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Chapter 3:
Analysis of closed-loop
systems
Control Automático
3º Curso. Ing. IndustrialEscuela Técnica Superior de Ingenieros
Universidad de Sevilla
Control of SISO systems
Control around an operation point (u0,y0)
Process
+u0
u(t)
-
y(t)∆u(t)
Controllerr(t) e(t)
(Reference is not a deviation
variable )
Automatic control
Controllere(t)
∆u(t)
Controller
How to chose the value of the controller parameters (Kp and Ti) in a way such that the
closed-loop system has an appropriate performance?
PI
Control of SISO systems
Heuristic design Tuning based on experiments
Real system Model
Design based on tables Tuning based on a set of experiments and a table that determines the
value of each parameter Zieger-Nichols (Chapter 6)
Mathematical design The tuning is based on a mathematical analisys of the closed-loop system
and provides guaranteed properties Transient response Steady state response Robustness
Analitic design techniques
Root–locus design techniques
Loop-shaping design techniques
Control of SISO systems
Process
+u0
u(t)
-
y(t)∆u(t)
Controllerr(t)e(t)
Closed-loop system
It is hard to analize the dynamics of the closed-loop system if the process or
the controller are nonlinear systems
The design techniques studied in this course are based on the linearized model
of the system around a given operation point (u0,y0)
Incremental variables model
The linearized model can be compared with the incremental variables model for a given operating point which is defined as follows:
System+ -
Incremental variables model (u0,y0)
Remark: The model
depends on the operating
point
Assumption: The initial state
is the operation point
Incremental variables model
Linearized model around (u0,y0)
Linearized model
-
∆u(t)Controller
e(t)
Analisys of the closed-loop system. In this case, both the controller and the system are
LTI systems.
Assumption: Zero initial conditions.
Teoría de sistemas
Assumption: The properties of this (simplified) system are similar to the ones of the real
closed-loop system if the trajectories are close to the operating point
- Speed of the transient
- Tracking of time-varying references
- Disturbance rejection
Teoría de sistemas
TEMA 1. Introducción y fundamentos. Sistemas dinámicos. Conceptos básicos. Ecuaciones y evolución temporal. Linealidad en los
sistemas dinámicos. TEMA 2. Representación de sistemas. Clasificación de los sistemas. Clasificación de comportamientos. Señales de prueba.
Descripción externa e interna. Ecuaciones diferenciales y en diferencias. Simulación. TEMA 3. Sistemas dinámicos lineales en tiempo continuo. Transformación de Laplace. Descripción externa de los sistemas dinámicos. Función de
transferencia. Respuesta impulsional. Descripción interna de los sistemas dinámicos. TEMA 4. Modelado y simulación. Modelado de sistemas. Modelado de sistemas mecánicos. Modelado de sistemas hidráulicos.
Modelado de sistemas eléctricos. Modelado de sistemas térmicos. Linealización de modelos no lineales. Modelos lineales. Álgebra de bloques. Simulación.
TEMA 5. Respuesta temporal de sistemas lineales. Sistemas dinámicos lineales de primer orden. Ejemplos. Sistemas dinámicos lineales de
segundo orden. Respuesta ante escalón. Sistemas de orden n. TEMA 6. Respuesta frecuencial de sistemas lineales. Función de transferencia en el dominio de la frecuencia. Transformación de Fourier.
Representación gráfica de la función de transferencia. Diagramas más comunes. Diagrama de Bode.
TEMA 7. Estabilidad. Estabilidad de sistemas lineales. Criterios relativos a la descripción externa de los sistemas
dinámicos. Criterio de Routh-Hurwitz. Criterio de Nyquist. Criterios relativos a la descripción interna.
Analysis of linear time invariant (LTI) systems
Closed-loop transfer function
Laplace transform (assuming zero initial conditions)
Remark: From now on, the variables “y” and “u” denote incremental values; that is,
the deviation of the input and the output from the operating point
Linear time invariant systems (LTI)
Properties used: Linearity, transform of the time derivative
Transfer function
Proportional term
Controllere(t)
u(t)
The value of the input u(t) is proportional to the error
Transfer function
C(s)E(s)
U(s)
Time domain
Frequency domain
Desing parameter: Kp
Integral term
Controllere(t)
u(t)
The value of the input u(t) is proportional to the error and its integral
Transfer function
C(s)E(s)
U(s)
Temporal domain
Frequency domain
Desing parameter: Kp, Ti
Lag compensation net
Controller with properties
similar to the PI
Derivative term
Controllere(t)
u(t)
Transfer function
C(s)E(s)
U(s)
Time domain
Frequency domain
Design parameters: Kp, Td
Lead compensator net
The value of the input u(t) is proportional to the error and its time derivative
Controller with properties
similar to the PD
PID controller
Controllere(t) u(t)
Widely used in industry
The value of the input u(t) is proportional to the error, its time derivative and its integral
PID controller
Controllere(t)u(t)
Transfer function
C(s)E(s)U(s)
Time domain
Frequency domainDesign parameter: Kp, Td, Ti
Lead-lag compensator
The value of the input u(t) is proportional to the error, its time derivative and its integral
Controller with properties similar to the PID
Index
Closed Loop Transfer Function
Tuning a Controller
Stability analysis of system
Steady-state response of a closed-loop systems
Transient response of a stable system.
Closed-loop poles and zeros vs. Controller parameters
Closed-loop transfer function
Block algebra (Ogata 3.3, Tema 3, Teoría de sistemas)
+
-
S1(s)
S2(s)
S3(s) = S1(s)-S2(s) S1(s) S2(s) = S1(s)
S3(s) = S1(s)
Signal sum Signal bifurcation
G(s)S1(s) S2(s)=G(s)S1(s)
LTI system
Closed-loop transfer function
G(s)
-Y(s)C(s)
E(s) U(s)R(s)
R(s) Y(s)Gbc(s) Gbc(s) models how the ouput of the closed-loop system reacts to changes in the references
The properties of a controller are defined based on the response of the closed-loop system
Closed-loop system
Other transfer functions
C(s) G(s)
H(s)
R(s)
Ym(s)
U(s)
Y(s)
Sensor dynamics
C(s) G(s)
H(s)
R(s)
Ym(s)
U(s)
Y(s)
Gd(s)
D(s)Disturbances
-
+
+
+
+
-
Index
Closed Loop Transfer Function
Tuning a Controller
Stability analysis of system
Steady-state response of a closed-loop systems
Transient response of a stable system.
Closed-loop poles and zeros vs. Controller parameters
Controller tuning
Obtain a set of controller parameter (C(s)) in order to guarantee that the closed-loop systems satisfies a given set of conditions (specifications)
SpecificationsStabilityRaise time (step reference change)Steady state error
Specifications on the linearized model are
relevant to the behaviour of the real closed-loop system
Mathematical designTuning based on the anlisys of the closed-loop dynamics (which depend on the controller parameters)
Gbc(s) is not defined if parameters of C(s) are not fixed
Example
Closed-loop system
Closed-loop response? Depends on the value of Kp
Three poles that depend on KpStatic gain depens on Kp
Reference signal: Step change - Simulations performed in Simulink/Matlab
Example
Kp=0.1
Kp=1
Kp=10
Kp=15
Example
0 10 20 30 40 50 600
0.5
1Kp = 0.1, Td = 0, 1/Ti = 0
y(t)
0 10 20 30 40 50 600
0.05
0.1
u(t)
0 10 20 30 40 50 600
0.5
1
e(t)
0 10 20 30 40 50 600
20
40
∫ 0t e(τ
)dτ
Respuesta del sistema en BC
0 10 20 30 40 50 600
0.5
1
1.5Kp = 1, Td = 0, 1/Ti = 0
y(t)
0 10 20 30 40 50 60−0.5
0
0.5
1
u(t)
0 10 20 30 40 50 60−0.5
0
0.5
1
e(t)
0 10 20 30 40 50 600
5
10
∫ 0t e(τ
)dτ
Respuesta del sistema en BC
0 10 20 30 40 50 600
1
2Kp = 10, Td = 0, 1/Ti = 0
y(t)
0 10 20 30 40 50 60−10
0
10
u(t)
0 10 20 30 40 50 60−1
0
1
e(t)
0 10 20 30 40 50 60−0.5
0
0.5
1
∫ 0t e(τ
)dτ
Respuesta del sistema en BC
0 10 20 30 40 50 60−10
0
10
20Kp = 15, Td = 0, 1/Ti = 0
y(t)
0 10 20 30 40 50 60−200
0
200
u(t)
0 10 20 30 40 50 60−20
−10
0
10e(
t)
0 10 20 30 40 50 60−5
0
5
10
∫ 0t e(τ
)dτ
Respuesta del sistema en BC
Example
0 10 20 30 40 50 600
0.5
1Kp = 0.1, Td = 0, 1/Ti = 0
y(t)
0 10 20 30 40 50 600
0.05
0.1
u(t)
0 10 20 30 40 50 600
0.5
1
e(t)
0 10 20 30 40 50 600
20
40
∫ 0t e(τ
)dτ
Respuesta del sistema en BC
Example
0 10 20 30 40 50 600
0.5
1
1.5Kp = 1, Td = 0, 1/Ti = 0
y(t)
0 10 20 30 40 50 60−0.5
0
0.5
1
u(t)
0 10 20 30 40 50 60−0.5
0
0.5
1
e(t)
0 10 20 30 40 50 600
5
10
∫ 0t e(τ
)dτ
Respuesta del sistema en BC
Example
0 10 20 30 40 50 600
1
2Kp = 10, Td = 0, 1/Ti = 0
y(t)
0 10 20 30 40 50 60−10
0
10
u(t)
0 10 20 30 40 50 60−1
0
1
e(t)
0 10 20 30 40 50 60−0.5
0
0.5
1
∫ 0t e(τ
)dτ
Respuesta del sistema en BC
Example
0 10 20 30 40 50 60−10
0
10
20Kp = 15, Td = 0, 1/Ti = 0
y(t)
0 10 20 30 40 50 60−200
0
200
u(t)
0 10 20 30 40 50 60−20
−10
0
10
e(t)
0 10 20 30 40 50 60−5
0
5
10
∫ 0t e(τ
)dτ
Respuesta del sistema en BC
Types of behaviour
Unit step response Clasification of the output signal ∆y(t) depending on the input signal
Unit Step (the most widely used). Ramp. Sinusoidal.
Provides information about the dynamic proterties of the system
Model expressed in error variables (u0,y0)
Assumption: Initial conditions in the operating point.
Unit Step: Behaviours:• Overdamped• Underdamped• Unstable• Oscilatory
Types of behaviour
Overdamped
0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Step Response
Time (sec)
Am
plitu
de
Delay LGain KRaise Time ts
Lts
K
Raise Time: Time to reach 63% of the steady state value.
Delay: Time of reaction for the output with respect to a change in the input.
Gain: Quotient of the output and the input values.
Types of behaviour
Underdamped
K
tetpts
Delay LGain KRaise Time tsPeak Time tpSettling Time teOvershoot Mp
Raise Time : Time to reach the steady state value for the first time.Peak Time : Time to reach the maximum value.Settling Time : Time to confine the output within a band of 5% arounf the steady state value.Overshoot : Percentage Increment of the peak value with respect to the steady state value..
Mp
Types of behaviour
Unstable
Index
Closed Loop Transfer Function
Tuning a Controller
Stability analysis of system
Steady-state response of a closed-loop systems
Transient response of a stable system.
Closed-loop poles and zeros vs. Controller parameters
Stability (Chapter 7. Stability)
Stability Criterion:Gbc(s) is stable if and only if all poles are located on the left-half complex plane.
Closed loop poles are the roots of (depend on C(s))
A closed-loop system might become unstable if the controller is not properly designed.
The controller design must guarantee closed loop stability
Example: Kp=15
Poles: -5.65, 0.0500 + 1.8272i, 0.0500 - 1.8272i
Stability
Analitic Procedure (Try & Error)• Evalute closed loop poles for every combination of the controller parameters (Kp, Td, Ti) using the model of the system.
Routh-Hurwitz stability criterion
• A tool to evaluate if a polinomial has roots on the right-half complex plane. • The method prevents form computing the whole set of roots of a higher
order polinomial• It can be used to evaluate stability conditions
Nyquist Stability Criterion (to be studied in chapter 5)
Routh-Hurwitz Stability Criterion
Allows to determine if there exists a root in the right-half complex plane
Important: Note the notation
1 – If there exists a negative parameter, then the polinomial has at least one root in the right-half plane.
2 – Build the Routh-Hurwitz table. If there exists a negative component on the first column, then the polinomial has at least one root in the right-half plane.
There are special rules to deal with degenerate cases (See Chapter 7)
Example
Closed loop system
The poles are the solution of the following equation (depends on Kp)
Gains range
Example
Closed loop system
The poles are the solution of the following equation (depend on Kp y Ti)
Not vey useful for multipleparameters
Index
Closed Loop Transfer Function
Tuning a Controller
Stability analysis of systems
Steady-state response of a closed-loop system
Transient response of a stable system.
Closed-loop poles and zeros vs. Controller parameters
Steady State Response
Analysis of system response as time tends to infinity (We assume the closed loop system is stable)
Steady state error
Final Value Theorem (property of Laplace transform)
Important: Depends on R(s)Different references define different steady-state error parameters.
Error for Step input
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
Error for a constant steady-state input
Position error constant
All stable systems have bounded steady-state errorsFor the error to be null (The system reaches the reference)
Error for Ramp input
Steady state error for ramp input
Velocity Error
Bounded Velocity Error ⇔ Null position Error(C(s)G(s) has at least one integrator)
For the velocity error to be null (the system reached the reference)
0 1 2 3 4 5 6 70
1
2
3
4
5
6
7
Error for parabolic input
Steady state error for parabolic input
Acceleration error constant
Bounded acceleration error ⇔ Null position error ⇔ Null velocity error (C(s)G(s) has at least two integrators)
For the parabolic error to be null (The system reaches the reference)
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Error Table
Type of a system = Number of integrators
0 1 2
Step 0 0
Ramp 0
Parabolic
ErrorType
Example
Type I System
Proportional Controller
Proportional Controller affects the Bode Gain of the system, but can not change its Type.
Improves (quantitatively) steady state behaviour.
Depends on Kp.
Example
0 10 20 30 40 50 600
0.5
1
1.5Kp = 1, Td = 0, 1/Ti = 0
y(t)
0 10 20 30 40 50 60−0.5
0
0.5
1
u(t)
0 10 20 30 40 50 60−0.5
0
0.5
1
e(t)
0 10 20 30 40 50 600
5
10
∫ 0t e(τ
)dτ
Respuesta del sistema en BC
Position error. Constant reference (Step)
Example
Velocity error. Increasing reference (ramp)
0 10 20 30 40 50 600
20
40
60Kp = 1, Td = 0, 1/Ti = 0
y(t)
0 10 20 30 40 50 600
5
10
u(t)
0 10 20 30 40 50 600
5
10
e(t)
0 10 20 30 40 50 600
100
200
300
∫ 0t e(τ
)dτ
Respuesta del sistema en BC
Example
Type I system
PI Controller
P controllers affect the Bode gain of the system and increases the system type
Improves (qualitatively) steady state behaviour
Depends on Kp and Ti
(Lag controllers allow to increase Bode gain)
Example
Position error. Constant reference (Step)
0 10 20 30 40 50 600
0.5
1
1.5Kp = 1, Td = 0, 1/Ti = 0.1
y(t)
0 10 20 30 40 50 60−1
0
1
2
u(t)
0 10 20 30 40 50 60−0.5
0
0.5
1
e(t)
0 10 20 30 40 50 600
1
2
3
∫ 0t e(τ
)dτ
Respuesta del sistema en BC
Example
Velocity error. Increasing reference (ramp)
0 10 20 30 40 50 600
20
40
60Kp = 1, Td = 0, 1/Ti = 0.1
y(t)
0 10 20 30 40 50 600
5
10
u(t)
0 10 20 30 40 50 600
1
2
3
e(t)
0 10 20 30 40 50 600
50
100
∫ 0t e(τ
)dτ
Respuesta del sistema en BC
The integral term is introduced to improve steady state response.
(It might unstabilize the system.Ex. Try simulation with Kp=1, Ti=1)
Index
Closed Loop Transfer Function
Tuning a Controller
Stability analysis of systems
Steady-state response of a closed-loop system
Transient response of a stable system .
Closed-loop poles and zeros vs. Controller parameters
Transient Response
Response to unit step input Classification of output signal ∆y(t) depending on the input signal.
Unit Step (the most widely used). Ramp. Sinusoidal.
Provides information about the dynamical properties of the system
Y(s)G(s)
-C(s)
E(s) U(s)R(s)
Response in y(t) when a reference r(t) is applied
Reference signal: Unit Step signal. Shows the speed of response of the system
(In general the reference signal will be different than the unit step)
Transient Response
TEMA 5. Respuesta temporal de sistemas lineales. Sistemas dinámicos lineales de primer orden. Ejemplos. Sistemas dinámicos lineales de segundo orden. Respuesta ante escalón. Sistemas de orden n.
We are interested in the output y(t) as r(t) varies in time (Closed-loop behavior)
The trnasient response of a LTI system depends on the closed-loop transfer function (Gbc(s))
One option: Try & ErrorGiven a system, simulate or aplly inverde Laplace transformIt is difficult to characterize propoerties as raise time or overshootIdentify the effect of parameters in the response.
First Order Systems
s
K
yuKydt
dy
τ
τ
+==
==+
1U(s)
Y(s)G(s)
0)0( ,
units) in time (measured Constant Time :
output) &input toaccording (units u
yGain Static :K
τ∞
∞
∆∆
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300
1
2
3
4
5
tiempo
2=∆u
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
tiempo
y
6=∆y
ττττ
78.363.0 =∆⋅ y
Second order Systems
rad/s) (frequency Natural :
nal)(adimensiot Coefficien Damping :
U)Y/dim (dimGain Static :K
2
n
222
2
1212
2
ωδ
ωωωδ uKydt
dy
dt
yd
ubyadt
dya
dt
yd
nnn =++
=++
22
2
2U(s)
Y(s)G(s)
nn
n
ss
K
ωωδω
++==
Second order Systems
1:Poles 2 −±− δωωδ nn
Im
Re
<=>
:1
:1
:1
δδδ Overdamped
Critically damped.
Underdamped
Im
Re
Second order Systems
21 δωαπ−
−=n
st
21 δωπ
−=
n
pt
21100.(%). δπδ
−−
⋅= eOS
δωnet
3=
Underdamped system
00
Tiempo
y(t)
)(
)()(..
∞∞−
=y
ytyOS p
)(∞y
etptst
)(lim gain static theisK where
)]1cos()1([)(
)2()(
)('1)(
0
22
11
22
11
1
sGK
tctsenbeeaKty
ssps
csk
ssY
s
kkkk
r
k
tt
j
tpj
kkk
r
kj
t
j
i
m
i
kkj
→
=
−
=
−
==
=
=
⋅−+⋅−++=
++∏+∏
+∏⋅=
∑∑ δωδω
ωωδ
ωδ
Higher order Systems
nnnn
mmm
mmm
m
m
m
nnn
n
n
n
asasas
bsbsbsG
tubdt
tdub
dt
tudb
dt
tudbya
dt
tdya
dt
tyda
dt
tyd
+++++++=
++++=++++
−−
−
−−
−
−−
−
11
1
110
11
1
1011
1
1
...
...)(
)()(
...)()()(
...)()(
• In practice, some poles have more influence in the response than others. These poles are called dominant poles
• The dominant poles are those yielding the slowest reponse
• The response speed is given by the exponent of the exponential terms (the real part of the pole). Remember:
Dominant Poles
Dominant dynamics: poles with the slowest response
In practice, the dominat poles are determined from their relative distance to the imaginary axis.
Re
Imp1
p’1
p2
p’2
d2
d1
Re
Im
p1
p2
p’2
d2
d1
p1 is dominant if d2/d1>5
The static gain must remain the same
Dominant Poles
1
2
)17)(16)(1(
544
)17)(16)(1(
544)(
+=
+≈
+++=
ssssssG
-1 is el dominant. The remaining poles are neglected
Re
Im
-1-16
-17
Tiempo(s)0 1 2 3 4 5 6
0
0.5
1
1.5
2
2.5
y(t)
Effect of zeros in the output
0 0.5 1 1.50
1
2
3
4
5
6 Step Response
Time (sec)
Am
plitu
de
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
1.5
2
2.5Step Response
Time (sec)
Am
plitu
de
Zeros have an influence in the response
Effect of zeros in the output
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
1
2
3
4
5
6
Step Response
Time (sec)
Am
plitu
de
y(t)dy(t)/dtyc(t)
Qualitatively:
Effect of adding a zero
Non-minimum phase zeros
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2Step Response
Time (sec)
Am
plitu
de
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2Step Response
Time (sec)
Am
plitu
de
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
1.5
2
2.5Step Response
Time (sec)
Am
plitu
de
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
1
2
3
4
5
6Step Response
Time (sec)
Am
plitu
de
-20 -15 -10 -5 0 5-1
0
1
x xo o o o
Non-minimum phase zeros
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2Step Response
Time (sec)
Am
plitu
de
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
-0.5
0
0.5
1
1.5
2Step Response
Time (sec)A
mpl
itude
-20 -15 -10 -5 0 5-1
0
1
x x o o
Dynamics cancellation
-7 -6 -5 -4 -3 -2 -1 0-1
0
1
x xo
The closer the zero is to the pole, the less it influences system response Affects the dominant dynamics (in transient regime) Settling time is not significantly affected.
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
Step Response
Time (sec)
Am
plitu
de
Design Hypothesis
Quick review of concepts• Time response of first order systems• Time response of second order systems• Time response of higher order systems• Effect of zeros
It is difficult to obtain explicit results in general
Design Hypothesis• Explicit expressions for the effect of controller parameters on the
transient response for step inputs are required.• A pair of conjugate complex poles dominate the closed loop
response• Zeros are difficult to deal with, in general. Not considered.
Two design tools:• Root Locus design• Frequency domain design
Index
Closed Loop Transfer Function
Tuning a Controller
Stability analysis of system
Steady-state response of a closed-loop systems
Transient response of a stable system.
Closed-loop poles and zeros vs. Controller parameter s
Poles and Zeros of Gbc(s)
Closed-loop transfer function
Zeros of the closed-loop system The same zeros of the open-loop plant plus those of the controller
Poles of the closed-loop system Depend on the design parameter
• In some cases it is possible to get explicit expressions for 2nd order systems with P and PD controllers
• Not possible in generalApproximate methodologies
ExampleP Controller
The poles depend on Kp
Complex plane representation
Root Locus
Example
−6 −5 −4 −3 −2 −1 0 1−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Kp=0.1
Kp=1
Kp=10
Kp=15
Illustratuve example:
Magnetic levitation system
Description Value
Ball material Steel
Ball diameter 25 mm
Coil diameter 80 mm
Winding turns 2850
Resistence 22 Ω
Inductance 277 mH a 1 kHz442 mH a 120 kHz
Illustratuve example:
Magnetic levitation system
Nonlinear model of the system
2
2
X
IkmgXm −=&&
m : Ball massg : Gravity constantX : Distance between ball and coil
(magnitudes to be controlled)I : Coil current (control action)K : constant coefficient
X Fm
Fg
System Linealization
We assume the operating point X0 with control action I0 and consider error variables
XXX
III
∆+=∆+=
0
0
The incremental variables depend on theselected operating point.
System Linearization
System Linearization
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