the amplitude spectral density gives a continuous display of the amplitude density spectrum, which in this case is in the form of impulses, rather than just a number.. Similarly the Four
Trang 1the amplitude spectral density gives a continuous display of the amplitude density spectrum, which in this case is in the form of impulses, rather than just a number
Similarly the Fourier transform of a train of impulses of the form
(23.27)
is given by
(23.28)
Energy and Power Spectral Density 6
Suppose x(t) is an aperiodic signal with a Fourier transform X( f ), then its energy is given by
(23.29)
This is Parseval’s theorem and it shows that the principle of conservation of energy in the time and
frequency domains holds The amplitude spectrum X( f ) can be expressed as
TABLE 23.5 Properties of the Fourier Transform
Evenness and oddness
Time differentiation Frequency differentiation t n x
Integration
Parseval’s theorem
t
x( ) –t =x t( )
x( ) –t = x( )t X f( ) 2 0 x t( )cos(2πft)d t
∞
∫
=
X f( ) 2 x t( ) sin ( 2πft )d t
0
∞
∫
=
at
a
- X f
a
f
( )
t
t
t
x t ( )e j2 πf0t
f–f0
d n
dt n
- x t( ) (j2pf)n
X f( )
t
df n
- X f( )
x( ) τ dτ
∞ –
∞
j2 πf - X f( ) 1
2
X 0( )δ f
+
R xy( ) τ y t ( )x t( – τ) t d
∞ –
∞
∫
x t( ) 2
t d
∞ –
∞
f d
∞ –
∞
∫
p t( ) δ t nT( – )
n= ∞ –
∞
∑
=
P f( ) 1
T δ f kF( – s ), F s
k= ∞ –
∞
T
E R xx( )0 x t( )2
t d
∞ –
∞
f d
∞ –
∞
∫
X f( ) = X f ( ) X f∠ ( )
Trang 2the amplitude spectral density gives a continuous display of the amplitude density spectrum, which in this case is in the form of impulses, rather than just a number
Similarly the Fourier transform of a train of impulses of the form
(23.27)
is given by
(23.28)
Energy and Power Spectral Density 6
Suppose x(t) is an aperiodic signal with a Fourier transform X( f ), then its energy is given by
(23.29)
This is Parseval’s theorem and it shows that the principle of conservation of energy in the time and
frequency domains holds The amplitude spectrum X( f ) can be expressed as
TABLE 23.5 Properties of the Fourier Transform
Evenness and oddness
Time differentiation Frequency differentiation t n x
Integration
Parseval’s theorem
t
x( ) –t =x t( )
x( ) –t = x( )t X f( ) 2 0 x t( )cos(2πft)d t
∞
∫
=
X f( ) 2 x t( ) sin ( 2πft )d t
0
∞
∫
=
at
a
- X f
a
f
( )
t
t
t
x t ( )e j2 πf0t
f–f0
d n
dt n
- x t( ) (j2pf)n
X f( )
t
df n
- X f( )
x( ) τ dτ
∞ –
∞
j2 πf - X f( ) 1
2
X 0( )δ f
+
R xy( ) τ y t ( )x t( – τ) t d
∞ –
∞
∫
x t( ) 2
t d
∞ –
∞
f d
∞ –
∞
∫
p t( ) δ t nT( – )
n= ∞ –
∞
∑
=
P f( ) 1
T δ f kF( – s ), F s
k= ∞ –
∞
T
E R xx( )0 x t( )2
t d
∞ –
∞
f d
∞ –
∞
∫
X f( ) = X f ( ) X f∠ ( )
Trang 3
The Discrete Fourier Transform
Consider a finite length sequence that is zero outside the interval 0 ≤k≤N− 1 Evaluation of the z transform X(z) at N equally spaced points on the unit circle z= exp(iωk T) = exp[i(2π/NT)kT] for
k= 0, 1,…, N− 1 defines the discrete Fourier transform (DFT) of a signal x with a sampling period h and
N measurements:
(23.60)
Notice that the discrete Fourier transform is only defined at the discrete frequency points
(23.61)
In fact, the discrete Fourier transform adapts the Fourier transform and the z transform to the practical requirements of finite measurements Similar properties hold for the discrete Laplace transform with z = exp(sT), where s is the Laplace transform variable
The Transfer Function
Consider the following discrete-time linear system with input sequence {u k} (stimulus) and output sequence {y k} (response) The dependency of the output of a linear system is characterized by the convolution-type equation and its z transform,
(23.62)
where the sequence {v k} represents some external input of errors and disturbances and with Y(z) ={y},
U(z) ={u}, V(z) ={v} as output and inputs The weighting function h(kT) = , which is zero for negative k and for reasons of causality is sometimes called pulse response of the digital system (compare
impulse response of continuous-time systems) The pulse response and its z transform, the pulse transfer function,
(23.63)
determine the system’s response to an input U(z); see Fig 23.18 The pulse transfer function H(z) is obtained
as the ratio
(23.64)
FIGURE 23.18 Block diagram with an assumed transfer function relationship H(z) between input U(z), disturbance
V(z), intermediate X(z), and output Y(z).
{x k}k N=0−1
X k DFT x kT{ ( )} x lexp(–iw k lT)
l=0
N−1
{X k}k N=0−1
NT
- k, for k 0, 1,…, N 1–
y k h m u k −m+v k
m=0
∞
m=−∞
k
Y z( ) = H z ( )U z ( ) V z+ ( )
{h k}k∞=0
H z( ) h kT{ ( )} h k z–k
k=0
∞
∑
H z( ) X z( )
U z( )
-=
V(z) Σ H(z)
0066_Frame_C23 Page 33 Wednesday, January 9, 2002 1:55 PM
Trang 4State Space Analysis and System Properties 24.1 Models: Fundamental Concepts
24.2 State Variables: Basic Concepts
Introduction • Basic State Space Models • Signals and State Space Description
24.3 State Space Description for Continuous-Time Systems
Linearization • Linear State Space models • State Similarity Transformation • State Space and Transfer Functions
24.4 State Space Description for Discrete-Time and Sampled Data Systems
Linearization of Discrete-Time Systems • Sampled Data Systems • Linear State Space Models • State Similarity Transformation • State Space and Transfer Functions
24.5 State Space Models for Interconnected Systems
24.6 System Properties
Controllability, Reachability, and Stabilizability
• Observability, Reconstructibility, and Detectability
• Canonical Decomposition • PBH Test
24.7 State Observers
Basic Concepts • Observer Dynamics • Observers and Measurement Noise
24.8 State Feedback
Basic Concepts • Feedback Dynamics • Optimal State Feedback The Optimal Regulator
24.9 Observed State Feedback
Separation Strategy • Transfer Function Interpretation for the Single-Input Single-Output Case
24.1 Models: Fundamental Concepts
An essential connection between an engineer/scientist and a system relies on his/her ability to describe the system in a way which is useful to understand and to quantify its behavior
Any description supporting that connection is a model In system theory, models play a fundamental role, since they are needed to analyze, to synthesize, and to design systems of all imaginable sorts There is not a unique model for a given system Firstly, the need for a model may obey different purposes For instance, when dealing with an electric motor, we might be interested in the electro-mechanical energy conversion process, alternatively, we might be interested in modelling the motor either
as a thermal system, or as a mechanical system to study vibrations, the strength of the materials, and so on
Mario E Salgado
Universidad Técnica Federico Santa María
Juan I Yuz
Universidad Técnica Federico Santa María