Identification of distributed parameter system with recurrent trajectory via deterministic learning

The main aim of this paper was to find the correct method of calculating equations of heat and mass transfer for the adsorption process and to calculate it numerically in reasonable time and with proper accuracy. An adsorption heat pump with a silica gel adsorbent and water adsorbate is discussed. We developed a mathematical model of temperature and uptake changes in the adsorber/desorber comprising the set of heat and mass balance partial differential equations (PDEs), together with the initial and boundary conditions and solved it by the numerical method of lines (NMOL). Spatial discretization was performed with equally spaced axial nodes and the PDEs were reduced to a set of ordinary differential equations (ODEs). We focused on the comparison of results obtained when the set of heat and mass balance ODEs for an adsorber was solved using: (1) the Runge–Kutta fixed step size fourth-order method (RKfixed), (2) the Runge–Kutta–Fehlberg 4.5th-order method with a variable step size (RK45), and (3) the Gear Backward Differentiation Formulae numerical (Gear BDF) methods. In our experience, all three types of ODE numerical methods (RKfixed, RK45, and Gear BDF) can be applied in simple models to model an adsorber with attention on their limitations. The Gear BDF method usually requires much fewer steps than the RK45 method for almost the same calculating time. RK methods require many more steps to obtain results, and the calculating time depends on accuracy or defined time step. Moreover, one should pay attention to the number of nodes or possible oscillations.

The dynamic modeling and experimental validation of the model response of the poly(methyl methacrylate) (PMMA) cell-cast process for plastic sheet production is presented. It is based on the straightforward initiation, propagation, and termination reaction polymerization mechanism. The sheet molding process was modeled by a two-dimensional dynamic mathematical model able to predict conversion, temperature, and molecular weight averages. The mathematical model is cast as a partial differential equations (PDEs) system that is discretized using the numerical method of lines. The resulting set of ordinary differential equations, representing the heat and mass balances for this polymerization system, are then solved by standard ordinary differential equations (ODE) solvers. Comparison of the model prediction capabilities against experimental temperature measurements taken at the extremes of the PMMA sheet being produced are presented.

Regions of asymptotic stability for distributed parameter systems

In the paper, the method of straight lines approximately solves one class of optimal control problems for systems, the behavior of which is described by a nonlinear equation of parabolic type and a set of ordinary differential equations. Control is carried out using distributed and lumped parameters. Distributed control is included in the partial differential equation, and lumped controls are contained both in the boundary conditions and in the right-hand side of the ordinary differential equation. The convergence of the solutions of the approximating boundary value problem to the solution of the original one is proved when the step of the grid of straight lines tends to zero, and on the basis of this fact, the convergence of the approximate solution of the approximating optimal problem with respect to the functional is established. A constructive scheme for constructing an optimal control by a minimizing sequence of controls is proposed. The control of the process in the approximate solution of a class of optimization problems is carried out on the basis of the Pontryagin maximum principle using the method of straight lines. For the numerical solution of the problem, a gradient projection scheme with a special choice of step is used, this gives a converging sequence in the control space. The numerical solution of one variational problem of the mentioned type related to a one-dimensional heat conduction equation with boundary conditions of the second kind is presented. An inequality-type constraint is imposed on the control function entering the right-hand side of the ordinary differential equation. The numerical results obtained on the basis of the compiled computer program are presented in the form of tables and figures. The described numerical method gives a sufficiently accurate solution in a short time and does not show a tendency to «dispersion». With an increase in the number of iterations, the value of the functional monotonically tends to zero

Conjugated heat transfer in a concentric annular pipe

The objective of this work is to present a three-dimensional analysis of the elasto-plastic response of a center-cracked rectangular panel by using an alternate method, the method of lines (MOL). The problem of a finite thickness rectangular plate containing a through-thickness central crack under uniaxial tension is studied in detail. An incremental procedure is used to seek the 3-D elasto-plastic stress and strain distribution. The Prandtl-Reuss equations and the associated von Mises flow rule are employed. The material is assumed to be elastic-plastic with nonlinear work-hardening. Effective stress and effective strain concepts are used. The MOL is used to reduce the governing equations on displacements to a coupled set of ordinary differential equations (ODE) along x, y, z lines. These are integrated by Peano-Baker method. Since the ODE are of constant coefficient the coefficient matrix is diagonalized and the matrizon is easily evaluated. Numerical results on stresses, displacements, the growth of plastic zone and the crack surface opening displacements are reported and their variation along the thickness are presented in graphs. The variation of the stress intensity factor K 1, along the thickness direction is given. It is found that the computation time needed for determining the distribution of stresses and strains is much less than what has been reported by using other methods. A comparison of results on SIF with those obtained by other methods is also given.

ABSTRACTThe problem of feedback control of bandwidth allocation for Internet links is analysed in this article. It is shown that the procedure for numerical solution of a fluid flow partial differential equation (PDE) which describes the bandwidth allocation on a link results into a set of ordinary differential equations (ODEs) and into an associated state-space equations model. For the local subsystems, into which the flow PDE is decomposed, it becomes possible to apply boundary-based feedback control. The solution of the control problem proceeds by showing that the state-space model of the flow PDE stands for a differentially flat system. Next, for each subsystem which is related to an ODE, a virtual control input is computed, which can invert the subsystem’s dynamics and can eliminate the subsystem’s tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the data-flow PDE is found. This control input contains recursively all virtual control inputs which were computed for the individual ODE subsystems associated with the previous rows of the state-space equation. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the data flow PDE of the communication network so as to assure that all its state variables will converge to the desirable set points.

A feedforward controller for the automatic regulation of chemical composition of molten steel in the tundish of a continuous casting machine is proposed in this work. The flow of molten steel inside the tundish is modeled as a distributed parameter system, and the resulting partial differential equation is transformed into a set of ordinary differential equations by means of the finite differences technique. From the above set and using a proper boundary condition, a feedforward control law is synthesized. No experimental tests are reported, however, the dynamic performance of the controller is illustrated by means of numerical simulations.

Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed

A fault detection and isolation scheme is addressed for a class of linear distributed parameter systems (DPS) described by partial differential equations (PDE). In contrast to using ordinary differential equations (ODE) for describing DPS, in this paper, a filter based observer based on the linear PDE representation is proposed with an output measurement. A fault is declared active when the magnitude of the detection residual exceeds a predefined threshold. Upon detection, an actuator and a sensor fault isolation estimators are activated to identify the fault type when their isolation residual is below a predefined threshold and the other is above the threshold. When both actuator and sensor fault isolation estimator residuals are above their isolation thresholds, a state fault is considered to have occurred. Upon isolation, the magnitude of the fault parameter is identified. Finally, the performance of the fault detection and isolation scheme is demonstrated on a heat reactor system which is represented by linear parabolic PDEs.

Parameter Estimation of Hyperbolic Model of Counterflow Heat Exchanger

The paper considers financial derivatives and option pricing models which are described with the use of diffusiontype partial differential equations (e.g. Black-Scholes models). Using this approach a new filtering method for distributed parameter systems is developed, for estimating option prices variations without knowledge of initial conditions. The proposed filtering method is the so-called Derivative-free nonlinear Kalman Filter and is based on a decomposition of the nonlinear partial-differential equation of the financial system into a set of ordinary differential equations with respect to time. Next, each one of the local models associated with the ordinary differential equations is written in the linear canonical form through a transformation which is based on differential flatness theory. This transformation provides a model of the nonlinear dynamics of the option pricing model for which state estimation is possible by applying the standard Kalman Filter recursion. Based on the obtained state estimate, validation of the Black-Scholes PDE model can be performed and the existence of inconsistent parameters in the Black-Scholes PDE model can be concluded.

Fast and scalable computation of shape-morphing nonlinear solutions with application to evolutional neural networks

In the last decade, the Computational Intelligence tools (CI), including Artificial Neural Networks (ANN) and Fuzzy Systems (FS), applying soft computing, became universal means for many applications. Because of their approximation and learning capabilities, the ANNs have been widely employed to dynamic process modeling, identification, prediction and control, (Boskovic & Narendra, 1995; Haykin, 1999; Bulsari & Palosaari, 1993; Deng & Li, 2003; Deng et al., 2005; Gonzalez-Garcia et al., 1998; Padhi & Balakrishnan, 2003; Padhi et al., 2001; Ray,1989). Many applications have been done for identification and control of biotechnological plants too, (Padhi et al., 2001). Among several possible neural network architectures the ones most widely used are the Feedforward NN (FFNN) and the Recurrent NN (RNN), (Haykin, 1999). The main NN property namely the ability to approximate complex non-linear relationships without prior knowledge of the model structure makes them a very attractive alternative to the classical modeling and control techniques. Also, a great boost has been made in the applied NN-based adaptive control methodology incorporating integral plus state control action in the control law, (Baruch et al., 2004; Baruch & Garrido, 2005; Baruch et al., 2007). The FFNN and the RNN have been applied for Distributed Parameter Systems (DPS) identification and control too. In (Pietil & Koivo, 1996), a RNN is used for system identification and process prediction of a DPS dynamics an adsorption column for wastewater treatment of water contaminated with toxic chemicals. In (Deng & Li, 2003; Deng et al., 2005) a spectral-approximation-based intelligent modeling approach, including NNs for state estimation and system identification, is proposed for the distributed thermal processing of the snap curing oven DPS that is used in semiconductor packaging industry. In (Bulsari & Palosaari, 1993), it is presented a new methodology for the identification of DPS, based on NN architectures, motivated by standard numerical discretization techniques used for the solution of Partial Differential Equation (PDE). In (Padhi & Balakrishnan, 2003), an attempt is made to use the philosophy of the NN adaptivecritic design to the optimal control of distributed parameter systems. In (Padhi et al., 2001) the concept of proper orthogonal decomposition is used for the model reduction of DPS to form a reduced order lumped parameter problem. The optimal control problem is then solved in the time domain, in a state feedback sense, following the philosophy of adaptive critic NNs. The control solution is then mapped back to the spatial domain using the same basis functions. In (Pietil & Koivo, 1996), measurement data of an industrial process are

The calculus of differential forms is used to discuss the local integration theory of a general set of autonomous first order ordinary differential equations. Geometrically, such a set is a vector field ${\bf V}$ in the space of dependent variables. Integration consists of seeking associated geometric structures invariant along ${\bf V}$: scalar fields (first integrals), forms (such as symplectic tensors), vectors (CT generators, Lie’s invariance generators), integrals over subspaces (which are not limited to Poincaré’s absolute and relative invariants). Most classical results are easily understood as simple relations between these. It is shown first that to any field ${\bf V}$ can be associated a Hamiltonian structure of forms (if, when dealing with an odd number of dependent variables, an arbitrary equation of constraint is also added). Families of integral invariants are an immediate consequence. Poisson brackets are isomorphic to Lie products of associated CT-generating vector fields. Hamilton’s variational principle follows from the fact that the maximal regular integral manifolds of a closed set of forms (here a Hamiltonian structure) must include the characteristics of the set (here the trajectories of ${\bf V}$). Auxiliary vector fields are also carefully discussed: the hierarchy of invariance generators of (i) the vector field ${\bf V}$, of (ii) its trajectories, of (iii) any associated Hamiltonian structure of forms. An Appendix summarizes the manipulation of vectors and forms, as a tool for applied mathematics.