Control of Reaction Systems via Rate Estimation and Feedback Linearization Diogo Rodrigues, Julien Billeter, Dominique Bonvin Laboratoire d’Automatique Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL) PSE-2015/ESCAPE-25 June 3, 2015 Laboratoire d’Automatique - EPFL Control of Reaction Systems via Rate Estimation and Feedback Linearization June 3, 2015 1 / 18
Outline Introduction Decoupling dynamic effects Controlling reaction systems Description of the reaction system Mole and heat balance equations Transformation to reaction-variant states Control problem Estimation of reaction rates Feedback linearization Feedback control of the temperature Simulated CSTR Conclusions Laboratoire d’Automatique - EPFL Control of Reaction Systems via Rate Estimation and Feedback Linearization June 3, 2015 2 / 18
Introduction Decoupling dynamic effects Efficient control of reaction systems typically requires kinetic models, whose identification can be difficult and time consuming. One can infer reaction rates from measurements, without a kinetic model, if the rates are decoupled. 1 Reaction variants/invariants decouple reaction rates, thereby facilitating analysis and control. 2 More generally, variant/invariant states can decouple dynamic effects via a linear transformation to vessel extents. 3 1 Mhamdi, A.; Marquardt, W. In ADCHEM 2003, Hong Kong, China , 2004, pp 171–176. 2 Asbjørnsen, O. A.; Fjeld, M. Chem. Eng. Sci. 1970 , 25 , 1627–1636. 3 Rodrigues, D. et al. Comp. Chem. Eng. 2015 , 73 , 23–33. Laboratoire d’Automatique - EPFL Control of Reaction Systems via Rate Estimation and Feedback Linearization June 3, 2015 3 / 18
Introduction Controlling reaction systems Various control strategies for open reactors are based on reaction variants and extensive variables. 4 There is no systematic control method that takes advantage of multiple measurements, in particular without a kinetic model. The control of chemical reactors without kinetic models is possible, by (i) estimating reaction rates from concentration and temperature via the concept of variants, (ii) using feedback linearization and these estimated rates to effectively control the temperature by manipulating the exchanged heat. 4 Georgakis, C. Chem. Eng. Sci. 1986 , 41 , 1471–1484; Farschman, C. A. et al. AIChE J. 1998 , 44 , 1841–1857. Laboratoire d’Automatique - EPFL Control of Reaction Systems via Rate Estimation and Feedback Linearization June 3, 2015 4 / 18
Description of the reaction system Mole and heat balance equations Open homogeneous reactor with S species, R independent reactions, p inlet streams and 1 outlet stream. The S -dimensional vector of numbers of moles n , and W in , ˇ T in , u in � � the heat energy Q = mc p T − T ref are state variables. Mole and heat balance equations: 5 � ˙ � � � � � � W in � � � N T n ( t ) n ( t ) 0 S = r v ( t ) + q ex ( t ) + u in ( t ) − ω ( t ) , ˙ T T ∆ H ) T 1 ˇ Q ( t ) Q ( t ) ( − ∆ ∆ in z N � �� � � �� � � �� � � �� � q ex r v � �� � ˙ b z ( t ) z ( t ) C ∆ H ∆ ∆ A z (0) = z 0 . Time-variant signals r v ( t ) R reaction rates, q ex ( t ) exchanged heat power, z , ω u in ( t ) p inlet flowrates, ω ( t ) inverse of residence time. Structural information N ( R × S ) stoichiometry, ∆ H R heats of reaction, W in ( S × p ) inlet composition, ˇ T in p inlet specific heats. 5 Rodrigues, D. et al. Comp. Chem. Eng. 2015 , 73 , 23–33 Laboratoire d’Automatique - EPFL Control of Reaction Systems via Rate Estimation and Feedback Linearization June 3, 2015 5 / 18
Description of the reaction system Transformation to reaction-variant states If rank ( A ) = R , there exists an R × ( S + 1) transformation matrix T such that T A = I R , � N T � where A = . ∆ ∆ H ) T ( − ∆ Apply T to the balance equations and define y r ( t ) := T z ( t ): ˙ y r ( t ) = r v ( t ) + ( T b ) q ex ( t ) + ( T C ) u in ( t ) − ω ( t ) y r ( t ) , y r (0) = T z 0 . The transformed states y r are reaction variants, with each state y r , i ( i = 1 , . . . , R ) depending on the corresponding rate r v , i . 6 6 Asbjørnsen, O. A.; Fjeld, M. Chem. Eng. Sci. 1970 , 25 , 1627–1636. Laboratoire d’Automatique - EPFL Control of Reaction Systems via Rate Estimation and Feedback Linearization June 3, 2015 6 / 18
Control problem Objective and method Objective: control the heat Q (indirectly temperature) to the setpoint Q s by manipulating q ex . Method: Rate Estimation z ( t ) , q ex ( t ) , u in ( t ) , ω ( t ) ˆ r v ( t ) Feedback Q s ( t ) Feedback v ( t ) q ex ( t ) Q ( t ) Plant Control Linearization ˙ Q ( t ) = v ( t ) Laboratoire d’Automatique - EPFL Control of Reaction Systems via Rate Estimation and Feedback Linearization June 3, 2015 7 / 18
Control problem Estimation of reaction rates Estimation of r v via differentiation of y r that is obtained by transformation of z , and the knowledge of q ex , u in and ω . Reformulate the dynamic equations of y r : r v ( t ) = ˙ y r ( t ) − ( T b ) q ex ( t ) − ( T C ) u in ( t ) + ω ( t ) y r ( t ) . Rate Estimation ˆ r v ( t ) z ( t ) , q ex ( t ) , u in ( t ) , ω ( t ) The transformation T requires that at least R elements of z be measured. Different transformations T satisfy T A = I R , e.g. T = A † (Moore-Penrose). With noisy measurements of z , a maximum-likelihood estimator is obtained with Σ − 1 A ) − 1 A T Σ Σ − 1 , where Σ T = ( A T Σ Σ Σ Σ Σ is the variance-covariance matrix. Laboratoire d’Automatique - EPFL Control of Reaction Systems via Rate Estimation and Feedback Linearization June 3, 2015 8 / 18
Control problem Feedback linearization Feedback linearization (linear, first-order relationship between v and Q ). Define the new input v as the right-hand side of the heat balance equation: ˙ T r v ( t ) + q ex ( t ) + ˇ ! T ∆ Q ( t ) = ( − ∆ ∆ H ) T in u in ( t ) − ω ( t ) Q ( t ) = v ( t ) . ˆ r v ( t ) z ( t ) , q ex ( t ) , u in ( t ) , ω ( t ) Feedback v ( t ) q ex ( t ) Linearization The relationship between the new input v and q ex is known: r v ( t ) − ˇ T ˆ T q ex ( t ) = v ( t ) − ( − ∆ ∆ ∆ H ) T in u in ( t ) + ω ( t ) Q ( t ) . Laboratoire d’Automatique - EPFL Control of Reaction Systems via Rate Estimation and Feedback Linearization June 3, 2015 9 / 18
Control problem Feedback control of the temperature Design of a feedback controller for the system ˙ Q ( t ) = v ( t ), using pole placement Q ( s ) or loop shaping (closed-loop transfer function Q s ( s ) = 1). The feedback controller using the control law v ( t ) = ˙ � � Q s ( t ) + γ Q s ( t ) − Q ( t ) forces the error e ( t ) := Q s ( t ) − Q ( t ) to converge exponentially to zero at a rate γ : e ( t ) = − γ e ( t ) , ˙ e (0) = Q s (0) − Q (0) . Q s ( t ) Feedback v ( t ) Control Q ( t ) The output of the feedback controller is v , which determines q ex according to r v ( t ) − ˇ T ˆ T ∆ q ex ( t ) = v ( t ) − ( − ∆ ∆ H ) T in u in ( t ) + ω ( t ) Q ( t ) . Laboratoire d’Automatique - EPFL Control of Reaction Systems via Rate Estimation and Feedback Linearization June 3, 2015 10 / 18
Control problem Objective and method Objective: control the heat Q (indirectly temperature) to the setpoint Q s by manipulating q ex . Method: Rate Estimation z ( t ) , q ex ( t ) , u in ( t ) , ω ( t ) ˆ r v ( t ) Feedback Q s ( t ) Feedback v ( t ) q ex ( t ) Q ( t ) Plant Control Linearization ˙ Q ( t ) = v ( t ) Laboratoire d’Automatique - EPFL Control of Reaction Systems via Rate Estimation and Feedback Linearization June 3, 2015 11 / 18
Simulated CSTR Physical description Acetoacetylation of pyrrole in a homogeneous CSTR: 7 W in , ˇ T in , u in S = 4 species (A: pyrrole; B: diketene). R = 2 reactions (A + B → 2-acetoacetylpyrrole, 2B → dehydroacetic acid). p = 2 inlets (of A and B). 1 outlet (flowrate adjusted to keep constant volume). z Constant heat capacity mc p . N q ex r v Heat exchange only with the jacket. ∆ ∆ ∆ H Reaction rates are complex and unknown. The system is initially at a steady state corresponding z , ω � ¯ u in , A � to ¯ q ex and ¯ u in = . u in , B ¯ Control objective: Reject effect on the temperature T of 15 kg min − 1 step disturbance in u in , B u in , B = 15 kg min − 1 ) by manipulating q ex . (with ¯ 7 Ruppen, D. et al. Comp. Chem. Eng. 1998 , 22 , 185–189. Laboratoire d’Automatique - EPFL Control of Reaction Systems via Rate Estimation and Feedback Linearization June 3, 2015 12 / 18
Simulated CSTR Data treatment Following values are assumed to be known: W in , ˇ T in , u in Stoichiometry N . Heats of reaction ∆ ∆ ∆ H . Inlet composition W in . Inlet specific enthalpies ˇ T in . Measurements of z , q ex , u in and ω are available at the z N sampling time h s = 0 . 4 s. q ex r v ∆ ∆ ∆ H Standard deviation of added measurement noise n : 0.5% (relative to maximum value for each species). Q : 0 . 5 K. z , ω Savitzky-Golay differentiation filter (of order 1 and window size q = 25) is used. 8 Benchmark comparison: FL control with convergence rate γ = 5 min − 1 . PI control with gain K p = 5 min − 1 and integral time constant τ I = 0 . 2 min. 8 Savitzky, A.; Golay, M. Anal. Chem. 1964 , 36 , 1627–1639. Laboratoire d’Automatique - EPFL Control of Reaction Systems via Rate Estimation and Feedback Linearization June 3, 2015 13 / 18
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