PPT Mathematical Modelling of Systems - Control Systems - Electrical Engineering

 Page 1


Mathematical Modeling
of Systems
Page 2


Mathematical Modeling
of Systems
Objectives
? In this lecture, we lead you through a study of mathematical models of 
physical systems.
? After completing the chapter , you should be able to
? Describe a physical system in terms of differential equations.
? Understand the way these equations are obtained.
? Realize the use of physical laws governing a particular system such as
? N e w to n ’ s law for mechanical systems and K i r c h h o f f ’ s laws for electrical 
systems.
? Realize that deriving mathematical models is the most important part of 
the entire analysis of control systems.
2
Page 3


Mathematical Modeling
of Systems
Objectives
? In this lecture, we lead you through a study of mathematical models of 
physical systems.
? After completing the chapter , you should be able to
? Describe a physical system in terms of differential equations.
? Understand the way these equations are obtained.
? Realize the use of physical laws governing a particular system such as
? N e w to n ’ s law for mechanical systems and K i r c h h o f f ’ s laws for electrical 
systems.
? Realize that deriving mathematical models is the most important part of 
the entire analysis of control systems.
2
Mathematical Model
? Mathematical modeling of any control system is the first and foremost task that a
control engineer has to accomplish for design and analysis of any control engineering
problem.
? A mathematical model of a dynamic system is defined as a set of differential equations
that represents the dynamics of the system accurately, or at least fairly well.
? Note that a mathematical model is not unique to a given system. A system may be
represented in many different ways and, therefore, may have many mathematical
models, depending on o n e ’ s perspective. For example,
? In optimal control problems, it is good to use state-space representations.
? On the other hand, for the transient-response or frequency-response analysis of single-
input-single-output, linear, time-invariant systems, the transfer function
representation may be more convenient than any other.
3
Page 4


Mathematical Modeling
of Systems
Objectives
? In this lecture, we lead you through a study of mathematical models of 
physical systems.
? After completing the chapter , you should be able to
? Describe a physical system in terms of differential equations.
? Understand the way these equations are obtained.
? Realize the use of physical laws governing a particular system such as
? N e w to n ’ s law for mechanical systems and K i r c h h o f f ’ s laws for electrical 
systems.
? Realize that deriving mathematical models is the most important part of 
the entire analysis of control systems.
2
Mathematical Model
? Mathematical modeling of any control system is the first and foremost task that a
control engineer has to accomplish for design and analysis of any control engineering
problem.
? A mathematical model of a dynamic system is defined as a set of differential equations
that represents the dynamics of the system accurately, or at least fairly well.
? Note that a mathematical model is not unique to a given system. A system may be
represented in many different ways and, therefore, may have many mathematical
models, depending on o n e ’ s perspective. For example,
? In optimal control problems, it is good to use state-space representations.
? On the other hand, for the transient-response or frequency-response analysis of single-
input-single-output, linear, time-invariant systems, the transfer function
representation may be more convenient than any other.
3
Different Mathematical Models
? Commonly used mathematical models are
? Differential equation model (Time Domain).
? Transfer function model (S-Domain).
? State space model (Time Domain).
? Use of the models depends on the application. For example, to find the transient or
steady state response of SISO (Single Input Single Output) LTI (Linear Time Invariant)
system transfer function model is useful. On the other hand for optimal control
application state space model is useful.
4
Page 5


Mathematical Modeling
of Systems
Objectives
? In this lecture, we lead you through a study of mathematical models of 
physical systems.
? After completing the chapter , you should be able to
? Describe a physical system in terms of differential equations.
? Understand the way these equations are obtained.
? Realize the use of physical laws governing a particular system such as
? N e w to n ’ s law for mechanical systems and K i r c h h o f f ’ s laws for electrical 
systems.
? Realize that deriving mathematical models is the most important part of 
the entire analysis of control systems.
2
Mathematical Model
? Mathematical modeling of any control system is the first and foremost task that a
control engineer has to accomplish for design and analysis of any control engineering
problem.
? A mathematical model of a dynamic system is defined as a set of differential equations
that represents the dynamics of the system accurately, or at least fairly well.
? Note that a mathematical model is not unique to a given system. A system may be
represented in many different ways and, therefore, may have many mathematical
models, depending on o n e ’ s perspective. For example,
? In optimal control problems, it is good to use state-space representations.
? On the other hand, for the transient-response or frequency-response analysis of single-
input-single-output, linear, time-invariant systems, the transfer function
representation may be more convenient than any other.
3
Different Mathematical Models
? Commonly used mathematical models are
? Differential equation model (Time Domain).
? Transfer function model (S-Domain).
? State space model (Time Domain).
? Use of the models depends on the application. For example, to find the transient or
steady state response of SISO (Single Input Single Output) LTI (Linear Time Invariant)
system transfer function model is useful. On the other hand for optimal control
application state space model is useful.
4
Control systems Classifications
5
Non-Linear System OR Linear System
Time Varying System OR Time Invariant System
Single Variable Control OR Multivariable Control
Classical Representation 
(Classical Control)
OR
State Space Representation 
(Modern Control)
Manual Control System OR Automatic Control System
Open-Loop Control system OR Closed-Loop Control system