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Introduction to Control Systems

A control system measures its own output and uses the error to correct itself. That single loop is what lets a thermostat hold a temperature and a robot hold a position despite the unexpected.

01

Readiness check

This opening chapter needs algebra and a feel for systems. Tick only what you can do closed-notes.

  • Simplify a ratio of rational expressions.
  • Identify an input, an output, and a process.
  • Read a block diagram with arrows.
  • Recall what a Laplace-domain variable s represents.
  • Describe what makes a system stable in plain words.
0 or 1 weak itemsContinue with this chapter.
2 weak itemsSkim Laplace transforms in Math: Laplace.
3 or more weak itemsReview differential equations before continuing.
02

The core idea

A closed-loop system compares its output to a reference, forms an error, and acts on that error. Feedback makes the result depend on the loop, not just the plant.

E = R − HYT = Y/R = G/(1 + GH)closed loop reshapes the poles

In open-loop control, the command drives the plant blindly, and any disturbance or modeling error shows up directly in the output. Closing the loop feeds the measured output back, subtracts it from the reference to form an error, and drives the plant with that error. The closed-loop transfer function T = G/(1 + GH) shows the payoff: feedback moves the poles, so it can make a system faster and far less sensitive to changes in the plant. That robustness is why almost every real machine is closed-loop.

The skill works when: you identify the loop, form the error, and use T = G/(1 + GH) instead of the plant alone.
The skill breaks down when: the open-loop plant G is analysed as if the loop were not closed.
The concept. The reference R enters a summing junction, the measured output H·Y is subtracted to form the error E, and G acts on it. The closed-loop response is T = G/(1 + GH).
03

The skills, taught in order

Five skills set the vocabulary of feedback, the closed-loop formula, and why feedback is worth the trouble.

1.1 Open-loop versus closed-loop

An open-loop system applies a command and hopes: a toaster runs for a set time regardless of how brown the bread is. A closed-loop system measures the output and corrects: an oven with a thermostat holds temperature whatever the load. Measurement and correction are what distinguish the two.

1.2 The parts of a control system

A loop has a reference (the desired value), a comparator that forms the error, a controller, an actuator, the plant being controlled, and a sensor in the feedback path. Naming each part keeps a messy real system tractable as a block diagram.

AspectOpen loopClosed loop
Uses the output?noyes, as feedback
Disturbance rejectionpoorgood
Sensitivity to plant changeshighlow
Can become unstable?no (if plant is stable)yes, if mis-tuned

1.3 The closed-loop transfer function

For a forward path G and a feedback path H, the closed-loop transfer function is T = G/(1 + GH). The denominator 1 + GH = 0 is the characteristic equation, whose roots are the closed-loop poles that set speed and stability.

1.4 What feedback buys

Feedback can speed up a sluggish plant by moving its poles, reject disturbances, and reduce sensitivity to plant variation. The price is the risk of instability if the loop gain is pushed too far, the tension that the rest of the course manages.

1.5 Sensitivity

The sensitivity of the closed loop to a plant change is S = 1/(1 + GH). A large loop gain GH makes S small, so a closed loop barely notices changes that would swing an open-loop output one-for-one.

Engineering connection: every later tool, transient response, Routh, root locus, and Bode, is a way to place or check the closed-loop poles hidden in 1 + GH.

04

Worked example 1: closing the loop

A plant G(s) = 10/(s + 2) is placed in unity negative feedback (H = 1). Find the closed-loop transfer function and compare the open- and closed-loop poles.

Figure 1. Closing the unity loop adds the forward gain to the pole: the open-loop pole at −2 moves to −12, six times faster.
  1. ProblemFind T(s) for the loop in Figure 1 and compare poles.
  2. Given / findG(s) = 10/(s + 2), H = 1. Find T(s) and the closed-loop pole.
  3. AssumptionsLinear, time-invariant; unity negative feedback.
  4. ModelUse T = G/(1 + GH) and simplify the rational expression.
  5. EquationsT = G/(1 + GH)
  6. SolveT = [10/(s + 2)] / [1 + 10/(s + 2)] = 10/[(s + 2) + 10] = 10/(s + 12). The open-loop pole at −2 becomes a closed-loop pole at −12.
  7. CheckThe pole moved left by exactly the forward gain (10), so the closed loop responds six times faster. The DC gain dropped from 5 to 10/12 = 0.83, the usual trade for speed.
  8. ConclusionFeedback reshaped the dynamics: a faster pole at the cost of some DC gain. The denominator 1 + GH is where that pole came from.
Result. T = 10/(s + 12); the pole moves from −2 to −12.
05

Worked example 2: feedback reduces sensitivity

For the same loop, the loop gain at low frequency is GH = 10. If the plant gain drifts by 10 percent, by how much does the closed-loop output change? Use the sensitivity S = 1/(1 + GH).

Figure 2. A plant change passes straight through an open loop, but the closed loop divides it by 1 + GH, so the same drift barely shows in the output.
  1. ProblemFind the closed-loop output change for a 10 percent plant drift in Figure 2.
  2. Given / findGH = 10, plant change 10 percent. Find S and the output change.
  3. AssumptionsSmall change, so the sensitivity is the local fractional ratio; low-frequency loop gain GH = 10.
  4. ModelThe closed-loop fractional change is S times the plant fractional change, with S = 1/(1 + GH).
  5. EquationsS = 1/(1 + GH)ΔT/T = S · ΔG/G
  6. SolveS = 1/(1 + 10) = 1/11 = 0.091. The output change is 0.091 × 10 percent = 0.9 percent.
  7. CheckAn open-loop output would shift the full 10 percent; the closed loop cuts it to under 1 percent. Larger loop gain would reduce the sensitivity further.
  8. ConclusionFeedback trades raw gain for robustness: a high loop gain makes the closed loop almost indifferent to plant variation. This is the deepest reason to close the loop.
Result. S = 1/11, so a 10 percent plant drift becomes a 0.9 percent output change.
06

Misconceptions and diagnostics

MistakeSymptomDiagnostic questionCorrection
Analysing G with the loop closedWrong poles and response"Is the loop closed here?"Use T = G/(1 + GH) once feedback is present.
Thinking feedback only helpsPushing gain until it oscillates"Could the closed loop go unstable?"Feedback can destabilise; the poles of 1 + GH must stay in the left half-plane.
Confusing reference and errorDriving the plant with R, not E"What signal actually reaches the plant?"The plant sees the error E = R − HY, not the reference.
Ignoring the sensor HLoop gain miscounted"Is there an H in the feedback path?"Include H in GH; only unity feedback has H = 1.
07

Practice ladder

Level 1 · Direct skill

A plant G = 5/(s + 1) is in unity feedback. Find the closed-loop transfer function.

Show answer

T = G/(1 + G) = [5/(s + 1)]/[1 + 5/(s + 1)] = 5/(s + 6). The pole moves from −1 to −6.

Level 2 · Mixed concept

For the worked-example loop, what loop gain GH would cut the sensitivity to 1 percent?

Show answer

S = 1/(1 + GH) = 0.01 needs 1 + GH = 100, so GH = 99. Driving sensitivity down by ten needs roughly ten times the loop gain.

Level 3 · Independent problem

A plant G = 8/(s + 4) is in feedback with a sensor H = 0.5. Find the closed-loop transfer function and pole.

Show answer

T = G/(1 + GH) = [8/(s + 4)]/[1 + 4/(s + 4)] = 8/(s + 8). The pole moves from −4 to −8; the sensor gain enters through GH = 4/(s + 4).

Level 4 · Transfer to real engineering

Pick a real device (cruise control, a 3D-printer heater, a drone altitude hold). Identify the reference, sensor, controller, and plant, and say what feedback buys it.

What good work looks like

Each block of the loop named, and a concrete benefit (disturbance rejection or insensitivity to load) tied to feedback rather than to the plant alone.

08

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I used T = G/(1 + GH) and not just the plant."
"Give me four systems; I will label each open or closed loop."
"What is the closed-loop transfer function?" Forming and simplifying it yourself is the skill.
"Does feedback help here?" Reasoning about poles and sensitivity is the point.

Portfolio task

Draw the block diagram of one real feedback system, write its closed-loop transfer function, and state one benefit feedback gives it, with the sensitivity if you can estimate the loop gain.

Must include: a labelled loop, a closed-loop transfer function, and a named benefit of feedback.
09

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. What distinguishes closed-loop from open-loop control?

Closed-loop measures the output and corrects with feedback; open-loop does not use the output.

2. Write the closed-loop transfer function.

T = G/(1 + GH), with 1 + GH = 0 the characteristic equation.

3. What signal drives the plant?

The error E = R − HY, not the reference directly.

4. State the sensitivity of the closed loop.

S = 1/(1 + GH); large loop gain makes it small.

5. What is the price of feedback?

Possible instability if the loop gain is pushed too far.

TodayFinish this quiz and Levels 1 and 2 of the ladder.
+1 dayRe-derive a closed-loop transfer function and sensitivity from a blank page.
+3 daysClassify three new systems as open or closed loop.
+7 daysCarry the loop into transfer functions, Chapter 2.
+30 daysReuse 1 + GH as the characteristic equation in stability and root locus.
10

Textbook mapping

ItemMapping
Primary sourceOgata, Modern Control Engineering, Chapter 1 (Introduction to Control Systems)
Cross-referenceNise, Ch. 1 · Dorf and Bishop, Ch. 1
Core topics1.1 Open vs closed loop · 1.2 Parts of a control system · 1.3 Closed-loop transfer function · 1.4 What feedback buys · 1.5 Sensitivity
Engineering connectionEvery later method places or checks the poles of 1 + GH.
Read nextChapter 2: Laplace Transforms and Transfer Functions.