Control Systems, Lecture 05 İbrahim Beklan Küçükdemiral Yıldız Teknik Üniversitesi 2015 1 / 33
Laplace Transform Solution of State Equations In previous sections, systems were modeled in state space, where the state-space representation consisted of a state equation and an output equation. In this section, we use the Laplace transform to solve the state equations for the state and output vectors. Consider an LTI system governed by state and output equations x = Ax + Bu, x = x0 y = Cx + Du Taking the Laplace transform of both sides of the state equation yields sx (s) x(0) = AX (s) + BU(s) [si A]X (s) = x(0) + BU(s) 2 / 33
Then X (s) = [si A] 1 x(0) + [si A] 1 BU(s) adj(si A) = [x(0) + BU(s)] si A Finally, Y (s) = CX (s) + DU(s) y (t) = L{Y (s)} If initial conditions are zero, i.e., x(0) = 0 then Y (s) = C [si A] 1 B + D U(s) where C [si A] 1 B + D is called the Transfer Function of the system. 3 / 33
Example Given the system represented in state space by 0 1 0 0 0 1 x(t) + 0 e t 1(t) x (t) = 0 24 26 9 1 y (t) = 1 1 0 x(t) Solve the preceding state equation and obtain the output for the given exponential input when all initial conditions are zero. 4 / 33
Note that (si A) 1 s 1 0 s 1 (si A) = 0 24 26 s + 9 2 s + 9s + 26 s +9 1 1 24 s 2 + 9s s = 3 2 s + 9s + 26s + 24 24s (26s + 24) s 2 1+s U(s) + + 26s + 24 s +1 1 Y (s) = 3 (s + 9s 2 + 26s + 24) (s + 1) Y (s) = C (si A) 1 BU(s) = s3 9s 2 0.5 1 0.5 + + s +2 s +3 s +4 y (t) = 0.5e 2t e 3t + 0.5e 4t 1(t) Y (s) = 5 / 33
Block Diagrams and Reduction of Multiple Sub-systems A subsystem is represented as a block with an input, an output, and a transfer function Many systems are composed of multiple subsystems, When multiple subsystems are interconnected, a few more schematic elements must be added to the block diagram. These new elements are summing junctions and pickoff points. 6 / 33
Fundamental Components in a Block Diagram In Matlab we use the command num=[...]; den=[...]; System=tf(num,den); 7 / 33
Interconnections Cascade Connection In MATLAB: G1=tf(num1,den1); G2=tf(num2,den2); G3=tf(num3,den3); Ge=G1*G2*G3; 8 / 33
Interconnections Parallel Connection In MATLAB G1=tf(num1,den1); G2=tf(num2,den2); G3=tf(num3,den3); Ge=G1+G2+G3; 9 / 33
Feedback Connection In MATLAB: feedback(g,h) 10 / 33
Moving Blocks 11 / 33
Moving Blocks 12 / 33
Example Question Reduce the block diagram shown in Figure to a single transfer function. 13 / 33
Solution... 14 / 33
Solution... 15 / 33
Example Reduce the system shown in Figure to a single transfer function 16 / 33
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Signal Flow Diagrams Signal-flow graphs are an alternative to block diagrams. Unlike block diagrams, which consist of blocks, signals, summing junctions, and pickoff points, a signal-flow graph consists only of branches, which represent systems, and nodes, which represent signals. A system is represented by a line with an arrow showing the direction of signal flow through the system. Adjacent to the line we write the transfer function. A signal is a node with the signal s name written adjacent to the node. 20 / 33
A Signal Flow Diagram V (s) = G1 (s)r1 (s) G2 (s)r2 (s) + G3 (s)r3 (s) C1 (s) =? C2 (s) =? C3 (s) =? 21 / 33
Example Convert the block diagram of figure to a signal-flow graph. 22 / 33
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Mason s Rule Earlier in this chapter, we discussed how to reduce block diagrams to single transfer functions. We are ready to discuss a technique for reducing signal-flow graphs to single transfer functions that relate the output of a system to its input. The block diagram reduction technique we studied in previous section requires successive application of fundamental relationships in order to arrive at the system transfer function. Mason s1 rule for reducing a signal-flow graph to a single transfer function requires the application of one formula. 1 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, 1953) 24 / 33
Mason s Rule... Definitions Loop gain The product of branch gains found by traversing a path that starts at a node and ends at the same node, following the direction of the signal flow, without passing through any other node more than once. Forward-path gain The product of gains found by traversing a path from the input node to the output node of the signal-flow graph in the direction of signal flow. 25 / 33
Mason s Rule... Definitions Nontouching loops Loops that do not have any nodes in common. Nontouching-loop gain The product of loop gains from nontouching loops taken two, three, four, or more at a time. 26 / 33
Example: Loop Gain G2 (s)h1 (s) G4 (s)h2 (s) G4 (s)g6 (s)h3 (s) G4 (s)g5 (s)h3 (s) 27 / 33
Example: Forward-path Gain G1 (s)g2 (s)g3 (s)g4 (s)g5 (s)g7 (s) G1 (s)g2 (s)g3 (s)g4 (s)g6 (s)g7 (s) 28 / 33
Example: Nontouching Loops G2 H1 and G4 H2 G2 H1 and G4 G5 H3 G2 H1 and G4 G6 H3 29 / 33
Example: Nontouching Loop Gains G2 (s)g4 (s)h1 (s)h2 (s) G2 (s)g4 (s)g5 (s)h1 (s)h3 (s) G2 (s)g4 (s)g6 (s)h1 (s)h3 (s) 30 / 33
Mason s Rule The transfer function, C (s)/r(s) of a system represented by a signal-flow graph is C (s) G (s) = = R(s) PN k=1 Tk k k = number of forward paths Tk = the kth forward-path gain P P = 1loop gains+ nontouching-loop gains taken P two at a timep nontouching-loop gains taken three at a time + nontouching-loop gains taken four at a time... P k = loop gain terms in that touch the kth forward path 31 / 33
Example Find the transfer function C (s)/r(s) 32 / 33
Example Find the transfer function C (s)/r(s) 33 / 33