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Flatness-based optimization of batch processes
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
589
Flatness-based optimization of batch processes M.E. van Wissen, S.Palanki* and J. Grievink, Delft University of Technology, Department of Chemical Technology, Julianalaan 136, 2628 BL Delft, The Netherlands. * Florida A&M University- Florida State University, Department of Chemical Engineering, Tallahassee,FL,32310-6046, USA.
Abstract A flatness optimization framework is proposed, to deal with optimization of batch processes under uncertainty. Via the concept of differential flatness we transform the problem, such that the terminal-cost optimization problem can be solved in a cascade optimization scheme. The results have been tested on an example of a batch bioreactor.
1. Introduction A wide variety of specialty chemicals are made in batch reactors. In batch process operations, the variables change considerably with time and thus there is no constant setpoint around which the process can be regulated. Because there is no steady state, the objective is to optimize some objective function, which expresses the system performance. The optimal operating policy for a given batch process is usually calculated under the assumption of a perfect model. However, realistic applications are subject to uncertainty in initial conditions, model mismatch, and process disturbances, all of which affect the optimal solution. This provides the economic drive for on-line calculation and implementation of the optimal operating policy. Normally, the optimization of batch processes leads to a piece wise, discontinuous solution. A cascade optimization scheme is proposed by Visser et al. (2000) to implement such an optimal trajectory despite disturbances and uncertainty. It is very often the case, that the optimal solution of such a problem lies on the input bounds, or state constraints, or a combination of both. In the proposed method, we make explicitly use of this information. Also uncertainty is treated in the framework of cascade optimization. The classical approach is via the Hamilton-Jacobi-Bellman (HJB) formulation. In batch reactor modeling, a lot of systems have been found, which have the so-called flatness property. For this class of systems, we propose a new theory which does not rely on the HJB-formulation. Via the concept of differential flatness, which is basically a method to dispose of the differential equations, we have developed an approach for which flat systems can be treated in the cascade optimization scheme. Moreover, the feedback controller, needed in the cascade optimization is now easy to obtain. The proposed scheme is illustrated on a simulation of an optimization problem in batch bioreactors.
590
2. Optimization Problem Problem formulation without uncertainty Batch optimization problems typically involve both dynamic and static constraints and fall under the class of dynamic optimization problems. It is true, that in most batch chemical processes, the inputs are flow-rates that enter the system equations in an affme manner (Srinivasan et al. (1997), Srinivasan et al. (2000), Visser et al. (2000)), but quite a few processes have been found for which this is not the case (Rouchon and Rudolph(2000)). The terminal-cost optimization of general dynamical systems can be stated as follows: P:
minJ(x(tf),u(tf))
(la)
u(t)
subject to x(t) = f(x(t),u(t)),tG[to,tf), 0 = x(to)-Xo.
(lb)
0 < c ( x ( t ) , u ( t ) ) , t G [ t o , t , ) , 0
(Ic)
where jc is the n-vector of state variables with known initial conditions XQ, U the mvector of control variables. We exploit the degree of freedom in u(t) to minimize an (economical) objective function P subject to path and endpoint constraints in x(t) and u(t). The right-hand side of the nonlinear dynamical system is defined as the n-vector/ The constraints to be enforced during the process and at the final time tf are given by c and Cp respectively. For a more detailed description of this problem, see Kumar and Daoutidis(1995). The classical solution of (1) can be found by solving the well-known Hamilton-JacobiBellman (HJB) formulation, which is easier to solve when the system is in controlaffine form, (Palanki et al. (1991), Visser et al. (2000)). Problem formulation with uncertainty For many realistic applications, we can assume that the model structure is known, but the model parameters are unknown or only known within bounds. For this case, the terminal-cost optimization of general dynamical systems is as follows: P:
minJ(x(t,,e),u(t,,e))
(2a)
u(t)
s.t. x(t) = f (x(t),u(t),e) + d(t), 0 = x(to) - X,
(2b)
0
(2c)
in which 6 is the /?-vector 0 of uncertain parameters and d is the function representing the unknown disturbances. Here, we choose to have probabilistic parametric uncertainty for 6 , in which the objective function P is the expected value of a random variable, i.e.
591 P: minE[J(x(tf,0),u(tf,0))].
(^^
u(t)
In realistic cases, one can think of the expected product quality or quantity to be maximized, or the expected loss to be minimized. The resulting optimization problem can be found in Srinivasan et al. (2000b), p. 14.
3. Differential flatness Differential flatness has been introduced by Fliess et al. (1995) in their studies of the feedback linearization problem in the context of differential algebra. A system is flat if we can find a set of outputs (equal in number to the number of inputs) such that all states and inputs can be determined from these outputs without integration. More to the point, if the system has states x (n-vector), and inputs u (m-vector), then the system is flat if we can find outputs y (m-vector) of the form y = y(x,u,u,...,u''*),
(4)
such that, x = x(y,y,...,y^''^),
(5a)
u = u(x,u,ti,...,u^''^).
(5b)
The outputs y are called/Zar (or linearizing) outputs. Differentially flat systems are useful in situations where explicit trajectory tracking generation is required. The reason for this, is that the behaviour of flat systems is determined by the flat outputs, and hence we can plan trajectories in output space, and then map these appropriate inputs. This property can be quite useful, when dealing with only a few flat outputs in comparison with the number of states and the number of inputs. For optimization problems the main advantage lies in the reduction of the computational effort, as there is no need to numerically solve the sensitivity differential equations in the nonlinear program (NLP). For the dynamic optimization problem P, we replace the equations (1) (or (2)) with the expressions containing all the flat outputs and higher derivatives. Another useful property of flat systems that we can use, called endogenous dynamic feedback (Fliess et al. (1995)) which is a dynamic feedback of the form (v is the mvector of new inputs) z = a(x,z,v),
(6a)
u = b(x,z,v),
(6b)
592 such that z satisfying equation (6a) can be expressed as a function of x and u and a finite number of their derivatives: z = a(x,u,...,u'),
(7a)
The endogenous dynamic feedback can be used in the cascade optimization scheme as proposed by Visser et al. (2000), which combines notions of feedback control with notions from the field of optimization under uncertainty, and hence can be seen as a practical implementation strategy for problems containing uncertainty, see Visser et al. (1997). For a detailed description of trajectory planning of differentially flat systems and applications, see Faiz et al. (2001) and Veeraklaew and Agrawal (2001).
4. Cascade optimization scheme In the cascade optimization framework proposed by Visser et al. (2000), the optimization problem is transformed into a tracking problem by use of so-called invariant signals obtained via the HJB-approach. In short, in the cascade optimization a 'low level' tracking controller is used, which guarantees that the system stays close to the optimal trajectory. After that, 'a high level' optimizer is used to guarantee optimality despite disturbances. We use the same framework, but instead of Visser et al. (2000), obtain the tracking controller for free, using the differential flatness approach.
5. Results Illustrating example The proposed methodology will be applied to a nonlinear model of fermentation of whey lactose to lactic acid by Lactobacillum bulgaricus in a batch bioreactor (Agrawal et al., (1989)). The mass balances are given by:
x, =^i(x)Xi - u x i / x ^
(8a)
X2 =u(Sf-X2)/x4-|a(x)Xj/Yx/s X3 =(a^i(x) + P)Xi -UX3/X4
(^^) (8c)
x,=u
(8d) K^^+x^+x^'/K. '
0
(8f)
^93 Table 1. Initial states and parameters Symbol State/parameters Biomass concentration Xl Substrate concentration X2 Product concentration X3 Volume X4 Substrate feed cone. Sf Substrate saturation const. Km Substrate inhibition const. Ki Product inhibition const. Cell mass yield Yx/s a Growth assoc. prod, yield Non-growth assoc.pr. yield P Max. spec, growth rate Final time
Value lg/1 Og/1 0.5 g/1 21 15 g/1 1.2 g/1 22 g/1 50 g/1 0.4 g/g 2.2 g/g 0.2 h-^ 0.48 h*^ 11.6h
_tf
The states in the model (8) are related by the following equation: (9) x,(x, + Y^,{x, -S,)) = C = x°,(xj + Y^3(x^ -S,)) In Mahadevan et al. (2001) the flat output for this problem has been derived, using (9). This has been used in the optimization problem of maximization the amount of product formed at the end of the batch, such that equations (8a)-(8e) are satisfied. In Figure 1 the trajectories and inputs of the optimal solution are plotted. Note that the input does not change too much.
Conclusions We proposed an optimization approach of general batch processes using the concept of differential flatness. Especially, in the cascade optimization scheme, the construction of the feedback controller is easy, once the flat outputs are obtained. In comparison with classical methods (e.g., the JB-formulation), the proposed method could also lead to reductions in computing time on-line, because of the transformation to a NLP. A disadvantage of the proposed method is that, beforehand, one does not know if a system is differentially flat. Moreover, even if one knows that a given system is flat, the flat ouputs are not always easy to obtain and might involve long algebraic manipulations. However, for a large group of reactor models for instance, it has been shown, see Rouchon and Rudolph (2000), that these are differentially flat and the flat outputs have been calculated. It is interesting to see if for all these models, the terminalcost optimization problem under parametric uncertainty can be dealt with using differential flatness. Another subject under investigation is the robustness of the method against parametric uncertainty.
594 state 5
^^^-^
4 3
x4
2
x3^^
X
^.,^-^'5(2
1 0
Xl
()
2
4
6 t Input
8
10
1
-
0.14 0.12
-
0.1 008
1
_j,^_
, _ _ _^
—
_
1
10 Optimal solution for batchbio :
12
objective: 11
Figure I. The states and inputs of the optimal solution of the batch bio reactor.
References Agrawal,P.,G.Koshy and M.Ramseier, 1989, An algorithm for operating a fed-batch fermenter at optimum specific growth rate, Biotechn. Bioengg., 33, 115. Faiz, N.,S.K. Agrawal and R.M. Murray, 2001, Trajectory planning of differentially flat systems with dynamics and inequalities, J. of Guid., Cont. Dyn., 24(2),219. Fliess, M., J. Levine, P. Martin and P. Rouchon, 1995, Flatness and defect of non-linear systems: introductory theory and examples. Int. J. Cont. 61(6), 1327. Kumar, A. and P. Daoutidis, 1995, Feedback control of nonlinear differential-algebraicequation systems, AIChE J. 41(3), 619. Mahadevan, R., S.K. Agrawal and F.J. Doyle (2001), Differential flatness based nonlinear predictive control of fed batch bioreactors, Cont. Engg. Pract 9, 889 Palanki, S., C.Kravaris and H.Y. Wang, 1991, Synthesis of state feedback laws for endpoint optimization in batch processes, Chem. Eng. Sc. 48(1), 135. Rouchon, P. and J. Rudolph, 2000, Reacteurs chimiques differentiellement plats: planification et suivi de trajectories, to be published. Srinivasan, B., E. Visser and D. Bonvin, 1997, Optimization-based control with imposed feedback structures, IF AC ADCHEM'97, 635. Srinivasan, B., S. Palanki and D. Bonvin, 2000a, A tutorial on the optimization of batch processes: I. Characterization of the optimal solution, to be published. Srinivasan, B., S. Palanki and D. Bonvin, 2000b, A tutorial on the optimization of batch processes: II. Handling uncertainty using measurements, to be published. Veeraklaew, T. and S.K. Agrawal, 2001, New computional framework for trajectory optimization of higher-order dynamic systems, J. of Guid., Cont. Dyn., 24(2),228. Visser, E., B. Srinivasan, S. Palanki and D. Bonvin, 2000, A feedback-based implementation scheme for batch process optimization, J. Proc. Cont. 10(5), 399.
589
Flatness-based optimization of batch processes M.E. van Wissen, S.Palanki* and J. Grievink, Delft University of Technology, Department of Chemical Technology, Julianalaan 136, 2628 BL Delft, The Netherlands. * Florida A&M University- Florida State University, Department of Chemical Engineering, Tallahassee,FL,32310-6046, USA.
Abstract A flatness optimization framework is proposed, to deal with optimization of batch processes under uncertainty. Via the concept of differential flatness we transform the problem, such that the terminal-cost optimization problem can be solved in a cascade optimization scheme. The results have been tested on an example of a batch bioreactor.
1. Introduction A wide variety of specialty chemicals are made in batch reactors. In batch process operations, the variables change considerably with time and thus there is no constant setpoint around which the process can be regulated. Because there is no steady state, the objective is to optimize some objective function, which expresses the system performance. The optimal operating policy for a given batch process is usually calculated under the assumption of a perfect model. However, realistic applications are subject to uncertainty in initial conditions, model mismatch, and process disturbances, all of which affect the optimal solution. This provides the economic drive for on-line calculation and implementation of the optimal operating policy. Normally, the optimization of batch processes leads to a piece wise, discontinuous solution. A cascade optimization scheme is proposed by Visser et al. (2000) to implement such an optimal trajectory despite disturbances and uncertainty. It is very often the case, that the optimal solution of such a problem lies on the input bounds, or state constraints, or a combination of both. In the proposed method, we make explicitly use of this information. Also uncertainty is treated in the framework of cascade optimization. The classical approach is via the Hamilton-Jacobi-Bellman (HJB) formulation. In batch reactor modeling, a lot of systems have been found, which have the so-called flatness property. For this class of systems, we propose a new theory which does not rely on the HJB-formulation. Via the concept of differential flatness, which is basically a method to dispose of the differential equations, we have developed an approach for which flat systems can be treated in the cascade optimization scheme. Moreover, the feedback controller, needed in the cascade optimization is now easy to obtain. The proposed scheme is illustrated on a simulation of an optimization problem in batch bioreactors.
590
2. Optimization Problem Problem formulation without uncertainty Batch optimization problems typically involve both dynamic and static constraints and fall under the class of dynamic optimization problems. It is true, that in most batch chemical processes, the inputs are flow-rates that enter the system equations in an affme manner (Srinivasan et al. (1997), Srinivasan et al. (2000), Visser et al. (2000)), but quite a few processes have been found for which this is not the case (Rouchon and Rudolph(2000)). The terminal-cost optimization of general dynamical systems can be stated as follows: P:
minJ(x(tf),u(tf))
(la)
u(t)
subject to x(t) = f(x(t),u(t)),tG[to,tf), 0 = x(to)-Xo.
(lb)
0 < c ( x ( t ) , u ( t ) ) , t G [ t o , t , ) , 0
(Ic)
where jc is the n-vector of state variables with known initial conditions XQ, U the mvector of control variables. We exploit the degree of freedom in u(t) to minimize an (economical) objective function P subject to path and endpoint constraints in x(t) and u(t). The right-hand side of the nonlinear dynamical system is defined as the n-vector/ The constraints to be enforced during the process and at the final time tf are given by c and Cp respectively. For a more detailed description of this problem, see Kumar and Daoutidis(1995). The classical solution of (1) can be found by solving the well-known Hamilton-JacobiBellman (HJB) formulation, which is easier to solve when the system is in controlaffine form, (Palanki et al. (1991), Visser et al. (2000)). Problem formulation with uncertainty For many realistic applications, we can assume that the model structure is known, but the model parameters are unknown or only known within bounds. For this case, the terminal-cost optimization of general dynamical systems is as follows: P:
minJ(x(t,,e),u(t,,e))
(2a)
u(t)
s.t. x(t) = f (x(t),u(t),e) + d(t), 0 = x(to) - X,
(2b)
0
(2c)
in which 6 is the /?-vector 0 of uncertain parameters and d is the function representing the unknown disturbances. Here, we choose to have probabilistic parametric uncertainty for 6 , in which the objective function P is the expected value of a random variable, i.e.
591 P: minE[J(x(tf,0),u(tf,0))].
(^^
u(t)
In realistic cases, one can think of the expected product quality or quantity to be maximized, or the expected loss to be minimized. The resulting optimization problem can be found in Srinivasan et al. (2000b), p. 14.
3. Differential flatness Differential flatness has been introduced by Fliess et al. (1995) in their studies of the feedback linearization problem in the context of differential algebra. A system is flat if we can find a set of outputs (equal in number to the number of inputs) such that all states and inputs can be determined from these outputs without integration. More to the point, if the system has states x (n-vector), and inputs u (m-vector), then the system is flat if we can find outputs y (m-vector) of the form y = y(x,u,u,...,u''*),
(4)
such that, x = x(y,y,...,y^''^),
(5a)
u = u(x,u,ti,...,u^''^).
(5b)
The outputs y are called/Zar (or linearizing) outputs. Differentially flat systems are useful in situations where explicit trajectory tracking generation is required. The reason for this, is that the behaviour of flat systems is determined by the flat outputs, and hence we can plan trajectories in output space, and then map these appropriate inputs. This property can be quite useful, when dealing with only a few flat outputs in comparison with the number of states and the number of inputs. For optimization problems the main advantage lies in the reduction of the computational effort, as there is no need to numerically solve the sensitivity differential equations in the nonlinear program (NLP). For the dynamic optimization problem P, we replace the equations (1) (or (2)) with the expressions containing all the flat outputs and higher derivatives. Another useful property of flat systems that we can use, called endogenous dynamic feedback (Fliess et al. (1995)) which is a dynamic feedback of the form (v is the mvector of new inputs) z = a(x,z,v),
(6a)
u = b(x,z,v),
(6b)
592 such that z satisfying equation (6a) can be expressed as a function of x and u and a finite number of their derivatives: z = a(x,u,...,u'),
(7a)
The endogenous dynamic feedback can be used in the cascade optimization scheme as proposed by Visser et al. (2000), which combines notions of feedback control with notions from the field of optimization under uncertainty, and hence can be seen as a practical implementation strategy for problems containing uncertainty, see Visser et al. (1997). For a detailed description of trajectory planning of differentially flat systems and applications, see Faiz et al. (2001) and Veeraklaew and Agrawal (2001).
4. Cascade optimization scheme In the cascade optimization framework proposed by Visser et al. (2000), the optimization problem is transformed into a tracking problem by use of so-called invariant signals obtained via the HJB-approach. In short, in the cascade optimization a 'low level' tracking controller is used, which guarantees that the system stays close to the optimal trajectory. After that, 'a high level' optimizer is used to guarantee optimality despite disturbances. We use the same framework, but instead of Visser et al. (2000), obtain the tracking controller for free, using the differential flatness approach.
5. Results Illustrating example The proposed methodology will be applied to a nonlinear model of fermentation of whey lactose to lactic acid by Lactobacillum bulgaricus in a batch bioreactor (Agrawal et al., (1989)). The mass balances are given by:
x, =^i(x)Xi - u x i / x ^
(8a)
X2 =u(Sf-X2)/x4-|a(x)Xj/Yx/s X3 =(a^i(x) + P)Xi -UX3/X4
(^^) (8c)
x,=u
(8d) K^^+x^+x^'/K. '
0
(8f)
^93 Table 1. Initial states and parameters Symbol State/parameters Biomass concentration Xl Substrate concentration X2 Product concentration X3 Volume X4 Substrate feed cone. Sf Substrate saturation const. Km Substrate inhibition const. Ki Product inhibition const. Cell mass yield Yx/s a Growth assoc. prod, yield Non-growth assoc.pr. yield P Max. spec, growth rate Final time
Value lg/1 Og/1 0.5 g/1 21 15 g/1 1.2 g/1 22 g/1 50 g/1 0.4 g/g 2.2 g/g 0.2 h-^ 0.48 h*^ 11.6h
_tf
The states in the model (8) are related by the following equation: (9) x,(x, + Y^,{x, -S,)) = C = x°,(xj + Y^3(x^ -S,)) In Mahadevan et al. (2001) the flat output for this problem has been derived, using (9). This has been used in the optimization problem of maximization the amount of product formed at the end of the batch, such that equations (8a)-(8e) are satisfied. In Figure 1 the trajectories and inputs of the optimal solution are plotted. Note that the input does not change too much.
Conclusions We proposed an optimization approach of general batch processes using the concept of differential flatness. Especially, in the cascade optimization scheme, the construction of the feedback controller is easy, once the flat outputs are obtained. In comparison with classical methods (e.g., the JB-formulation), the proposed method could also lead to reductions in computing time on-line, because of the transformation to a NLP. A disadvantage of the proposed method is that, beforehand, one does not know if a system is differentially flat. Moreover, even if one knows that a given system is flat, the flat ouputs are not always easy to obtain and might involve long algebraic manipulations. However, for a large group of reactor models for instance, it has been shown, see Rouchon and Rudolph (2000), that these are differentially flat and the flat outputs have been calculated. It is interesting to see if for all these models, the terminalcost optimization problem under parametric uncertainty can be dealt with using differential flatness. Another subject under investigation is the robustness of the method against parametric uncertainty.
594 state 5
^^^-^
4 3
x4
2
x3^^
X
^.,^-^'5(2
1 0
Xl
()
2
4
6 t Input
8
10
1
-
0.14 0.12
-
0.1 008
1
_j,^_
, _ _ _^
—
_
1
10 Optimal solution for batchbio :
12
objective: 11
Figure I. The states and inputs of the optimal solution of the batch bio reactor.
References Agrawal,P.,G.Koshy and M.Ramseier, 1989, An algorithm for operating a fed-batch fermenter at optimum specific growth rate, Biotechn. Bioengg., 33, 115. Faiz, N.,S.K. Agrawal and R.M. Murray, 2001, Trajectory planning of differentially flat systems with dynamics and inequalities, J. of Guid., Cont. Dyn., 24(2),219. Fliess, M., J. Levine, P. Martin and P. Rouchon, 1995, Flatness and defect of non-linear systems: introductory theory and examples. Int. J. Cont. 61(6), 1327. Kumar, A. and P. Daoutidis, 1995, Feedback control of nonlinear differential-algebraicequation systems, AIChE J. 41(3), 619. Mahadevan, R., S.K. Agrawal and F.J. Doyle (2001), Differential flatness based nonlinear predictive control of fed batch bioreactors, Cont. Engg. Pract 9, 889 Palanki, S., C.Kravaris and H.Y. Wang, 1991, Synthesis of state feedback laws for endpoint optimization in batch processes, Chem. Eng. Sc. 48(1), 135. Rouchon, P. and J. Rudolph, 2000, Reacteurs chimiques differentiellement plats: planification et suivi de trajectories, to be published. Srinivasan, B., E. Visser and D. Bonvin, 1997, Optimization-based control with imposed feedback structures, IF AC ADCHEM'97, 635. Srinivasan, B., S. Palanki and D. Bonvin, 2000a, A tutorial on the optimization of batch processes: I. Characterization of the optimal solution, to be published. Srinivasan, B., S. Palanki and D. Bonvin, 2000b, A tutorial on the optimization of batch processes: II. Handling uncertainty using measurements, to be published. Veeraklaew, T. and S.K. Agrawal, 2001, New computional framework for trajectory optimization of higher-order dynamic systems, J. of Guid., Cont. Dyn., 24(2),228. Visser, E., B. Srinivasan, S. Palanki and D. Bonvin, 2000, A feedback-based implementation scheme for batch process optimization, J. Proc. Cont. 10(5), 399.