25 Response of Dynamic Systems 25.1 System and Signal Analysis Continuous Time Systems • Discrete Time Systems • Laplace and z-Transform • Transfer Function Models 25.2 Dynamic Respons
Trang 1
25
Response of Dynamic
Systems
25.1 System and Signal Analysis
Continuous Time Systems • Discrete Time Systems
• Laplace and z-Transform • Transfer Function Models
25.2 Dynamic Response
Pulse and Step Response • Sinusoid and Frequency Response
25.3 Performance Indicators for Dynamic Systems
Step Response Parameters • Frequency Domain Parameters
25.1 System and Signal Analysis
In dynamic system design and analysis it is important to predict and understand the dynamic behavior
of the system Examining the dynamic behavior can be done by using a mathematical model that describes the relevant dynamic behavior of the system in which we are interested Typically, a model is formulated
to describe either continuous or discrete time behavior of a system The corresponding equations that govern the model are used to predict and understand the dynamic behavior of the system
A rigorous analysis can be done for relatively simple models of a dynamic system by actually computing solutions to the equations of the model Usually, this analysis is limited to linear first and second order models Although limited to small order models, the solutions tend to give insight in the typical responses
of a dynamic system For more complicated, higher order and possibly nonlinear models, numerical simulation tools provide an alternative for the dynamic system analysis
In the following we review the analysis of linear models of discrete and continuous time dynamic systems The equations that describe and relate continuous and discrete time behavior are presented For the analysis of continuous time systems extensive use is made of the Laplace transform that converts
discrete time systems
Continuous Time Systems
Models that describe the linear continuous time dynamical behavior of a system are usually given in the
equation of a time invariant linear continuous time model has the general format
(25.1)
a j d
j
dt j
- y t( )
j=0
n a
∑ b j d
k
dt j
- u t( )
j=0
n b
∑
=
Raymond de Callafon
University of California
0066_Frame_C25 Page 1 Wednesday, January 9, 2002 7:05 PM
Trang 2
25
Response of Dynamic
Systems
25.1 System and Signal Analysis
Continuous Time Systems • Discrete Time Systems
• Laplace and z-Transform • Transfer Function Models
25.2 Dynamic Response
Pulse and Step Response • Sinusoid and Frequency Response
25.3 Performance Indicators for Dynamic Systems
Step Response Parameters • Frequency Domain Parameters
25.1 System and Signal Analysis
In dynamic system design and analysis it is important to predict and understand the dynamic behavior
of the system Examining the dynamic behavior can be done by using a mathematical model that describes the relevant dynamic behavior of the system in which we are interested Typically, a model is formulated
to describe either continuous or discrete time behavior of a system The corresponding equations that govern the model are used to predict and understand the dynamic behavior of the system
A rigorous analysis can be done for relatively simple models of a dynamic system by actually computing solutions to the equations of the model Usually, this analysis is limited to linear first and second order models Although limited to small order models, the solutions tend to give insight in the typical responses
of a dynamic system For more complicated, higher order and possibly nonlinear models, numerical simulation tools provide an alternative for the dynamic system analysis
In the following we review the analysis of linear models of discrete and continuous time dynamic systems The equations that describe and relate continuous and discrete time behavior are presented For the analysis of continuous time systems extensive use is made of the Laplace transform that converts
discrete time systems
Continuous Time Systems
Models that describe the linear continuous time dynamical behavior of a system are usually given in the
equation of a time invariant linear continuous time model has the general format
(25.1)
a j d
j
dt j
- y t( )
j=0
n a
∑ b j d
k
dt j
- u t( )
j=0
n b
∑
=
Raymond de Callafon
University of California
0066_Frame_C25 Page 1 Wednesday, January 9, 2002 7:05 PM
Trang 3The Root Locus Method
26.1 Introduction
26.2 Desired Pole Locations
26.3 Root Locus Construction
Root Locus Rules • Root Locus Construction
• Design Examples
26.4 Complementary Root Locus
26.5 Root Locus for Systems with Time Delays
Stability of Delay Systems • Dominant Roots of a Quasi-Polynomial • Root Locus Using Padé Approximations
26.6 Notes and References
26.1 Introduction
The root locus technique is a graphical tool used in feedback control system analysis and design It has been formally introduced to the engineering community by W R Evans [3,4], who received the Richard
E Bellman Control Heritage Award from the American Automatic Control Council in 1988 for this major contribution
In order to discuss the root locus method, we must first review the basic definition of bounded input
dynamics
The feedback system is said to be stable if none of the closed-loop transfer functions, from external inputs
checking feedback system stability becomes equivalent to checking whether all the roots of
(26.1) are in the open left half plane, The roots of (26.1) are the closed-loop system poles We would like to understand how the closed-loop system pole locations vary as functions of a real
1
Here we consider the continuous time case; there is essentially no difference between the continuous time case and the discrete time case, as far as the root locus construction is concerned In the discrete time case the desired closed-loop pole locations are defined relative to the unit circle, whereas in the continuous time case desired pole locations are defined relative to the imaginary axis.
+ := s { Œ : Re s( ) ≥0}
1 G s+ ( ) =0
- := s : Re(s){ ∈ <0} Hitay Özbay
The Ohio State University