Modeling, linear control system

The main conclusion to be drawn from these studies is that for most practical purposes the linear rate model provides an adequate approximation and the use of the more cumbersome and computationally time consuming diffusing models is generally not necessary. The Glueckauf approximation provides the required estimate of the effective mass transfer coefficient for a diffusion controlled system. More detailed analysis shows that when more than one mass transfer resistance is significant the overall rate coefficient may be estimated simply from the sum of the resistances (7) ... 

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 ... 

The program can solve both steady-state problems as well as time-dependent problems, and has provisions for both linear and nonlinear problems. The boundary conditions and material properties can vary with time, temperature, and position. The property variation with position can be a straight line function or or a series of connected straight line functions. User-written Fortran subroutines can be used to implement more exotic changes of boundary conditions, material properties, or to model control systems. The program has been implemented on MS DOS microcomputers, VAX computers, and CRAY supercomputers. The present work used the MS DOS microcomputer implementation. 

This section is a review of the properties of a first order differential equation model. Our Chapter 2 examples of mixed vessels, stined-tank heater, and homework problems of isothermal stirred-tank chemical reactors all fall into this category. Furthermore, the differential equation may represent either a process or a control system. What we cover here applies to any problem or situation as long as it can be described by a linear first order differential equation. 

Equation 23.25 has the same basic form as the linear Willans Line Equation used to model steam turbines. The basic assumption behind the use of Equation 23.25 is that the gas turbine would need to have a control system that would maintain a fixed fuel-to-air ratio and steam-to-air ratio at part-load. 

As seen previously for some specific applications such as wastewater treatment plants, software sensors can be envisaged to provide on-line estimation of non-measurable variables, model parameters or to overcome measurement delays [81-83]. Software sensors have been developed mainly for monitoring bioprocesses because the control system design of bioreactors is not straightforward due to [84] significant model uncertainty, lack of reliable on-line sensors, the non-linear and time-varying nature of the system or slow response of the process. 

Almasy, G., and Sztano, T. (1975). Checking and correction of measurements on the basis of linear system model. Probl. Control Inf. Theory 4, 57. 

This chapter is organized in the following way. First, the general model of the CSTR process, based on first principles, is derived. A linearized approximate model of the reactor around the equilibrium points is then obtained. The analysis of this model will provide some hints about the appropriate control structures. Decentralized control as well as multivariable (MIMO) control systems can be designed according to the requirements. 

Proposition 4. Since the AD model (2) is a minimum-phase system and has a well-defined relative degree r under NOC. Then, the following input-output linearizing controller will make converge exponentially the total concentration of organic substrate St to a desired value for all t > 0... 

The values of Km and T2d from Eq.(36) can be obtained from the transfer function of the linearized model at the equilibrium point, applying conventional methods from the linear control theory (see [1]). In order to investigate the self-oscillating behavior, one can determine the linearized system at the equilibrium point, and the corresponding complex eigenvalues with zero real part, when the parameters Km and of the PI controller are varied. For example, taking into account Eq.(34), the Jacobian matrix of the linearized system at dimensionless set point temperature xs is the following ... 

For continuous process systems, empirical models are used most often for control system development and implementation. Model predictive control strategies often make use of linear input-output models, developed through empirical identification steps conducted on the actual plant. Linear input-output models are obtained from a fit to input-output data from this plant. For batch processes such as autoclave curing, however, the time-dependent nature of these processes—and the extreme state variations that occur during them—prevent use of these models. Hence, one must use a nonlinear process model, obtained through a nonlinear regression technique for fitting data from many batch runs. 

In simplifying the packed bed reactor model, it is advantageous for control system design if the equations can be reduced to lit into the framework of modern multivariable control theory, which usually requires a model expressed as a set of linear first-order ordinary differential equations in the so-called state-space form ... 

Even after linearization, the state-space model often contains too many dependent variables for controller design or for implementation as part of the actual control system. Low-order models are thus required for on-line implementation of multivariable control strategies. In this section, we study the reduction in size, or order, of the linearized model. 

As with scale-up, two levels of implementation are possible. The first level only entails the ability to sense, and a directional characterization of the effect of variables. PAT methods can be extremely effective for this purpose by generating large datasets of process inputs and outputs that can then be correlated to generate statistical or polynomial control models. Provided that (i) deviations from desired set-points are small, (ii) interactions between inputs are weak, and (Hi) the response surface does not depart too much from linearity, such systems can provide the basis of an initial effort to control a system. 

A. Almtey and . Sztand, Checking and correction of measurements an the basis of linear system models. Probl. Control. Inf. Theory, 4 (1975) 59-69. 

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. 

The most general approach to model-based nonlinear control is the so-called Feedback Linearization (FL) [35], In fact, FL control approaches use the model of the plant to achieve a global linearization of the closed-loop systems, so as well-established linear controllers can be adopted for the globally linearized model. In... 

Several process control design methods, such as the Generic Model Control (GMC) [41], the Globally Linearizing Control (GLC) [37], the Internal Decoupling Control (IDC) [7], the reference system synthesis [8], and the Nonlinear Internal Model Control (NIMC) [29], are based on input-output linearization. 

As previously mentioned, the glucose-insulin control system is often regarded as a simple system to keep the plasma glucose concentration within narrow limits. In this context it has been compared to technical control systems and described by simple, often linear or linearized models. Section 6.2 of this chapter gives an outline of classic control and underlines some of the peculiarities of biological control systems. 

Listing C.2. S-function illustrating the use of the reduced-order model of the slow dynamics in the derivation of a control system (including an input-output linearizing temperature controller) for the reactor-feed effluent heat exchanger system in Section 6.6... 

LeLann et al. [6] discuss tendency modeling (using approximate stoichiometric and kinetic models for a reaction) and the use of model predictive control (linear and nonlinear) in batch reactor operation. Studies of a hybrid heating-cooling system on a 16-L pilot plant are presented. 

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