Supplement material: Signal-Flow Graphs (SFGs) A signal-flow graph (SFG) is a diagram consisting of nodes that are connected by several directed branches and is a graphically representation of a set of linear relations. 1 G1 G2 G3 G4 -H1 -H2 -H3 -H4 -1 SFGs: is the abbreviation of signal flow graphs.

SFG may be regarded as a simplified version of a block diagram. By using Mason’s formula, one is able to obtain the transfer function C(s)/R(s) no matter how complex the block diagram may be.

a b 1. Signal-flow graph 1.Node and Branch Node: A node is a point representing a variable or signal. Branch: A branch is a directed line segment joining two nodes. The gain of a branch is a transfer function. A branch is A directed line segment, where a and b denote the variables and G is called branch transfer function. A signal can transmit through a branch only in the direction of the arrow. The SFG represents an algebra: b=Ga. G(s) a b b=Ga

-H4 -H3 Example. 1 G1 G2 G3 G4 -H1 -H2 -1 In this example, we have 8 nodes and 11 braches. Note that, the first (last) two nodes are equivalent since the transfer function is 1. We can combine the first two nodes in one.

a b c G1 G2 2. Path and path transfer function A path is any collection of continuous succession of branches traversed in the same direction. a b c G1 G2 traverse 遍历 The product of the branch transfer functions encountered in traversing a path is called the path transfer function; for instance, P=G1G2

u c 3. Forward path and forward path transfer function A forward path is a path that starts at an input node and end at an output node and along which no node is traversed more than once. P=G1G2G3 G1 G2 G3 u c

-H4 -H3 Example. 1 G1 G2 G3 G4 -H1 -H2 -1 By definition, for this example, we have only 1 forward path with path transfer function G1G2G3G4. How many forward paths do we have in this example?

Example. By definition, for this example, we have two forward paths with path transfer functions being G1G2G3G4 and G5G6G7G8, respectively.

u c 4. Loop and loop transfer function A loop is a path that originates and terminates on the same node and along which no other node is encountered more than once. Loop transfer function is defined as the path transfer function of a loop, where sign must take into account. It represents a feedback loop. G1 G2 G3 u c 

Nontouching loops: Loops are nontouching if they do not possess any common nodes. (L1, L3) and (L1, L4) are nontouching loops. However, (L1, L2) and (L1, L4) are touching loops.

= the determinant of the graph =1Li+LiLj LiLjLk+; 5. The Mason’ Formula = the determinant of the graph =1Li+LiLj LiLjLk+; Li=transfer function of the ith loop; LiLj=product of transfer functions of two non-touching loops; Li: Loop transfer function; \Sigma Li: sum of all different loop gains; \Sigma LiLj: sum of the products of all combinations of two nontouching loops. LiLjLk=product of transfer functions of three non-touching loops; LiLjLkLl…..;

N=total number of forward paths; Pk=transfer function of kth forward path; k= the cofactor of the kth forward path with the loops touching the kth forward path removed.

Example. There are four loops:

Example. There are four loops: Touching loops: if the same signal flows two loops, then the two loops are called touching loops. Otherwise, the two loops are called non-touching loops.

Example. There are four loops:

Example. There are four loops: Loops L1 and L2 do not touch L3 and L4. Therefore, the determinant is

Example. Are L1 and L2 (or L3 and L4) touching loops? Path 1:

Example. Delta Path 1: Path 2:

The cofactor for path 1 is: Example. Cofactor余子式; \Delta 1=\Delta Path 1: Path 2: The cofactor for path 1 is: The cofactor for path 2 is:

Example. The transfer function is:

Example. Find the transfer function C(s)/R(s). G1 G2 G3 H2 H1 H3 R C Solution: First of all, find all the single loops: Let's count how many single loops the system has. In many cases, it is not necessary to draw the SFG for a given block diagram.

Then, from the diagram it is shown that, no non-touching loops exist Then, from the diagram it is shown that, no non-touching loops exist. Therefore, the determinant of the graph is The only forward path is The corresponding cofactor is From Mason’s formula,