1. Proportional control Consider forward path gain A Feedback and Control If the size of the loop gain is large, that is if |A  >> 1, then or

2. Consider an amplifier: Feedback and Control If the loop gain AR 2 /(R 1 +R 2 ) is large, AR 2 >>R 1 +R 2 : In the above,  =R 2 /(R 1 +R 2 ), so if loop gain is large: e.g. A=10 5, R 1 =6k , R 2 =4k  : AR 2 =4 x 10 8 >>R 1 +R 2 =10 4. V O /V I = 10/4=2.5

3. Consider an actual amplifier: Assuming T = 0.001 s Feedback and Control

4. Second order systems: [Unless told otherwise, assume D = 0] Determine values for C in order to make the system Overdamped Critically damped Underdamped Higher gain -> faster but more oscillatory response If too oscillatory - very long time before final value. We cannot, therefore, just increase controller gain

5. Feedback and Control Second order systems, non-unity feedback: M affects transient response: if M = 0.5, C can be doubled for same amount of damping. M sets the steady state value: if M = 0.5, this equals 2.

6. Feedback and Control Second order systems, advanced control: Suppose the system is one with controller C and plant K (at steady state) with no integrator in the plant. {Plant may be of form K/(1+sT)}. The steady state output is If CK is large, the steady state error is reduced: ~ 1. But cannot always increase C too much: get oscillatory response If an integrator is put in so C = C1 + C2/s, steady state errors are removed!

7. Controllers which cancel part of Plant Plant a. Sketch the argand diagram and time response if T 1 is too large for a required fast acting system: b. Sketch the argand diagram and time response if T 2 is too small for a slow reacting system. [Assume other time constant was set appropriately]

8. Controllers which cancel part of Plant Plant Make T e equal to largest of T 1 and T 2, say T 2, to speed up system. Then Gives Lead Lag Controller

9. Controllers which cancel part of Plant The above is an example of pole zero cancellation. A pole is a 1+sT term on denominator, a zero is such a term on numerator. Above shows system where T1=8 and T2 = 15, with and without pole zero cancellation: d.c. gain of controller is 9. System has been speeded up.

10. Controllers Another form of controller is the P+I controller Its transfer function can be written as Here C = P and T = I/C. P and I can be chosen so the 1+sT term (controller zero) cancels Plant pole. Suppose Plant If apply P+I to this Plant, and make T = T 2, then So Note, the I term means that the steady state value is 1.

11. Controllers P+D controller Make 1+s T cancel Plant pole. If Plant is and P+D is applied, then so PID controller - can cancel two lags in a plant: then In all these examples, by careful arrangement, systems is first/second order. Cancellation may not give best response, but analysis of systems is easier!

12. The transfer function relating the output position, P, to the armature voltage V of an armature controlled motor is: where K is 4 Nm/A and T is 0.1s. The motor forms part of a position control system, as shown below, where the controller gain C, R required position. Derive the transfer function relating the R and P. If R is a unit height step applied at time t = 0, sketch graphs of P and R against time for the following three cases: a) C = 8b) C = 10c) C = 12 In each case show the calculations which you have used to determine the form of the response. Example Exam Question

13. 1)For the system below: a) Sketch the step response when C is 40: label your sketch suitably. Assume D is 0 b) If C is 40, by how much would  o change if the plant becomes 0.2/(1+0.25s) c) If C is 40, and D is a unit step, by how much does  o change? d) Sketch the step response if C = 15. 2) In a central heating system, the transfer function of the heater is where V is the signal into the heater, T is the temperature. Sketch variation of T if V is a unit height step. The heater is put in a feedback system- controller gain 50. Find the closed loop transfer function, and sketch variation of T if the input is a step. Comment on the effect on the system of feedback. Exercises

14. 1)For a position control system for which Km = 10 and Tm = 0.2, sketch the response when the input is a unit step, if a) C = 9 b) C = 12.5 c) C = 15. 2) A field controlled motor has the following parameters: K = 2 Nm/A, R = 5W, L = 10H, J = 0.06 kgm2 & F = 0.02 Nm per rad/s. Sketch the open loop and the closed loop step response when the controller has a gain of 1. 3) A motor with transfer function is connected in a feedback loop with controller gain C. Find the closed loop transfer function. Sketch the step response of the closed loop system when: a)C = 1.25b) C = 2 c) C = 1 d) C = 5 Show all calculations used in producing these sketches. Exercises