data was then found with the MATLAB software.13 The transfer-function between the applied inputvoltage Vxt and the output of the inductive sensor yt was found to be 23.134 with units of
Trang 1data was then found with the MATLAB software.13 The transfer-function between the applied input
voltage Vx(t) and the output of the inductive sensor y(t) was found to be
(23.134)
with units of V/V Equation (23.135) was scaled by the inductive sensor gain (30 Å/V) and the transfer-function between the applied voltage Vx(t) and the actual displacement of the piezo-tube xp(t) is given by
(23.135) with units of Å/V
Time Scaling of a Transfer-Function Model
We present below an approach for rescaling time for G2(s) from seconds [s] to milliseconds [ms] We
briefly recall the time scaling property of the Laplace transform presented in [1, Chapter 3, section 1.4]
Let F(s) be the Laplace transform of f(t), i.e.,
(23.136)
where L denotes the Laplace transform operator Now, consider a new time scale defined as ,
where a is a constant The Laplace transform of (at) is given by
(23.137)
Using relation (23.137), we can reduce the coefficients of G2(s) by changing the time units of both the input signal u(t) and output signal y(t) as follows:
(23.138)
Therefore, to rescale time for G2(s) from seconds [s] to millisecond [ms], we choose
and the new rescaled transfer (s) becomes
(23.139)
13
The MATLAB command invfreqs gives real numerator and denominator coefficients of experimentally determined frequency response data.
G1( )s Y s( )
V x( )s
-=
5.544×105s4–7.528×109s3+1.476×1015s2–4.571×1018s+9.415×1023
s6+1.255×104s5+1.632×109s4+1.855×1013s3+6.5×1017s2+6.25×1021s+1.378×1025
-=
G2( )s X p( )s
V x( )s
-=
1.663×107s4–2.258×1011s3+4.427×1016s2–1.371×1020s+2.825×1025
s6+1.255×104s5+1.632×109s4+1.855×1013s3+6.5×1017s2+6.25×1021s+1.378×1025
-=
L
f t( )→F s( )
tˆ = at f(tˆ) = f
L
f tˆ( ) f at( ) 1
a
-F s
a
G ˆ s( ) Yˆ s( )
Uˆ s( )
- Y s( /a )/ a
U s( /a )/ a
- Y s( /a)
U s( /a)
-=
a
tˆ = at = 0.001t
Gˆ2
Gˆ2( )s G2 s
a
a=0.001
=
G2(1000s)
=
Gˆ2( )s 16.63s4–225.8s3+4.427×104s2–1.371×105s+2.825×107
s6+12.55s5+1.632×103s4+1.855×104s3+6.5×105s2+6.25×106s+1.378×107
-=
Trang 2data was then found with the MATLAB software.13 The transfer-function between the applied input
voltage Vx(t) and the output of the inductive sensor y(t) was found to be
(23.134)
with units of V/V Equation (23.135) was scaled by the inductive sensor gain (30 Å/V) and the transfer-function between the applied voltage Vx(t) and the actual displacement of the piezo-tube xp(t) is given by
(23.135) with units of Å/V
Time Scaling of a Transfer-Function Model
We present below an approach for rescaling time for G2(s) from seconds [s] to milliseconds [ms] We
briefly recall the time scaling property of the Laplace transform presented in [1, Chapter 3, section 1.4]
Let F(s) be the Laplace transform of f(t), i.e.,
(23.136)
where L denotes the Laplace transform operator Now, consider a new time scale defined as ,
where a is a constant The Laplace transform of (at) is given by
(23.137)
Using relation (23.137), we can reduce the coefficients of G2(s) by changing the time units of both the input signal u(t) and output signal y(t) as follows:
(23.138)
Therefore, to rescale time for G2(s) from seconds [s] to millisecond [ms], we choose
and the new rescaled transfer (s) becomes
(23.139)
13
The MATLAB command invfreqs gives real numerator and denominator coefficients of experimentally determined frequency response data.
G1( )s Y s( )
V x( )s
-=
5.544×105s4–7.528×109s3+1.476×1015s2–4.571×1018s+9.415×1023
s6+1.255×104s5+1.632×109s4+1.855×1013s3+6.5×1017s2+6.25×1021s+1.378×1025
-=
G2( )s X p( )s
V x( )s
-=
1.663×107s4–2.258×1011s3+4.427×1016s2–1.371×1020s+2.825×1025
s6+1.255×104s5+1.632×109s4+1.855×1013s3+6.5×1017s2+6.25×1021s+1.378×1025
-=
L
f t( )→F s( )
tˆ = at f(tˆ) = f
L
f tˆ( ) f at( ) 1
a
-F s
a
G ˆ s( ) Yˆ s( )
Uˆ s( )
- Y s( /a )/ a
U s( /a )/ a
- Y s( /a)
U s( /a)
-=
a
tˆ = at = 0.001t
Gˆ2
Gˆ2( )s G2 s
a
a=0.001
=
G2(1000s)
=
Gˆ2( )s 16.63s4–225.8s3+4.427×104s2–1.371×105s+2.825×107
s6+12.55s5+1.632×103s4+1.855×104s3+6.5×105s2+6.25×106s+1.378×107
-=
Trang 324 State 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