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Feedback Control Systems (FCS) Lecture-14-15 Block Diagram Representation of Control Systems Dr. Imtiaz Hussain email: [email protected] URL :http://imtiazhussainkalwar.weebly.com/ Introduction • A Block Diagram is a shorthand pictorial representation of the cause-and-effect relationship of a system. • The interior of the rectangle representing the block usually contains a description of or the name of the element, or the symbol for the mathematical operation to be performed on the input to yield the output. • The arrows represent the direction of information or signal flow. x d dt y Introduction • The operations of addition and subtraction have a special representation. • The block becomes a small circle, called a summing point, with the appropriate plus or minus sign associated with the arrows entering the circle. • The output is the algebraic sum of the inputs. • Any number of inputs may enter a summing point. • Some books put a cross in the circle. Introduction • In order to have the same signal or variable be an input to more than one block or summing point, a takeoff point is used. • This permits the signal to proceed unaltered along several different paths to several destinations. Example-1 • Consider the following equations in which x1, x2, x3, are variables, and a1, a2 are general coefficients or mathematical operators. x 3 a1 x1 a 2 x 2 5 Example-1 • Consider the following equations in which x1, x2, x3, are variables, and a1, a2 are general coefficients or mathematical operators. x 3 a1 x1 a 2 x 2 5 Example-2 • Consider the following equations in which x1, x2,. . . , xn, are variables, and a1, a2,. . . , an , are general coefficients or mathematical operators. x n a1 x1 a 2 x 2 a n 1 x n 1 Example-3 • Draw the Block Diagrams of the following equations. (1 ) (2) x 2 a1 dx 1 x 3 a1 d dt 2 dt 1 b x2 2 x1 dt 3 dx 1 dt bx 1 Canonical Form of A Feedback Control System Characteristic Equation • The control ratio is the closed loop transfer function of the system. C(s) R( s ) G(s) 1 G ( s )H ( s ) • The denominator of closed loop transfer function determines the characteristic equation of the system. • Which is usually determined as: 1 G ( s )H ( s ) 0 Example-4 B( s ) 1. Open loop transfer function E(s) 2. Feed Forward Transfer function C(s) 3. control ratio R( s ) 4. feedback ratio 5. error ratio B( s ) R( s ) E(s) R( s ) G ( s )H ( s ) C(s) G(s) E(s) G(s) G(s) 1 G ( s )H ( s ) G ( s )H ( s ) 1 G ( s )H ( s ) H (s) 1 1 G ( s )H ( s ) 6. closed loop transfer function C(s) R( s ) G(s) 1 G ( s )H ( s ) 7. characteristic equation 1 G ( s ) H ( s ) 0 8. closed loop poles and zeros if K=10. Reduction techniques 1. Combining blocks in cascade G2 G1 G 1G 2 2. Combining blocks in parallel G1 G2 G1 G 2 Example-5: Reduce the Block Diagram to Canonical Form. Example-5: Continue. However in this example step-4 does not apply. However in this example step-6 does not apply. Example-6 • For the system represented by the following block diagram determine: 1. 2. 3. 4. 5. 6. 7. 8. Open loop transfer function Feed Forward Transfer function control ratio feedback ratio error ratio closed loop transfer function characteristic equation closed loop poles and zeros if K=10. Example-6 – First we will reduce the given block diagram to canonical form K s 1 Example-6 K s 1 K G 1 GH 1 s 1 K s 1 s Example-6 B( s ) 1. Open loop transfer function E(s) 2. Feed Forward Transfer function C(s) 3. control ratio R( s ) 4. feedback ratio 5. error ratio B( s ) R( s ) E(s) R( s ) G ( s )H ( s ) C(s) G(s) E(s) G(s) G(s) 1 G ( s )H ( s ) G ( s )H ( s ) 1 G ( s )H ( s ) H (s) 1 1 G ( s )H ( s ) 6. closed loop transfer function C(s) R( s ) G(s) 1 G ( s )H ( s ) 7. characteristic equation 1 G ( s ) H ( s ) 0 8. closed loop poles and zeros if K=10. Example-7 • For the system represented by the following block diagram determine: 1. 2. 3. 4. 5. 6. 7. 8. Open loop transfer function Feed Forward Transfer function control ratio feedback ratio error ratio closed loop transfer function characteristic equation closed loop poles and zeros if K=100. Reduction techniques 3. Moving a summing point behind a block G G G 3. Moving a summing point ahead of a block G G 1 G 4. Moving a pickoff point behind a block G G 1 G 5. Moving a pickoff point ahead of a block G G G 6. Eliminating a feedback loop G G 1 GH H G G 1 G H 1 7. Swap with two neighboring summing points A B B A Example-8 H 2 _ R +_ + + G1 + H1 C G2 G3 Example-8 H2 G1 _ R +_ + + + C G1 H1 G2 G3 Example-8 H2 G1 _ R +_ + + C + G 1G 2 H1 G3 Example-8 H2 G1 _ R +_ + C + G 1G 2 + H1 G3 block diagram: reduction example H2 G1 _ R +_ + G 1G 2 1 G 1G 2 H 1 C G3 block diagram: reduction example H2 G1 _ R +_ + G 1G 2 G 3 1 G 1G 2 H 1 C block diagram: reduction example R +_ G 1G 2 G 3 1 G 1G 2 H 1 G 2 G 3 H 2 C Example-8 R G 1G 2 G 3 1 G 1G 2 H 1 G 2 G 3 H 2 G 1G 2 G 3 C Example 9 Find the transfer function of the following block diagrams G4 R (s ) Y (s) G1 G2 G3 H H1 2 I G4 R (s) B G1 G2 A Y (s) G3 H H1 G2 Solution: 1. Moving pickoff point A ahead of block 2. Eliminate loop I & simplify B G 4 G 2G 3 G2 2 G4 R (s) G1 G A G 2GG 3 G 4 B 2 Y (s) 3 H H 1G 2 2 G 4 G 2G 3 3. Moving pickoff point B behind block II R (s) G1 B G 4 G 2G 3 H H 1G 2 2 1 /( G 4 G 2 G 3 ) C Y (s) 4. Eliminate loop III R (s) G1 GG4 4GG2 G2 G 3 3 C C Y (s) 1 H 2 ( GH4 2 G 2 G 3 ) G2H1 G 4 G 2G 3 Using rule 6 G1 (G 4 G 2 G 3 ) R (s) Y (s) 1 G 1G 2 H 1 H 2 ( G 4 G 2 G 3 ) T (s) Y (s) R (s) G1 (G 4 G 2 G 3 ) 1 G 1G 2 H 1 H 2 ( G 4 G 2 G 3 ) G 1 ( G 4 G 2 G 3 ) Example 10 Find the transfer function of the following block diagrams R (s) G1 G2 H1 H H3 2 Y (s) Solution: 1. Eliminate loop I R (s) G1 G2 A I G2 B Y (s) B Y (s) 1 GH2 H 2 H1 2 H3 G2 2. Moving pickoff point A behind block R (s) G1 H1 1 G2H 2 G2 A 1 G2H 2 1 G2H 2 G2 H3 II H 3 H1( 1 G2H 2 Not a feedback loop G2 ) 3. Eliminate loop II R (s) G 1G 2 Y (s) 1 G2H 2 H3 H 1 (1 G 2 H 2 ) G2 Using rule 6 T (s) Y (s) R (s) G 1G 2 1 G 2 H 2 G 1G 2 H 3 G 1 H 1 G 1G 2 H 1 H 2 Example 11 Find the transfer function of the following block diagrams H 4 R (s) Y (s) G1 G2 G3 H3 H 2 H1 G4 Solution: 1. Moving pickoff point A behind block G4 I H R (s) 4 Y (s) G1 G2 G3 H H3 H 2 3 1 G4 G4 1 H 2 G4 G 4 H1 A G4 B 2. Eliminate loop I and Simplify R (s) II G 2 G 3G 4 G1 Y (s) B 1 G 3G 4 H 4 H 3 G4 H2 III G4 H1 II feedback III Not feedback G 2 G 3G 4 H 2 G4H1 1 G 3G 4 H 4 G 2 G 3 H 3 G4 3. Eliminate loop II & IIII R (s) G 1G 2 G 3 G 4 Y (s) 1 G 3G 4 H 4 G 2 G 3 H 3 H 2 G4H1 G4 Using rule 6 T (s) Y (s) R (s) G 1G 2 G 3 G 4 1 G 2 G 3 H 3 G 3 G 4 H 4 G 1G 2 G 3 H 2 G 1G 2 G 3 G 4 H 1 Example 12 Find the transfer function of the following block diagrams H R (s) G2 G1 H1 G4 A 2 G3 Y (s) B Solution: 1. Moving pickoff point A behind block G3 H R (s) G2 G1 2 A G3 1 H1 H1 G4 I G3 1 G3 B Y (s) 2. Eliminate loop I & Simplify H 2 G3 G2 H1 R (s) 1 H1 G3 G3 G1 B G 2G3 B H2 II G 2G 3 1 G 2 H 1 G 2G 3 H 2 H1 G3 G4 Y (s) 3. Eliminate loop II R (s) Y (s) G 1G 2 G 3 1 G 2 H 1 G 2 G 3 H 2 G 1G 2 H 1 G4 T (s) Y (s) R (s) G4 G 1G 2 G 3 1 G 2 H 1 G 2 G 3 H 2 G 1G 2 H 1 Example-13: Simplify the Block Diagram. Example-13: Continue. Example-14: Reduce the Block Diagram. Example-14: Continue. Example-15: Reduce the Block Diagram. (from Nise: page-242) Example-15: Continue. Example-16: Reduce the system to a single transfer function. (from Nise:page-243). Example-17: Simplify the block diagram then obtain the closeloop transfer function C(S)/R(S). (from Ogata: Page-47) Example-18: Multiple Input System. Determine the output C due to inputs R and U using the Superposition Method. Example-18: Continue. Example-18: Continue. Example-19: Multiple-Input System. Determine the output C due to inputs R, U1 and U2 using the Superposition Method. Example-19: Continue. Example-19: Continue. Example-20: Multi-Input Multi-Output System. Determine C1 and C2 due to R1 and R2. Example-20: Continue. Example-20: Continue. When R1 = 0, When R2 = 0, Block Diagram of Armature Controlled D.C Motor Ra La c Va ia eb T J L a s Js R a I a (s) K b (s) V a (s) c (s) K ma I a (s) Block Diagram of Armature Controlled D.C Motor L a s R a I a (s) K b (s) E a (s) Block Diagram of Armature Controlled D.C Motor Js c (s) K ma I a (s) Block Diagram of Armature Controlled D.C Motor Block Diagram of liquid level system q1 h1 h 2 q2 R1 C1 h2 R2 C2 dh 1 dt dh 2 dt q q1 q1 q 2 Block Diagram of liquid level system L q1 Q1 ( s ) q2 L h1 h 2 C1 R1 H 1( s ) H 2 ( s ) R1 C2 Q2 (s) H 2(s) R2 dt q q1 L C 1 sH 1 ( s ) Q ( s ) Q1 ( s ) h2 R2 dh 1 dh 2 dt q1 q 2 L C 2 sH 2 ( s ) Q1 ( s ) Q 2 ( s ) Block Diagram of liquid level system Q1 ( s ) H 1( s ) H 2 ( s ) Q2 (s) C 1 sH 1 ( s ) Q ( s ) Q1 ( s ) R1 H 2(s) R2 C 2 sH 2 ( s ) Q1 ( s ) Q 2 ( s ) Block Diagram of liquid level system To download this lecture visit http://imtiazhussainkalwar.weebly.com/ END OF LECTURES-14-15