Chapter 2: Introduction to the Control of SISO Systems · Chapter 2: Introduction to the Control of...
Transcript of Chapter 2: Introduction to the Control of SISO Systems · Chapter 2: Introduction to the Control of...
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Chapter 2:
Introduction to the
Control of SISO Systems
Control Automático
3º Curso. Ing. IndustrialEscuela Técnica Superior de Ingenieros
Universidad de Sevilla
(Some of the illustrations are borrowed from : Modern Control Systems (Dorf and Bishop)
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Outline of the presentation
1. Dynamical Systems
2. Single Input-Single Output Systems (SISO Systems)
3. Identification of Dynamic Systems
4. Equilibrium points. Steady state characteristic
5. Linearization
6. Control scheme
7. Basic control actions
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Dynamical systems
� System: object composed by a number of interrelated parts. Theproperties of the system are determined by the relationshipsbetween its different parts.
� Dynamical: its state varies with time
� Signal or variable: every magnitude that evolves with time
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Basic notions
� We understand the system to be part of thereal world with a boundary with the outsideenvironment.
� Types of signals:
� Input signals: they act upon the system and are responsible for its future evolution.
� Output signals: they are the signals to be measured (and controlled). They represent the effect of the system on its environment.
� Internal variables: all the remaining
variables
� Examples:
xx
x
x xxx
xx
xx x
x
x
states
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Basic notions
� Types of inputs: (from a technological point of view )� Manipulated variables: their evolution can be manipulated and fixed to a
desired value � Disturbances: are often regarded as uncontrolled being determined by the
environment in which the system resides (weather variations, process feed quality variations, …)
� Parameters of the system: magnitudes that characterize the system. They
allow one to distinguish between systems with similar structural and functional characteristics.
� Example: distinguish between parameters and signals of the systems
corresponding to the illustrations above.
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Basic notions
� Models:
� Representation of the system that enables
its study.
� Physical representation (scaled-models)
� Mathematical representation (dynamic equations)
� Purposes of a model:
� Prediction of the evolution of the system
� Analysis of the behavior of the system
� Analysis of the effect of the variation of a parameter
� Analysis of the effect of the inputs on the evolution of the system
Modeling error
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Modelling of Dynamical Systems
� Trade off between the accuracy of the model and its simplicity
� The type of model should be chosen according to the desired functionalities and purposes � Analysis
� Objective: cualitative analysis of the system’s behaviour.� This analysis can be a difficult task. � The model should be as simple as possible, but reflecting the main characteristics
and properties of the dynamics.
� Simulation� Objective: prediction of the evolution of the system.� This is normally a simpler task than the analysis (it can be solved by means of
numerical integration).� The model should have a degree of detail capable of yielding small prediction
errors.
ErrorComplexity
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Simulation of systems
� Numerical Integration of the differential equations
� Discretization of time {t0, t1, t2,…}
� Integration step
� Computation of the outputs {y0, y1, y2,…}
� Example: Euler Method
−= )()(
1)( ty
A
Ktq
Aty p
&
−+= −−− 1k
p
1k1kk yA
Kq
A
1hyy
� Initialization : y0=y(0)
� For k=1 to N
� tk=k h
�
� End
Model
Input Output
Initial conditions
SIMULATOR
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System Representation
• Inputs• Manipulated inputs:
• Cold water valve xf• Hot water valve xc
• Disturbances• Ambient temperatureTa• Temperatures Tc y Tf• Pressure at the pipes
of cold and hot water• Outputs
• Temperature of tank T• Water level in tank h
• Measurements:• Metal resistance termometer • Pressure sensor
Tc
xc qf
Tf
qc
qsT
h T
Tm
hm
xf
Ta
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System Representation
xc
xf
Tah
T Tm
hm
System SensorsActuator
qcqf
∆Pvr
Tm
xc qf
Tf
qc
qsT
h T
hm
xf
Ta
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Single Input-Single Output
Systems
1. Dynamical Systems
2. Single Input-Single Output Systems (SISO Systems)
3. Identification of Dynamic Systems
4. Equilibrium points. Steady state characteristic
5. Linearization
6. Control scheme
7. Basic control actions
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Linear systems representation
• Differential equation: it models the dynamics of a lumped parameter linear system in continuous time.
• Laplace transform:
mnmodelsCausal
nequationtheofOrder
tubdt
tdub
dt
tudb
dt
tudbya
dt
tdya
dt
tyda
dt
tydmmm
m
m
m
nnn
n
n
n
≥
++++=++++ −−
−
−−
−
:
:
)()(
...)()()(
...)()(
11
1
1011
1
1
systemu(t) y(t)
G(s)U(s) Y(s)G(s)U(s)
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Frequency response
� Steady-state output for sinusoidal input
� G(jw) characterizes the frequency response of the system
� Fourier Series expansion ⇒ G(jw) characterizes the system
systemu(t) y(t)
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Graphic plots
� Objective: Graphic plot of
� Bode Diagram:
2 semi-logarithmic scalar plots
� Magnitude
� Phase
-120
-100
-80
-60
-40
-20
0
Mag
nitu
de (
dB)
10-2
10-1
100
101
102
103
-180
-135
-90
-45
0
Pha
se (
deg)
Bode Diagram
Frequenc y (rad/s ec )
-120
-100
-80
-60
-40
-20
0
Mag
nitu
de (
dB)
10-2
10-1
100
101
102
103
-180
-135
-90
-45
0
Pha
se (
deg)
Bode Diagram
Frequenc y (rad/s ec )
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Example
qc
Tm
Ta
T
Caldera
xc
-
-
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Identification of Dynamic Systems
1. Dynamical Systems
2. Single Input-Single Output Systems (SISO Systems)
3. Identification of Dynamic Systems
4. Equilibrium points. Steady state characteristic
5. Linearization
6. Control scheme
7. Basic control actions
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Identification
� Obtaining a model from the temporal response of the system � Model parameters (for a given structure of the model)� Parametric model
� Structure and parameters (unknown model)� Black box identification
� Analysis of the system’s output corresponding to
different test input signals
� Impulse response
� Step response
� Sinousoidal response
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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
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
6
7
8
9
10
tiempo
u
Step input signal Output of the system
G(s)?
Identification based on the step response
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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
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
6
7
8
9
10
tiempo
u
Step input signal Output of the system
Characteristic response of a first order system:Exponential evolution with non zero slope at the instant corresponding to the step jump
Identification based on the step response
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Candidate Transfer Function
sK
sGττττ+
=1
)(
Two parameters:K?
?ττττ
Identification based on the step response
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K: it is obtained from the steady state :
32
6
13
28 ==−−=
∆∆=
uy
K
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
2=∆u
6=∆y
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
6
7
8
9
10
tiempo
u
Identification based on the step response
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τ : it is obtained from the transitory response
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
0.51
1.52
2.53
3.54
4.55
5.56
6.57
7.58
8.59
9.510
tiempo
y
6=∆y
ττττ
78.363.0 =∆⋅ y
Identification based on the step response
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Frequency based identification
� G(s) can be determined from the experimental Bode Diagram
� Determination of the frequency range:� Step response: Characteristic time constant of the system
� Other factors:� Frequency range of noise
� Sampling time
systemu(t) y(t)
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Frequency identification of a tank
Operating point:
Qs
H
k
k
h(t)
Válvulah
To Workspace
Sine Wave
Scopesqrt
MathFunction
1s
Integrator
h0 Constant1
q0 Constant
1/A
1/A
Qs
H
k
k
h(t)
Válvulah
To Workspace
Sine Wave
Scopesqrt
MathFunction
1s
Integrator
h0 Constant1
q0 Constant
1/A
1/A
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25
10-4
10-3
10-2
10-1
100
10
15
20
25
30
35
10-4
10-3
10-2
10-1
100
-100
-80
-60
-40
-20
0
Frequency identification of a tank
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10-4
10-3
10-2
10-1
100
10
15
20
25
30
35
10-4
10-3
10-2
10-1
100
-100
-80
-60
-40
-20
0
Bode ExperimentalBode sistema aprox.
1/τ
Ke (dB)
Frequency identification of a tank
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Equilibrium points. Steady state
characteristic
1. Dynamical Systems
2. Single Input-Single Output Systems (SISO Systems)
3. Identification of Dynamic Systems
4. Equilibrium points. Steady state characteristic
5. Linearization
6. Control scheme
7. Basic control actions
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0 5 10 15 20 25 30 35 400
0.5
1
1.5
2
0 5 10 15 20 25 30 35 400
0.5
1
1.5
2
Transitory and steady state response
Steady state Transitory responseresponse
Steady state responseTransitory response
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Equilibrium point
The equilibrium point is reached when the derivative of vs is zero. That is, when ve = vs
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Equilibrium point
Uniqueness of the equilibrium point for linear systems:
• Given an input, for example ve= 1 volt, the system will evolve till it
reaches a unique equilibrium point that corresponds to theoutput v
s=1 volt.
•If the input is ve= 2 volts, then the system evolves till it reaches an
equilibrium point that corresponds in this case to an output vs=2
volts.
• For a given input, there is only one equilibrium point.
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Steady state characteristic
Relationship between the input and the output in the steady state regimen.
Example:
ve
vs
In steady state:
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R+ _
V
Steady state characteristic
The steady state characteristic can be often obtained in an experimental way:
For example: DC Motor
Input: Applied voltage V (volts)
Output: Angular velocity (r.p.s.) revolutions per second
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Steady state characteristic
V(v) R(r.p.s.)
0 0
1 0
2 0.2
3 1.3
4 3.2
5 5.1
6 6.5
7 7.2
8 7.4
9 7.4
Applying different voltages at the input and measuring the revolutions per second in steady state:
R+ _
V
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Steady state characteristic
Graphic representation of the steady state characteristic
1 2 3 4 5 6 7 8 9
1
2
3
4
5
6
7
8
9R
V
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Steady state characteristic
Some considerations for the analysis of the steady state characterisitic
1 2 3 4 5 6 7 8 9
1
2
3
4
5
6
7
8
9R
V
Zone of linear behaviour
Zone of non linear behaviour
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Static gain
u
yK static ∆
∆=
The static gain allows one to determine which is the final increment at the output of the system due to a given increment in the input.
systemu(t) y(t)
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Static gain
0 5 10012
3456
78
Consider the following data, obtained from the step response of the system. Which is the static gain ?
0 5 10012
3456
78
?staticKsystemu(t) y(t)
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Static Gain
1=∆u
0 5 10012
3456
78
0 5 10012345678u y
3=∆y
1=∆u
1
5
2
53
1
3
12
25 ≠≠==−−=
∆∆= staticstaticstatic KK
u
yK
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Static Gain
• The steady state characteristic of a system allows one to determine which is its static gain at each operating point (equilibrium point): It is given by the slope of the curve.
1 2 3 4 5 6 7 8 9
1
2
3
4
5
6
7
8
9y
u
u
yK static ∆
∆=
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Static gain• In the zone corresponding to a linear behaviour, the static gain characteristic has a constant slope. Therefore, in this zone the static gain is constant regardless of the operating point
1 2 3 4 5 6 7 8 9
1
2
3
4
5
6
7
8
9y
u
Linear zone: same static gain Kstatic for every operating point
Zones of non linear behaviour: Kstatic depends on the operating point
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Linearization
1. Dynamical Systems
2. Single Input-Single Output Systems (SISO Systems)
3. Identification of Dynamic Systems
4. Equilibrium points. Steady state characteristic
5. Linearization
6. Control scheme
7. Basic control actions
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Linear dynamic systems:
Superposition principle
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3
3.5
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3
3.5
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3
3.5
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3
3.5
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3
3.5
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3
3.5
u1
y1
u2 y2
u1+u2
y1+y2
Linear system
Linear system
Linear system
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Superposition principle (it is not applicable for non linear systems)
0 5 10 15 20 25 300
2
4
6
8
10
12
0 5 10 15 20 25 300
2
4
6
8
10
12
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3
3.5
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3
3.5
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3
3.5
u1
u2 y2
ut=u1+u2
0 5 10 15 20 25 300
2
4
6
8
10
12
y1
yt=y1+y2/
Non Linear system
Non Linear system
Non Linear system
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System Linearization
� Objective:
� Obtaining approximated linear models from non linear ones.
� Operating poing:
� Equilibrium point at which the linearization is done.
� Properties:
� It represents in a correct way the system in a neigborhod of the
equilibrium point.
� Outside of the region of applicability of the linearized model, the error
might be too large.
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Linealización de sistemas
Las variables incrementales dependendel punto de funcionamiento elegido
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Linealización de sistemas
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Example
Operating point:
Defining incremental variables
Modeling error
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Illustrative example
� Good approximation around the
equilibrium point
� For larger deviations, the linear
model might incurr in large errors
� All the signals evolve around their
value at the equilibrium point
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49
Control scheme
1. Dynamical Systems
2. Single Input-Single Output Systems (SISO Systems)
3. Identification of Dynamic Systems
4. Equilibrium points. Steady state characteristic
5. Linearization
6. Control scheme
7. Basic control actions
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Feedback control
Controller
Manipulatedvariable
Controlled outputActuator System
Sensor
Measured signal
-y(t)
error
e u
Reference
Negative feedback:
↑↑↑↑e � ↑↑↑↑y � ↓↓↓↓e Compensation for the error(if not, unstable)
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Controller gain
� The controller should guarantee a positive gain, that is, ↑↑↑↑e � ↑↑↑↑y � Positive gain:
� If ↑↑↑↑u � ↑↑↑↑y, then ↑↑↑↑e � ↑↑↑↑u
� Negative gain:
� If ↑↑↑↑u � ↓↓↓↓y, then ↑↑↑↑e � ↓↓↓↓ u
h
h
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Linearization and control
Linearizedmodel
u(t) y(t)
Plant
+u0
U(t)
-
Y(t)
y0
y(t)u(t)
u(t)u0
U(t)y(t)
y0
Y(t)
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Control of linearized systems
e(t) = (R(t)-y0)-(Y(t)-y0)= R(t)-Y(t)
Plant
+u0
U(t)
-
Y(t)u(t)Controller
R(t) e(t)
Equivalent (linear) control system
Controller
G(s)C(s)
Gs(s)
Ga(s)
Plant
Sensor
-
+R E U Y
Ym
VG(s)C(s)
Gs(s)
Ga(s)
Sensor
-
+G(s)G(s)C(s)C(s)
Gs(s)Gs(s)
Ga(s)Ga(s)
Actuator
Sensor
-
+R E U Y
Ym
V
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Basic control actions
1. Dynamical Systems
2. Single Input-Single Output Systems (SISO Systems)
3. Identification of Dynamic Systems
4. Equilibrium points. Steady state characteristic
5. Linearization
6. Control scheme
7. Basic control actions
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Basic control terms
� Relay based control
� Proportional term
� Integral term
� Derivative term
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Relay based control
� On-Off control
� Control law
� If e(t)>0, u(t)=umax
� If e(t)<0, u(t)=umin
� Oscillatory behavior
� Drives the system to the reference point
� Relay control with hysteresis
� Reduces oscillatory behavior
� Increasing the band of the
hysteresis reduces the frequency and
increases the amplitude
SystemU(t)
-
Y(t)R(t) e(t)
Relay
e
uumax
umin
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Level Control of a vessel
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9Histéresis de anchura 0.04
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9Histéresis de anchura 0.08
Qs
H
10
k
Válvulah
To Workspace
Step1
Step
Scope
Rele
r
Referencia
sqrt
MathFunction
1s
Integrator
1/5
1/A
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Proportional term
� Control law
Proportionalband
umax
umin
e
u
u0
System
+u0
U(t)
-
Y(t)Kp
R(t) e(t)
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Proportional term
� Properties:
� Reduces oscillatory behavior
� BP=0% � Relay control
� It eliminates the tracking error of the step-response for the equilibrium reference u0
� In general it does not eliminate the tracking error of the step response for arbitrary references
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Proportional level control of a Vessel
Qs
H
10
k
Válvulah
T o Workspace
Step1
Step
Scope
r
Referencia
sqrt
M athFunction
1s
Integrator
10
Gain
7.0711 Constant
1/5
1/A
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Kp=10
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Kp=10
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Kp=100
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Integral term
� PI control law
SystemU(t)
-
Y(t)PI
R(t) e(t) • Eliminates the tracking error of the step-response for arbitrary references
• Increases oscillatory behavior (may lead to instability)
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Integral term
� Adapts the value of u0
� If the closed-loop system is stable then
u(t) bounded � bounded � e(t) → 0
System
+ u0
U(t)
-
Y(t)
KpR(t)
e(t)
1er order(K=1, t=Ti)
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PI level control of a vessel
Qs
H
10
k
Válvula
1
s+1
Transfer Fcn
h
To Workspace
Step1
Step
Scope
r
Referencia
sqrt
MathFunction
1s
Integrator
100
Gain
1/5
1/A
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Kp=100 T
i=1
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Kp=100, Ti=0.1
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Derivative term
� PD control law
� Predicts future evolution of the error
� May improve transient
� Amplifies high-frequency noise
System
+u0
U(t)
-
Y(t)PD
R(t) e(t)