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Modeling & Simulation of Dynamic Systems Lecture-2 Block Diagram & Signal Flow Graph 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,. . . , 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 Exercise-1 • 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-2 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 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-3 • 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-3 – First we will reduce the given block diagram to canonical form K s 1 Example-3 K s 1 K G 1 GH 1 s 1 K s 1 s Example-3 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. Exercise-2 • 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. Example-4 H 2 _ R +_ + + G1 + H1 C G2 G3 Example-4 H2 G1 _ R +_ + + + C G1 H1 G2 G3 Example-4 H2 G1 _ R +_ + + C + G 1G 2 H1 G3 Example-4 H2 G1 _ R +_ + C + G 1G 2 + H1 G3 Example-4 H2 G1 _ R +_ + G 1G 2 1 G 1G 2 H 1 C G3 Example-4 H2 G1 _ R +_ + G 1G 2 G 3 1 G 1G 2 H 1 C Example-4 R +_ G 1G 2 G 3 1 G 1G 2 H 1 G 2 G 3 H 2 C Example-5 Find the transfer function of the following block diagrams H 4 R (s) Y (s) G1 G2 G3 H3 H 2 H1 G4 Example-5 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 Example-5 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 Example-5 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 Y (s) R( s ) G1G 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-5 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) Example-5 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 Superposition of Multiple Inputs Multiple Input System. Determine the output C due to inputs R and U using the Superposition Method. Example-6 Example-6 Example-6 Exercise-3: Multi-Input Multi-Output System. Determine C1 and C2 due to R1 and R2. Introduction • Alternative method to block diagram representation, developed by Samuel Jefferson Mason. • Advantage: the availability of a flow graph gain formula, also called Mason’s gain formula. • A signal-flow graph consists of a network in which nodes are connected by directed branches. • It depicts the flow of signals from one point of a system to another and gives the relationships among the signals. 37 Fundamentals of Signal Flow Graphs • Consider a simple equation below and draw its signal flow graph: y ax • The signal flow graph of the equation is shown below; x a y • Every variable in a signal flow graph is designed by a Node. • Every transmission function in a signal flow graph is designed by a Branch. • Branches are always unidirectional. • The arrow in the branch denotes the direction of the signal flow. Signal-Flow Graph Models Example-7: R1 and R2 are inputs and Y1 and Y2 are outputs Y1 ( s ) G11 ( s ) R 1 ( s ) G12 ( s ) R 2 ( s ) Y2 ( s ) G21 ( s ) R 1 ( s ) G22 ( s ) R 2 ( s ) Signal-Flow Graph Models Exercise-4: r1 and r2 are inputs and x1 and x2 are outputs a 11 x1 a 12 x2 r1 x1 a 21 x1 a 22 x2 r2 x2 Signal-Flow Graph Models Example-8: xo is input and x4 is output x1 ax0 bx1 cx 2 x 2 dx1 ex 3 f c x0 a x1 d x2 x3 g x 3 fx0 gx 2 x4 hx 3 b e h x4 Construct the signal flow graph for the following set of simultaneous equations. • There are four variables in the equations (i.e., x1,x2,x3,and x4) therefore four nodes are required to construct the signal flow graph. • Arrange these four nodes from left to right and connect them with the associated branches. • Another way to arrange this graph is shown in the figure. Terminologies • An input node or source contain only the outgoing branches. i.e., X1 • An output node or sink contain only the incoming branches. i.e., X4 • A path is a continuous, unidirectional succession of branches along which no node is passed more than ones. i.e., X1 to X2 to X3 to X4 X1 to X2 to X4 X2 to X3 to X4 • A forward path is a path from the input node to the output node. i.e., X1 to X2 to X3 to X4 , and X1 to X2 to X4 , are forward paths. • A feedback path or feedback loop is a path which originates and terminates on the same node. i.e.; X2 to X3 and back to X2 is a feedback path. Terminologies • A self-loop is a feedback loop consisting of a single branch. i.e.; A33 is a self loop. • The gain of a branch is the transmission function of that branch. • The path gain is the product of branch gains encountered in traversing a path. i.e. the gain of forwards path X1 to X2 to X3 to X4 is A21A32A43 • The loop gain is the product of the branch gains of the loop. i.e., the loop gain of the feedback loop from X2 to X3 and back to X2 is A32A23. • Two loops, paths, or loop and a path are said to be non-touching if they have no nodes in common. Consider the signal flow graph below and identify the following Example-9: a) b) c) d) e) f) g) Input node. Output node. Forward paths. Feedback paths (loops). Determine the loop gains of the feedback loops. Determine the path gains of the forward paths. Non-touching loops Consider the signal flow graph below and identify the following Example-9: • There are two forward path gains; Consider the signal flow graph below and identify the following Example-9: • There are four loops Consider the signal flow graph below and identify the following Example-9: • Nontouching loop gains; Consider the signal flow graph below and identify the following Example-10: a) b) c) d) e) f) g) Input node. Output node. Forward paths. Feedback paths. Self loop. Determine the loop gains of the feedback loops. Determine the path gains of the forward paths. Input and output Nodes Example-10: a) Input node b) Output node (c) Forward Paths Example-10: (d) Feedback Paths or Loops Example-10: (d) Feedback Paths or Loops Example-10: (d) Feedback Paths or Loops Example-10: (d) Feedback Paths or Loops Example-10: (e) Self Loop(s) Example-10: (f) Loop Gains of the Feedback Loops Example-10: (g) Path Gains of the Forward Paths Example-10: Mason’s Rule (Mason, 1953) • The block diagram reduction technique requires successive application of fundamental relationships in order to arrive at the system transfer function. • On the other hand, Mason’s rule for reducing a signal-flow graph to a single transfer function requires the application of one formula. • The formula was derived by S. J. Mason when he related the signal-flow graph to the simultaneous equations that can be written from the graph. Mason’s Rule: • The transfer function, C(s)/R(s), of a system represented by a signal-flow graph is; n C(s) R( s ) Pi i i 1 Where n = number of forward paths. Pi = the i th forward-path gain. ∆ = Determinant of the system ∆i = Determinant of the ith forward path • ∆ is called the signal flow graph determinant or characteristic function. Since ∆=0 is the system characteristic equation. Mason’s Rule: n C(s) R( s ) Pi i i 1 ∆ = 1- (sum of all individual loop gains) + (sum of the products of the gains of all possible two loops that do not touch each other) – (sum of the products of the gains of all possible three loops that do not touch each other) + … and so forth with sums of higher number of non-touching loop gains ∆i = value of Δ for the part of the block diagram that does not touch the ith forward path (Δi = 1 if there are no non-touching loops to the i-th path.) Systematic approach 1. Calculate forward path gain Pi for each forward path i. 2. Calculate all loop transfer functions 3. Consider non-touching loops 2 at a time 4. Consider non-touching loops 3 at a time 5. etc 6. Calculate Δ from steps 2,3,4 and 5 7. Calculate Δi as portion of Δ not touching forward path i 62 Example-11: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph Therefore, C R P1 1 P2 2 There are three feedback loops L1 G1G 4 H 1 , L 2 G1 G 2 G 4 H 2 , L 3 G1 G 3 G 4 H 2 Example-11: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph There are no non-touching loops, therefore ∆ = 1- (sum of all individual loop gains) 1 L1 L 2 L 3 1 G1G 4 H 1 G1G 2 G 4 H 2 G1G 3 G 4 H 2 Example-11: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph Eliminate forward path-1 ∆1 = 1- (sum of all individual loop gains)+... ∆1 = 1 Eliminate forward path-2 ∆2 = 1- (sum of all individual loop gains)+... ∆2 = 1 Example-11: Continue Exercise-5: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph 67 Exercise-6 • Find the transfer function, C(s)/R(s), for the signal-flow graph in figure below. Example-12: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph There are three forward paths, therefore n=3. 3 C(s) R( s ) Pi i i 1 P1 1 P2 2 P3 3 Example-12: Forward Paths P3 A42 A54 A65 A76 P1 A32 A43 A54 A65 A76 P2 A72 Example-12: Loop Gains of the Feedback Loops L1 A32 A 23 L 2 A43 A34 L 3 A54 A45 L 4 A65 A56 L 5 A76 A67 L 6 A77 L 7 A42 A34 A23 L 8 A65 A76 A67 L 9 A72 A57 A45 A34 A23 L10 A72 A67 A56 A45 A34 A23 Example-12: two non-touching loops L1 L 3 L2 L4 L3 L5 L4 L6 L1 L 4 L 2 L5 L3 L6 L4 L7 L1 L 5 L2 L6 L1 L 6 L 2 L8 L1 L8 L5 L7 L 7 L8 Example-12: Three non-touching loops L1 L 3 L2 L4 L3 L5 L4 L6 L1 L 4 L 2 L5 L3 L6 L4 L7 L1 L 5 L2 L6 L1 L 6 L 2 L8 L1 L8 L5 L7 L 7 L8 From Block Diagram to Signal-Flow Graph Models Example-13: H1 R(s) E(s) X1 G1 － － － G2 X2 G3 X3 G4 H2 H3 R(s) 1 E(s) G1 X1 G2 －H2 －H3 X2 G3 －H1 X3 G4 C(s) C(s) From Block Diagram to Signal-Flow Graph Models Example-13: R(s) 1 －H1 E(s) G1 X1 G2 X2 G3 G4 X3 1 C(s) －H2 －H3 1 (G1G2G3G4 H 3 G2G3 H 2 G3G4 H 1 ) P1 G1G2G3G4 ; G C ( s) R( s ) 1 1 G1G2G3G4 1 G1G2G3G4 H 3 G2G3 H 2 G3G4 H 1 Exercise-7 － R(s) E(s) － － X1 G1 Y1 + + －X 2 + － G2 Y2 C(s) To download This Lecture Visit :http://imtiazhussainkalwar.weebly.com END OF LECTURE