Block-Diagram Algebra & Closed-Loop Transfer Functions

A correct block diagram names the signal at every arrow and makes the physical comparison, measurement, actuation, disturbance, and noise locations unambiguous. Algebraic reduction is then ordinary elimination of internal variables. Drawing blocks first and assigning meaning later is a common source of wrong disturbance paths and wrong signs.

Opening the loop at different locations produces different return ratios. Keep signal locations explicit before calculating any margin.

Primitive interconnections #

For compatible SISO blocks:

• series: $G_{eq}=G_2G_1$;
• parallel sum: $G_{eq}=G_1+G_2$ with signs inherited from the junction;
• negative feedback around forward $G$ and return $H$:

$$G_{cl}=\frac{G}{1+GH};$$

• positive feedback changes the denominator to $1-GH$.

The order of scalar LTI blocks in series can be exchanged algebraically. Physical blocks cannot always be exchanged: moving a saturation, sampler, noise source, or disturbance changes the system. For MIMO matrices, multiplication does not commute even when every block is linear.

Derive the loop from signal equations #

For the diagram shown, let measurement be $y_m=Hy+n$, error $e=r-y_m$, control $u=Ce$, and plant output $y=P(u+d_i)+d_o$. Then

$$y=PC(r-Hy-n)+Pd_i+d_o.$$

Collecting $y$ gives

$$y=\frac{PC}{1+PCH}r+\frac{P}{1+PCH}d_i+\frac{1}{1+PCH}d_o-\frac{PC}{1+PCH}n.$$

With loop transfer $L=PCH$,

$$\frac{Y}{R}=\frac{PC}{1+L},\quad \frac{Y}{D_i}=\frac{P}{1+L},\quad \frac{Y}{D_o}=\frac{1}{1+L},\quad \frac{Y}{N}=-\frac{PC}{1+L}.$$

The same denominator appears, but disturbance location changes the numerator. Saying “feedback rejects disturbances” without naming where the disturbance enters is incomplete.

Moving summing and pickoff points #

A summing junction may be moved across a block only by transforming the side path so the signal equation stays identical. Moving a summing point from before $G$ to after $G$ multiplies the moved side input by $G$; moving it backward requires $G^{-1}$. A pickoff moved across $G$ requires the complementary transformation.

This can introduce an inverse that is improper, unstable, noncausal, or undefined. The algebra may remain formal while the new diagram is not a realizable architecture. Use transformations to simplify derivation, not to claim that sensors or disturbances can be physically relocated.

Signal-flow graphs and Mason's rule #

A signal-flow graph represents linear equations as directed branches. Mason's gain formula gives transfer from source to sink:

$$T=\frac{\sum_k P_k\Delta_k}{\Delta},$$

where $P_k$ are forward-path gains, $\Delta$ contains loop gains and products of non-touching loops with alternating signs, and $\Delta_k$ excludes loops touching path $k$. It is useful for densely interconnected scalar graphs, but direct matrix elimination is usually clearer and safer for state-space or MIMO models.

Before using the rule, ensure each node is a signal variable and each branch equation is causal. A pretty graph does not repair an incorrect physical model.

The return ratio depends on where the loop is broken #

Stability margins are computed for a particular loop opening while preserving loading and the rest of the interconnection. In a simple unity-feedback SISO loop, $L=PC$ is obvious. With sensor dynamics, nested loops, feedforward paths, or multiloop control, “the open-loop transfer function” is not unique.

Document:

• the broken signal and injection/measurement directions;
• whether other loops remain closed;
• the sign convention used by the Nyquist criterion;
• whether the opening changes block loading or operating point.

Loop gain is a property of an interconnection at a defined break, not a label copied from a block.

Internal stability and hidden cancellations #

An external transfer function can be stable while an internal signal diverges. Suppose an unstable plant pole is cancelled by a controller zero. The selected $r\to y$ expression may simplify, but plant state, controller state, sensor noise, or model mismatch can still excite the unstable mode.

Internal stability requires every relevant closed-loop transfer between exogenous inputs and internal signals to be stable, with a well-posed interconnection. Check the full realization or a coprime formulation; do not infer it from one simplified ratio.

Worked two-degree-of-freedom form #

Let feedback controller $C$ regulate error and prefilter $F$ shape reference:

$$u=C(Fr-Hy).$$

Then

$$\frac{Y}{R}=\frac{PCF}{1+PCH},$$

while disturbance rejection and robustness remain governed by $1+PCH$. Choosing $F$ can improve nominal command following without changing the characteristic equation. It cannot stabilize an unstable feedback loop or reject unmodeled disturbances. This separation is why reference shaping and regulation should not be conflated.

Reduction checklist #

1. Label every arrow with variable, units, and direction.
2. Write the signal equations before collapsing blocks.
3. Derive separate maps for reference, each disturbance, noise, and control effort.
4. Preserve delays, nonlinearities, samplers, and constraints as distinct blocks.
5. Check properness and realizability after any junction movement.
6. Inspect internal stability before accepting a cancellation.
7. Verify the reduced equations against the unreduced simulation or state-space interconnection.

The value of block-diagram algebra is not the smallest possible picture. It is a traceable map from physical paths to closed-loop equations.