Empirical linear Dynamic Models

Many empirical methods exist where an a priori model stracture has to be chosen and subsequently the model parameters are identified by a suitable technique. The method employed could for example be a least squares method. 

Consider a process with an input u(k) and an outputwhere k is the discrete time value. Assuming that the signals can be related by a linear process, we can write  

Process Dynamics and Control Modeling for Control and Prediction. Brian Roffel and Ben Betlem. 2006 John WUey Sons Ltd. 

MATLAB is a suitable environment to identify the model parameters. The identification toolbox provides an exeellerrt rrser irrletface for these types of model. 

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The most direct way of obtaining an empirical linear dynamic model of a process is to find the parameters (deadtime, time constant, and damping coefficient) that fit the experimentally obtained step response data. The process being identified is usually openloop, but experimental testing of closedloop systems is also possible. 

A key featui-e of MPC is that a dynamic model of the pi ocess is used to pi-edict futui e values of the contmlled outputs. Thei-e is considei--able flexibihty concei-ning the choice of the dynamic model. Fof example, a physical model based on fifst principles (e.g., mass and energy balances) or an empirical model coiild be selected. Also, the empirical model could be a linear model (e.g., transfer function, step response model, or state space model) or a nonhnear model (e.g., neural net model). However, most industrial applications of MPC have relied on linear empirical models, which may include simple nonlinear transformations of process variables. 

Implementation Issues A critical factor in the successful application of any model-based technique is the availability of a suitaole dynamic model. In typical MPC applications, an empirical model is identified from data acquired during extensive plant tests. The experiments generally consist of a series of bump tests in the manipulated variables. Typically, the manipulated variables are adjusted one at a time and the plant tests require a period of one to three weeks. The step or impulse response coefficients are then calculated using linear-regression techniques such as least-sqiiares methods. However, details concerning the procedures utihzed in the plant tests and subsequent model identification are considered to be proprietary information. The scaling and conditioning of plant data for use in model identification and control calculations can be key factors in the success of the apphcation. 

There is strong interest to analytically describe the fzme-dependence of polymer creep in order to extrapolate the deformation behaviour into otherwise inaccessible time-ranges. Several empirical and thermo-dynamical models have been proposed, such as the Andrade or Findley Potential equation [47,48] or the classical linear and non-linear visco-elastic theories ([36,37,49-51]). In the linear viscoelastic range Findley [48] and Schapery [49] successfully represent the (primary) creep compliance D(t) by a potential equation ... 

Johansen, T.A. and Foss, B.A. (1995). Semi-empirical modeling of non-linear dynamic systems through identification of operating regimes and locals models. In Neural Network Engineering in Control Systems, K Hunt, G Irwin and K Warwick, Eds., pp. 105-126, Springer-Verlag. 

Most work on the development of dynamic process models has been empirical this work is usually referred to as process identification. As mentioned earlier, two classes of empirical identification techniques are available one uses deterministic (step, pulse, etc.) functions, the other stochastic (random) identification functions. With either technique, the process is perturbed and the resulting variations of the response are measured. The relationship between the perturbing variable and the response is expressed as a transfer function. This function is the process model. Empirical identification of process models by the deterministic method has been reported by various workers [55-58]. A drawback of this method is the difficulty in obtaining a measurable response while restricting the process to a linear response (small perturbation). If the perturbation is large, the process response will be nonlinear and the representations of the process with a linear process model will be inaccurate. 

As already discussed in the previous chapters, process behavior is usually non-Unear. Whether or not the empirical model to be developed should also be non-linear depends on the operating range in which the model will be used. If the process is controlled and the operating range is small, a linear process model may be an adequate approximation of reality. The application of the model will determine whether the model needs to be dynamic or static. For control and prediction type applications, models are usually dynamic. 

The MFC predictions are made using a dynamic model, typically a linear empirical model such as a multivariable version of the step response or difference equation models that were introduced in Chapter 7. Alternatively, transfer function or state-space models (Section 6.5) can be employed. For very nonhnear processes, it can be advantageous to predict future output values using a nonlinear dynamic model. Both physical models and empirical models, such as neural networks (Section 7.3), have been used in nonlinear MFC (Badgwell... 

Let us define a computer experiment as integration of the dynamic model [Eq. (2.2)] corresponding to the 11-reaction mechanism. The outcome of these experiments is the vector of predicted values of the actual responses. Assuming the logarithms of the rate constants as the controllable variables of the computational process, let us begin the empirical modeling of this process (Section 3.2) by considering the linear form... 

To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

Various theoretical methods and approaches have been used to model properties and reactivities of metalloporphyrins. They range from the early use of qualitative molecular orbital diagrams (24,25), linear combination of atomic orbitals to yield molecular orbitals (LCAO-MO) calculations (26-30), molecular mechanics (31,32) and semi-empirical methods (33-35), and self-consistent field method (SCF) calculations (36-43) to the methods commonly used nowadays (molecular dynamic simulations (31,44,45), density functional theory (DFT) (35,46-49), Moller-Plesset perturbation theory ( ) (50-53), configuration interaction (Cl) (35,42,54-56), coupled cluster (CC) (57,58), and CASSCF/CASPT2 (59-63)). 

The studies of Ertl and co-workers showed that the reason for self-oscillations [142, 145, 185-187] and hysteresis effects [143] in CO oxidation over Pt(100) in high vacuum ( 10 4 Torr) is the existence of spatio-temporal waves of the reversible surface phase transition hex - (1 x 1). The mathematical model [188] suggests that in each of the phases an adsorption mechanism with various parameters of CO and 02 adsorption/desorption and their interaction is realized, and the phase transition is modelled by a semi-empirical method via the introduction of discontinuous non-linearity. Later, an imitation model based on the stochastic automat was used [189] to study the qualitative characteristics for the dynamic behaviour of the surface. 

Empirical Model Identification. In this section we consider linear difference equation models for characterizing both the process dynamics and the stochastic disturbances inherent in the process. We shall discuss how to specify the model structure, how to estimate its parameters, and how to check its adequacy. Although discussion will be limited to single-input, single-output processes, the ideas are directly extendable to multiple-input, multiple-output processes. 

At the core of the model is a dynamic relationship between product price and the industry s operating rate, that is, its balance of supply and demand. It is the fact that this empirically observed relationship is non-linear that leads to short-term price fly-ups separated by long troughs in which, typically, few players make money. 

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