Goals for today 2.004

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1 Goals for today Block diagrams revisited Block diagram components Block diagram cascade Summing and pickoff junctions Feedback topology Negative vs positive feedback Example of a system with feedback Derivation of the closedloop transfer function Specification of the transient response by selecting the feedback gain The opamp in feedback configuration

2 Block diagram components Transfer Function =. Images removed due to copyright restrictions. Please see: Fig. 5.2 in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, 2004.

3 Cascading subsystems Images removed due to copyright restrictions. Please see: Fig. 5.3 in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, Note: to cascade two subsystems, we must ensure that the second subsystem does not load the first subsystem

4 Loading and cascade Images removed due to copyright restrictions. Please see: Fig. 5.4 in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, 2004.

5 Cascading with an opamp I 3 R 4 V i R 1 R V 3 3 _ V = 0 I I _ 1 3 C I 1 C1 V 4 I 2 R 2 C 2 V 2 V 2 V i = We can show that 1 R 1 C 1 s 1 R 1 C 1 1 R 3 C 1 µ R 4 R 3 1 R 2 C 2 s 1 R 2 C 2. 1 R 1 C 1 s 1 R 1 C 1 µ R 4 R 3 1 R 2 C 2 s 1 R 2 C 2 if R 3 À R 1.

6 Parallel subsystems Images removed due to copyright restrictions. Please see: Fig. 5.5 in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, 2004.

7 Negative feedback E = H Input E Actuating signal (error) Plant and controller H Feedback Output = E h i = H = 1H. : Closed loop TF H : Open loop TF. 1 H Input Output equivalent system Figure 5.6

8 Positive feedback Input E Actuating signal (error) Plant and controller Output E = H = E h i = H = 1 H. H Feedback : Closed loop TF H : Open loop TF. Figure 5.6 Input 1 _ H Output equivalent system Generally, positive feedback is dangerous: it may lead to unstable response (i.e. exponentially increasing) if not used with care

9 A more general feedback system Images removed due to copyright restrictions. Please see: Fig. 5.6 in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, Plant: the system we want to control (e.g., elevator plant: input=voltage, output=elevator position) Controller: apparatus that produces input to plant (i.e. voltage to elevator s motor) Transducers: converting physical quantities so the system can use them (e.g., input transducer: floor button pushed voltage; output transducer: current elevator position voltage) Feedback: apparatus that contributes current system state to error signal (e.g., in elevator system, error=voltage representing desired position voltage representing current position

10 Transient response of a feedback system 25 s(s 5) Figure 5.15 Plant & Controller 25 = s(s 5), Gain K =25, Feedback H = 1. Open loop TF H 25 = s(s 5) ; Closed loop TF 1H = 25 s 2 5s 25.

11 Transient response of a feedback system 25 s(s 5) Figure 5.15 T = 1H = 25 s 2 5s 25. Recall ω 2 n s 2 2ζω n s ω 2 n ω n = 25 = 5 rad/sec; 2ζω n =5 ζ =0.5; The feedback system is underdamped.

12 Transient response of a feedback system 25 s(s 5) Figure 5.15 T = 1H = 25 s 2 5s 25. π Peak time T p = p ω =0.726 sec, n 1 ζ 2 Ã! %OS = exp p ζπ 100 = 16.3, 1 ζ 2 Settling time T s = 4 ζω n =1.6 sec.

13 Adjusting the transient by feedback K s(s 5) Figure 5.16 T = 1H = K s 2 5s K. We wish to reduce overshoot to %OS=10% or less. ω n = K; 2ζω n =5 ζ = 5 2 K. For 10% overshoot or less, we need ζ =0.591 or more. Therefore, K =17.9 or less.

14 The opamp as a feedback system V i Z 1 I 1 V 1 I a Z 2 I 2 v o V i 1 Z 1 I 2 I 1 Z a A V 1 I a 1 Z 2 V o Figure 2.10(c) I 1 = V i V 1 Z 1 I 2 = V o V 1 Z 2 V i Z 1 if V 1 0 V o Z 2 if V 1 0 V o = AV 1 = AI a Z a =(I 1 I 2 ) Z a µ Vi A V o Z a. Z 1 Z 2 µ V o 1 AZ a Z a = AV i. Z 2 Z 1 V o V i = = AZ a Z 1 1 AZ a Z Z 1 AZ a 1 Z 2 Z 2 Z 1 if A.

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