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FIGURE 18-28 Dead-time compensation for Example 18.4.
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II Dead-time compensation, K,= 4
I Conventional control, K,= 2
FIGURE
18-29
Comparison of response for conventional control with response for dead-time compensation for
Example 18.4.
in Fig. 18.27. If one fits the step response of (S + 1)-4 to a first-order with dead-time model, one obtains 1 e-1.5s 3s + 1 This model was obtained from a unit-step response using a least squares fit procedure. We can now draw the diagram for the dead-time compensation system as shown in Fig. 18.28. The system shown in Fig. 18.28 was simulated by computer in order to compare the responses of the two control systems as shown in Fig. 18.29. Using a K, of 2.0 (the Ziegler-Nichols value) for the conventional control we see from Curve I that the response is quite oscillatory and has an offset of 0.333 as required for this value of gain. Using a K, of 4.0 for the dead-time compensation, we see that the response is less oscillatory and the offset is 0.20. It should be noted that if a K, of 4.0 were applied to the conventional control system, the system would be on the verge of instability since a K, of 4.0 is the ultimate gain. In conclusion, the dead-time compensation has permitted the use of a higher value of Kc, reduced the offset, and produced a less oscillatory response. The deadtime compensation response shown in Fig. 18.29 can be improved by adding integral action to the controller and tuning the controller parameters. To successfully apply dead-time compensation to the control of a process, one must have an accurate model of the process, such as a first-order with dead-time model. The parameters in this model (r and 7~) can be considered as controller parameters along with the controller parameters of G,(s). For the case of dead-time compensation with proportional control in Example 18.4, we actually have three controller parameters: Kc, 713, and 7. If the process dynamics [G&)] changes, all three parameters may need adjustment in order to achieve good control.
INTERNAL MODEL CONTROL
Internal model control (IMC) has been the subject of intense research since about 1980. This method of control, which is based on an accurate model of the pro-
PROCESS APPLICATIONS
cess, leads to the design of a control system that is stable and robust. A robust control system is one that maintains satisfactory control in spite of changes in the dynamics of the process. In applying the IMC method of control system design, the following information must be specified: Process model Model uncertainty Type of input (step, ramp, etc.) Performance objective (integral square error, overshoot, etc.) In many industrial applications for control systems, none of the above items is available, with the result that the system usually performs in a less than optimum manner. Determining the mathematical model and its uncertainty can be a difficult task. When the process is not sufficiently understood to obtain a mathematical model by applying fundamental principles, one must obtain a model experimentally. A discussion on the modeling of a process is presented in the next chapter. The choice of a performance objective is subjective and often arbitrary. In the IMC method, the integral square error is implied. A simple description of the IMC method will be presented here. The interested reader is advised to consult the book by Morari and Zafiriou (1989) for a full treatment of internal model control. The literature on IMC is difficult to understand without a good foundation in control theory and mathematics. A full treatment of IMC is beyond the scope of this text. It is hoped that the simple treatment given here will stimulate interest in this important new area of process control.
Internal Model Control Structure
A block diagram of an IMC system is shown in Fig. 18.30~. Notice that the diagram is similar to the diagram for the Smith predictor method shown in Fig. 18.25~. In this diagram, G is the transfer function of the process and G, is the model of the process. Although G and G, are called the transfer functions of the process, they actually include the valve and the process. The transfer function of the measuring element is taken as 1.0. The portion of the diagram that is implemented by the computer includes the IMC controller and the model; this portion is surrounded by the dotted boundary. In order to compare the IMC structure of Fig. 18.30~ with the conventional control structure, the diagram of Fig. 18.30~ has been rearranged as shown in Fig. 18.30b. For convenience, the transfer function through which the load U passes has been omitted. We show only the output from the load block (Ui). We may use the structure in Fig. 18.30b to relate the IMC controller to the conventional controller. Replacing the inner loop of Fig. 18.30b with a single block gives the structure shown in Fig. 18.30~. Since this structure is the conventional single-
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