# Efficient Moving Horizon Estimation of DAE Systems John D. - PowerPoint PPT Presentation

## Efficient Moving Horizon Estimation of DAE Systems John D. Hedengren Thomas F. Edgar The University of Texas at Austin TWMCC Feb 7, 2005 Outline Introduction: Moving Horizon Estimation (MHE) Explicit Solution to the MHE Problem

1. Efficient Moving Horizon Estimation of DAE Systems John D. Hedengren Thomas F. Edgar The University of Texas at Austin TWMCC – Feb 7, 2005

2. Outline • Introduction: Moving Horizon Estimation (MHE) • Explicit Solution to the MHE Problem • Observability of Index-1 DAE Systems • Flash Column Example • Conclusions and Future Work

3. Introduction • Explicit MHE of nonlinear systems (Ramamurthi, Sistu, and Bequette, 1993) • MHE definitively shown to outperform EKF (Haseltine and Rawlings, 2004). Price of improvement is greater computational expense of MHE. • This presentation incorporates the recent MHE advances in an explicit solution form • Motivation: State estimation of large-scale, nonlinear DAE systems • Strategy: Combine elements of existing technologies in new ways to solve large-scale problems • Everything has been thought of before, but the problem is to think of it again. – Goethe (via Jim Rawlings/via Tom Badgwell)

4. Moving Horizon Estimation Nonlinear DAE model (implicit form) ɺ 0 = f x x u ( , , ) DAE model - implicit form y = g x ( ) System measurements Discretized linear time-varying form  y   y  model ,0 meas ,0 x + = A x + B u k 1 k k k k     Y = ⋮ , Y = ⋮ , n = horizon length model meas     y = C x + D u k k k k k     y y model n , meas n ,     MHE optimization (least squares approach) T ( ) ( ) minimize Y − Y Q Y − Y meas model meas model subject to the model equations

5. Explicit MHE (Previous Work) Explicit solution to the least squares MHE problem (Ramamurthi, Sistu, and Bequette, 1993 – (Bequette UT PhD grad 1988, now at RPI) j − 1 k − 1 k   ∏ ∑ ∏ ω = C A ψ = C A B u + D u k k j k k  i k − − j  k − j k − j k k j = 0 j = 1 i = 1   Y = ω x + ψ model 0 − 1 T T [ ] ˆ   x = ω Q ω ω Q Y − ψ 0 y y meas   Forgetting factor ( α ) adds infinite horizon approximation by incorporating previous state estimates (Haseltine and Rawlings, 2004)  x   x ˆ  model ,0 prev ,0     ˆ X = ⋮ , X = ⋮ , n = horizon length model prev         ˆ x x model n , prev n ,     T T ( ) ( ) ( ) ( ) ˆ ˆ minimize Y − Y Q Y − Y + α X − X X − X meas model meas model prev model prev model subject to the model equations

6. Example 1: Explicit MHE Solution (Unconstrained, Linear) Discrete/State Space Form .8144 − 0.0905 0.0905     1 x = x + u k + 1 k k     G s = ( ) 2 0.0905 0.9953 0.0047     s + 2 s + 1 [ ] y = 0 1 x + v k k k Sampling time: 0.1 sec ν k (output noise) is normally distributed with µ = 0 and σ = 0.1 Initial conditions: Actual = [0 0] T Predicted = [1 1] T MHE values: forgetting factor α = 0.5 (weighting on x 0,est )

7. Example 1: Results 1 Actual 0.8 Measured (2nd state only) Explicit MHE 0.6 MHE x 1 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1.5 1 x 2 0.5 0 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time

8. MHE for Real Systems • Upper and lower bounds that represent physical limits on state variables (e.g., mole fractions are between 0 and 1) • Variable measurement frequencies (e.g., temperatures at 1 sample/sec and concentrations at 1 sample/minute) • Corrupt or missing data • Large-scale, nonlinear, rigorous DAE models • Solve MHE subject to real-time constraints • The new approach in this presentation: – Explicitly solves the least squares MHE problem subject to upper and lower bounds on the states – Is able to meet real-time constraints for large-scale problems – Is flexible to handle variable measurement frequencies and missing data

9. Incorporate Constraints (Upper and Lower Bounds) • Iteratively add/remove measurements that add constraint information • Uses strategies from the active set and penalty methods from nonlinear programming – Active set strategy – optimizer only deals with active inequality constraints ( x = a or x = b ) and ignores inactive constraints ( a < x < b ) – Penalty method – cost added to the objective function when constraints are violated T ( ) ( ) minimize Y − Y Q Y − Y meas model meas model subject to the model equations subject to a ≤ x ≤ b

10. Adding/Removing Constraints • For active constraints – Define the Lagrange multiplier (shadow price) ( λ lower = Q ( a - Y model ) or λ upper = Q ( Y model - b )) – If λ < 0, remove constraint from the active set • For inactive constraints – If x > b add upper limit constraint with a measurement ( y meas = b ) – If x < a add lower limit constraint with a measurement ( y meas = a ) • Pseudo-code: Do – Compute Explicit MHE – If λ < 0 remove constraint – If x > b add measurement y meas =b – If x < a add measurement y meas =a Loop Until No Active Set Change

11. Constrained Explicit MHE (Other Enhancements) Forgetting factor ( α ) adds infinite horizon approximation by incorporating previous state estimates (Haseltine and Rawlings, 2004)  ˆ   x  x model ,0 prev ,0     ˆ X = ⋮ , X = ⋮ , n = horizon length model prev         ˆ x x model n , prev n ,     T T ( ) ( ) ( ) ( ) ˆ ˆ minimize Y − Y Q Y − Y + α X − X X − X meas model meas model prev model prev model subject to the model equations Explicit solution (new) T ( ) Q = C Q C x k , k y k , k j − 1 k − 1 k   ∏ ∑ ∏ ω = A ψ = A B u + D u k j k  i k − − j  k − j k − j k k j = 1 j = 0 i = 1   − 1 T ( T T T ) ˆ   ˆ     x = ω Q + α I ω ω Q C Y − ψ + αω X − ψ ( ) 0 x x meas prev      

12. Constrained Explicit MHE (Other Enhancements) Augment system with input disturbance variables ( d ) (Muske and Badgwell, 2002) x A B x B         = + u         k d 0 1 d 0         k + 1 k k k x   [ ] y = C 0 k   k d   k Estimate input disturbances and states in one explicit step instead of two iterative steps as in Ramamurthi, Sistu, and Bequette (1993)

13. Example 2: (Constrained Version of Example 1) 1 Actual Upper bound: x 1 <0.2 Measured Lower bound: x 1 >0.0 Physical constraint: x(1) < 0.2 Unconstrained MHE x 1 0.5 Constrained MHE 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 2 Upper bound: x 2 < ∞ 1 Lower bound: x 2 > -∞ x 2 0 -1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 d -0.5 -1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time

14. Example 2: Final Solution 1 Actual Measured Note x 1 is now below the upper bound CE MHE 0.5 x 1 MHE 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 2 1 x 2 0 -1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 -0.5 d -1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time

15. Horizon = 50, Iteration 1 Actual 0.4 Measured CE MHE MHE x 1 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 0.5 x 2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.2 -0.25 x 3 -0.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time

16. Horizon = 50, Iteration 2 Actual 0.4 Measured CE MHE MHE x 1 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 0.5 x 2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.2 -0.25 x 3 -0.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time

17. Horizon = 50, Iteration 3 Actual 0.4 Measured CE MHE MHE x 1 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 0.5 x 2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.2 -0.25 x 3 -0.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time

18. Horizon = 50, Iteration 4 Actual 0.4 Measured CE MHE MHE x 1 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 0.5 x 2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.2 -0.25 x 3 -0.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time

19. Horizon = 50, Iteration 5 Actual 0.4 Measured CE MHE MHE x 1 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 0.5 x 2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.2 -0.25 x 3 -0.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time

20. Constrained Explicit MHE Properties • No optimizers involved • Explicit solution at each iteration = fast computation, reliable solution • Incorporate constraints by iteratively adding or removing measurements based on an active set strategy • Constraints are restricted to upper and lower variable bounds • Computational costs for the previous example (horizon=50) – Constrained Explicit MHE (6,831 flops) – MHE (122,048,803 flops)