· este exemplar foi revisado e alterado em relação à versão original, sob responsabilidade...

LUZ ADRIANA ALVAREZ TORO

Strategies with guarantee of stability for the integration of Model

Predictive Control and Real Time Optimization

Tese apresentada à Escola Politécnica de

São Paulo da Universidade de São Paulo

para obtenção do Título de Doutor em

Engenharia.

SÃO PAULO

2012

LUZ ADRIANA ALVAREZ TORO

Strategies with guarantee of stability for the integration of Model

Predictive Control and Real Time Optimization

Tese apresentada à Escola Politécnica de

São Paulo da Universidade de São Paulo

para obtenção do Título de Doutor em

Engenharia.

Área de concentração: Engenharia

Química

Orientador: Prof. Dr. Darci Odloak

SÃO PAULO

2012

Este exemplar foi revisado e alterado em relação à versão original, sob responsabilidade única do autor e com a anuência de seu orientador. São Paulo, 11 de abril de 2012. Assinatura do autor ____________________________ Assinatura do orientador _______________________

FICHA CATALOGRÁFICA

FICHA CATALOGRÁFICA

Alvarez Toro, Luz Adriana

Strategies with guarantee of stability for the inte gration of model predictive control and real time optimization / L.A. Alvarez Toro. -- ed.rev. -- São Paulo, 2012.

173 p.

Tese (Doutorado) - Escola Politécnica da Universida de de São Paulo. Departamento de Engenharia Química.

1. Controle preditivo 2. Tempo-real I. Universidade de São Paulo. Escola Politécnica. Departamento de Engenhar ia Química II. t.

Strategies with guarantee of stability for the integration of MPC and RTO ___________________________________________________________________________________________________________

ACKNOWLEDGEMENTS (In Portuguese)

Ao Professor Darci Odloak pela competência com que orientou e acompanhou esta Tese.

Para ter me dado a oportunidade de trabalhar em Controle Preditivo e de fazer uma

pesquisa enriquecedora.

Aos membros da Banca Examinadora: os Professores Luis Claudio Oliveira, Claudio

Garcia, Oscar Sotomayor e Dr. Antonio Carlos Zanin, pela leitura, correção e sugestões

de melhora desta Tese.

Aos Professores Galo Le Roux e Roberto Guardani do LSCP e ao Dr. Lincoln Moro da

Petrobrás pelas críticas e comentários sobre uma primeira versão deste trabalho.

Ao Professor Oscar A. Z. Sotomayor pelas discussões e pela sua colaboração nos

primeiros meses quando recém cheguei no Brasil.

Ao Professor Reinaldo Giudici pela resposta a algumas dúvidas na parte de polimerização.

Aos Professores Hernán Alvarez e Jairo Espinosa da Universidad Nacional de Colombia

por terem me motivado a continuar pesquisando em controle.

À Professora Cristina Borba da EP-USP por todas as ferramentas fornecidas no curso de

redação acadêmica em inglês, foram muito úteis para escrever esta Tese.

Aos companheiros e o pessoal do LSCP e do Departamento de Engenharia Química.

À minha familia na Colômbia e meu noivo Christophe.

Às agências de fomento FAPESP (Proc. 08/57511-9) e CNPq (Proc. 141787/2008-2) pelo

apoio financeiro.


ABSTRACT

The aim of this Thesis is the development of predictive controllers (MPC) with guarantee

of stability and that are part of a control structure where Real Time Optimization (RTO) is

present and produces optimizing targets for the predictive controller. The approaches of

two and three-layer are considered. Three different strategies are presented: the first

strategy is developed for integrating systems; it consists on an infinite horizon MPC

algorithm in two extended versions. This controller is designed to be implemented in the

two-layer structure. Simulation results in a linear system with integrating and stable modes

show the ability of the MPC controller to follow input targets from the RTO layer even

when there are targets for integrating systems. As a second strategy, it is developed an

algorithm that guarantees nominal stability of the MPC controller when it interacts with the

intermediary layer of the three-layer structure. In order to produce a robust structure, an

extension to uncertain systems is also developed. This approach is tested with both a linear

and a nonlinear system. For the nonlinear system, which is an industrial process, the full

structure that includes the RTO with the robust algorithm is simulated. Results show that

the structure is capable to follow target changes when disturbances affect the process.

Finally, the last strategy proposed in this Thesis consists on the inclusion of a convex

function in the MPC controller to follow the RTO targets. The gradient of this convex

function is considered in a quadratic term in the objective function of the MPC controller.

This controller is also simulated with both a linear system and a nonlinear system. For the

MPC with gradient, an extension to the case of uncertain systems is developed in order to

provide a more robust controller. For the strategies developed in this Thesis, theorems that

guarantee the recursive feasibility and the convergence of the closed-loop system are

provided.


RESUMO

O propósito desta Tese é desenvolver controladores preditivos (MPC) com garantia de

estabilidade e que são parte de uma estrutura onde a otimização em tempo real (RTO)

produz targets otimizantes para o controlador preditivo. As aproximações de duas e três

camadas foram consideradas. Três diferentes estratégias são apresentadas: A primeira

estratégia é desenvolvida para sistemas com pólos integradores; esta consiste em um

algoritmo MPC de horizonte infinito em duas versões estendidas. Este controlador é

formulado para implementação em uma estrutura de duas camadas. Os resultados de

simulação em um sistema linear com modos estáveis e integradores mostram a capacidade

do controlador MPC para seguir targets nas entradas vindos da camada RTO mesmo

quando há targets para sistemas integradores. Como segunda estratégia, foi desenvolvido

um algoritmo que garante estabilidade nominal do controlador MPC quando este interage

com a camada intermediária da estrutura em três camadas. Além da versão nominal, é

desenvolvida uma extensão deste controlador para sistemas com incerteza, o controlador

resultante tem estabilidade robusta. Esta aproximação é testada com um sistema linear e

um sistema não linear. Para o sistema não-linear, que é um processo industrial, é simulada

a estrutura completa incluindo a RTO com o algoritmo robusto. Os resultados mostram que

a estrutura é capaz de seguir mudanças nos targets quando os distúrbios afetam o processo.

Finalmente, a última estratégia proposta nesta Tese consiste na inclusão de uma função

convexa no controlador MPC para seguir os targets. O gradiente desta função convexa é

considerada num termo quadrâtico na função objetivo do controlador MPC. Este

controlador é simulado com um sistema linear e um sistema não linear. Com o objetivo de

desenvolver uma versão robusta, o MPC com gradiente foi estendido ao caso de sistemas

com incertezas. Para todas as estratégias apresentadas em esta Tese, foram formulados

teoremas que garantem a viabilidade recursiva e a convergência do sistema em malha

fechada.

Strategies with guarantee of stability for the integration of MPC and RTO

_______________________________________________________________________

NOMENCLATURE

0ny Matrix with entries equal to zero and dimension ny

A State matrix of the state-space model in discrete time

B Input matrix of the state-space model in discrete time

C Output matrix of the state-space model in discrete time

Cu Weight of the deviation of the input targets in the objective function of the Target

Calculation layer (Chapter 3)

Cy Weight of the deviation of the output targets in the objective function of the

Target Calculation layer (Chapter 3)

Cε Weight of the slack variable in the objective function of the Target Calculation

layer (Chapter 3)

d Measured disturbance (Chapter 1)/ gradient vector of the economic function or

convex function (Chapter 4)

0D Static gain matrix defined in the OPOM model for the states xs, see Appendix B

0Dɶ Matrix constructed from D0 appearing in the terminal state constraint of the

infinite horizon MPC

0iD Static gain matrix of the model i

0nD Static gain matrix of the nominal model

0tD Static gain matrix of the true model

dD Matrix defined in the OPOM model for the states xd, see Appendix B

diD Matrix D

d of the OPOM model (Appendix B) for the model i

dnD Matrix D

d of the OPOM model (Appendix B) for the nominal model


_______________________________________________________________________

dtD Matrix D

d of the OPOM model (Appendix B) for the true model

iD Matrix defined in the OPOM model for the states xi, see Appendix B

1imD Matrix constructed from D

i appearing in the terminal constraint for the integrating

state xi of the infinite horizon MPC for integrating systems (Chapter 2)

2imD Matrix constructed from D

i appearing in the terminal state constraint of the

infinite horizon MPC for integrating systems (Chapter 2)

D0 Zero-th order moment of the dead-polymer

D1 First order moment of the dead-polymer

D2 Second order moment of the dead-polymer

0f Hypothetic fixed value for the economic function

feco Economic function

fi Initiator efficiency in the styrene polymerization process

F Matrix defined for the OPOM model, see Appendix B

Fe Economic function or convex function

Fi Matrix F of the OPOM model (Appendix B) for the model i

Fn Matrix F of the OPOM model (Appendix B) for the nominal model

Ft Matrix F of the OPOM model (Appendix B) for the true model

G Hessian matrix in the gradient calculation of the economic function or convex

function

( )sG Transfer function matrix

hA Overall heat transfer coefficient of the styrene reactor

I Initiator specie of the styrene polymerization process

Iɶ Matrix constructed from Inu appearing in the input target constraint of MPC

[I] Concentration of specie I in the styrene reactor

[I f] Concentration of specie I in the feed of the styrene reactor

nuI Identity matrix with dimension nu

nyI Identity matrix with dimension ny


_______________________________________________________________________

*nyI Matrix defined in the OPOM model for integrating systems, see Appendix B

ki Rate constant for initiation reaction in the styrene polymerization process

kd Rate constant for initiator decomposition in the styrene polymerization process

kp Rate constant for propagation reaction in the styrene polymerization process

kt Rate constant for termination reaction in the styrene polymerization process

Kui Parameter of the convex function of the targets for the input i

Kyi Parameter of the convex function of the targets for the output i

L Number of models that define the uncertainty

m Control horizon of the MPC

M Monomer specie in the styrene reactor in the styrene polymerization process

[M] Concentration of specie M in the styrene reactor

[M f] Concentration of specie M in the feed of the styrene reactor

mM Molecular weight of the monomer specie

nM Number-average molecular weight of the polymer

wM Weight-average molecular weight of the polymer

na Number of poles of the linear system

nd Total number of stable poles in the multivariable system, it is the product nu.ny.na

nu Number of inputs

nut Number of targets for the inputs

ny Number of outputs

nyt Number of targets for the outputs

N Matrix defined for the OPOM model, see Appendix B

P Weight of the gradient of the convex function, see Chapter 4

[P] Total concentration of live-polymers in the styrene reactor

Pn Live-polymer chain with size n in the styrene polymerization process

PD Polydispersity of the polymer

Q Weight of the output predition error in the objective function of MPC

Q Terminal weight of the infinite horizon MPC


_______________________________________________________________________

Qc Flow rate of cooling jacket fluid in the styrene reactor

Qi Initiator flow rate in the styrene reactor

iQ Nominal value for the initiator flow rate in the styrene reactor

Qm Monomer flow rate in the styrene reactor

mQ Nominal value for the monomer flow rate in the styrene reactor

Qs Solvent flow rate in the styrene reactor

Qt Total flow rate in the styrene reactor

Qu Weight of the input target in the objective function of MPC

Qy Weight of the output prediction error in the objective function of MPC

R Radical specie of the styrene polymerization process

R Weight that penalizes the input changes in the objective function of MPC

S Weight of the slack variable in the objective function of MPC

Si Weight of the slack for the integrating state in the objective function of MPC

Su Weight of the input slack in the objective function of MPC

Sy Weight of the output slack in the objective function of MPC

T Temperature of the styrene reactor

Tc Temperature of the cooling jacket fluid in the styrene reactor

Tcf Temperature of the cooling jacket fluid in the feed of the styrene reactor

Tf Temperature of styrene reactor feed

Tn Dead-polymer chain with size n in the styrene polymerization process

u Input of the process system

u0 Initial value of the input

udes Input targets calculated by the Target Calculation layer

iu Nominal value of the input i

umax Maximum bound for the input

umin Minimum bound for the input

uRTO Input targets calculated by the RTO layer

uS,max Maximum value of the input for scaling


_______________________________________________________________________

uS,min Minimum value of the input for scaling

uscaled Scaled value of the input

U Feasible set of the inputs

V Volume of the styrene reactor

cV Volume of the cooling jacket of the styrene reactor

Vk Objective function at time step k

x State of the process system

xi Integrating component of the state of the OPOM model, see Appendix B

xd Stable component of the state of the OPOM model, see Appendix B

xs Integrated state component produced by the incremental form of the OPOM

model, see Appendix B

0sx Initial value of the state xs

y Output of the process system

ˆ( )y ∞ Stationary prediction of the output

ˆ ( )∞y Vector of stationary prediction of the outputs

y0 Initial value of the output

ydes Output targets calculated by the Target Calculation layer

ymax Maximum bound for the output

ymin Minimum bound for the output

yRTO Output targets calculated by the RTO layer

yS,max Maximum value of the output for scaling

yS,min Minimum value of the output for scaling

yscaled Scaled value of the output

ysp Set-point for the output

Greek letters

δ Slack variable of the infinite horizon MPC

iδ Slack variable for the integrating state


_______________________________________________________________________

sδ Slack variable for the outputs in the infinite horizon MPC

uδ Slack variable for the inputs

yδ Slack variable for the outputs in the infinite horizon MPC

rH∆ Heat of polymerization

t∆ Sample time

( )u k∆ Single input move at time step k

ku∆ Vector of input moves from time step k to 1m −

u∆ Total move of the input vector from time step 1k − to 1k m+ −

maxu∆ Maximum move of the manipulated input

ε Slack variable of the Target Calculation layer

η Intrinsic viscosity of the polymer

θ Set of matrices of the OPOM model for which the uncertainty is defined

tθ Set of matrices with uncertainty of the OPOM model for the true model

nθ Set of matrices with uncertainty of the OPOM model for the nominal model

iθ Set of matrices with uncertainty of the OPOM model of the model i

pCρ Mean heat capacity of styrene reactor fluid

c pcCρ Heat capacity of cooling jacket fluid of the styrene reactor

u uς +∆ Second order approximation of the gradient

Φ Matrix defined for the OPOM model, see section 3.2

Ψ Matrix defined for the OPOM model, see Appendix B

Ω Set of possible plants when there is model uncertainty

Strategies with guarantee of stability for the integration of MPC and RTO ______________________________________________________________________________________________________________

CONTENTS

Page

CHAPTER 1. INTRODUCTION………………………………………………………….1

1.1 Real Time Optimization and Model Predictive Control………………………...1

1.2 Stability of MPC controllers…………………………………………………….4

1.3 Zone Control…………………………………………………………………….6

1.4 Outline of the Thesis………………………………………………………….....8

1.5 Publications……………………………………………………………………...8

CHAPTER 2. STABLE MODEL PREDICTIVE CONTROL WITH INPUT

TARGETS FOR INTEGRATING SYSTEMS…………………………………………….10

2.1 Introduction……………………………………………………………………10

2.2 Prediction model for the integrating system……………………………………11

2.3 Stable MPC with input targets in one step……………………………………..12

Theorem 2-1………………………………………………………………15

2.4 Stable MPC with input targets in two steps…………………………………24

Theorem 2-2………………………………………………………………..26

2.5 Example: The desiobutanizer distillation column……………………………...28

2.6 Conclusions…………………………………………………………………….36

CHAPTER 3. ROBUST INTEGRATION OF MODEL PREDICTIVE CONTROL

AND REAL TIME OPTIMIZATION……………………………………………………..38

3.1 Introduction……………………………………………………………………38

3.2 Structure of the prediction model……………………………………………...39

3.3 Stable RTO/MPC……………………………………………………………..40

Theorem 3-1………………………………………………………………..43

3.4 Robust RTO/MPC…………………………………………………………….46

Theorem 3-2………………………………………………………………..49


3.5 Example. Linear system………………………………………………………..51

3.5.1 Simulation of Stable RTO/MPC……………………………………...51

3.5.2 Simulation of Robust RTO/MPC…………………………………….58

3.6 Example. Nonlinear system……………………………………………………61

Process description…………………………………………………………61

Control system……………………………………………………………...65

3.6.1 Simulation of Stable RTO/MPC……………………………………...67

3.6.2 Simulation of Robust RTO/MPC…………………………………….71

3.6.3 Comparison to a typical MPC controller……………………………..75

Case A………………………………………………………………76

Case B………………………………………………………………80

3.6.4 Complete RTO with the robust structure……………………………..83

3.7 Conclusions…………………………………………………………………….89

CHAPTER 4. STABLE MODEL PREDICTIVE CONTROL WITH GRADIENT

OF A CONVEX FUNCTION OF THE TARGETS……………………………………….91

4.1 Introduction…………………………………………………………………….91

4.2 MPC with infinite horizon and input targets…………………………………...92

4.3 Integration of RTO and MPC through the economic gradient………………....95

4.3.1 Gradient of the economic function…………………………………...95

4.3.2 Model predictive control with real time optimization………………..96

4.4 Stable MPC with gradient of a convex function……………………………….97

Theorem 4-1………………………………………………………………100

Remark 1………………………………………………………………….109

Remark 2………………………………………………………………….111

4.5 Example: Linear system………………………………………………………111

4.6 Example: Polymerization reactor……………………………………………..116

4.6.1 Control system………………………………………………………116

4.6.2 Simulation results…………………………………………………...117

4.7 Extension to uncertain systems: Robust MPC………………………………..124

Theorem 4-2………………………………………………………………126


4.8 Example: System with polytopic uncertainty………………………………...129

4.9 Conclusions…………………………………………………………………...134

CHAPTER 5. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK…………135

REFERENCES……………………………………………………………………………139

APPENDIX A: Derivation of the model equations for the styrene reactor ……..……….144

APPENDIX B: Output Prediction Oriented Model (OPOM)…………………………….152


LIST OF FIGURES

Page

CHAPTER 1

Figure 1.1. Integration of MPC and RTO in a two-layer structure………………………….2

Figure 1.2. Integration of MPC and RTO in a three-layer structure………………………...3

Figure 1.3. Integration of MPC and RTO in an one-layer structure………………………...4

Figure 1.4. Process Output controlled by zones…………………………………………….7

CHAPTER 2

Figure 2.1. Schematic diagram of the deisobutanizer column…………………………….29

Figure 2.2. Inputs Targets (− − −) and inputs of the deisobutanizer column with

Controller I (− · − ·−) and Controller II (——)……………………………………………33

Figure 2.3. Outputs of the deisobutanizer column with Controller I (− · − ·−) and

Controller II (——)……………………………………………………………………...34

Figure 2.4. Objective functions of Controller I (− · − ·−) and Controller II (——) for

the deisobutanizer column……………………………………………………………….35

Figure 2.5 Slack variable for the integrating output of the deisobutanizer column with

Controller I (− · − ·−) and Controller II (——)………………………………………….36

CHAPTER 3

Figure 3.1. System outputs () and computed targets ydes (⋅⋅⋅⋅⋅⋅) for reachable

optimizing target ( )…………………………………………………………….53

Figure 3.2. System inputs () and computed targets udes (⋅⋅⋅⋅⋅⋅) for reachable

optimizing target ( )………………………………………………………….....53

Figure 3.3. Objective function of problems P3-1a and P3-1b for reachable optimizing

target……………………………………………………………………………………...54

Figure 3.4. System outputs () and computed targets ydes (⋅⋅⋅⋅⋅⋅) for unreachable



Figure 3.5. System inputs () and computed targets udes (⋅⋅⋅⋅⋅⋅) for unreachable


Figure 3.6. Objective function of problems P3-1a and P3-1b for unreachable optimizing

target……………………………………………………………………………………...56

Figure 3.7. System outputs () and computed targets ydes (⋅⋅⋅⋅⋅⋅) with model

mismatch…………………………………………………………………………………57

Figure 3.8. System inputs () and computed targets udes (⋅⋅⋅⋅⋅⋅) with model

mismatch…………………………………………………………………………………57

Figure 3.9. Uncertain system: outputs (), computed output targets ydes for model 1

( ⋅ ⋅) and model 2 (⋅⋅⋅⋅⋅⋅) and optimizing output target ( )………………59

Figure 3.10. Uncertain system: inputs (), computed targets udes (⋅⋅⋅⋅⋅⋅) and

optimizing input target ( )……………………………………………………...60

Figure 3.11. Uncertain system objective function of problems P3-2a and P3-2b, model 1

() and model 2 ( )……………………………………………………....60

Figure 3.12. Process diagram of the styrene polymerization reactor……………………...62

Figure 3.13. Stable RTO/MPC simulation. Outputs (black solid line), calculated output

targets (blue dashed line), RTO output target (green dashed line) and output zone (red

dashed line)………………………………………………………………………………69

Figure 3.14. Stable RTO/MPC simulation. Inputs (black solid lines), calculated input

targets (blue dashed lines), RTO input target (green dashed line)………………………70

Figure 3.15. Stable RTO/MPC simulation. Objective functions of P3-1a and P3-1b…….70

Figure 3.16. Output values T and η for the steady states where each linear model MN,

M1, M2 and M3 where obtained………………………………………………………...72

Figure 3.17. Robust RTO/MPC simulation. Outputs (black solid lines), calculated

output targets (blue dashed lines), RTO output target (green dashed line) and output

zone (red dashed lines)…………………………………………………………………..74

Figure 3.18. Robust RTO/MPC simulation. Inputs (black solid lines), calculated input

targets (blue dashed lines), RTO input target (green dashed line)………………………74

Figure 3.19. Robust RTO/MPC simulation. Objective functions of P3-2a and P3-2b.......75

Figure 3.20. Process outputs with the robust structure in Case A. Red dashed line:


Output bounds. Blue dashed line: calculated targets. Green dashed line: RTO targets.

Solid line: Process outputs………………………………………………………………78

Figure 3.21. Process inputs with the robust structure in Case A. Blue dashed line:

calculated input targets. Green dashed line: RTO input targets. Solid line: Process

inputs…………………………………………………………………………………….78

Figure 3.22. Process outputs for conventional MPC controller in the Case A. Red

dashed line: Output bounds. Blue dashed line: RTO targets. Solid line: Process

outputs…………………………………………………………………………………..79

Figure 3.23. Process inputs for conventional MPC in Case A. Blue dashed line: RTO

input targets. Solid line: Process inputs…………………………………………………79

Figure 3.24. Outputs with robust structure in Case B. Red dashed line: Output bounds.

Blue dashed line: calculated targets. Green dashed line: RTO targets. Solid line:

Process outputs…………………………………………………………………………..81

Figure 3.25. Process inputs with robust structure in Case B. Blue dashed line: calculated

input targets. Green dashed line: RTO input targets. Solid line: Process inputs………...81

Figure 3.26. Process outputs for conventional MPC in Case B. Red dashed line: Output

bounds. Blue dashed line: RTO targets. Continuous line: Process outputs……………...82

Figure 3.27. Process inputs for conventional MPC in Case B. Blue dashed line: RTO

input targets. Solid line: Process inputs………………………………………………….82

Figure 3.28. Process outputs with the RTO operating in presence of disturbances. Red

dashed line: Output bounds. Blue dashed line: calculated targets. Green dashed line:

RTO targets. Solid line: Process outputs………………………………………………...87

Figure 3.29. Process inputs with RTO operating in presence of disturbances. Blue

dashed line: calculated input targets. Green dashed line: RTO input targets. Solid line:

Process inputs…………………………………………………………………………....87

Figure 3.30. Objective functions of the robust algorithm with the RTO operating in

presence of disturbances………………………………………………………………....88

Figure 3.31. Production rate of the process with the RTO operating in presence of

disturbances………………………………………………………………………………88

Figure 3.32. Polydispersity of the produced polymer with the RTO operating in presence

of disturbances…………………………………………………………………………...89


CHAPTER 4

Figure 4.1. System outputs and zones of the linear system controlled by the stable MPC

with gradient of a convex function……………………………………………………...113

Figure 4.2. System inputs and input targets of the linear system controlled by the stable

MPC with gradient of a convex function……………………………………………….114

Figure 4.3. Gradient of the convex function when linear system is controlled by the

stable MPC with gradient…………………………………………………………….....114

Figure 4.4. Objective function of the the stable MPC with gradient of a convex

function…………………………………………………………………………………115

Figure 4.5. Process Outputs of the gradient controller (blue solid line) and output zone

limits (red dashed line)………………………………………………………………….119

Figure 4.6. Process Inputs of the gradient controller……………………………………..119

Figure 4.7. Evolution of the economic function f of the process with gradient MPC……120

Figure 4.8. Evolution of the components of the gradient vector d of the economic

function…………………………………………………………………………………120

Figure 4.9. Process Outputs of the gradient controller (blue solid line), the target

controller (pink solid line) and output zone limits (red dashed line)…………………...122

Figure 4.10. Process Inputs of the gradient controller (blue solid line), the target

controller (pink solid line) and input targets (red dashed line)…………………………123

Figure 4.11. Evolution of the economic function f of the process with gradient MPC

(blue line) and target MPC (pink line)………………………………………………….123

Figure 4.12. System outputs (solid line) and outputs set-points (dashed line) for the

Robust MPC with gradient of a convex function of the targets………………………..131

Figure 4.13. System Inputs (solid line) and Input targets (dashed line) for the Robust

MPC with gradient of a convex function of the targets………………………………...132

Figure 4.14. Objective function of the nominal model for the Robust MPC with gradient

of a convex function of the targets……………………………………………………...133

Figure 4.15. Components of the gradient of the convex function for the Robust MPC

with gradient of a convex function of the targets……………………………………….133


LIST OF TABLES

Page

CHAPTER 3

Table 3.1. Process parameters for the polymerization reactor……………………………..65

Table 3.2. Steady-state operational condition for the polymerization reactor……………..66


1

CHAPTER 1

INTRODUCTION

Model Predictive Control (MPC) refers to a class of computer control algorithms that

utilize an explicit process model to predict the future response of a plant. At each control

interval an MPC algorithm attempts to optimize future plant behavior by computing a

sequence of future manipulated variable adjustments. The first input in the optimal

sequence is then sent to the plant, and the entire calculation is repeated at subsequent time

intervals. Predictive control was originated in the late seventies and has been developed

considerably over the last few years, both within the research control community and in

industry. The reason for this success can be attributed to the fact that Model Predictive

Control is, perhaps, the most general way of posing the process control problem in the

time domain (Camacho and Bordons, 2004). It can be used to control a great variety of

processes, from those with relatively simple dynamics to other more complex ones. MPC

is also the most popular strategy to deal with multivariable control problems, and presents

a series of advantages, for instance, process constraints can be handled naturally. Another

interesting feature of the predictive control is the natural way of introducing feedforward

control to compensate for measured disturbances and to combine this with the feedback,

as it takes the last output measure to predict the future behavior of the plant. Originally

developed to meet the specialized control needs of power plants and petroleum refineries,

the MPC technology can now be found in a wide variety of application areas in the

process industry.

1.1. REAL TIME OPTIMIZATION AND MODEL PREDICTIVE CONTROL

Today, in a globalized market, the purpose of the control system cannot be just to keep

the controlled variables at their set-points or to nicely track dynamic set-point changes,


2

but to operate the plant such that the net return of investment is maximized in the

presence of disturbances and uncertainties, exploiting the available measurements (Engell,

2007). The economic objective is actually specified through “set-points” or “targets” for

some process variables. The control system has to bring those process variables to those

optimal values. Thus, in practice, MPC controllers are implemented as part of a

multilevel hierarchy of control functions (Ying and Joseph, 1999; Kassmann et al., 2000;

Tatjewski, 2008; Darby et al., 2011), as depicted in Figure 1.1.

RTO(Nonlinear steady-state model)

MPC(Dynamic model)

Process System

,RTO RTOy u

( )u k ( )y k

Figure 1.1. Integration of MPC and RTO in a two-layer structure

Here, the upper layer is a plant-optimizer which executes a Real Time Optimization

(RTO). It consists of a nonlinear optimization in which the economic objective is

optimized subject to process constraints, including the rigorous nonlinear steady-state

model of the process. The RTO determines the optimal economic steady-state settings for

the process units of the chemical plant. These settings may be represented as targets to

some or all of the system inputs as well as set-points to some of the system outputs. The

MPC controller receives these targets and its task is to move and maintain the plant as

close as possible to these “optimal” values. The frequency of execution of the RTO layer

is lower than the MPC layer, operating as a cascade structure. Notice that in this case the


3

presence of disturbances can shift the economic optimum from the initial point, computed

by the real time optimizer. Since the RTO routine is not executed at the same frequency

as the MPC controller, the process may operate at suboptimal conditions until the next

updating of the RTO targets. This is the reason to insert a steady-state target optimizer

between the RTO and the MPC layers, which results in the three-layer structure shown in

Figure 1.2.

, ,,des k des ky u

RTO(Nonlinear steady-state model)

Target Calculation(Linear steady-state model)

Process System

,RTO RTOy u

( )u k ( )y k

MPC(Dynamic model)

( )d k

Figure 1.2. Integration of MPC and RTO in a three-layer structure

This intermediary routine is executed at the same sampling frequency as the MPC

controller. At each time step, it computes feasible steady-state operating points, assuming

that the RTO layer produces piecewise constant optimizing reference inputs. In this way,

the three-layer structure allows the control system to track changes in the optimum

caused by disturbances, guaranteeing offset free performance. The three-layer structure is

the most traditional way to integrate RTO with MPC (Ying and Joseph, 1999). Although

still not frequently implemented in practice, a multi-layer approach can also be followed


4

to integrate the Dynamic Real Time Optimization (D-RTO) and MPC (Würth et al., 2009,

2011).

RTO + MPC(Dynamic model)

Process System

( )u k ( )y k

Figure 1.3. Integration of MPC and RTO in an one-layer structure

On the other hand, some studies (Zanin et al., 2002; Biegler and Zavala, 2009; De Souza

et al., 2010; Ochoa et al., 2010) have attempted to integrate the RTO layer to the MPC

controller, in such a way that both the economic objective and the dynamic regulation are

solved in one single optimization routine. This approach is denoted as one-layer structure

and is shown in Figure 1.3. The strategies proposed in this Thesis for the integration of

RTO and MPC considers the two and three-layer approaches.

1.2. STABILITY OF MPC CONTROLLERS

An essential issue in the application of MPC control to industrial processes is stability. In

comparison to the classical control, the stability of MPC controllers should be

independent of the parameters of the controller. A popular approach to obtain a stable

MPC consists in adopting an infinite prediction horizon (Rawlings and Muske, 1993). For

stable systems, the infinite-horizon open-loop objective function is reduced to a finite-

horizon objective by defining a terminal state weight, which is computed as the solution

of a discrete-time Lyapunov equation. The stability of this controller was demonstrated

for the regulator case of systems with constraints on the inputs and states. So it deals only

with the regulator problem where the system steady-state lies at the origin. On the other


5

hand, the regulator of Rawlings and Muske does not include the case of unmeasured

disturbances. The use of a model in the positional form described by eq. (1.1) can result

in offset.

( 1) ( ) ( )

( ) ( )

x k Ax k Bu k

y k Cx k

+ = +=

(1.1)

To handle this, in Odloak (2004), it is suggested to use a model in the incremental form

as defined in eq. (1.2).

( 1) ( ) ( )

( ) ( )

x k Ax k B u k

y k Cx k

+ = + ∆=

(1.2)

This work is based on an infinite horizon controller, which uses for prediction a model

representation denoted as OPOM (Output Prediction Oriented Model). A terminal state

constraint is also incorporated and to soften this constraint, slack variables are included in

the MPC optimization problem. This controller has nominal stability and was also

extended to uncertain systems.

In practice, when the predictive controller is integrated to a plant-optimizer, the MPC

algorithm has to be designed for target tracking, which can compromise the stability of

the MPC controller. Furthermore, disturbances can turn the RTO targets infeasible or

unreachable for the MPC, mainly when the process operates under the two-layer structure.

Infeasible references for constrained systems were initially treated by adding a

compensating system called reference governor that modifies, when necessary, the

reference in such a way that constraints are satisfied (Bemporad et al., 1998; Gilbert and

Kolmanovsky, 1999; Casavola et al, 2000). For the case where stability is forced through

the endpoint constraint, Rossiter et al. (1996) propose to base this constraint on a slack

rather than on the actual set-point if the controller is infeasible. While the controller

remains infeasible, the control objective is to minimize the slack, and when feasibility is

achieved, the control cost based on the true set-point is minimized. Rawlings et al. (2008)

defines a steerable set-point located at the minimum distance from the unreachable set-

point, and the endpoint constraint is based on the steerable set-point. For the tracking of


6

unreachable steady-states, Limon et al. (2008) define an artificial steady-state for the

system inputs and outputs that are considered as additional decision variables of the

control problem. The endpoint constraint is written in terms of the artificial steady-state

and the cost is extended with a term that penalizes the deviation between the artificial

steady-state and the desired one. A similar approach is followed by Shead and Rossiter

(2007) who define the reachable steady-state in terms of a vector of unknown parameters

that become additional decision variables of the dual mode MPC. In Gonzalez and

Odloak (2009), the controller assumes that the process system has targets for some of the

inputs and outputs, and the remaining outputs are controlled by zones.

Usually, the real processes are not rigorously represented through linear models, but only

an approximated representation is obtained. In that case, the stability is only guaranteed if

the model used for prediction is exactly the same plant model, this is known as nominal

stability. In order to improve the stability of the closed-loop, the model uncertainty

should be included in the MPC controller. For several classes of uncertainty, many robust

approaches have been proposed (Badgwell, 1997; Rodrigues and Odloak, 2003; Odloak,

2004) and apparently they can be applied in practice without major difficulties (Porfirio

et al., 2003). In this Thesis the robust stability of the integration of MPC and RTO is

adressed in Chapters 3 and 4.

1.3. ZONE CONTROL

In some chemical processes, the exact value of the controlled outputs is not important, as

long as they remain within specified boundaries, or zones, as shown in Figure 1.4. This is

what is called zone control (Maciejowski, 2002). This strategy is desired, for instance,

when the aim is to drive the feed rate to its maximum value subject to constraints. Also,

the zone control is adopted in some systems, where there are highly correlated outputs to

be controlled and where there are no inputs enough to control them independently. When

the zone control is considered, the aim of the MPC is to maintain the outputs inside

appropriate zones instead of guiding all of them to fixed set-points.


7

In practice, one way to implement zone control in predictive controllers is setting to zero

the term corresponding to the output prediction error in the objective function of the MPC

when the controller outputs are inside the zone (Zanin et al., 2002; De Souza et al., 2010).

If any output overpasses the desired zone, the contribution of the output error is activated

for this output and the value of the set-point is the one corresponding to the exceeded

bound, ymax or ymin.

Figure 1.4. Process Output controlled by zones

The infinite horizon MPC was also proposed to stable systems with zone control for the

system outputs (Gonzalez and Odloak, 2009). To implement the zone control, the output

set-point is considered as an additional variable of the control problem. This new decision

variable of the control problem needs to be restricted to satisfy the minimum and

maximum bounds of the controlled variables. To implement the input target in the infinite

output horizon context, inputs that have targets are treated as outputs, and the control

objective function is extended to consider the error of these new outputs with respect to

their targets. To guarantee that this new term of the cost function will be bounded,

additional constraints need to be included in the control problem. The algorithms


8

proposed in this work adopt the zone control strategy as well as the infinite prediction

horizon to guarantee stability.

1.4. OUTLINE OF THE THESIS

This Thesis presents and analyzes stable MPC strategies that can be inserted in a control

structure where RTO is included and sends targets to the MPC controller. In Chapter 2,

an MPC controller with optimizing targets that deals with integrating systems is

described and tested. Then, a three-layer RTO/MPC structure that guarantees robust

stability for the intermediary layer and the MPC controller is proposed in Chapter 3.

Chapter 4 presents an MPC algorithm which uses the gradient of a convex function to

follow the RTO targets. Stability is proved for all the proposed controllers, which

operates under zone control. The controllers described in Chapters 3 and 4 are tested for

both linear and nonlinear systems. As Chapters 2, 3 and 4 present different control

strategies, they can be read separately. The general conclusions are finally presented in

Chapter 5.

1.5. PUBLICATIONS

Some of the results of this Thesis appear in the following journals and events:

International Journals

• Alvarez, L.A.; Francischinelli, E.; Santoro, B.; Odloak, D. (2009) Stable model

predictive control for integrating systems with optimizing targets. Industrial &.

Engineering Chemistry Research, 48, pp. 9141-9150.

• Alvarez, L.A.; Odloak, D. (2010) Robust integration of Real Time Optimization

with linear Model Predictive Control. Computers & Chemical Engineering, 34, pp.

1937-1944.


9

• Alvarez, L.A.; Odloak, D. (2012) Optimization and Control of a Continuous

Polymerization Reactor. Brazilian Journal of Chemical Engineering. Accepted

for publication.

International Congresses

• Alvarez, L.A.; Odloak, D. (2010) A simplified RTO/MPC algorithm with infinite

horizon for industrial process operation. 19th International Congress of Chemical

and Process Engineering, CHISA 2010. Prague. 28 August - 1 September 2010.

• Alvarez, L.A.; Odloak, D. (2011) Optimizing the operation of a polymerization

reactor via multimodel MPC. 11th international chemical and biological

engineering conference, CHEMPOR 2011. Lisbon, Portugal. September 5-7 2011.


10

CHAPTER 2

STABLE MODEL PREDICTIVE CONTROL WITH INPUT

TARGETS FOR INTEGRATING SYSTEMS

2.1. INTRODUCTION

Stability of the MPC was demonstrated by Rawlings and Muske (1993) for the regulator

case of systems with constraints in the inputs and states. For the case of stable systems,

the strategy consists of adopting an infinite prediction horizon and reducing the infinite

horizon through the inclusion of a terminal state weight, which is computed as the

solution to a discrete-time Lyapunov equation. Carrapiço and Odloak (2005) proposed an

extension of the method to the output tracking case for systems with stable and

integrating modes, by defining a modified cost function that includes suitable slacks to

enlarge the feasibility region of the controller. To guarantee that the cost function will be

bounded, a set of constraints related to the cancellation of the integrating modes of the

system at the end of the control horizon is included in the control problem. Also, to

remove output offset in the set-point tracking problem, the process model is considered in

the incremental form, which produces other constraints related to the additional

integrating modes that need to be included in the control problem to guarantee that the

control objective function remains bounded. The zone control strategy for stable systems

presented in Gonzalez and Odloak (2009) is also adopted. Thus, in this chapter, the

methods presented in Carrapiço and Odloak (2005) and Gonzalez and Odloak (2009) are

combined to produce a stable MPC controller that receives targets from the RTO layer in

the two-layer approach. The feasibility of the optimizing targets is compromised when

disturbances affect the process, as discussed earlier in section 1.2. Here, the controller

assumes that the process system has targets for some of the inputs and outputs, and the

remaining outputs are controlled by zones. The controllers proposed in this chapter are

also described in (Alvarez et al., 2009).


11

This chapter is organized as follows. The prediction model for integrating systems is

presented in section 2.2. The first algorithm that solves the control problem in one step is

described in section 2.3. Then, the optimization problem presented in section 2.3 is

decomposed in two steps. The first step is related to the integrating states and the second

step is related to the stable states. The two-step algorithm is presented in section 2.4.

Stability proofs for the two versions of the MPC controller are also provided. The

controllers are simulated in a distillation system and results are discussed in section 2.5.

Finally the conclusions of this chapter are presented.

2.2. PREDICTION MODEL OF THE INTEGRATING SYSTEM

In this section, the state-space model that represents the process system with integrating

modes is described. It is a minimal order model with a particular structure that simplifies

the development of a stable MPC with infinite prediction horizon and the integration of

the controller with the real time optimization of the process plant. Here, it is assumed that

the system has nu inputs and ny outputs. The state-space model adopted here is the

OPOM model (Output Prediction Oriented Model) and it is described as follows

(Carrrapiço and Odloak, 2005):

)()(

)()()1(

kCxky

kuBkAxkx

=∆+=+

(2.1)

where

=)(

)(

)(

)(

kx

kx

kx

kxi

d

s

;

*

*

00 00 0

ny ny

ny

I t IA F

I

∆ =

;

0 i

d

i

D tDB D FN

D

+ ∆ =

(2.2)

0ny nyC I = Ψ ; ( ) ; ( ) ; ( )s ny d nd i nyx k x k x k∈ ∈ ∈ℂR R

t∆ is the sampling time *nyI is a diagonal matrix of dimension ny×ny whose entries are 1 for the integrating

outputs and 0 for the stable outputs.


12

When a transfer function model is available, a method to obtain matrices F, D0, Di, Dd

andΨ is described in the Appendix B. Observe that in the model defined in eq. (2.1), it is

assumed that the system may have integrating outputs (when 0iD ≠ ). When this happens,

the three state components defined in eq. (2.2) can be interpreted as follows: sx is the

state component corresponding to the integrating modes created by the incremental form

of the model; dx corresponds to the stable modes of the system and ix is the state

component corresponding to the true integrating outputs of the system. Since the model

defined in eq. (2.1) is incremental in the inputs, it is offset free and there is no need to

include an intermediary target calculation layer as in Figure 1.1, which is normally used

to eliminate offset in the usual MPC implementation (Kassman et al., 2000; Muske and

Rawlings, 1993).

2.3. STABLE MPC WITH INPUT TARGETS

The MPC considered here assumes that the outputs of the process system are controlled

inside zones ( )min max,y y or at specific optimizing targets ( )RTOy , and the inputs may

also have targets ( )RTOu that would be defined by the RTO layer of the control structure. Assuming that k is the present time instant, the optimization problem that defines the

controller considered here is the following:

Problem P2-1

( ) ( )

( ) ( )

, , , ,1,

, , , ,

, , , , , ,0

, ,0

1

, , , , ,0

min

( / ) ( / )

( / ) ( / )

( / ) ( / )

k y k sp k u k i kk

u y

Tsp k y k i k y sp k y k i k

j

TRTO u k u RTO u k

j

mT T TT

y k y y k u k u u k i k ij

V

y k j k y j t Q y k j k y j t

u k j k u Q u k j k u

u k j k R u k j k S S S

∆ δ δ δ

∞

=

∞

=

−

=

=

+ − − δ + ∆ δ + − − δ + ∆ δ

+ + − − δ + − − δ

+ ∆ + ∆ + + δ δ + δ δ + δ δ

∑

∑

∑ ,i k

(2.3)


13

subject to:

( / )u k j k U∆ + ∈ (2.4)

0, 2 ,( ) [ ] 0s i

sp k m k y kx k y D D u− + − ∆ − δ =ɶ (2.5)

,( 1) 0nu k RTO u ku k I u u− + ∆ − − δ = (2.6)

1 ,( ) 0i i

m k i kx k D u+ ∆ + δ = (2.7)

min , maxsp ky y y≤ ≤ (2.8)

where:

≥=+∆

≤+∆+−∆≤

∆≤+∆≤∆−

+∆= ∑=

mjkjku

ukikukuu

ukjkuu

kjkuUj

i

,0)/(

)/()1(

)/(

)/( max0

min

maxmax

[ ]TTTTk kmkukkukkuu )/1()/1()/( −+∆+∆∆=∆ … m

nu nuI I I =

��ɶ ⋯ ; 0 0 0

m

D D D =

ɶ ⋯��

;

1 [ ]

m

i i i imD D D D=��

⋯ , 2 0 ( 1)i i imD t D m t D = ∆ − ∆ ⋯

The first term of the objective function defined in eq. (2.3) is the output error with respect

to the set-point that is also a variable of the control problem in the zone control strategy.

To compute the output error, it is considered an infinite sum since stability is to be

assured. It can be shown that the infinite term can be reduced via a terminal weight Q

(Odloak, 2004). The second term is the input quadratic error with respect to the input

optimizing target coming from the optimization layer. The third term of eq. (2.3) is

intended to minimize the control effort, m is defined as the control horizon.


14

In order to enlarge the region where the controller is feasible, slack variables ,y kδ , ,i kδ ,

,u kδ for the outputs, integrating states and inputs, respectively, are included in the control

problem. These variables should be small enough to reduce the deviation of the original

constraints, so they are penalized in the objective function using weights Sy, Si and Su.

Input constraints (2.4) are the typical MPC constraints. The constraints defined in eqs.

(2.5) and (2.6) imply that the output and input errors should be minimized at the steady-

state. Constraint (2.7) sets to zero the integrating states at the end of the control horizon m.

Constraint (2.8) defines the range where the output set-points should lie. For those

outputs with optimizing targets, the controller considers min max RTOy y y= = .

The tuning parameters of the controller resulting from the solution to Problem P2-1 are

the weight matrices Qy, and R and control horizon m as in the conventional MPC and may

follow the same tuning rules as in the conventional MPC. Additional tuning parameters

of the proposed controller are Qu and the slack weight matrices Sy, Su and Si that are

related to the slack variables introduced in the control problem. These weights should be

large enough in order to guarantee that the Hessian matrix of the control objective is

strictly positive definite, otherwise the solution of the control problem may not be unique

and the control objective will not have a minimum. Other considerations about the

numerical values of these weights will be presented later in this Thesis.

Notice that the use of slacks in Problem P2-1 makes this problem always feasible, even

for a set of targets that can not be reached by the process inputs and outputs. For the

process with optimizing targets, it is assumed that the steady-state defined by

( ),RTO RTOu y is an admissible steady-state, when the inputs and outputs that have optimizing targets are at their desired values, the remaining inputs and outputs of the

system will be inside their bounds (or all the slacks will be equal to zero). Here, it is

implicitly assumed that the number of variables (inputs plus outputs) that have targets is

equal to the number of degrees of freedom of the process system. In this case, it is

possible to show that, when the slack vector ,i kδ can be reduced, which corresponds to


15

the integrating modes, to zero, then, the sequential solution of Problem P2-1 will force

the closed-loop system to converge to the desired steady-state.

Theorem 2-1. Feasibility and Convergence of the one-step MPC

For a system with stable and integrating outputs that can be stabilized at a desired steady-

state, if at time k the optimal solution to Problem P2-1 results in a slack vector

corresponding to the integrating poles (,i kδ ) equal to zero, then for the undisturbed

system the solution of Problem P2-1 at any subsequent time step k+j is feasible

with , 0i k jδ + = and it is possible to find a solution to Problem P2-1 at the subsequent time

steps that leads the system in closed loop to the desired steady-state.

Proof

Recursive feasibility

For the optimal solution to Problem P2-1 at time k, that results in a slack vector

corresponding to the integrating poles (,i kδ ) equal to zero, the constraint corresponding to

eq. (2.7) is feasible. Also, suppose that at time step k the optimal solution to Problem P2-

1 is designated as ( )* * * * *, , , ,, , , ,k y k sp k u k i ku yδ δ δ∆ , and at time k+1 consider the solution defined by ( )1 , 1 , 1 , 1 , 1, , , ,k y k sp k u k i ku yδ δ δ+ + + + +∆ ɶ ɶ ɶɶ ɶ , which is computed as follows:

* *

1

* * * *, 1 , , 1 , , 1 , , 1 ,

( 1/ ) ( 1/ ) 0

, , , 0

TT Tk

y k y k sp k sp k u k u k i k i k

u u k k u k m k

y yδ δ δ δ δ δ+

+ + + +

∆ = ∆ + ∆ + − = = = = =

ɶ ⋯

ɶ ɶ ɶɶ (2.9)

Since this solution is inherited from the solution at the previous time step, it is clear that

( / 1)u k j k U∆ + + ∈ɶ and min , 1 maxsp ky y y+≤ ≤ɶ . Also, assuming that the input optimizing

targets are fixed, it is easy to show that


16

1 , 1( ) 0k RTO u ku k I u u+ ++ ∆ − − δ =ɶɶ ɶ

Then, constraint (2.6) is satisfied by the solution proposed in eq. (2.9). Analogously, it

can be shown that

1 1( 1) 0i i

m kx k D u ++ + ∆ =ɶ

The above expression shows that constraint (2.7) is also satisfied by the solution

proposed in (2.9). Then, it remains to verify if constraint (2.5) is satisfied by the solution

defined in (2.9). At time step k+1, with the proposed solution, the left hand side of eq.

(2.5) can be written as follows

0, 1 2 1 , 1( 1) [ ]s i

sp k m k y kLHS x k y D D u+ + += + − + − ∆ − δɶɶɶ ɶ (2.10)

Using the model expressions defined in eqs. (2.1), (2.2) and the relations contained in eq.

(2.9), the expression of the eq. (2.10) can be written as follows

( )0 * * * 0 *, ,0 * * *

( ) ( ) ( / ) ( 1/ )

( 1/ ) ( 2 / ) ( 2) ( 1/ )

s i isp k y k

i i

LHS x k t x k D t D u k k y D u k k

D u k m k t D u k k m t D u k m k

= + ∆ + + ∆ ∆ − − δ + ∆ + +

+ ∆ + − − ∆ ∆ + − − − ∆ ∆ + −

⋯

⋯

(2.11)

Since eq. (2.7) is satisfied at time step k, it can be written that

* * * ,( ) ( / ) ( 1/ )i i i

i kx k D u k k D u k m k= − ∆ − − ∆ + − − δ⋯ (2.12)

Now, using the eq. (2.12) to eliminate the integrating state ( )ix k , eq. (2.11) becomes


17

0 * * * * *, , 2 ,( )

s ik sp k y k m k i kLHS x k D u y D u t= + ∆ − − δ − ∆ − ∆ δɶ

Observe that if *, 0i kδ = , the right hand side of the above equation becomes equal to the

left side of eq. (2.5) at time k and consequently eq. (2.5) is also satisfied by the solution

defined in eq. (2.9). Then, the solution defined in eq. (2.9) is feasible.

Convergence of the objective function

To show the convergence of the objective function, let the objective function of Problem

P2-1 corresponding to the optimal solution at time step k be designated *1,kV and the value

of the objective function at time step k+1 corresponding to ( )1 , 1 , 1 , 1, , , ,0k y k sp k u ku yδ δ+ + + +∆ ɶ ɶɶ ɶ be designated 1, 1kV +ɶ . It is clear that

( ) ( )( ) ( )

* * * * *1, 1, 1 , , , ,

* * * * * *, ,

( / ) ( / )

( / ) ( / ) ( / ) ( / )

T

k k sp k y k y sp k y k

T TRTO u k u RTO u k

V V y k k y Q y k k y

u k k u Q u k k u u k k R u k k

δ δ

δ δ

+= + − − − −

+ − − − − + ∆ ∆

ɶ

As matrices Qy, Qu and R are assumed to be positive definite, if any of the last three terms

of the right hand side of the above equation is different from zero, then *1, 1 1,k kV V+


18

1, , , , ,T T

y y y u u uV S S∞ ∞ ∞ ∞ ∞= δ δ + δ δ (2.13)

Observe also that at this steady-state, constraints (2.5) to (2.7) can be written as follows

, ,( ) 0s

sp yx y δ∞ ∞∞ − − =

,( ) 0RTO uu u δ ∞∞ − − = (2.14)

( ) 0ix ∞ = (2.15)

Observe that if 1, 0V ∞ = , then the system will stabilize at the desired steady-state given by

( ) RTOu u∞ = and ,( ) spy y ∞∞ = with min , maxspy y y∞≤ ≤ .

It can be shown that if the desired steady-state (optimizing inputs and outputs at their

targets and the remaining inputs and outputs inside their bounds) is reachable and weight

matrix Su is selected appropriately, then the closed loop system with the proposed

controller will not stabilize at the steady-state where , 0yδ ∞ ≠ or , 0uδ ∞ ≠ . For this

purpose, suppose that when the system reaches the above steady-state, Problem P2-1 is

solved again. Then, the resulting control move u∞∆ will have to satisfy the constraint

defined in eq. (2.6) that, at this time step, can be written as follows

',( ) 0RTO uu u I u δ∞ ∞∞ − + ∆ − =ɶ (2.16)

Now, considering eq. (2.14), the expression of eq. (2.16) can be written as follows

', , 0u uI uδ δ∞ ∞ ∞+ ∆ − =ɶ (2.17)

To simplify the analysis, suppose also that the inputs are unconstrained. Then, u∞∆ can

be chosen according to

, 0u I uδ ∞ ∞+ ∆ =ɶ (2.18)


19

and consequently, from eq. (2.17) it results that ' , 0uδ ∞ = . Then, from eq. (2.16) follows

that

'( ) ( ) RTOu u I u u∞∞ = ∞ + ∆ =ɶ

This means that the new control input will be exactly at the desired value. Corresponding

to this input, the output will reach a new steady-state that is given by

02'( ) ( )

s imy x D D u∞ ∞ = ∞ + − ∆

ɶ , which by assumption is inside the output control zone.

Then, the constraint defined in eq. (2.5) becomes

' 0 ', 2 ,( ) 0

s isp m yx y D D u δ∞ ∞ ∞ ∞ − + − ∆ − = ɶ

As the predicted output '( )y ∞ is inside the control zone, ' ,spy ∞ can be chosen such that

' 0, 2( )

s isp my x D D u∞ ∞ = ∞ + − ∆

ɶ (2.19)

The combination of the two above equations produces ' , 0yδ ∞ = . Now, consider constraint

(2.7) that will be also satisfied by control action u∞∆ . When solving Problem P2-1 at the

steady-state, eq. (2.7) will be written as follows

'1 ,( ) 0

i im ix D u δ∞ ∞∞ + ∆ + = (2.20)

However, since from eq. (2.15) ( )ix ∞ is null, and in the successive solution of Problem

P2-1, it is assumed that the slack vector corresponding to the integrating modes is also

kept null, eq. (2.20) reduces to

1 0imD u∞∆ = (2.21)


20

Then, the control action should be such that both eqs. (2.15) and (2.21) are satisfied. So,

some care must be taken when defining, in the design stage of the controller, which

inputs will receive target signals. For instance, the condition defined in eq. (2.21) could

not be satisfied if all the inputs that are integrated by a given output have independent

targets.

Now observe the objective defined in eq. (2.13). Using eq. (2.18), it is concluded that

1, , ,TT T

u u u uV S u I S I uδ δ∞ ∞ ∞ ∞ ∞≥ = ∆ ∆ɶ ɶ

At this point the control objective '1,V ∞ corresponding to the proposed solution to Problem

P2-1 can be determined. To simplify the analysis, it is assumed that the control horizon

(m) is equal to 2. In this case, the control objective corresponding to the proposed

solution to Problem P2-1 is given by

( )'1, 1 2 3TV u G G G R u∞ ∞ ∞= ∆ + + + ∆

where

{ } { }{ } { }

0 0 0 01 2 2

2 22 2

Ti i d i i dm u u y m u u

Ti i i d d i i i d dm y m

G D D D tD N D N Q D D D tD N D N

D tD tD FD D Q D tD tD FD D

ψ ψ

ψ ψ

= − − + + ∆ + − − + + ∆ +

+ + ∆ ∆ + + ∆ ∆ +

ɶ ɶ

[ ]0u nuN I=

2

Td d d dG FD D Q FD D =

T T TyQ F QF F Q Fψ ψ− =

( ) ( )3 Tu u uG N I Q N I= − −ɶ ɶ

=

��⋯

m

RRdiagR


21

Then, if

1 2 3T

uG G G R I S I+ + + < ɶ ɶ (2.22)

the proposed solution to Problem P2-1 has a control objective lower than 1,V ∞ , that is

'1, 1,V V∞ ∞< , and the steady-state where , 0uδ ∞ ≠ is not optimal. Thus, the successive

solution of Problem P2-1 will force the system to converge to the desired steady-state.

Condition to guarantee the convergence of the closed loop to the targets

To guarantee the convergence of the process system to the desired steady-state, the

controller tuning parameters should satisfy the eq. (2.22), as shown above. Notice that G1,

G2 and G3 do not depend on the weight matrix Sy corresponding to the slack of the output

steady-state. This does not mean that Sy can be any positive definite matrix. This is so

because the proposed controller should also provide nominal stability for the case in

which the controller is applied in the conventional MPC scheme where there are no input

targets and the outputs are controlled with fixed set-points. Problem P2-1 can be easily

adapted to this case by considering Qu=0 and Su=0, and there is no need to include

constraint (2.7).

Then, if ,i kδ is reduced to zero, the solution to Problem P2-1 produces a stabilizing

control law that leads the system inputs and outputs to the optimizing targets. For the case

in which the controller is operating in the conventional approach (set-points for the

outputs and no targets for the inputs), the successive application of this control law will

lead the system to a steady-state where

1, , ,Ty y yV Sδ δ∞ ∞ ∞= (2.23)

,( ) 0sp yy y δ ∞∞ − − = (2.24)

( ) 0ix ∞ = (2.25)


22

Now, Sy should be selected such that, at this steady-state slack ,yδ ∞ is null. For this

purpose suppose that Problem P2-1 is solved when the system reaches the steady-state

where conditions (2.23) and (2.24) are satisfied. To simplify the analysis assume that the

control horizon (m) is equal to 2 and the system input is unconstrained. In this case,

Problem P2-1 has only two constraints:

0 '2 ,( ) 0

s isp m yx y D D u δ∞ ∞ ∞ − + − ∆ − = ɶ (2.26)

1 0imB u∞∆ = (2.27)

As the input is assumed to be unconstrained, it is possible to find u∞∆ such that

02 ,im yD D u δ∞ ∞ − ∆ = ɶ (2.28)

Now, substituting eq. (2.28) in eq. (2.23) it is easy to see that

0 01, 2 2

TT i im y mV u D D S D D u∞ ∞ ∞ = ∆ − − ∆

ɶ ɶ (2.29)

Then, following the same steps as the last case, it is possible to show that the control

objective of Problem P2-1 at the steady-state defined above is reduced to the expression

( )'1, 1 2TV u G G R u∞ ∞ ∞= ∆ + + ∆ (2.30)

From eqs. (2.29) and (2.30), it is clear that, in order to force slack ,y kδ to converge to

zero, Sy has to satisfy the following condition

0 02 2 1 2

Ti im y mD D S D D G G R − − > + +

ɶ ɶ (2.31)


23

The expression (2.31) provides the minimum bound for Sy to guarantee stability for the

closed loop system. Now, observing the state matrix of the model defined in eq. (2.1), it

is easy to see that the system cannot reach a steady-state where the state component xi is

not equal to zero. If this happens, state component xs will grow unbounded and so the

output will be unlimited. From constraint (2.7), at steady-state, follows that ,( )i

ix δ ∞∞ = ,

then the slack of the integrating state will also be zero otherwise the output would be

unbounded. Then, the controller behaves differently in terms of the tuning parameter Si

when compared to Sy or Su. As long as a positive definite Si is adopted, the convergence

of the system to the steady-state will not depend on the numerical value of Si.□

In Carrapiço and Odloak (2005), it is proposed to include the following contracting

constraint in the infinite horizon MPC of integrating systems

, , , 1 , 1T Ti k i i k i k i i kS Sδ δ δ δ− −≤ ɶ ɶ (2.32)

where , 1i kδ −ɶ is computed so as to satisfy the eq. (2.7) in the previous time step

, 1 1 1( 1)i i

i k m kx k D u− −δ = − − − ∆ɶ ɶ

( 1) ( ) ( 1)i i ix k x k D u k− = − ∆ −ɶ

The problem that arises when the constraint (2.32) is included into the Problem P2-1 to

force the decrease of the norm of the slack of the integrating output to zero is that this is

not a linear constraint. Using the Schur complement, it can be converted in a linear

matrix inequality (LMI):

*,

, , 1 , 1

0ny i i k

T Ti k i i k i i k

I S

S S

δ

δ δ δ− −

≥

ɶ ɶ


24

Although, the above inequality can be included in Problem P2-1 and the resulting

problem can be converted into a LMI problem (linear objective and linear matrix

inequalities as constraints), to solve the resulting LMI problem requires a LMI solver that

is usually not as robust as the available QP solvers. An alternative form to include the

contracting constraint (2.32) in Problem P2-1, but keeping the control problem a QP, is

described in the next section of this work.

2.4. STABLE MPC WITH INPUT TARGETS IN TWO STEPS

Since recursive convergence of the closed loop system with the controller resulting from

Problem P2-1 can only be guaranteed after reduction of the slack corresponding to the

integrating output to zero, a two steps approach can be adopted, where in the first step the

predicted control moves are used to minimize ,i kδ . In the second step, the remaining

degrees of freedom of the control system are used to minimize the distance between the

predicted steady-state and the optimum steady-state. One of the constraints of this second

problem is that the proposed control action should not increase slack ,i kδ . Then, the

proposed two steps MPC is based on the following problems:

Problem P2-2a

, ,2 , , ,

,mina k i k

Ta k i k i i k

uV S

δδ δ

∆= (2.33)

subject to:

( / )au k j k U∆ + ∈

1 , ,( ) 0i i

m a k i kx k D u+ ∆ + δ = (2.34)

where [ ]TTaTaTaka kmkukkukkuu )/1()/1()/(, −+∆+∆∆=∆ …

≥=+∆

≤+∆+−∆≤

∆≤+∆≤∆−

+∆= ∑=

mjkjku

ukikukuu

ukjkuu

kjkuUj

i

,0)/(

)/()1(

)/(

)/( max0

min

maxmax


25

Let the optimum solution to Problem P2-2a be designated as ( )* *, ,,a k i ku δ∆ . This solution is then passed to a second optimization problem that is solved within the same sampling

step. This second problem is defined as follows:

Problem P2-2b

( ) ( )

( ) ( ), , , ,

2 , , , , ,, , , 0

, ,0

1

, , , ,0

min ( / ) ( / )

( / ) ( / )

( / ) ( / )

b k y k sp k u k

Tb k sp k y k y sp k y k

u y j

Tb RTO u k u b RTO u k

j

m T TTb b y k y y k u k u u k

j

V y k j k y Q y k j k y

u k j k u Q u k j k u

u k j k R u k j k S S

δ δδ δ

δ δ

δ δ δ δ

∞

∆ =

∞

=

−

=

= + − − + − −∑

+ + − − + − −∑

+ ∆ + ∆ + + +∑

(2.35)

subject to:

( / )bu k j k U∆ + ∈

*1 , 1 ,i im b k m a kD u D u∆ = ∆ (2.36)

0, 2 , ,( ) [ ] 0

s isp k m b k y kx k y D D u δ− + − ∆ − =ɶ (2.37)

, ,( 1) 0b k RTO u ku k I u u δ− + ∆ − − =ɶ (2.38)

min , maxsp ky y y≤ ≤

where: [ ]TTbTbTbkb kmkukkukkuu )/1()/1()/(, −+∆+∆∆=∆ …

≥=+∆

≤+∆+−∆≤

∆≤+∆≤∆−

+∆= ∑=

mjkjku

ukikukuu

ukjkuu

kjkuUj

i

,0)/(

)/()1(

)/(

)/( max0

min

maxmax

The purpose of including equality constraint defined in eq. (2.36) is to guarantee that the

optimum solution to Problem P2-2b will not allow an increase of the objective function

of Problem P2-2a.


26

The controller resulting from the solution to Problem P2-1 is here designated as

Controller I, while the controller resulting from the solution to problems P2-2a and P2-2b

is designated as Controller II. In the practical implementation of Controller II, at time

step k, Problem P2-2a is solved first, then, Problem P2-2b is solved using the control

move ( * ,a ku∆ ) corresponding to the optimal solution to P2-2a. From the optimal solution

to P2-2b, the control sequence * ,b ku∆ becomes available and the first control move

* ( / )bu k k∆ is injected in the real process. At time state k+1, the procedure is repeated

starting from Problem P2-2a. The successive solution of problems P2-2a and P2-2b

forces the system to the desired steady-state as shown in the theorem that follows.

Theorem 2-2. Feasibility and Convergence of the two steps MPC

For a system with stable and integrating outputs that can be stabilized at a desired steady-

state, the control law resulting from the successive solution of problem P2-2a and P2-2b,

where the solution of P2-2b is implemented in the real process, leads the undisturbed

system asymptotically to the desired steady-state.

Proof

Recursive Feasibility

Observe that Problem P2-2a is always feasible because slack ,i kδ is not limited. Also,

once the optimal solution to Problem P2-2a has been found, a solution where

*, ,b k a ku u∆ = ∆ and slacks ,y kδ and ,u kδ are computed such that (4.37) and (2.38) are

satisfied, is a feasible solution to Problem P2-2b. So, problems P2-2a and P2-2b are

always feasible.


27

Convergence

Assume that at time k Problem P2-2a is solved and the optimal solution is represented by

( )* *, ,,a k i ku δ∆ and let the corresponding value of the objective function be designated *1,kV . Also, at the same time step and using this solution to Problem P2-2a, Problem P2-2b is

solved and the optimal solution is represented by ( )* * * *, , , ,, , ,b k y k sp k u ku yδ δ∆ . Then, the first control action * ( / )bu k k∆ is injected into the true system. At time step k+1, problems P2-

2a and P2-2b need to be solved again.

In the solution of Problem P2-2a at time step k+1, consider the solution represented by

( ), 1 , 1,a k i ku δ+ +∆ ɶɶ where:

* *, 1 ( 1/ ) ( 1/ ) 0

T

a k b bu u k k u k m k+ ∆

· este exemplar foi revisado e alterado em relação à versão original, sob responsabilidade...

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