Appearance
6.1 — Open Loop, Closed Loop and Block Diagrams
A toaster runs for two minutes and stops. It has no idea whether the bread is brown. An oven set to 180 °C measures its own temperature and switches the element on and off to hold it there. It knows.
That difference — whether the output is measured and fed back — is the whole of control engineering, and it changes everything about how the system behaves, what can go wrong, and what mathematics you need.
Control theory is Chapter 4.5's Laplace transform put to work, and the systems it acts on are the LTI systems of Chapter 4.2. This chapter introduces the one structural idea those two chapters do not contain: feeding the output back to the input.
1. Open loop
An open-loop system applies a fixed input and hopes. There is no measurement of what actually happened.
\text{input} \to \text{controller} \to \text{plant} \to \text{output}
The plant is the thing being controlled — a motor, a heater, a chemical reactor. The word comes from process engineering and has stuck.
Where open loop works well:
- The relationship between input and output is known and stable.
- Disturbances are small or absent.
- Errors are tolerable.
A toaster. Two minutes at full power. If the bread is frozen, or the mains voltage is low, or the previous slice left the element hot, the result differs — and the toaster neither knows nor cares.
A stepper motor. Send 200 pulses and the shaft turns exactly one revolution, because each pulse advances it by one physical step. No feedback is needed as long as the load never exceeds the motor's torque. If it does, steps are lost silently and the position is wrong from then on — which is why 3D printers occasionally produce a print with a layer shifted sideways.
A washing machine's timer. Fill for 90 seconds, wash for 12 minutes, spin for 5. It does not measure cleanliness.
The advantages are real: simple, cheap, and it cannot become unstable, because there is no loop to go unstable.
2. Closed loop
A closed-loop or feedback system measures the output, compares it with what was wanted, and acts on the difference.
e(t)=r(t)-y(t)
where r is the reference or setpoint, y is the measured output, and e is the error. The controller acts on the error, not on the reference.
What feedback buys you, and it is a lot:
- Disturbance rejection. The oven does not care whether you opened the door — the temperature drops, the error grows, the element turns on.
- Insensitivity to the plant. If the heater's power output changes by 20% with age, the loop compensates. This is the same desensitivity that Chapter 2.6 derived for amplifiers, and the mathematics is identical.
- Accuracy without precision components. A cheap actuator inside a good loop beats an expensive actuator with no loop.
- Changed dynamics. Feedback can make a slow system fast, or a ringing system calm — you are not stuck with the plant you were given.
What it costs:
- A sensor, with its own noise, drift and cost. The loop can never be more accurate than the sensor, because it drives the measured output to the setpoint, not the actual one.
- Complexity.
- The possibility of instability. This is the big one, and it is why Chapters 6.4 and 6.5 exist.
3. Why feedback can destabilise
The intuition is straightforward once stated. Feedback works by opposing the error. But every real system takes time to respond, and if the response arrives late enough, the correction is applied at the wrong moment and makes things worse.
Push a swing at the wrong point in its cycle and you fight it. Delay the push by half a period and you are pushing when you should be pulling — the negative feedback has become positive.
Formally: if the loop introduces 180° of phase shift at some frequency, and the gain at that frequency is 1 or more, then the Barkhausen condition of Chapter 2.6 is satisfied and the system oscillates.
Every real plant accumulates phase lag with frequency, so at high enough frequency there is always 180° of lag. The only question is whether the gain has fallen below 1 by the time you get there. That is what Chapter 6.5's gain and phase margins measure.
Everyday instances of this happening:
- A shower with a long pipe. Adjust the tap, wait, still too cold, adjust more — then it is scalding. The delay between action and effect turns a person into an unstable controller.
- A car driven by someone overcorrecting on a slippery road.
- Audio feedback howl when a microphone hears its own loudspeaker.
4. Block diagram algebra
Real systems are drawn as interconnected blocks, and reducing them to a single transfer function is mechanical.
Series (cascade):
G_{total}=G_1G_2
Parallel:
G_{total}=G_1+G_2
Feedback — the important one:
\boxed{T(s)=\frac{Y(s)}{R(s)}=\frac{G(s)}{1+G(s)H(s)}}
for negative feedback, and with + replaced by - in the denominator for positive feedback.
Derivation, in three lines. The error is E = R - HY. The output is Y = GE. Substitute:
Y=G(R-HY) \;\Rightarrow\; Y(1+GH)=GR \;\Rightarrow\; \frac YR=\frac{G}{1+GH}
The vocabulary that goes with it, because every later chapter uses it:
- G(s) — the forward path, controller and plant together.
- H(s) — the feedback path, usually the sensor.
- GH — the loop gain, or open-loop transfer function. This is the single most important quantity in control, and Chapters 6.4 and 6.5 study it almost exclusively.
- 1+GH — the characteristic polynomial. Setting it to zero gives the closed-loop poles.
Unity feedback means H=1, so the sensor is perfect and the output is compared directly with the setpoint. Most textbook analysis assumes it, and any loop can be rearranged into that form.
Reading the formula
When |GH| \gg 1:
T \approx \frac{G}{GH}=\frac1H
The closed-loop behaviour depends on the sensor and not on the plant at all. That is the desensitisation result, and it is why feedback is worth the trouble. It also says the sensor's accuracy is now the system's accuracy — you have traded dependence on the actuator for dependence on the measurement.
When |GH| \ll 1: T \approx G, and the feedback is doing nothing.
When GH = -1: the denominator is zero, T is infinite, and the system produces output with no input. That is instability, and the whole of stability analysis is about staying away from that point.
5. Modelling — where the transfer functions come from
Before analysing a loop you need G(s), and it comes from physics.
A DC motor
Two equations. The electrical side, from Chapter 1.5:
V=Ri+L\frac{di}{dt}+K_e\omega
The last term is the back-EMF: a spinning motor is also a generator, and it produces a voltage opposing the supply, proportional to speed.
The mechanical side, from Newton:
J\frac{d\omega}{dt}+b\omega=K_ti
where J is the rotor inertia, b the friction, and K_t the torque constant. In SI units K_t = K_e numerically, which is not a coincidence — it follows from conservation of energy.
Laplace transform both and eliminate i:
\frac{\Omega(s)}{V(s)}=\frac{K_t}{(Ls+R)(Js+b)+K_tK_e}
A second-order system. And because electrical time constants are usually far shorter than mechanical ones (L/R of a millisecond against J/b of a hundred milliseconds), the electrical pole is far away and the system behaves as first order:
\frac{\Omega(s)}{V(s)}\approx\frac{K_t/(R b + K_tK_e)}{\tau_ms+1}
That simplification is standard practice and worth recognising: when one pole is ten times faster than another, drop it.
A heated tank
Energy balance: heat in minus heat lost equals heat stored.
C\frac{dT}{dt}=P-\frac{T-T_{ambient}}{R_{th}}
with C the thermal capacitance and R_{th} the thermal resistance to the surroundings. Laplace:
\frac{T(s)}{P(s)}=\frac{R_{th}}{R_{th}Cs+1}
A first-order lag with time constant R_{th}C — structurally identical to the RC circuit of Chapter 1.5, which is why thermal engineers borrow the electrical vocabulary wholesale.
Delay
Some plants have genuine transport delay: material moving along a conveyor, fluid in a pipe, a signal crossing a network.
G(s)=e^{-sT_d}
Delay is the enemy of feedback. It contributes phase lag -\omega T_d that grows without limit as frequency rises, while contributing no attenuation at all. A system with significant delay must have low loop gain, or it will oscillate — which is exactly the shower problem of section 3.
The Padé approximation turns the exponential into a rational function so the algebra stays manageable:
e^{-sT_d}\approx\frac{1-sT_d/2}{1+sT_d/2}
Note the zero in the right half plane. That is not an artifact — it correctly captures the fact that a delayed system initially responds in the wrong direction when you look at the approximation's step response, which is the mathematical signature of dead time.
6. The vocabulary of performance
Five numbers describe how a loop behaves, and every design specification is written in them.
Rise time — how long to get from 10% to 90% of the final value. Speed.
Overshoot — how far past the target it goes, as a percentage. Aggressiveness.
Settling time — how long until it stays within ±2% of final. Usually the number that matters most, because it is when the system is actually usable.
Steady-state error — how far off it settles, permanently.
Disturbance rejection — how much a disturbance moves the output, and how fast it recovers.
These fight each other. More gain gives faster response and less steady-state error, and more overshoot and less stability margin. The whole of control design is navigating that trade, and Chapters 6.3 and 6.6 do it quantitatively.
7. Feedforward, and why you should use both
Feedback is reactive: it only acts after an error exists. If a disturbance is measurable in advance, you can act before the error appears.
Feedforward measures the disturbance and applies a correction directly, without waiting for the output to move.
A concrete example. A furnace heating a metal strip that moves through it. When the strip speeds up, more heat is needed. A pure feedback controller waits for the temperature to fall, then responds. A feedforward term reads the line speed and increases the power immediately, so the temperature barely moves.
The trade:
- Feedforward is fast and requires an accurate model. If your model of the disturbance's effect is wrong, the correction is wrong, and nothing detects it.
- Feedback is slower and self-correcting. It does not need a model, and it fixes anything, including the feedforward's mistakes.
The right answer is nearly always both: feedforward for the bulk of the correction, feedback to clean up what the model got wrong. Every well-designed process control loop is built this way.
A familiar instance. Cruise control that reads the road gradient from a map and increases throttle before the hill, with the ordinary speed feedback loop still running underneath. The car holds speed on a hill far better than feedback alone can manage.
Chapter 6.2 turns these block diagrams into transfer functions and pole positions, so that "will it be stable and how will it behave" becomes a calculation rather than a guess.
Every formula above, built from scratch
None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.
Sensitivity
S=\frac{\partial T/T}{\partial G/G}=\frac{1}{1+GH}, \qquad T_{comp}=\frac{GH}{1+GH}
S+T_{comp}=1 \qquad\text{always, at every frequency}
Bode's integral:
\int_0^\infty\ln|S(j\omega)|\,d\omega=0
The waterbed effect: area of sensitivity reduction equals area of amplification. Improving one frequency band necessarily worsens another.
What the next chapter fixes
Talking about a loop in words gets you only as far as "it corrects itself". To predict how it corrects, and whether it corrects at all, the plant needs to be written as mathematics. Chapter 6.2 does that with the transfer function, and shows that two small sets of numbers — the poles and the zeros — contain everything about how a system will behave.