Topic #8 Signal Flow Graphs Reference textbook : Control Systems, - PowerPoint PPT Presentation

ME 779 Control Systems Topic #8 Signal Flow Graphs Reference textbook : Control Systems, Dhanesh N. Manik, Cengage Publishing, 2012 1

Control Systems: Signal Flow Graphs Learning Objectives • Definition • Canonical feedback system • Mason’s formula • Examples 2

Control Systems: Signal Flow Graphs Definition A signal flow graph is a diagram consisting of nodes that are connected by several directed branches, each node representing a variable of the system . Y s ( ) X s  G s ( ) ( ) 3

Control Systems: Signal Flow Graphs Definition A forward path is any path which goes from the input node to the output node along which no node is traversed more than once . A loop is any path which originates and terminates at the same node along which no node is traversed more than once 4

Control Systems: Signal Flow Graphs Canonical feedback system   E s ( ) R s ( ) C s H s ( ) ( ) forward path is the one that goes from R(s) — E(s) — C(s) G(s) is the product of all the gains in the forward path -G(s)H(s) is the product of all the gains in the loop Loop: E(s)-C(s)-E(s) 5

Control Systems: Signal Flow Graphs Mason’s formula T k is the gain of the k th forward path  n C s T ( )   k k from the input node R(s) to the output  R s ( ) node C(s)  k 1 ∆ =1-(sum of all individual loop gains)+(sum of gain products of all combinations of two non-touching loops)-(sum of gain products of all combinations of three non-touching loops)+.. ∆ k =determinant of graph in which all loops touching the k th forward path are removed 6

Control Systems: Signal Flow Graphs Mason’s formula Canonical feedback system    T 1 G s ( ) G s ( ) 1      L G s H s  1 1 ( ) ( ) n C s T G s ( )  ( ) 1   k k   R s ( ) 1 G s H s ( ) ( )  k 1 ∆ 1 =1 7

Control Systems: Signal Flow Graphs Example signals: R(s), E 1 (s), E 2 (s), C 1 (s), C(s) 8

Control Systems: Signal Flow Graphs Example R(s) E 1 (s) E 2 (s) C 1 (s) C(s) 1 Loop1: E 2 (s)-C 1 (s)-E 2 (s) 2 Touching loops     T s ( ) G s G s G s ( ) ( ) ( ) G s ( ) 1 1 2 3 4 Loop2: E 1 (s)-E 2 (s)-C 1 (s)-C(s)-E 1 (s) 9

Control Systems: Signal Flow Graphs Example           G s G s H s G s G s G s G s H s 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 1 2 3 4 2 ∆ =1-(sum of all individual loop gains)+(sum of gain products of all combinations of two non-touching loops)-(sum of gain products of all combinations of three non-touching loops)+.. ∆ 1 =1 after removing all the loops in the forward path 10

Control Systems: Signal Flow Graphs Example      T s ( ) G s G s G s ( ) ( ) ( ) G s ( ) n C s T ( )  1 1 2 3 4  k k       G s G s H s ( ) ( ) ( ) R s ( )      1 2 1 k 1  1       G s G s G s ( ) ( ) ( ) G s ( ) H ( ) s   1 2 3 4 2 ∆ 1 =1    G s G s G s ( ) ( ) ( ) G s ( ) C s ( )    1 2 3 4 T s ( )     R s ( ) 1 G s G s H s ( ) ( ) ( ) G s G s G s ( ) ( ) ( ) G s ( ) H ( ) s 1 2 1 1 2 3 4 2 11

Control Systems: Signal Flow Graphs Example signals: Q 1 (s), E 1 (s), H 1 (s), E 2 (s), Q 2 (s), E 3 (s), H 2 (s), Q 3 (s) 12

Control Systems: Signal Flow Graphs Example Q 1 (s) E 1 (s) H 1 (s) E 2 (s) Q 2 (s) E 3 (s) H 2 (s) Q 3 (s) 3 2 1 Loops 1 and 2 are non-touching Loop1: E 1 (s)-H 1 (s)-E 2 (s)-Q 2 (s)-E 1 (s) Loop2: E 3 (s)-H 2 (s)-Q 3 (s)-E 3 (s) Loops 1 and 3 are touching Loops 2 and 3 are touching 13 Loop3:E 2 (s)-Q 2 (s)-E 3 (s)-H 2 (s)-E 2 (s)

Control Systems: Signal Flow Graphs ∆ =1-(sum of all individual loop gains)+ Example (sum of gain products of all combinations of two non-touching loops 1  T       1 1 1 1 1 1 2 R C R C s                1 1 1 2 2       R C s R C s R C s R C s R C s Δ 1 =1 1 1 1 2 2 2 1 1 2 2 1 1 1 1      1 2 R C s R C s R C s R C R C s 1 1 1 2 2 2 1 1 2 2 1 2 Q s ( ) R C R C s   1 1 1 2 2 n C s T ( )   1 1 1 1 Q s ( ) k k     1 3  2 R C s R C s R C s R C R C s R s ( )  1 1 1 2 2 2 1 1 2 2 k 1 1        2 R C R C s s R C ( R C R C ) 1   1 1 2 2 1 1 2 2 2 1 14