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 f x u ɺ (1) Original ODE model ( ) = y h x 1 1 ( − ) ( − ( ), ) = x T T f T Tx u Determine a similarity ɺ (2a) transform to optimally 1 ( − ( )) = y h T Tx reduce the model states _ __ _ ( , ) = x f x u ɺ x = Tx Transformed states (2b) __ _ ( ) = y h x
Model Reduction Binary distillation model reduction shows the relative weighting of the 32 original states in the top 3 transformed states. x 1 Inputs States Distillate x 2 _ RR x 9 . 1 0.015 x 1 1 _ ⋯ 49 . 5 - 0.060 = x (3) Feed 2 x 17 ⋯ ⋮ _ 4 . 9 - 0.202 − x x 3 32 ⋯ x 31 Bottoms x 32
Model Reduction Truncation Residualization __ _ __ _ _ _ ( , ) ( , ) f x u f x u x x 1 1 1 1 ɺ _ __ _ __ _ ɺ _ x ( , ) ( , ) f x u f x u x 2 2 2 2 ɺ _ ɺ __ _ _ __ _ x ( , ) ( , ) f x u = f x u 3 x = 3 ɺ 3 3 _ ɺ __ _ x 0 0 4 ( , ) f x u ɺ 4 _ ⋮ ⋮ ⋮ __ _ x 0 ⋮ 0 32 ( , ) f x u ɺ 32
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 Reduced Storage and Balanced ISAT Principles model retrieval of Covariance Model reduced Matrices model integrations
ISAT Introduction φ f Desired Integration Approximation Error Αδφ 0 φ 0 u φ = x φ f δφ 0 ISAT 0 φ 0 ISAT Nearby ISAT Record Fig. 1. Approximation of the desired integration final state with a nearby ISAT record.
ISAT Search • Binary Tree Architecture – Search times are O(log 2 (N)) compared with O(N) for a sequential search = φ − φ v 2 1 φ φ 2 φ + φ α = T 2 1 v 2 T φ > α v φ 1 T φ < α v
Binary Trees Branch φ 2 Cutting Plane φ 1 v φ = α φ 2 φ 1 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 Before After φ 2 φ 1 φ 2 φ 1 φ 3 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 φ T M ( ) ( ) φ − φ φ − φ ≤ ε 1 1 tol φ 1
ISAT Integration • Scenario #2: Outside the region of accuracy but within the error tolerance φ T M ( ) ( ) φ − φ φ − φ > ε 1 1 tol Compute M new so that the new region is a symmetric, minimum φ 1 volume ellipsoid that includes φ
ISAT Integration • Scenario #3: Outside the region of accuracy and outside the error tolerance Define cutting plane = φ − φ v φ 1 φ + φ α = T 1 v 2 φ 1 Find a conservative estimate for the region of accuracy around φ
Application #1: Binary Distillation 32 state ODE 5 state Storage and Balanced ISAT model of reduced retrieval of Covariance binary model integrations Matrices distillation Fig. 4. Model and computational reduction flowchart.
Closed-loop Response set point 0.94 5 states/ISAT 5 states Distillate Composition (x A ) 32 states 32 states/Linear 0.93 0.92 0.91 0 5 10 15 20 25 Time (min) 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 120 5 states/ISAT 5 states 100 32 states 32 states/Linear Speed-up Factor 80 0.26 sec avg 60 40 0.77 sec avg 20 9.3 sec avg 22.2 sec avg 0 1 2 3 4 5 Optimization # 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 • Dual CSTR model Feed V 1 V 2 Reaction Q T 1 T 2 A B C A1 C A2 q Product 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 Layer 1 Layer 2 6 Hyperbolic Linear O I tangent transfer u n sigmoid function t p transfer p u 6 neurons function u t t s 20 neurons 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 460 440 420 Temperature (K) Actual Neural Net 400 ISAT 380 ISAT Retrieval ISAT Growth 360 ISAT Addition 340 0 1 2 3 4 5 6 7 8 9 10 Time (min) Fig. 9. The error control of ISAT indicates that additional records must be added, thereby avoiding extrapolation error.
Closed-loop Response #1 454 set point 6 states/ISAT Reactor #2 Temperature (K) 452 6 states 6 states/Neural Net 450 448 446 444 0 0.5 1 1.5 2 Time (min) Fig. 10. Small closed loop set point change within the training domain.
Closed-loop Response #2 455 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) 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
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