiments with exponential, normal and lognormal distributions of the batch sizes are presented and analyzed. The effects of process parameters on the design are also investigated.

On fluidification of Petri Nets: from discrete to hybrid and continuous models

Petri Nets (PNs) is a well-known modelling paradigm for discrete event systems (DES). As in other paradigms, hybrid and continuous PN formalisms have appeared in the literature, some of them being used in different engineering application domains. Hybridization may be obtained for example “by direct addition” of capabilities to model continuous subsystems. The approach adopted in this work is different: hybrid and continuous models appear because natural variables of a PN–DES model are transformed into non-negative reals. This relaxation may be quite reasonable when very populated or high traffic systems are considered. It is a classical relaxation applied to fight against the state explosion problem appearing when dealing with the analysis and synthesis of models. In tune with this, the paper presents “a biased” view of works in the hybrid PN arena. Partly an overview, this work revisits hybrid and continuous PNs, all of them being in essence hybrid models. Limitations to (partial) fluidification, and analysis and synthesis problems in this evolving field are considered. Several optimization problems (at design and at control) are also introduced here.

Coordinating feedback and switching for control of hybrid nonlinear processes

AbstractA robust hybrid control strategy for a broad class of hybrid nonlinear processes with actuator constraints and uncertain dynamics is proposed. These variable‐structure processes comprise a finite family of constrained uncertain continuous nonlinear dynamical subsystems, together with discrete events that trigger the transition between the continuous subsystems. The proposed control strategy is predicated on the idea of coordinating the hierarchical tasks of lower‐level feedback‐controller synthesis and upper‐level switching logic design. Using multiple Lyapunov functions, a family of bounded robust, nonlinear feedback controllers are initially designed to robustly stabilize the constituent modes of the hybrid process, subject to uncertainty and constraints. The region of guaranteed closed‐loop stability is then explicitly characterized for each mode in terms of the magnitude of actuator constraints and the size of the uncertainty. A set of stabilizing switching laws that track the energy evolution of the constituent modes are then derived to orchestrate safe transitions between the stability regions of the constituent modes and their respective controllers, in a way that respects actuator constraints and guarantees robust stability of the overall uncertain hybrid closed‐loop system. This hybrid control method is applied through computer simulations to robustly stabilize an exothermic chemical reactor with switched dynamics, model uncertainty, and actuator constraints at an unstable steady‐state and to design a fault‐tolerant control system for chemical reactors through switching between multiple constrained control configurations.

On Fluidification of Petri Nets: From Discrete to Hybrid and Continuous Models

Petri Nets (PNs) is a well-known modelling paradigm for Discrete Event Systems (DES). As in other paradigms, hybrid and continuous PN formalisms have appeared in the literature, some of them being used in different engineering application domains. Hybridization may be obtained for example “by direct addition” of capabilities to model continuous subsystems. The approach adopted in this work is different: hybrid and continuous models appear because natural variables of a PN-DES model are transformed into non-negative reals. This relaxation may be quite reasonable when very populated or high traffic systems are considered. It is a classical relaxation applied in order to fight against the state explosion problem appearing when dealing with the analysis and synthesis of models. In tune with this, the paper presents “a biased” view of works in the hybrid PN arena. Partly an overview, this work revisits hybrid and continuous PNs, all of them being in essence hybrid models. Limitations to (partial) fluidification, and analysis and synthesis problems in this evolving field are considered.

Optimal Control of Hybrid Systems with an Infinite Set of Discrete States

Hybrid control systems are described by a family of continuous subsystems and a set of logic rules for switching between them. This paper concerns a broad class of optimization problems for hybrid systems, in which the continuous subsystems are modelled as differential inclusions. The formulation allows endpoint constraints and a general objective function that includes “transaction costs” associated with abrupt changes of discrete and continuous states, and terms associated with continuous control action as well as the terminal value of the continuous state. In consequence of the endpoint constraints, the value function m ay be discontinuous. It is shown that the collection of value functions (associated with all discrete states) is the unique lower semicontinuous solution of a system of generalized Bensoussan-Lions type quasi-variational inequalities, suitably interpreted for nondifferentiable, extended valued functions. It is also shown how optimal strategies and value functions are related. The proof techniques are system theoretic, i.e., based on the construction of state trajectories with suitable properties. A distinctive feature of the analysis is that it permits an infinite set of discrete states.

Simulation of Sliding Motion of Charge-Density-Wave by Considering Dipole and Screening Effects

Long-length scale interaction model proposed by Littlewood has been extended to study dipole and screening effects on sliding motion of one-dimensional charge-density-wave (CDW). The system in the extended model consists of a site-by-site main-system fixed at pinning centers and the continuous sub-systems in each pinning site. For strong pinning case, we have demonstrated these effects on the CDW phase, CDW current, and narrow band noise spectrum. The dipole effect suppresses the pinning potential and gives rise to a change from deformable “plastic flow” like to rigid “moving solid” like CDW condenstates, while the screening effect acts to recover the former.

Robust adaptive regulation of potentially inversely unstable first-order hybrid systems

This paper presents a robust adaptive stabilization scheme for a class of hybrid time-invariant linear system involving both continuous and discrete signals. The continuous subsystem and the discrete one are both of first-order. The usual assumptions of inverse stability of the plant and knowledge of the high-frequency gain are not required for this first-order hybrid system. The design philosophy used relies on the separation of the continuous and discrete dynamics by generating two additive terms to build the control action. The estimation scheme is unified in the sense that both continuous and discrete estimated parameters are generated from the same adaptation error. The controllability of the nominally estimated plant model is maintained by using an hysteresis switching function under the controllability of the nominal plant. This allows relaxing the usual assumption of stability of the plant inverse. Also, an adaptation dead zone is used for robust stabilization by using a known overbounding function of the contribution of the unmodelled dynamics.

Transfer matrix method for the electrochemical impedance of inhomogeneous porous electrodes and membranes

The method presented here is based on the two-phase model of a porous system with two continuous subsystems, electrons in the porous material and ions in the pore electrolyte. Both are continuously interconnected via the pore surfaces e.g. by the double layer capacity and/or the charge transfer resistance. The equivalent circuit for this system is the transmission line model. The method applies to systems with parameters which are not constant across the layer. The layer is divided into a number of slabs and in each slab all parameters are replaced by their mean values. The potentials and the currents of two adjacent slabs are connected by a matrix, in the general case a 4×4 matrix. The potential propagation in the whole layer is determined by the product matrix. The impedance for both a layer coating a metallic current collector and a porous membrane embedded in the electrolyte (or the porous layer with electrolyte-filled pores in between two metallic current collectors) can be expressed by the elements of the product matrix. The matrix is reduced to a 2×2-form if one of the resistivities is negligible. In this case for a system of two homogeneous sublayers an analytical formulation is given. The method is applied to a system with an interconnection consisting of double layer capacity, charge transfer resistance and its hindrance by finite diffusion (applicable to polymers). Here the inhomogeneity gradients of the resistivities are considered. It is demonstrated that they can result in significant qualitative modifications of the impedance. This concerns especially the low frequency pseudo-capacitive behaviour which is transformed into a dependence resembling the well known empirical description by constant phase elements often used to interpolate experimental data.

THE USE OF CONVENTIONAL AND MINPLUS ALGEBRA FOR THE MODELING OF HYBRID PETRI NETS

In this paper we are interested in a mathematical description of the class of hybrid systems, which can be modeled by hybrid Petri nets HPNs . A state space model using conventional algebra for the continuous subsystem and minplus algebra for the discrete subsystem is given. The interface between the discrete and continuous part appears in our model. This model is illustrated through an example. We introduce the concept of discrete control of HPNs.

On the hyperstability of a class of hybrid systems

Sufficient conditions of the hyperstability of a class of single-input single-output hybrid systems, composed of interconnected continuous and discrete subsystems, are given. Hyperstability is guaranteed by that of a discrete extended system which is built by incorporating samples of the continuous substrate. Asymptotic hyperstability is guaranteed by that of one discrete system, together with that of a continuous one. Both systems are related to an augmented state representation of the linear part of the overall system for all the nonlinear output-feedback time-varying actuators that satisfy generalized Popov-type inequalities.

Constraint nets: a semantic model for hybrid dynamic systems

Hybrid dynamic systems are systems consisting of a nontrivial mixture of discrete and continuous components, such as a controller realized by a combination of digital and analog circuits, a robot composed of a digital controller and a physical plant, or a robotic system consisting of a computer-controlled robot coupled to a continuous environment. Hybrid dynamic systems are more general then traditional real-time systems. The former can be composed of continuous subsystems in addition to discrete and event-controlled components. In this paper, we develop a semantic model, constraint nets (CN), for hybrid systems. CN captures the most general structure of dynamic systems so that systems with discrete and continuous time, discrete and continuous variables, and asynchronous as well as synchronous event structures, can be modeled in a unitary framework. Using aggregation operators, a system can be modeled hierarchically in CN; therefore, the dynamics of the environment as well as the dynamics of the plant and the dynamics of the controller can be modeled individually and then integrated. Based on abstract algebra and topology, CN supports multiple levels of abstraction, so that a system can be analyzed at different levels of detail. CN also provides a rigorous formal programming semantics for the design of hybrid real-time embedded systems.

A hybrid model for intelligent control systems

Hybrid continuous-time and discrete-event system models are developed for applications to intelligent robot and autonomous control systems. Certain general properties, including the interface structure between the discrete event part and the continuous part of hybrid systems, are studied. Petri nets are proposed as the model for the discrete event sub-system (DES), and state equations with a set of binary parameters as the model for the continuous time sub-system (CTS) in a hybrid system. The quantized CTS is formulated in prime event structures, which can also be modeled by a condition/event Petri net. Therefore, a complete hybrid system can be fit into an extended Petri net formalism, which provides a unified framework for hybrid system description. The validity of the proposed model is demonstrated by modeling a two-tank system.

The statistics of power flow between two continuous one-dimensional subsystems

The power flow between two continuous subsystems is examined using a wave solution [1]. One of the subsystems is excited by time-harmonic forces, and both subsystems can support only one power-transmitting wave mode. Uncertainty exists in the subsystems, which are assumed to be drawn at random from ensembles. Those ensembles members that have a natural frequency at the excitation frequency are determined. The particular members that are resonant are seen to depend on the strength of coupling. The statistics of power flow are considered; in particular, the maximum, minimum, ensemble arithmetic and geometric means, the variance and the cumulative probability distribution of the power flow. The maximum power flow undergoes a qualitative change at a particular coupling strength. For weak coupling the peak power flows occur at the uncoupled natural frequencies, while for stronger coupling they occur at the coupled frequencies. The ensemble average power flow may be significantly less than that expected from a “normal” SEA approach due to power re-radiation, especially for a lightly damped receiving subsystem and strong coupling. Conversely, it may be much greater due to power re-injection, which occurs particularly for a lightly damped excited subsystem and weak coupling. The power flow variance is largest for small damping and weak coupling, under which circumstances the cumulative probability distribution has an infrequent but dominant tail of very large power flows.

Resonance characteristics of connected subsystems: Theory and simple configurations

The dynamic properties of connected continuous subsystems are examined with use of analytical expressions for the modal properties. The analysis begins with a Lagrange multiplier formulation to develop a characteristic equation in terms of subsystem mobilities and impedances. The complexity of the problem is examined in terms of the order of the polynomial expressions in the characteristic equation. To obtain insight into the system's complicated dynamic characteristics, the complexity of the problem is reduced. It is shown that the reduction of complexity can be obtained only with a reduction of accuracy, but by retaining the dominant effects of the dynamics problem, the loss of accuracy is not excessive. The reduced problem is examined further to develop simple, yet powerful expressions for the modal properties which provide insight into the resonance characteristics of the connected subsystem problem. The results are useful as a complement to existing computational techniques for understanding and interpreting dynamic analysis results. In this paper, the theory and simplest configurations are examined, and in the companion paper, more general configurations are studied.