Introduction to Control Systems
Almost every machine that holds a speed, tracks a target, or keeps a temperature steady is running a control system. Before any mathematics, we need the central idea: measure what a system is doing, compare it with what we want, and feed the difference back. This chapter builds that vocabulary — plants and controllers, open versus closed loop, and the feedback equation the rest of the course rests on.
- What a control system is, and the four roles inside every one: plant, controller, actuator, and sensor.
- The difference between an open-loop and a closed-loop system, and why feedback changes everything.
- How to read a block diagram — reference, error, manipulated variable, controlled output, and feedback.
- The single most important formula of the course, the closed-loop transfer function \( C/R = G/(1+GH) \), and how to derive it.
- Why negative feedback improves accuracy, disturbance rejection, and sensitivity — and what it costs.
- How control systems are classified: linear/nonlinear, time-invariant/varying, continuous/discrete, SISO/MIMO, servo/regulator.
What Is a Control System?
A control system is an arrangement of components connected to command, direct, or regulate the behaviour of itself or another system so that a chosen output follows a desired value. The thing being controlled is the plant (or process) — a motor, an oven, an aircraft, a chemical reactor. The quantity we care about is the controlled output \(c(t)\): a speed, a temperature, a position. The value we want it to take is the reference or setpoint \(r(t)\).
Between the command and the plant sit two more roles. The controller decides what action to take; the actuator is the muscle that applies that action to the plant (a power amplifier, a valve, a heating element). In a feedback system a fourth role appears — the sensor, which measures the output and reports it back. Almost every control problem is some combination of these four parts.
The engineering goal is always the same: make \(c(t)\) track \(r(t)\) quickly, accurately, and stably, even when the plant is disturbed or its parameters drift. The rest of this book is a toolbox for achieving exactly that.
Everything that follows — modeling, time response, stability, controller design — exists to shape that arrow from \(r\) to \(c\) so the output behaves the way we want.
Open-Loop Systems
In an open-loop system the control action does not depend on the output. The controller is calibrated in advance; it applies a fixed action and simply trusts that the plant will respond as expected. There is no measurement of what actually happened — no feedback.
A bread toaster runs for a set time regardless of how brown the bread is; a washing machine steps through a fixed timed cycle; a simple traffic light switches on a clock. Each ignores its own result. For a single block of gain \(G\), the output is simply
Open-loop control is simple, cheap, and inherently stable (it cannot oscillate from feedback because there is none). Its weakness is decisive: it cannot correct for disturbances (a power dip, a draught, an extra load) or for changes in the plant (a worn motor, a different bread). Accuracy depends entirely on calibration, and calibration drifts.
Closed-Loop (Feedback) Systems
A closed-loop system measures the output, feeds it back, and compares it with the reference. Their difference is the error, and the controller acts on that error to drive it toward zero. Because the signal returns and subtracts, this is called negative feedback — the central mechanism of the entire subject.
Read the loop signal by signal. The output is the forward gain times the error, \(C = G\,E\); the feedback signal is the output scaled by the sensor, \(B = H\,C\); and the summing junction forms \(E = R - B\). Substitute the last two into the first and solve for \(C\):
The product \(G(s)H(s)\) is the loop gain (open-loop transfer function). When the feedback is unity (\(H = 1\)) this reduces to \(C/R = G/(1+G)\). Almost every analysis in this course begins by writing down this one ratio. For positive feedback the sign flips to \(G/(1 - GH)\), which is exactly why positive feedback tends toward instability.
Open vs Closed Loop — the Trade-offs
Feedback is not free. It buys accuracy and robustness at the price of complexity and the risk of instability — a badly designed loop can oscillate or run away. The whole of classical control (Parts 3 and 4) is about claiming the benefits while guaranteeing the loop stays stable.
| Property | Open-loop | Closed-loop (feedback) |
|---|---|---|
| Output measured? | No | Yes — sensor in feedback path |
| Disturbance rejection | Poor | Strong (error is corrected) |
| Sensitivity to plant changes | High | Reduced by factor \(1/(1+GH)\) |
| Accuracy | Depends on calibration | High; error drives the controller |
| Stability | Inherently stable | Must be designed for |
| Cost & complexity | Low | Higher (sensor + electronics) |
| Typical use | Toaster, timed washer | Cruise control, thermostat, servo |
Block-Diagram Vocabulary
A block diagram is the engineer's shorthand for how signals flow through a system. Each block holds a transfer function; arrows are signals; a summing junction (the circle) adds or subtracts; a take-off point (the dot) splits one signal to two destinations without changing it. The names below recur on every diagram in the book.
| Symbol | Name | Meaning |
|---|---|---|
| \(R(s)\) | Reference / setpoint | The desired value of the output |
| \(E(s)\) | Error (actuating) signal | \(E = R - B\); what the controller acts on |
| \(C(s)\) | Controlled output | The quantity being regulated |
| \(B(s)\) | Feedback signal | \(B = HC\); the measured output |
| \(G(s)\) | Forward-path transfer function | Controller × actuator × plant |
| \(H(s)\) | Feedback-path transfer function | Sensor / measurement dynamics |
| \(G(s)H(s)\) | Loop gain | Open-loop transfer function |
Driving \(E\) toward zero is the controller's job. In Part 2 we will see that the steady-state value of \(E\) measures how accurately the system tracks its command — the first hard performance number of the course.
Control Systems Around Us
Once you look for the loop, it is everywhere. A room thermostat compares measured temperature with the dial and switches the heater; cruise control compares road speed with the set speed and adjusts the throttle; a DC-motor servo compares shaft position with a command and drives the armature; an aircraft autopilot holds heading and altitude against wind. Even reaching for a cup is biological feedback: your eyes sense the gap, your brain commands, your arm acts.
Classification of Control Systems
Control systems are sorted along a few independent axes. Knowing where a system sits tells you which mathematical tools apply — and which chapters of this book you will need.
A linear system obeys superposition and homogeneity; a nonlinear one does not (saturation, dead-zone, \(c=r^2\)). Most of this course assumes linearity; Chapter 29 tackles the rest.
Time-invariant parameters are constant; time-varying ones change (a rocket losing fuel mass). We work mostly with LTI systems.
Continuous-time signals exist at every instant; discrete/digital systems sample at intervals and use the \(z\)-transform (Part 7).
Single-input single-output versus multi-input multi-output. MIMO is handled naturally by the state-space methods of Part 6.
A servomechanism tracks a changing command (a robot arm following a path); a regulator holds an output constant against disturbance (voltage or temperature control).
Deterministic signals are known functions of time; stochastic ones involve random noise, calling for probabilistic design (beyond this first course).
Worked Examples
Problem. Classify each: (a) a toaster on a timer, (b) an oven held at 180 °C by a thermostat.
Solution. (a) The toaster applies heat for a fixed time and never checks the result — open-loop. (b) The oven measures its temperature and switches the element to hold the setpoint — closed-loop. The deciding test is always: is the output measured and fed back?
Problem. A unity-feedback system has \(G(s) = \dfrac{10}{s+2}\), \(H(s)=1\). Find \(C/R\) and its pole.
Solution. Apply the master formula:
The open-loop pole at \(s=-2\) moves to \(s=-12\): feedback has made the system six times faster. Relocating poles like this is the heart of design.
Problem. A forward gain \(G=10{,}000\) sits in a loop with \(H=0.1\). Find the closed-loop gain, then the new gain if \(G\) falls by half to \(5000\).
Solution. With \(GH \gg 1\) the loop gain dominates the denominator:
A 50% collapse in plant gain changes the closed-loop gain by about 0.1%. When \(GH\gg1\), \(T\approx 1/H\) — the output is set by the (reliable) sensor, not the (drifting) plant. This is why feedback amplifiers are built from huge gain tamed by precise feedback.
Problem. Show how much feedback reduces sensitivity, and evaluate it for \(GH = 24\).
Solution. The sensitivity of \(T=G/(1+GH)\) to changes in \(G\) is
So a 1% change in the plant produces only a 0.04% change in the closed-loop response — feedback divides sensitivity by \(1+GH\). The open-loop system, by contrast, has sensitivity exactly 1.
Problem. For unity feedback with constant forward gain \(G=49\), find the fractional error \(E/R\).
Solution. With \(H=1\), \(E = R - C\) and \(C = \dfrac{G}{1+G}R\), so
Larger loop gain shrinks the error — a preview of the steady-state-error analysis in Chapter 9.
Problem. An autopilot samples sensors every 20 ms, controls pitch and roll together, and its equations have constant coefficients. Classify it.
Solution. Sampling ⇒ discrete/digital; two outputs driven together ⇒ MIMO; constant coefficients ⇒ time-invariant; tracking commanded angles ⇒ a servomechanism. It is therefore best analysed with the digital (Part 7) and state-space (Part 6) tools.
Chapter Summary
Every control system has a plant, a controller, an actuator, and (if closed-loop) a sensor.
No measurement of the output: \(C = GR\). Simple and stable, but blind to disturbances and drift.
Output measured and subtracted: \(E = R - HC\). Negative feedback corrects error automatically.
\( \dfrac{C}{R} = \dfrac{G}{1+GH}\); the loop gain \(GH\) governs speed, accuracy, and sensitivity.
It cuts sensitivity and disturbance effect by \(1/(1+GH)\) — at the cost of complexity and a stability requirement.
Linear/nonlinear, time-invariant/varying, continuous/discrete, SISO/MIMO, servo/regulator.
Problems
For each item, first decide whether feedback is present and write the block diagram, then apply \(C/R = G/(1+GH)\) where needed. Difficulty rises down the list.
- Classify as open- or closed-loop: (a) a timed washing machine, (b) a thermostat-controlled oven, (c) a fixed-cycle traffic signal, (d) car cruise control.
- Identify the plant, controller, actuator, and sensor in a domestic water-heater (geyser) that holds a set temperature.
- For \(G(s)=\dfrac{5}{s+1}\) with unity feedback, find the closed-loop transfer function and its pole.
- A forward path has gain \(G=200\) with unity feedback. Find the closed-loop gain, and the percentage change if \(G\) drops to \(150\).
- Derive \(C/R\) for a positive-feedback loop and explain what happens as \(GH \to 1\).
- Write the error \(E\) in terms of \(R\), \(C\), and \(H\) for a non-unity feedback loop, then express \(E/R\) using \(G\) and \(H\).
- Compute the sensitivity \(S^{T}_{G}\) for \(GH = 24\) and state, in words, what it means.
- For unity feedback with \(G=99\), find the fractional steady-state error \(E/R\) to a constant reference.
- A loop has \(G(s)=\dfrac{K}{s(s+4)}\) and \(H=1\). Write \(C/R\) and identify its characteristic equation.
- Two blocks \(G_1\) and \(G_2\) are in series in the forward path, with feedback \(H\) around the pair. Find \(C/R\).
- List two advantages and two disadvantages of open-loop control, giving a real device for each.
- A satellite attitude controller fires thrusters based on samples taken every 0.1 s, regulates three axes at once, and its torque is proportional to a clipped (saturating) command. Classify it on every axis — linear/nonlinear, continuous/discrete, SISO/MIMO, servo/regulator — and justify each call.