There are many different forms of mathematical models used in control systems. Out of them, transfer function models and state space representation are two main forms. Depending on the particular system and the particular circumstances, one mathematical model may be better suited than the other models. In this course, we are going to concentrate on transfer function methods only.
But, in both methods, the first step on developing a mathematical model is to apply the fundamental physical laws of science and engineering. By doing this we can obtain a set of differential equations which describes the input output relationship with the system variables. If these equations can be linearized then the Laplace Transform can be used to simplify the method of solution.
Although the set of differential equations relates the system to its input and output, it is not a satisfying representation from a systems perspective. We will then proceed to the input output relationship for systems in the form of transfer functions. We use methods of block diagrams and signal flow graphs to depict transfer functions graphically.
Steps used in Modelling
When obtaining a mathematical model for a system we can use the following steps.
1. Understand the physical system and its components.
2. Make appropriate simplifying assumptions.
3. Use basic principles to formulate the mathematical model.
4. Write differential equations and algebraic equations describing the model.
5. Check the model for validity.
Differential Equations of linear systems
But, in both methods, the first step on developing a mathematical model is to apply the fundamental physical laws of science and engineering. By doing this we can obtain a set of differential equations which describes the input output relationship with the system variables. If these equations can be linearized then the Laplace Transform can be used to simplify the method of solution.
Although the set of differential equations relates the system to its input and output, it is not a satisfying representation from a systems perspective. We will then proceed to the input output relationship for systems in the form of transfer functions. We use methods of block diagrams and signal flow graphs to depict transfer functions graphically.
Steps used in Modelling
When obtaining a mathematical model for a system we can use the following steps.
1. Understand the physical system and its components.
2. Make appropriate simplifying assumptions.
3. Use basic principles to formulate the mathematical model.
4. Write differential equations and algebraic equations describing the model.
5. Check the model for validity.
Differential Equations of linear systems
We can obtain the differential equations describing the dynamic performance of a physical system by using physical laws of the process. We can apply this approach equally well to various types of processes such as electrical, mechanical, fluid and thermodynamic systems. In the next session we show how to write the differential equations for electrical and mechanical systems.
To get the solution of differential equations describing the process, we can use the classical methods such as the use of integrating factors and the method of undetermined coefficients. if the system is linear we can easily use Laplace transformations to obtain the solution. A majority of physical systems are linear within some range of variables. Unlike most of the electrical and mechanical elements thermal and fluid elements are more frequently non linear. However these elements can be linearized by assuming small signal conditions.
As a summary, we can get the time response solution by the following operations:
1. Obtain the differential equations.
2. Obtain the Laplace Transformation of the differential equations.
3. Solve the resulting algebraic transform of the variable of interest.
4. Use Inverse Laplace Transformation for getting the time domain solution.
To get the solution of differential equations describing the process, we can use the classical methods such as the use of integrating factors and the method of undetermined coefficients. if the system is linear we can easily use Laplace transformations to obtain the solution. A majority of physical systems are linear within some range of variables. Unlike most of the electrical and mechanical elements thermal and fluid elements are more frequently non linear. However these elements can be linearized by assuming small signal conditions.
As a summary, we can get the time response solution by the following operations:
1. Obtain the differential equations.
2. Obtain the Laplace Transformation of the differential equations.
3. Solve the resulting algebraic transform of the variable of interest.
4. Use Inverse Laplace Transformation for getting the time domain solution.