Appendix 5.1: Simulink Diagram for the Control
7.3 Controller Design Under Uncertainty: Batch Reactor
As a second process to investigate the control design under uncertainty, a batch reactor is considered. As discussed in the previous chapter, it is rather frequent to approximate the behavior of industrial processes by FOPDT models, which is also an approximation of the real system. Thus, it is important to consider the presence of Fig. 7.4 Absolute error inx-variable. Blue line: C0 error; green line: C1 error; cyan line: C3 error
Fig. 7.5 Absolute error iny-variable. Blue line: C0 error; green line: C1 error; cyan line: C3 error
112 7 Uncertainty Treatment
model uncertainty in the controller design (Mayne et al.,2000; Michalska & Mayne, 1993).
In this section, an analysis for the perturbed systems is performed, assuming the disturbances as well as model mismatches as transient ones. An unknown additive uncertainty is introduced into the model of the system, and (6.19) takes the form:
x1,nþ1 x2,nþ1
ẳ x1,n
x2,n
þT 0 1 KB KA
x1,n
x2,n
þ 0 KKB
u1,n
þ 0
1 En ð7:25ị
If the disturbances are not considered, the so-called proportional controller leads to the disturbed error equation (6.32), that is,
e1,nþ1 e2,nþ1
ẳ k1 T 0 k2
e1,n e2,n
þ 0
1 En ð7:26ị
showing the effect of the disturbance in the tracking errors.
Note that the initial state of variable y1 (called y0 in Chap. 6), is no longer subtracted. It is assumed as an uncertainty to be compensated by the controller (Sardella, Serrano, Camacho, & Scaglia,2019).
The mathematical formulation of the uncertainty that represents model mis- matches can be done as follows:
Starting from the equation of the system (6.12):
τy t_ð ỵt0ị ỵy tð ỵt0ị ẳK u tð ị ð7:27ị In order to deal with the delay, a Taylor approximation (see Appendix A.4.1) is applied. Thence,
y tð ỵt0ị ẳy tð ị ỵy t_ð ịt0ỵ €y tð ỵλt0ịt02
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}2
Complementary termẳH tð ị
; 0<λ<1 ð7:28ị
and _
y tð ỵt0ị ẳy t_ð ị ỵ€y tð ịt0ỵ . . .y tð ỵλt0ịt02
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}2
Complementary termẳdH tdtð ị
; 0<λ<1 ð7:29ị
replacing (7.28) and (7.29) into (7.27), the following expression is obtained:
τt0€y tð ị ỵðτỵt0ị_y tð ị ỵy tð ị ẳKu tð ị H tð ị ỵτdH tð ị dt
ð7:30ị
and taking into account (6.15), it yields
€yỵKAy_ỵKByỵKB H tð ị ỵτH tð ị dt
ẳK KBu ð7:31ị
Thus, applying Euler DT approximation, (7.25) is obtained, and the uncertainty term takes the form:
Enẳ KBT H nTð ị ỵτdH tð ị dt
tẳnT
ð7:32ị
By this term, all the model uncertainties are collected, including the time delay approximation, the uncertainty in the initial model parameters, as well as some additive extra disturbances.
Applying the LAB CD methodology to (7.25), including an integral action, the control law is:
unẳ 1 K KB
y2ref,nþ1k2y2ref,ny2,n
y2,nþK1Unþ1
T þKBy1,nþKAy2,n
ð7:33ị where
Unỵ1 ẳUnỵ
Z ðnỵ1ịT
nT
e2ð ịt dtffi Unỵe2,nT ð7:34ị
The controlled plant model is obtained by replacing (7.33) in (7.25):
y2nỵ1ẳy2,nỵy2ref,nỵ1k2e2,ny2,nỵK1Unỵ1KAy2,nTKBy1,nTỵKAy2,nTỵKBy1,nTỵEn
y2nỵ1ẳy2,ref,nỵ1k2e2,nỵK1Unỵ1ỵEn
y2ref,nþ1y2,nþ1
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
e2,nþ1
ẳk2e2,nK1Unỵ1En
And after some simple operations, it yields
114 7 Uncertainty Treatment
e2,nỵ2ðk2K1Tỵ1ịe2,nỵ1ỵk2e2,nẳE|fflfflfflfflfflffl{zfflfflfflfflfflffl}nỵ1En δEn
ð7:35ị
To ensure the stability of the linear system, the zeros of the characteristic polynomial of equation (7.35) should comply:
0j jri <1,iẳf1, 2g
Thene2,n+ 1!0 asn! 1ande1,n+ 1ẳk1e1,n+Te2,n!0 asn! 1.
This means that the error will tend to zero despite uncertainties, if they are constant.
Remark 7.1 Note that the integral term, added to correct uncertainties effects, is defined overe2instead ofe1, to avoid overshoot in the response variable. Whene1is used, the expression ofy2, ref,nshould be defined as:
y2ref,nẳy1ref,nỵ1k1e1,ny1,nỵK1Unỵ1 T
andUn+ 1takes the form:Unỵ1ẳUnỵRðnỵ1ịT
nT e1ð ịt dtffi Unỵe1,nT. And the expression fore1, when there are no perturbations on the system, is:
e1nỵ1k1e1,nỵT e2nK1Unỵ1ẳ0 ð7:36ị Working under nominal conditions, e1 and e2should tend to zero. So, to satisfy (7.25), the integral term should tend to zero too. To accomplish this condition,e1will take positive and negative values, leading always to an overshoot ony1.
The integral overe2produces the same effect ony2, which is the derivative ofy1, but a change in the derivative sign does not imply an overshoot on y1. This improvement allows the adjustment of the integral term of the controller, avoiding overshoot in the response.
Experimental results demonstrate the superiority of this methodology, applied to the experimental batch reactor cited in Chap. 6. Figure 7.6 shows the system response for a variable time temperature profile, when the process is under nominal operation conditions and under uncertainties. These results are compared with the system controlled by the original LABC. It can be seen that the tracking error is very low when there are minor uncertainties, but it is unable to correct significant mis- matches as the ones included in this work (Chap.6). When the improved controller was used, the controller performance was as good as when no significant mismatches were present.
References
Mayne, D. Q., Rawlings, J. B., Rao, C. V., & Scokaert, P. O. M. (2000). Constrained model predictive control: stability and optimality.Automatica, 36, 789–814.
Michalska, H., & Mayne, D. Q. (1993). Robust receding horizon control of constrained nonlinear systems.IEEE Transactions on Automatic Control, 38, 1623–1633.
Sardella, M. F., Serrano, M. E., Camacho, O., & Scaglia, G. (2019). Design and application of a linear algebra based controller from a reduced-order model for regulation and tracking of chemical processes under uncertainties.Industrial & Engineering Chemistry Research Pub- lisher: American Chemical Society, 1, 2019.https://doi.org/10.1021/acs.iecr.9b01257.
Scaglia, G. J. E., Mut, V. A., Jordan, M., Calvo, C., & Quintero, L. (2009). Mobile robot control based on robust control techniques.Journal of Engineering Mathematics, 63(1), 17–32.https://
doi.org/10.1007/s10665-008-9252-0.
Serrano, M. E., Godoy, S. A., Gandolfo, D., Mut, V. A., & Scaglia, G. J. E. (2018). Nonlinear trajectory tracking control for marine vessels with additive uncertainties.Information Technol- ogy and Control, 47(1), 118–130.https://doi.org/10.5755/j01.itc.47.1.18021.
Zhou, K., Doyle, J. C., & Glover, K. (1996).Robust and optimal control(Vol. 40, p. 146). New Jersey: Prentice hall.
Fig. 7.6 Experimental results for LABC applied to the system under nominal and operation and under uncertainty
116 7 Uncertainty Treatment
Linear Algebra-Based Controller Implementation Issues
In the previous chapters, the Linear Algebra-Based Control Design methodology has been presented and applied to a number of processes, showing its applicability to a variety of models. But, obviously, this is not the panacea to design the control for any plant. There are some constraints but also some clear benefits.
In this chapter, the advantages in using this methodology to design the control will befirst summarized and later on, some issues and critical drawbacks will be discussed, providing simple guidelines to implement the designed controller.