Laplace transform, process transfer control

Chapters 2 and 3 have considered dynamic models in the form of ordinary differential equations (ODE). In this chapter, we introduce an alternative model form based on Laplace transforms the transfer function model. Both types of models can be used to determine the dynamic behavior of a process after changes in input variables. The transfer function also plays a key role in the design and analysis of control systems, as will be considered in later chapters. 

Our primary use of Laplace transformations in process control involves representing the dynamics of the process in terms of "transfer functions." These are output-input relationships and are obtained by Laplace-transfonning algebraic... 

J0i Use Laplace transforms to prove mathematically that a P controller produces steadystate ofiMt and that a PI controller does not. The disturbance is a step change in the load variable. The process openloop transfer functions, Gm and G[, are both liist-order lags with dUTerent gains but identical time constants. 

Notice that the right-hand side of Eq. (34) is equal to the ratio of the transformed concentration at the second measurement point to the transformed concentration at the first measurement point. In the terminology of control engineering, this quantity is the transfer function of the system between Xo and Xm- The Laplace-transform method is possible because the diffusion equation is a linear differential equation. Thus, the right-hand side of Eq. (34) could in principle be used in a control-system analysis of an axial-dispersion process. 

Block Diagram Analysis One shortcoming of this feedforward design procedure is that it is based on the steady-state characteristics of the process and, as such, neglects process (Ramies (i.e., how fast the controlled variable responds to changes in the load and manipulated variables). Thus, it is often necessary to include dynamic compensation in the feedforward controller. The most direct method of designing the FF dynamic compensator is to use a block dir rram of a general process, as shown in Fig. 8-34, where G, represents the disturbance transmitter, (iis the feedforward controller, Cj relates the disturbance to the controlled variable, G is the valve, Gp is the process, G is the output transmitter, and G is the feedback controller. All blocks correspond to transfer fimetions (via Laplace transforms). 

Consider the block diagram of a direct digital feedback control loop shown in Figure 29.9. Such loops contain both continuous- and discrete-time signals and dynamic elements. Three samplers are present to indicate the discrete-time nature of the set point j/Sp( ), control command c(z), and sampled process output y(z). The continuous signals are denoted by their Laplace transforms [i.e., y(s), Jn(s), and d(s)]. Furthermore, the continuous dynamic elements (e.g., hold, process, disturbance element) are denoted by their continuous transfer functions, H(s), Gp(s), and GAs), respectively. For the control algorithm, which is the only discrete element, we have used its discrete transfer function, D(z). 

In this chapter we demonstrate the significant computational and notational advantages of Laplace transforms. The techniques involve finding the transfer function of the openloop process, specifying the desired performance of the closedloop system (process plus controller), and finding the feedback controller transfer function that is required to do the job. 

Openloop process transfer function. These three ODEs are linear, so we do not have to linearize. Converting to perturbation variables, Laplace transforming, and solving for the transfer function between the controlled variable T, and the manipulated variable Q give... 

Experience in the process industries indicates that there are a limited number of expected dynamic behaviors that actually influence the controller design step. These behaviors can be categorized using the step response and are based on a transfer function representation of the process model, which is assumed to be linear or a linear approximation of a nonlinear model. A transfer function is found by taking the Laplace transform of the ordinary differential equation that describes the system the mathematical definition of the Laplace transform is... 

Laplace transform used in the development of transfer functions, which are the most widely used model form in process control studies. The Laplace transform converts an ordinary dilferential equation (ODE) to an algebraic equation and, likewise, converts a partial dilferential equation (PDE) into an ordinary dilferential equation (ODE). 

It follows from the last example that when controlling the potential, its value is to be related to i and surface concentration of EAC. The latter term appears in the kinetic equation. For this purpose, it is necessary to know mechanism and kinetic parameters of the charge transfer process. However, this complicated way may be avoided, if the experimentally obtained i(t)-function is utilized. Then, to find concentration profiles, the Laplace transform of this function is to be obtained ... 

In previous chapters, Laplace transform techniques were used to calculate transient responses from transfer functions. This chapter focuses on an alternative way to analyze dynamic systems by using frequency response analysis. Frequency response concepts and techniques play an important role in stability analysis, control system design, and robustness analysis. Historically, frequency response techniques provided the conceptual framework for early control theory and important applications in the field of communications (MacFarlane, 1979). We introduce a simplified procedure to calculate the frequency response characteristics from the transfer function of any linear process. Two concepts, the Bode and Nyquist stability criteria, are generally applicable for feedback control systems and stability analysis. Next we introduce two useful metrics for relative stability, namely gain and phase margins. These metrics indicate how close to instability a control system is. A related issue is robustness, which addresses the sensitivity of... 

A transfer function is the relationship between the input and output of a system. In classical control systems literature that makes use of Laplace transforms, extensive use is made of Laplace transfer functions. Table 3.2 presents the transfer functions of common process systems dynamics. 

For decades, the subject of control theory has been taught using transfer functions, frequency-domain analysis, and Laplace transform mathematics. For linear systems (like those from the electromechanical areas from which these classical control techniques emerged) this approach is well suited. As an approach to the control of chemical processes, which are often characterized by nonlinearity and large doses of dead time, classical control techniques have some limitations. 

In Chap. 18 we will define mathematically the sampling process, derive the z transforms of common functions (learn our German vocabulary) and develop transfer functions in the z domain. These fundamentals are then applied to basic controller design in Chap. 19 and to advanced controllers in Chap. 20. We will find that practically all the stability-analysis and controller-design techniques that we used in the Laplace and frequency domains can be directly applied in the z domain for sampled-data systems. 

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