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

Transcription

1 Efficient Moving Horizon Estimation of DAE Systems John D. Hedengren Thomas F. Edgar The University of Teas at Austin TWMCC Feb 7, 25

2 Outline Introduction: Moving Horizon Estimation (MHE) Eplicit Solution to the MHE Problem Observability of Inde- DAE Systems Flash Column Eample Conclusions and Future Work

3 Introduction Eplicit MHE of nonlinear systems (Ramamurthi, Sistu, and Bequette, 993) MHE definitively shown to outperform EKF (Haseltine and Rawlings, 24). Price of improvement is greater computational epense of MHE. This presentation incorporates the recent MHE advances in an eplicit solution form Motivation: State estimation of large-scale, nonlinear DAE systems Strategy: Combine elements of eisting 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) = f u ( ɺ,, ) DAE model - implicit form y = g ( ) System measurements Discretized linear time-varying form A B u + = + k k k k k = + y C D u k k k k k y model, meas, model =, meas =, = horizon length Y Y n y y model, n meas, n y MHE optimization (least squares approach) T minimize ( Y Y ) Q( Y Y ) meas model meas model subject to the model equations

5 Eplicit MHE (Previous Work) Eplicit solution to the least squares MHE problem (Ramamurthi, Sistu, and Bequette, 993 (Bequette UT PhD grad 988, now at RPI) ω Y k k j = C k k A ψ = C j k k Ai k B u D u j k j k + j k k j= j= i= model ω = + ψ [ ] T T ˆ = ω Q ω ω Q Y ψ y y meas Forgetting factor (α) adds infinite horizon approimation by incorporating previous state estimates (Haseltine and Rawlings, 24) ˆ model, prev, ˆ =, =, = horizon length model prev X X n minimize ˆ model, n prev, n T T ( Y Y ) Q( Y Y ) + ˆ ˆ α meas model meas model ( X X prev model ) ( X X prev model ) subject to the model equations

6 Eample : Eplicit MHE Solution (Unconstrained, Linear) Discrete/State Space Form G s ( ) = s 2 + 2s k + = + k k u [ ] y = + v k k k Sampling time:. sec ν k (output noise) is normally distributed with µ = and σ =. Initial conditions: Actual = [ ] T Predicted = [ ] T MHE values: forgetting factor α =.5 (weighting on,est )

7 Eample : Results Actual Measured (2nd state only) Eplicit MHE MHE time

8 MHE for Real Systems Upper and lower bounds that represent physical limits on state variables (e.g., mole fractions are between and ) Variable measurement frequencies (e.g., temperatures at sample/sec and concentrations at 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: Eplicitly 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 fleible 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 ( = a or = b) and ignores inactive constraints (a < < b) Penalty method cost added to the objective function when constraints are violated T minimize ( Y Y ) Q( Y Y ) subject to the model equations subject to a b meas model meas model

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

11 Constrained Eplicit MHE (Other Enhancements) Forgetting factor (α) adds infinite horizon approimation by incorporating previous state estimates (Haseltine and Rawlings, 24) ˆ model, prev, ˆ =, =, = horizon length model prev X X n minimize ˆ model, n prev, n T T ( Y Y ) Q( Y Y ) + ˆ ˆ α meas model meas model ( X X prev model ) ( X X prev model ) subject to the model equations Eplicit solution (new) T ( ) Q C Q C ω =, k k y, k k k k j = k A ψ = j k Ai k B u D u j k j k + j k k j= j= i= ( ) ˆ ( meas prev ) T T T T ˆ Q I Q C Y X ω α ω ω ψ αω ψ = + +

12 Constrained Eplicit MHE (Other Enhancements) Augment system with input disturbance variables (d) (Muske and Badgwell, 22) A B B = + d d k + k k k y k = [ C ] k d k u k Estimate input disturbances and states in one eplicit step instead of two iterative steps as in Ramamurthi, Sistu, and Bequette (993)

13 Eample 2: (Constrained Version of Eample ).5 Upper bound: <.2 Lower Physical bound: constraint: >. () <.2 Actual Measured Unconstrained MHE Constrained MHE Upper bound: 2 < Lower bound: 2 > d time

14 Eample 2: Final Solution Actual.5 Note is now below the upper bound Measured CE MHE MHE d time

15 .4.2 Horizon = 5, Iteration Actual Measured CE MHE MHE time

16 .4.2 Horizon = 5, Iteration 2 Actual Measured CE MHE MHE time

17 .4.2 Horizon = 5, Iteration 3 Actual Measured CE MHE MHE time

18 .4.2 Horizon = 5, Iteration 4 Actual Measured CE MHE MHE time

19 .4.2 Horizon = 5, Iteration 5 Actual Measured CE MHE MHE time

20 Constrained Constrained Eplicit MHE Properties No optimizers involved Eplicit 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 eample (horizon=5) Eplicit MHE (6,83 flops) MHE (22,48,83 flops)

21 Local Observability of Inde- DAE States Recall for discrete linear time-invariant (LTI) ODE models: The system is observable iff Γ o [A,C] (observability matri) is full rank For inde- DAE models The system is observable if every dynamic mode in A is connected to the output through C A A B ɺ 2 = + A A B y C C Du = [ 2 ] + 2 u Semi-eplicit DAE form (linearized)

22 Local Observability of Inde- DAE States Define how the dynamic modes (differential states) connect to the output (measurements) ɶ A = e ( A A2 A22 A2 ) t = ɶ C C C A A ( A C) ɶ System is observable if and only if, is full rank Γ ɶ o

23 Eample 3: Composition Estimation Eample Can the composition of a hydrocarbon miture be reconstructed with temperature and flow rate measurements of a flash column? Measure feed and flash temperature Measure vapor and liquid flow rates Rigorous nonlinear flash column model 7 state model (reduced from 57 states) 5 differential equations / 2 algebraic equations Vapor Hydrocarbon miture with unknown composition C 4 H C 5 H 2 C 6 H 4 C 7 H 6 C 8 H 8 Flash column Liquid

24 Object Oriented Modeling Plant-wide dynamic models Advanced Models distillation columns, etc. Variables. Name 2. Value 3. d(value)/dt 4. Default Scaling 5. Upper limit 6. Lower limit 7. Etc Basic Models vessel, splitter, mier, compressor, flash column, distillation stage Vessel Accumulation Mier Stream Equations. Name 2. Residual Value 3. Default Scaling 4. Variable Sparsity 5. Analytic Derivatives 6. Etc Variables - pressure, temperature, mole fractions, mass fractions, concentrations, moles, mass, volume, enthalpy Fundamental dynamic equations Thermo-physical properties database (DIPPR)

25 Model Reduction: Eample 3 Flowsheet Feed Vessel Flash Column Streams (6) pressure (), temperature (), mole fractions (5), mass fractions (5), concentrations (5), molar flow rate (), mass flow rate (), volumetric flow rate (), density (), enthalpy () Accumulations () pressure (), temperature (), mole fractions (5), mass fractions (5), concentrations (5), moles (), mass (), molar volume (), density (), enthalpy () Additional Vessel Variables vessel volume (), heat added () Additional Flash Column Variables heat added () Total variables/equations: 57

26 Model Reduction: Eample 3 Lower triangular block diagonalization of the sparsity pattern reveals variables that can be removed from the implicit set eplicit transformation of equations (reduction of 9 variables) Merge stream objects instead of defining connection equations (reduction of 2 variables) Specify feed stream, vessel volume, heat addition to the vessel, and heat addition to the flash column (specify 9 variables) Resulting model size: 7 variables (5 differential/2 algebraic)

27 Composition Estimation Eample Measurements every sec T tank σ noise =.5 dn/dt vapor σ noise =.2 T flash σ noise =.5 dn/dt liquid σ noise =.2 Estimation parameters α=.5 (forgetting factor weight only on ) Epanding horizon (up to 9 as new measurements are available) Initial state estimates all have a +. absolute error Relative initial state errors vary (5% for mole fractions) (.3% for temperatures) Observability criteria Dynamic modes (5) States: Temperature (), Mole Fractions (4) Observability matri rank (3) Conclusion: Not fully observable, but some information may be reconstructed to give estimates that are better than the initial guesses

28 Composition Estimation Measurements p u ld o H Temperature (K) r o p a V h s la F Flow Rate (mol/sec) Actual Measured Estimated h s la F Time (sec) id u iq L h s la F Time (sec)

29 .3 H.2 C 4 H 2 C 5 H 4 C 6 H 6 C 7 H 8 C 8 Estimated Actual Partially Observable Compositions Time (sec) The results are better than the initial estimates but, as indicated by the rank deficiency of the observability matri (3/5), the system is not fully observable all states deviate from correct values, some more than others

30 Constraints and Observability Suppose you know that the mole fraction of C 7 H 6 should not be above.22 Incorporate this knowledge into the state estimation by adding a constraint to the least squares problem Because the constraint is active, the system observability (detectability) is improved

31 Constraint Improves System Observability.3 H.2 C 4 H 2 C 5 H 4 C 6 H 6 C 7 H 8 C Estimated Constraint Actual λ= Time (sec) The state estimates are improved, but there is still some deviation from the correct values because the observability matri is still not full rank (4/5)

32 Horizon Length Scaling (7 State Model) 2 Linear Constrained Eplicit MHE Nonlinear Constrained Eplicit MHE ) 5 ( s p l o f : R 2 = : R 2 = horizon length

33 Model Size Scaling (Horizon=5) 9 8 Linear Constrained Eplicit MHE Nonlinear Constrained Eplicit MHE 7 ) 5 ( s p o l f model size

34 Conclusions New solution approach for MHE problems with upper and lower bounds on the variables Applicable for state estimation of any process described by an ODE or inde- DAE Constrained Eplicit MHE scaling to large scale problems Nonlinear model O(n H2 ) with increasing horizon length O(n v2 ) with number of variables Linear model O(n H ) with increasing horizon length O(n v2 ) with number of variables Eample demonstrates >, speed up over MHE solved with optimization software Future work Simultaneous state and parameter estimation