TRAJECTORY FOLLOWING AND REGULATION OF CHEMICAL BATCH REACTORS VIA - - PowerPoint PPT Presentation

SYSTEMS YSTEMS ENGINEER NGINEERING ING LAB ABORATORY ATORY

IFAC IFAC ALSIS ALSIS 200 2006, 6, PAGE

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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:

• Application of particle filtering to

solving an optimal control problem Outline:

• Background
• GDT algorithm and some properties
• Numerical illustrations
• Regulation using GDT
• Discussion

SYSTEMS YSTEMS ENGINEER NGINEERING ING LAB ABORATORY ATORY

IFAC IFAC ALSIS ALSIS 200 2006, 6, PAGE

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BACKGROUND

• Optimal filtering
• Bayesian approach: Estimate the evolving posterior

distribution recursively in time (prediction + updating)

• Kalman filter (linear Gaussian)
• Particle filtering aka Sequential Monte Carlo (non-linear

non-Gaussian)

• 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)

• Duality between optimal filtering and regulation

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PROBLEM FORMULATION Model:

( )

n n n n

F U X X ,

1 −

= ; X

( )

n n n

h X Y = Control objective:

( )

∑ ∑

= =

− + =

T n ref n n T n n T T

n n

J

1 2 1 2 2 1

,..., ,

B A

Y Y U U U U T length of trajectory (horizon)

n

A ,

n

B control and error costs (covariances) Find the sequence of control actions that will minimize the control objective for open-loop control.

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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: Initialize recursions: ˆ X X =

i

, N i , 1 = , n = 1. Generate iid controls:

( )

n i n

A U , ~ N . Evaluate model:

( )

i n i n n i n

F U X X , ˆ

1 −

= ;

( )

i n n i n

h X Y = . Weight according to

∑

=

      − −       − − = N

j n j n n i n i n

n n

p

1 2 ref 2 ref

2 exp 2 exp

B B

Y Y Y Y β β . Resample controls from

( )

∑

=

=

N i i n n

i n

p p

1 U

u δ for each N j , 1 = which leads to:

( )

i n i n n j n

F U X X , ˆ ˆ

1 −

= ;

( )

j n n j n

h X Y ˆ =

{ }

N i j , 1 each for ∈

. Repeat for T n , 2 = .

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GENEALOGICAL DECISION TREE Interpretation as a genetic particle evolution model Interpret state

j n 1

ˆ

−

X as the parent of individual

i n

X ˆ :

• denote

j n i n n 1 , 1

ˆ ˆ

− −

= X X , etc. Solution at n = T :

( )

i n n i n i n n N i

J I

, , 2 , 1 , 1

ˆ ,..., ˆ , ˆ inf arg U U U

=

= Ancestral lines:

i n i n n j n i n n k n i n n i n

X X X X X X X ˆ ˆ ˆ ˆ ˆ ˆ ... ˆ

, 1 , 1 2 , 2 ,

= ← = ← = ← ←

− − − − i n i n n j n i n n k n i n n i n

U U U U U U U ˆ ˆ ˆ ˆ ˆ ˆ ... ˆ

, 1 , 1 2 , 2 , 1

= ← = ← = ← ←

− − − −

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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:

( )

n n n n

F W X X ,

1 −

= ;

( )

n n n n

h V X Y + = where W and V are Gaussian random vectors with covariances An and Bn. 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:

( ) ( )

{ }

( ) ( )

n n n n n n k k k k k n k k k n n n n n

d d J Z d d h Z d w w w w w w X Y w Y Y Y Y w w W W

B A

... ,..., 2 exp 1 ... 2 exp 1 ,..., ,..., ,..., Pr

1 1 1 1 2 ref 1 2 ref ref 1 1 1 1

      − =             − + − = = = ∈

∑ ∑

= =

β β

3. Convergence of actions to optimal actions (as N → ∞).

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NUMERICAl EXAMPLES (1) ‘ABC’-batch plant

Plant nonlinear equations:

( ) ( ) ( ) ( ) ( ) ( ) ( )u

T b b T a a c T k c T k dt dT c T k c T k dt dc c T k dt dc

B A B A B A A 2 1 2 1 2 2 2 1 1 2 2 1 2 1

+ + + + + = − = − = γ γ

temperature target trajectory:

( )

t T ref 02 . exp 20 − =

GDT

• ptimize sequence of ∆u’s

A = 22 (‘tolerated dev. on input’) B = 0.22 (tolerated dev. on output’) β = 1, N = 2500 (# particles) Results design specs fulfilled randomness apparent

10 20 30 40 50 60 70 80 90 5 10 15 20 T, T

ref, cB

10 20 30 40 50 60 70 80 90 2 4 6 8 u time

temperature (output) temperature reference (target) dimensionless scaling (input)

SYSTEMS YSTEMS ENGINEER NGINEERING ING LAB ABORATORY ATORY

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NUMERICAL EXAMPLES (2) RTP-repetitive plant

Plant nonlinear equations:

( ) ( ) ( )

4 4 3 2 4 4 1 P F P A F P F u F

T T c dt dT T T c T T c u b dt dT − = − − − − =

temperature target trajectory consisting of ramp and constant phases GDT

• ptimize sequence of u’s

A = 1 (‘tolerated dev. on input’) B = 1 (tolerated dev. on output’) β = 1, N = 500 (# particles) Results design specs fulfilled randomness apparent

2 4 6 8 10 12 14 16 200 400 600 800 1000 TF, TP, Tp

ref

TF TP 2 4 6 8 10 12 14 16 1 2 3 4 u time

part temperature heating intensity More examples available: 3x3 power plant, 2-joint robot arm,

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GDT-BASED REGULATION

‘Assumptions’:

• Accuracy can be increased by making

the discretization more dense (for non- chaotic plants)

• Given a minimal finite horizon Tmin<T,

each sequence contains a number of

• ptimal sub-sequences
• Regulation problems (=setpoint

trajectory) + time-invariant plants make the approach feasible Feedback:

• 1. Add (SISO linear) feedback based on
• utput deviation (PI, for example)
• suitable, e.g., for partially measured
• utput/state trajectories
• 2. Receding-horizon MPC
• solve optimization problem from

current (disturbed) state

• computationally heavy => discretized

approximation with precomputed solutions

Algorithm

• ff-line:
• 1. Solve optimal trajectories of length T

from K initial states x0

• 2. Store all sequences.
• n-line:
• 3. a) if measurement of x is available:

Compare state x with K*(T-Tmin+1) states in memory and find the closest match x*. Set next and future controls equal to controls in the selected solution from point x* forward. b) if no new measurements: Select next control from the sequence.

• 4. Apply control to plant.
• 5. Return to Step 3.

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NUMERICAL EXAMPLE (3): van der Vusse-regulation

20 40 60 80 100 8 10 12 14 16 18 20 V’/VR

regulated

• pen−loop

20 40 60 80 100 1.05 1.07 1.09 1.11 1.13 1.15 cB

RMSE=0.161779 RMSE=0.258119 setpoint regulated

• pen−loop

∆cA ∆cB

Open-loop GDT vs. state disturbances GDT regulation vs state disturbances

20 40 60 80 100 8 10 12 14 16 18 20 V’/VR

GDT MDP

20 40 60 80 100 1.05 1.07 1.09 1.11 1.13 1.15 cB

RMSE=0.161779 RMSE=0.121665 setpoint GDT MDP

MDP-based optimal control GDT-based regulation with input constraints van der Vusse CSTR: A→B→C non-monotone ss-gain non-minimum phase dyn. Simulations (dbase): isothermal simulations T = 100 (traj. length) controlled: cB manipulated: V’/VR GDT parameters: A = 0.52, B = 0.012 β = 1, N = 2000 ∆u optimized GDT-regulation: K = 300 random init. states: cA(0)=2.126 ± 10% cB(0)=1.09 ± 10% JT < 700, Tmin = 60 finite state dbase of 5760 state entries Simulations (regulation): impulse disturb. in states

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DISCUSSION Why (..do we need the approach)?

• an optimization technique suitable for

solving (potentially) difficult problems

• nonlinear, discontinuous, sequential
• pen loop trajectory problems
• main drawback: a noiseless state-space

model is required/assumed

(Selection of) GDT-algorithm parameters ?

• not always simple (trial and error)
• future directions: non-gaussian non-

diagonal distributions in A and B

(Large) number of particles N needed ?

• Computations are realizable on office PC

(at least for small dimensional problems)

(Theoretical..) properties ?

duality btw. optimal control and optimal filtering => ...

(How to assess potential..) usefulness of GDT ?

• What to compare with?
• MDP (finite state MC + Bellman equation)
• what else would be fair / interesting?
• Real industrial applications ?

(How to use GDT in..) feedback control ?

• Linear feedback for disturbances
• e.g., GDT + PI
• Receding-horizon model predictive control
• approximate on-line solutions to open

loop problems

• requires state measurements and/or

state estimators

SYSTEMS YSTEMS ENGINEER NGINEERING ING LAB ABORATORY ATORY

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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

For more info, see:

http://cc.oulu.fi/~iko/MGDT.htm

• MATLAB-code
• a users’ guide with examples

T Th ha an nk k y yo

• u

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