Feedback Control Systems
feedback control - 8.1
8. FEEDBACK CONTROL SYSTEMS
Topics:
• Transfer functions, block diagrams and simplification
• Feedback controllers
• Control system design
Objectives:
• To be able to represent a control system with block diagrams.
• To be able to select controller parameters to meet design objectives.
8.1 INTRODUCTION
Every engineered component has some function. A function can be described as a transformation of inputs to outputs. For example it could be an amplifier that accepts a signal from a sensor and amplifies it. Or, consider a mechanical gear box with an input and output shaft. A manual transmission has an input shaft from the motor and from the shifter. When analyzing systems we will often use transfer functions that describe a system as a ratio of output to input.
8.2 TRANSFER FUNCTIONS
Transfer functions are used for equations with one input and one output variable.
An example of a transfer function is shown below in Figure 8.1. The general form calls for output over input on the left hand side. The right hand side is comprised of constants and the ’D’ operator. In the example ’x’ is the output, while ’F’ is the input.
The general form
An example
output
---------------- = f ( D ) input x
4+D
-- = -------------------------------2
F
D + 4D + 16
Figure 8.1
A transfer function example
feedback control - 8.2
If both sides of the example were inverted then the output would become ’F’, and the input ’x’. This ability to invert a transfer function is called reversibility. In reality many systems are not reversible.
There is a direct relationship between transfer functions and differential equations.
This is shown for the second-order differential equation in Figure 8.2. The homogeneous equation (the left hand side) ends up as the denominator of the transfer function. The nonhomogeneous solution ends up as the numerator of the expression.
f2
·
·· x + 2ζωn x + ω n x = ---M
2
2 fxD +
8. FEEDBACK CONTROL SYSTEMS
Topics:
• Transfer functions, block diagrams and simplification
• Feedback controllers
• Control system design
Objectives:
• To be able to represent a control system with block diagrams.
• To be able to select controller parameters to meet design objectives.
8.1 INTRODUCTION
Every engineered component has some function. A function can be described as a transformation of inputs to outputs. For example it could be an amplifier that accepts a signal from a sensor and amplifies it. Or, consider a mechanical gear box with an input and output shaft. A manual transmission has an input shaft from the motor and from the shifter. When analyzing systems we will often use transfer functions that describe a system as a ratio of output to input.
8.2 TRANSFER FUNCTIONS
Transfer functions are used for equations with one input and one output variable.
An example of a transfer function is shown below in Figure 8.1. The general form calls for output over input on the left hand side. The right hand side is comprised of constants and the ’D’ operator. In the example ’x’ is the output, while ’F’ is the input.
The general form
An example
output
---------------- = f ( D ) input x
4+D
-- = -------------------------------2
F
D + 4D + 16
Figure 8.1
A transfer function example
feedback control - 8.2
If both sides of the example were inverted then the output would become ’F’, and the input ’x’. This ability to invert a transfer function is called reversibility. In reality many systems are not reversible.
There is a direct relationship between transfer functions and differential equations.
This is shown for the second-order differential equation in Figure 8.2. The homogeneous equation (the left hand side) ends up as the denominator of the transfer function. The nonhomogeneous solution ends up as the numerator of the expression.
f2
·
·· x + 2ζωn x + ω n x = ---M
2
2 fxD +