Multiple Model approach toMulti-Parametric Model Predictive

Control of a Nonlinear Process a simulation case study

Boštjan Pregelj, Samo GerkšičJožef Stefan Institute, Ljubljana, Slovenia

bostjan.pregelj@ijs.si, samo.grerksic@ijs.si

10th PhD Workshop on Systems and ControlSeptember 2009, Hluboka nad Vltavou, Czech Republic

Introductionwith explicit solution the MPC is

expanding its application area to low-level control• disturbance rejection• offset-free tracking• output feedback (states usually not measurable)

» controller – estimator interplay• complexity (significant offline computation burden)

hybrid mp-MPC methods• control of hybrid or nonlinear systems• hybrid estimator required• controller and estimator model stitching/switching• extremly demanding computation & complex partition

multiple-model approach• simplified, suboptimal solution

Outlinemulti-parametric MPCtracking controller and offset removalcase study plant

• pressure control in wire annealer• nonlinear simulation model

controller design• PWA process model• controller & Kalman filter tuning

resultsremarks & conclusions

Model predictive controller, an MPC

linear system defined by a SS model

state and input constraintsMPC optimisation problem =

CFTOC

s.t.:

Pkukx

fkuBkxAkx

)()(

if)()()1(

cMkLukEx )()(

ikuTik

N

iik

TikNkN

TNkkN uRuQxxxQxxUJ

1

0

);(

);(min11 ,...,, kkuuu

xJuNkkkk

uu

)0(,

,

,~if

0

maxmin

max1min

1

xxuuuxxx

Pux

fuBxAx

k

k

k

kkkk

Explicit solution of MPCu(k) = function of current state!PWA on polyhedra control law

• where describes i -th region (polyhedron)

properties:• regions have affine boundaries• value function J*k is convex, continuous,

piece-wise quadratic function of x(k), • optimizer: x*k is affine function of x(k), possibly

discontinuous (at some types of boundaries)

kik

ik

ik

ik NiKxHgkxfkxu ,...,1,if)())((*

ik

ik KxH

State controller -> Tracking contrl.

offset-free reference tracking»velocity form augmentation

elimination of offset due to disturbance

»tracking error integration»disturbance estimation

output feedback»Kalman filter observer»additional integrating disturbance state d(k)»additional KF tuning possibilities

> responce tuning with disturb. on states, inputs> input/output step disturbance model

)(0)()(

1)(

)(0)(

)(0

0)1()1(

kuDkdkx

Cky

kuB

kdkx

IA

kdkx

)(0)()1(

)(00)(

)(0)(

)1()(

1000100

)1()(

)1(

ref

refref

kuky

kuk

Cky

kuIB

kykukBA

kykuk

x

xx

Process: pressure control in annealer

nonlinear high-order process, disturbancesactuators:

• pump – slow response, large operating range• valve – fast response, small operating range

two input single output constrained system• additional DOF• constraints 0 < u1 < 50 [s-1], 0 < u2 < 100 [%],

-5 < Δu1 < 5 [s-2], -50 < Δu2 < 50 [%/s]. 0 < p < 133 [mbar]

Process: nonlinear simulation model

2nd order linear dynamics

static input nonlinearities• u1: polynomial function y = f(u1)• u2: affine function

> y = ki u2 + ni

> i = f (u1)

• u2 nonlinearity»narrow the input constraint limit to linear range

00,1010,

0.05270.4622

00

00

0.00630.0600

,

0.94730.1362000.4622-0.420900

000.99370.1769000.0600-0.7762

DCBA

f(u1)

f(u2)

Control design: hybrid PWA model

augment the original linear model with data from other operating points

model switching» f(x2) » f(x2, x4)

boundary lines:

OP u1 [HZ] u2 [%] u1 gain u2 gain

1 (low extreme) 15 30 -0.3203 -1.0057

2 (high extreme) 10 30 -1.0010 -2.4136

3 (intermediate) 12.5 30 -0.7007 -1.7096

)4()4(

)()2()2()4()4(

)()2()2(

32

23123)3,2(2

21

12112)2,1(2

CCggxCCx

CCggxCCx

b

b

Control design: PWA process model

gains for each local dynamical model defined in output equation(Wiener model)

continuous transitions between models desiredcontroller implementation

active controller takes current state and computes control action

ii gDuxCy

PWA dynamic (i)OUTPUT (GAIN)

MATRIX CIoffset (gi)

1 [ 0 -1.0010 0 -2.4136 ] 4.24082 [ 0 -0.7007 0 -1.7096 ] -1.24973 [ 0 -0.3203 0 -1.0057 ] -8.5920

Control design: tuningcontroller parameter tuning

• guide: reasonable computation time of controller• tuning using LLA (Local Linear Analysis)

» root loci of dominant controller poles» parameters: N = 6, Nu = 2, Rdu = diag([0.1 0.05]), Ru = diag([10-6 0.02])

KF tuning• extended LLA of closed loop system• parameters:QK = diag([10-6 10-6 10-6 10-6 1])RK = 10-3

Results: simulation studiesMM mp-MPC

(N=6,Nu=2) vs linear mp-MPC (N=6, Nu=2)

tracking reference signal steps along three local dynamical models)

linear model (black) from intermediate OP

controller partition composed of 3x100 reg.

(hybrid mp-MPC 200k)

Results: simulation studiesMM mp-MPC

(N=27,Nu=2) vs linear mp-MPC (N=27, Nu=2)

improved performance due to longer horizons.

controller resuling in ~3x300 regions

hybrid mp-MPC not really feasible

Conclusions improved performance due do reduced

plant-to-model mismatch low computation demand & complexityemphasis to nonlinear PWA plane matchingsuboptimal solution

• controller does not anticipate switch in prediction• controller sellection via scheduling variable

better results achievable• other suboptimal approaches (current & future

work)» simplified hybrid mp-MPC» restrict switching among dynamics in prediction» keeps higher level of optimality

Thank you!

Multiple Model approach toMulti-Parametric Model Predictive

Control of a Nonlinear Process a simulation case study

Boštjan Pregelj, Samo GerkšičJožef Stefan Institute, Ljubljana, Slovenia

bostjan.pregelj@ijs.si, samo.grerksic@ijs.si

10th PhD Workshop on Systems and ControlSeptember 2009, Hluboka nad Vltavou, Czech Republic