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1 Contents Chapter Topic Page Chapter- Chapter- Chapter-3 Chapter-4 Introduction Transfer Function, Block Diagrams and Signal Flow Graphs Mathematical Modeling Control System 35 Time Response Analysis of No

2 India s No IES Academy Contents Chapter-5 Chapter-6 Chapter-7 Chapter-8 Chapter-9 Frequency Analysis Stability Analysis of Control System Root Locus Technique Compensators Industrial Controllers Chapter-0 Introduction to State Space Variable Page- 5, st Floor, Jia Sarai, Near IIT. New Delhi-6 Ph: ,

3 India s No IES Academy Chapter Contents for this chapter Introduction. Introduction. Open Loop System 3. Mathematical Model for Open-loop 4. Closed-Loop 5. Mathematical Model for Closed-Loop System 6. Comparison of Open Loop and Closed Loop 7. Laplace Transform 8. Basic Laplace Transform Theorem 9. Summary. Introduction Theory at a Glance (For IES, GATE, PSU & JTO) Control system is a combination of elements arranged in a planned manner. Where each element causes an effect to produce a desire output. Example of control systems. System for the control of position.. System for the control of velocity.. Open Loop System. No feedback in open loop system is used.. Control system (open-loop) depends only on the accuracy of input calibration. Example of open-loop control system. Traffic signal light. Electric lift 3. Automatic washing machine 3. Mathematical Model for Open-loop C G R Page- 5, st Floor, Jia Sarai, Near IIT. New Delhi-6 Ph: ,

4 India s No IES Academy Chapter Where, G gain of system C o/p of system R input Points:. Feedback system is not used for improving stability.. An open-loop system may become unstable when we used negative feedback. 4. Closed-Loop In a closed loop control system the output has an effect on control action through a feedback. Example of closed-loop system:. D.C. Motor speed control. Radar tracking system 3. Auto pilot system 5. Mathematical Model for Closed-Loop System C R G +GH * Here feedback is negative. This form is also called control canonical form From figure C(S) G(S) E(S) As a Forward path transfer function B(S) H(S) C(S) As a feedback transfer function The o/p of summing point E(S) [ R(S) B(S) ] ; C(S) R(S) B(S) ; G(S) C(S) R(S) C(S) H(S) ; G(S) C(S) R(S) G(S) G(S) C(S) H(S) ; C(S) [ +G(S) H(S) ] R(S) G(S) Page-3 5, st Floor, Jia Sarai, Near IIT. New Delhi-6 Ph: ,

5 India s No IES Academy Chapter C(S) G(S) R(S) +G(S) H(S) 6. Comparison of Open Loop and Closed Loop Open Loop System. The accuracy of an open loop system depends on the calibration of the i/p. Closed Loop System. As the error between the reference input and the output is continuously measuredthrough feedback.. The open loop system is more stable.. The closed loop system is less stable. 3. It is less accurate. 3. It is more accurate. 4. It is cheap and less complex. 4. It is expensive and more complex circuit. 5. Effect of Noise and disturbance is more in open loop control system. 7. Laplace Transform Laplace transformation is very great tool in control system. The mathematical expression for laplace transforms LF(t) st F(S) F(t) e dt 0 5. Effect of Noise and disturbance is less in closed loop control system. F(S) The term laplace transform of F(t) is used for the letter LF(t). 8. Basic Laplace Transform Theorem Basic theorems of laplace transform are given below Theorem : Multiplication by a constant Let k be a constant and F(S) be the laplace transform of F(t), then [ Kf(t) ] Theorem : Sum and difference KF(S) Let F(S) and F(S) be the laplace transform of f ( t ) f ( t) ( ) ( ) ±, then f t ± f t F(S)± F (S) Page-4 5, st Floor, Jia Sarai, Near IIT. New Delhi-6 Ph: ,

6 India s No IES Academy Chapter Theorem 3: Differentiation df(t) i. L SF(S)-F(0) dt [ ] df(t) ii. L S F(S)-F(0)-f (0) dt df(0) where, F (0) dt In general, for higher order derivatives or F(t) n d F() t L sfs n s f O s f f dt n n n () ( n ) ( ) ( ) (0) (0) Where, F (0) denotes the i th order derivative of f(t) with respect to t, Theorem 4: Integration i. L F(t) + S F(S) F (0) ii. L F(t) + + S S S S F(S) F (0) F (0) Theorem 5: Shift in time The laplace transform of F(t) delayed by time T is equal to the laplace transform F(t) multiplied by e ST that is -ST L [ F(t T)u s(t T) ] e F(S) Where US(t T) denotes the unit step function that is shifted in time to the right by T. Theorem 6: Complex shifting The laplace transform of F(t) multiplied by transform F(S), with S replaced by ( S ± α ) L e αt Theorem 7: Initial-value theorem If the laplace transform of F(t) is F(S), then t 0 αt e, where α is a constant is equal to the laplace that is F(t) F(S±α) lim F( t) lim SF( S ) S Page-5 5, st Floor, Jia Sarai, Near IIT. New Delhi-6 Ph: ,

7 India s No IES Academy Chapter Theorem 8: Final value theorem lim F( t) lim SLF( t) t S 0 lim F( t) lim SF( S) t S 0 Point to be Remember If the denominator of SF(S) has any root having real part as zero or positive, then final value theorem is not valid. [GATE 007] USEFUL TRANSFORM (LAPLACE) PAIR F(t) F(S) LF(t) δ(t) unit impulse 3 U(t) S 4 U(t T) st e S 5 t S t S n t n s + at e at e at te at te 3 n s+ a s a ( s+ a) ( s a) 3 h αt n+ te n/ ( s+ a ) Page-6 5, st Floor, Jia Sarai, Near IIT. New Delhi-6 Ph: ,

8 India s No IES Academy Chapter ω 4 sinωt s +ω αt 5 e cosωt αt 6 e sinωt 7 sin hα t 8 cos hα t s + α ( ) s + α + ω ω ( ) s + α + ω α s α s s α 9. Summary. Open loop control system no feedback used.. In closed loop control system we used feedback. 3. Open loop system is more stable. 4. Closed loop system is more accurate. 5. Final value theorem can not used if denominators of SF(S) have real part as a zero or positive. Page-7 5, st Floor, Jia Sarai, Near IIT. New Delhi-6 Ph: ,

9 India s No IES Academy Chapter ASKED OBJECTIVE QUESTIONS (GATE, IES) Basic Laplace Transform Theorem GATE-. If the Laplace Transform of a signal y(t) is Y() s, ss ( ) then its final value is: [GATE-007] (a) - (b) 0 (c) 0 (d) Unbounded t GATE-. The unit impulse response of a system is f () t e, t 0 [GATE-006] For this system, the steady-state value of the output for unit step input is equal to (a) - (b) 0 (c) (d) Closed-Loop IES-. When a human being tries to approach an object, his brain acts as (a) An error measuring device (b) A controller [IES-999] (c) An actuator (d) An amplifier IES-. Assertion (A): Feedback control systems offer more accurate control over open-loop systems. [IES-000] Reason (R): The feedback path establishes a link for input and output comparison and subsequent error correction. (a) Both A and R are true and R is the correct explanation of A (b) Both A and R are true but R is NOT the correct explanation of A (c) A is true but R is false (d) A is false but R is true IES-3. Consider the following statements: [IES-000]. The effect of feedback is to reduce the system error. Feedback increases the gain of the system in one frequency range but decreases in another 3. Feedback can cause a system that is originally stable to become unstable Which of these statements are correct? (a), and 3 (b) and (c) and 3 (d) and 3 IES-4. Consider the following statements which respect to feedback control systems: [IES-006] Page-8 5, st Floor, Jia Sarai, Near IIT. New Delhi-6 Ph: ,

10 India s No IES Academy Chapter. Accuracy cannot be obtained by adjusting loop gain.. Feedback decreases overall gain. 3. Introduction of noise due to sensor reduces overall accuracy. 4. Introduction of feedback may lead to the possibility of instability of closed loop system. Which of the statements given above are correct? (a),, 3 and 4 (b) Only, and 4 (c) Only and 3 (d) Only, 3 and 4 IES-5. A negative-feedback closed-loop system is supplied to an input of 5V. The system has a forward gain of and a feedback gain of a. What is the output voltage? [IES-009] (a).0 V (b).5 V (c).0 V (d).5 V Basic Laplace Transform Theorem ω IES-6. Consider the function F(s) where F(s) is the Laplace transform s + ω of f(t). What is the steady-state value of f(t)? [IES-009] (a) Zero (b) One (c) Two (d) A value between - and + IES-7. The transfer function of a linear-time-invariant system is given as. What is the steady-state value of the unit-impulse response? ( s + ) [IES-009] (a) Zero (b) One (c) Two (d) Infinite Page-9 5, st Floor, Jia Sarai, Near IIT. New Delhi-6 Ph: ,

11 India s No IES Academy Chapter Answers with Explanation (Objective) GATE-. Ans. (d) Y() s SS ( + ) Final value of Y(s) LT ( Y( s)) LT LT + ( + ) SS S S Yt () e t ut () final value t GATE-. Ans. (c) Unit impulse response of a system is f() t e t t 0 f() s S + O/P for unit step I/P S + S SS ( + ) ( Cs ( )) lim S t s 0 SS ( + ) IES-. Ans. (b) IES-. Ans. (a) IES-3. Ans. (d) Feedback is applied to reduce the system error. Consider the example. Cs ( ) Gs ( ) Rs GsHs ( ) ( ) ( ) s + s s+ Thus, we see that the closed loop system is unstable while the open loop system is stable. IES-4. Ans. (d) Page-0 5, st Floor, Jia Sarai, Near IIT. New Delhi-6 Ph: ,

12 India s No IES Academy Chapter IES-5. Ans. (d) Output voltage Vin A AB 5x.5V + + ( xx ) IES-6. Ans. (d) This is the Laplace transform of sin t. So, f(t) sin t Steady-state value of f(t) is undetermined because poles of F(s) are not in LHS of s-plane. Therefore, steady-state value will vary between - and +. IES-7. Ans. (a) Steady state value lims 0 s 0 s+ ( ) Page- 5, st Floor, Jia Sarai, Near IIT. New Delhi-6 Ph: ,

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