MÉTODOS DE SINTONÍA Y AUTOSINTONÍA DE PIDs FRACCIONARIOS · 2 MÉTODOS DE SINTONÍA Y...

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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

Cálculo FraccionarioControl FraccionarioSintonía de controladores PID fraccionariosAutosintoníaConclusiones

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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

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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.

• Autosintonía: Compensación atraso-adelanto.

µλ sk

skksC d

ip ++=)(

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SINTONÍA DE CONTROLADORES PID FRACCIONARIOSEspecificaciones

Phase Margin (φm) and Gain Crossover Frequency (ωcg)mcgcgcg jGjCArgjFArg φπωωω +−== ))()(())((

dBjGjCjFdBcgcgdBcg 0)()()( == ωωω

Robustness to Variations in the Gain of the Plant( ) 0))((

== cg

djFArgd

ωωωω

Output Disturbance Rejection (Sensitivity Function)

sec/,)()(1

1)( radBdBjGjC

jS sdB

dBωω

ωωω ≤∀≤

+=

)()(1 sTsS =−High Frequency Noise RejectionComplementary

Functionssec/,)()(1

)()()( radAdBjGjC

jGjCjT tdB

dBωω

ωωωωω ≥∀≤

+=

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SINTONÍA DE CONTROLADORES PID FRACCIONARIOS

Diseño

Set of Five Nonlinear Equations with Five Unknown Parameters(kp,kd,ki,λ,µ)

NonlinearNonlinear MinimizationMinimization ProblemProblem

Main Function to Minimize0)( =dBcgjF ω

0))(( =−+ mcgjFArg φπω

( ) 0))((=

= cgd

jFArgd

ωωωω

0)( =−dBt AjT ω

0)( =−dBs BjS ω

FMINCON(Matlab)Constraints

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Ejemplo: Simulación

Design Specifications:First-Order Plant Plus Integrator:

degº5.48)2 =mφsec/1)1 radcg =ω

)1(25.0

)1()(

+=

+=

ssssksG

τ3) Flat Phase

sec/10,20)()4 raddBjT tdB=≥∀−≤ ωωω

sec/01.0,20)()5 raddBjS sdB=≤∀−≤ ωωω

Fractional PIλDµ Controller:

8090.06264.0 2195.21199.28159.3)( s

ssk

skksC d

ip ++=++= µ

λ

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Control of a First-Order Plant Plus an Integrator

3) Flat Phasesec/1)1 radcg =ω degº5.48)2 =mφ

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sec/10,20)()4 raddBjT tdB=≥∀−≤ ωωω

sec/01.0,20)()5 raddBjS sdB=≤∀−≤ ωωω

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Ejemplo: Experimento

LsesksG −

+=

1)(

τ

3146.07333.0 4747.10469.00469.0)( s

ssC ++=

Design Specifications:

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SINTONÍA DE CONTROLADORES PID FRACCIONARIOSOtros métodos

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SINTONÍA DE CONTROLADORES PID FRACCIONARIOSOtros métodos

Controladores PID – Optimización (Barbosa, Machado, 2003)

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AUTOSINTONÍA

The idea: automatic tuning of a controller, generally a PID, so that anunknown plant can be controlled, fulfilling several design especifications.

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 theimplementation problem (computer/PLC).

skskksC d

ip ++=)(

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AUTOSINTONÍA

Auto-Tuning Method for a Fractional PIλDµ Controller.

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 tocontrol, due to its reliability and simplicity.

µλ sk

skksC d

ip ++=)(

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AUTOSINTONÍA

Relay Auto-Tuning Scheme:

Condition for Oscillation:

Process: G(s) ( ))(

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)(,)(argaNd

ajGjG ==+−=πωθωπω

ωc: Frequency of interest

12121

1 )( −−−−−

− +−−−

= nnnnn

ncn θθθ

ωωωωθ

n: Curent iteration number

cacc

How to select the right value of θa which corresponds to ωc?

Iterative Process

(θ-1, θ0 ); (ω-1,ω0): Two Initial Values of the Delay and Their Corresponding Frequencies

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AUTOSINTONÍAMétodo

SPECIFICATIONS OF DESIGN:Crossover Frequency ωcPhase Margin ϕmRobustness Property (flat phase)

Lag Compensator with pole at the origin=PI

Lead Compensator:Noise Filter

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1) Robustness Constraint: flat phase of the open-loop system

Slope of the phase of the plant, estimated by the relay test

Constant overshoot for gain variations

λ

λ

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2) * Crossover frequency especification, ωc* Phase margin especification, ϕm

Lead Compensator

+ ROBUSTNESS CONSTRAINT (AGAIN)

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LEAD COMPENSATOR (α>0)

αα

ωωωω

α λωλω

=

++

===

= xxsC

xksC

m

m

c mm

11)(1)()()(

2

2'

+−

== −= x

xsCArg mm 11sin))(( 1' αφωω

ℜ∈<<

++

=

++

= αλλ

λλ α

αα

,10,1

1/1/1)( x

sxsxk

xssksC cc

•α: Flexibility in the Design Modulation of the Phase Curve

•φm: Does Not Impose the Value of xxm λ

ω 1=λω /1=zero

λω xpole /1=

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LEAD COMPENSATOR (α>0)

α≥αmin Lead Region(α,a1,b1) (α,x,λ); kc=kss/kxααmin xmin: Maximum Distance

Between Zero-Pole(Phase Flatness)

cbbaabaa

ax

ωλ

2

2

)1()1(1

+−=

+−−

=Doing Some Calculations:

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• 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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AUTOSINTONÍAEjemplo: Simulación

UNKNOWN PLANT:

sess

sG 05.0

)16.0(55.0)( −

+=

SPECIFICATIONS OF DESIGN:Robustness ConstraintCrossover Frequency ωc=2.3rad /secPhase Margin ϕm=72o

RELAY TEST:Slope of the phase of the plant=0.2566

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UNKNOWN PLANT:

sess

sG 05.0

)16.0(55.0)( −

+=



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UNKNOWN PLANT:

sess

sG 05.0

)16.0(55.0)( −

+=



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UNKNOWN PLANT:

sess

sG 05.0

)16.0(55.0)( −

+=


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UNKNOWN PLANT:

sess

sG 05.0

)16.0(55.0)( −

+=


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AUTOSINTONÍAEjemplo: Experimento

)1()(

+=

ssksG

τ8486.014348.0)(

+

=s

ssPI λ8160.0

10039.010350.4)(

++

=sssPDµ

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CONCLUSIONES Y TRABAJO FUTURO

ConclusionesMétodo de sintonía: optimización: determinar el valor de los 5parámetros para satisfacer 5 especificaciones de diseño.Método de autosintonía: especificaciones en frecuencia: Test del relé+ compensación en atraso (conseguir fase plana) + compensaciónen adelanto (conseguir especificaciones respetando robustez).Implantaciones en PC y PLC.

Trabajo futuroOtras especificacionesAproximaciones adecuadas para autosintonía: pulsar un botón.

MÉTODOS DE SINTONÍA Y AUTOSINTONÍA DE PIDs FRACCIONARIOS · 2 MÉTODOS DE SINTONÍA Y...

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