In Situ Adaptive Tabulation for Real-Time Control J. D. Hedengren - - PowerPoint PPT Presentation

In Situ Adaptive Tabulation for Real-Time Control

• J. D. Hedengren
• T. F. Edgar

The University of Texas at Austin 2004 American Control Conference Boston, MA

Outline

• Model reduction and computational

reduction

• Introduction to ISAT
• ISAT theory
• Application #1: Combined Approach
• Application #2: ISAT vs. Neural Nets
• Conclusions

Model Reduction

• Optimally reduce the number of model

variables

• Linear combination of states that retain the

most important dynamics

• Methods

– Proper Orthogonal Decomposition (or PCA) – Balanced Covariance Matrices

Model Reduction

) ( ) , ( x h y u x f x = = ɺ

)) ( ( ) ), ( ( ) (

1 1 1

Tx T h y u Tx T f T T x

− − −

= = ɺ

) ( ) , (

_ __ _ __ _

x h y u x f x = = ɺ

Original ODE model Determine a similarity transform to optimally reduce the model states

(1) (2a) (2b)

Transformed states

Tx x =

Model Reduction

                    − =            

32 1 3 _ 2 _ 1 _

0.202

• 9

. 4 0.060

• 5

. 49 0.015 1 . 9 x x x x x ⋮ ⋯ ⋯ ⋯ Binary distillation model reduction shows the relative weighting of the 32

• riginal states in the top 3 transformed states.

x32 x1 Inputs States RR x17 x31 x2 Feed Distillate Bottoms

(3)

Model Reduction

                      =                       ) , ( ) , ( ) , (

_ 3 __ _ 2 __ _ 1 __ 32 _ 4 _ 3 _ 2 _ 1 _

⋮ ɺ ⋮ ɺ ɺ ɺ ɺ u x f u x f u x f x x x x x                       =                     ) , ( ) , ( ) , ( ) , ( ) , (

_ 32 __ _ 4 __ _ 3 __ _ 2 __ _ 1 __ 3 _ 2 _ 1 _

u x f u x f u x f u x f u x f x x x ⋮ ⋮ ɺ ɺ ɺ

Truncation Residualization

Computational Reduction

• Retain all the of dynamics
• Storage and retrieval to reduce the

computational cost

• Methods

– Artificial neural networks – In situ adaptive tabulation (ISAT)

Combined Approach

• Combined approach for NMPC

– Model reduction first – Computational reduction second

First Principles Model Reduced model Storage and retrieval of reduced model integrations

Balanced Covariance Matrices ISAT

ISAT Introduction

Nearby ISAT Record Desired Integration

Approximation Error

φ0 φf δφ0 Αδφ0 φ0

ISAT

φf

ISAT

      = x u φ

• Fig. 1. Approximation of the desired integration final state with a nearby

ISAT record.

ISAT Search

• Binary Tree Architecture

– Search times are O(log2(N)) compared with O(N) for a sequential search φ2 φ1 φ

α φ <

T

v

      + = 2

1 2

φ φ α

T

v

1 2

φ φ − = v α φ >

T

v

Binary Trees

v φ = α φ2 φ1 Cutting Plane φ1 φ2 Branch Leaves

• Fig. 2. An illustration of the binary tree structure in the cutting plane

format (on the left) and the tree format (on the right).

Binary Tree Growth

φ1 φ2 Before φ2 After φ3 φ1

• Fig. 3. Binary tree growth. A tree with one branch and two leaves is

grown to include another leaf.

Binary Trees

• To increase the accuracy of the binary tree

search, multiple binary trees are searched.

• This increases the probability of finding a better

record.

• Number of binary trees is a tuning parameter

that balances search speed with search accuracy.

ISAT Integration

• Scenario #1: Inside the region of accuracy

φ1 φ

( ) ( )

tol T M

ε φ φ φ φ ≤ − −

1 1

ISAT Integration

φ1

• Scenario #2: Outside the region of accuracy but within

the error tolerance φ

( ) ( )

tol T M

ε φ φ φ φ > − −

1 1

Compute Mnew so that the new region is a symmetric, minimum volume ellipsoid that includes φ

ISAT Integration

φ1

• Scenario #3: Outside the region of accuracy and
• utside the error tolerance

φ Define cutting plane

      + = 2

1

φ φ α

T

v

1

φ φ − = v

Find a conservative estimate for the region

• f accuracy around φ

Application #1: Binary Distillation

32 state ODE model of binary distillation 5 state reduced model Storage and retrieval of integrations

Balanced Covariance Matrices ISAT

• Fig. 4. Model and computational reduction flowchart.

Closed-loop Response

5 10 15 20 25 0.91 0.92 0.93 0.94 Distillate Composition (xA) Time (min) set point 5 states/ISAT 5 states 32 states 32 states/Linear

• Fig. 5. Closed loop response comparison for nonlinear MPC with ISAT

with 5 states, nonlinear MPC with 5 states, nonlinear MPC with 32 states, and linear MPC.

CPU times

1 2 3 4 5 20 40 60 80 100 120 Speed-up Factor Optimization # 5 states/ISAT 5 states 32 states 32 states/Linear 0.26 sec avg 0.77 sec avg 9.3 sec avg 22.2 sec avg

• Fig. 6. Speed-up factor for each of the optimizations shown in Fig. 5.

The number above each curve indicates the average optimization cpu time on a 2 GHz processor.

Application #2: ISAT vs. neural net

CA1 Feed A B Reaction T1 Product q Q CA2 T2 V1 V2

• Dual CSTR model
• Fig. 7. Diagram of two CSTRs in series with a first order reaction.

The manipulated variable is the cooling rate to the first CSTR

Artificial Neural Network

7 I n p u t s Layer 1 Hyperbolic tangent sigmoid transfer function 20 neurons Layer 2 Linear transfer function 6 neurons 6 O u t p u t s

• Fig. 8. Neural net with one hidden layer and one output layer. The

hidden layer is a hyperbolic tangent function and the output layer is a linear function. This neural net relates 7 inputs to 6 outputs.

Open-loop Response

1 2 3 4 5 6 7 8 9 10 340 360 380 400 420 440 460 Temperature (K) Time (min) Actual Neural Net ISAT ISAT Retrieval ISAT Growth ISAT Addition

• Fig. 9. The error control of ISAT indicates that additional records

must be added, thereby avoiding extrapolation error.

Closed-loop Response #1

0.5 1 1.5 2 444 446 448 450 452 454 Reactor #2 Temperature (K) Time (min) set point 6 states/ISAT 6 states 6 states/Neural Net

• Fig. 10. Small closed loop set point change within the training

domain.

Closed-loop Response #2

0.5 1 1.5 2 435 440 445 450 455 Reactor #2 Temperature (K) Time (min) set point 6 states/ISAT 6 states 6 states/Neural Net

• Fig. 11. Large closed loop set point change outside of the

training domain.

Summary and Conclusions

• Combined approach includes model reduction

followed by computational reduction

• ISAT is a storage and retrieval method
• With a 32 state binary distillation, the CPU time

for NMPC is reduced by 85 times

Summary and Conclusions

• ISAT indicates when the retrieval is outside of

the storage domain

• ISAT incorporates automatic error control to

avoid extrapolation errors