Block diagrams and feedback

A robot joint is not a single transfer function but a chain of them: a controller drives a power amplifier, which drives a motor, which turns a gearbox and a link, whose angle is measured by an encoder and compared with the command. Block diagrams make such chains visible, and the algebra of transfer functions lets us collapse them into a single input-output relation. The most important structure in any block diagram is the feedback loop, in which the output is measured and used to correct the input. Feedback is the reason robots can hold position against gravity, track trajectories despite friction and keep working when parameters change, and it is also the reason they can oscillate or go unstable when tuned badly. This lesson develops block-diagram algebra, derives the closed-loop transfer function, and explains what feedback buys: disturbance rejection, reduced sensitivity to parameter changes, and the ability to move poles. Designing the controller itself is the subject of ROB-300; here the aim is to understand what a loop does to the signals passing through it.

Block diagram elements and algebra

A block diagram has three elements: blocks, each containing a transfer function that multiplies its input; summing junctions, which add or subtract signals according to the signs marked on them; and pickoff points, where a signal is copied to two destinations without being changed. Three rules reduce most diagrams.

• Series: blocks \( G_1 \) and \( G_2 \) in cascade combine into \( G_1G_2 \). This follows from the convolution property of lesson 5 and requires that the second block does not load the first, which holds for signals in software and for buffered electronics but needs care with mechanical and electrical connections.
• Parallel: blocks sharing an input with outputs added combine into \( G_1 + G_2 \).
• Feedback: a forward block G with a feedback block H subtracted at the input combines into \( G/(1 + GH) \), derived below.

Pickoff points and summing junctions can also be moved past blocks, provided compensating blocks are inserted, which helps when untangling complicated diagrams. With these rules any diagram built from linear blocks reduces to a single transfer function, although for large systems the state-space methods of lesson 12 are more systematic.

The closed-loop transfer function

Consider the standard loop: a reference R, an error \( E = R - HY \), a controller C and plant G in the forward path, so \( Y = CGE \), and a sensor H in the feedback path. Substituting the error,

The product \( L(s) = C(s)G(s)H(s) \), the gain around the loop, is called the loop transfer function or open-loop transfer function. It is the single most important quantity in feedback design. The denominator \( 1 + L(s) \) sets the closed-loop poles: they are the roots of \( 1 + L(s) = 0 \), the closed-loop characteristic equation. Feedback therefore moves poles. A plant with poles in an awkward place, or even in the right half-plane, can be given well-damped closed-loop poles by choosing C. When the loop gain is large, \( Y/R \approx 1/H \): the output follows the reference scaled by the sensor, almost regardless of the plant. This is the essential magic of feedback, and also its main requirement: the sensor must be accurate, because the loop faithfully tracks whatever the sensor reports.

Steady-state error and system type

How well a loop tracks a reference in steady state depends on how many integrators, poles at s = 0, the loop transfer function contains. With unity feedback the error is \( E = R/(1 + L) \), and the final value theorem gives the steady error for standard inputs:

This explains a classic observation. A position loop on a motor naturally contains one integrator, because angle is the integral of speed, so proportional position control reaches a step target exactly (in the absence of friction and gravity) but lags a ramp, such as a moving conveyor, by a constant amount. Adding integral action raises the type and removes that lag.

Disturbance rejection

Robots are constantly pushed by disturbances: gravity on a link, friction in a gearbox, a payload picked up, wind on a drone. In the standard loop, a disturbance D added at the plant input reaches the output through

Compared with the open-loop effect G, feedback divides the disturbance by \( 1 + L \). Wherever the loop gain is large, typically at low frequencies, disturbances are strongly suppressed. A joint holding position against a slowly varying load hardly moves; a sudden fast push still gets through, because no practical loop has high gain at high frequency. The frequency-domain view of lesson 7 makes this precise: disturbance rejection is good below the loop's bandwidth and absent above it.

Sensitivity to parameter changes and to noise

Plants change: batteries sag, loads vary, friction rises in the cold. The sensitivity of the closed-loop transfer function T to changes in the plant G is

so a 10 % change in the plant produces only a \( 10/(1 + L) \) % change in closed-loop behaviour. With L = 9, a 10 % plant change becomes a 1 % change in the loop. This robustness is the original reason feedback amplifiers were invented in 1927, and it is why a well-tuned robot joint behaves consistently even though its motor and gearbox vary from unit to unit. The function \( S = 1/(1 + L) \) is called the sensitivity function, and its partner \( T = L/(1 + L) \) the complementary sensitivity; note that \( S + T = 1 \).

Feedback has a price. Sensor noise N enters the loop at the same point as the reference, so the output responds to noise through T, just as it responds to commands. A loop with high gain at high frequency faithfully reproduces the noise of its sensor, making motors hum and links chatter. Because S + T = 1, we cannot make both small at the same frequency: we want S small at low frequencies for tracking and disturbance rejection, and T small at high frequencies to ignore noise. Managing this trade-off is the core of loop shaping in ROB-300, and it is one reason sensor filtering (lesson 11) must be designed together with the controller.

Feedback, delay and instability

Feedback can also destabilize. If the loop gain is high and the signal coming back around the loop is delayed or phase-shifted enough, the correction arrives late and pushes in the wrong direction, reinforcing the error instead of reducing it. Every real loop contains such delays: sample-and-hold in the controller (lesson 9), sensor filtering, communication latency, motor inductance and structural flexibility. Lesson 7 shows how to measure the margin of safety in terms of gain and phase. A simple experiment makes the danger concrete: the code below closes a proportional loop around a pure integrator with a small delay, and increases the gain until the response begins to oscillate and then grows.

import numpy as np
h, T, delay = 1e-3, 3.0, 0.05          # 50 ms loop delay
nd = int(delay/h)
for K in [5, 15, 30, 40]:
    y = 0.0
    u_hist = [0.0]*nd                   # buffer of delayed commands
    out = []
    for k in range(int(T/h)):
        u = K*(1.0 - y)                 # proportional control, unit step command
        u_hist.append(u)
        y += h*u_hist.pop(0)            # plant: integrator driven by delayed command
        out.append(y)
    out = np.array(out)
    tail = out[int(2/h):]
    print(f"K = {K:2d}: peak {out.max():6.2f}, late swing {tail.max()-tail.min():8.3f}")

K =  5: peak   1.00, late swing    0.000
K = 15: peak   1.26, late swing    0.000
K = 30: peak   2.02, late swing    0.740
K = 40: peak 37534.78, late swing 64323.848

At K = 5 the loop settles smoothly; at K = 15 it overshoots by about a quarter; at K = 30, just below the critical value, it is still swinging two seconds later; and at K = 40 the oscillation has grown enormously. Beyond about K = 31, where the loop's phase lag from the 50 ms delay reaches 180° at the frequency where the loop gain is one, the oscillation grows without bound. The critical gain predicted by frequency-response analysis is \( K = \pi/(2 \times 0.05) \approx 31 \) s⁻¹, which lesson 7 explains.