1 Teaching Innovation - Entrepreneurial - Global The Centre for Technology Enabled Teaching & Learning, MGI, India DTEL DTEL (Department for Technology Enhanced Learning)

DEPARTMENT OF MECHANICAL ENGINEERING VI -SEMESTER CONTROL SYSTEMS ENGINEERING 1 CHAPTER NO.1 Transfer Function system Representation through Block Diagram and Signal Flow Graph

CHAPTER 2:- SYLLABUSDTEL Block Diagram representation Reduction Techniques for single and multiple input/output Conversion of Block Diagram into Signal Flow Graph Conversion of algebraic equation into Block Diagram and Signal Flow Graph Transfer function through Block Diagram Simplification using Masons Gain Formula.

CHAPTER-2 SPECIFIC OBJECTIVE / COURSE OUTCOMEDTEL To familiarize the students with concepts related to the operation. 1 To make understanding of various control systems 2 3 The student will be able to:

DTEL 4 LECTURE 1:- Block Diagram is a pictorial representation of the given system. It is a very simple way of representing the complicated practical system. – Any Block Diagram has following five basic elements associated with it: Blocks Transfer functions of elements shown inside the blocks. Summing points. Take off points. Arrows Introduction

DTEL 5 LECTURE 1: Rule 1 – Associative law R1 R2 R3 R2 R R1-R2 R1-R2+R3 R1+R3 Rules for Block Diagram Reduction

DTEL 6 LECTURE 1: Rule 2 – For blocks in series Rule 3 – For blocks in parallel G1 - G2 + G3 C(s) R(s) G1 G2 - G3G3 R(s) C(s) R(s) - C(s) G1 G2 G3G3 G1G2G3 C(s) R(s) Rules for Block Diagram Reduction

DTEL 7 LECTURE 1: Rule 4 – Shifting a summing point behind the block Rule 5 – Shifting a summing point beyond the block R(s) G1G1 G C(s) + y 1 /G G1 G C(s) + y R(s) G1G1 G C(s) + R(s) y G1 G C(s) + y G1G1 G R(s) Rules for Block Diagram Reduction

DTEL 8 LECTURE 1: Rule 6 – Shifting a takeoff point behind the blocks Rule 7 – Shifting a take off point beyond the blocks G1 G C(s) y R(s) G1G1 G C(s) G1G1 G y R(s) C(s) y G G1 G C(s) y G1G1 R(s) 1 /G Rules for Block Diagram Reduction

DTEL 9 LECTURE 1: Rule 8 – Removing minor feedback loop R1 C’(s) G’(s ) H’(s) R1(s ) C’(s) Rule 9 – For multiple input system use superposition theorem System R1 C R2 Rn Consider only one input at a time treating all other as zero Consider R1’ R2 = R3 =………… Rn = 0 and find output C1’ Then consider R2’R1 = R3 =………… Rn = 0 and find output C2’ Total outputC= C1+C2+……….+ Cn Rules for Block Diagram Reduction

DTEL 10 LECTURE 1: Critical Rules Rule 10 – Shifting take off point after a summing point. z R1 y y x=R1 + R1 y x=R1 Rules for Block Diagram Reduction

DTEL 12 LECTURE 2 :- Procedure to solve block diagram reduction problems 1)Reduce the blocks connected in series. 2)Reduce the blocks connected in parallel. 3)Reduce the minor internal feedback loops. 4)As far as possible try to shift take off point towards right and summing points to the left. 5)Repeat steps 1 to a till simple form is obtained. 6)Using standard TF of simple closed loop system obtain the closed loop TF of overall system

DTEL 13 LECTURE 3:- 1.1 solve the following problem R(s) Y(s) Numerical

DTEL 14 LECTURE 3:- Move H 2 behind G 4 Eliminate loop G 3 G 4 H 1 Eliminate loop containing H 2 /G 4 Reduce the loop containing H 3 Solution Numerical

DTEL 15 LECTURE 4:- y Problem 1.2 Solution Numerical

DTEL 16 LECTURE 4:- Numerical

DTEL 17 LECTURE 5:- For complex systems, the block diagram method can become difficult to complete. By using the signal-flow graph model, the reduction procedure (used in the block diagram method) is not necessary to determine the relationship between system variables. Fig. 2.1 Signal flow graph of DC motor Signal Flow Graph V(s) Ѳ (s)

DTEL 18 LECTURE 5:- Signal Flow Graph Fig. 2.2 Problem On Signal Flow Graph

DTEL 19 LECTURE 6:- Convert Block Dia. Into SFG Step 1:- Single node Covert all summing points and take off points into the nodes Nodes Fig. 2.3 Problem on Block Reduction

DTEL 20 LECTURE 6:- Step 2:- Simplified SFG Convert Block Dia. Into SFG R(s) G1(s) G2(s)G3(s) 1 -H3(s) -H1(s) -H2(s) Fig.2.4 Signal Flow Graph

DTEL 21 LECTURE 7 :- Figure 2.5 Problem on Mason’s Gain Introduction Using Numerical

DTEL 22 LECTURE 7 :- Some Important Definitions Loop Gain:-The product of branch gain found by traversing a path that starts at the node and end at the same node without passing through any other node more than once and following the direction of the signal flow. Figure 2.1 has four loops which are shown below Introduction Using Numerical 1.G 2 (s)H 1 (s) 2.G 4 (s)H 2 (s) 3.G 4 (s) G 5 (s)H 3 (s) 4.G 4 (s) G 6 (s)H 3 (s)

DTEL 23 LECTURE 7 :- Forward path gain:-The product of gains found by traversing a path from the input to the output node of the signal flow graph in the direction of signal flow. There are two forward path gains for figure 2.1 Introduction Using Numerical 1. G 1 (s)G 2 (s)G 3 (s)G 4 (s)G 5 (s)G 7 (s) 2. G 1 (s)G 2 (s)G 3 (s)G 4 (s)G 6 (s)G 7 (s)

DTEL 24 LECTURE 8:- Non touching loops:- Loops that do not have any node in common Non touching loop gain:- the product of loop gains from non-touching loop taken two, three, four etc. at a time. Introduction Using Numerical 1. [G 2 (s)H 1 (s)] [G 4 (s)H 2 (s)] 2. [G 2 (s)H 1 (s)][G 4 (s)G 5 (s)H 3 (s)] 3. [G 2 (s)H 1 (s)][G 4 (s)G 6 (s)H 3 (s )]

DTEL 25 LECTURE 8:- Mason’s Rule:- Introduction Using Numerical G(s)= [∑ ( T k Δ k)] / Δ Where, K= number of forward paths T k = the K th forward path gain Δ= 1-{∑loop gains + ∑ non-touching loop gains taken two at a time - ∑non-touching loop gains taken three at a time + ∑ non-touching loop gains taken four at a time - ……………}

DTEL 26 THANK YOU

DTEL References Books: 1.Automatic control system by Farid Golnaraghi. 2.Modern control System Engineering by Katsuhiko Ogata 3.Feedback Control System by R. A. Barapate 4.Automatic control system by Benjamin 27