multiple model approach to multi-parametric model predictive control of a nonlinear process a...
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Multiple Model approach toMulti-Parametric Model Predictive
Control of a Nonlinear Process a simulation case study
Boštjan Pregelj, Samo GerkšičJožef Stefan Institute, Ljubljana, Slovenia
[email protected], [email protected]
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
Outline
multi-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.:
Pku
kxfkuBkxAkx
)(
)(if)()()1(
cMkLukEx )()(
ikuTik
N
iik
TikNkN
TNkkN uRuQxxxQxxUJ
1
0
);(
);(min11 ,...,,
kkuuu
xJuNkkkk
uu
)0(
,
,
,~
if
0
maxmin
max1min
1
xx
uuu
xxx
Pu
xfuBxAx
k
k
k
kkkk
Explicit solution of MPC
u(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(
kuDkd
kxCky
kuB
kd
kx
I
A
kd
kx
)(0
)(
)1(
)(
00)(
)(
0)(
)1(
)(
100
010
0
)1(
)(
)1(
ref
refref
ku
ky
ku
k
Cky
kuI
B
ky
ku
kBA
ky
ku
k
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,
0.0527
0.4622
0
0
0
0
0.0063
0.0600
,
0.94730.136200
0.4622-0.420900
000.99370.1769
000.0600-0.7762
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
CC
ggxCCx
CC
ggxCCx
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: tuning
controller 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 studies
MM 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 studies
MM 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
[email protected], [email protected]
10th PhD Workshop on Systems and ControlSeptember 2009, Hluboka nad Vltavou, Czech Republic