Control Systems — Transfer Function and Block Diagrams

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

GATE Weightage: ~6–10 marks/year (Electrical/Instrumentation branches); block diagram reduction and Mason’s gain formula are perennial favorites.

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Transfer Function G(s) = Laplace of output / Laplace of input (with zero initial conditions)

• Standard form: G(s) = K·(s+z₁)(s+z₂)… / (s+p₁)(s+p₂)…
• Poles = roots of denominator; Zeros = roots of numerator
• Order = degree of denominator (highest power of s)

Block Diagram Algebra

Block in series	G₁·s G₂ = G₁G₂
Block in parallel (summing point)	G₁ ± G₂
Feedback loop	G/(1 ± GH) (negative/positive feedback)
Moving a pickoff point	Introduce/remove equal G in path
Moving a summing point	Account for new paths created

Mason’s Gain Formula:

M = (1/Δ) · Σ P_k · Δ_k

Where: P_k = kth forward path gain, Δ = 1 – Σ(L₁) + Σ(L₂) – Σ(L₃) + … (determinant), Δ_k = Δ with loops touching P_k removed.

⚡ GATE Tips

• Always reduce the inner loops first before simplifying the outer structure
• For positive feedback, sign in denominator is (+GH); for negative feedback it’s (1 – GH) or (1 + GH) depending on convention
• Mason’s formula is faster than block reduction for complex diagrams — learn it well

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Transfer Function — Definition and Properties

The transfer function G(s) of a linear time-invariant (LTI) system is defined as:

G(s) = Y(s) / R(s) with all initial conditions = 0

Where Y(s) = Laplace transform of output y(t), R(s) = Laplace transform of input r(t).

Properties of Transfer Functions

1. System order = degree of denominator polynomial (highest power of s)
2. Proper transfer function: degree of numerator ≤ degree of denominator
3. Strictly proper: degree of numerator < degree of denominator
4. Poles = values of s that make G(s) → ∞ (denominator roots)
5. Zeros = values of s that make G(s) → 0 (numerator roots)
6. dc gain (static gain) = G(s) evaluated at s = 0

Standard Transfer Function Forms

First-order system: G(s) = K / (τs + 1) = ω_n / (s + ω_n) for normalized form

Second-order system (standard form):

G(s) = ω_n² / (s² + 2ζω_n s + ω_n²)

Where ζ = damping ratio, ω_n = natural frequency.

Type number = number of poles at s = 0 (integrators in forward path)

Block Diagram Reduction Techniques

Basic Building Blocks

Element	Input	Output
Summing junction	Take algebraic sum of inputs	Single output
Pickoff point (branch)	One input	Multiple outputs (same magnitude)
Forward path	Input → System → Output
Feedback path	Output → Feedback element H → Summing junction

Step-by-Step Reduction Method

1. Identify all feedback loops (closed paths)
2. Combine blocks in series — multiply transfer functions
3. Combine blocks in parallel — add/subtract at summing points
4. Reduce inner feedback loops using G/(1 ± GH)
5. Move pickoff/summing points if needed to expose series/parallel combinations
6. Repeat until single G(s) remains

Common mistake: Students forget that moving a summing point or pickoff creates new paths that must be accounted for. When moving a pickoff ahead, multiply the moved path by G; when moving a pickoff behind, multiply by 1/G.

Signal Flow Graph vs Block Diagram

Block diagrams show subsystems with arrows; signal flow graphs (SFG) show individual signals as nodes and systems as branches.

• Node = signal variable
• Branch = system (direction matters)
• Forward path = path from input to output node without passing any node twice
• Loop = closed path returning to starting node

Mason’s Gain Formula

For finding the overall transfer function from SFG directly:

T(s) = (Σ P_k Δ_k) / Δ

Where:

• P_k = product of branch gains along the kth forward path
• Δ (system determinant) = 1 – (sum of individual loop gains) + (sum of products of 2 nontouching loops) – (sum of products of 3 nontouching loops) + …
• Δ_k = cofactor of P_k; value of Δ with loops touching P_k removed

Loop Types

Loop type	Description
Individual loop	L_i = product of branches in one closed loop
Two nontouching loops	Product of 2 loops with no common node
Three nontouching loops	Product of 3 pairwise nontouching loops

Key rule: Loops are nontouching only if they share NO common node.

Feedback and Sensitivity

Open-loop: G(s) (no feedback path)

Closed-loop (negative feedback): T(s) = G/(1 + GH)

Sensitivity S_M of parameter M: S_M = (dT/T) / (dM/M) = (d ln T) / (d ln M)

For closed-loop system:

• Sensitivity to G variations: S_G = 1/(1 + GH) (reduced by feedback!)
• Sensitivity to H variations: S_H = –GH/(1 + GH) (shows H affects system more)

** Steady-state error:** e_ss = 1/(1 + K_p) for step input, etc. (covered in Topic 18)

Example Problem — Block Diagram Reduction

Reduce to find C(s)/R(s):

R →[G1]→ (+) →[G2]→ C
         ↑        |
         | [H1]  |
         +-------+

Negative feedback with H1.

Solution:

• Forward path: G1·G2
• Feedback path: H1
• Loop gain: G1·G2·H1
• Closed-loop: C/R = G1·G2 / (1 + G1·G2·H1)

Example Problem — Mason’s Gain Formula

SFG with: input X, output Y. Forward path P₁ = G1·G2·G3. One loop L₁ = G1·G2·H1 (touches P₁). No nontouching loops.

Solution:

• Δ = 1 – L₁ = 1 – G1·G2·H1
• Δ₁ = 1 (loop touches P₁, so remove it → just 1)
• T = P₁·Δ₁/Δ = G1·G2·G3 / (1 – G1·G2·H1)

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Mason’s Gain Formula — Detailed Derivation

The formula arises from solving simultaneous equations representing all node relationships. For a system with N nodes:

Y_i = Σ_j G_ij · Y_j (sum of all incoming signals)

Solving these using Cramer’s rule and Gaussian elimination yields the Mason’s formula structure.

Multiple Forward Paths

When k forward paths exist:

• List all P₁, P₂, P₃… (all distinct paths from input to output)
• Each P_k gets its own Δ_k
• Total T = (P₁Δ₁ + P₂Δ₂ + …) / Δ

Nontouching Loop Groups

Δ computation rules:

1. Start with 1
2. Subtract sum of all individual loop gains (L₁, L₂, L₃…)
3. Add sum of products of all pairs of nontouching loops
4. Subtract sum of products of all triples of nontouching loops
5. Continue alternating signs

GATE Trick: When asked “how many loops are nontouching?” — draw the SFG and physically check which loops share nodes.

Block Diagram to SFG Conversion

Converting a block diagram to SFG:

1. Replace each block arrow with a branch (gain = block transfer function)
2. Place a node at each summing junction output
3. Place a node at each pickoff point
4. Label all intermediate nodes
5. Draw branches following the original signal flow

Special Transfer Function Forms

Time Delay Systems

G(s) = e^(–τs) — pure time delay of τ seconds

• Cannot be expressed as rational polynomial
• Approximated by Padé: e^(–τs) ≈ (1 – τs/2) / (1 + τs/2) (first-order Padé)

Systems with Time Delay in Feedback

When G(s) has e^(–τs), closed-loop characteristic equation becomes transcendental: 1 + G(s)H(s)e^(–τs) = 0. Requires numerical/approximate methods.

Minimum Phase vs Non-Minimum Phase

• Minimum phase: All poles and zeros in left-half plane (LHP)
• Non-minimum phase: Has a zero in RHP or time delay
• All-pass systems have magnitude = 1 but phase shift

Controller Block Diagrams

In feedback control systems:

G(s) = G_c(s) · G_p(s) — controller × plant

H(s) = sensor/feedback dynamics

Standard form of closed-loop transfer function:

• T(s) = G_c·G_p / (1 + G_c·G_p·H) for negative feedback
• Characteristic equation: 1 + G_c·G_p·H = 0

System Type and Error Constants

Type number = number of poles at origin in forward path G(s):

Type	Position error K_p	Velocity error K_v	Acceleration error K_a
Type 0	K_p = lim G(s)	0	0
Type 1	∞	K_v = lim s·G(s)	0
Type 2	∞	∞	K_a = lim s²·G(s)

Higher type → better steady-state tracking, but harder to stabilize.

GATE Exam Strategy — Transfer Functions & Block Diagrams

Expected question types:

1. Reduce block diagram → find C/R
2. Draw SFG from given system equations, then apply Mason’s formula
3. Find transfer function C/R from complex interconnected system
4. Determine poles/zeros/gain from given G(s)
5. Find type of system and steady-state error

Common GATE mistakes:

• Confusing positive vs negative feedback sign in denominator
• Forgetting that moving summing points changes paths
• Miscounting nontouching loops in Mason’s Δ
• Writing characteristic equation with wrong sign for feedback

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