PFC Predictive Functional Control

Prof. Cesar de Prada Dpt. of Systems Engineering and Automatic Control

University of Valladolid, Spain prada@autom.uva.es

Outline

As simple as possible – Motivation – PFC main ideas – An introductory example

Motivation

Predictive control is a widely used control technique in the process industry combined with an optimization layer

Most of the implementations use internal linear models (step responses) and LP optimizers

u(t)

t t+1 t+2 ...

time

past future

tiempo

Set point

Ootput prediction CV, PV

MV, OP

SP

LP, economic optimizer

Sets optimal MVs

Set target

DMC no constraints

Computes MVs

DMC

Motivation

When a significant non-linear process is faced, a non-linear controller based on a non-linear plant model is required.

Current approaches based on first-principles models, NN, volterra series, etc. lead to a non-linear optimization problem that must be solved on-line

This is a heavy burden, both from the implementation and computational load

Non-linear Predictive Control

Process Optimizer u w y v

Predictor y(t+j)

[ ] [ ]

uj)u(tu yj)y(ty))t(u,x(t)(g y(t) )u(t),t)(x(fx

j)u(tj)w(tj)(tyu

Jmin2N

1Nj

1Nu

0j

22

≤+≤≤+≤==

+∆β++−+=∆

∑ ∑=

−

=

Solving the dynamic NLP problem

Process

Simulation from t to t+N2 for computing J(u,x(t))

Optimizer

u J

w

u(t) y(t)

State constraints

Gradients

Aim

The aim of Parametric Predictive Control (PPC) is to facilitate the implementation of the controller while retaining the main non-linear characteristics in its internal model

Good compromise between speed of execution, easy of implementation and performance

Target: Embedded controller in a DCS

PPC Main ideas

Combines first principles models with MPC ideas and generates a simplified solution that is updated every sampling time.

Parametric Predictive Control (PPC) (J. Richalet, 1996) was developed for, and successfully applied to, temperature control of batch reactors.

A. Assandri, A. Rueda, PhD students Three steps:

– Basic ideas presented in the linear case – Extended to the non-linear case with a chemical reactor CSTR – Industrial application to the bottom temperature control of a

distillation column heated with a furnace-reboiler.

Basic concepts

Consigna

Proceso

w

py

p∆

dE

t + Nt

m∆Modelo

my

futuropresentepasado

SP

Process

Model

Past t Future

∆p

∆m

Aim: Decrease the error in the future until a certain percentage of the current error w – y(t)

This implies to change the process output by ∆p

Use the model to compute the control u(t) that provides a change in the model output ∆m (u) = ∆p

Key design equation: ∆m (u) = ∆p

Basic concepts

Consigna

Proceso

w

py

p∆

dE

t + Nt

m∆Modelo

my

futuropresentepasado

SP

Process

Model

Past t Future

∆p

∆m

1 0 ))t(yw(E pN

d <λ<−λ=

The future error at t+N must be a fraction of the current error w- y(t)

))t(yw)(-1( pN

p −λ=∆

Example: First order model

)t(uk)t(ydt

)t(dy=+τ

Basic concepts (Monoreg)

Example: First order model

)t(uk)t(ydt

)t(dy=+τ

Assuming that u(t) is kept constant along the prediction horizon, that is Nu = 1, and with initial condition yp(t):

Ts sampling time

N prediction (o coincidence) horizon

−+=+ τ

−τ

− ss TN

p

TN

sm e1)t(uk)t(ye)TNt(y

u(t) time

−−

−=∆ τ

−τ

−1e)t(uk)t(y1ey

ss TN

p

TN

m

Design equation

Consigna

Proceso

w

py

p∆

dE

t + Nt

m∆Modelo

my

futuropresentepasado

SP

Process

Model

Past t Future

∆p

∆m

Consigna

Proceso

w

py

p∆

dE

t + Nt

m∆Modelo

my

futuropresentepasado

SP

Process

Model

Past t Future

∆p

∆m

design equation: ∆m (u) = ∆p

−−

−=∆ τ

−τ

−1e)t(uk)t(y1ey

ss TN

p

TN

m

))t(yw)(-1( pN

p −λ=∆

( )[ ]

−

−λ−−

−

=τ

−

τ−

1eK

)t(y)t(w1)t(y1e)t(u

s

s

TN

pN

p

TN

Explicit solution for the control signal

Tuning parameters

( )[ ]

−

−λ−−

−

=τ

−

τ−

1eK

)t(y)t(w1)t(y1e)t(u

s

s

TN

pN

p

TN N prediction horizon (number of sampling periods required to decrease the current error by λN)

λ Reduction factor (0,..,1)

A discrete first order system with pole λ will give a free response as the one desired at t+NTs

1q11

−λ− x

Example. Ideal case

1s51)s(G+

=

Model = Process

K = 1

τ = 5 min.

N = 20

λ= 0.8

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

TIME

wy

Process output and SP

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

TIME

uControl signal

hours

Extension to Nu > 1

Example: First order model

)t(uk)t(ydt

)t(dy=+τ

−+++=+

−+=+

τ−

−τ−

−

τ−

τ−

ss

ss

T)1N(

ssm

T)1N(

sm

T

p

T

sm

e1)Tt(uk)Tt(ye)TNt(y

e1)t(uk)t(ye)Tt(y

u(t)

time

Nu = 2

Ts

Predictions are more complex

Two unknows: u(t), u(t+Ts), then two coincidence points are required

u(t+Ts)

ym(t+Ts)

Robustness

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

TIME

wy

1s5.51)s(G+

=

Model ≠ Process

N = 20

λ = 0.8

Process output and SP

10% change in the parameters

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

TIME

uControl signal

A bit slower

hours Model:

Robustness

1s59.0)s(G+

=

Model ≠ Process

N = 20

λ = 0.8

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

TIME

wy

Steady state error

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

TIME

uControl signal

Solution: error model or explicit integrator added

hours

Model:

Incorporating errors

Model:

v)t(uk)t(ydt

)t(dy+=+τ

Assuming also that v does not change along the prediction horizon:

u(t) time

v disturbance

−+

−+=+ τ

−τ

−τ

− sss TNTN

p

TN

s e1ve1)t(uK)t(ye)NTt(y

−+

−+

−=∆ τ

−τ

−τ

− sss TNTN

p

TN

m e1ve1)t(uK)t(y1ey

Design equation

Consigna

Proceso

w

py

p∆

dE

t + Nt

m∆Modelo

my

futuropresentepasado

SP

Process

Model

Past t Future

∆p

∆m

Consigna

Proceso

w

py

p∆

dE

t + Nt

m∆Modelo

my

futuropresentepasado

SP

Process

Model

Past t Future

∆p

∆m

design equation: ∆m (u) = ∆p

))t(yw)(-1( pN

p −λ=∆

−+

−+

−=∆ τ

−τ

−τ

− sss TNTN

p

TN

m e1ve1)t(uK)t(y1ey

( ) ( )[ ]

−

−λ−−−

−

=τ

−

τ−

1eK

)t(y)k(w1v)t(y1e)t(u

s

s

TN

pN

p

TNController equation.

v is not known and needs to be estimated every sampling time

Estimating v

t

t

t -Ts

t -Ts

u(t-1)

y(t-Ts)

y(t)

−+

−−+−= τ

−τ

−τ

− sss TT

ssp

T

p e1ve1)Tt(uK)Tt(ye)t(y

−

−−−−−

=τ

−

τ−

τ−

s

ss

T

T

ssp

T

p

e1

e1)Tt(uK)Tt(ye)t(yv

v is estimated from the process model in order to cancel the difference between the measured process output at t and the prediction made with values at t - Ts

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

TIME

u

Example

Control signal 1s5.5

9.0)s(G+

=

Model ≠ Process

N = 20

λ = 0.8

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

TIME

wy

The error estimation compensates

the modelling errors

0.00

0.05

0.10

0.15

0.20

0.25

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

TIME

v

Estimated disturbance

hours Model:

Adding an integrator

))Tt(y)Tt(w(TT)Tt(i)Tt(e

TT)Tt(i)t(i sps

i

sss

i

ss −−−+−=−+−=

( )[ ]( ) ( )s

i

ssTN

pN

p

TN

TteTTTti

1eK

)t(y)t(w1)t(y1e)t(u

s

s

−+−+

−

−λ−−

−

=τ

−

τ−

PPC

∫=t

i

p deTk

ti ττ )()(

u(t) w e

yp(t)

Example

1s5.59.0)s(G+

=

Model ≠ Process

N = 20

λ= 0.8

Ti = 0.1 min

Ts = 20 sec.

hours

No se puede mostrar la imagen en este momento.

Control signal

Model:

First order + delay

Model:

v)dt(uk)t(ydt

)t(dy+−=+τ

v disturbance

d delay

d = DTs

u(t) time

yp

d

∆p

∆m

Process

Model ym

t t+N ( )[ ])t(y)t(w1 p

DNp −λ−=∆ −

First order + delay

( )

−+

−+

+−

−+

−=∆

τ−

τ−

−

−+

=

τ−

ττ−

τ−

∑ss

ssss

TNTDN

1Dt

tm

T)tm(TTN

p

TN

m

e1ve1K)t(u

e)Dm(u1eKe)t(y1ey

( )[ ])t(y)t(w1 pDN

p −λ−=∆ −

( )

( )[ ]

−−−

−−

−

−−−λ−

−=

τ−−+

=

τ−

ττ−

τ−−

−

τ−

−

∑ssss

ss

TN1Dt

tm

T)tm(TTN

p

TN

pDN

1TDN

e1ve)Dm(u1eKe

)t(y1e)t(y)t(w1e1K)t(u

N > D

Estimating v

t t -Ts

t –(D+1)Ts

u(t-(D+1)Ts)

y(t-Ts)

y(t)

t –DTs

DTs

−

−+−−−−

=τ

−

τ−

τ−

s

ss

T

T

ssp

T

p

e1

e1)T)1D(t(uK)Tt(ye)t(yv

v is estimated from the process model in order to cancel the difference between the measured process output at t and the prediction made with values at t - Ts

+ smoothing filter

0.5

1.0

1.5

2.0

2.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

TIME

u

0.5

1.0

1.5

2.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

TIME

wy

Example. Ideal case

1s5e)s(G

s10

+=

−

Model = Process

K = 1

τ= 5 min.

d = 10 min.

N = 40

λ= 0.8

Ts = 1 min.

Process output and SP

Control signal

hours

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

TIME

wy

Robustness Model ≠ Process

N = 40

λ = 0.8

The error estimation compensates

the modelling errors hours

1s5.5e9.0)s(G

s9

+=

−

0.0

1.0

2.0

3.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

TIME

uControl signal

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

TIME

v

Estimated disturbance

Model: