Block diagrams and closed-loop transfer functions

Why it matters

A block diagram turns a P&ID loop (sensor, transmitter, controller, valve, process, disturbance) into algebra. From it you get the closed-loop transfer functions that predict set-point tracking, disturbance rejection, offset and stability, which is what every later topic (P/PI/PID, Routh, root locus, frequency response, tuning) works with.

Key ideas

Elements of a feedback loop. In deviation variables and the Laplace domain:

• Controller G_c(s): acts on the error E = R − B (set point minus measured value). The comparator is part of the controller.
• Final control element G_v(s): usually the valve with its actuator.
• Process G_p(s): from the manipulated variable M to the controlled variable C.
• Load (disturbance) path G_d(s): from the disturbance L to C.
• Measuring element G_m(s): sensor plus transmitter, giving B from C.
• Sometimes a set-point transducer K_sp converts the set point into the same units as the measurement; for consistency K_sp = K_m (the steady-state gain of G_m).

Block-diagram algebra.

• Blocks in series multiply: G = G₁·G₂ (valid when the second block does not load the first).
• Blocks in parallel add: G = G₁ ± G₂.
• A negative feedback loop with forward path G and feedback H reduces to G/(1 + G·H).
• Summing points and take-off points can be moved across blocks if the moved signal is multiplied or divided by that block's transfer function.

Closed-loop transfer functions (negative feedback). With G_OL = G_c·G_v·G_p·G_m (the open-loop transfer function, i.e. the product of all blocks around the loop):

• Servo problem (set-point change, L = 0): C/R = K_sp·G_c·G_v·G_p/(1 + G_OL).
• Regulator problem (load change, R = 0): C/L = G_d/(1 + G_OL).
• The general rule: output/input = (product of blocks in the forward path from that input to that output)/(1 + product of all blocks around the loop).
• Both transfer functions share the same denominator. Its roots, from the characteristic equation 1 + G_OL(s) = 0, are the closed-loop poles and decide stability and speed for both problems.

What feedback does. It makes the response faster and less sensitive to the process gain and to disturbances, but it can also make a stable process oscillate or go unstable, and with proportional action alone it leaves an offset. Breaking the loop (for example, putting the controller in manual) returns the system to the open-loop process dynamics.

Offset from the final-value theorem. For a unit step in set point, offset = 1 − lim(s→0) C/R (with K_sp = K_m = 1). For a P controller on a self-regulating process, offset = 1/(1 + K_OL), where K_OL is the steady-state open-loop gain.

Formulas

G_OL(s) = G_c·G_v·G_p·G_m

• Open-loop transfer function (product around the loop).

C/R = K_sp·G_c·G_v·G_p / (1 + G_OL)

• Servo (set-point) response; K_sp set-point gain (often 1 when G_m = 1).

C/L = G_d / (1 + G_OL)

• Regulator (load) response.

1 + G_OL(s) = 0

• Characteristic equation; closed-loop poles.

G/(1 + G·H), G₁·G₂, G₁ + G₂

• Feedback, series and parallel reductions; G forward path, H feedback path.

Offset = 1/(1 + K_OL)

• Unit set-point step, proportional control, unity measurement, self-regulating process; K_OL = K_c·K_v·K_p·K_m (dimensionless).

Worked examples

Example 1 (standard): P control of a first-order process. G_c = K_c = 4, G_v = 1, G_p = 2/(5s + 1) (time in min), G_m = 1. Find C/R, the closed-loop gain and time constant, and the offset for a unit set-point step.

1. G_OL = 4·2/(5s + 1) = 8/(5s + 1).
2. C/R = G_OL/(1 + G_OL) = 8/(5s + 1 + 8) = 8/(5s + 9).
3. Normalise: C/R = (8/9)/((5/9)s + 1). Gain = 0.889; τ_CL = 0.556 min (the open-loop τ was 5 min).
4. Offset = 1 − 8/9 = 1/9 = 0.111 (equal to 1/(1 + 8)).

C/R = 8/(5s + 9); τ_CL ≈ 0.556 min; offset ≈ 0.111 per unit step

Example 2 (GATE level): measurement lag in the loop. G_c = K_c = 4, G_v = 1, G_p = G_d = 1/(2s + 1), G_m = 1/(s + 1) (min), K_sp = 1. Find the servo and regulator transfer functions, the closed-loop damping factor, and the steady-state responses to unit steps.

1. G_OL = 4/((2s + 1)(s + 1)).
2. Characteristic equation: (2s + 1)(s + 1) + 4 = 2s² + 3s + 5 = 0.
3. Servo: C/R = K_c·G_p/(1 + G_OL) = 4(s + 1)/(2s² + 3s + 5). Note the zero at s = −1: the measurement lag appears in the numerator because C is the true output, not the measured one.
4. Regulator: C/L = G_d/(1 + G_OL) = (s + 1)/(2s² + 3s + 5).
5. Normalise the denominator: 0.4s² + 0.6s + 1, so τ = √0.4 = 0.632 min and 2ζτ = 0.6 gives ζ = 0.474 (underdamped, although each element alone is overdamped).
6. Final values (s → 0): servo 4/5 = 0.8, so offset 0.2; regulator 1/5 = 0.2.

C/R = 4(s + 1)/(2s² + 3s + 5), C/L = (s + 1)/(2s² + 3s + 5); ζ ≈ 0.474; offset 0.2 for both problems

Common mistakes

• Using G/(1 + G) when the measuring element is not unity; the denominator must contain the whole loop product including G_m.
• Putting G_m in the numerator of C/R; the numerator is the forward path from R to C only.
• Writing a different denominator for the servo and regulator problems; it is always 1 + G_OL.
• Using a positive sign in the denominator for positive feedback (it should be 1 − G·H).
• Forgetting K_sp, so the "set point" is in different units from the measurement.
• Applying the final-value theorem without first checking that the closed loop is stable.

For GATE CH

Expect block-diagram reduction to C/R or C/L, the characteristic equation, the closed-loop gain, time constant or damping factor, and offset from the final-value theorem. Questions often add a measurement lag or a load path, and then ask about stability (leading to Routh) or the value of K_c giving a specified damping. Practise reducing diagrams with inner loops.

Quick check

1. Give the closed-loop transfer function for G = 6/(s + 1) with unity negative feedback.
2. What is the offset for a unit set-point step when K_OL = 4 under P control?
3. Do the servo and regulator transfer functions share numerators or denominators?
4. With G = 10/(s + 3) and H = 0.5, what is G/(1 + GH)?

Answers: 1. 6/(s + 7). 2. 1/5 = 0.2. 3. Denominators (1 + G_OL). 4. 10/(s + 8).