Métodos de Sintonía y Autosintonía de PIDs Fraccionarios

Transcription

Jornadas de Ingeniería de Control
Pamplona, 14 y 15 de Marzo de 2006
MÉTODOS DE SINTONÍA Y AUTOSINTONÍA
DE PIDs FRACCIONARIOS
Blas M. Vinagre, Concepción A. Monje
*Escuela de Ingenierías Industriales. Universidad de Extremadura (Badajoz), Spain.
e-mail: [email protected]; [email protected]
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MÉTODOS DE SINTONÍA Y
AUTOSINTONÍA DE PIDs
FRACCIONARIOS
ÍNDICE
X Cálculo Fraccionario
Y Control Fraccionario
Z Sintonía de controladores PID fraccionarios
[ Autosintonía
\ Conclusiones
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CÁLCULO FRACCIONARIO
• Definiciones (Riemann – Liouville):
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CÁLCULO FRACCIONARIO
• Definición de Gründwald – Letnikov:
• Otras definiciones: Weyl (Potencial), Caputo
(Condiciones iniciales interpretables).
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CONTROL FRACCIONARIO
5
CONTROL FRACCIONARIO
Acciones básicas de control
Acción integral
Acción derivativa
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CONTROLADORES PID FRACCIONARIOS
• Dos estructuas:
• Sintonía: Encontrar los valores de los parámetros para satisfacer
5 especificaciones de diseño.
ki
C ( s) = k p + λ + kd s µ
s
• Autosintonía: Compensación atraso-adelanto.
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SINTONÍA DE CONTROLADORES PID
FRACCIONARIOS
Especificaciones
Phase Margin (φm) and Gain Crossover Frequency (ωcg)
Arg ( F ( jωcg )) = Arg (C ( jωcg )G ( jωcg )) = −π + φm
F ( jωcg )

dB
= C ( jωcg )G ( jωcg )
dB
= 0dB
Robustness to Variations in the Gain of the Plant
d ( Arg ( F ( jω )) )
=0
dω
ω =ωcg

Output Disturbance Rejection (Sensitivity Function)
S ( jω ) dB =

1
1 + C ( jω )G ( jω ) dB
≤ BdB, ∀ω ≤ ω s rad / sec
High Frequency Noise Rejection
T ( jω ) dB
C ( jω )G ( jω )
=
≤ AdB, ∀ω ≥ ωt rad / sec
1 + C ( jω )G ( jω ) dB
1 − S ( s) = T ( s)
Complementary
Functions
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FRACCIONARIOS
Diseño
Set of Five Nonlinear Equations with Five Unknown Parameters
(kp,kd,ki,λ,µ)
Nonlinear Minimization Problem
F ( jωcg )
dB
=0
Main Function to Minimize
Arg ( F ( jωcg )) + π − φm = 0
d ( Arg ( F ( jω )) )
=0
dω
ω =ωcg
Constraints
FMINCON
(Matlab)
S ( jω s ) − B dB = 0
T ( jωt ) − A dB = 0
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FRACCIONARIOS
Ejemplo: Simulación
First-Order Plant Plus Integrator:
k
0.25
=
G (s) =
s (τs + 1) s ( s + 1)
Design Specifications:
1) ωcg = 1rad / sec
2) φm = 48.5º deg
3) Flat Phase
4) T ( jω ) dB ≤ −20dB, ∀ω ≥ ωt = 10rad / sec
5) S ( jω ) dB ≤ −20dB, ∀ω ≤ ω s = 0.01rad / sec
Fractional PIλDµ Controller:
ki
2.1199
µ
C ( s ) = k p + λ + k d s = 3.8159 + 0.6264 + 2.2195s 0.8090
s
s
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FRACCIONARIOS
Control of a First-Order Plant Plus an Integrator
1) ωcg = 1rad / sec
2) φm = 48.5º deg
3) Flat Phase
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FRACCIONARIOS
4) T ( jω ) dB ≤ −20dB, ∀ω ≥ ωt = 10rad / sec
5) S ( jω ) dB ≤ −20dB, ∀ω ≤ ω s = 0.01rad / sec
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FRACCIONARIOS
Ejemplo: Experimento
Design Specifications:
G (s) =
k
e − Ls
τs + 1
C ( s ) = 0.0469 +
0.0469
s 0.7333
+ 1.4747 s 0.3146
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FRACCIONARIOS
Otros métodos
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FRACCIONARIOS
Otros métodos
Controladores PID – Optimización (Barbosa,
Machado, 2003)
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AUTOSINTONÍA
The idea: automatic tuning of a controller, generally a PID, so that an
unknown plant can be controlled, fulfilling several design especifications.
ki
+ kd s
C (s) = k p +
s
Two parts: a) information of the plant (relay test).
b) tuning of the controller with that information.
There are different auto-tuning methods for conventional PID controllers,
currently working on industrial environments.
The more complex the controller design method, the more difficult the
implementation problem (computer/PLC).
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AUTOSINTONÍA
Auto-Tuning Method for a Fractional PIλDµ Controller.
C (s) = k p +
ki
µ
+
k
s
d
sλ
The introduction of the orders λ and µ allows the fulfillment of a robustness
constraint without increasing the complexity of the design method (simple
equations).
The method uses the relay test to obtain the information of the plant to
control, due to its reliability and simplicity.
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AUTOSINTONÍA
Relay Auto-Tuning Scheme:
Process: G(s)
Condition for Oscillation:
πa
1
=
arg(G ( jωc ) ) = −π + ωcθ a , G ( jωc ) =
4d N ( a )
ωc: Frequency of interest
How to select the right value of θa
which corresponds to ωc?
Iterative Process
θn =
ωc − ωn −1
(θ n −1 − θ n − 2 ) + θ n −1
ωn −1 − ωn − 2
n: Curent iteration number
(θ-1, θ0 ); (ω-1,ω0): Two Initial Values of the Delay and Their Corresponding
Frequencies
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AUTOSINTONÍA
Método

Lag Compensator with
pole at the origin=PI
SPECIFICATIONS OF DESIGN:
Crossover Frequency ωc
Phase Margin ϕm
Robustness Property (flat phase)
Lead Compensator:
Noise Filter
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AUTOSINTONÍA
Método
1) Robustness Constraint: flat phase of the open-loop system
Constant overshoot for gain variations
λ
λ
Slope of the phase of the plant,
estimated by the relay test
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AUTOSINTONÍA
Método
2) * Crossover frequency especification, ωc
* Phase margin especification, ϕm
Lead Compensator
+ ROBUSTNESS CONSTRAINT (AGAIN)
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AUTOSINTONÍA
Método
LEAD COMPENSATOR (α>0)
α
α
 s +1/ λ 
α  λs + 1 
C ( s) = kc 
 = kc x 
 , 0 < x < 1, α ∈ ℜ
+
s
x
x
s
+
1
/
λ
λ
1




C(s)
= C' (s)
α
ω=ωm
kc x ω=ω
m
α
 (λω )2 +1   1 α
m
 = 
=
2
 ( xλωm ) +1   x 


1− x 
Arg (C ' ( s )) ω =ω m = φ m = α sin −1 

1+ x 
ω zero = 1 /λ
ω pole = 1 / xλ
ωm =
1
λ x
•α: Flexibility in the Design
Modulation of the Phase Curve
•φm: Does Not Impose the Value of x
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AUTOSINTONÍA
Método
LEAD COMPENSATOR (α>0)
α≥αmin
(α,a1,b1)
αmin
x=
Doing Some Calculations:
Lead Region
(α,x,λ); kc=kss/kxα
xmin: Maximum Distance
Between Zero-Pole
(Phase Flatness)
a −1
a (a − 1) + b 2
a(a − 1) + b 2
λ=
bωc
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AUTOSINTONÍA
Método
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AUTOSINTONÍA
Método
• The relay test: *magnitude and phase of the plant at ω1, ωc, ω2
*estimation of the slope of the phase of the plant
• PIλ(s): cancell the slope of the phase of the plant, giving the
minimum lag phase possible=robustness constraint
• PDµ(s): fulfills the frequency especifications of ωc and ϕm,
following a robustness constraint
• The parameters of the fractional PIλDµ(s) controller are obtained by
simple equations that can be solved by a PLC (industrial application)
• The implementation of the fractional controller: we are currently
working on it
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AUTOSINTONÍA
UNKNOWN PLANT:
G (s) =
0.55
e −0.05 s
s (0.6 s + 1)

Robustness Constraint
Crossover Frequency ωc =2.3rad /sec
Phase Margin ϕm =72o
RELAY TEST:

Slope of the phase of the plant=0.2566
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AUTOSINTONÍA
UNKNOWN PLANT:
G (s) =
0.55
e −0.05 s
s (0.6 s + 1)

RELAY TEST:

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AUTOSINTONÍA
UNKNOWN PLANT:
G (s) =
0.55
e −0.05 s
s (0.6 s + 1)

RELAY TEST:

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AUTOSINTONÍA
UNKNOWN PLANT:
G (s) =
0.55
e −0.05 s
s (0.6 s + 1)

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AUTOSINTONÍA
UNKNOWN PLANT:
G (s) =
0.55
e −0.05 s
s (0.6 s + 1)

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AUTOSINTONÍA
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AUTOSINTONÍA
Ejemplo: Experimento
G (s) =
k
s (τs + 1)
 0.4348s + 1 
PI ( s ) = 

s


λ
0.8486
 4.0350 s + 1 
PD µ ( s ) = 

s
+
0
.
0039
1


0.8160
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CONCLUSIONES Y TRABAJO FUTURO
Conclusiones
Método de sintonía: optimización: determinar el valor de los 5
parámetros para satisfacer 5 especificaciones de diseño.
Método de autosintonía: especificaciones en frecuencia: Test del relé
+ compensación en atraso (conseguir fase plana) + compensación
en adelanto (conseguir especificaciones respetando robustez).
Implantaciones en PC y PLC.
Trabajo futuro
Otras especificaciones
Aproximaciones adecuadas para autosintonía: pulsar un botón.
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