On the use of shape constraints for state estimation in reaction systems S. Srinivasan 1 , D. M. Darsha Kumar 2 , J. Billeter 1 , S. Narasimhan 2 , D. Bonvin 1 1 Laboratoire d’Automatique, EPFL, Lausanne, Switzerland 2 Indian Institute of Technolgy, Madras June, 2016 (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 1 / 23
Outline Motivation System representation Shape constraints using concentrations 1 using extents 2 State estimation via RNK filter Simulated case study Conclusion (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 2 / 23
Motivation Problem definition Measurements are usually corrupted with both systematic and random errors Models of the reaction system also contain some uncertainity Problem definition Given a process model and measurements up to time t h , what are the best estimates of the state variables at t h ? The estimated variables can then be used for process monitoring and control (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 3 / 23
System representation Material balance equations Consider a reaction system with S species, R reactions, p inlets and one outlet stream System representation in terms of numbers of moles: Material balance equations - All species and invariants T r v ( t ) + W in u in ( t ) − ω ( t ) n ( t ) (Species) n ( t ) = N ˙ n (0) = n 0 P + [ N T W in n 0 ] = 0 q P + n ( t ) = 0 q (Invariants) where ω ( t ) := u out ( t ) m ( t ) is the inverse residence time d = R + p + 1 is the number of variant states and q = S − d is the number of invariants Note: d = R + p for semi-batch and d = R for batch reactor (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 4 / 23
System representation Material balance equations Consider a reaction system with S species, R reactions, p inlets and one outlet stream System representation in terms of numbers of moles: Material balance equations - Independent and dependent species (Independent) n 1 ( t ) = N ˙ 1 r v ( t ) + W in , 1 u in ( t ) − ω ( t ) n 1 ( t ) T n 1 (0) = n 01 n 2 ( t ) = − ( P 2 ) P + (Dependent) 1 n 1 ( t ) d differential equations and q algebraic equations (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 5 / 23
System representation Vessel extents equations An alternative representation is based on the concept of extents 1 For a chemical reactor with S species, R reactions, p inlets and one outlet stream: there are d variant states called extents and q invariant states Vessel extents equations x r ( t ) = r v ( t ) − ω ( t ) x r ( t ) ˙ x r (0) = 0 R x in ( t ) = u in ( t ) − ω ( t ) x in ( t ) ˙ x in (0) = 0 p x ic ( t ) = − ω ( t ) x ic ( t ) ˙ x ic (0) = 1 x iv ( t ) = 0 q T x r ( t ) + W in x in ( t ) + n 0 x ic ( t ) n ( t ) = N 1 Rodrigues et al., Variant and Invariant States for Chemical Reaction Systems , Comp & Chem Eng. 73, p. 23-33, 2015 (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 6 / 23
Example Semi-batch reactor Consider the following two-reaction system: Reaction system R 1 : A + B → C r 1 = k 1 c A c B R 2 : A + C → D r 2 = k 2 c A c C The reaction system is operated in a semi-batch reactor with an inlet stream of B The number of independent species is equal to d = R + p = 3 Species A, B and D are chosen as the independent species (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 7 / 23
Example System representation For the reaction system in a semi-batch reactor R 1 : A + B → C r 1 = k 1 c A c B R 2 : A + C → D r 2 = k 2 c A c C Material balance equations n A ( t ) = − V ( t ) r 1 ( t ) − V ( t ) r 2 ( t ) ˙ n A (0) = n A 0 n B ( t ) = − V ( t ) r 1 ( t ) + w in , B u in ( t ) ˙ n B (0) = n B 0 n D ( t ) = V ( t ) r 2 ( t ) ˙ n C (0) = n C 0 n C ( t ) = n A 0 + n C 0 + 2 n D 0 − n A ( t ) − 2 n D ( t ) (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 8 / 23
Example System representation For the reaction system in a semi-batch reactor R 1 : A + B → C r 1 = k 1 c A c B R 2 : A + C → D r 2 = k 2 c A c C Vessel extent equations x r , 1 ( t ) = V ( t ) r 1 ( t ) ˙ x r , 1 (0) = 0 x r , 2 ( t ) = V ( t ) r 2 ( t ) ˙ x r , 2 (0) = 0 x in ( t ) = u in ( t ) ˙ x in (0) = 0 n ( t ) = N T x r ( t ) + W in x in ( t ) + n 0 (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 9 / 23
Shape constraints Numbers of moles - Generally valid constraints Numbers of moles are affected by various rate processes - Hard to impose shape constraints Batch reactor If a species appears only as reactant (product) in an irreversible reaction, then the corresponding number of moles is monotonically decreasing (increasing) Semi-batch reactor If a species appears only as reactant (product) in an irreversible reaction and is not added via an inlet stream, then the corresponding number of moles is monotonically decreasing (increasing) (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 10 / 23
Shape constraints Vessel extents - Generally valid constraints (batch and semi-batch reactor) Each vessel extent is affected by a single rate process - Easier to impose shape constraints Vessel extents of inlet Nonnegative monotonically increasing functions Convex (concave) if the corresponding inlet flowrates are monotonically increasing (decreasing) Vessel extents of reactions Nonnegative monotonically increasing functions, Concave (convex) if the corresponding reaction rates are monotonically decreasing (increasing). (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 11 / 23
Shape constraints Vessel extents - generally valid constraints (reactors with outlet) Each vessel extent is affected by a single rate process and also by the outlet flow rate - There are very few generally valid constraints Vessel extents of initial conditions The extent of initial conditions is a nonnegative monotonically decreasing function Constraints on other extents need to be inferred from measurements (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 12 / 23
Shape constraints Constraints from measurements Shape constraints based on measurements Select a time window T of size N Compute the extents ˜ x ( t h ) = T ˜ n ( t h ) in the time window T from the measured numbers of moles ˜ n ( t h ) Calculate the first and second derivatives of each extent using the analytical expressions of the kinetic models Monotonicity constraints based on the sign of the estimated first derivatives: increasing (+) / decreasing (-) Design shape constraints based on the sign of the estimated second derivatives: convex (+) / concave (-) Note that measurement-based constraints can also be applied to numbers of moles (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 13 / 23
State estimator Receding-horizon nonlinear Kalman filter (RNK) The RNK filter is a nonlinear filter based on the prediction and update steps of a Kalman filter The system representation with process and measurement noises can be written as: System representation - Vessel extents x r ( t ) = f r = r v ( t ) − ω ( t ) x r ( t ) + w r ( t ) ˙ x r (0) = 0 R x in ( t ) = f in = u in ( t ) − ω ( t ) x in ( t ) + w in ( t ) ˙ x in (0) = 0 p x ic ( t ) = f ic = − ω ( t ) x ic ( t ) + w ic ( t ) ˙ x ic (0) = 1 T x r ( t ) + W in x in ( t ) + n 0 x ic ( t ) + v y ( t ) y ( t ) = f y = N where w r , w in , w ic , v y are Gaussian random variables with zero-mean and constant variance-covariances Q r , Q in , q ic and R y (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 14 / 23
State estimator RNK - Prediction step Given the state vector x ( t h | t h ), compute the a priori estimate x T | t h = [ x ( t h +1 | t h ) , . . . , x ( t h + N | t h )] for the time window T The elements of the covariance matrix P T | t h are estimated from P ( t h | t h ) using the following iterative relationships A priori covariance estimation P t h + N | t h = A T t h + N − 1 P t h + N − 1 | t h A t h + N − 1 + Q x P ( t h + N − 1 )( t h + N ) | t h = P ( t h + N − 1 )( t h + N − 1 ) | t h A T t h + N − 1 0 0 Q r and A t h := exp { ∂ f x where Q x = 0 Q in 0 ∂ x | x ( th | th ) } 0 0 q ic (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 15 / 23
State estimator RNK - Update step y ( t h +1 ) T , . . . , y ( t h + N ) T � T , the � Given the N measured outputs y T := update step is formulated as an optimization problem Update step α T P − 1 T | t h α + β T R − 1 min y β x T | th + N s.t. α := x T | t h + N − x T | t h � � β := y T − f y x T | t h h ( x T | t h + N ) ≤ 0 m x T | t h + N ≥ 0 where h ( · ) denotes the m applicable shape constraints (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 16 / 23
State estimator RNK - Update step The a posteriori estimate of the covariance matrix is computed as: A posteriori covariance estimation T | t h + R y ) − 1 K T | t h + N = P T | t h C T | t h ( C T | t h P T | t h C T P T | t h + N = ( I − K T | t h + N C T | t h ) P T | t h where C T | t h is the linearized measurement equation obtained at x T | t h (Laboratoire d’Automatique – EPFL) State estimation using extents June, 2016 17 / 23
Recommend
More recommend
