TRAJECTORY FOLLOWING AND REGULATION OF CHEMICAL BATCH REACTORS VIA GENEALOGICAL DECISION TREES Enso Ikonen Systems Engineering Laboratory Department of Process and Environmental Engineering, University of Oulu, Finland Eduardo Gomez-Ramirez Universidad La Salle, Mexico City, Mexico Kaddour Najim Process Control Laboratory, E.N.S.I.A.C.E.T., Toulouse, France Contents : o Application of particle filtering to solving an optimal control problem Outline : o Background o GDT algorithm and some properties o Numerical illustrations o Regulation using GDT o Discussion PAGE 1 S YSTEMS YSTEMS E NGINEER ING L AB ATORY IFAC ALSIS IFAC ALSIS 200 2006, 6, PAGE NGINEERING ABORATORY
BACKGROUND o Optimal filtering o Bayesian approach: Estimate the evolving posterior distribution recursively in time (prediction + updating) o Kalman filter (linear Gaussian) o Particle filtering aka Sequential Monte Carlo (non-linear non-Gaussian) o Importance Sampling & Resampling Step 0. Initialization (Set initial particle positions) Step 1. Importance Sampling • Predict (using model) • Evaluate importance weights (using observation) Step 2. Resampling (Sample from weighted distribution) o Duality between optimal filtering and regulation PAGE 2 S YSTEMS YSTEMS E NGINEER ING L AB ATORY IFAC IFAC ALSIS ALSIS 200 2006, 6, PAGE NGINEERING ABORATORY
PROBLEM FORMULATION Model: ( ) = X F X , U ; X − n n n 1 n 0 ( ) Y = h X n n n Control objective: T T 2 ( ) 2 ref = + − J U , U ,..., U U Y Y ∑ ∑ T 1 2 T n n n A B = n = n 1 n 1 n T length of trajectory (horizon) A , B control and error costs (covariances) n n Find the sequence of control actions that will minimize the control objective for open-loop control. PAGE 3 S YSTEMS YSTEMS E NGINEER ING L AB ATORY IFAC IFAC ALSIS ALSIS 200 2006, 6, PAGE NGINEERING ABORATORY
OPTIMIZATION OF THE CONTROL SEQUENCE Idea: Associate Gaussian distributions to the norms of the control actions and tracking errors, and translate the cost function as the likelihood of a conditional probability Algorithm: i ˆ X = = Initialize recursions: X , i 1 , N , n = 1. 0 0 ( ) i ~ N Generate iid controls: U 0 , A . n n ( ) ( ) i i i i i ˆ = Y = Evaluate model: X F X , U ; h X . − n n n 1 n n n n β 2 i ref − − exp Y Y n n 2 B i = N n Weight according to p . n β 2 j ref − − ∑ exp Y Y n n 2 B = j 1 n N ( ) i = δ = Resample controls from p u ∑ p for each j 1 , N n n i U = i 1 n which leads to: ( ) ( ) for each j j i i j j ˆ ˆ ˆ = = X F X , U ; Y h X . { } − n n n 1 n n n n ∈ i 1 , N = Repeat for n 2 , T . PAGE 4 S YSTEMS YSTEMS E NGINEER ING L AB ATORY IFAC IFAC ALSIS ALSIS 200 2006, 6, PAGE NGINEERING ABORATORY
GENEALOGICAL DECISION TREE Interpretation as a genetic particle evolution model j ˆ Interpret state X as the parent of − n 1 i ˆ : individual X n j i ˆ ˆ = X o denote X , etc. − n 1 , n − n 1 Ancestral lines: i i ˆ ˆ ← ← X ... U ... 0 , n 1 , n i k i k ˆ ˆ ˆ ˆ ← = ← = X X U U − − − − n 2 , n n 2 n 2 , n n 2 i j i j ˆ ˆ ˆ ˆ ← = ← = X X U U − − n 1 , n − n 1 , n − n 1 n 1 i i i i ˆ ˆ ˆ ˆ ← = ← = X X U U n , n n n , n n Solution at n = T : ( ) i i i ˆ ˆ ˆ = I arg inf J U , U ,..., U n 1 , n 2 , n n , n = i 1 , N PAGE 5 S YSTEMS YSTEMS E NGINEER ING L AB ATORY IFAC ALSIS IFAC ALSIS 200 2006, 6, PAGE NGINEERING ABORATORY
CONVERGENCE Idea : - Associate Gaussian distributions to the norms in the cost function - Translate the cost function as the likelihood of a conditional probability - Duality between control and filtering problems Corresponding filtering problem: ( ) ( ) = = + X F X , W ; Y h X V − n n n 1 n n n n n where W and V are Gaussian random vectors with covariances A n and B n . We can show the following (see works with P. Del Moral): 1. To find control actions which minimize the control objective, it is equivalent to look for most likely W. 2. The conditional probability mass of W is concentrated around the optimal control sequence: { } ( ) ( ) ref ref ∈ = = Pr W ,..., W d w ,..., w Y Y ,..., Y Y 1 n 1 n 1 1 n n − β 1 n n 2 ( ) 2 = + ref − exp w Y h X d w ... d w ∑ ∑ k k k k 1 n A Z 2 k B = = k k 1 k 1 n − β 1 ( ) = exp J w ,..., w d w ... d w n 1 n 1 n Z 2 n 3. Convergence of actions to optimal actions (as N → ∞ ). PAGE 6 S YSTEMS YSTEMS E NGINEER ING L AB ATORY IFAC ALSIS IFAC ALSIS 200 2006, 6, PAGE NGINEERING ABORATORY
NUMERICAl EXAMPLES (1) ‘ABC’-batch plant Plant 20 nonlinear equations: 15 dc ( ) 2 A = − k T c 1 A dt ref , c B dc ( ) ( ) 10 2 B = − k T c k T c T, T 1 A 2 B dt dT ( ) ( ) ( ) ( ) u 2 = γ + γ + + + + k T c k T c a a T b b T 5 1 1 A 2 2 B 1 2 1 2 dt temperature target trajectory: 0 ( ) 0 10 20 30 40 50 60 70 80 90 T ref = − 20 exp 0 . 02 t 8 GDT 6 optimize sequence of ∆ u’s A = 2 2 (‘tolerated dev. on input’) 4 u B = 0.2 2 (tolerated dev. on output’) β = 1, 2 N = 2500 (# particles) 0 0 10 20 30 40 50 60 70 80 90 Results time temperature (output) design specs fulfilled temperature reference (target) randomness apparent dimensionless scaling (input) PAGE 7 S YSTEMS YSTEMS E NGINEER ING L AB ATORY IFAC IFAC ALSIS ALSIS 200 2006, 6, PAGE NGINEERING ABORATORY
NUMERICAL EXAMPLES (2) RTP-repetitive plant Plant 1000 nonlinear equations: ( ) ( ) 800 dT 4 4 F = − − − − b u c T T c T T u 1 F P 2 F A ref dt T F , T P , T p T F ( ) dT 600 4 4 P = − c T T 3 F P dt temperature target trajectory T P 400 consisting of ramp and constant 200 phases 0 2 4 6 8 10 12 14 16 GDT 4 optimize sequence of u’s A = 1 (‘tolerated dev. on input’) 3 B = 1 (tolerated dev. on output’) 2 u β = 1, N = 500 (# particles) 1 Results 0 0 2 4 6 8 10 12 14 16 design specs fulfilled time part temperature randomness apparent heating intensity More examples available: 3x3 power plant, 2-joint robot arm, PAGE 8 S YSTEMS YSTEMS E NGINEER ING L AB ATORY IFAC ALSIS IFAC ALSIS 200 2006, 6, PAGE NGINEERING ABORATORY
GDT-BASED REGULATION Feedback: Algorithm 1. Add ( SISO linear ) feedback based on output deviation (PI, for example) off-line: • suitable, e.g., for partially measured 1. Solve optimal trajectories of length T output/state trajectories from K initial states x 0 2. Receding-horizon MPC • solve optimization problem from 2. Store all sequences. current (disturbed) state • computationally heavy => discretized on-line: 3. a) if measurement of x is available : approximation with precomputed solutions Compare state x with K *( T - T min +1) states in memory and find the closest match x *. ‘Assumptions’: Set next and future controls equal to • Accuracy can be increased by making controls in the selected solution from the discretization more dense (for non- point x * forward. chaotic plants) b) if no new measurements : • Given a minimal finite horizon T min < T , Select next control from the sequence. each sequence contains a number of optimal sub-sequences 4. Apply control to plant. • Regulation problems (=setpoint 5. Return to Step 3. trajectory) + time-invariant plants make the approach feasible PAGE 9 S YSTEMS YSTEMS E NGINEER ING L AB ATORY IFAC ALSIS IFAC ALSIS 200 2006, 6, PAGE NGINEERING ABORATORY
NUMERICAL EXAMPLE (3): van der Vusse-regulation van der Vusse CSTR : setpoint setpoint A → B → C 1.15 regulated GDT 1.15 open−loop MDP ∆ c A non-monotone ss-gain 1.13 ∆ c B 1.13 non-minimum phase dyn. 1.11 1.11 1.09 c B c B 1.09 Simulations (dbase): 1.07 isothermal simulations 1.07 RMSE=0.161779 1.05 RMSE=0.258119 T = 100 (traj. length) RMSE=0.161779 1.05 RMSE=0.121665 controlled: c B manipulated: V’/V R 0 20 40 60 80 100 0 20 40 60 80 100 GDT parameters : A = 0.5 2 , B = 0.01 2 20 20 β = 1, N = 2000 regulated GDT open−loop MDP ∆ u optimized 18 18 16 16 GDT-regulation : V’/V R V’/V R K = 300 random init. states: 14 14 c A (0)=2.126 ± 10% 12 12 c B (0)=1.09 ± 10% J T < 700, T min = 60 10 10 � finite state dbase of 8 8 5760 state entries 0 20 40 60 80 100 0 20 40 60 80 100 Open-loop GDT vs. state disturbances MDP-based optimal control Simulations (regulation): GDT regulation vs state disturbances GDT-based regulation impulse disturb. in states with input constraints 10 S YSTEMS YSTEMS E NGINEER ING L AB ATORY IFAC ALSIS IFAC ALSIS 200 2006, 6, PAGE PAGE 10 NGINEERING ABORATORY
Recommend
More recommend
