Nonlinear dynamics and control of process systems with recycle

Process systems with material and energy recycle are well-known to exhibit complex dynamics and to present significant control challenges, due to the feedback interactions induced by the recycle streams. In this paper, we address the dynamic analysis and control of such process systems. Initially, we establish, through an asymptotic analysis, that (i) small recycle flowrates induce a weak coupling among individual processes, whereas (ii) large recycle flowrates induce a time scale separation, with the dynamics of individual processes evolving in a fast time scale with weak interactions, and the dynamics of the overall system evolving in a slow time scale where these interactions become significant; these slow dynamics is usually nonlinear and of low order. Motivated by this, we present (i) a model reduction methodology for deriving nonlinear low-order models of the slow dynamics induced by large recycle streams, and (ii) a controller design framework consisting of properly coordinated controllers in the fast and the slow time scales. The theoretical results are illustrated in a reaction-separation network with a large recycle compared to the throughput.     

Three‐Time‐Scale Singular Perturbation Stability Analysis of Three‐Phase Power Converters

This paper analyzes the stability of the well‐known three‐phase two‐level power converter. Focusing on the rectifier operating mode, the dynamics of the system, when the instantaneous power and dc‐link voltage controllers are included, are described by a set of complex equations that results in a nonlinear autonomous singularly perturbed system. Hence, the closed‐loop system can be studied under the assumption of separate time scales. The analysis proposed in this work follows a novel three‐time‐scale approach, where the fast time scale corresponds with the instantaneous power dynamics, the mid‐range time scale is related to the dc‐link voltage dynamics, and the slow time scale is associated with the dc‐link voltage regulator dynamics. In this way, the analysis leads to the decomposition of the closed‐loop system into three simpler subsystems: fast, medium, and slow subsystems. These subsystems approximate the closed‐loop system behavior over the three different time scales. Finally, since the equilibrium point of each subsystem is exponentially stable and some other conditions are satisfied, it is shown that the equilibrium point of the closed‐loop system also presents exponential stability. Experimental results for a synchronous three‐phase power rectifier prototype are included to corroborate the analysis carried out.     

Control of singularly perturbed hybrid stochastic systems

We study a class of optimal stochastic control problems involving two different time scales. The fast mode of the system is represented by deterministic state equations whereas the slow mode of the system corresponds to a jump disturbance process. Under a fundamental “ergodicity” property for a class of “infinitesimal control systems” associated with the fast mode, we show that there exists a limit problem which provides a good approximation to the optimal control of the perturbed system. Both the finite- and infinite-discounted horizon cases are considered. We show how an approximate optimal control law can be constructed from the solution of the limit control problem. In the particular case where the infinitesimal control systems possess the so-called turnpike property, i.e., characterized by the existence of global attractors, the limit control problem can be given an interpretation related to a decomposition approach     

A Hybrid Implicit-Explicit Adaptive Multirate Numerical Scheme for Time-Dependent Equations

We develop a hybrid implicit and explicit adaptive multirate time integration method to solve systems of time-dependent equations that present two significantly different scales. We adopt an iteration scheme to decouple the equations with different time scales. At each iteration, we use an implicit Galerkin method with a fast time-step to solve for the fast scale variables and an explicit method with a slow time-step to solve for the slow variables. We derive an error estimator using a posteriori analysis which controls both the iteration number and the adaptive time-step selection. We present several numerical examples demonstrating the efficiency of our scheme and conclude with a stability analysis for a model problem.     

Synchronization and desynchronization in a network of locallycoupled Wilson-Cowan oscillators

A network of Wilson-Cowan (WC) oscillators is constructed, and its emergent properties of synchronization and desynchronization are investigated by both computer simulation and formal analysis. The network is a 2D matrix, where each oscillator is coupled only to its neighbors. We show analytically that a chain of locally coupled oscillators (the piecewise linear approximation to the WC oscillator) synchronizes, and we present a technique to rapidly entrain finite numbers of oscillators. The coupling strengths change on a fast time scale based on a Hebbian rule. A global separator is introduced which receives input from and sends feedback to each oscillator in the matrix. The global separator is used to desynchronize different oscillator groups. Unlike many other models, the properties of this network emerge from local connections that preserve spatial relationships among components and are critical for encoding Gestalt principles of feature grouping. The ability to synchronize and desynchronize oscillator groups within this network offers a promising approach for pattern segmentation and figure/ground segregation based on oscillatory correlation.     

System decomposition technique for spray modelling in CFD codes

A new decomposition technique for a system of ordinary differential equations is suggested, based on the geometrical version of the integral manifold method. This is based on comparing the values of the right hand sides of these equations, leading to the separation of the equations into ‘fast’ and ‘slow’ variables. The hierarchy of the decomposition is allowed to vary with time. Equations for fast variables are solved by a stiff ODE system solver with the slow variables taken at the beginning of the time step. The solution of the equations for the slow variables is presented in a simplified form, assuming linearised variation of these variables for the known time evolution of the fast variables. This can be considered as the first order approximation for the fast manifold. This technique is applied to analyse the explosion of a polydisperse spray of diesel fuel. Clear advantages are demonstrated from the point of view of accuracy and CPU efficiency when compared with the conventional approach widely used in CFD codes. The difference between the solution of the full system of equations and the solution of the decomposed system of equations is shown to be negligibly small for practical applications. It is shown that in some cases the system of fast equations is reduced to a single equation.     

Robustness of exponential stability to singular perturbations and delays

Singularly perturbed nonlinear differential equations with small time delays in the slow variables are considered. Averages of the fast variables are used in order to obtain a sufficient condition under which the exponential stability of the slow subsystem is robust to singular perturbations and delays.     

Model reduction and control of reactor–heat exchanger networks

This paper focuses on the dynamics and control of process networks consisting of a reactor connected with an external heat exchanger through a large material recycle stream that acts as an energy carrier. Using singular perturbation arguments, we show that such networks exhibit a dynamic behavior featuring two time scales: a fast one, in which the energy balance variables evolve, and a slow time scale that captures the evolution of the terms in the material balance equations. We present a procedure for deriving reduced-order, non-stiff models for the fast and slow dynamics, and a framework for rational control system design that accounts for the time scale separation exhibited by the system dynamics. The theoretical developments are illustrated with an example and numerical simulation results.     

Discrete waves and phase-locked oscillations in the growth of a single-species population over a patchy environment

A system of retarded functional differential equations is proposed as a model of the growth of a single-species population distributed over a ring of identical patches where the dispersion from one patch to others occurs only in nearest neighbors. It is shown that the temporal delay and spatial dispersion can give rise to a special type of discrete wave, the so-called phase-locked oscillation in which a population in one patch oscillates just like others except in different phases. Such oscillations cannot be observed in the absence of dispersion. Our results are based on the theory of symmetric Hopf bifurcations from multiple eigenvalues of differential delay equations recently developed by Geba and the authors.This research was supported by NSERC-Canada.     

A Taylor Collocation Method for the Solution of Linear Integro-Differential Equations

In this study, a matrix method called the Taylor collocation method is presented for numerically solving the linear integro-differential equations by a truncated Taylor series. Using the Taylor collocation points, this method transforms the integro-differential equation to a matrix equation which corresponds to a system of linear algebraic equations with unknown Taylor coefficients. Also the method can be used for linear differential and integral equations. To illustrate the method, it is applied to certain linear differential, integral, and integro-differential equations and the results are compared.     

Second-order predictor-corrector schemes for nonlinear distributed-order space-fractional differential equations with non-smooth initial data

ABSTRACT

We propose second-order linearly implicit predictor-corrector schemes for diffusion and reaction-diffusion equations of distributed-order. For diffusion equations of distributed order, we propose an analytical solution based on the spectral representation of the fractional Laplacian. Numerically, we approximate the integral term of the equation by the midpoint quadrature rule to obtain a multi-term space-fractional differential equation. The matrix transfer technique is used for spatial discretization of the resulting differential equation and methods based on Padé approximations to the exponential function are used in time. In particular, we discuss the (0,2)- and (1,1)-Padé approximations to the exponential function. The method based on the (1,1)-Padé approximation to the exponential function are seen to produce oscillations for some time steps and we propose a constraint on the choice of the time step to avoid these unwanted oscillations. Stability and convergence of the schemes are discussed. Numerical experiments are performed to support our theoretical observations.     

A Green-function-based multiscale method for uncertainty quantification of finite body random heterogeneous materials

Classical continuum theories are formulated based on the assumption of large scale separation. For scale-coupling problems involving uncertainties, novel multiscale methods are desired. In this study, by employing the generalized variational principles, a Green-function-based multiscale method is formulated to decompose a boundary value problem with random microstructure into a slow scale deterministic problem and a fast scale stochastic one. The slow scale problem corresponds to common engineering practices by smearing out fine-scale microstructures. The fast scale problem evaluates fluctuations due to random microstructures, which is important for scale-coupling systems and particularly failure problems. Two numerical examples are provided at the end.     

Multirate extrapolation methods for differential equations with different time scales

A multirate extrapolation method is developed for the integration of differential equations whose components evolve at different time scales. Numerical work is focused on fast components. The partitioning into different levels of slow to fast components is obtained automatically during the extrapolation process. The method has been implemented in the Fortran code MURX.     

A new operator splitting method for the numerical solution of partial differential equations II

We extend the applications of a new method for splitting operators in partial differential equations introduced by us (A. Rouhi and J. Wright, A new operator splitting method for the numerical solution of partial differential equations, Comput. Phys. Commun. 85 (1995) 18–28, and Spectral implementation of a new operator splitting method for solving partial differential equations, Comput. Phys. (1995), to be published.) to equations in two spatial dimensions, and show how the method allows the use of explicit time stepping methods in some instances when other methods require implicit time stepping. This odd-even splitting method also enables one to increase the order of accuracy of time stepping in a straightforward manner. Our main examples will be the two-dimensional Navier-Stokes equations and the shallow water equations. In the first example we show how the pressure term can be dealt with in simple geometries. We will then discuss the treatment of the diffusion term. Next we will discuss how fast waves can be treated by explicit methods using the odd-even splitting, while retaining all stability and accuracy advantages of usual implicit methods. Our example here will be the shallow water equations in two dimensions.     

Non-linear filtering for distributed parameter systems having a small parameter

A general non-linear filter is derived for dynamic systems described by partial differential equations containing a small parameter multiplying the time derivatives of some state variables. By allowing this small parameter to approach zero, it is possible to approximate the system filter equations by low-order slow and fast filters. As an example, a fixed-bed chemical reactor is used to demonstrate the application of the present theory.     

Two-time scale control and observer design for trajectory tracking of two cooperating robot manipulators moving a flexible beam

In this paper, we present two-time scale control design for trajectory tracking of two cooperating planar rigid robots moving a flexible beam, which does not require vibration measurement for the beam. First, the kinematics and dynamics of the robots and the object are derived. Then, using the relations between different forces acting on the object by the manipulators’ end-effectors, dynamics equations of the robots and the object are combined. The resulting equations show that the coupled dynamics including beam vibration and the rigid motion take place in two different time domains. By applying two-time scale control theory on the combined dynamics, a composite control scheme is elaborated which makes the beam orientation and its center of mass position track a desired trajectory while suppressing the beam vibration. For the controller algorithm, first a slow controller is utilized for the slow (rigid) subsystem and then a fast stabilizing controller is considered for the fast (flexible) subsystem. To avoid requiring measurement of beam vibration for the fast control law, a linear observer is also designed. The simulation results show the efficiency of the proposed control scheme.     

A Neural Paradigm for Motion Understanding

The main aim of this paper is to propose a new neural algorithm to perform a segmentation of an observed scene in regions corresponding to different moving objects, by analysing a time-varying image sequence. The method consists of a classification step, where the motion of small patches is recovered through an optimisation approach, and a segmen-tation step merging neighbouring patches characterised by the same motion. Classification of motion is performed without optical flow computation. Three-dimensional motion parameter estimates are obtained directly from the spatial and temporal image gradients by minimising an appropriate energy function with a Hopfield-like neural network. Network convergence is accelerated by integrating the quantitative estimation of the motion parameters with a qualitative estimate of dominant motion using the geometric theory of differential equations.     

Three-dimensional Poisson solver for a charged beam with large aspect ratio in a conducting pipe

In this paper, we present a three-dimensional Poisson equation solver for the electrostatic potential of a charged beam with large longitudinal to transverse aspect ratio in a straight and a bent conducting pipe with open-end boundary conditions. In this solver, we have used a Hermite-Gaussian series to represent the longitudinal spatial dependence of the charge density and the electric potential. Using the Hermite-Gaussian approximation, the original three-dimensional Poisson equation has been reduced into a group of coupled two-dimensional partial differential equations with the coupling strength proportional to the inverse square of the longitudinal-to-transverse aspect ratio. For a large aspect ratio, the coupling is weak. These two-dimensional partial differential equations can be solved independently using an iterative approach. The iterations converge quickly due to the large aspect ratio of the beam. For a transverse round conducting pipe, the two-dimensional Poisson equation is solved using a Bessel function approximation and a Fourier function approximation. The three-dimensional Poisson solver can have important applications in the study of the space-charge effects in the high intensity proton storage ring accelerator or induction linear accelerator for heavy ion fusion where the ratio of bunch length to the transverse size is large.     

Singular Perturbations and Lindblad-Kossakowski Differential Equations

We consider an ensemble of quantum systems described by a density matrix, solution of a Lindblad-Kossakowski differential equation. We focus on the special case where the decoherence is only due to a highly unstable excited state and where the spontaneously emitted photons are measured by a photo-detector. We propose a systematic method to eliminate the fast and asymptotically stable dynamics associated with the excited state in order to obtain another differential equation for the slow part. We show that this slow differential equation is still of Lindblad-Kossakowski type, that the decoherence terms and the measured output depend explicitly on the amplitudes of quasi-resonant applied field, i.e., the control. Beside a rigorous proof of the slow/fast (adiabatic) reduction based on singular perturbation theory, we also provide a physical interpretation of the result in the context of coherence population trapping via dark states and decoherence-free subspaces. Numerical simulations illustrate the accuracy of the proposed approximation for a 5-level systems.