Observer-Based Backstepping Controller for Microalgae Cultivation

Article pubs.acs.org/IECR

Observer-Based Backstepping Controller for Microalgae Cultivation Masood Khaksar Toroghi, Guillaume Goffaux, and Michel Perrier* Department of Chemical Engineering, École Polytechnique Montréal, Montréal, Canada, H3C 3A7 ABSTRACT: Microalgae are microscopic plants that exist in an aquatic environment. They are involved in the production of high value compounds and also have applications in energy production. In this work, the regulation of biomass concentration in a bioreactor is investigated by using an observer-based backstepping control approach. The process is controlled to operate in a constant biomass concentration mode, in order to maintain the culture at a desired concentration and to sustain high biomass production levels. Combined with the backstepping controller, a nonlinear Lipschitz observer is proposed. On the basis of the Droop model, which describes the dynamic behavior of the microalgae process and based on the biomass measurements, the stability of the state error dynamics can be guaranteed. Finally, simulations and comparisons with a PI controller are developed to show the performance of the nonlinear observer-based controller in set point tracking and load rejection in the presence of parameter uncertainties.

■

INTRODUCTION Bioprocesses play an effective role in the production of valueadded products in the pharmaceutical industry, and among the various types of cultivation, microalgae are one of the most applicable types of cultures. One of its applications is in the production of high-value compounds because this biomass is a great source of fatty acids, vitamins, and pigments. Another domain of interest is energy production (biofuel, bioethanol). Indeed, two advantages are that microalgae grow very quickly in comparison with other terrestrial crops1 and that they contain between 20 and 50% of oil under normal conditions.2 Moreover, biomass from microalgae can be burned to produce electricity and heat, and they can produce hydrogen under specific environmental conditions.3,4 In addition, microalgae cultures provide opportunities for environmental applications such as wastewater treatment and carbon dioxide fixation5 or decomposition of different classes of toxic compounds.6 Consequently, keeping the culture under control at a desired value is an important objective using a suitable control method. This is why the applications of process control have been popularized for microalgae culture.7 Generally speaking, the control of bioprocesses has a number of challenging problems. For instance, the presence of living organisms and the interaction between them give rise to interdependent complex mathematical models. In addition, only a few measurements are available and measurement devices can provide unreliable measurements. In order to circumvent this problem with measuring devices, observers (also called state estimator or software sensor) can be used. They are dynamical systems which are used to estimate important process variables by means of accessible measured variables. Their design and their application in process control have been an active research area over the past decades, especially in bioprocess applications.8−11 In the context of the microalgae applications, Bernard et al.12 designed a high gain observer based on the Droop model to monitor phytoplankton. They applied the proposed software sensor to a real experimental setup, and they showed the validity and the efficiency of the observer. Following this work, Goffaux et al.13,14 © XXXX American Chemical Society

applied continuous-discrete interval observers for this culture. Moreover, other studies have been carried on such as the design of an extended Kalman filter to estimate biomass from dissolved oxygen measurements and a simple unstructured process model15 or the design of a moving horizon estimator in the context of lipid production optimization.16 However, these methodologies do not take into account the global convergence of the estimation error. That is why, in this study, another observer is considered to estimate the state variables. On the basis of the Lipschitz properties of the chosen microalgae model, a Lipschitz observer is designed and a proof of the stability of the error dynamics is given. In order to compensate the nonlinearity and the complexity of this kind of processes, several nonlinear control strategies were developed such as optimization-based approaches,17 adaptive approaches,18−20 sliding mode control,21 exact linearization approach,7 and backstepping approach.22 In the case of microalgae, Becerra et al.7 used an input−output linearization technique to keep the biomass at a constant value for continuous cultivation. Also, in Hafidi et al.,17 a nonlinear model predictive controller is applied to the culture in order to maintain the culture at optimal population density in a constant high biomass density mode. Furthermore, in Abdollahi et al., an interior point optimization and a model predictive control with moving-horizon observer are used to maximize and regulate the lipid production in a real time fed-batch microalgae cultivation.16 The estimator and the controller design are based on a set of linearized models in the microalgae growth process. In this work, among the possible robust control strategies, a backstepping approach is chosen and designed to control the biomass concentration. Robustness is one of its advantages, which cannot be obtained by a traditional controller scheme. This approach has been known for more than two decades, but its application for bioprocess is not common. Received: September 25, 2012 Revised: May 1, 2013 Accepted: May 7, 2013

A

dx.doi.org/10.1021/ie302615j | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

Article

Lipschitz Observer. Consider the nonlinear system described by

This paper is organized as follows. In section 2, the description of the microalgae model, called the Droop model, is presented. Section 3 contains the procedure to design the nonlinear observer, and in section 4, the methodology to derive the control law is explained. Section 5 is devoted to numerical simulations, and finally, some conclusions are provided in section 6.

x ̇ = Ax + ϕ(x , u) y = Cx

■

(2)

where ϕ(x, u) is a nonlinear vector field with a Lipschitz constant γ, the vector x ∈ n stands for the state variables, and the input u ∈  represents the manipulated variable. The measured output is represented by the vector y ∈ m. The pair (A, C) is considered observable, and the following inequality is valid for the nonlinear part.

MODEL DESCRIPTION The Droop model23 is a simple and widely used model that can represent the growth of microalgae. It includes three state variables: the biomass concentration X, the internal quota QN, which is defined as the quantity of intracellular nitrogen per unit of biomass, and the substrate concentration S. The dynamics mass balance equations of the Droop model are given by

∥ϕ(x , u) − ϕ(x ̂ , u)∥ ≤ γ∥(x − x)̂ ∥

where γ is a Lipschitz constant and ∥·∥ is the Euclidean norm. The observer is assumed to be of the form

Ẋ (t ) = −D(t )X(t ) + μ(Q N (t ))X(t )

x ̂ ̇ = Ax ̂ + ϕ(x ̂ , u) + L(y − Cx)̂

Q̇ N (t ) = ρ(S(t )) − μ(Q N (t ))Q N (t ) S(̇ t ) = D(t )(Sin − S(t )) − ρ(S(t ))X(t )

with ρ(S) = ρm(S(t)/(S(t) + Ks)) as the substrate uptake rate and μ(QN) = μ̅(1 − (KQ/QN(t))) as the specific growth rate. In eq 1, D is the dilution rate (F/V) and is considered as a process input. Sin is the input substrate concentration, ρ the substrate uptake rate, and μ the growth rate. In the uptake rate equation, Ks and ρm represent, respectively, a half-saturation constant for the substrate and the maximum uptake rate. Similarly, in the growth rate equation, μ̅ is the theoretical maximum growth rate obtained for an infinite internal quota and KQ the minimum internal quota allowing growth. Finally, the dilution rate is considered the input variable for control purposes. The parameters of the model13 used in this study are given in Table 1.

e ̇ = (A − LC)x ̂ + [ϕ(x , u) − ϕ(x ̂, u)]

unit

value

μmol L−1 μmol L−1 d−1 μmol μm−3 μmol μm−3 d−1

100 0.105 2 1.8 9.3

(5)

In this structure, L is the observer gain and is obtained from a pole placement technique, giving some convergence for the linear part only. To take into account the whole structure, the following result can be used. Theorem 129 If a gain matrix L can be chosen such that the Lipschitz constant γ (eq 3) is upper bounded, γ
0 and P = PT ≥ 0 satisfy the Riccati equation (A − LC)TP + P(A − LC) = −Q, then eq 4 leads to asymptotically stable estimates for eq 2. In eq 6, λmin(Q) and λmax(P) represent the smallest and largest eigenvalues of Q and P, respectively. In the following theorem, the existence of a solution for a Riccati equation is presented. Theorem 230 Consider the following Riccati equation

Table 1. Model Parameters Sin Ks μ̅ KQ ρm

(4)

By subtracting eq 4 from eq 2, the estimation error dynamics (e = x − x̂) is given by

(1)

parameter

(3)

(T P + P ( + PRP + Q = 0

If P ≥ 0 is a solution of the Riccati equation, then the following conditions need to be true

Properties of the Droop Model. There are two important properties for the Droop model. First, the trajectories of the Droop model are bounded and KQ ≤ QN ≤ QNmax. Second, the Droop model is uniformly input observable with y = X if X ≠ 0. The proof of these properties can be found in Bernard et al.24

λmin(R )tr(Q ) − nλmin 2(:) < 0

λmin(:) < 0,

■

OBSERVER DESIGN One of the difficulties in designing an observer is the proof of global convergence of the estimation error. Generally speaking, there are several approaches to designing a state observer for nonlinear systems such as the high-gain observer,25 the movinghorizon observer,26 the extended Kalman filter,11 and the extended Luenberger observer.27 However, a proof of convergence is not usually possible or is provided with restricted conditions.28 The Lipschitz observer is another nonlinear observer for a specific class of nonlinear systems satisfying the Lipschitz property. It can guarantee the stability of the error dynamics, globally or locally depending on the Lipschitz property of the nonlinearities. In the following section, the Lipschitz observer is described and applied to the Droop model.

:=

(( + (T ) 2

(7)

(8)

where tr(X) and λmin(X) are the trace and the smallest eigenvalue of matrix X, respectively . Remark 1 If A is Hurwitz, then λmin(: ) < 0. In the context of the Lipschitz observer, as L is chosen by a pole placement technique, ( = A − LC is Hurwitz and eq 8 is satisfied. Consequently, the following procedure can be considered. On the basis of the pole placement technique, a gain matrix L is computed satisfying some dynamics of the linear part of the estimation error. Then, the related Riccati equation is solved by trial and error thanks to eq 2 in order to find suitable matrices Q and P in accordance with eq 6. Actually, the tuning matrices Q and R are first chosen in order to satisfy the condition in eq 7. After that, the Riccatti equation is solved and eq 6 is checked. If no matrix can be obtained, then another gain matrix L should be determined. B



Article

Figure 1. Estimation of biomass with measurement noise.

Figure 2. Estimation of the internal quota with measurement noise.

Remark 2 Matrix R is given by the identity matrix in order to satisfy the form of the Riccati equation in eq 1. Then, a choice of a matrix Q can be made satisfying eq 7. In the sequel, the Lipschitz observer is applied to the Droop model. Representation of the Droop Model. By adding and subtracting extra terms to the Droop model, it is possible to obtain the nonlinear model combination of a linear and a nonlinear part satisfying the observability conditions of the Lipschitz observer (i.e., the pair (A, C) is observable).

where ⎛ μ̅ − D μ̅ 0 ⎞ ⎜ ⎟ −μ ̅ −ρm ⎟ , A(D) = ⎜ 0 ⎜ ⎟ −D ⎠ 0 ⎝−ρm ⎛ μ̅ KQ X ⎞ ⎜− μ ̅ Q N − ⎟ QN ⎟ ⎜ ⎟ ϕ(x) = ⎜ ⎜ K Q μ ̅ + Sρm + ρ ⎟ ⎜ ⎟ ⎝ DSin + X(ρm − ρ)⎠

x ̇ = A(D)x + ϕ(x) y=X

(9) C



Article

Figure 3. Estimation of substrate with measurement noise.

As can be observed in eq 9, the nonlinear part with respect to the state variables is continuous and differentiable. Moreover, on the basis of the properties of the Droop model, ϕ is bounded. Therefore, one can conclude that ϕ satisfies the Lipschitz property and that a Lipschitz observer can be designed. In order to solve the resulting Riccati equation, the identity matrix is chosen for matrix R and matrix Q. Furthermore, gain matrix L is such that ( = A(D) − LC is Hurwitz and one has λmin(: ) < 0. Consequently, eq 8 is satisfied. Moreover, eq 7 can be rewritten by the following inequality

1 < λmin 2(:)

Table 2. Simulation Parameters parameter

value

γ λ1 λ2 λ3 D

19.0 −8.0 −6.0 −4.0 1.3

Table 3. Parameters of Eigenvalues

(10)

parameter

value

α1 α2 α3 b1 b2 b3 ε1 ε2 ε3

−8.00 −6.00 −4.00 −0.069 −0.053 −0.04 −0.60 −0.80 −0.90

In conclusion, knowing L and P, eqs 10 and 6 have to be satisfied. If not, L is modified by trial and error. Remark 3 One can note that the systems given by eqs 2 and 9 dif fer in the state af f ine part as a function of the process input. Therefore, λmin(: ) is a function of the dilution rate and eqs 10 and 6 should be checked for the possible values of D.

■

CONTROLLER DESIGN A main objective of the control is to guarantee that the process will produce the desired amount of biomass during cultivation. This can be done by appropriately changing the flow rate to the bioreactor. In this paper, the regulation of biomass concentration is based on a nonlinear backstepping control strategy using the dilution rate. In the following paragraph, the control algorithm is briefly described. Backstepping Controller. Backstepping is a recursive methodology to obtain a feedback control law and an associated Lyapunov function in a systematic manner. This technique provides a powerful design tool for nonlinear systems in the pure and strict feedback form.31 Robustness is one of the advantages, which cannot be obtained by a traditional controller scheme. In backstepping, the design is more flexible because of the possibility of choosing the nonlinear damping terms. Therefore, additional robustness is obtained. More information about the general design of a backstepping controller can be

found in Krstic et al.32 The key procedures of the backstepping design for the microalgae process are summarized to obtain the control law.33 At first, the control problem is formulated such that the integral action goes to zero. In this context, let us define the error variable Z1 as Z1 =

∫0

t

(y − yd ) dt

and

Z1̇ = y − yd

(11)

where yd is the set point for y. Moreover, another error variable Z2 is defined, which is a combination of the control error y − yd and the integral action Z1: 1 Z 2 = (y − yd − α), η > 0 η (12) D



Article

Figure 4. Estimation of biomass with a measurement noise and a time-varying gain matrix.

Figure 5. Estimation of the internal quota with a measurement noise and a time-varying gain matrix.

To compute the derivative of the second part, V2 = (1/2)Z22, Ż 2 is obtained as

In the backstepping formalism, α = −c1Z1 (c1 > 0) and β = −c2Z2 (c2 > 0) are the stabilizing functions or damping terms. On the basis of these considerations, Ż 1 can be rewritten by Z1̇ = −c1Z1 + ηZ 2

Ż 2 =

(13)

In order to prove the convergence of the error variables Z1 and Z2 to zero, a Lyapunov function is chosen: V = (1/2)Z12 + (1/2) Z22 and its derivative is computed. The first part of the Lyapunov function is V1 = (1/2)Z12, and its derivative is given by

V1̇ = −c1Z12 + ηZ1Z 2

∂α ̇ ⎞ 1⎛ Z1⎟ ⎜y ̇ − yḋ − η⎝ ∂Z1 ⎠

(15)

where ẏ is Ẋ = −DX + μX in the context of the Droop model. Thus, we have the following equation for Ż 2 Ż 2 =

(14) E

∂α ̇ ⎞ 1⎛ Z1⎟ ⎜ −DX + μX − yḋ − η⎝ ∂Z1 ⎠

(16)



Article

Figure 6. Estimation of substrate with a measurement noise and a time-varying gain matrix.

Figure 7. Control of biomass concentration.

Therefore, the derivative of the Lyapunov function V would be V̇ = V1̇ + Z 2Ż2

D=u=−

(17)

X (19)

For the stabilization of eq 17 using the second damping term β = −c2Z2, the backstepping controller law determines the expression of the dilution rate and indirectly the input flow rate such that the derivative of the Lyapunov function V has the following form, proving stability: V̇ = V1̇ + Z 2Ż2 = −c1Z12 − c 2Z 2 2

Z1( − η2 + c12) − Z 2(c 2η + c1η) − μX + yḋ

where c1, c2, and η are positive constants. From eq 19, one can notice that the internal quota is necessary for the control law, illustrating the importance of a state observer.

■

SIMULATION RESULTS In order to show the effectiveness of the observer−controller scheme, several simulations are performed. First, the observer performance in the presence of measurement noise is illustrated. Then, the

(18)

After some computations, one can show that the dilution rate is given by F



Article

Figure 8. Control action (dilution rate).

with 0 as mean and 1 as standard deviation. Therefore, the measurement is performed according to relative noise with respect to the biomass concentration. In this context, a high amount of biomass is less precisely measured than low concentration. In a practical point of view, this behavior is related to the optical sensor used to measure the cell concentration in microalgae culture. Indeed, in the case of a high biomass concentration, the analysis is more complicated to distinguish the cells correctly. Observer Performance. In Figures 1−3, the observer performance in the presence of measurement noise is shown. Simulation parameters are presented in Table 2 with λ1, λ2, and λ3 the eigenvalues of the linear part of the error dynamics.

Table 4. Controller’s Parameters parameter

value

c1 c2 η Kc KI

0.6 0.1 0.1 −0.037 −0.057

backstepping controller is coupled with the designed observer in order to maintain the biomass concentration at a constant level. The noisy measurements are built on the basis of the following equation: y = X(1 + 0.1ε), with ε as a Gaussian white noise and

Figure 9. Control of biomass concentration with modeling error. G



Article

Figure 10. Control action with modeling error.

Figure 11. Estimation of internal quota with modeling error.

• By increasing the amount of biomass, the measure noise increases and small values of the observer gain are required. Consequently, the value of the eigenvalues will change according to the following time dependent function.

As is illustrated, the performance decreases with time. This can be explained by the relative biomass measurement noise. Indeed, as the time evolves, the biomass concentration increases along with its measurement noise, penalizing the observer estimation. In order to improve the performance, time-varying eigenvalues for the linear part of the error dynamics are considered. The trajectory of the time-varying eigenvalues is selected on the basis of the following considerations. • At the beginning of the simulation, the biomass concentration is low and large values for the observer gain are possible, as the measurement noise is low. The convergence is fast in the beginning.

λi(t ) = αi exp(bit ) + εi

(20)

The parameters of the selected functions were chosen on the basis of simulation performance (Table 3). It can be observed in Figures 4−6 that this technique has a better effect on performance. H



Article

Figure 12. Estimation of substrate with modeling error.

Controller Performance. Finally, the performance of the observer-based backstepping control is illustrated. At first, a PI controller based on the linearized model using a pole placement technique around the operating point is designed and then is compared with the nonlinear controller in Figures 7 and 8. The controller’s parameters are given in Table 4. As can be observed, the performance is almost the same for both strategies. However, when the operating point changes, as expected, the PI controller is unable to properly control the process and the performance of the backstepping controller is better than the linear controller. This property can give an advantage to the nonlinear controller during transient operation. In addition, in order to show the performance and the robustness of the backstepping controller with modeling error, 10% parameter uncertainty is considered for the theoretical maximum growth rate value μ̅. In Figures 9−12, the performance of the observer-based controller is depicted. As illustrated, the controller has a good performance in the presence of modeling error. Also, the designed observer would be stable with nonzero steady state values as is expected. Finally, larger sampling periods for the measurements have been tested until 1 h. Simulations have again shown good results if the model is calibrated with 10% parameter uncertainty.

as a future work, the proof of the stability of the whole observer− controller closed loop will be one of the main challenges.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

REFERENCES

(1) Chisti, Y. Biodiesel from microalgae. Biotechnol. Adv. 2007, 25, 294−306. (2) Metting, F. Biodiversity and application of microalgae. J. Ind. Microbiol. 1996, 17, 477−489. (3) Benemann, J. R. Hydrogen production by microalgae. J. Appl. Phycol. 2000, 12, 291−300. (4) Hallenbeck, P. C.; Benemann, J. R. Biological hydrogen production; fundamentals and limiting processes. Int. J. Hydrogen Energy 2002, 27, 1185−1193. (5) Benemann, J. R. CO2 mitigation with microalgae systems. Energy Convers. Manage. 1997, 38, 475−479. (6) Boussiba, S.; Leu, S. Microalgal biotechnology for environmental remediation, 2007. (7) Becerra-Celis, G.; Joannis-Cassan, C.; Isambert, A.; Siguerdidjane, H. Control strategy for continuous microalgae cultivation process in a photobioreactor, 2008. (8) Doyle, F. Nonlinear inferential control for process applications. J. Process Control 1998, 8, 339−353. (9) Oisiovici, R. M.; Cruz, S. L. State estimation of batch distillation columns using an extended Kalman filter. Chem. Eng. Sci. 2000, 55, 4667−4680. (10) Dochain, D. State and parameter estimation in chemical and biochemical processes: a tutorial. J. Process Control 2003, 13, 801−818. (11) Khodadadi, H.; Jazayeri-Rad, H. Applying a dual extended Kalman filter for the nonlinear state and parameter estimations of a continuous stirred tank reactor. Comput. Chem. Eng. 2011, 35, 2426− 2436. (12) Bernard, O.; Sciandra, A.; Sallet, O. G. A non-linear software sensor to monitor the internal nitrogen quota of phytoplanktonic cells. Oceanol. Acta 2001, 24, 435−442.

■

CONCLUSION In this work, the problem of an observer-based controller is addressed to regulate the biomass concentration in microalgae cultivation. On the basis of the properties of the Droop model, a Lipschitz observer with guaranteed stability of error dynamics was designed. Results are obtained with a time-varying gain with increasing measurement noise. In addition, a backstepping controller with Lyapunov stability was applied in order to control the biomass concentration. Finally, the observer and controller were coupled and simulation results show that the proposed observer−controller scheme has an acceptable performance over a wide range of operation. Also, the controller and observer have a good performance in the presence of a modeling error. Finally, I



Article

(13) Goffaux, G.; Vande Wouwer, A.; Bernard, O. Continuous discrete interval observers for monitoring microalgae cultures. Biotechnol. Prog. 2009, 3 (3), 667−675. (14) Goffaux, G.; Vande Wouwer, A.; Bernard, O. Improving continuous discrete interval observers with application to microalgaebased bioprocesses. J. Process Control 2009, 19 (7), 1182−1190. (15) Li, J.; Xu, N. S.; Su, W. W. Online estimation of stirred-tank microalgal photobioreactor cultures based on dissolved oxygen measurement. Biochem. Eng. J. 2003, 14, 51−65. (16) Abdollahi, J.; Dubljevic, S. Lipid production optimization and optimal control of heterotrophic microalgae fed-batch bioreactor. Chem. Eng. Sci. 2012, 84, 619−627. (17) Becerra-Celis, G.; Hafidi, G.; Tebbani, S.; Dumur, D. Nonlinear predictive control for continuous microalgae cultivation process in a photobioreactor, 2008. (18) Marcos, N. I.; Guay, M.; Dochain, D. Output feedback adaptive extremum seeking control of a continuous stirred tank bioreactor with Monod’s kinetics. J. Process Control 2004, 14, 807−818. (19) Cougnon, P.; Dochain, D.; Guay, M.; Perrier, M. On-line optimization of fedbatch bioreactors by adaptive extremum seeking control. J. Process Control 2011, 21, 1526−1532. (20) Mailleret, L.; Bernard, O.; Steyer, J. P. Nonlinear adaptive control for bioreactors with unknown kinetics. Automatica 2004, 40, 1379− 1385. (21) Selisteanu, D.; Petre, E.; Rasvan, V. Sliding mode and adaptive sliding-mode control of a class of nonlinear bioprocesses. Int. J. Adapt. Control Signal Process. 2007, 21, 795−822. (22) Dochain, D.; Perrier, M. Adaptive backstepping nonlinear control of bioprocesses, 2004. (23) Lemesle, V.; Mailleret, L. A mechanistic investigation of the algae growth “Droop model”. Acta Biotheor. 2008, 56, 87−102. (24) Bernard, O.; Gouzé, J. L. Transient behavior of biological loop models with application to the Droop model. Math. Biosci. 1995, 127, 19−43. (25) Biagiola, S.; Figueroa, J. L. A high gain nonlinear observer: Application to the control of an unstable nonlinear process. Comput. Chem. Eng. 2004, 28, 1881−1898. (26) Rao, C. V.; Rawlings, J. B. Constrained process monitoring: Moving-horizon approach. AIChE J. 2002, 48, 97−109. (27) Quintero-Marmol, E.; Luyben, W.; Georgakis, C. Application of an extended Luenberger observer to the control of multicomponent batch distillation. Ind. Eng. Chem. Res. 1991, 30, 1870−1880. (28) Scala, B. F. L.; Bitmead, R. R.; James, M. R. Conditions for stability of the extended Kalman filter and their application to the frequency tracking problem. Math. Control, Signals, Syst. 1995, 8, 1−26. (29) Thau, F. Observing the state of non-linear dynamic systems. Int. J. Control 1973, 17, 471−479. (30) Zhu, Y.; Pagil, R. A note on the necessary conditions for the algebraic Riccati equation. IMA J. Math. Control Inf. 2005, 22, 181−186. (31) Khalil, H. Nonlinear systems, 2nd ed.; Prentice-Hall: Upper Saddle River, NJ, 1996. (32) Krstic, M.; Kanellakopoulos, I.; Kokotovic, P. V. Nonlinear and adaptive control design; Wiley-Interscience: New York, 1995. (33) Lee, T. C.; Yang, D. R.; Lee, K. S.; Yoon, T. W. Indirect adaptive backstepping control of a pH neutralization process based on recursive prediction error method for combined state and parameter estimation. Ind. Eng. Chem. Res. 2001, 40, 4102−4110.

J


Observer-Based Backstepping Controller for Microalgae Cultivation

Recommend Documents