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Introduction to Control Systems

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System Fundamentals

Every control system has four core elements:

Controller — computes corrective action from error
Actuator — converts signals into physical forces
Plant — the physical process being controlled
Sensor — measures output for feedback

Open vs Closed Loop

Feature	Open	Closed
Accuracy	Lower	Higher
Disturbance	Can't correct	Self-correcting
Complexity	Simple	More complex

Analogy: The Toaster

A toaster runs blindly on a timer (open-loop) without checking toast darkness — it can't self-correct.

The Feedback Difference

Closed-loop systems use a feedback path to compare actual output against the desired reference.

Error Signal

e(t) = r(t) − y(t)

James Watt Innovation

The flyball governor used mechanical feedback to maintain constant steam engine speed.

PID Controller

PID Equation

u(t) = Kp·e(t) + Ki·∫e(t)dt + Kd·de/dt

P — proportional to current error
I — accumulated past error (eliminates offset)
D — rate of change (reduces overshoot)

Laplace Transform

Differential → Algebraic translation.

Converts differential equations into algebraic equations in the s-domain.

The Derivative Property

d/dt → × s

Transfer Function G(s)

Represents the ratio of output to input.

Definition

G(s) = Y(s) / U(s)

Series: G₁·G₂ | Parallel: G₁+G₂ | Feedback: G/(1+GH)

Mind Maps

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Mechanical Building Blocks

Element	Property	Energy Role
Mass (m)	Inertia	Stores Kinetic
Spring (k)	Stiffness	Stores Potential
Damper (c)	Friction	Dissipates Heat

Degrees of Freedom

Define independent generalized coordinates.

Identify the minimum number of independent coordinates to completely describe system motion.

System Identification

Building models from measured data.

Obtaining mathematical models by analyzing experimental input-output results.

Configuration Form

Scalar

mq̈ + cq̇ + kq = f

Matrix

[M]{q̈} + [C]{q̇} + [K]{q} = {f}

The Mathematical Model

A set of equations representing system dynamics.

Usually differential equations derived from physical laws like Newton's or Kirchhoff's.

The "Map" Analogy

Models are not the physical territory.

A model captures essential features for navigation while remaining simple enough to use.

System Representation Methods

TFTransfer Functions

Classical input-output modelling via Laplace transform ratio, assuming zero initial conditions.

G(s) = Y(s) / U(s)

Properties: Order, Type, Rank — unique to system parameters

Stability: All poles must have negative real parts

Example: 2ẍ + 12ẇ + 18x = 6f → ζ=1, ωn=3 rad/s (critically damped)

SSState-Space Representations

Modern approach for complex MIMO systems using vector-matrix form.

ẋ = Ax + Bu , y = Cx + Du

Matrices: A (state), B (input), C (output), D (feedthrough)

Stability: Eigenvalues of A → characteristic roots

Bridge: G(s) = C(sI−A)⁻¹B + D

BDBlock Diagrams

Graphical representation showing component relationships and functions.

Series: G₁G₂ | Parallel: G₁+G₂ | Feedback: G/(1±GH)

Elements: Functional blocks, summing junctions, pick-off points

Stability: From characteristic equation 1 + G(s)H(s) = 0

Example: Multi-loop reduction → overall C(s)/R(s)

SFGSignal Flow Graphs

Network of nodes and directed branches — simplified block diagram alternative.

Mason's: M = (1/Δ) Σ(Pk·Δk)

Elements: Nodes, branch gains, forward paths, loop gains

Stability: Determinant Δ and loop gains via Mason's formula

Advantage: No step-by-step reduction needed

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Transfer Function — Classical

Definition

G(s) = Output(s) / Input(s)

The ratio of Laplace-transformed output to input, assuming zero initial conditions.

Analogy: The Vending Machine

You provide an input (coin) to get a specific output (soda) without understanding the internal mechanics.

State-Space — Modern

Equations

ẋ = Ax + Bu , y = Cx + Du

First-order differential equations for complex MIMO systems using matrices.

Analogy: GPS Navigation

State variables act like coordinates — tracking the system's full internal state at any moment.

Stability (Characteristic Eq)

Setting the denominator = 0 reveals poles that determine stability.

Re(s) < 0

Stable

Re(s) > 0

Unstable

Bridge Formula

Convert State-Space to Transfer Function:

G(s) = C(sI − A)⁻¹B + D

Critical Matrices

[A]State — dynamics & stability

[B]Input — relates states to inputs

[C]Output — states to output

[D]Feedforward — direct path

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System Performance

1% Settling Time

Time required for response to remain within 1% of final value.

Steady-State Error (e_ss)

Use the Final Value Theorem to calculate error for step or ramp inputs.

Damping Ratio & Oscillation

As ζ approaches zero, the system becomes more oscillatory and less stable.

ζ ≈ 1

Overdamped

ζ = 0.5

Underdamped

ζ → 0

Oscillatory

Root Locus Workflow

Map Poles (n) and Zeros (m)

Identify roots to determine starting points of all branches.

Asymptote & Departure Angles

Calculate intersection point (σ_a) and angles (φ) to guide branches.

Critical Gain (K_cr)

Max gain before roots cross into the unstable region.

Exceeding K_cr causes instability — verify with Routh-Hurwitz.

Impact of Loop Gain (K)

K = 0ζ = 1.000 · Overdamped / Stable

K = 1ζ = 0.866 · Faster / Stable

K = 10ζ = 0.433 · Oscillatory / Unstable

Resources

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Bode Plots — Component Blueprint

Dual-Graph Frequency Mapping

Displays Magnitude (dB) and Phase (degrees) on log scales to reveal individual pole/zero effects.

The ±20 dB Rule

Each pole drops slope by −20 dB/decade. Each zero adds +20 dB/decade.

Positive Margins = Stability

GM > 0 dB AND PM > 0°

Nyquist Diagrams — Stability Map

Single Polar Perspective

Combines magnitude and phase into one continuous curve to visualize the entire system's behavior.

Critical Point (−1, j0)

Stability confirmed if the plot does not encircle the critical point clockwise.

The "Dangerous Cliff" Analogy

The critical point is a cliff edge — Gain Margin is your buffer zone before falling into instability.

Smaller margins = closer to the edge. Always design with adequate safety buffer.

Comparison

Aspect	Bode Plot	Nyquist Diagram
Format	Two graphs (Mag + Phase)	Single polar plot
Best For	Controller design	Stability verification
Log Scale	Yes	No (parametric)
Key Feature	Individual pole/zero contributions	Encirclement of critical point
Stability	GM & PM from crossovers	Count encirclements of (−1, j0)

MEC524 Control Engineering · Fakulti Kejuruteraan Mekanikal, UiTM Shah Alam · Dr. Mohd Hanif Mohd Ramli