Sem.
2
Introduction and Mathematic Preliminaries 2015/2016
Definitions
Control is a process of making system variables to conform to desired values such that the
desired performance characteristics are realized. Variables may be speed, temperature, etc.
Plant: The system to be controlled is referred to as plant.
Sensor: Is that which measures the quantity to be controlled.
Actuator: Is that which receives control signal to affect the plant.
Disturbance: Is unknown signal influencing the plant.
Controller: Is that which processes sensor information to drive the actuator
Control Goals: Design a controller Gs(s) in order to achieve the desired performance
characteristic of plant Gp(s).
Stabilization: Stabilize unstable plant
Regulation: Plant to operate about some desired operation point
Tracking: To follow a given commanded signal
Disturbance rejection: Rejection of system disturbances and/or measurement
noises
Dr.-Ing. Jackson G. Njiri EMT 2339 L-1: Introduction 0
Sem. 2
Introduction and Mathematic Preliminaries 2015/2016
Open-loop control: Do not compensate for disturbances and is simplify commanded by input
signal. No mechanism of checking the accuracy of output or controlled variable.
Closed-loop control: Normally referred to as feedback control. Higher accuracy compared to
open-loop, less sensitive to measurement noise and disturbances. Steady state error and
transient response can be easily controlled .
Dr.-Ing. Jackson G. Njiri EMT 2339 L-1: Introduction 1
Sem. 2
Introduction and Mathematic Preliminaries 2015/2016
Maths Review and Laplace Transformation
Euler’s identity
Polar representation
Magnitude:
Phase:
Matrices
The inverse of a matrix A is given by
Dr.-Ing. Jackson G. Njiri EMT 2339 L-1: Introduction 2
Sem. 2
Introduction and Mathematic Preliminaries 2015/2016
Laplace transformation
Definition: The Laplace transformation of a function is given by
where is a complex variable , normally called Laplace operator
Inverse Laplace transformation
Dr.-Ing. Jackson G. Njiri EMT 2339 L-1: Introduction 3
Sem. 2
Introduction and Mathematic Preliminaries 2015/2016
Properties of Laplace transformation
Derivatives: The Laplace transformation is given by
where are initial value
Constant multiplication
Real shift theorem
Initial value theorem
Task: Derive the Laplace transform
Final value theorem for the following functions
1. Impulse function
2. Unit step function
3. Parabola function
4. Ramp function
5. Sinusoidal functions
Dr.-Ing. Jackson G. Njiri EMT 2339 L-1: Introduction 4
Sem. 2
Introduction and Mathematic Preliminaries 2015/2016
Table of Common Laplace transforms
Dr.-Ing. Jackson G. Njiri EMT 2339 L-1: Introduction 5
Sem. 2
Introduction and Mathematic Preliminaries 2015/2016
Partial fraction expansion
An important tool in solving inverse Laplace transformation problem
Case 1: Real and distinct roots
where
Constants are given by
Then inverse Laplace transformation is expressed as
Case 2: Real and repeated roots
Dr.-Ing. Jackson G. Njiri EMT 2339 L-1: Introduction 6
Sem. 2
Introduction and Mathematic Preliminaries 2015/2016
First, find the function
Then, the general expression for multiple roots is given by
Case 3: Complex conjugate roots
Completing the square gives
Task: Do practice on inverse Laplace
transformation using partial fraction
technique
Dr.-Ing. Jackson G. Njiri EMT 2339 L-1: Introduction 7
