Control Predictivo: pasado, presente y futurointranet.ceautomatica.es/old/actividades/jornadas/XXVII/... · 2014. 2. 22. · pasado, presente y futuro Eduardo F. Camacho Universidad

1

Control Control PredictivoPredictivo: : pasado, presente y futuropasado, presente y futuro

Eduardo F. CamachoUniversidad de Sevilla

JJAA Almeria'06 Eduardo F. Camacho Control Predictivo 2

MPC successful in industry.

Many and very diverse and successful applications:

Refining, petrochemical, polymers, Semiconductor production scheduling,Air traffic controlClinical anesthesia,….Life Extending of Boiler-Turbine Systems via Model Predictive Methods, Li et al (2004)

Many MPC vendors.


MPC successful in Academia

Many MPC sessions in control conferences and control journals, MPC workshops.4/8 finalist papers for the CEP best paper award were MPC papers (2/3 finally awarded were MPC papers)


Why is MPC so successful ?

MPC is Most general way of posing the control problem in the time domain:

Optimal controlStochastic controlKnown referencesMeasurable disturbancesMultivariableDead timeConstraintsUncertainties


tt+1t+2

t+N t t+1 t+2 …….. t+N

u(t)

Only the first control move is applied

Errors minimized over a finite horizon

Constraints taken into account

Model of process used for predicting


t+2 t+1

t+N

t+N+1

t t+1 t+2 …….. t+N t+N+1

u(t)

Only the first control move is applied again


MPC

PID: u(t)=u(t-1)+g0 e(t) + g1 e(t-1) + g2 e(t-2)

vs. PID


Real reason of success: EconomicsMPC can be used to optimize operating points (economic objectives). Optimum usually at the intersection of a set of constraints.Obtaining smaller variance and taking constraints into account allow to operate closer to constraints (and optimum).Repsol reported 2-6 months payback periods for new MPC applications.

P1 P2

Pmax


Flash

Línea 2

Línea 1 Lavado

Contacto 1

Contacto 3

Contacto 2

ESQUEMA GENERAL CIRCUITO DE GASES


- 5

- 4

- 3

- 2

- 1

0

1

2

1 1 0 0 1 9 9 2 9 8 3 9 7 4 9 6 5 9 5 6 9 4 7 9 3 8 9 2

S e r i e 1


GuiónUn poco de historiaMPC: conceptos fundamentalesMPC nolineal

ModeladoIdentificación y estimación de estadoEstabilidad y robustezImplementación

Algunas aplicacionesConclusiones


A little bit of history: the beginning

Kalman, LQG (1960) Propoi, “Use of LP methods ...” (1963)Richalet et al, Model Predictive Heuristic Control (MPHC) IDCOM (1976, 1978) (150.000 $/year benefits because of increased flowrate in the fractionator application)Cutler & Ramaker, DMC (1979,1980)Cutler et al QDMC (QP+DMC) (1983) Clarke et al GPC (1987)

First book: Bitmead et al, (1990)


The impulse of the 90s. A renewed interest from Academia (stability)

Stability was difficult to prove because of the finite horizon and the presence of constraints (non linear controller, no explicit solution, …)

A breakthrough produced in the field. As pointed out by Morari: ”the recent work has removed this technical and to some extent psychological barrier (people did not even try) and started wide spread efforts to tackle extensions of this basic problem with the new tools”. (Rawlings & Muske, 1993)Many contributions to stability and robustness of MPC: Allgower, Campo, Chen, Jaddbabaie, Kothare, Limon, Magni, Mayne, Michalska, Morari, Mosca, de Nicolao, de Olivera, Scattolini, Scokaert…


The new millenium

Linear MPC is a mature discipline. More than 4500 industrial applications (not counting licensed technology companies app.) Qinand Badgwell’03. The number of applications seems to duplicate every 4 years.Some vendors have NMPC products: Adersa (PFC), Aspen Tech (Aspen Target), Continental Control (MVC), DOT Products (NOVA-NLC), Pavilon Tech. (Process Perfecter)Efforts to developed MPC for more difficult situations:

Multiple and logical objectives (Morari, Floudas)Hybrid processes (Morari, Bemporad, Borrelli, De Schutter, van den Boom …)Nonlinear (Alamir, Alamo, Allgower, Biegler, Bock, Bravo, Chen, De Nicolao, Findeisen, Jadbadbadie, Limon, Magni, …)


MPC strategy

Consider a discrete time system:x+=f(x,u), x ∈ Rn, u ∈ Rm

The system is subject to hard constraintsx ∈ X, u ∈ U

Let u={u(0),...,u(N-1) } be a sequence of N control inputsapplied at x(0)=x,

the predicted state at i isx(i)=Φ(i;x, u)=f(x(i-1), u(i-1) )


MPC strategy (2)

1. Optimization problem PN(x,Ω):

u*= arg minu Σ(i=0,...,N-1) l(x(i),u(i)) + F(x(N))

Operating constaints .x(i) ∈ X, u(i) ∈ U, i=0,...,N-1

Terminal constraint (stability): x(N) ∈ Ω

2. Apply the receding horizon control law: KN(x)=u*(0).


Linear MPC

f(x,u) is an affine function (model)X,U,Ω are polyhedra (constraints)l and F are quadratic functions (or 1-norm or ∞-norm functions)

⇓QP or LP


Otherwise

If f(x,u) is not an affine functionOr any of X,U,Ω are not polyhedraOr any of l and F are not quadratic functions(or 1-norm or ∞-norm functions)

⇓Non linear MPC (NMPC)Non linear (non necessarily convex) optimization problem much more difficult to solve.


MPC and nonlinear processes

Most processes are non-linear, Linear approximations works for small perturbations around the operating point (well in most cases)There are processes with

continuous transitions (startups, shutdowns, etc.) and spend a great deal of time away from a steady-state operating region or never in steady-state operation (i.e. batch processes, solar plants), where the whole operation is carried out in transient mode. severe nonlinearities (even in the vicinity of steady states) hybrid


NMPC vs. LMPC

Better predictions should be obtained from more accurate models.Better predictive control should be obtain with better predictions.Is that so ?


MPC turns out to be a linear controller with a feedback of the prediction of y(t+D)

H C Process

Predictory(t+D)^

u(t) y(t)

R

(Camacho & Bordons, 1995)


Falatious congeture

An optimal predictor plus an optimal controller is going to produce the “best” closed loop behaviour.J. Normey et al (2000) showed that Smith predictor (and other DTC structures) produce “better” (more robust) controllers than optimal predictor.Optimal state estimator + optimal controller (LQG/LTR)Optimal identifier + optimal controller does not produce the optimal adaptive control (identification for control)Prediction for Control: A fundamental issue: The best model is the one that produces the “best” close loop controlnot necessarily best predictions..






Modeling:

Fundamental Models (first principle) Empirical Models (fixed structure, parameters determined from data)

State space x(t+1)=f(x(t),u(t)), y(t)=g(x(t))

Input-output (NARMAX)y(t+)= Φ(y(t), ..., y(t-ny), u(t),...,u(t-nu),e(t),...,e(t-ne+1))

Volterra (FIR, bilineal)y(t+1)= y0+Σ {i=0..N}h1(i) u(k-i)+ Σ {i=0..M} Σ {i=0..M} h2(i,j)u(t-i)u(t-j)

HammersteinWienerNN


Local Model Networks

A set of models to accommodate local operating regimesThe output of each submodel is passed through a processing function that generates a window of validity.

y(t+1)=F(Ψ(t), Φ(t)) = Σ {i=1.M}fi(Ψ(t), Φ(t)), ρi (Φ(t))

local models are usually linear multiplied by basis functions ρi (Φ(t)) chosen to have a value close to 1 in regimes where is a good approximation and a value close to 0 in other cases.

Piece Wise Affine (PWA) systems


PWA. (Sontag, 1981)

Nonlinear System:

PWA Approx.

Some hybrid processes can be modeled by PWA system

x(t+1)=f(x(t),u(t))

y(t)=g(x(t))


Function


PWA Approximation


Sistemas híbridos

Variables discretas

Enclavamientos

Modelos:- Ecuaciones diferenciales...

Modelos:Automata, redes de Petri nets, etc.

13

45

2C

AB

BB

C

CC

A))(),(()(

))(),(()1(kkgk

kkfkuxy

uxx=

=+

Variables continuas

Regulación


3

2CB

BB

CA

1

A5C

4

Hybrid Systems: Finite state machine approach

X={1,2,3,4,5}U={A,B,C}

output

1

dx/dt=f1(x)

X=1

reference control system

5

X=5

dx/dt=f2(x)


4

X=4

dx/dt=f3(x)


C

U=C, t=t1 U=A, t=t2

AState machine embedding dynamics


Control engineering approach

x(k+1) = f(x(k),u(k))y(k) = g(x(k),u(k))

state x = [xcT, xd

T]T

output y = [ycT,yd

T]T

input u = [ucT,ud

T]T,


Control engineering approach

discretecontrol command

generator

dynamicalsystem

reference

output

1 - Piecewise Affine (PWA) Systems2 - Mixed Logical Dynamical (MLD) Systems 3 - Linear Complementary (LC) Systems 4 - Extended Linear Complementary (ELC) Systems5 - Max-Min-Plus-Scaling (MMPS) Systems6 - etc


Equivalence Results

Figure 1: Graphical representation of the links between the classes of hybrid systems. An arrow going from class A to class B means that A is a subset of B. The label next to each arrow corresponds to the result that states this relation. An arrow with a star (*) require conditions to establish the indicate inclusion.

PWA

MLD

LC MMPS

ELC

Prop. 9*

Prop. 6Prop. 5*

Prop. 1Prop. 2*Prop. 4*

Prop. 3

Prop. 7

Prop. 8*

Cor. 2Cor. 1*Cor. 3*Rem. 3

Rem. 4






Nonlinear Identification is more difficult

Lack of a superposition principleA high number of plant tests required

tests with many different size steps Multivariable processes the difference in the number of tests required is even greater.

Optimization problem for parameter estimation is more difficult (offline)


State estimation

The observer error must be small to guarantee stability of the closed-loop and in general little can be said about the necessary degree of smallness.No general valid separation principle for nonlinear systems exists. Nevertheless observers are applied successfully in many NMPC applications.Extended Kalman Filter (EKF), High gain estimators.


State estimation (2)

Moving Horizon Estimation (MHE).Moving window looking backwardA dual problem to MPC: Control moves: known, process state: unknown, horizon: backwards. (Kwon et al’83), (Zimmer’94), (Mishalska and Mayne’95) ..IDCOM (Richalet’76) was developed as the dual to identification !!!

Set-membership estimationCompute a set of all states consistent with the measured output and the given noise parameters.Ellipsoidal bounding (Schweppe’68), (Kurzhanski and Valyi’96)Polyhedron bounding (Kuntsevich and Lychak’85)Interval arithmetics, zonotopes (Kieffer et al’01), (Alamo et al’03)


NMPC stabilityInfinite horizon, the objective function can be considered a Lyapunovfunction, providing nominal stability. Cannot be implemented: an infinite set of decision variables.Terminal cost. Bitmead et al’90 (linear uncostrained), Rawling & Muske’93 (linear contrained).Terminal state equality constraint. Kwon & Pearson’77 (LQR constraints), Keerthi and Gilbert’88, x(k+N)= xS

xS

x(t)

x(t+1) x(t+2)

x(t+N)


NMPC stability: Terminal region

Dual control. Michalska and Mayne(1993) x(N) ∈ ΩOnce the state enters Ω the controller switches to a previously computed stable linear strategy.

Quasi-infinite horizon. Chen and Allgower (1998). Terminal region and stabilizing control, but only for the computation of the terminal cost. The control action is determined by solving a finite horizon problem without switching to the linear controller even inside the terminal region. The term (|| x(t+N)||P)2 added to the cost function and approximates the infinite- horizon cost to go.

x(t)

x(t+1) x(t+2)

x(t+N)Ω


NMPC stability:all ingredients

Asymptotic stability theorem (Mayne 2001)The terminal set Ω is a control invariant set.The terminal cost F(x) is an associated Control Lyapunov function such thatmin{u ∈ U} {F(f(x,u))-F(x) + l(x,u) | f(x,u)∈Ω} ≤0 ∀ x∈Ω

Then the closed loop system is asymptotically stable in XN(Ω )


Removing the terminal constraint maintaining stability

The optimization problem is simplified (especially when the state is not constrained)

u*= arg minu Σ(i=0,...,N-1) l(x(i),u(i)) + F(x(N)) s.t. x(i) ∈ X, u(i) ∈ U, i=0,...,N-1 and to the terminal constraint: x(N) ∈ Ω

There is a strong interplay between infinite horizon-terminal cost and terminal regions :

A prediction horizon N and a quadratic terminal cost stabilizes the system in a neighborhood of the origin. (Parisini and Zoppoli’95)

The unconstrained (no terminal constraints) MPC satisfies the terminal constraint in a neighborhood of the origin. (Jadbabaie et al’01)

Given a terminal cost F(x) that is a CLF in Ω , define Fs(x)= F(x) if x ∈ Ω and Fs(x)= α if x ∉ Ω where Ω = {x ∈ Rn: F(x) ≤ α}. The MPC with Fs(x) as terminal cost is stabilizing for all initial state in the region where the optimal solution to PN(x,X) reaches the terminal region. (Hu and Linnemann’02)

Procedure for removing the terminal constraint while maintaining asymptotic stability and computing the domain of attraction. (Limon et al’03). Suboptimality


RobustnessNonlinear uncertain system: x+=f(x,u, θ ), x ∈ Rn, u ∈ Rm θ ∈ Rp

With bounded uncertainties θ∈Θ and subject to hard constraints x ∈ X, u ∈ U

The uncertain evolution sets: X(i)=Γ(i;x, u)= {z ∈ Rn | ∃ θ∈Θ , y ∈ X(i-1), z=f(y, u(i-1), θ)}

and X(0)=x

t t+1 t+2 … t+N

y(t)

u(t)


Robustness (2)The stability conditions has to be satisfied for all possible values of the uncertainties.

The terminal set Ω is a robustcontrol invariant set. (i.e. ∀ x∈Ω, ∀θ∈Θ ∃ u ∈ U | f(x,u, θ)∈Ω)

The terminal cost F(x) is an associated Control Lyapunovfunction such thatmin{u ∈ U} {F(f(x,u,θ))-F(x) + l(x,u) |

f(x,u,θ)∈Ω} ≤0 ∀ x∈Ω, ∀ θ∈Θ


Computation of regions in robust NMPC

Invariant regions,domain of attraction, uncertain evolution setsbounding sets (state estimation)bounding sets (identification) …

−0.4 0−1

0

x1

x 2

Γ3 (λ=1)

X3

Ω

Γ3 (λ=10)

Continuous stirred tank reactor (CSTR)

•Relatively easy if regions are polyhedron and linear transformations (f(x,u, θ) ), Kerrigan’00

•NMPC more complex and approximations are normally used.


Example: continuos stirred tank reactor (CSTR) Zonotopes: (Bravo et al’03)

( )

( ) ( )

0

0

exp

exp

AAf A A

f A cp p

dC q EC C k Cdt V RT

H kdT q E U AT T C T Tdt V C RT V Cρ ρ

⎛ ⎞= ⋅ − − ⋅ − ⋅⎜ ⎟⎝ ⎠

Δ ⋅ ⋅⎛ ⎞= ⋅ − − ⋅ − ⋅ + ⋅ −⎜ ⎟⋅ ⋅ ⋅⎝ ⎠

Sampling period 0.03.

A normalized additive uncertainty is added. It is bounded by w1=(-6.5*10-3, 6.5*10-3) and w2=(-1.2*10-3, 1.2*10-3).

A terminal robust positively invariant set is calculated.

A prediction horizon of N = 11 is used.






NMPC implementation

Solving a Nonlinear (non QP), possibly nonconvex. Real time and no convexity >>> suboptimal solutions

Sequential Quadratic Programming (SQP)Simultaneous approach (Findeisen and Allgower’02)Using a sequential approach with successive linearization around the previous trajectory.PWA >>>> Mixed Integer Programming Problem.


MPC and PWA systems

PWA Model:

Where is a polyhedral partition ofstates and input space


PWA approximations

Xkyk=Ckxk+gk

xk+1=Ak x k+Bk uk+f k?

Xk+1yk+1=Ck+1xk+1+gk+1

xk+2=Ak+1xk+1+Bk+1uk+1+fk+1

?Xk+2

yk+2=Ck+2xk+2+gk+2

xk+3=Ak+2xk+2+Bk+2uk+2+f k+2

The resulting optimization problem

U = {u(k), u(k+1), u(k+2), …,u (k+N-1)} realI = {I(k), I(k+1), I(k+2),…, I(k+N-1)} Integer

Mixed Integer-Real

Optimization Problem


I = {I(k), I(k+1), I(k+2),…, I(k+N-1)}

Index can be written as

The predicted vector is written as

The problem reduces to a QP Problem

Problem when I is fixed


...2 s1 ... 2 s1 ...2 s1 ... [Ik, Ik+1, Ik+2]

N = 2

State Transition Graph

Ik [Ik]

QP (LP) problem[Ik ,Ik+1 ,Ik+2 ]

1 s2

N = 1

[Ik, Ik+1]...


Computationalcosts

If the number of sub-systems is san the prediction horizon is N

Then the number of QP (or LP) to solve is

NRP = sN

We need to reduce the Computational Problem


2 - Use the systems information to prune (1)A - Reach and controllable sets

xk+1=Ak x k+Bk uk+f k

Xkyk=Ckxk+gk



xk+2=Ak+1xk+1+Bk+1uk+1+fk+1 xk+3=Ak+2xk+2+Bk+2uk+2+f k+2

? ?

Ik

...

[Ik,*,*]

1

2 s1 ...

s

2 s1 ...

[Ik,Ik+1,*]

[Ik,Ik+1,Ik+2]

2

2 s1 ...

Reach and controllable sets


Definition 1: (Kerrigan 2000) Reach set (RS)

Definition 2: n-Step Reachable Neighbors

Definition 3: Index set of n-SRN

The STG can be pruned as Ij+k should belong to Nk

j

Pruning using Reach Sets

2 - Use the systems information to prune (2)A - Reach and controllable sets


2 - Use the systems information to prune (3)

A - Reach and controllable sets Reach Set

-3.5 -3 -2.5 -2 -1.5 -1

2

3

4

5

10

11

12

13

18

19

21

x

-6

-4

-2

0

2

x 2

11

11

N111 = {10, 11, 12, 13, 14, 16, 18, 19 ,21};

N211 = {3, 4, 5, 8, 10, 11, 13, 14, 12, 16, 18, 19, 21, 22};

11

N311 = {3, 4, 5, 8, 10, 11, 13, 14, 12, 16, 18, 19, 21, 22}

1


Definition 4: (Kerrigan 2000) The robust one-step controllable set

Definition 5: The robust one-step controllable set of the subsystem j over the subsystem i

Pruning using Controllable Sets

If xk∉Q1Ik+1|Ik is satisfied then the xk+1 state does not belong to Ik+1

transition; therefore, the transition from Ik to Ik+1is not allowed


-2.2 -2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5

-10

-8

-6

-4

-2

0

2

9

10

11

12

13 16

17

18

19

2122

x 1

x2

1 12R (X )

Controllable sets

int(R1(X12),X 9)

Q29/12Q1

9/12 Q39/12


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k

rootnode 0depth 0

k+N-2k+1 k+2 k+Nlevel 0 level 1 level 2 level N-2 level N

k+N-1level N-1

x(k) x(k+1) x(k+2) x(k+N-2) x(k+N)x(k+N-1)

node 1,1depth 1branch 1

node 1,sdepth 1branch s

node 2,1depth 2branch 1

node 2,s2

depth 2branch s2

leave N,1depth Nbranch 1

leave N,sN

depth Nbranch sN

+ >Branch & Bound


where

It can be rearranged as

Bound on the objective function

If the sequence of subsystems is defined

then, the state and input fulfill that


It is possible to obtain a minimum bound of Ji

This is a QP Problem

then

This is a lower bound of the index and can be used in a Branch & Bound algorithm

Somewhat conservative but there are more tricks


Prediction horizon vs. Nº QP evaluation

2 4 6 8 10100

105

1010

1015

Prediction Horizon

Log(Number of QP evaluation)

Enumerative QPevaluation = SN-1

Max Number of QP evaluation

Min Number of QP evaluationMean Number of QP evaluation


NMPC implementation (2)

Although there are many clever tricks to alleviate the situation, these algorithms take time. This is a major obstacle. Only slow or small processes. Approximations and simplifications

Using short horizonsPrecomputation of solution over a grid in the state space (only small systems)


Explicit solution

The control law can be write as u(x(k)) = Fi x(k) + Gi if ∈ x(k) ∈ Pi

- Multi-parametric Problems- Computational Expensive (off-line)- Only useful for simple systems


qc3

qc4h3

h4

h1h2

qo1 qo2

u2u1

JJAA Almeria'06 Eduardo F. Camacho Control Predictivo 65 JJAA Almeria'06 Eduardo F. Camacho Control Predictivo 66





Application of NMPC to a mobile robot: Path tracking in an unstructured environment

Problem Future trajectory known (computed by planner)Unexpected obstaclesControl signals and state are constrainedSystem model is highly nonlinearThe objective function: position error, the acceleration, robot angular velocity and the proximity between the robot and the obstacles (detected with an ultrasound proximity system) Unexpected obstacles makes the objective function more complex.

(Gómez-Ortega et al, 1994)


Objective function

J= Σ {i=1..N} {|x(t+i)-xd(t+i)|2 + l (x(t+i)) +

|u(t+i-1)-u(t+i-2)|2 + |u(t+i-1)-ud(t+i-1)|2 }

l (x(.))

Potencial function (twoobstacles)

J(u)

N=1, One obstacle


NMPC for mobile robot


Controller training


Mobile robot NMPC


NMPC at a Solar plant Almeria


Distributed collectors


Metal: ρm CmAm∂Tm/ ∂t= ηo I G – G Hl(Tm-Ta) – LHt(Tm-Tf)

Fluid: ρf CfAf∂Tf/ ∂t + ρf Cf q ∂Tm/ ∂x = LHt(Tm-Tf)

Process model

Simulink model can be downloaded from:http://www.esi.us.es/~eduardo/libro-s/libro.html


NMPC computation

(Berenguel et al, 1997)


Plant results: a clear day

Very fast setpoint tracking, little overshoot


Plant results: a cloudy day


HYCON Network of ExcellenceFP6 – IST- 511368

Fuel cell: PWA model


Fuel cell: safety objective

Evaluate the possibility of starvation

Sistema

Ist

λo2



Obtained model is of the form:

where x is the state, r the ratio input, w the uncertainty, u the control action.

Suitable for MPC:1. It is possible to obtain the greatest robust (control) invariant set.2. It is possible (using LMIs) to design a controller of the form:

3. It is not difficult to implement a MPC generalizing the results of Kothare et al Automatica 1996.



Fuel cell: identification results

Example: a random current ratio

Output Error

10 20 30 40 50 60 70 80 90

2

2.05

2.1

2.15

2.2

2.25

2.3

2.35

2.4

2.45

2.5

Time [s]

Lam

bda o2

LambdaIdIdint

10 20 30 40 50 60 70 80 90

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Time [s]

Erro

r

Error IdError Id Int



HYCON benchmark exercise

Solar cooling plant9 groups participatingHYCON benchmark exercise award



Detailed simulink model andprocess data available

HYCON WP2 websitenyquist.us.es/hycon/


Conclusiones

MPC notable éxitoGrandes expectativas (híbrido y nolineal)Contribuciones notables pero muchostemas abiertos.


[email protected]://www.esi.us.es/~eduardo

Documents

Control Predictivo: pasado, presente y futurointranet.ceautomatica.es/old/actividades/jornadas/XXVII/... · 2014. 2. 22. · pasado, presente y futuro Eduardo F. Camacho Universidad