1 Approaches to increase the range of use of Model predictive control Miguel Rodriguez Advisor:...

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Approaches to increase the range of use

of Model predictive control Miguel Rodriguez

Advisor: Cesar De Prada Systems Engineering and Automatic Control Department

University of Valladolid, Spain

Pisa, October 2008.

2

Outline

• Motive

• Control explicit using Multiparametric Programming

• Approaches for using NMPC in Hybrid Systems– Mixed continuous-batch processes

– Hybrid system

• Reduced Order Model

• Conclusions

Approaches to increase the range of use of Model predictive control

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Motive

• MPC is used to controlling a wide range of process industrials.

• MPC is capable of operating without expert intervention for long periods time.

• Centralized control, Multi-level, complex plants.

• Constraint handling, Input saturation, states constraints, etc.

• But– MPC to require a time of calculation to find the optimal control signal.

– The time of calculation is increased when the systems are Hybrid or Nonlinear.

– If optimization time is higher that the response time, MPC is impossible to apply.


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Motive

• The main objective is find techniques that decreased the optimization time and retain all benefits of MPC approach.

• Three approach are presented in this work– Control explicit using Multiparametric Programming

– Approaches for NMPC to Hybrid Systems• Mixed continuous-batch processes

• Hybrid system

– Reduced Order Model


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Control explicit using Multiparametric Programming

• Linear MPC without constraints

Using the steady states model and making predictions to the horizon prediction


2Nt

0t

22

21

u)t(u)t(w)t(yJmin

12

0

122

1

0

12

)(~~~

)(ˆ~~

)(~~~

)(ˆ~~

)(~~

)(ˆ~~

)|2(ˆ

)|2(ˆ)|1(ˆ

N

i

iNN

i

i

ikuBACkxAC

ikuBACkxAC

kuBCkxAC

kNky

kkykky

Y

6


• Linear MPC without constraints Then

Explicit Solution:where


HUkxFY )(ˆ

RUUWYQWYJ TT )()(

))(ˆ()( 121 kxFWHRQHHU TT

TNukukukuU )]1()1()([ Tkukxkx )]1()([)(ˆ

7


• Linear MPC with constraints

Using multiparametric programming, z is dependent variable of the current states x and the system constraints.

with the KKT conditions, we can found, of way iterative, the explicit solution into the region where it solution is valid.


),(min)(* xzVxVz

0),(. xzgts

0),( xzh

8


• Linear MPC without constraints

finally, we have an explicit solution for each region CRi


]')(...)1()([

)(

nCRi

CRiCRiCRi

HtututuU

FtxHU

CR0

CR1

CR2

CR3

CR4

CR5

CR6

CR7

X

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• DC-DC Converter (Buck-Boost type)

Average Model (continuous conduction mode)

Search method: binary search tree (Tondel, Johansen and Bemporad, 2002)


Control

mp-QP

EcoSimPro

Set point

V0 , IL

d, R

MatLab

States estimator

States 00

~VV

d

Disturbance model

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• DC-DC Converter (Buck-Boost type) Controller partition with 51 regions


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• DC-DC Converter Comparison between Slide Model Control and mp-MPC.


0 0.005 0.01 0.015 0.02 0.025 0.030

12

20

Vo

0 0.005 0.01 0.015 0.02 0.025 0.03

40

80

120

R

0 0.005 0.01 0.015 0.02 0.025 0.030

100

200

Vo

0 0.005 0.01 0.015 0.02 0.025 0.03

40

80

120

R

0 0.005 0.01 0.015 0.02 0.025 0.03

12

100

150

Vo

0 0.005 0.01 0.015 0.02 0.025 0.0320

40

60

R

0 0.005 0.01 0.015 0.02 0.025 0.03

12

100

150

Vo

0 0.005 0.01 0.015 0.02 0.025 0.03100

120

140

Time (sec)

R

0 0.005 0.01 0.015 0.02 0.025 0.030

12

20

Vo

0 0.005 0.01 0.015 0.02 0.025 0.03

40

80

120

R

Sliding Mode Control mp-MPC

0 0.005 0.01 0.015 0.02 0.025 0.030

100

200

Vo

0 0.005 0.01 0.015 0.02 0.025 0.03

40

80

120

R

0 0.005 0.01 0.015 0.02 0.025 0.03

12

100

150

Vo

0 0.005 0.01 0.015 0.02 0.025 0.0320

40

60R

0 0.005 0.01 0.015 0.02 0.025 0.03

12

100

150

Vo

Sim

ula

tion 4

Sim

ula

tion 3

Sim

ula

tion 2

Sim

ula

tion 1

0 0.005 0.01 0.015 0.02 0.025 0.03100

120

140

Time (sec)

R

Load (Ohms)

Ref (Volts)

V0 (Volts)

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• Feasibility of implementing the controller.– Ts=0.1 ms

– State Estimator ( 16 multiplications + 32 assignations ) ≈ 50 cycles

– Signal capture ( 2 input, Voltage and Current) ≈ 4 cycles

– Search of region (51 regions X 5 operations ) ≈ 205 cycles

– Calculation control signal (3 multiplications + 3 assignations) ≈ 6 cycles

– Output PWM (3 assignations) ≈ 6 cycles

– Total of cycles ≈ 270 cycles

Controller frequency

Standard floating point DSP controller (Texas, Microchip, etc)Approaches to increase the range of use of Model predictive control

0,27)(1.0

º)(

segTs

cyclesNMhzF

13

Approaches for NMPC to Hybrid Systems.

• Mixed continuous-batch processes– Parallel Production Line

• Hybrid system– Solar Air conditioning plant


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Parallel Production Line

• The benchmark is a chemical process proposed by UCL, Belgium


Reactant

Hot steam

Cool water

AC1

AC2

Storage Tank FoutST

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• Aims of control– Maximize the productivity in the presence of uncertainties and disturbances.

– Maximize the output flow of storage tank and hold transfer continuously to downstream processing stage.

– To avoid the total discharge in the storage tank.

• Decision variables– Standby times for filling, heating and discharging of both autoclaves.

– Outflow of B product from storage tank (FoutST)

• Non-measured disturbances– Change in the temperature of hot steam (Th)


dtTSTF

VVVVVVJ

tConstrainsout

STSTSTST

predictionend

STrefST

)()/1(...

...)()()(

54

2max3

2min2

_

0

21

16

Tcontraints=Toverlap fill+Toverlap Heat


Tconstraints


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Parallel Production Line (Simulation Results. Overview)


Values of the parameters of Objective Function:

Values of the weights

Almost, for each batch unit, 3 batches are predicted, 2 of them are controlled. So, Np=3 and Ncb1=Ncb2=2.

4 changes for classical continuous variables FoutST

VSTref VSTmax VSTmin FoutST Tconstraints

Weight (ai) 0.1 1 1 1 100

VSTref VSTmax VSTmin

25 40 10

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Parallel Production Line (Simulation Results. Overview)


Manipulated variableControlled variable

Batch sequences

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Solar Air conditioning plant (Description)

• The absorption machine• Energy supply systems

– solar collector – gas heater – Accumulation tank

• Aims of control– Maintaining the chilled

water temperature (75º-95º)

– Minimize the gas used

• Decision variables– Continuous

vB1, vm3

– Discretemode of operation (set of on/off valves)


Problem MINLP very complex

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Solar Air conditioning plant (Embedded Logic Control)

• Fictitious variable u to represent the energy supply to the plant

• Definition of a set of rules

• Integration of the rules and the fictitious variable

• Solution of the associated optimization problem every sampling period

• Objective function


Embedded logic control rules of Operation

Hu

i

Hp

refgiU

iuTT1

2

0

2)(min

22

Sequential approach to dynamic optimization

Solar Air conditioning plant (Controller implementation)


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Solar Air conditioning plant (Results)

• Simulation results


Manipulated variables

Controlled variable

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Solar Air conditioning plant (Results)

• Test real plant


Manipulated variables

Controlled variable

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Reduced Order Model (The open plate reactor)

It combines heat exchanger and micro-reactor: more efficient but more difficult to control

Controlled variables: γ conversion

Ti temperature along the reactorManipulated variables:

uB1, uB2, feed flows rates of B TfeedA, temperature of reactant A, Tcool the cooling temperature.

A + B → C

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The open plate reactor (Dynamics and Aims)

Aim: Finding a reduced dynamic model that facilitates the use of NMPC

Start-up of the plate reactor avoiding ‘hot spots’

Highly non-linear distributed process

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The open plate reactor (Proper Orthogonal Decomposition POD)

A field x can be represented as a complete series of orthonormal globally defined functions i

A projection is made on a subspace retaining % of the energy of the signals, allowing to obtain a model with a smaller number of ODE’s

1)()(),(

iii ztcztx

DFM

POD

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The open plate reactor (NMPC)

NMPC based on continuous time formulation with the POD model and a sequential approach for the NLP problem

POD model

Results of the reactor start-up

Inpu

ts

Out

puts

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Conclusions

• Approaches to apply NMPC in Fast-System and hybrid system have been presented.

• Transformation of problem discrete variables to Continuous variables.

• To use NLP approach to solver Mixed integer no linear problem.

• A study of feasibility to implementing mp-MPC in controllers has been presented.


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Reference

• M. Rodríguez, D. Sarabia, and C. de Prada: Hybrid Predictive Control of a Simulated Chemical Plant. Taming Heterogeneity and Complexity of Embedded Control Systems , International Scientific & Technical Encyclopedia (ISTE), London, Editors: F. Lamnabhi-Lagarrigue, S. Laghrouche, A. Loria and E. Panteley, pp. 617-634, 2007

• M. Rodríguez, C. De Prada, F. Capraro, S. Cristea, and R. M. C. De Keyser: Hybrid Predictive Control of a Solar Air Conditioning Plant. 17th IFAC World Congress, Seoul, Korea, 2008.

• D. Sarabia, C. de Prada, and S. Cristea: A Mixed Continuous-Batch Process: Implementation of Hybrid Predictive Controllers. Proc. 7th IFAC Symp. on Advances in Control Education, 2006, Paper-ID: 148.

• M. Rodríguez, C. de Prada, A. A. Alonso, C. Vilas and M. García: A nonlinear model predictive controller for the start-up of a open plate reactor, International Workshop on Assessment and Future Directions Of Nonlinear Model Predictive Control, Pavia, Italy, 2008.


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Questions?

• Thank you.


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