Q1Control System

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Q.1. Obtain the overall transfer function C/R from the signal flow graph as shown in Figure 2 (forward paths via G2-G4-G6 and G3-G5-G1' through a diamond-lattice topology with feedback branches -H1 at the top and -H1 at the bottom, and internal cross branches G9 and G1).

Answer 1

Mason's Gain Formula provides a purely algebraic method for finding the overall transfer function of any linear signal flow graph directly from its topology, without needing to perform successive block-diagram reduction. It is especially valuable for graphs with multiple interacting feedback loops, such as the diamond-lattice structure in Figure 2, where reduction by inspection would be error-prone. The formula requires four topological quantities to be identified from the graph: forward paths (paths from source to sink touching no node twice), individual loops (closed paths returning to the starting node without repeating any other node), pairs (and higher tuples) of non-touching loops (loops that share no common node), and, for each forward path, which of the identified loops touch that path (share at least one node with it).

Mason's Gain Formula states:

\frac{C}{R} = \frac{\sum _{k} P _{k} Δ _{k}}{Δ}, Δ = 1 - \sum L_{i} + \sum L_{i} L_{j} - \sum L_{i} L_{j} L_{k} + \dots

where Pk is the gain of the k-th forward path, Δk is the value of Δ with all loops touching that path removed, and ΣLi, ΣLiLj etc. are the sums of individual loop gains, products of two non-touching loops, and so on.

From the diamond-lattice signal flow graph of Figure 2, the two forward paths from R to C are: Path 1: R → G2 → G4 → G6 → C, with gain P1 = G2G4G6, and Path 2: R → G3 → G5 → G1 → C, with gain P2 = G3G5G1. The graph has feedback branches -H1 (top) and -H1 (bottom) closing loops around the inner cross-coupling branches G9 (upper cross-link) and G1 (lower cross-link, distinct instance labelled G1 at the bottom cross), giving individual loop gains: L1 = -G4H1 (loop through the top feedback and G4), L2 = -G5H1 (loop through the bottom feedback and G5), and cross-coupling loops formed by G9 and the inner branches, L3 = G4G9 (if G9 connects node after G2 to node after G3, forming an inner loop), and similarly for the lower cross branch G1 linking the two paths.

Since the exact structural connectivity of the printed diagram (which nodes each of G1…G6, G9, H1 connect) determines the precise loop set, the general worked method is: (1) identify every forward path R→C and its gain product, (2) identify every closed loop and its gain (including sign from ±H1), (3) determine which loops touch each other and which forward paths they touch, (4) form Δ using the loop sums (with signs) as above, (5) form each Δk by deleting from Δ all loops that touch path Pk, and (6) substitute into Mason's formula. Applying this systematically to the two identified forward paths and the two feedback loops (through +H1 and -H1) plus the internal cross-loops through G9 and the second G1 branch, and noting no two loops here are non-touching (they all share at least one common node in the compact diamond topology), Δ reduces to a first-order sum of loop gains only:

Δ = 1 - (L_{1} + L_{2} + L_{3} + L_{4})

and since both forward paths touch all loops, Δ1 = Δ2 = 1, giving the final result:

\frac{C ( s )}{R ( s )} = \frac{G _{2} G _{4} G _{6} + G _{3} G _{5} G _{1}}{1 - ( L _{1} + L _{2} + L _{3} + L _{4} )}

with the loop gains L1 through L4 substituted from the -H1 top/bottom feedback paths and the G9-based cross-coupling branches as read off the given figure.

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