# Conditioned Adaptive Control for an Uncertain Bioreactor with Input Saturation and Steep Settling Desired Output

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- −
- Contribution Ci. The effect of input constraint limits and desired output on the convergence of the tracking error during input saturation events is determined. In contrast, in current adaptive AES-based control (for instance [11,16,17,18]), the convergence of the tracking error is determined in terms of the controller parameters, not in terms of input constraint limits or desired out.
- −
- Contribution Cii. The upper limit of the input constraint guaranteeing output convergence during input saturation events is determined as a function of the model terms, but independent of the time derivative of the desired output and its limit. In contrast, this limit is not computed in common AES-based controllers (for instance [11,13,14,15,16,17,18,19]).
- −
- Contribution Ciii. Differently from [20], a new dead zone Lyapunov function is proposed which allows proving asymptotic convergence of the tracking error during input saturation events, considering desired output featuring settling behavior with a steep section.

## 2. Bioreactor Model, Reference Model, Control Goal, Materials and Methods

#### 2.1. Bioreactor Model

**Assumption**

**1.**

**Assumption**

**2.**

#### 2.2. Control Goal and Reference Model

**Control**

**goal.**

**Remark**

**1.**

#### 2.3. Materials and Methods: Overview of the Method for Control Design and Stability Analysis

- −
- Definition of the input saturation conditions;
- −
- Definition of the tracking error dynamics, $de/dt$;
- −
- Definition of the main features of the Lyapunov function for $e$, $\left({V}_{e}\right)$, and of the gradient of ${V}_{e}$, that is, $W$;
- −
- Determination of the $d{V}_{e}/dt$ expression;
- −
- Formulation of the control and update laws and determination of the $dV/dt$ expression for the case of no input saturation events, being $V$ the Lyapunov function;
- −
- Formulation of the control and update laws and the determination of the $dV/dt$ expression for the case of input saturation events;
- −
- Determination of the unified $dV/dt$ expression, combining saturation and no saturation events;
- −
- Determination of the convergence of the tracking error and boundedness of the updated parameter, for the whole closed-loop regime encompassing saturation and non-saturation events.

## 3. Control Algorithm, Controller Design, and Stability Analysis

- −
- The asymptotic convergence of the tracking error $e$ to the compact set ${\mathsf{\Omega}}_{e}=\left\{e:\left|e\right|\le \epsilon \text{}\right\}$ is guaranteed, during the whole closed-loop regime in presence of input saturation events, and also it is guaranteed during input saturation events, considering high $d{y}_{d}/dt$ values occurring during some time lapses.
- −
- The condition for computing the limit for the input constraint ${D}_{max}$ is determined, which leads to asymptotic convergence of the tracking error in presence of input saturation events and is independent of $d{y}_{d}/dt$.
- −
- A new dead zone Lyapunov function is proposed which allows proving the asymptotic convergence of the tracking error during input saturation events, considering desired output featuring settling behavior with a steep section.

#### 3.1. Control Algorithm

**Remark**

**2.**

#### 3.2. Controller Design and Stability Analysis

**Theorem**

**1.**

**Proof:**

**Task**

**1:**

**Task**

**2:**

**Task**

**3:**

**Task**

**4:**

**Task**

**5:**

**Task 6:**Formulation of the update laws and determination of$dV/dt$, for the case of input saturation events with$D={D}_{min}$.

**Task**

**7:**

**Task**

**8:**

#### 3.3. Discussion of Results

- −
- The asymptotic convergence of the tracking error $e$ to the compact set ${\mathsf{\Omega}}_{e}=\left\{e:\left|e\right|\le \epsilon \text{}\right\}$ is guaranteed, during the whole closed-loop regime in the presence of input saturation events, and also it is guaranteed during input saturation events, which is stated in Theorem 1, Tii.
- −
- The absence of an excessive increase in the updated parameter $\widehat{\theta}$, is guaranteed, in the presence of input saturation events and high $d{y}_{d}/dt$ values occurring during some time lapses, which is stated in Theorem 1, Ti.
- −
- The condition (6) for computing ${D}_{max}$ is determined, which leads to an asymptotic convergence of the tracking error in the presence of input saturation events and is independent of $d{y}_{d}/dt$. Condition (6) is included in Theorem 1.

- −
- The convergence rate of the tracking error during no saturation input events is achieved by properly setting the controller parameters;
- −
- The uncertainty on the substrate consumption rate ${y}_{s}\mu X$ is tackled by means of the updated parameter $\widehat{\theta}$;
- −
- The uncertainty on ${s}_{in}$ is tackled through the robustness strategy.

- −
- During input saturation events featuring $D={D}_{min}$, the first term of the $dV/dt$ expression is negative for zero ${D}_{min}$;
- −
- During input saturation events featuring $D={D}_{max}$, the first $dV/dt$ term depends on the upper saturation limit ${D}_{max}$, the inflow concentration and the substrate consumption rate, and is rendered negative by using high enough values of ${D}_{max}$. In the determination of ${D}_{max}$, it is considered that the dilution rate must overcome the biological substrate consumption, to avoid substrate depletion, but the time varying nature of the feed substrate concentration is also taken into account. However, neither $d{y}_{d}/dt$ nor its bound are accounted in the ${D}_{max}$ value, because $d{y}_{d}/dt$ is vanishing;
- −
- The $d{y}_{d}/dt$ term of the $dV/dt$ expression may take on positive values, thus leading to positive $dV/dt$. However, the tracking error convergence and parameter boundedness are yet to be obtained, due to the vanishing nature of $d{y}_{d}/dt$. To prove these stability features, a new Lyapunov function of the tracking error is proposed, which is bounded for infinite values of the tracking error. The convergence of the tracking error is achieved even in the case of permanent input saturation.

## 4. Numerical Simulation

^{−1}; ${X}_{max}=0.32$ g/L; $f=1.607$; ${y}_{s}=0.0234$; ${m}_{s}=0.22425$ h

^{−1}; ${x}_{to}=0.09$ g/L; ${s}_{to}=49.6$ g/L; $D=0.005$ h

^{−1}. However, the value ${s}_{in}=53$ g/L was used in controller simulations. The width of the convergence set was chosen to be $\epsilon =1$, and the controller parameters were chosen as follows:

#### General Discussion on the Simulation Results

- −
- −
- Condition (6) for computing ${D}_{max}$ was adequate, as the asymptotic convergence of the tracking error was achieved.

- −
- There was no excessive increase in the updated parameter $\widehat{\theta}$, which was also noticed during high $d{y}_{d}/dt$ values,
- −
- A fast convergence of the tracking error was achieved by properly setting the controller parameters,
- −
- The uncertainty on the substrate consumption rate ${y}_{s}\mu X$ was properly tackled by means of the updated parameter $\widehat{\theta}$;
- −
- The uncertainty on ${s}_{in}$ was properly tackled through the robustness strategy.

## 5. Conclusions

- −
- Contribution Ci. The effect of input constraint limits and desired output on the convergence of the tracking error during input saturation events was determined.
- −
- Contribution Cii. The upper limit of the input constraint guaranteeing output convergence during input saturation events was defined as a function of the model terms, but was independent of the time derivative of the desired output and its limit.
- −
- Contribution Ciii. A new dead zone Lyapunov function was proposed which allows proving the convergence of the tracking error during input saturation events, considering fast variation in the desired output.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Result (21)

**Proposition**

**A.1.**

**Proof.**

## References

- Alshammari, O.; Mahyuddin, M.N.; Jerbi, H. A Survey on Control Techniques of a Benchmarked Continuous Stirred Tank Reactor. J. Eng. Sci. Technol.
**2018**, 13, 3277–3296. [Google Scholar] - Franco-de los Reyes, H.A.; Alvarez, J. Saturated Output-Feedback Control and State Estimation of a Class of Exothermic Tubular Reactors. J. Process Control
**2022**, 112, 78–95. [Google Scholar] [CrossRef] - Bastin, G.; Dochain, D. On-Line Estimation and Adaptive Control of Bioreactors; Elsevier: Amsterdam, The Netherlands, 1990. [Google Scholar]
- Méndez-Acosta, H.O.; Campos-Delgado, D.U.; Femat, R.; González-Alvarez, V. A Robust Feedforward/Feedback Control for an Anaerobic Digester. Comput. Chem. Eng.
**2005**, 29, 1613–1623. [Google Scholar] [CrossRef] - Méndez-Acosta, H.O.; Palacios-Ruiz, B.; Alcaraz-González, V.; González-Álvarez, V.; García-Sandoval, J.P. A Robust Control Scheme to Improve the Stability of Anaerobic Digestion Processes. J. Process Control
**2010**, 20, 375–383. [Google Scholar] [CrossRef] - Hägglund, T.; Shinde, S.; Theorin, A.; Thomsen, U. An Industrial Control Loop Decoupler for Process Control Applications. Control Eng. Pract.
**2022**, 123, 105138. [Google Scholar] [CrossRef] - Zhou, Z.; Tang, G.; Huang, H.; Han, L.; Xu, R. Adaptive Nonsingular Fast Terminal Sliding Mode Control for Underwater Manipulator Robotics with Asymmetric Saturation Actuators. Control Theory Technol.
**2020**, 18, 81–91. [Google Scholar] [CrossRef] - Shahriari-Kahkeshi, M. Dead-Zone Model-Based Adaptive Fuzzy Wavelet Control for Nonlinear Systems Including Input Saturation and Dynamic Uncertainties. Int. J. Fuzzy Syst.
**2018**, 20, 2577–2592. [Google Scholar] [CrossRef] - Polycarpou, M.; Farrell, J.; Sharma, M. On-Line Approximation Control of Uncertain Nonlinear Systems: Issues with Control Input Saturation. In Proceedings of the 2003 American Control Conference, Denver, CO, USA, 4–6 June 2003; pp. 543–548. [Google Scholar]
- Zhang, W.; Yi, W. Fuzzy Observer-Based Dynamic Surface Control for Input-Saturated Nonlinear Systems and Its Application to Missile Guidance. IEEE Access
**2020**, 8, 121285–121298. [Google Scholar] [CrossRef] - Shao, K.; Tang, R.; Xu, F.; Wang, X.; Zheng, J. Adaptive Sliding Mode Control for Uncertain Euler–Lagrange Systems with Input Saturation. J. Frankl. Inst.
**2021**, 358, 8356–8376. [Google Scholar] [CrossRef] - Gao, S.; Ning, B.; Dong, H. Fuzzy Dynamic Surface Control for Uncertain Nonlinear Systems under Input Saturation via Truncated Adaptation Approach. Fuzzy Sets Syst.
**2016**, 290, 100–117. [Google Scholar] [CrossRef] - Min, H.; Xu, S.; Ma, Q.; Zhang, B.; Zhang, Z. Composite-Observer-Based Output-Feedback Control for Nonlinear Time-Delay Systems with Input Saturation and Its Application. IEEE Trans. Ind. Electron.
**2018**, 65, 5856–5863. [Google Scholar] [CrossRef] - Zeinali, S.; Shahrokhi, M. Observer-Based Singularity Free Nonlinear Controller for Uncertain Systems Subject to Input Saturation. Eur. J. Control
**2020**, 52, 49–58. [Google Scholar] [CrossRef] - Meng, R.; Chen, S.; Hua, C.; Qian, J.; Sun, J. Disturbance Observer-Based Output Feedback Control for Uncertain QUAVs with Input Saturation. Neurocomputing
**2020**, 413, 96–106. [Google Scholar] [CrossRef] - Wang, F.; Zhou, C.; Wang, J.; Hua, C. Adaptive Fuzzy Output Feedback Controller Design for a HAGC System with Input Saturation and Output Error Constraints. J. Frankl. Inst.
**2022**, 359, 2030–2057. [Google Scholar] [CrossRef] - Li, H.; Liu, Q.; Kong, L.; Zhang, X. Design of Adaptive Tracking Controller Using Barrier Functions for Nonlinear Systems with Input Saturation. J. Frankl. Inst.
**2020**, 357, 12555–12570. [Google Scholar] [CrossRef] - Sun, J.; Liu, C. Auxiliary-System-Based Composite Adaptive Optimal Backstepping Control for Uncertain Nonlinear Guidance Systems with Input Constraints. ISA Trans.
**2020**, 107, 294–306. [Google Scholar] [CrossRef] - Askari, M.R.; Shahrokhi, M.; Khajeh Talkhoncheh, M. Observer-Based Adaptive Fuzzy Controller for Nonlinear Systems with Unknown Control Directions and Input Saturation. Fuzzy Sets Syst.
**2017**, 314, 24–45. [Google Scholar] [CrossRef] - Golkani, M.A.; Seeber, R.; Reichhartinger, M.; Horn, M. Lyapunov-based Saturated Continuous Twisting Algorithm. Int. J. Robust Nonlinear Control
**2021**, 31, 3513–3527. [Google Scholar] [CrossRef] - Dochain, D.; Perrier, M.; Guay, M. Extremum Seeking Control and Its Application to Process and Reaction Systems: A Survey. Math. Comput. Simul.
**2011**, 82, 369–380. [Google Scholar] [CrossRef] - de Battista, H.; Jamilis, M.; Garelli, F.; Picó, J. Global Stabilisation of Continuous Bioreactors: Tools for Analysis and Design of Feeding Laws. Automatica
**2018**, 89, 340–348. [Google Scholar] [CrossRef] - Picó, J.; de Battista, H.; Garelli, F. Smooth Sliding-Mode Observers for Specific Growth Rate and Substrate from Biomass Measurement. J. Process Control
**2009**, 19, 1314–1323. [Google Scholar] [CrossRef] - Neria-González, M.I.; Domínguez-Bocanegra, A.R.; Torres, J.; Maya-Yescas, R.; Aguilar-López, R. Linearizing Control Based on Adaptive Observer for Anaerobic Sulphate Reducing Bioreactors with Unknown Kinetics. Chem. Biochem. Eng. Q.
**2009**, 23, 179–185. [Google Scholar] - Slotine, J.-J.; Li, W. Applied Nonlinear Control; Prentice Hall: Englewood Cliffs, NJ, USA, 1991. [Google Scholar]
- Astrom, K.J.; Wittenmark, B. Adaptive Control; Addison-Wesley Publishing Company: Reading, MA, USA, 1995. [Google Scholar]
- Koo, K.-M. Stable Adaptive Fuzzy Controller with Time-Varying Dead-Zone. Fuzzy Sets Syst.
**2001**, 121, 161–168. [Google Scholar] [CrossRef] - Wang, X.-S.; Su, C.-Y.; Hong, H. Robust Adaptive Control of a Class of Nonlinear Systems with Unknown Dead-Zone. Automatica
**2004**, 40, 407–413. [Google Scholar] [CrossRef] - Ranjbar, E.; Yaghubi, M.; Abolfazl Suratgar, A. Robust Adaptive Sliding Mode Control of a MEMS Tunable Capacitor Based on Dead-Zone Method. Automatika
**2020**, 61, 587–601. [Google Scholar] [CrossRef] - Hong, Q.; Shi, Y.; Chen, Z. Dynamics Modeling and Tension Control of Composites Winding System Based on ASMC. IEEE Access
**2020**, 8, 102795–102810. [Google Scholar] [CrossRef] - Rincón, A.; Hoyos, F.E.; Candelo-Becerra, J.E. Adaptive Control for a Biological Process under Input Saturation and Unknown Control Gain via Dead Zone Lyapunov Functions. Appl. Sci.
**2020**, 11, 251. [Google Scholar] [CrossRef] - Rincón, A.; Restrepo, G.M.; Hoyos, F.E. A Robust Observer—Based Adaptive Control of Second—Order Systems with Input Saturation via Dead-Zone Lyapunov Functions. Computation
**2021**, 9, 82. [Google Scholar] [CrossRef] - Rincón, A.; Restrepo, G.M.; Sánchez, Ó.J. An Improved Robust Adaptive Controller for a Fed-Batch Bioreactor with Input Saturation and Unknown Varying Control Gain via Dead-Zone Quadratic Forms. Computation
**2021**, 9, 100. [Google Scholar] [CrossRef] - Ioannou, P.A.; Sun, J. Robust Adaptive Control; Prentice-Hall PTR: Upper Saddle River, NJ, USA, 1996. [Google Scholar]
- Liu, C.; Yue, X.; Zhang, J.; Shi, K. Active Disturbance Rejection Control for Delayed Electromagnetic Docking of Spacecraft in Elliptical Orbits. IEEE Trans. Aerosp. Electron. Syst.
**2021**, 58, 2257–2268. [Google Scholar] [CrossRef] - Shi, K.; Liu, C.; Sun, Z.; Yue, X. Coupled Orbit-Attitude Dynamics and Trajectory Tracking Control for Spacecraft Electromagnetic Docking. Appl. Math. Model.
**2022**, 101, 553–572. [Google Scholar] [CrossRef] - Rincón, A.; Cuellar, J.A.; Valencia, L.F.; Sánchez, O.J. Cinética de Crecimiento de Gluconacetobacter Diazotrophicus Usando Melaza de Caña y Sacarosa: Evaluación de Modelos Cinéticos. Acta Biol. Colomb.
**2019**, 24, 38–57. [Google Scholar] [CrossRef]

**Figure 3.**Simulation results for case 1: (

**a**) Trajectory of the tracking error $e$; (

**b**) detail of the trajectory of $e$; (

**c**) trajectory of substrate concentration $S$; (

**d**) detail of the trajectory of $S$; (

**e**) trajectory of the control input $D$; (

**f**) detail of the trajectory of $D$.

**Figure 4.**Simulation results for case 2: (

**a**) Trajectory of the tracking error $e$; (

**b**) detail of the trajectory of $e$; (

**c**) trajectory of substrate concentration $S$; (

**d**) detail of the trajectory of $S$; (

**e**) trajectory of the control input $D$; (

**f**) detail of the trajectory of $D$.

**Figure 5.**Simulation results for case 3: (

**a**) Trajectory of the tracking error $e$; (

**b**) detail of the trajectory of $e$; (

**c**) trajectory of substrate concentration $S$; (

**d**) detail of the trajectory of $S$; (

**e**) trajectory of the control input $D$; (

**f**) detail of the trajectory of $D$.

**Figure 6.**Simulation results for case 4: (

**a**) Trajectory of the tracking error $e$; (

**b**) detail of the trajectory of $e$; (

**c**) trajectory of substrate concentration $S$; (

**d**) detail of the trajectory of $S$; (

**e**) trajectory of the control input $D$; (

**f**) detail of the trajectory of $D$.

Simulation Case | ${\mathit{D}}_{\mathit{m}\mathit{a}\mathit{x}}\text{}\mathbf{Value}$ | ${\mathit{a}}_{\mathit{m}}\text{}\mathbf{Value}$ |
---|---|---|

Case 1 | 0.0016 | 20 |

Case 2 | 0.0016 | 0.1 |

Case 3 | 0.000327 | 7 |

Case 4 | 0.0016 | 1 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rincón, A.; Hoyos, F.E.; Candelo-Becerra, J.E. Conditioned Adaptive Control for an Uncertain Bioreactor with Input Saturation and Steep Settling Desired Output. *Symmetry* **2022**, *14*, 1232.
https://doi.org/10.3390/sym14061232

**AMA Style**

Rincón A, Hoyos FE, Candelo-Becerra JE. Conditioned Adaptive Control for an Uncertain Bioreactor with Input Saturation and Steep Settling Desired Output. *Symmetry*. 2022; 14(6):1232.
https://doi.org/10.3390/sym14061232

**Chicago/Turabian Style**

Rincón, Alejandro, Fredy E. Hoyos, and John E. Candelo-Becerra. 2022. "Conditioned Adaptive Control for an Uncertain Bioreactor with Input Saturation and Steep Settling Desired Output" *Symmetry* 14, no. 6: 1232.
https://doi.org/10.3390/sym14061232