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If we introduce these approximations for the integral and the derivative into Eqs. (27.47) and (27.48), we obtain after subtracting Eq. (27.48) from Eq. (27.47) m(nT) - m[(n - l)T] = K, e(nT) - e[(n - l)T] + $e(nT) i
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e(nT) - 2e[(n - l)T] + e[(n - 2)T]
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(27.50)
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Converting this equation to the z-domain and solving for M(z)/E(z), which is D(z), finally gives the algorithm:
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D(z) =
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where p =
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KJz* - Pz + rl PZ(Z - 1)
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(27.51)
CT + 27Dh T* + Tq + TITD
71 + 70
T* + Tq + TITD
TTI P= T* + Tq + TjTD
The nature of the response for a unit-step change in input for K, = 71 = T = 1 and ro = 2 is shown in Fig. 27.14. The details of obtaining this result are left as an exercise for the reader. The response of the continuous PID controller to a unit-step change in input for the same parameters (Kc,71,7~) is also shown in Fig. 27.14. Notice that the impulse at t = 0 for the continuous response is replaced by a pulse during the first sampling period that reaches a value of 4.0 instead of infinity. After t = 1, the sampled-data response is the same as for the PI sampled-data response shown in Fig. 27.13. As 70 is increased, the pulse during the first sampling period will become larger, thereby approximating more closely the jump to infinity for the continuous response. The simple backward difference formula used to approximate the derivative term in the PID control algorithm can be replaced by a higher order difference
continuous I responseiI m(4for
I :r
FIGURE 27-14 Comparison of sampled-data response and continuous response for a PID controller subjected to a step change in input.
DESIGN OF SAMPLED-DATA COmOLLERS
approximation to give an alternate version of D(z). In fact, many alternate difference approximations for the integral term and the derivative term can be used to give a variety of forms for D(z). As the sampling period T is reduced, the response of the control system using different forms of D(z) for PI or PID control should approach the response for continuous versions of the algorithms. One of the problems at the end of this chapter involves the calculation of the response of a system that uses the D(z) for a PI controller given by Eq. (27.45). In general, the replacement of a continuous controller by its equivalent sampled-data version will give a less stable response for the same set of controller parameters (Kc,r~,ro).
SUMMARY
In this chapter a systematic procedure for the design of direct-digital control algorithms was described. The procedure requires that a model for the process be known and that the location of the disturbance (set point or load) and the type of disturbance (step, ramp, etc.) be specified. These requirements are similar to those for designing a controller by the internal model control procedure discussed in Chap. 18. The design. procedure presented here gives the designer a wide choice of the desired response of the control system; this choice is usually based on knowledge of the response of the process model. The minimal prototype response is an ideal response that reduces the error (at sampling instants) to zero in the least time. The control algorithm D(z) obtained by the design procedure can be written in a form that can be used by a digital computer to control the process. The need to test a proposed algorithm for a disturbance other than the one used to design the algorithm was emphasized and illustrated by examples. The equivalent sampled-data control algorithms for conventional (PI and PID) control were derived and the open-loop response for each algorithm was compared to the response for the corresponding continuous algorithms. As the sampling period T decreases, the response of the digital algorithm approaches that of the continuous algorithm.
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