An Introduction to ISAT for NMPC by Civet

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```									In Situ Adaptive Tabulation for
Nonlinear MPC
J. D. Hedengren
T. F. Edgar
The University of Texas at Austin

22 Sept. 2003
Outline
•   Introduction to ISAT
•   ISAT Theory
•   NMPC with ISAT
•   ISAT vs. Neural Nets
•   Future Directions
In Situ Adaptive Tabulation (ISAT)
• Developed by Pope1 in 1997 for turbulent
combustion simulations.
• ASCI C-SAFE application2 at the
University of Utah in 2002.
• Integrated with FLUENT3, popular
computational fluid dynamics software.
Computational Reduction Idea
Desired Integration          φf
Approximation
Error

Αδφ0
φ0
u 
δφ0                              φfISAT        φ= 
 x0 
φ0ISAT         Nearby ISAT Record

In reacting flow simulations, the integration of the chemistry
model can occur >108 times. ISAT is a storage and retrieval
method for the integrations.
ISAT Search
• Binary Tree Architecture
– Search times are O(log2(N)) compared with O(N) for
a sequential search

φ    v = φ 2 − φ1
φ2
 φ2 + φ1 
α =v T

 2 
φ1
vT φ < α
ISAT Integration
• Scenario #1: Inside the region of accuracy

φ
(φ − φ1 )
T
M (φ − φ1 ) ≤ ε tol
φ1
ISAT Integration
• Scenario #2: Outside the region of accuracy but
within the error tolerance
φ
(φ − φ1 )
T
M (φ − φ1 ) > ε tol

Compute Mnew so that the new
φ1        region is a symmetric, minimum
volume ellipsoid that includes φ
ISAT Integration
• Scenario #3: Outside the region of accuracy
and outside the error tolerance
Define cutting plane
φ                              v = φ − φ1
 φ + φ1 
α =v 
T

 2 
φ1     Find a conservative
estimate for the region
of accuracy around φ
Nonlinear MPC

-N     -N+1         -1           0        1     N-1       N

Dynamic Data Reconciliation4
def    −1
min Φ ( x,η , y ) =
x ,η
∑ [C ( x
k =− N
k   , y k ) + Ξ (η k )] + C ( x 0 , y 0 ) + Ξ (η 0 ) s.t.

y given, u given, xk +1 = F ( xk , uk ), Gxk − η k ≤ g , η k ≥ 0
Dynamic Optimization5
min Φ( x, u,η ) s.t.
x ,u ,η

x0 given, xk +1 = F ( xk , uk ), Duk ≤ d , Gxk − η k ≤ g , η k ≥ 0
Nonlinear MPC
Given: Continuous DAE model

x = f1 ( x, u )
&
-N     -N+1         -1   0       1     N-1       N
0 = f 2 ( x, u )
Dynamic Data Reconciliation1
def    −1            Need: Discrete DAE model
min Φ ( x,η , y ) = ∑ [C ( x k , y k ) + Ξ (η k )] + C ( x 0 , y 0 ) + Ξ (η 0 ) s.t.
x ,η
k =− N

y given, u given, xk +1 = F ( xk , uk ), Gxk − η k ≤ g , η k ≥ 0
Dynamic Optimization2
min Φ( x, u,η ) s.t.
x ,u ,η

x0 given, xk +1 = F ( xk , u k ), Duk ≤ d , Gxk − η k ≤ g , η k ≥ 0
ISAT with NMPC
• ISAT replaces the DAE integrator and
sensitivity calculator

u, xinitial

Optimizer    xfinal, A      ISAT
NMPC Example with ISAT
32 state binary distillation
Inputs
x1                 column model
States               Distillate
x2                       MV: reflux ratio
RR
CV: distillate composition
Feed
x17                      Simplex optimizer
Soft constraint on the MV
x31                      Control Horizon = 10 min
Bottoms
x32                Prediction Horizon = 15 min
Closed Loop Response
70
set point                                                                   32 states/ISAT
0.94                       32 states/ISAT                                                              32 states
60
32 states                                                                   32 states/Linear
Distillate Composition (x A )

32 states/Linear                          50
0.28 sec average

Speed-up Factor
0.93
40

30
0.92
0.84 sec average
20

0.91                                                                 10
12.6 sec average
0
0   5    10        15      20          25                          1      2          3              4              5
Time (min)                                                           Optimization #
ISAT Performance
• Successful with ODE and DAE models
• Computational speedup 20 – 500 times
• Storage <100MB for 96 state DAE model
with εtol = 10-3
ISAT vs. Neural Nets
Feed
6 state dual CSTR model

V1                   V2
MV: cooling rate of CSTR 1
Reaction
Q   T1                   T2                 A    B
CA1                  CA2
CV: product temperature
q   Product
ISAT and Neural Net used
the same training data
7         Layer 1      Layer 2          6
Hyperbolic    Linear           O              Compared in open loop and
I
n
tangent
sigmoid
transfer
function
u
t
closed loop simulations
p
transfer                      p
u                     6 neurons
t
function                      u              Control Horizon = 0.4 min
t
s        20 neurons
s
Prediction Horizon = 0.6 min
Open Loop (ISAT vs. Neural Net)
460

440

420
Temperature (K)

Actual
400                Neural Net
ISAT

380               ISAT Retrieval
ISAT Growth

340
0   1   2       3         4       5      6   7   8   9   10
Time (min)
Closed Loop (ISAT vs. Neural Net)
455
set point
6 states/ISAT
Reactor #2 Temperature (K)

6 states
450                          6 states/Neural Net

445

440

435
0   0.5       1              1.5             2
Time (min)
Future Directions
• Develop ISAT in
C++/Fortran (currently in
MATLAB)
• Integrate ISAT with
NMPC toolbox for
Octave
• Control of Reactive
Distillation
• Other applications?
• Questions?
References
[1] S. B. Pope, Pope, S. B. Computationally Efficient Implementation of Combustion
Chemistry Using In Situ Adaptive Tabulation. Combustion Theory Modeling
vol. 1, pp. 41-63, 1997.

[2] C-SAFE, Center for the Simulation of Accidental Fires and Explosions,
URL: http://www.csafe.utah.edu/, Accessed Sept. 2003.

[3] FLUENT, Reacting Flows, URL:
http://www.fluent.com/software/fluent/focus/reacting.htm, Accessed Sept. 2003.

[4] M. J. Liebman, T. F. Edgar, and L. S. Lasdon, Efficient data reconciliation and
estimation for dynamic processes using nonlinear programming techniques,
Comp. Chem. Eng., 16:963-986, 1992.

[5] M. J. Tenny, S. J. Wright, and J. B. Rawlings, Nonlinear model predictive control
via feasibility-perturbed sequential quadratic programming, Texas-Wisconsin
Modeling and Control Consortium, Report TWMCC-2002-02, 2002.

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