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Controls and Systems · Chapter 3 of 10 · Intermediate

System Modeling and Block Diagrams

Real systems are tangles of springs, dampers, resistors, and capacitors. Block-diagram algebra reduces that tangle to one transfer function, and the standard second-order form gives the two numbers that matter.

Readiness check

This chapter builds and simplifies models. Tick only what you can do closed-notes.

Write Newton's second law for a mass with spring and damper.
Recall a transfer function from a differential equation.
Multiply and add rational functions.
Use the feedback formula G/(1 + GH).
Recognise a second-order denominator.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsRevisit transfer functions in Chapter 2.

3 or more weak itemsReview vibration models in Dynamics and Vibrations.

The core idea

Each physical element becomes a transfer-function block, and three algebra rules, series, parallel, and feedback, collapse any block diagram to a single transfer function.

series: G₁G₂feedback: G/(1 + GH)2nd order: ω_n²/(s² + 2ζω_ns + ω_n²)

A mass, spring, and damper, or a resistor, inductor, and capacitor, each contributes a term to a differential equation that transforms into a block. Blocks in cascade multiply, blocks in parallel add, and a block in a feedback loop becomes G/(1 + GH). Applying these rules repeatedly reduces a complicated diagram to one transfer function. Writing that result in the standard second-order form ω_n²/(s² + 2ζω_ns + ω_n²) exposes the natural frequency and damping ratio, the two numbers that set the response.

The skill works when: you reduce inner loops first, then cascade outward, and read ω_n and ζ from the standard form.

The skill breaks down when: a feedback loop is treated as a simple cascade, or signs in the loop are dropped.

The concept. A cascade G₁ feeding a minor feedback loop around G₂. Reducing the inner loop to G₂/(1 + G₂H) and then cascading with G₁ gives a single equivalent block.

The skills, taught in order

Five skills model the physical elements, build the diagram, and reduce it to a standard form.

3.1 Modeling mechanical systems

Newton's second law on a mass with a spring and damper gives mẍ + cẋ + kx = f(t), which transforms to the transfer function 1/(ms² + cs + k). Rotational systems follow the same pattern with inertia J, torsional stiffness, and rotational damping.

3.2 The electrical analogy

Electrical networks obey the same form: a resistor, inductor, and capacitor map onto a damper, mass, and spring. This force-voltage analogy means the control tools developed for one domain transfer directly to the other.

Mechanical	Electrical (force-voltage)	Role
mass m	inductance L	stores kinetic energy
damper c	resistance R	dissipates energy
spring k	1/capacitance, 1/C	stores potential energy

3.3 Block-diagram algebra

Three rules cover everything: cascaded blocks multiply (G₁G₂), parallel blocks add (G₁ + G₂), and a feedback loop becomes G/(1 + GH), with a plus sign for negative feedback. Summing junctions and pickoff points can also be moved with care.

3.4 Reducing a diagram

Work from the inside out: reduce the innermost loop first, replace it with its single block, and repeat until one block remains. Reducing inner loops before outer ones avoids the most common errors.

3.5 The standard second-order form

Writing a second-order transfer function as ω_n²/(s² + 2ζω_ns + ω_n²) reads off the natural frequency ω_n = √(k/m) and the damping ratio ζ = c/(2√(km)). These two numbers, developed in Chapter 4, determine overshoot and settling time.

Engineering connection: the reduced transfer function and its ω_n and ζ feed directly into the transient-response analysis of the next chapter.

Worked example 1: reducing a block diagram

A forward block G₁ = 5 cascades into an inner feedback loop with forward block G₂ = 1/(s + 2) and feedback H = 2. Reduce the diagram to a single transfer function.

Figure 1. The inner loop reduces to 1/(s + 4); cascading the forward gain of 5 gives the single block 5/(s + 4).

ProblemReduce the diagram in Figure 1 to one transfer function.
Given / findG₁ = 5, G₂ = 1/(s + 2), H = 2 around G₂. Find the overall T.
AssumptionsNegative feedback on the inner loop; linear blocks.
ModelReduce the inner loop to G₂/(1 + G₂H), then cascade with G₁.
Equationsinner = G₂/(1 + G₂H)T = G₁ · inner
Solveinner = [1/(s + 2)]/[1 + 2/(s + 2)] = 1/[(s + 2) + 2] = 1/(s + 4). Then T = 5 × 1/(s + 4) = 5/(s + 4).
CheckThe inner feedback moved the pole from −2 to −4, and the cascade kept the single pole. The DC gain is 5/4 = 1.25.
ConclusionReducing the inner loop first turned a two-block diagram with feedback into one clean transfer function, ready for response analysis.

Result. T = 5/(s + 4), a single pole at −4.

Worked example 2: a mechanical system in standard form

A mass-spring-damper has m = 2 kg, c = 6 N·s/m, and k = 50 N/m. Find its transfer function X/F, then read off the natural frequency and damping ratio from the standard form.

Figure 2. The three mechanical parameters set the standard-form numbers: the spring and mass fix the natural frequency, and the damper sets the damping ratio.

ProblemFind X/F and the values of ω_n and ζ for the system in Figure 2.
Given / findm = 2 kg, c = 6 N·s/m, k = 50 N/m. Find the transfer function, ω_n, and ζ.
AssumptionsLinear elements; zero initial conditions.
ModelTransfer function 1/(ms² + cs + k); match to ω_n²/(s² + 2ζω_ns + ω_n²).
EquationsX/F = 1/(2s² + 6s + 50) = 0.5/(s² + 3s + 25)ω_n = √(k/m), ζ = c/(2√(km))
SolveMatching, ω_n² = 25 so ω_n = 5 rad/s, and 2ζω_n = 3 so ζ = 3/10 = 0.3. The check ζ = 6/(2√(2·50)) = 6/20 = 0.3 agrees.
Checkζ = 0.3 is between 0 and 1, so the system is underdamped and will overshoot. Both formulas for ζ give the same value, confirming the standard-form match.
ConclusionTwo numbers, ω_n and ζ, now summarise the whole mechanical system. Chapter 4 turns them into overshoot and settling time.

Result. X/F = 0.5/(s² + 3s + 25), with ω_n = 5 rad/s and ζ = 0.3.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Reducing outer loops first	Tangled, wrong result	"Which loop is innermost?"	Reduce the innermost loop first, then work outward.
Treating feedback as cascade	Missing the 1 + GH denominator	"Is this a loop or a series path?"	A loop becomes G/(1 + GH), not a product.
Forgetting to normalise	ω_n and ζ misread	"Is the s² coefficient 1?"	Divide through so the denominator matches the standard form.
Dropping a sign in the loop	Stability flips unexpectedly	"Is the feedback positive or negative?"	Negative feedback gives 1 + GH; positive gives 1 − GH.

Practice ladder

Level 1 · Direct skill

Two blocks G₁ = 3 and G₂ = 1/(s + 1) are in cascade. Find the combined transfer function.

Show answer

Cascaded blocks multiply: G₁G₂ = 3/(s + 1).

Level 2 · Mixed concept

A mass-spring-damper has m = 1, c = 2, k = 16. Find ω_n and ζ.

Show answer

ω_n = √(16/1) = 4 rad/s. ζ = c/(2√(km)) = 2/(2√16) = 2/8 = 0.25. Underdamped.

Level 3 · Independent problem

Reduce a unity-feedback loop whose forward path is G = 10/(s(s + 3)). Find the closed-loop transfer function.

Show answer

T = G/(1 + G) = [10/(s(s+3))]/[1 + 10/(s(s+3))] = 10/(s² + 3s + 10). Then ω_n = √10 ≈ 3.16, 2ζω_n = 3 so ζ ≈ 0.47.

Level 4 · Transfer to real engineering

Model a real second-order system (a quarter-car suspension, an RLC filter) and write it in standard form, identifying ω_n and ζ.

What good work looks like

A correct transfer function, a clean standard-form match, and ω_n and ζ with an interpretation of whether the system is under-, over-, or critically damped.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I reduced the inner loop before cascading."

"Give me three diagrams; I will identify series, parallel, and feedback parts."

"Reduce this block diagram." Working inside out yourself is the skill.

"What are ω_n and ζ?" Matching the standard form is the point.

Portfolio task

Take a real multi-element system, draw its block diagram, reduce it to one transfer function, and write that result in standard second-order form where it applies.

Must include: a block diagram, a reduction to one transfer function, and ω_n and ζ if second order.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. What is the transfer function of a mass-spring-damper?

1/(ms² + cs + k), from mẍ + cẋ + kx = f(t).

2. Give the three block-diagram rules.

Series multiply, parallel add, feedback becomes G/(1 + GH).

3. In what order do you reduce a diagram?

Inside out: innermost loop first, then outward.

4. Write the standard second-order form.

ω_n²/(s² + 2ζω_ns + ω_n²).

5. How do m, c, k set ω_n and ζ?

ω_n = √(k/m) and ζ = c/(2√(km)).

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive a block reduction and a standard-form match from a blank page.

+3 daysReduce two new diagrams.

+7 daysCarry ω_n and ζ into transient response, Chapter 4.

+30 daysReuse block reduction to find closed-loop poles for stability.

Textbook mapping

Item	Mapping
Primary source	Ogata, Modern Control Engineering, Chapters 2 and 3 (Modeling and Mechanical/Electrical Systems)
Cross-reference	Nise, Ch. 2 and 5 · Electrical Circuits and Sensors
Core topics	3.1 Mechanical modeling · 3.2 Electrical analogy · 3.3 Block-diagram algebra · 3.4 Reduction · 3.5 Standard second-order form
Engineering connection	The reduced transfer function and its ω_n, ζ feed the response analysis.
Read next	Chapter 4: Transient Response Analysis.