Dynamic linear modeling

The performance and properties of multiscale Bayesian rectification are compared with those of other methods in the following examples. The examples compare the performance for rectification of Gaussian errors with steady-state and dynamic linear models for stochastic and deterministic underlying signals. 

Nonlinear versus Linear Models If F, and k are constant, then Eq. (8-1) is an example of a linear differential equation model. In a linear equation, the output and input variables and their derivatives only appear to the first power. If the rate of reac tion were second order, then the resiilting dynamic mass balance woiild be ... 

Simulation of Dynamic Models Linear dynamic models are particularly useful for analyzing control-system behavior. The insight gained through linear analysis is invaluable. However, accurate dynamic process models can involve large sets of nonlinear equations. Analytical solution of these models is not possible. Thus, in these cases, one must turn to simulation approaches to study process dynamics and the effect of process control. Equation (8-3) will be used to illustrate the simulation of nonhnear processes. If dcjdi on the left-hand side of Eq. (8-3) is replaced with its finite difference approximation, one gets ... 

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. 

To model this highly complex and nonlinear dynamics properly, we need the heat and mass balances. In classical control, however, we would replace them with a linearized model that is the sum of two functions in parallel ... 

A control algorithm has been derived that has improved the dynamic decoupling of the two outputs MW and S while maintaining a minimum "cost of operation" at the steady state. This algorithm combines precompensation on the flow rate to the reactor with state variable feedback to improve the overall speed of response. Although based on the linearized model, the algorithm has been demonstrated to work well for the nonlinear reactor model. 

Some attempts to exploit sensor dynamics for concentration prediction were carried out in the past. Davide et al. approached the problem using dynamic system theory, applying non-linear Volterra series to the modelling of Thickness Shear Mode Resonator (TSMR) sensors [4], This approach gave rise to non-linear models where the difficulty to discriminate the intrinsic sensor properties from those of the gas delivery systems limited the efficiency of the approach. 

Exercise 6. Show that the equilibrium point of the model defined by Eq.(34) and the simplified model R given by Eq.(35), i.e. when the dynamics of the jacket is considered negligible, are the same. Deduce the Jacobian of the system (35) at the corresponding equilibrium point. Write a computer program to determine the eigenvalues of the linearized model R at the equilibrium point as a function of the dimensionless inlet flow 4 50. Values of the dimensionless parameters of the PI controller can be fixed at Ktd = 1-52 T2d = 5. The set point dimensionless temperature and the inlet coolant flow rate temperature are Xg = 0.0398, X40 = 0.0351 respectively. An appropriate value of dimensionless reference concentration is C g = 0.245. Does it exist some value of 2 50 for which the eigenvalues of the linearized system R at the equilibrium point are complex with zero real part Note that it is necessary to vary 2 50 from small to great values. Check the possibility to obtain similar results for the R model. 

The MPC strategy can be summarized as follows. A dynamic process model (usually linear) is used to predict the expected behavior of the controlled output variable over a finite horizon into the future. On-line measurement of the output is used to make corrections to this predicted output trajectory, and hence provide a feedback correction. The moves of the manipulated variable required in the near future are computed to bring the predicted output as close to the desired target as possible without violating the constraints. The procedure is repeated each time a new output measurement becomes available. 

For start-up and disturbance simulations, the linearized model does predict an eventual return to the steady state around which the system was linearized. However, for step-input changes where the final steady state differs from the original, some minimal loss in accuracy is apparent in the final steady state reached using dynamic simulations of the linear model from the original steady state. This difficulty can easily be circumvented in the case of step changes by relinearizing about the new final steady-state conditions somewhere during the simulation. 

It is evident that, of the 30 modes of the full linear model (with N = 6), 18 are very fast in comparison to the remaining 12 (by 2 orders of magnitude or more). Thus direct modal reduction to a 12th-order model using Davison s method should provide good dynamic accuracy. However, by simply neglecting the non-dominant modes of the system, the contribution of these modes is also absent at steady state, thus leading to possible (usually minor) steady-state offset. Several identical modifications (Wilson et al, 1974) to Davison s... 

We will then compare the dynamics of the radial diffusion model with a first-order exchange model which gives the same half-life as the radial model (Eq. 18-37). A preview of this comparison is given in Fig. 18.7b, which shows that the linear model underpredicts the exchange at short times and overpredicts it at long times. 

An important advance in the understanding of microscopic solvation and Onsager s snowball picture has recently been made through the introduction of the linearized mean spherical approximation (MSA) model for the solvation dynamics around ionic and dipolar solutes. The first model of this type was introduced by Wolynes who extended the equilibrium linearized microscopic theory of solvation to handle dynamic solvation [38]. Wolynes further demonstrated that approximate solutions to the new dynamic MSA model were in accord with Onsager s predictions. Subsequently, Rips, Klafter, and Jortner published an exact solution for the solvation dynamics within the framework of the MSA [43], For an ionic solute, the exact results from these author s calculations are in agreement with Onsager s inverted snowball model and the previous numerical calculations of Calef and Wolynes [37]. Recently, the MSA model has been extended by Nichols and Calef and Rips et al. [39-43] to solvation of a dipolar solute. 

In the high crack velocity regime three different values of Kid can be assigned to one rate of crack propagation depending on the state of crack acceleration. This behaviour was ascribed to inertia effects associated with crack acceleration and deceleration. Such a hypothesis is corroborated by the computed K data (also shown in Fig. 9), which were obtained from a finite element model, taking into consideration the mentioned transient dynamic linear elastic effects [35]. 

To best achieve the benefits of hybrid systems, improved dynamic system models are needed. Much of the opportunity for innovation and ultimate commercial success for this technology lies in the area of system dynamics and control. To achieve commercial success, it is critical that the technical issues surrounding system dynamics are identified. Dynamic models can play a helpful role in that regard. Chapter 9 describes dynamic modeling of the primary device, the SOFC itself. This chapter s section will expand on this to discuss a full non-linear hybrid system dynamic model. 

Typically, a non-linear system dynamic model is made up of individual lumped models of the components which at a minimum conserve mass and energy across the given component, but may also have a momentum equation if pressure drops must also be analyzed. For most dynamic problems of interest in hybrid studies, however, the momentum equation may be taken as quasi-steady (unless the solver requires the dynamic form to perform the numerical solution). Higher fidelity individual models or reduced order models (ROMs) can also be used, where the connection to the system model would be made at each subcomponent boundary. Since dynamic systems modeling is not as common as steady-state modeling, some discussion of modeling approaches will be given. There are two primary methods used to provide solutions for the pressure-flow dynamics of a system model. 

The data of heating rate and the temperature at which the maximum rate of reaction occurs, Tpeak, was plotted, and fitted to a linear model. The activation energy, E, of a silicone rubber was calculated with the data from four dynamic scanning rates. The slope of the line corresponds to the negative ratio of the activation energy and the universal gas constant R (8.3145 J/gmol/K) as can be seen in Fig. 7.20. 

Early applications of MPC took place in the 1970s, mainly in industrial contexts, but only later MPC became a research topic. One of the first solid theoretic formulations of MPC is due to Richalet et al. [53], who proposed the so-called Model Predictive Heuristic Control (MPHC). MPHC uses a linear model, based on the impulse response and, in the presence of constraints, computes the process input via a heuristic iterative algorithm. In [23], the Dynamic Matrix Control (DMC) was introduced, which had a wide success in chemical process control both impulse and step models are used in DMC, while the process is described via a matrix of constant coefficients. In later formulations of DMC, constraints have been included in the optimization problem. Starting from the late 1980s, MPC algorithms using state-space models have been developed [38, 43], In parallel, Clarke et al. used transfer functions to formulate the so-called Generalized Predictive Control (GPC) [19-21] that turned out to be very popular in chemical process control. In the last two decades, a number of nonlinear MPC techniques has been developed [34,46, 57],... 

Considerable insight into the dynamic behavior of the system can be gained by exploring the effects of various parameters on a linearized version of the system equations. Dynamic features such as damping, speed of response, and stability are clearly revealed using a linear model. 

The linear model permits the use of all the linear analysis tools available to the process control engineer. For example, the poles and zeros of the openloop transfer function reveal the dynamics of the openloop system. A root locus plot shows the range of controller gains over which the system will be closedloop-stable. 

In a later section we show results of rigorous dynamic simulations of this process. However, it may be useful at this point to show the predictions of a linear model of this type of FEHE-reactor system. 

McAvoy (1999) advanced the use of optimization calculations at the controller design stage, proposing the synthesis of plant-wide control structures that ensure minimal actuator movements. The initial work relying on steady-state models (McAvoy 1999) was recast into a controller synthesis procedure based on linear dynamic plant models (Chen and McAvoy 2003, Chen et al. 2004), whereby the performance of the generated plant-wide control structures was evaluated through dynamic simulations. 

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