Control predictivo no lineal: Estado del arte y nuevos resultados

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1 Control predictivo no lineal:

Estado del arte

y nuevos resultados

Daniel Limón

Grupo de Estimación, Predicción, Optimización y Control

Dpto. Ingeniería de Sistemas y Automática

Escuela Técnica Superior de Ingeniería

Universidad de Sevilla

XIII Simposio CEA de ingeniería de control

D. Limon. MPC no lineal: estado del arte y nuevos resultados. 2

Outline



Stabilizing design of predictive controllers



Tracking model predictive control



Economic model predictive control

(2)

D. Limon. MPC no lineal: estado del arte y nuevos resultados.

Optimal operation of a plant

3

Larger

s

a

m

p

lin

g

t

im

e

Economic

Planning & Scheduling

Real Time Optimization

Advanced Control (MPC)

Low-level Control

PLANT

Market

Cost functions

and constraints

Economic set points

Low-level set-points

Manipulable inputs

Stationary plant model

Dynamic plant model

(3)

3 Model Predictive Control

Steady-state target

optimization

Model Predictive Control

MPC targets

Economic set points

Low-level set-points

Two layer MPC



RTO disadvantages

[Engell, 2007]



Slow reaction to process variations



Mismatches between the models of RTO and MPC

The RTO may provide inconsistent set points to the MPC

Steady-state Target Optimization

(4)

Problem statement

7

(5)

5 Lyapunov function

9

Lyapunov function

(6)

Inherent robustness

(7)

7 ISS for constrained systems

Inherent robustness

1

2

3

0 2 4 6 8 10 0 1 2 3 4 5

(8)

Inherent robustness

1

2

3

0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

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9 The optimal controller

17

The optimal controller

(10)

The optimal controller

19

Stabilizing nominal MPC

Stabilizing

Terminal

Ingredients

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11 Stabilizing nominal MPC

21

Stabilizing nominal MPC

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Stabilizing nominal MPC

23

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13 Stabilizing nominal MPC

25

Stabilizing nominal MPC

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Stabilizing nominal MPC

27

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15 Stability with terminal equality constraint

29

Stabilizing MPC without terminal constraint

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Stabilizing MPC without terminal constraint

31

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17 Prediction and control horizon

33

Local optimality

(18)

Local optimality

35

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19 Inherent robustness of MPC

Illustrative example

(20)

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21 ISS of nominal MPC

Outline



Stabilizing design of predictive controllers



Tracking model predictive control



Economic model predictive control

(22)

Economic

Planning & Scheduling

Real Time Optimization

SSTO

Low-level Control

PLANT

Market

Cost functions

and constraints

Economic set points

Low-level set-points

Manipulable inputs

MPC

MPC targets



Objective: regulate the system to the MPC target





The optimal predicted sequence is

computed



Receding horizon

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23 Changing set-points

45



Market-driven production



Production goals



Cost functions



Operation constraints



Real Time Optimization



Disturbances



Estimation errors



Model parameters

Frequent target changes

Larger

sa

m

p

lin

g

tim

e

Economic

Planning &

Scheduling

Real Time

Optimization

Advanced Control

(MPC)

Low-level Control

PLANT

Market

Cost functions and constraints

Economic set points

Low-level set-points

Manipulable inputs

Stability issues



Stability loss:



Redesign of the terminal conditions



Feasibility loss:

46

Initial state

Initial target

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MPC for tracking

47

MPC for regulation

MPC for tracking

Example:

Consider the discrete time LTI system:

Subject to the following hard constraints:

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25 Consider the

target steady

state as a

decision variable

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Adding a

penalization term

between the

artificial steady

state and the real

one

(27)

27

(28)

(29)

29

(30)

(31)

31 X

_N

Proj

x

(



(32)

MPC for tracking

63

Offset cost function

Extended terminal constraint

Terminal cost function

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33 Stability theorem

65



Stability for any change of the target

66

Properties of the controller

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

Larger domain of attraction

CSTR

Properties of the controller

Equilibrium points

DoA of MPC for regulation (N=17)

DoA of MPC for tracking (N=2)

DoA of MPC for tracking (N=10)

DoA of MPC for tracking (N=17)

67



The solutions of MPC and MPC for tracking may differ (when

both are feasible)



Reason: the artificial reference



Then the local optimality property of MPC may be lost



This can be solved by taking

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35 Outline



Stabilizing design of predictive controllers



Robustness and robust design



Set-point tracking predictive control



Economic predictive control



Conclusions

Economic optimization



Economic optimization of the steady

operation

(Economic set point)



The transient control problem is posed

as a

tracking control problem to the

MPC target

70

Economic

Planning & Scheduling

Real Time Optimization

SSTO

Low-level Control

PLANT

Market

Cost functions

and constraints

Economic set points

Low-level set-points

Manipulable inputs

MPC

MPC targets

Is this the economically optimal

operation of the plant?

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Economic optimization



Tracking control to the MPC target



Economic operation of the plant

71

The economic cost function

should be used to measure

the performance of the transient

Integration of SSTO in the MPC

Steady-state target

optimization

Model Precitve Control

Set-points

Economic targets

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37 Economic MPC



MPC for regulation to the setpoint



Economic MPC



The economic stage cost function is not positive definite



Existing Lyapunov stability results can not be used



Stability issues

73

Economic MPC

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Economic MPC for changing targets

75



If the economic set point changes, feasibility may be lost



Motivation:

guaranteed feasibility (MPC for tracking) + economic

optimality (Economic MPC)



MPC problem

:

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39 Example: the double integrator

Double integrator

Cost functions:

Controller parameters:

Constraints:

77

D. Limon, MPC for changing economic targets.

Example: the double integrator

78



Economic optimality:



N=10



Feasibility:

N = 3

E-MPCT vs. MPCT

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Future directions and open problems



Relaxation of the stabilizing design



MPC without terminal constraint (Grüne)



Economic MPC based on turnpike property (Faulwasser)



Economic MPC for non-steady operation



Reference tracking predictive control



Design of robust predictive controllers



Stochastic predictive controller



Scenario-based predictive controller

Future directions and open problems



Distributed nonlinear MPC



Multiobjective predictive control



Specialized optimization algorithms for MPC



Embedded implementation



Novel applications

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41 Algunas referencias



D.Q. Mayne et al. Constrained model predictive control: Stability

and optimality, Automatica, 2000



D.Q. Mayne, Model predictive control: Recent developments and

future promise. Automatica 2014



J.B. Rawlings and D.Q Mayne. Model Predictive Control: Theory

and Design. Nob Hill Publishing 2009



D. Limon. Control predictivo de sistemas no lineales sujetos a

restricciones: Estabilidad y robustez. Tesis doctoral. Univ. Sevilla



D. Limon et al. Input-to-State Stability: a Unifying Framework for

Robust Model Predictive Control. NMPC’08



D. Limon et al. Model Predictive Control for changing economic

targets. NMPC’12



Rawlings, J. B. et al. Fundamentals of economic model predictive

control. IEEE Conference on Decision and Control (CDC), 2012.

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¡Muchas gracias!