Process Dynamics and Control_Ch05 - Chapter 5 Dynamic...

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Chapter 5 Dynamic Response Characteristics of More Complicated Processes In Chapter 4 we discussed the dynamics of relatively simple processes, those that can be modeled as either first- or second-order transfer functions or as an integrator. Now we consider more complex transfer function models that include additional time constants in the denominator and/or functions of s in the numerator. We show that the forms of the numerator and denominator of the transfer function model influence the dynamic behavior of the process. We also introduce a very important concept, the time delay, and consider the approximation of complicated transfer function models by simpler, low-order models. Additional topics in this chapter include interacting processes, state-space models, and processes with multiple inputs and outputs. 5.1 POLES AND ZEROS AND THEIR EFFECT ON PROCESS RESPONSE An important feature of the simple process elements discussed in Chapter 4 is that their response characteristics are determined by the factors of the transfer function denominator. For example, consider a transfer function, where 0 ≤ ζ < 1. Using partial fraction expansion followed by the inverse transformation operation, we know that the response of system (5-1) to any input will contain the following functions of time: A constant term resulting from the s factor An e t/τ1 term resulting from the (τ 1 S + 1) factor Additional terms determined by the specific input forcing will also appear in the response, but the intrinsic dynamic features of the process, the so-called response modes or natural modes, are determined by the process itself. Each of the above response modes is determined from the factors of the denominator polynomial, which is also called the characteristic polynomial (cf. Section A.3). The roots of these factors are S 1 = 0 Roots S 3 and S 4 are obtained by applying the quadratic formula. Control engineers refer to the values of s that are roots of the denominator polynomial as the poles of transfer function G(s). Sometimes it is useful to plot the roots (poles) and to discuss process response characteristics in terms of root locations in the complex s plane. In Fig. 5.1 the ordinate expresses the imaginary part of each root; the abscissa expresses the real part. Figure 5.1 is based on Eq. 5-2 and indicates the presence of four poles: an integrating element (pole at the origin),
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Case i: 0 < τ 1 < τ a α = 8) Case ii: 0 < τ a < τ 1 a = 1, 2) Case iii: τ α < 0 a = –1, –4) Case i: τ a Case ii: 0 < τ Case iii: τ a Figure 5.1 . The real pole is closer to the imaginary axis than the complex pair, indicating a slower response mode ( e –t/τ1 decays slower than e –ζ t /τ2 ). In general, the speed of response for a given mode increases as the pole location moves farther away from the imaginary axis.
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  • Fall '16
  • naile
  • Laplace, state-space models

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