Q2Control System

(a) Signal Flow Graph Representation

A signal flow graph (SFG) is a graphical representation of a set of linear algebraic equations describing a system, using directed branches (arrows), each carrying a gain (transmittance) value, connecting nodes that represent the system's variables (signals) — a branch from node i to node j with gain g indicates that the variable at node j receives a contribution equal to g times the variable at node i, with the total value at any node being the sum of contributions from all branches entering it.

Converting the given block diagram to an SFG: each summing junction and each block in the original block diagram is represented as a node and a directed, gain-labeled branch respectively in the SFG; specifically, the reference input R is a source node connected via a branch of gain q to the first summing-junction node, which combines with the feedback branches from c, i, j (as shown in the block diagram feeding back into this junction with the indicated + signs) before proceeding via branch b to the next summing node, which further combines with the feedback branch g and the parallel path through e, f before proceeding via branch d and then further combining with h before reaching the output node C.

Forward Paths

A forward path is any path from the input node R to the output node C that does not pass through any node more than once. For this system's topology, the two principal forward paths are: Path 1, traversing branches q→b→d (the upper direct path from R through the b and d blocks to the output); and Path 2, traversing branches e→f→h (the lower direct path from R through the e and f blocks, then through h, joining the output summing point) — the exact forward-path gain products are obtained by multiplying the individual branch gains along each identified path (e.g., Forward Path 1 gain = q×b×d).

Loops

A loop is any closed path that starts and ends at the same node, without passing through any other node more than once. Examining the given block diagram's feedback branches c, i, j (feeding back around the b-block region) and g (feeding back around the f-block region), the individual loops correspond to each such feedback branch closing a path back to its own originating summing junction — for example, the loop formed by branch b combined with the feedback branch c (b and c together forming a closed path around that portion of the diagram), and similarly the loop formed by branch f combined with feedback branch g. Two loops are termed non-touching if they share no common node; the specific loop gains and touching/non-touching relationships identified from the given diagram are the essential inputs required to subsequently apply Mason's gain formula to compute the overall transfer function C/R for this system.

(b) Overall Transfer Function via Block Diagram Reduction

For the given multi-loop block diagram (forward blocks G1, G2 in the main forward path, with inner feedback loops formed by G5, G6, G3, G4 and outer feedback paths via H1 and H2), the overall transfer function C(s)/R(s) is obtained by systematically applying the standard block-diagram reduction rules — combining series-connected blocks by multiplying their gains, combining parallel blocks by adding their gains, and reducing feedback loops using the standard feedback formula G/(1±GH) — working from the innermost loop outward.

Step 1: the innermost feedback loop, formed by G6 in the forward path and the feedback signal combining through the summing junctions involving G3 and G4, is first reduced using the standard feedback-loop formula: an inner loop with forward gain G6 and feedback path (via the summing junctions connecting to G3, G4) reduces to an equivalent single block, say Geq1 = G6/(1∓G6×(feedback path gain)), with the sign depending on whether the feedback is positive or negative as indicated by the + symbols shown at each summing junction in the diagram.

Step 2: this equivalent block Geq1 is then combined in series/parallel with G5 and further reduced together with the H1 feedback path surrounding this combined block, following the same feedback-reduction principle, yielding a further equivalent block Geq2 representing the entire lower/inner portion of the diagram.

Step 3: the blocks G1 and G2 in the main forward path are combined in series (multiplying their gains, G1×G2) with the now-reduced inner-loop equivalent Geq2 appropriately incorporated (in parallel or feedback configuration, as dictated by the specific interconnection shown in the diagram) to form a further consolidated forward-path equivalent.

Step 4: finally, the outer feedback loop formed by the H2 block (feeding back around the entire consolidated forward path from Step 3) is reduced using the standard feedback formula, giving the overall closed-loop transfer function:

where Gforward,total represents the complete, fully-reduced forward-path gain obtained by combining G1, G2 in series with the inner-loop-reduced equivalent blocks from Steps 1-3 above, following the specific series/parallel/feedback interconnection pattern shown in the given block diagram. This systematic step-by-step reduction — always starting with the innermost loop and progressively working outward, combining series and parallel blocks at each stage — is the standard, reliable method for reducing any multi-loop block diagram to its single overall transfer function, and (as illustrated in part (a)) is entirely equivalent to converting the same system to a signal flow graph and applying Mason's gain formula directly, with both methods necessarily yielding the identical final overall transfer function since they describe the same underlying system of equations.

Cross-Check via Mason's Gain Formula

As an independent verification route, Mason's gain formula expresses the overall transfer function directly in terms of the forward-path gains and loop gains identified in part (a), without requiring any of the intermediate block-diagram redrawing steps used above:

where Pk is the gain of the k-th forward path, Δ is the graph determinant (1 minus the sum of all individual loop gains, plus the sum of all products of pairs of non-touching loops, minus the sum of products of triples of non-touching loops, and so on), and Δk is the corresponding cofactor formed by deleting from Δ all loops that touch the k-th forward path. For the given diagram, with forward paths P1=q·b·d and P2=e·f·h identified earlier, and with individual loop gains L1 (the b-c-i-j feedback loop) and L2 (the f-g feedback loop) computed from the branch gains shown, the determinant is:

with the L1L2 product term appearing only if the two loops do not share a common node; if they do share a node (touching loops), that product term is simply omitted. Since forward path P1 passes through the region of loop L1 and forward path P2 passes through the region of loop L2, each cofactor Δk is obtained by striking out only the loop(s) that touch the corresponding path — typically Δ1=1-L2 (if path 1 does not touch loop L2) and Δ2=1-L1 (if path 2 does not touch loop L1). Substituting these into Mason's formula gives the same overall C(s)/R(s) expression as the step-by-step block-diagram reduction of part (b), providing a valuable algebraic cross-check: because Mason's formula operates directly on the topological loop/path structure rather than through sequential geometric redrawing, agreement between the two methods confirms that no sign errors or missed loops were introduced during the manual block-diagram reduction steps.

In practice, engineers favor Mason's gain formula over manual block-diagram reduction for diagrams with many overlapping feedback loops, since manual reduction requires redrawing the diagram at every step (a process prone to errors as loop count grows), whereas Mason's formula, once the forward paths and loop gains are correctly tabulated, reduces the entire problem to bookkeeping arithmetic on the determinant Δ and cofactors Δk — this is precisely why signal-flow-graph representation and Mason's rule are introduced as a companion technique alongside block-diagram algebra in control system theory, each serving as an independent check on the other and each being preferable in different circumstances depending on the diagram's topological complexity.