paris’09 daniel r. ramirez & eduardo f. camacho min ... eduardo camacho/wednesday...4...

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1Paris’09 Daniel R. Ramirez & Eduardo F. Camacho Min-max MPC

Min-Max Model PredictiveControl

Implementation StrategiesDaniel R. Ramirez & Eduardo F. Camacho


Outline

Why Min-Max Model Predictive Control ?

Min-Max Model Predictive Control using open-

loop predictions.

Min-Max Model Predictive Control using closed-

loop predictions.



Model Predictive control is a successboth the industrial community andacademia:

4600 Applications (Qin y Badgwell, 2002).

67 % Refining and petrochemical industry.

Hundreds of paper published.



• Pros:Multivariable from the beginning.Deadtime compensation.Optimal design.Guaranteed stability (academic version).Able to consider constraints.

• Cons:Computational burden.Need of a model.

Modelling errors (uncertainties, disturbances) can degrade the closed loop performance.Modelling errors (uncertainties, disturbances) can degrade the closed loop performance.

RobustModelPredictiveControl


Why Min-Max MPC ?

• Robust MPC:– Robust stability guaranteed:

• Keep the state within a region.• Robust constraint satisfaction.

– Improve performance in presence of uncertaintiesand disturbances:

• Improving robustness.• Take into account uncertainties in the design.

Min-Max formulation: worstcase optimized designMin-Max formulation: worstcase optimized design

0 1 2 3 4 5 6-0.2

0

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Min-Max MPC (MMMPC)Campo y Morari, 1987



0 2 4 6 8 10-0.6

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1MPC

segundos

y

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1MMMPC

segundos

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Better performance againstuncertainties More RobustnessBetter performance againstuncertainties More Robustness

2



0 50 100 150 200 250 300 350 400 450 5002

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muestras

y k

MMMPC

0 50 100 150 200 250 300 350 400 450 5002

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muestras

y k

MPC

This increase in robustness alsoappear in real world applicationsThis increase in robustness alsoappear in real world applications

Control Engineering Practice, 2005



10 20 30 40 50 60 70 80 90-1.5

-1

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1

muestras

Δ u k

10 20 30 40 50 60 70 80 900

0.2

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1

1.2

muestras

y kMore robustness can also helpavoding feasibility problems:

Constraint violation due tomodelling errors.

Unfeasible optimization problem.

More robustness can also helpavoding feasibility problems:

Constraint violation due tomodelling errors.

Unfeasible optimization problem.



4 5 6 710-1

100

101

102

103

horizonte de predicción

Tiem

po (s

eg.)

Even when there are advantagesthe number of reported applicationsis very low

Even when there are advantagesthe number of reported applicationsis very low

¿?

ComputationalBurden issues

Berenguel et al. 1997Kim et al. 1998Álvarez et al. 2003



• Desiderable properties of an Min-Max MPC:– Far lower computational burden, complexity growing

with prediction horizon at non exponential rate.– Use efficient numerical algorithms (such as in QP) to

compute control signal.– Flexible implementation, able to change parameters in

real-time.– Able to use constraints.– Low error if an approximation of the optimizer is used. – Robust stability guaranteed.


Types of Min-Max MPC

• A broad division of Min-Max MPC can be established between thosewhich use open-loop predictions and those with closed-looppredictions (note that despite its type all of them are feedback controllers).

• Open-loop prediction based controllers are the most maturealgorithms dating back to 1987 (Campo and Morari).

• They suffer from conservatism in the predicted evolution of the processthat can lead to poor performance or feasibility problems. However, properly tuned they work well as proved later.

• Although its computational burden grows exponentially with certainparameters, there exists techniques to implement it in a wide class ofsystems.


Types of Min-Max MPC

• On the other hand, Closed-loop prediction based controllers havelower conservatism and better feasibility properties, because they takeinto account the fact that the control law is applied in a feedback manner.

• The price to be paid is an even greater computational burden and fewerimplementation options (in fact there is no any single reportedapplication to a real process).

• These controllers date from 1997 (Lee and Yu) and 1998 (Scokaert et al) and usually rely in recursive min-max problems or the optimizationof a set control policies instead of a sequence of control signal values.

• Finally there is a comprimise between these two types of controllerscalled the semi-feedback approach, that can be used with open-loopprediction controllers to add a certain degree of closed-loop behavior in the predictions.

3


Open loop vs close loop prediction





Paris’09 Daniel R. Ramirez & Eduardo F. Camacho Min-max MPC

Min-Max Model Predictive Control basedon Open-Loop Predictions


MMMPC using Open-Loop Predictions

Robust Model Predictive ControlExplicit implementation

Implementation using a nonlinear bound

Implementation using a QP based bound

Applications and Examples


Predictive Control

Output predictions

Manipulated

y(t+k)

variables

t t+1 t+N

futurepast

u(t+k)

Apply the first component of u*(t)

Solve a finite time optimal control problem to obtain u*(t)

Get new measures y(t)

Estimate x(t)

At each sampling time t:

4


Robust Predictive Control

Output Predictions

VariablesManipulated

t t+1 t+N

futurepast

u(t+k)

y(t+k)

Based on models that take into account uncertainties:

UNCERTAIN PREDICTIONS


Robust Model Predictive ControlSystem model:

Bounded additive uncertainties

Two strategies to consider u(t):

Open-loop predictions:

Semi-feedback predictions:Computed by the controller


Robust MPC

Outputpredictions

VariablesManipulated

t t+1 t+N

futurepast

u(t+k)

y(t+k)

min-maxMPC Plant C

K

w v u xy

-+

The inner loop pre-stabilizesthe nominal system


Min-max MPC (∞-∞ norm)

• Campo and showed that by using an ∞-∞norm themin-max problem reduces to a linear programmingproblem.

• Although the algorithm was developed for thetruncated impulse response, it can easily be extended to the other models used.

• The objective function is now described as





5





A LP (with many constraints: the vertices of the uncertainty polytope)


Min-max MPC (1-norm)


Min-max MPC (1-norm)


Min-max MPC (quadratic cost)

Cost Function:

Constraints:


The Min-Max Strategy:

Robust constraintSatisfaction

V*(x,u) is convex

Dependence on uncertainty in constraints can be removed offline

(Lofberg 03,Kerrigan 03)The “max” function:

Convex function

The maximum is attained at least at one of the vertices of Θ.(Bazaraa and Shetty 79)


6



Thus, the max function can be computed as:

Number of vertices: Exponential complexity

NP hard problem

As a result, the min-max problem is of the NP hard kind and in general it cannot be solved in real time except for slow processes and/or small horizons.


Real time implementation of Min-Max MPC

Use an explicit implementation, in which the min-

max problem is precomputed for feasible states.

Solve an equivalent reduced min-max problem.

Substitute the worst case cost for a low

computational burden close approximation.

How to get an Min-Max MPC that can be computed in real time ??

Some suggestions:



Robust Model Predictive Control

Explicit implementationImplementation using a nonlinear bound




Explicit implementation of Min-Max MPC

MMMPC not solvable on line

An alternative: Multi parametric programming

Explicit form control laws:


Multi-parametric Programming

Successfully applied to MPC:

YESYESYESLP based norms

NOYesYESQuadratic norms

Min Max Closed Loop predictions

Min MaxOpen Loop predictions

Nominal

(Bemporad et.al. AUT 02, Bemporad et.al. TAC 03)

(Sakizlis et.al. AUT 04, Wan et.al. TAC 04)

Approximate closed loop explicit solutions


Geometrical Approach

(Ramirez and Camacho Automatica 2006); (Muñoz de la Peña et.al. CEP 05, SCL 2006)

• Based on properties of cost function and the concept of active vertices (i.e., the vertices in which the worst case is attained at the solution).

• It allows:

• To prove that the control law is PWA on the states.

• To find an explicit solution using a constructive algorithm.

Works well, but… difficult to understand, complex specially with constraints, heuristic if need to be efficient.

7


Min-max as a Quadratic Program

The min-max problem can be rewritten as:

Added and subtracted V(x,v,0)

Taking into account the cost definition:

Quadratic dependence on wAffine dependence on x and v



Min max problem:

Epigraph approach:

Maximization



Min max problem:

Quadratic Programming problemExponential number of constraintsSemi-definite positive function

MULTIPARAMETRIC THEORY

(Tondel et.al. CDC 03)

linear constraints


Example

Consider the double integrator with bounded additive uncertainties:

The uncertainty satisfies:

The state and control input are constrained:

The control performance objectives are described by

A linear feedback law is considered:


Example

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

6

8

10

x1

x 2

N = 1

I=1 A=10

I=1 A=[ ]

I=2 A=[ ]

I=1 A=9

I=2 A=10

I=2 A=9

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

6

8

10

x1

x 2

N = 1

I=1 A=10

I=1 A=[ ]

I=2 A=[ ]

I=1 A=9

I=2 A=10

I=2 A=9

N=1


Example

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

−8

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−4

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0

2

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6

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x1

x 2

N=3

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

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−2

0

2

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6

8

10

x1

x 2

N=3

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

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−2

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2

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6

8

10

x1

x 2

N=5

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

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−2

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2

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6

8

10

x1

x 2

N=5

N=3

N=5

8


Example

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

6

8

10

x1

x 2

N = 1

I=1 A=10

I=1 A=[ ]

I=2 A=[ ]

I=1 A=9

I=2 A=10

I=2 A=9

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

6

8

10

x1

x 2

N = 1

I=1 A=10

I=1 A=[ ]

I=2 A=[ ]

I=1 A=9

I=2 A=10

I=2 A=9

Set of active verticesis very small


Feedback PT-326

Heat Stored

Heat In

Heat Out

•Scaled laboratory process•2nd order system•Fast dynamics•Ts = 0.4 s.

(Explicit solution)


Results of the MMMPC

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10

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10

samples

MMMPC with linear feedback yk u

k

N=15, λ =1

N=20, λ =15

N=8, λ =10

N=8, λ =5

Different controllers

Applied with success

(Muñoz de la Peña et.al CEP 05)


Comparisons

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samples

y k

MPC

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samples

y k

MMMPC

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samples

y k

MMMPC with linear feedback

MPC

MMMPC

MMMPCwith linear feedback

Different positions of the inlet throttle from 20o to 100o


Explicit implementation: Pros and Cons

• Possibility to implement explicit min-max MPC control laws. using standard multi-parametric techniques.• Well suited for fast systems and embedded applications.• But…⌧ The exponential number of constraints limits the value of the prediction horizon.⌧ The implementation is not flexible, the user cannot change parameters without recomputing the controller.




Explicit implementation

Implementation using a nonlinear boundImplementation using a QP based bound


9


Using an upper bound of the worst case cost

A computationally cheaper upper bound is minimized instead of the worst case cost :

Previous results:LMI methods Kothare et.al 96, Lofberg 03BQP Algorithms Alamo el.al 04

Lowercomputationalburden

Strategy based in a nonlinear bound, Ramirez et al., JPC 2006

Strategy based in QP problems, Alamo et al., Automatica 2007

Strategy based in a nonlinear bound, Ramirez et al., JPC 2006

Strategy based in QP problems, Alamo et al., Automatica 2007


Strategy based on a nonlinear bound

Main Properties:Low computational burden.Based on matrix computations.Close to the worst case cost.

Main contribution:

New upper bound of the maximization problem not based on LMIs or complex algorithms.


Computing the nonlinear upper bound

The quadratic maximization problem can be rewritten as:

Asume that ε = 1, then:Binary optimization problem

Note that matrix H depends on (x,u)



The idea to obtain the bound is:

If T is a diagonal positive definite matrix such that:

then:

Computation of T: LMI orProposed method based on simple matrix computations



T will be obtained adding to H a series of matrices of the form:

The vi will be computed in such a way that allow to obtain a close bound with low complexity.

positive definite matrix



Vectors vi are computed so that matrix H is partially diagonalized:

If

This idea is applied recursively

Free parameter

10


cc


The error introduced at each diagonalization step must be minimum:


Nonlinear upper bound algorithm

Let

For k=1 to n-1

Let Hsub = [Tij] for i,j= k,...,n Compute α for Hsub

Form

Form

End For

When the procedure is over, the bound is computed as σ u(H) = trace(T)


Computing the control signal

• At each sampling time, solve:

and apply the first component of u*(x) using a receding horizon strategy.

If at any step of the diagonalization procedure, all the elements of Hsubare ≥ 0 the best bound will be

No need to continuethe computationloop


Accuracy of the nonlinear upper bound

-1-0.5

00.5

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20

30

40

500

10

20

30

40

media de los elementos de H0

Dimension matriz

Des

viac

ión

de la

cot

a LM

I (%

)

It will be compared against theLMI bound:

Nesterov et al, 2000

Deviation for a set of random matrices of the form H=H0'H0.

Typical matrices in MPC problems are outside this zone, because the mean of its elements are different from zero


Computational burden of the bound

Relative speed up from the LMI bound Solver LMI by F. Rendl

http://www.math.uni-klu.ac.at/or/Software.

-2-1

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Dimensión matriz

Ace

lera

ción

Typical matrices in MPC problems are outside this zone, because the mean of its elements are different from zero






Implementation using a QP based boundApplications and Examples

11


QP based strategy

• It has two desiderable properties:√ Robust stability guaranteed.√ Implementation based on an optimization problem similar to that

of conventional MPC Quadratic Programming (QP).• Outline of the strategy:

1. Obtain an initial estimation of the solution of the min-max problem.

2. Obtain a quadratic function that bounds the worst case cost.

2.1 Compute some parameters denoted αk for the initial estimation.

2.2 Use a diagonalization procedure to obtain the quadratic function.

3. Compute the control signal as the minimizer of the quadratic function.


QP based strategy

Robust stability guaranteed, but requires semi-feedback for non open-loop stable sytems:

The control law of the original MMMPC original is computed by solving at each step t:

Stability conditions imposedon Terminal Cost and Region

Terminal region

IncludesIncludes a a terminal terminal costcost

StabilizingdesignMayne, 2000


Stabilizing design

If there exists a solution for x(k)

Then existssolution for x(k+1)

Convergence to the origin(Optimal cost Lyapunovfunction)

Conditions on the terminal regionAdmissiblerobustinvariant set

Maximal robust invariant set computed for the systema controlled by:

Condition on the terminal cost

Guaranteesrobustfeasibility


QP problems based strategy

Step 1: Initial estimation of the solution of the min-max problem

The maximum cost can be computed as:

An easily computable (although not very good) upper bound is :

Initialestimation

QP equivalentSlack variables


Step 2: Computing the quadratic function

Step 2.1: Compute the parameters αk for the initial estimationTaking into account that if it is assumed ε=1:

Use the diagonalization algorithm of the nonlinear bound overto obtain the α k:

The resulting diagonal matrix, denoted by σu(v) verifies:


Step 2: Computing the quadratic function

• Step 2.2: Compute a matrix denoted by

12


Step 3: Computing the signal control• Let:

• The control signal will be computed solving:

• The suboptimality is bounded:


Stability• The proposed control law guarantees the closed loop robust stability. • Outline of the proof:

– Let J(x(t)) be the optimal cost of the original min-max problem.– It can be proved that using the proposed control law there exists γ > 0

such that:

– This implies that:

– That is, from any x(t) the state evolves into:

It can escape


Stability• The state is always steered into Φε, but it can be escape from it.• The set to which the state can evolve from Φε is:









A simulated example

F1

F2

FR

h1

h2

o1=a1h1

o2=a2h2

Two tank process:Ogunnaike y Ray, 1994

Tanks sections: 3 m2 and 2m2.a1 = a2 = 0.5 m2min-1. 40% of the liquid is pumped back

to tank 1.


A simulated example

Controller based on the nonlinear bound

ε = 0.02N=15, Nu=10, 230 vertices

0 20 40 60 80 100 1200

0.2

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nive

les

0 20 40 60 80 100 1200

0.2

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muestras

caud

ales

13


Computational burden

Average, max and min speed-up over the exact MMMPC.

4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

900


Ace

lera

ción

(med

ia y

máx

-mín

)

4 5 6 7 8 910

0

101

102

103

104

105

106


Ace

lera

ción

(med

ia y

máx

-mín

)

Average, max and min speed-up over the minimization of the LMI bound.


Accuracy of the nonlinear bound

Desviation from the exact optimal cost and from the optimal cost obtained using the LMI bound.


Accuracy of the nonlinear bound

1 20 40 60 80 100 1200

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7

muestras

cost

e óp

timo

1 20 40 60 80 100 1200

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10

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cost

e óp

timo

Optimal cost with the nonlinear bound and the LMI bound for a simulation (Nu=5,N=20).

Optimal cost with the nonlinear bound and the exact MMMPC for a simulation.


A simulated example

Controller based on QP problems

ε = 0.025N=7, Nu=7

0 20 40 60 80 1000

0.7

1

nive

les

0 20 40 60 80 1000

0.2

0.4ca

udal

es

10 20 30 40 50 60 70 80 90 100-0.03

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-0.01

0

muestras

desv

iaci

ón


A simulated example

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14

muestras

cost

es ó

ptim

os

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10

cost

es ó

ptim

os

10 20 30 40 50 60 70 80 90 1000

1

2

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4

muestras

desv

iaci

ón (%

)

Optimal cost of the exactMMMPC and the QP basedcontroller (Nu=N=7).

Optimal cost of the exact MMMPC when it is used both the exact solutionand that of the QP based controller, that is


A simulated example

4 5 6 7 8 9100

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105

106


Ace

lera

ción

(m

edia

y m

áx-m

ín)

Average, max and min speed-up for the QP basedcontroller over the exact MMMPC.

14


Application to a pilot plant

Inte

rcam

biad

or

de C

alor

Tank with a 14.4kW heater.

Recirculation through a heat exchanger.

Industrial instrumentation.

Connected to Simatic-IT.

It emulates a CSTR using the heater tosimulate the heat of an exothermic reaction.

Model by Santos et al., 2001



0 100 200 300 400 500 600

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TT5

o C

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Tiempo (minutos)

V8

%

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0

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erro

r

Multilevel Step sequence

Identification of a first ordermodel by Least Squares



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TT5

o C

0 25 50 75 100 125 15020

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V8

%

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0.2

0.3

Tiempo (minutos)

CA

Controller based on the nonlinear boundε = 0.25N=15, Nu=12

Sampling time: 60 seg.

Deadtime rounded up to 1.

Smith Predictor-like correction.Normey y Camacho, 07



0 20 40 60 80 100 12045

50

55

60

65

TT5

o C

0 20 40 60 80 100 12020

40

60

80

V8

%

0 20 40 60 80 100 1203040506070

u

0 20 40 60 80 100 1200.050.1

0.150.20.2

Tiempo (minutos)

CA

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45

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60

65

TT5

o C

0 20 40 60 80 10030

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50

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80

V8

%

0 20 40 60 80 1000.05

0.1

0.15

0.20.25

Tiempo (minutos)

CA

Disturbances in the input (15%)Disturbances in the inlet flow (25%)



Controller based on QP problems

ε = 0.25N=15, Nu=12

0 25 50 75 100 125 15035

40

45

50

55

60

65

T

0 25 50 75 100 125 15020

40

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tiempo

V8

%

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0.1

0.2

0.3

Tiempo (minutos)

CA



0 20 40 60 80 100 12040

45

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65

T

0 20 40 60 80 100 12030

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Tiempo (minutos)

v8

0 20 40 60 80 100 1200.05

0.1

0.15

0.2

0.25

Tiempo (minutos)

CA

0 25 50 75 100 125 15035

40

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60

65

T

0 25 50 75 100 125 15020

40

60

80

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tiempo

v8

MMMPC NLMPCMMMPC QPref

MMMPC NLMPCMMMPC QP

Disturbances in the inlet flow (25%)

Comparison for a set point trackingexperiment using:

MPC.MMMPC based on the nonlinear

bound.MMMPC based on QP problems.

15


Feedback PT-326

Heat Stored

Heat In

Heat Out

•Scaled laborat ory process•2nd order syst em•Fast dynamics•T s = 0.4 s.

( Explicit solut ion)


0 500 1000 15000

1

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9

10

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e k

0 500 1000 1500−0.15

−0.1

−0.05

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0.05

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samples

e k

One step identification error for ARX model

One step identification error for CARIMA model

System Identification

A(z−1)yk = B(z−1)uk−1−dARX

L east squares ident if icat ion met hod

Random binary input

Ident if icat ion set / Validat ion set

Error bounding CARIM A model

ΔA(z−1)yk = z−dB(z−1)Δuk−1−d+ θkCA RIM A

B est f it t ing

T s = 0.4D elay = 4


System Identification

A(z−1) = 1 −0.5510z−1− 0.4072z−2

B(z−1) = 0.0090+ 0.0127z−1+ 0.0105z−2

B est m odel:

D elay = 4error bound = 0.05

A(z−1)yk = B(z−1)uk−1−dARX ΔA(z−1)yk = z−dB(z−1)Δuk−1−d+ θkCA RIM A


Delay Compensation

Smit h Predict or St rat egy:

yk+d= yk+d|k+ (yk|k−d− yk)

PT -326PT -326

G( z)G( z)

z-dz-d+ -

+

+

ukyk

yk+ d,k

yk,k-d

yk+d


Reference Traking

In order t o evalut at e reference t raking, t he st at evect or is augm ent ed:

( B em porad et .al. 2002)

F ixed reference over t he predicit ion horizon

zk =hyk+d−1 yk+d−2 Δuk−1 Δuk−2 Δuk−3 rk

iT

Q uadrat ic crit erion(yk − rk)2 = zTk Qzk


Explicit control law

IF L zk �h4.265 0.232 3.03 3.28 3.28 3.03 0.232 4.265

iTTHEN Δuk =

h−0.006 −0.003 −0.273 0.135 0.104 0.034

izk

L=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1.501 0.727 78.847 −34.088 −28.186 −16.5740.023 0.011 1.156 −0.531 −0.426 −0.1990.020 0.010 1.248 −0.443 −0.400 −0.404−0.057 −0.030 −3.521 1.232 1.144 1.1440.057 0.03 3.521 −1.232 −1.144 −1.144−0.020 −0.010 −1.248 0.443 0.400 0.404−0.023 −0.011 −1.156 0.531 0.426 0.199−1.501 −0.727 −78.847 34.088 28.186 16.574

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

16


Results of the MMMPC

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samples0 100 200 300 400 500

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samples

MMMPC yk u

k

N=16, λ = 10,α=0.95

N=8, λ = 0.3,α=0.8

N=15, λ = 0.5,α=0.75

N=8, λ = 10,α=0.99

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y k

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u k

MMMPC N=16, λ = 10, α =0.9

“ W ell” t uned M M M PC

A gresive cont rol law

O pen looppredict ions

D if ferent cont rollers


MMMPC with Linear Feedback

2 4 6 8 10 12 14−3

−2

−1

0

1

2

3Worst case evolution of a system

Time step k

Out

put y

k

Nominal

Linear feedback

Δuk = Kzk+ vk

L inear feedback:

L inear cont roller

Feedback on t he predict ions wit hout complexit y increase( B emporad et ,al, 1998)


MMMPC with Linear Feedback

• ìΔuk = Kzk+ vk

L inear feedback:

PT -326PT -326Δυk yk

vk

K zK z

M M M PCM M M PC +

Pre-cont rolled syst em ( linear model)


Results of the MMMPC with linear feedback

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MMMPC with linear feedback yk u

k

N=15, λ =1

N=20, λ =15

N=8, λ =10

N=8, λ =5

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u k

MMMPC with linear feedback, N=8, λ = 20, α =1

“ W ell” t uned M M M PC

“ Soft er” cont rol law

Feedbackin t he

predict ions

D if ferent cont rollers


Comparisons

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y k

MPC

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y k

MMMPC

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y k


M PC

M M M PC

M M M PCwit h linear feedback

D if ferent posit ions of t he inlet t hrot t le f rom 20o t o 100o


Comparisons

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y k

MMMPC

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y k


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MMMPC


Cont rol Input

17

Paris’09 Daniel R. Ramirez & Eduardo F. Camacho Min-max MPC

Min-Max Model Predictive Control usingClosed-Loop Predictions


MMMPC using Closed-Loop Predictions

Feedback Min-Max MPCApproximate Multi-parametric Programming

Decomposition algorithm

Example


Feedback Min Max MPC

subject to:

Open Loop Min-Max MPC

Feedback Min-Max MPC

Notion of FEEDBACK(Receding horizon scheme)


Feedback Min Max MPC

subject to:


Hard problem

References:

•LMI approaches•Multi-parametric approaches•Large scale deterministic equivalents

(Bemporad et.al. TAC 2003)

(Kothare et.al. Automatica 1996)

(Scockaert et.al. TAC 1999)


How to implement it ??

• Approximate Multi-parametric Programming • Decomposition Algorithm

Clearly an open problem. Suggestions:


The Closed-Loop Min-Max MPC

• Dynamic Programming can be used to state theclosed-loop min-max MPC.

• The problem can be expressed as the recursiveproblem.

J*t (x (t)) Jt (x (t), u (t)) s.t.

Rx x (t) + Ruu (t) ≥ rF (x (t), u (t), Ø (t)) є X (t + 1)

Jt (x (t), u (t) L (x (t), u (t)) + J*t+1 (x (t + 1))

• Where X (t+1) is the region where functionJ*t+1 (x (t+1) is defined.

18


The Closed-Loop Min-Max MPC (2)

• x (t+1) = A (Ø (t) x (t) + B (Ø (t)) u (t) + E (Ø (t)), • with • A (Ø (t)) = Ao + Ai Øi (t), • B (Ø (t)) = B0 + BiØi (t), • E (Ø (t)) = E0 + EiØi (t).

• Where Øi (t) denotes the ith component of the uncertainty vector Ø(t) є Θ.

• The stage cost of the objective function defined as:L (x (t+j), u (t+j) ||Q x (t+j) ||p + ||Ru (t+j) ||p

• With the terminal cost defined asJ*t+N (x (t+N)) ||Px (t+N) ||p.

• The solution is a piece affine function of the state.(Bemporad et. al. 2003)



The demostration is based on the following

1. If J (x (t), u) is a convex piecewise affine function(i.e., J (x (t), u) = maxi=1,.. s {Liu + Hi x (t) + ki}),the problem:

is equivalent to the following mp-LP problem:



2. If J (u, x (t),θ) and g (u, x (t), θ) are convex functions in θ for all (u, x (t)) with θ є Θ, where Θis a polyhedron with vertices θ i (i= 1, …, NØ) then the problem

is equivalent to the problem:



3. If J (u, x (t), θ) is convex and piecewise in x (t) u(i.e. J (u, x (t), θ) = Li (θ) u + Hi (θ) x (t) + ki (θ))

4. and g (u, x (t), θ) is affine in x (t) and u (i.e. g (u, x (t), θ) = Lg (θ) u + Hg (θ) x (t) + kg (θ))

5. with Li, Hi, ki, Lg, Hg, kg convex functions, 6. Then the min-max Problem is equivalent to the

problem:



• Let us now consider the first step of thedynamic programming problem (11.36) with

L (x, u ) || Qx (t +j) ||p + || Ru (t+j) ||p• The terminal cost

• and the linear system



Appying the previous results, this is equivalent to the following mp-LP problem:

19


• J*N-1 is the solution of the mp-LP problem whichresults in a piecewise affine function of x (N-1).

• The corresponding control signal u* (N-1) is also a continuours piecewise affine function of the state.

• The feasible set X (N-1) is a convex polyhedron.• The same recursively for j= N-2, N-3, … 0,• We can conclude that u* (t) is a piecewise affine

function of state x (t). • Notice that the argument does not hold when the

objective funcion is quadratic as funtion (11.42) would not be piecewise affine convex with respect tothe maximization variables θ (N-1)



Fast Implementation of MPC and Dead Time Considerations

• The number of regions grows exponentiallywith the horizon, dimension of state vector and number of vertices of uncertainpolytope.

• The state vector dimension which can be very high for processes with long deadtimes. (as can be found frequently in industry)

• In the case of a dead time of d samplinginstants, an augmented state vector

• xa (t) = [x (t)t u (t-1)T… u(t-d)T].


Fast Implementation of MPC and Dead Time Considerations(2)

• To overcome these problems.– Use the predicted state

to compute instead of

• Consider a process modelled by the reactioncurve method with a dead time equal to itstime constant. If the sampling time chosen isone-tenth of the time constant, then dim (xa(t)) = 11 while dim (x (t))= 1




Approximate Multi-parametric ProgrammingDecomposition algorithm

Example


Multi-parametric Convex Programming

z=optimization variables, x=parameters

Convex constraints

Convex objective function

Feedback MPC can be often solved by convex optimization:

Approximate multiparametric convex programming solver:

A. Bemporad and C. Filippi, Approximate Multiparametric Convex Programming, CDC 2003


Multi-parametric Convex ProgrammingProperties:

piecewise affine (sub-)optimal solution

constraintsatisfaction

(sub-)optimalitywith guaranteederror bound

tree structuresolution

20



M.V.Kothare, V. Balakrishna, M.Morari Robust Constrained Model Predictive Control using Linear Matrix Inequalities, Automatica 33, 1996

Linear uncertain systems subject to constraints on (x,u)

Objective function

Application to:

Linear control lawThe gain matrix F is computed at each time step k



Kothare´s controller properties:

The control law robustly regulates the system to the origin while assuring robust constraint satisfaction.

Approximate control law:

The control law ultimately bounds the system in a region that contains the origin while assuring robust constraint satisfaction.

(Kothare et.al. Automatica 1996)

(Muñoz de la Peña et.al.TAC 2005)Very efficient implementation




Approximate Multi-parametric Programming

Decomposition algorithmExample


Decomposition Algorithm

subject to:


Linear systems with bounded uncertainties

Additive uncertaintiesPolytopic uncertainties



Explicit form control laws:

(Bemporad et.al. TAC 2003)Recursive application of mpLP solvers

Complexity depends on state dimension

Cost functions based on a LP problem:

Cost functions based on a QP problem:

No similar result


Decomposition AlgorithmDue to convexity of the prediction model and the cost function:

Only extreme realizations have to be taken into account(finite dimensional problem)

Large scale linear problem(deterministic)

•Solve on-line large LP

•Multi-parametric approach

(Sckocaert et.al. TAC 1998)

(Kerrigan et.al. IJRNC 2004)

Nodes of the tree grow exponentially with N

Time 0 1 2

w=1

w=-1

Root

w=1

w=1

w=-1

w=-1

w(t)=(-1,1)



No similar result

1



Equivalent problem. Multi-Stage Min Max Linear Problem.

Time 0 1 2

w=1

w=- 1

Root

w=1

w=1

w=- 1

w=- 1

w(t)=(-1,1)

Cost-to-go function Vj*

Each node:

Set of variables zj

Depends on uncertainty


zj



Main contribution

It allows on-line implementation of feedback min-max MPC.(For a broader family of systems)

Algorithm that solves

(Benders, NM 1962)

Based on the ideas introduced by Benders in 1962

MILP problems, LP problems, SP problems

(Muñoz de la Peña et.al. CDC 2005)



zi

k = 1k = 2k = 3

Function is PWA Algorithm convergesto optimum



Equivalent problem. Multi-Stage Min Max Quadratic Problem.

Time 0 1 2

w=1

w=- 1

Root

w=1

w=1

w=- 1

w=- 1

w(t)=(-1,1)

Allows implementation offeedback min max MPC withquadratic cost criterion.

Approximate solution


zj



zi

k = 1k = 2k = 3

Function is PWQ

Approximation error

Bound on theerror is a parameter




Approximate Multi-parametric Programming

Decomposition algorithm

Example

1


Numerical Examples

Quadruple tank process

nx = 4

Sampling time = 10s.

Model of a real plant


Numerical Examples

Computation times:

It can be implemented up to N=5

More than 2000 nodes

Only algorithm available in the literature.

Problem: Complexity still grows exponentially with N.

Good performance... N=10

Proposed solution: RECOURSE HORIZON – Still an open problem


Finally, some papers

D.R. Ramírez and E.F. Camacho, ON THE PIECEWISE NATURE OF MIN-MAX MODEL PREDICTIVE CONTROL WITH BOUNDED UNCERTAINTIES, IEEE CDC’01.

D.R. Ramírez and E.F. Camacho, PIECEWISE AFFINITY OF MIN-MAX MPC WITH BOUNDED ADDITIVE UNCERTAINTIES AND A QUADRATIC CRITERION, Automatica, 2006.

D. Muñoz de la Peña, T. Álamo, D.R. Ramírez y E.F. Camacho, MIN-MAX MODEL PREDICTIVE CONTROL BASED ON A QUADRATIC PROGRAM, IET Proc. Control Theory and Applications, 2007.

T. Álamo, D.R. Ramírez y E.F. Camacho, EFFICIENT IMPLEMENTATION OF CONSTRAINED MIN-MAX MODEL PREDICTIVE CONTROL WITH BOUNDED UNCERTAINTIES: A VERTEX REJECTION APPROACH, Journal of Process Control, 2005.

D.R. Ramírez, T. Álamo, E.F. Camacho y D. Muñoz de la Peña, MIN-MAX MPC BASED ON A COMPUTATIONALLY EFFICIENT UPPER BOUND OF THE WORST CASE COST, Journal of Process Control, 2006.

T. Álamo, D.R. Ramírez, T. Álamo, D. Muñoz de la Peña y E.F. Camacho, MIN-MAX MPC USING A TRACTABLE QP PROBLEM, Automatica, 2007.

D. Muñoz de la Peña, A. Bemporad, and C. Filippi, ROBUST EXPLICIT MPC BASED ON APPROXIMATE MULTI-PARAMETRIC CONVEX PROGRAMMING, IEEE Transactions on Automatic Control, 2006.

David Muñoz de la Peña Sequedo, Teodoro Alamo Cantarero, A. Bemporad, Eduardo Fernández Camacho,A Decomposition Algorithm for Feedback Min-Max Model Predictive Control, IEEE Transactions on Automatic Control, 2006.

paris’09 daniel r. ramirez & eduardo f. camacho min ... eduardo camacho/wednesday...4...

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