Transfer Function Representation

4 OPENLOOP AND CLOSEDLOOP SYSTEMS 15.4.1 Transfer Function Representation 

If the system is openloop, m can be changed independently. If the system is dosedloop, m is determined by the feedback controller. Substituting for m from Eq. (15.53) gives 

Bringing all the terms with x to the left side gives 

Equation (15.64) gives the effects of setpoint and load changes on tbe controlled variables in the closedloop multivariable environment. The matrix (of order N X N) multiplying the vector of setpoints is the closedloop servo transfer function matrix. The matrix (N x 1) multiplying the load disturbance is the closed-loop regulator transfer function vector. 

It is clear that the matrix equation [Eq. (15.64)] is very similar to the scalar equation describing a closedloop system derived back in Chap. 10 for SISO systems. 

Complex systems can often be represented by linear time-dependent differential equations. These can conveniently be converted to algebraic form using Laplace transformation and have found use in the analysis of dynamic systems (e.g., Coughanowr and Koppel, 1965, Stephanopolous, 1984 and Luyben, 1990). 

as shown in Fig. 2.17, the input-output transfer-function relationship, G(s), is algebraic, whereas the time domain is governed by the differential equation. 

Complex models are often slow in execution owing to the large number of equations involved and the large range of time constants. Under these circumstances it is often useful to approximate the transient behaviour of the full model by a simpler model representation which is faster to compute. Such simplifications are commonly achieved by a combination of first-order lags and time delays and are often represented in Laplace transform form, especially when the sub-model is to be used as part of a control engineering application. 

Model representations in Laplace transform form are mainly used in control theory. This approach is limited to linear differential equation systems or their 

Dynamic problems expressed in transfer function form are often very easily reformulated back into sets of differential equation and associated time delay functions. An example of this is shown in the simulation example TRANSIM. 

As shown in the preceding sections, the magnitude of various process time constants can be used to characterise the rate of response of a process resulting from an input disturbance. A fast process is characterised by a small value of the time constant and a slow process by large time constants. Time constants can therefore be used to compare rates of change and thus also to compare the relative importance of differing rate processes. 

Model representations in Laplace transform form are mainly used in control theory. This approach is limited to linear differential equation systems or their linearized approximations and is achieved by a combination of first-order lag function and time delays. This limitation together with additional complications of modelling procedures are the main reasons for not using this method here. Specialized books in control theory as mentioned above use this approach and are available to the interested reader. 

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The transfer function representation of a time delay discussed in Sec. 2.1.3 is given by... 

The limitation of transfer function representation becomes plain obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modem control, the method of choice is state space or state variables in time domain—essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modem control without a firm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, we will put the explanation in examples as much as possible. The actual state space control has to be delayed until after tackling classical transfer function feedback systems. 

Understand the how a state space representation is related to the transfer function representation. 

We will use this matrix of transfer function representation extensively in the rest of our work with multivariable processes. 

The time-domain differential equation description of systems can be used instead of the Laplace-domain transfer function description. Naturally the two are related, and we will derive these relationships later in this chapter. State variables are very popular in electrical and mechanical engineering control problems which tend to be of lower order (fewer differential equations) than chemical engineering control problems. Transfer function representations are more useful in practical process control problems because the matrices are of lower order than would be required by a state variable representation. 

State variable representation ean be transformed into transfer function representation by Laplace-transforming the set of M linear ordinary differential equations [Eq. (12.23)]. 

Matrix Properties /12.1.2 Transfer Function Representation / 12.1.3 State Variables... 

Oie transfer function representation of tlie syrstem oonsists of t o eqoationst... 

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... 

Figure 5. A double RC filter circuit, whose second-order transfer function representation models the response of a spin system in a pulsed saturation experiment.

The transfer function representation makes it easy to compare the effects of different inputs. For example, the dynamic model for the constant-flow stirred-tank blending system was derived in Section 4.1. 

A second advantage of the transfer function representation is that the dynamic behavior of a given process can be generalized easily. Once we analyze the response of the process to an input change, the response of any other process described by the same generic transfer function is then known. 

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