Transfer Functions
ECM2105 - Control Engineering
Dr Mustafa M Aziz (2010)
________________________________________________________________________________
TRANSFER FUNCTIONS AND BLOCK DIAGRAMS
1. Introduction
2. Transfer Function of Linear Time-Invariant (LTI) Systems
3. Block Diagrams
4. Multiple Inputs
5. Transfer Functions with MATLAB
6. Time Response Analysis with MATLAB
1. Introduction
An important step in the analysis and design of control systems is the mathematical modelling of the controlled process. There are a number of mathematical representations to describe a controlled process: Differential equations: You have learned this before.
Transfer function: It is defined as the ratio of the Laplace transform of the output variable to the
Laplace transform of the input variable, with all zero initial conditions.
Block diagram: It is used to represent all types of systems. It can be used, together with transfer functions, to describe the cause and effect relationships throughout the system.
State-space-representation: You will study this in an advanced Control Systems Design course.
1.1. Linear Time-Variant and Linear Time-Invariant Systems
Definition 1: A time-variable differential equation is a differential equation with one or more of its coefficients are functions of time, t. For example, the differential equation: d 2 y( t ) t2 + y( t ) = u ( t ) dt 2
(where u and y are dependent variables) is time-variable since the term t2d2y/dt2 depends explicitly on t through the coefficient t2.
An example of a time-varying system is a spacecraft system which the mass of spacecraft changes during flight due to fuel consumption.
Definition 2: A time-invariant differential equation is a differential equation in which none of its coefficients depend on the independent time variable, t. For example, the differential equation: d 2 y( t ) dy( t ) m +b
+ y( t ) = u ( t )
2
dt dt where the coefficients m and b are constants, is time-invariant
Dr Mustafa M Aziz (2010)
________________________________________________________________________________
TRANSFER FUNCTIONS AND BLOCK DIAGRAMS
1. Introduction
2. Transfer Function of Linear Time-Invariant (LTI) Systems
3. Block Diagrams
4. Multiple Inputs
5. Transfer Functions with MATLAB
6. Time Response Analysis with MATLAB
1. Introduction
An important step in the analysis and design of control systems is the mathematical modelling of the controlled process. There are a number of mathematical representations to describe a controlled process: Differential equations: You have learned this before.
Transfer function: It is defined as the ratio of the Laplace transform of the output variable to the
Laplace transform of the input variable, with all zero initial conditions.
Block diagram: It is used to represent all types of systems. It can be used, together with transfer functions, to describe the cause and effect relationships throughout the system.
State-space-representation: You will study this in an advanced Control Systems Design course.
1.1. Linear Time-Variant and Linear Time-Invariant Systems
Definition 1: A time-variable differential equation is a differential equation with one or more of its coefficients are functions of time, t. For example, the differential equation: d 2 y( t ) t2 + y( t ) = u ( t ) dt 2
(where u and y are dependent variables) is time-variable since the term t2d2y/dt2 depends explicitly on t through the coefficient t2.
An example of a time-varying system is a spacecraft system which the mass of spacecraft changes during flight due to fuel consumption.
Definition 2: A time-invariant differential equation is a differential equation in which none of its coefficients depend on the independent time variable, t. For example, the differential equation: d 2 y( t ) dy( t ) m +b
+ y( t ) = u ( t )
2
dt dt where the coefficients m and b are constants, is time-invariant