Post on 01-Aug-2018
transcript
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)
2
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
3
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
4
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
5
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.
6
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
7
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
8
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
9
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
10
System Representation
xc
xf
Tah
T Tm
hm
System SensorsActuator
qcqf
∆Pvr
Tm
xc qf
Tf
qc
qsT
h T
hm
xf
Ta
11
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
12
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)
13
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)
14
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 )
15
Example
qc
Tm
Ta
T
Caldera
xc
-
-
16
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
17
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
18
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
19
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
20
Candidate Transfer Function
sK
sGττττ+
=1
)(
Two parameters:K?
?ττττ
Identification based on the step response
21
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
22
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
23
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)
24
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
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
26
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
27
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
28
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
29
Equilibrium point
The equilibrium point is reached when the derivative of vs is zero. That is, when ve = vs
30
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.
31
Steady state characteristic
Relationship between the input and the output in the steady state regimen.
Example:
ve
vs
In steady state:
32
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
33
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
34
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
35
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
36
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)
37
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)
38
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
39
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 ∆
∆=
40
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
41
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
42
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
43
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
44
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.
45
Linealización de sistemas
Las variables incrementales dependendel punto de funcionamiento elegido
46
Linealización de sistemas
47
Example
Operating point:
Defining incremental variables
Modeling error
48
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
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
50
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)
51
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
52
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)
53
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
54
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
55
Basic control terms
� Relay based control
� Proportional term
� Integral term
� Derivative term
56
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
57
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
58
Proportional term
� Control law
Proportionalband
umax
umin
e
u
u0
System
+u0
U(t)
-
Y(t)Kp
R(t) e(t)
59
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
60
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
61
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)
62
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)
63
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
64
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)