KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Simulation)-Chapter_14

KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Simulation)-Chapter_14

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1 Chapter 14 Control System Design Based on Frequency Response Analysis Frequency response concepts and techniques play an important role in control system design and analysis. Closed-Loop Behavior In general, a feedback control system should satisfy the following design objectives: 1. Closed-loop stability 2. Good disturbance rejection (without excessive control action) 3. Fast set-point tracking (without excessive control action) 1. A satisfactory degree of robustness to process variations and model uncertainty 2. Low sensitivity to measurement noise
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2 Chapter 14 The block diagram of a general feedback control system is shown in Fig. 14.1. • It contains three external input signals: set point Y sp , disturbance D , and additive measurement noise, N . (14-1) 1 1 1 m c v p d c sp c c c K G G G G G G Y D N Y G G G G G G = - + + + + (14-2) 1 1 1 d m m m sp c c c G G G K E D N Y G G G G G G = - - + + + + (14-3) 1 1 1 d m c v m c v m c v sp c c c G G G G G G G K G G U D N Y G G G G G G = - - + + + + where G G v G p G m . @
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3 Chapter 14 Figure 14.1 Block diagram with a disturbance D and measurement noise N .
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4 Chapter 14 Example 14.1 Consider the feedback system in Fig. 14.1 and the following transfer functions: 0.5 , 1 1 2 p d v m G G G G s = = = = - Suppose that controller G c is designed to cancel the unstable pole in G p : 3 (1 2 ) 1 c s G s - = - + Evaluate closed-loop stability and characterize the output response for a sustained disturbance.
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5 Chapter 14 Solution The characteristic equation, 1 + G c G = 0, becomes: or 2.5 0 s + = In view of the single root at s = -2.5, it appears that the closed- loop system is stable. However, if we consider Eq. 14-1 for N = Y sp = 0, 3 (1 2 ) 0.5 1 0 1 1 2 s s s - + = + - ( 29 0.5 1 1 (1 2 )( 2.5) d c s G Y D D G G s s - + = = + - +
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6 Chapter 14 This transfer function has an unstable pole at s = +0.5. Thus, the output response to a disturbance is unstable. Furthermore, other transfer functions in (14-1) to (14-3) also have unstable poles. This apparent contradiction occurs because the characteristic equation does not include all of the information, namely, the unstable pole-zero cancellation. Example 14.2 Suppose that G d = G p , G m = K m and that G c is designed so that the closed-loop system is stable and | GG c | >> 1 over the frequency range of interest. Evaluate this control system design strategy for set-point changes, disturbances, and measurement noise. Also consider the behavior of the manipulated variable, U .
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7 Chapter 14 Solution Because | GG c | >> 1, 1 0 and 1 1 1 c c c G G G G G G s + + The first expression and (14-1) suggest that the output response to disturbances will be very good because Y/D ≈ 0. Next, we consider set-point responses. From Eq. 14-1, 1 m c v p sp c K G G G Y Y G G = + Because G m = K m , G = G v G p K m and the above equation can be written as, 1 c sp c G G Y Y G G = +
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8 Chapter 14 For | GG c | >> 1, 1 sp Y Y s Thus, ideal (instantaneous) set-point tracking would occur. Choosing
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KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Simulation)-Chapter_14

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