Error of the estimates of unknown rates

and is given by ˆru(t) = T ˜˙yU V(t) = Dq(˜yr, t) − T Wq(˜sa, t), (5.43) where T =LT_Σ−1_˙y L −1 LT_Σ−1_˙y , (5.44)

which satisfies Eq. (5.6), and ˜yr(t)are defined as in Eq. (5.9), using T from Eq. (5.44).

According to Eq. (5.43), the estimation of the unknown rates r_u(t) proceeds via appli- cation of a differentiation filter to the variants y_r(t)that are obtained via transformation of the available state vector y(t), and the knowledge of the previous values of the available rates sa(t). Note that Eq. (5.43) can be obtained from Eq. (5.8) if one replaces ru(t)by

ˆru(t), ˙yr(t)by Dq(˜yr, t), and sa(t)by Wq(˜sa, t).

5.4.3

Error of the estimates of unknown rates

The next proposition shows the final result of this section, which concerns the error and variance of the estimates of the unknown rates ru(t).

Proposition 5.3. Let the condition in Eq. (5.5) be satisfied, the transformation matrix T be

given by Eq. (5.44), ˜y_r(t) be defined as in Eq. (5.9), D_q(˜y_r, t) be defined as in Eq. (5.12),

W_q(˜s_a, t) be defined as in Eq. (5.33), and ˆr_u(t) be defined as in Eq.(5.43). Then,

ˆr_u(t) − ru(t) = −Rq(ru, t) + T Dq(dy, t), (5.45)

5.5. Conclusion and Varˆru(t)  = T Σ˙yTT=  LT_Σ−1_˙y L−1. (5.46)

Proof. One has to simply notice that, from Eqs. (5.32), (5.37), (5.43), and (5.44),

ˆru(t) − ru(t) = ˆru(t) − T L ru(t) = T €˜˙yU V(t) − ˙yU V(t)Š− T L Rq(ru, t) = −Rq(ru, t) + T Dq(dy, t) (5.47) and Varˆru(t) 

= TVar”˜˙yU V(t)—TT_{= T Σ˙y}TT=LT_Σ−1_˙y L

−1

. (5.48)

5.5

Conclusion

This chapter has shown how to compute the unknown rates r_u(t) from the available measurements, using knowledge about the structural relationship, given by the matrix L , between the available states y(t) on the one hand and the unknown rates ru(t) and the

available rates s_a(t)on the other hand. The variants y_r(t)are variant with respect to the unknown rates ru(t). The unknown rates ru(t)are estimated via numerical differentiation

of the variants y_r(t) that are computed from the available states y(t) via an appropriate linear transformation, without the use of any rate model. For this, the rank of L must be equal to the number of estimated unknown rates nr, which implies that the number of states

that are available has to be greater than or equal to nr. Only one parameter needs to be

tuned, namely, the parameter of the differentiation filter (the number of samples q in the case of the Savitzky-Golay filter) used for numerical differentiation of variants yr(t).

The implications of this estimation of rate signals without kinetic models for monitoring and diagnosis are obvious. Furthermore, the next two chapters present the implications of rate estimation with respect to control without kinetic models and estimation of plant steady state, which can then be used for real-time optimization.

6

Reactor Control

Part of this chapter is adapted from the postprint of the following article [111]:

D. Rodrigues, J. Billeter, and D. Bonvin. Control of reaction systems via rate estimation and feedback linearization. Comput. Aided Chem. Eng., 37:137–142, 2015.

Link: http://doi.org/10.1016/B978-0-444-63578-5.50018-9. Copyright © 2015 Elsevier B.V.

The author of this thesis contributed to that article by developing the main novel ideas, implementing the simulations, and writing a significant part of the text. Hence, the author retains the right to include the article in this thesis since it is not published commercially and the journal is referenced as the original source.

6.1

Introduction

Various control structures for homogeneous reactors based on reaction variants, exten- sive variables, and inventories have been proposed throughout the years. For example, Hammarström [28] claimed that the control of reaction and control variants in homoge- neous reactors is useful to reduce the number of controlled and measured variables. Geor- gakis [112] was among the first to suggest the use of extensive variables for efficient design of multivariable and nonlinear controllers for process units, namely reactors. Farschman et al. [113] proposed a structure called inventory control, whereby a certain type of ex- tensive variables called inventories are controlled efficiently via input-output feedback lin- earization, although the proposed structure implies the use of a kinetic model for estimation of reaction rates in the case of reactor control. Aggarwal et al. [41] expressed the model of multi-phase reaction systems operating at thermodynamic equilibrium in terms of reac- tion invariants and used this formulation for inventory control of these invariant quantities, labeled as invariant inventories. Hoang et al. [31] proposed to use the fact that the reac- tion invariants are exponentially stable to control only the reaction variants, although at the price of requiring the use of a kinetic model to compute the reaction rates. In an earlier and more restricted version of the developments presented in this chapter and also in the pre-

Chapter 6. Reactor Control

vious one, Rodrigues et al. [111] proposed temperature control in homogeneous reactors without the use of kinetic models, which was enabled by the estimation of the unknown reaction rates via numerical differentiation. Then, Zhao et al. [32] took advantage of this possibility to propose another scheme for control of reaction variants, but this time without the use of kinetic models since the reaction rates can be estimated from measurements with- out a kinetic model. However, there does not exist a systematic way of taking advantage of multiple measurements of variables that are not directly controlled to simplify the design of multiple-input multiple-output (MIMO) control of the temperature and concentrations (or their extensive counterparts, the heat and numbers of moles) in homogeneous reac- tors, in particular without the use of a kinetic model. The development of such systematic procedures is the main objective of this chapter.

This chapter starts by presenting in Section 6.2 a control approach that uses the kinetic model to achieve offset-free control of as many controlled variables as the number of manip- ulated variables and to set the closed-loop time constants of all the variables that represent the reaction system. Then, a feedback linearization approach that is based on the estimation of unknown rates using the concept of variants is presented in Section 6.3, thus allowing effective control without the use of rate models for these unknown rates. In particular, the possibility of controlling homogeneous reactors without the use of kinetic models is investi- gated. The reaction rates are estimated without the use of kinetic models and then used via a feedback-linearization scheme to control the reactor temperature and reactant concentra- tions by manipulating the amount of heat that is exchanged with the environment and the inlet flowrates in a homogeneous reactor. Finally, Section 6.4 concludes the chapter.