An overview of Model Predictive Control (MPC)

Prof. Cesar de Prada Dpt. of Systems Engineering and

Automatic Control University of Valladolid, Spain

Outline

UVA Introduction to MPC

– Motivation – Basic controllers – Constraints – Implementation methodology – Analysis: Stability, Robustness – Applications and industrial tools (Economic optimization)

Non linear MPC Hybrid MPC Industrial examples

Valladolid

•Capital of Castilla-León •Medium size town •Car industry, Renault

Madrid

Spain

France

Miguel de Cervantes

“El Quijote”

Cristobal Colombus

Valladolid-Madrid 1 h.

University of Valladolid

Second oldest in Spain ( XIII century ) All branches: Humanities, Law, Engineering, Medicine, … 26000 students

Santa Cruz Palace XV century

Vice-Chancellor offices

Dpt. of Systems Engineering and Automatic Control

Founded in 1973 Three locations

– School of Industrial Engineering – Politecnic School – Faculty of Sciences

Two Technology Centres – CARTIF (Automation and Robotics) – CTA (Centre for Sugar Technology)

PhD Course: Process and Systems Engineering (in cooperation with the Chemical Eng. Dpt.) Award of Excellence of the MEC

Erasmus Agreements

“Process control and supervision” Research group

The group 2 Professors 5 lecturers 3 doctoral contracts 12 research grants 2 technicians Research topics: Advanced Control, MPC Process Optimization Modelling and Simulation Fault detection and diagnosis

Web: www.isa.cie.uva.es

Develop new ideas and theory

Develop software tools

Industrial applications

Sede Mergelina

Industry / International cooperation

Int. Research Networks: HYCON2 NoE CITED Networks ALFA Projects European Embedded Control Institute (EECI)

Process Simulators for operators training

Plant-wide optimization

CTA (Centre for Sugar Technology) Joint Research Centre with the Spanish Sugar Industry

Repsol-YPF

CERN

LHC

Empresarios Agrupados

EcosimPro

Why MPC?: Industry Trends

Increasing demands in product quality, costs optimisation, productivity, flexibility to adapt to a wide range of operating conditions, security, care of the environment,..

Increasing pressure on the control systems to fulfil

new and stronger specifications. Increasing needs to rationalise upper level decisions

linked to plant economy, and to integrate and improve all different operating levels.

Conventional control

Reactor FT

FT

FC

FC

TT

AT

Temp

Comp.

Coolant

Product

TC

Feed

PID

Limited performance

Interaction

Saturation…

AC

Multivariable Predictive Control

Reactor

FT

FC

FC

TT AT

MPC

Comp.

Coolant Product

u1 u2

Feed

Temp

MPC considers all manipulated and controlled variables simultaneously, as well as the measured disturbances and constraints. It handles all interactions, disturbances and constraints using a process model in an optimal way improving the control performance.

TT

SP Temp SP Comp.

FT

Decision Hierarchy

In order to implement properly the actions taken in one level, the lower levels must work correctly.

In particular, if the control level does’nt work well, t is not possible to implement economic optimal decisions.

Level 0 Field Instrumentation

Level 1 Conventional Control

PID, DCS

Level 2 Advanced Control

Level 3 Optimization

Economic interest

Economic optimization and Control

Set Point

Set Point

Upper Constraint

Better control means smaller changes around the prescribed set points better product quality

Reduced variance allows to move set points respecting the constraints, giving room for optimization

Economic optimization and Control

Furnace

Dryer Air

Gas

pulp

Dried pulp

gases

MC MT

w

constraint

W1 W2

Moisture of the dryed pulp

SP

Improving control allows to move the moisture set point to a more profitable value, respecting the maximum moisture constraint

Model Predictive Control (MPC) MBPC

Control strategy based in the on-line use of a model to predict the future behaviour of the process output over a certain temporal horizon, as a function of the future control actions

The best control action is selected using an optimization procedure

Many methods sharing some common principles

Process Optimizer u w y v

Predictor y(t+j)

Model Predictive Control

u(t) u(t+j)

y(t+j|t) ^

t t+1 t+2 ...

time

past future

time

Set point

Output prediction CV

PV

MV

OP

SP

Model u y

MPC, Optimization

Process Optimizer u w y v

Predictor y(t+j)

[ ] [ ]∑ ∑=

−

=++∆β++−+=

2N

1Nj

1Nu

0j

22

),..1t(u),t(u)jt(u)jt(w)jt(yJmin

y(t+j) are related with u(t), u(t+1)…through the prediction equations computed with the process model

MPC, control implementation

u(t) u(t+j)

y(t+j|t) ^

t t+1 t+2 ...

time

past future

time

Set point

Output prediction CV

MV

SP

At time t, u(t) is applied to the process, but…

…at time t +1, the whole process is repeated with the information available Moving horizon policy

MPC Control implementation

t t+N2

w(t)

y (t)

r(t+j )y (t+j )

u(t-1)

t+NU

w

u

Y(t+1+j)

t+1

y(t+1)

u(t)

u(t-1)

MPC Key elements

Internal model to compute output predictions Set point Control variables structuring Optimization procedure to compute the best

control actions Moving horizon implementation

A bit of history…

J. Richalet (1978, ADERSA): IDCOM, MPHC, PFC, ... Prett, Gillete, Cutler, Ramaker (1979, Shell): DMC De Keyser, 1979: EPSAC D. Clarke, 1987: GPC Martín-Sánchez, 1984: APC Lemos, Mosca, 1985: MUSMAR .Morari Rawlings .Kouvaritakis, Scatolini, Allgower...

DMC Dynamic Matrix Control

Model: step response (linear) + disturbance

)t(n)it(ug)t(y1i

i +−∆= ∑∞

=

Predictions: )jt(n)ijt(ug)ijt(ug)jt(y1ji

i

j

1ii ++−+∆+−+∆=+ ∑∑

∞

+==

n t j n t y t g u t ip ii

( ) ( ) ( ) ( )+ = = − −=

∞

∑ ∆1

Disturbance model:

∑

∑∑∞

=

∞

+==

−∆−+

+−+∆+−+∆=+

1iip

1jii

j

1ii

)it(ug)t(y

)ijt(ug)ijt(ug)jt(yPredictions:

gi )1t(z)t(z)t(z −−=∆

DMC, Dynamic Matrix Control

∑∑∑∞

=

∞

+==

−∆−+−+∆+−+∆=+1i

ip1ji

i

j

1ii )it(ug)t(y)ijt(ug)ijt(ug)jt(y

∑

∑∑

∞

=+

++

∞

=

∞

+=

−∆−+=

−−∆−−∆−+−∆+−∆+=

=−∆−−+∆+=

1iiijpj

212j1jp

1ii

1jiipj

)it(u)gg()t(yp

....)2t(ug)1t(ug...)2t(ug)1t(ug)t(y

)it(ug)ijt(ug)t(yp

g g d N N Nj i i j i i+ +− ≅ − ≅ > =0 0 1 2 d i j, ,....,

In asymptotically stable systems:

∑=

+ −∆−+=N

1iiijpj )it(u)gg()t(yp Free response of the system

at time t

gi

DMC, Dynamic Matrix Control

( ) ( ) ( )y t j G q u t j pj j+ = + +−1 ∆

G q g q g qj jj( ) ......− − −= + +1

11

p free response

∑=

+ −∆−+=N

1iiijpj )it(u)gg()t(yp

p

u(t-1)

j

j

1ii p)ijt(ug)jt(y +−+∆=+ ∑

=

Predictions: Forced + free response

Notice that, in steady state, ∆u = 0 and pj = yp, so, the predictions correspond to the real process output, eliminating a possible offset

DMC, optimal choice of u

w

y(t+j)

N2

Prediction horizon

t time

u(t-1) u(t)

Nu Control horizon

t time

N1

∆u(t+j) = 0 j ≥ Nu Nu control horizon

Min Cost function:

[ ] [ ]J ( ) ( ) ( )= + − + + +∑∑=

−

=y t j w t j u t j

j

Nu

j N

N 2 2

0

1

1

2β∆

Control structuring:

( ) ( ) ( )y t j G q u t j pj j+ = + +−1 ∆

DMC

G =

+

− +

g gg g g

g g

N

N

N N Nu

1 1

1 1 2 1

2 2 1

0 00 0

0

... ...

... ...... ... ... ... ... ...... ... ... ... ... ... ...

... ... ... ... ...

It can be formulated in matrix form:

[ ]∆ ∆ ∆ ∆u( )' ( ), ( ),..., ( )t u t u t u t Nu= + + −1 1

min ∆u(t), ∆u(t+1),..

[ ]2N11N1N0 p)2Nt(w,...,p)11Nt(w,p)1Nt(w' −+−++−+= +e

[ ] [ ]J ( ) ( ) ( )= + − + + +∑∑=

−

=y t j w t j u t j

j

Nu

j N

N 2 2

0

1

1

2β∆

( ) ( ) ( )y t j G q u t j pj j+ = + +−1 ∆

[ ] 000 ')t('2)t(')t('J eeuGeuIGGu +∆−∆β+∆=

DMC

[ ] 000 ')t('2)t(')t('J eeuGeuIGGu +∆−∆β+∆=

[ ] 0uuGeG

0u

0 =∆β+∆−′−

=∆∂∂

22

J

[ ] 01 '')t( eGIGGu −β+=∆

If there are no constraints:

[ ] )('')t( j1 pwGIGGu −β+=∆ −

Integral action

Delay compensation Tuning: Horizons, β

Open loop stable processes

DMC offset

( ) ( ) ( )y t j G q u t j pj j+ = + +−1 ∆

p

pss

N

1iiijpj

yy

yp)it(u)gg()t(yp

=⇒

=⇒−∆−+= ∑=

+

In steady state:

DMC predictions:

And as

sspy =

So, the predictions in steady state are unbiased and, if the optimization drives the predictions to w, then the process output yp will be equal to the set point w, providing offset free control.

GPC Generalised Predictive Control

A q y t B q u t T q t( ) ( ) ( ) ( ) ( ) ( )− = − +−1 1 1

∆ ξ

Model: CARIMA

)t(y)q(T)q(F

)t(u)q(T)q(N

p p1

1j

1

1j

j −

−

−

−

+∆=

( ) ( ) ( )y t j G q u t j pj j+ = + +−1 ∆Predictions:

T qA q

E q qF q

A qjj j( )

( )( )

( )( )

−

−− −

−

−= +1

11

1

1∆ ∆B q E q

T qG q q

N qT q

jj

j j( ) ( )( )

( )( )( )

− −

−− −

−

−= +1 1

11

1

1

It can be used with open loop unstable systems

Tuning: N2, Nu, β, T(q-1), internal trajectory

PFC Predictive Functional Control

∑ −+=+

=+

+++=+

∑ −++=+

=

−

=

−

j

1i

1if

mj

L

fL

j

1i

1im

j

)1jt(BuCA)jt(y

)t(CA)jt(y

)jt(y)jt(y)jt(y

)1jt(BuA)t(AC)jt(y

m

m

x

x)t(Du)t(Cx)t(y

)t(Bu)t(Ax)1t(x

mm

mm

+=+=+

Model Predictions

∑µ=+=

bN

1iii )j(F)t()jt(u

Base functions

[ ]∑=

µµ+−+=

N

1j

2jj )ht(r)ht(yminJmin

)jt(n)j(y)t()t(CA)jt(y Bk

N

1kkm

jb

++∑µ+=+=

x

Hj coincidence points

r internal reference

∑ −==

−t

1ik

1iBk )1t(BFCA)t(y

Monoreg

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

Measurable disturbances

y t g u t i d v t i n tii

ii

( ) ( ) ( ) ( )= − + − +=

∞

=

∞

∑ ∑∆ ∆1 1

+ u

v

n

Measurable disturbance

Non-measurable disturbances

y

Controlled variable

DMC model:

Control action

Constraints

They appear naturally in the formulation of a practical control problem.

They can affect to the manipulated or controlled variables: – Saturation in the actuators – Smooth operation of a process – Variance of the controlled variable – Security or quality limits – ….

Constraints

Mm L)jt(yL ≤+≤

Lmax

Lmin

v deltav

Mm D)jt(uD ≤+∆≤

Mm U)jt(uU ≤+≤Range of control signals:

Rate of change of u:

Range of controlled variables:

Should be added to the computation of the optimal decisions

DMC with constraints

L y t j g u t k j p t j L

j N

m kk

j

M≤ + = − + + + ≤

==

∑ ( ) ( ) ( )

, . . . . ,

∆1

1 2

D u t j D

U u t u t j U

j Nu

m M

m Mi

j

≤ + ≤

≤ − + + ≤

==∑

∆

∆

( )

( ) ( )

, . . . ,

1

00

[ ] [ ]∑ ∑=

−

=

+∆β++−+=∆

2N

1Nj

1Nu

0j

22 )jt(u)jt(w)jt(yu

Jmin

u

Quadratic programming problem QP

QP

∆u(t)

∆u(t+1) 0x

bAx

xQ

≥≤

⋅⋅+⋅ xxc21 min

x

Efficient numerical software

A QP problem has to be solved on-line every sampling period

Feasibility

Constraints: Clipping?

∆ u(t+1)

∆u(t)

∆ umax

∆ umax

∆ u* ∆ u with clippling

Contours

Feasibility

Process u y

disturbances

Due to the process dynamics input constraints and disturbances, it may happens that there is no input able to maintain the output within its constraints. The QP problem has no solution.

Physical constraint Operating constraint Hard and soft

constraints

Feasibility (hard constraints)

Change the constraint

Constraint violation

Mm L)jt(yL ≤+≤ ε+≤+≤ Mm L)jt(yL

[ ] [ ] 22N

1Nj

1Nu

0j

22 )jt(u)jt(w)jt(y,uJmin ηε++∆β++−+=ε∆

∑ ∑=

−

=

New decision variable η weighting factor

Output constraint relaxation slack variable: ε ≥ 0

Feasibility

t

y

Prediction

t

u Future control

t

Prediction

t

Future control u

y

Input constraint relaxation

Instead of relaxing this…

Feasibility

Different feasibility polices Constraint Priorities Solved either as a separate problem or continuously

t+N3 t t+N4

Other option: changing the range of application of the constraint N3,N4

Feasibility, Penalty functions

Soft constraints: It is not critical if they are violated by a small amount

t

y

Prediction

[ ] [ ] ( )M

2N

1Nj

1Nu

0j

22 L)jt(yb)jt(u)jt(w)jt(yu

Jmin −+η++∆β++−+=∆

∑ ∑=

−

=

( )ML)jt(yb −+

)jt(y +LM

Multivariable formulation

A model relating the controlled variables with the manipulated ones and the measurable disturbances is needed

y ty t

y t

M MM M

M M

u tu t

u t

n tn t

n tn

m

m

n n nm m n

1

2

11 12 1

21 22 2

1 2

1

2

1

2

( )( )

....( )

...

( )( )

...( )

( )( )

...( )

=

+

M ... M ...

M ...

Reactor

FT

FC

FC

TT AT

MPC

Comp.

Coolant Product

u1 u2

Feed

Temp TT

SP Temp SP Comp.

FT

Multivariable MPC

y t g u t i g u t i g v t i n t

y t g u t i g u t i g v t i n t

y t g u t i g u t i g v t i n

ii

ii

ii

ii

ii

ii

ii

ii

ii

1 11 11

12 21

131

1

2 21 11

22 21

231

2

3 31 11

32 21

331

3

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) (

= − + − + − +

= − + − + − +

= − + − + − +

=

∞

=

∞

=

∞

=

∞

=

∞

=

∞

=

∞

=

∞

=

∞

∑ ∑ ∑

∑ ∑ ∑

∑ ∑ ∑

∆ ∆ ∆

∆ ∆ ∆

∆ ∆ ∆ t)

u1

u2

y1

y2

y3

v

modelo interno

Internal model

t

y yy t

t

t

y

t

y

t

y

y1

y2

u1 u2 v

Dynamic matrix

Multivariable MPC

J y t j r t j y t j r t j

u t j u t jj N

N

j

NU

= + − + + + − + +∑

+ + + +∑

=

=

−

γ γ

β β

1 1 12

2 2 22

1

2

1 12

2 22

0

1

( ( ) ( )) ( ( ) ( ))

( ) ( )∆ ∆

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

y t j g u t i j g u t i j p t j

y t j g u t i j g u t i j p t j

y t j g u t i j g u t i j p t j

ii

j

ii

j

ii

j

ii

j

ii

j

ii

j

1 11 11

12 21

1

2 21 11

22 21

2

3 31 11

32 21

3

+ = − +∑ + − +∑ + +

+ = − +∑ + − +∑ + +

+ = − +∑ + − +∑ + +

= =

= =

= =

∆ ∆

∆ ∆

∆ ∆

Predictions:

Min Cost function:

γ Equal concern errors β move suppression factors

With constraints: QP problem

Multivariable MPC

( ) ( )∑ ∑ ∑∑= =

−

==+∆+∆

+∆β++−+γ=N

1k

2N

1Nj

1Nu

0j

2kk

M

1K

2kkk)jt(u)jt(u

k

k

k

kk

)jt(u)jt(w)jt(yminJmin

with:

Dmk< ∆uk(t+j)< DMk Umk< uk(t+j)< UMk Lmk< yk(t+j)< LMk

)jt(p)ijt(ug)jt(y k

j

1isiks

n

1sk ++−+∆=+ ∑∑

==

MPC in state-space

[ ] [ ]∑ ∑=

−

=

+∆β++−+=+∆

2N

1Nj

1Nu

0j

22 )jt(u)jt(w)jt(y)jt(u

Jmin

)t(Cx)t(y)t(Bu)t(Ax)1t(x

=+=+

21Mm N,....,NjL)jt(yL =≤+≤

1N,....,0jD)jt(uD uMm −=≤+∆≤

1N,..,0jU)kt(u)1t(uUU)jt(uU uM

j

0kmMm −=≤+∆+−≤⇒≤+≤ ∑

=

uNj0)jt(u ≥=+∆

In order to evaluate J, it is necessary to compute the output predictions

Process model

Cost function

Computing output predictions

)t(Cx)t(y)t(Bu)t(Ax)1t(x

=+=+

∑

∑

=

−

=

−

−++=+

−++=+

+++++=+++=+

+++=+++=+

+=+

j

1k

kjj

j

1k

kjj

23

2

)1kt(BuCA)t(xCA)jt(y

)1kt(BuA)t(xA)jt(x

....)2t(Bu)1t(ABu)t(BuA)t(xA)2t(Bu)2t(Ax)3t(x

)1t(Bu)t(ABu)t(xA)1t(Bu)1t(Ax)2t(x)t(Bu)t(Ax)1t(x

But the state x(t) may be not accesible

Estimation of the model states

)t(Cx)t(y)t(Bu)t(Ax)1t(x

=+=+ As the model state x(t) normally it is not

measured, an state estimator or observer is needed

)t(xC)t(y))]1t(Bu)1t(xA(C)t(y[L)1t(Bu)1t(xA)t(x p

=

−+−−+−+−=

yp represents the measured process output

∑=

− −++=+j

1k

kjj )1kt(BuCA)t(xCA)jt(y

Output predictions are, then, computed by replacing the state x(t) by its estimation:

In steady state: Bu)A....AI(CxCABuCAxCA)jt(y 1jjj

1k

kjj −

=

− ++++=+=+ ∑

Observer design

)]t(xC)t(y[L)t(Ae)1t(e)t(x)t(x)t(e

)]t(xC)t(y[L)t(Bu)t(xA)1t(x

−+=+−=

−++=+

)t(Cx)t(y)t(Bu)t(Ax)1t(x

=+=+

There are many possible design criteria for choosing L. One of them considers the estimation error:

The model output is given by: )t(Cx)t(y =

)t(e)LCA()1t(e)t(LCe)t(Ae)]t(xC)t(Cx[L)t(Ae)1t(e

+=++=−+=+

L can be chosen so that the eigenvalues of (A+LC) are stable and provide the adequate speed of convergence to zero of the estimation error

How to estimate the model states x(t) measuring the model output y(t)?

In predictor form:

e(t) can be driven to zero by proper selection of L

Observer design in filter form

))]1t(Bu)1t(xA(C)t(y[L)1t(Ae)t(e)t(x)t(x)t(e

])1tt(xC)t(y[L)1tt(x)t(x)1t(Bu)1t(xA)1tt(x

−+−−+−=−=

−−+−=

−+−=−

)t(Cx)t(y)1t(Bu)1t(Ax)t(x

=−+−=

Model output is given by: )1t(CBu)1t(CAx)t(Cx)t(y −+−==

)1t(e)LCAA()t(e)1t(LCAe)1t(Ae)]1t(xCA)1t(CAx[L)1t(Ae)t(e

−+=−+−=−−−+−=

L can be chosen so that the eigenvalues of (A+ LCA) are stable and provide the adequate speed of convergence to zero of the estimation error

Predictor

Corrector

Estimating the states x(t) with information up to (t-1) plus y(t)

Estimation error

Time evolution of the estimation error

Offset free control

This formulation does not guarantee that, in steady state, the process output yp is equal to the set point w (being w feasible)

)]BuxA(Cy[LBuxAx p +−++=

The reason is that the observer, at steady state, will fulfill:

But this does not imply that )BuxA(Cy

BuxAx

p +=+=

p1jj yBu)A....AI(CxCA)jt(y ≠++++=+ −

So, the output predictions, in general, will be different from the process output

And the controller will lead the process to a wrong target, hence, an offset will appear

The model is not perfect!

Formulation of the cost function

[ ] [ ]∑ ∑=

−

=+∆+∆β++−+=

2N

1Nj

1Nu

0j

22

)jt(u)jt(u)jt(w)jt(yJmin

)t(u)1t(u)t(u)t(Cx)t(y

)t(Bu)t(Ax)1t(x

∆+−==

+=+The model and cost function must be consistently posed

[ ] [ ]∑ ∑=

−

=+∆+β++−+=

2N

1Nj

1Nu

0j

22

)jt(u)jt(u)jt(w)jt(yJmin

If the model is not perfect, the value of u that minimizes this cost function in steady state may be not be the one that makes the process output y equal to the set point w

J non-consistent

The value of ∆u=0 in steady state is compatible with any value of u that provides zero error in staeady state

Offset-free control

[ ] [ ]∑ ∑=

−

=+∆+∆β++−+=

2N

1Nj

1Nu

0j

22

)jt(u)jt(u)jt(w)jt(yJmin

)t(u)1t(u)t(u)t(Cx)t(y

)t(Bu)t(Ax)1t(x

∆+−==

+=+

)t(xC)t(y))]1t(Bu)1t(xA(C)t(y[L)1t(Bu)1t(xA)t(x p

=

−+−−+−+−=

In order to get offset-free control in spite of the model errors, two conditions are required: 1- The model and cost function must be consistently posed 2- The observer must include integral action in order to get unbiased output predictions

Bu)A...AI(CxCA)jt(y 1jj −++++=+At steady state, the output predictions are:

A biased estimation of y(t+j) leads to incorrect predictions in steady-state, hence to offset

a) Adding an integrating mode to the observer

[ ]

−∆

+

−−

−

+

+−∆

+

−−

=

)1t(uIB

)1t(u)1t(x

I0BA

0C)t(yLL

)1t(uIB

)1t(u)1t(x

I0BA

)t(u)t(x

pu

x

There are three ways of including an integrating mode to the observer: a) Expanding the observer by including u(t) in it b) Using a full velocity form in the model c) Adding a disturbance model

Estimating u(t), which is a known variable seems a bit unnatural, but using the estimated u for computing the output predictions leads to unbiased predictions

Where, in general, )t(u)t(u ≠

Adding an integrating mode to the observer

In steady state, with ∆u = 0:

[ ]

[ ][ ]

[ ]

uBxAx)uBxA(Cy

0)uBxA(CyL)uBxA(CyLuu

)uBxA(C)t(yLuBxAxux

I0BA

0C)t(yLL

ux

I0BA

ux

p

pu

pu

px

pu

x

+=

+=

=+−

+−+=

+−++=

−

+

=

And if Lu is full rank:

Providing unbiased estimations of the output in steady state if the predictions are computed with:

uB)A...AI(CxCA)jt(y 1jj −++++=+

b) Full velocity form of the model

)t(Cx)t(y)t(Bu)t(Ax)1t(x

=+=+

[ ]

)1t(u)t(u)t(u)1t(x)t(x)t(x

)t(y)t(x

I0)t(y

)t(uCBB

)t(y)t(x

ICA0A

)1t(y)1t(x

−−=∆−−=∆

∆=

∆

+

∆

=

++∆

[ ]I0CCBB

BICA0A

A

)t(y)t(x

z defining

vvv =

=

=

∆=

Is equivalent to:

The model can be written as:

)t(zC)t(y)t(uB)t(zA)1t(z

v

vv

=∆+=+

MPC in velocity form

( ) ( )

[ ]'w0....0w)t(zC)t(y

)t(uB)t(zA)1t(z

)jt(uR)'jt(uw)jt(zCQ'w)jt(zCJmin

v

v

vv

2N

1Nj

1N

0jvvvv

)jt(u

u

==

∆+=+

+∆+∆+−+−+= ∑ ∑=

−

=+∆

)jt(uB)A...AI(C)t(zAC)jt(y v1j

vvvjvv +∆++++=+ −

the target is to reduce to zero ∆x(t+j) and to drive y(t+j) as close as possible to w

∆=

)t(y)t(x

)t(zAs

The steady state output predictions will be:

[ ] [ ] [ ]ICA...CAICA...CA0A

I0ICA0A

I0AC jj

jjjvv ++=

++

=

=

with

plus constraints

MPC in velocity form

[ ] )jt(uB)A...AI(C)t(y)t(x

ICA...CA)jt(y v1j

vvvj +∆++++

∆++=+ −

y)jt(y0u,0x =+⇒→∆→∆

The estimator of z is:

[ ]

−∆

+

−−∆

−

+

+−∆

+

−−∆

=

∆

)1t(uCBB

)1t(y)1t(x

ICA0A

I0)t(yLL

)1t(uCBB

)1t(y)1t(x

ICA0A

)t(y)t(x

py

x

And as

MPC in velocity form

[ ]

[ ] [ ]yyLyyandyyL00u0x

uCBB

yx

ICA0A

I0yLL

uCBB

yx

ICA0A

yx

pypx

py

x

−+=−=→∆→∆

∆

+

∆

−

+∆

+

∆

=

∆

In steady state:

If Lx or Ly are full rank, then y)jt(y =+yyp = and as

pyy)jt(y ==+ The predictions are unbiased and the process output presents no offset provided the controller drives the model in z to the set point and reaches steady state

c) Adding a disturbance model

)t(u)1t(u)t(u)t(Cx)t(y

)t(Bu)t(Ax)1t(x

∆+−==

+=+Instead of the model: an expanded one including state and output disturbances is used:

[ ]

=

+

=

+++

)t(v)t(d)t(x

V0C)t(y

)t(u00B

)t(v)t(d)t(x

I000I00DA

)1t(v)1t(d)1t(x Here d(t) and v(t)

are disturbances in the states and the output respectively

Adding a disturbance model

[ ][ ])1t(vV))1t(dD)1t(Bu)1t(xA(C)t(y

LL

)1t(v)1t(d

)t(v)t(d

)1t(vV))1t(dD)1t(Bu)1t(xA(C)t(yL

)1t(dD)1t(Bu)1t(xA)t(x

pv

d

px

−−−+−+−−

+

−−

=

−−−+−+−−+

+−+−+−=

The observer equations are:

[ ][ ]

[ ] 0vV)dDBuxA(C)t(yLL

vV)dDBuxA(C)t(yLL

vd

vd

vV)udDBuxA(C)t(yLdDBuxAx

pv

d

pv

d

px

=−++−

−++−

+

=

−++−+++=In steady state:

so

Adding a disturbance model

[ ]

vVxC)t(ydDBuxAx

vV)dDBuxA(C)t(y

0vV)dDBuxA(C)t(yLL

p

p

pv

d

+=++=

+++=

=−++−

If [Ld`, Lv’]’ is full rank:

So, the extended model provides unbiased output estimation in steady state if the disturbances are incorporated to the model

The offset removal policy, in this case, uses fictitious targets, such that, when given to the MPC, the process reaches the desired set point w. It consists, then, of two stages: the computation of the fictitious targets and the computation of the modified MPC

and from the equation of x:

Modified steady state targets

( ) ( )

maxsmin

ss

sss

ssu,x

uuuvVCxy

dDBuAxx

ywQ'ywminss

≤≤+=

++=

−−

Given the estimated disturbances, we look for states xs and inputs us satisfying the steady state estimation equations and constraints and providing output estimations as close as possible to the desired set point w. They can be obtained solving the optimization problem:

Notice that, if the set point w is reachable, this problem will provide a steady state value ys = w and:

vVCxw s +=

Q weighting matrix

Modified MPC problem

R,S weighting matrixes

[ ]

1N,..,0j,u)jt(uu1N,..,0j,uu)jt(uu

)1jt(u)jt(u)jt(uNj,0)jt(uu)1t(u)1t(u

x)jt(x)jt(x)jt(uB)jt(xA)1jt(x

)jt(uS)'jt(u)jt(uR)'jt(u)jt(xQC'C)'jt(x(min

umaxmin

umaxsmin

us

s

0j)1Njt(u),..,1jt(u u

−=∆≤+∆≤∆−=≤++≤

−+−+=+∆≥=+−−=−

−+=++++=++

+∆+∆++++++∑∞

=−++++

If there is a stable, feasible solution, this modified problem will drive the optimal states and controls to zero, that is, in steady state:

0uuu,0xxx ss =−==−=

ps yvVxCvVCxw =+⇒+=And considering the solution of the observer and modified targets in ss:

Providing offset free control

MPC – Optimal control

u(t)u(t+j)

y(t+j|t)y(t+j|t)^

t t+1 t+2 ...

time

past future

time

Set point

Output predictionCV

PV

MV

OP

SP

Modelu y

MPC is NOT optimal control

An open loop optimization problem is solved every sampling time, starting in the current state

u(t), u(t+1), u(t+2) are considered independent variables in the optimization problem

ProcesoControlleruw y

SP

u(t)=kx(t)

In optimal control u(t+j) are not independent and the target is the control law (k)

Analysis - Stability

With finite horizon N2, there is no guarantee that, even with a perfect model, the application of the “optimally computed” control signals u(t) of MPC leads to a stable closed loop system

w

y(t+j)

N2

Prediction horizon

t

time

u(t-1) u(t)

Nu Control horizon

t time

N1

w

y(t+j)

N2

Prediction horizon

t time

N1

Stability

State at time t

Target

x1

x2

u(t-1) u(t)

Nu Control horizon

t time

State after ∆t

The key point to understand why with finite horizon it is possible to obtain closed loop instability, even if there is no disturbances and the model is perfect, is the fact that, if the predicted trajectory does not reach the target within the prediction horizon time, the solution of the open loop optimization starting at time t with N steps ahead, can be different from the solution of the problem starting at time t + ∆t with N steps ahead, hence, the state of the process at t +2∆t does not need to be in the original trajectory, etc.

State after 2∆t

Infinite horizon

State at time t

Target

x1

x2

u(t-1) u(t)

Nu Control horizon

t time

State after ∆t

State after 2∆t If the target can be reached within the prediction horizon, what happens when the prediction horizon is infinite, then one can apply the Bellman optimality principle to conclude that, as any part of an optimal trajectory is also optimal, the trajectories starting at the state at time t and t + ∆t must coincide. Hence, if the first open loop trajectory is stable, the closed loop trajectory is also stable.

State space model

Given the process x(t+1) =Ax(t)+Bu(t), at every sampling time t, compute the control signals v(0),.…v(N-1) that minimize:

And applying u(t) = v*(0,x(t)) bring the state to the origen starting at x0 at time 0. Q, R, P positive definite matrices

)t(x)0(z,)j(Bv)j(Az)1j(zwith

)j(Rv)'j(v)j(Qz)'j(z)N(Qz)'N(zmin))t(x(J1N

0jvN

=+=+

++= ∑

−

=

Stability

If (A, B) is stabilizable, the solution is unique and at every time t is given by: v(j)= KN-jz(j) with KN-j obtained from the Riccati equations then u(t)=v*(0,x(t))= KNx(t) and susbtituting in the process equation: x(t+1)=Ax(t)+B KNx(t) = (A+B KN)x(t) The stability is not guarantee, but depends on the values of A+B KN Tuning with Q, R, N

Use of JN as a Lyapunov function

As JN is positive definite and JN(0)=0, it can be used as a Lyapunov function for studying stability. Closed loop stability requires: JN(x(t)) > JN(x(t+1))

x1

x2

x0

JN along the system trayectories

++= ∑

−

=

1N

0jvN )j(Rv)'j(v)j(Qz)'j(z)N(Qz)'N(zmin))t(x(J

Optimality principle

J x t J x tx t Qx t u t Ru t J x t J x t

N N

N N

( ( )) ( ( ))( ) ( ) ( ) ( ) ( ( )) ( ( ))' '

− + =

+ + + − +−

11 11

This expression must be positive in order to guarantee stability

The term x t Qx t u t Ru t' '( ) ( ) ( ) ( )+ is positive

But the term J x t J x tN N− + − +1 1 1( ( )) ( ( )) is of unknown sign

Using the optimality principle, and substracting JN(x(t+1)) : t

t+1

x(t)

x(t+1) optimal trayectory from t

t+N

Terminal constraints

))1t(x(J)j(Rv)'j(v)j(Qz)'j(z

)1N(Rv)'1N(v)1N(Qz)'1N(zmin

0)N(z if

)j(Rv)'j(v)j(Qz)'j(z)N(Qz)'N(zmin))1t(x(J

1N2N

0j

v

1N

0jvN

+=

++

+−−+−−=

=

++=+

−−

=

−

=

∑

∑

because v(N-1) has no influence nor in z(N-1) neither in z(N-i) and the minimization should then give: v(N-1)=0 In this way, JN(x(t)) > JN(x(t+1)) and the closed loop system is asymptotically stable

Stability guarantee

w

y(t+j)

N1 N2 t time

Coincidence horizon

Terminal constraints

Clarke, Scatolini

The output predictions were forced to coincide with the set point w for n sampling times

)mt(ub....)1t(ub)nt(ya....)1t(ya)t(y m1n1 −++−+−−−−−=

[ ] [ ]minu

y t j r t j u t jj

Nu

j N

N

J ( ) ( ) ( )∆

= + − + + +=

−

=∑∑ 2 2

0

1

1

2

β∆

y(t+j) = Gj(q-1)∆u(t+j) + pj j = 1,…, N2+M y(t+j) = w j = N2,.....N2+M

Add these constraints

Feasibility?

Infinite horizon

If N→ ∞, ⇒ JN-1 (x(t+1)) - JN(x(t+1)) → 0 and the term x(t)’Qx(t)+u(t)’Ru(t) dominates over

JN-1 (x(t+1)) - JN(x(t+1)) so that is positive and the system is asymptotically stable How to solve the constrained problem with an infinite

number of constraints?

))1t(x(J))1t(x(J)t(Ru)t(u)t(Qx)t(x

))1t(x(J))t(x(J

N1N''

NN

+−+++

=+−

−

Stability: Infinite horizon

[v0, v1, v2, …., vN-1, 0,0,……]

[v0, v1, v2, …., vN-1, KzN, KzN+1 ,……]

w

y(t+j)

N1 N2 ⇒ ∞ Prediction horizon

t time

[ ] [ ]∑∑−

=

∞

=

+∆β++−+1Nu

0j

2

1Nj

2 )jt(u)j()jt(w)jt(yminu

Rawlings, Muske 1993 Input parameterization

Provides stability, but how to solve the constrained problem with an infinite number of constraints?

Rawlings and Muske 1993

QP using problem thesolve and sconstraint impose topossible isit

)j(Rv)'j(v)j(Qz)'j(z)N(Pz)'N(zmin))t(x(J

QA'AP :defining now,

)N(zQA'A)'N(z)N(zQA)')N(zA()j(Qz)'j(z

and )N(zA)jN(z implies this,Njfor 0= v(j)as

)j(Rv)'j(v)j(Qz)'j(z)j(Qz)'j(zmin

)j(Rv)'j(v)j(Qz)'j(zmin))t(x(J

1N

0jv

j

0j

j

j

0j

jj

0j

j

Nj

j

Nj

1N

0j

1N

0jv

0j

1N

0jv

++=

=

==

=+≥

=

++=

=

+=

∑

∑

∑∑∑

∑ ∑∑

∑ ∑

−

=

∞

=

∞

=

∞

=

∞

=

∞

=

−

=

−

=

∞

=

−

=

)j(Bv)j(Az)1j(z +=+

Computation of P

(dlyap) equation matrix Lyapunov QPA'AP:ngsubstracti

........QA'AQA'AQA'APA'A

.......QA'AQA'AQA'AQQA'AP

3322

3322j

0j

j

=−

+++=

++++== ∑∞

=

If A has stable eigenvalues Aj does not diverge and:

If A has some unstable eigenvalue, a terminal constraint must be added to eliminate the unstable modes It requires the origen be reachable in order to be finite

z j Qz jj N

( )' ( )=

∞

∑

Robust MPC

What happens if the model is not perfect?

θ≤θ≤θθρ∀≤ρ≤

≤≤

−ρρ

)( y),u(yy

uuu

w),u(ymax minu

The control actions that minimize the cost function J should be selected in order to minimize the worst possible prediction error depending on the changing or not precisely known parameters ρ(θ)

w

t+j

y

t

[ ] QR)t( )2Nt(),...2t(),1t( ∈θ+θ+θ+θ=ρ[ ])12Nt(u),...1t(u),t(uu −++=

Infinity norm

∞ norm corresponds to the largest deviation of the model output over the set point along the prediction horizon

|| y - w ||∞

w y

y - w for linear systems is a hyperplane function of θ, then, || y - w ||∞ is a convex hypersurface defined by the maximum of the hyperplanes. As θ is constrained to the convex set , then must be located in one of the 2q vertex of polytope Ω

θθθ ≤≤ ∞ρ−θρ w)(,u(ymax

Robust MPC

)ρ,v,u,jt(emaxmaxw)ρ,v,u(ymax ij,iρρ+=−

∞Minimize

is equivalent to maximize an upper bound of ei with respect to u over the vertex of Ω

Ω de vertices2 los de uno θ

wy)ρ,u(ewyuuu

0μμ)ρ,u(eμ

μmin

q

μ,u

−≤≤−

≤≤

≥≤≤−

Large LP programs

Economic optimization

Reactor

FT

FC

FC

TT AT

MPC

Comp.

Coolant Product

u1 u2

Feed

Temp TT

w Temp w Comp. Which is the most profitable operation point?

How to adjust the set points wTemp, wComp to this point?

FT

Economic optimization

MBPC

w

y SP

DCS Process

Economic Optimization

Most of commercial MPC incorporate an economic optimization layer

Trend: Merge both layers in a single MPC drive by an economic cost function

Economic optimization

Reactor

FT

FC

FC

TT AT

MPC

Comp.

Coolant Product

u1 u2

Feed

Temp TT

w Temp w Comp. Economic cost function:

(Product*concentration*price-

– row material*price –

-flow refrigerant*price)*

period of time

Max Benefit

under operating constraints and dynamical model

FT

Operating with constraints

Valve opening

Temperature

Turbine speed

∆ P in the column Pressure

Composition

Operator’s prefered region

Final region

New region of operation

Less variance

DMC Plus

min Linear function of u in steady state ∆u with the constraints: y = G ∆u+p in steady state range of u and y The solution provides ss SP for CV and IRV w = G ∆u*+p IRV = u*

t+N2

LP Optimization

MV1

MV2 Constraint: Values from the LP

IRV

IRV

unconstrained DMC control

DMC Plus

Predictions computation with the internal model

Field values

Unconstrained MPC with IRV settings

Static economic LP at t+N2

IRV SP

Constraints Prices

u

HITO Economic Optimization

Which is the best reachable state in the future? Local optimization around the operating point Control, disturbances and constraints are taken into account Local dynamic models The procedure is repeated every sampling time The targets are always feasible

x state in t Reachable states in t+N2

x optimal state in t+N2 state in t+1 x

x optimal state in t+1+N2

HITO

Predictions computation with the internal model

Field values

Constrained MPC

Economic QP Dynamic optimization

SP

Constraints Prices

u

Key to success…

Easy to understand by the users Can be applied to multivariable processes with

different number of manipulated and controlled variables

Compensates measurable disturbances and delays Takes into account constraints on MV and CV Can be applied to processes with difficult dynamics:

delays, inverse response, unstable systems, slow processes,…

Opens the door to economic optimization of the process operation

MPC

Without model

Controller tuning

Control specification

Modelling

Identification

Process analysis

...but it needs a dynamic process model

With model

Industrial acceptance

Typical application

FEED

PDI

FI

AI

FC

VAP 50#

FC

CONDENSATE

SP

CONTROLLER DMC

TI

AI

AI

SP

From several to dozens of MV, DV and CV

Distillation column

Fast MPC

,..

,min)(*

θ+≤

+θ+=θ

SbAzts

zczFHzzV TTTT

z

Multiparametric quadratic programming

–(Bemporad, Morari, Dua, y Pistikopoulos 2002)

Multiparametric MPC

It is possible to obtain a closed loop expression for the control action as a function of the current state

At each region an explicit control formula

Problems with the number of regions

BB converter

δ

−=

∆

+

δ

=

+δ+++

)()()()(

]1010[

0)()()()(

00000

)1()1()1()1(

ttdtvctil

Vo

dIB

ttdtvctil

IIBA

ttdtvctil

Changes in the input with disturbances δ = Vo(t)-Vo Est (t/t)

BB converter

0 0.005 0.01 0.015 0.02 0.0250

20

40

60

80

100

120

Time (seg)

Vo

(Vol

ts)

Respuesta del BBC

VoVref

Tiempo (mseg) V Referencia Carga R 0 -12 80 5 -100 80

10 -56 80 15 -56 40 20 -56 120

Non-linear MPC (NMPC)

u)jt(uu y)jt(yy

u(t))g(x(t),y(t) ))t(u),t(x(fdt

)t(dx

d))(u),(x(Fu

Jmin pTt

t

≤+≤≤+≤

==

τττ=∆

∫+

Uses an internal non-linear model:

First-principles

Volterra series

Neural network

Wiener, Hammestain,….

Non-linear MPC (NMPC)

u)jt(uu y)jt(yy

u(t))g(x(t),y(t) ))t(u),t(x(fdt

)t(dx

d))(u),(x(Fu

Jmin pTt

t

≤+≤≤+≤

==

τττ=∆

∫+ Dynamic

optimization problem solved every sampling time

How to solve the dynamic optimization problem?

How to estimate the state?

Sequential solution

Process

Simulation from N1to N2 to compute J(u,x(t))

NLP Optimizer

u J

w

u(t)

y(t)

Difficult to impose constraints on x or y

[ ] [ ]

u)jt(uu y)jt(yy

u(t)g(x(t),y(t) ))t(u),t(x(fdt

)t(dx

)jt(u)jt(w)jt(yu

Jmin2N

1Nj

1Nu

0j

22

≤+≤≤+≤

==

+∆β++−+=∆

∑ ∑=

−

=J is evaluated by dynamic simulation of the model

Simultaneous solution

[ ] [ ]

u)jt(uu y)jt(yy.....

1)))))u(t2),g(x(t2)y(t ))2t(u),2t(x(f)3t(x1))u(t1),g(x(t1)y(t ))1t(u),1t(x(f)2t(x

u(t)g(x(t),y(t) ))t(u),t(x(f)1t(x

)jt(u)jt(w)jt(yx,uJmin

2N

1Nj

1Nu

0j

22

≤+≤≤+≤

++=+++=+++=+++=+

==+

+∆β++−+=∆

∑ ∑=

−

=

Now the states x(t+1), x(t+2), ….. are decision variables in addition to u(t), u(t+1),….which results in a large NLP problem

It is easy to impose constraints on x or y (path constraints)

Can be solved with current SQP / IP methods

Discretization of a continuous stiff model can be difficult

Moving Horizon Estimation (MHE)

t-N t

Which initial state in time t-N and minimal disturbances d(t-i) would drive the process in the closest way to the actual output trayectory if the control actions were the ones actually applied ?

t-N

t u

y

d

x

[ ]

Vv

N

0i

22md,x

L)it(dL))t(u),t(x(g)t(y

))t(d),t(u),t(x(f)1t(x

)it(d)it(y)it(yminiNt

≤−≤=

=+

−γ+−−−∑=−

Moving Horizon Estimation (MHE)

t-N t

Once x(t-N) and d(t-N), d(t-N+1),…have been computed optimally, x(t) can be estimated by simulation of the model starting at x(t-N) and with inputs u(t-N), u(t-N+1),…. And disturbances d(t-N), d(t-N+1),…

t-N

t u

y

d

x

[ ]

Vv

N

0i

22md,x

L)it(dL))t(u),t(x(g)t(y

))t(d),t(u),t(x(f)1t(x

)it(d)it(y)it(yminiNt

≤−≤=

=+

−γ+−−−∑=−

Van der Vusse Reactor

DA2CBA

→→→

Highly non-linear reactor difficult to control

NMPC reactor

0.80

1.00

1.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

TIME

CbLinf[1]Lsup[1]refy[1]

0

200

400

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

TIME

FUinf[1]Usup[1]

-10000

-5000

00.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

TIME

QkUinf[2]Usup[2]

MV: Product flow MV: Heat removal

2.0

2.5

3.0

3.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

TIME

Cax_estim[1]

CV: Composition

370

375

380

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

TIME

T0w_estim[1]

Estimated Input temp.

Estimated CA

Stability guarantee

u)t(uu y)t(yyu(t))g(x(t),y(t) ))t(u),t(x(f)t(x

dRu'uQx'xu

Jmint

≤≤≤≤==

τ+=

•

∞

∫

Similar results that in the linear case. Nominal stability can be assured with infinite horizon and, under certain conditions, the NMPC with infinite horizon can be transformed into an equivalent MPC with terminal penalty

u)t(uu y)t(yyu(t))g(x(t),y(t) ))t(u),t(x(f)t(x

dRu'uQx'x)T(Px)'T(xu

JminT

t

≤≤≤≤==

τ++=

•

∫

Stability guarantee: Optimization of terminal constraints

w (set-point)

y(t+j)

t+N1 t+Np t

time Interval for stability constraints t+Np+Nm

d

(Prada et al., 1998)

• Terminal constraints to guarantee stability over the range [Np, Np+Nm] but not on the set point.

• Minimization of the distance d to the controller set-point, to assure a suitable response without stationary error.

Prediction horizon

Key idea: to split the problem in two parts:

Chemical reactor

mol/l 1C0K 380TK 340K 330TK 280

A

c

≤≤≤≤≤≤

Operation Constraints:

Significant nonlinearities and varying time constants

The objective is to control T and CA by manipulating Tc.

Tc=300+5 K

Tc=300-5 K

CSTR Tc

CA, T

CAf

Coolant stream temperature

T

CA

Concentration of A in the feed

Starting from the equilibrium point

=

=

=

K 300T K, 350T

,l/mol 5.0C

eqc

eq

eqA

Problem formulation

r

y(t+j)

t time

Interval for stability constraints

d

=++−++

=++−++

d)t/NNt(r)t/NNt(y

d)t/1Nt(r)t/1Nt(y

mpmp

pp

Mm

Mm

Mm

D)t/jt(uDU)t/jt(uUY)t/jt(yY

≤+∆≤≤+≤≤+≤

extra degree of freedom

t+N1 t+Np t+Np+Nm

d∈ℜ

[ ] [ ] 21Nu

0j

2tt

t

2

d,ud)jt(ud))(r)(y(min

pk

k

λ++∆β+ττ−τγ ∑∫−

=

+

∆

extra term

Simulation example: Tracking performance

5T ;K 305TK 270 cc ≤∆≤≤

Input constraints Prediction horizon: Np=15

Control horizon: Nu=2

NMPC Stable NMPC

T

CA

TC

T

CA

TC

Nm=3

Simulation example: Tracking performance

1T ;K 305TK 275 cc ≤∆≤≤

Input constraints Prediction horizon: Np=8

Control horizon: Nu=2

NMPC Stable NMPC

T

CA

TC

T

Nm=3