# An Internal Model Based—Sliding Mode Control for Open-Loop Unstable Chemical Processes with Time Delay

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Unstable Systems

#### 2.2. Internal Model Control

#### 2.3. SMC Fundamentals

## 3. Identification Procedure for Unstable Systems Using PI/PID Control

## 4. Dsmc Design Approach

#### 4.1. Controller Synthesis

#### 4.2. Stability Analysis

**Theorem**

**1.**

**Proof.**

#### 4.3. Internal P/PD Controller Design

## 5. Computer Simulations

#### 5.1. Biochemical Reactor Model

$B\left(t\right)$: | Biomass concentration |

$S\left(t\right)$: | Substrate concentration |

$D\left(t\right)$: | Dilution rate |

${S}_{f}$: | Substrate concentration on process feeding (main process disturbance) |

$\mu $: | Specific growth rate |

${\mu}_{max}$: | Maximum specific growth rate |

${k}_{m}$: | Substrate saturation constant |

${k}_{1}$: | Substrate inhibition constant |

$\gamma $: | Biomass to substrate mass yield |

#### 5.2. Control Performance Indices

#### 5.3. Tracking Performance Test

#### 5.4. Regulation Performance Test

#### 5.5. Regulation and Tracking Performance Tests

## 6. Conclusions

^{®}.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Huang, H.P.; Chen, C.C. Control-system synthesis for open-loop unstable process with time delay. IEEE Proc. Control Theory Appl.
**1997**, 144, 334–346. [Google Scholar] [CrossRef] [Green Version] - Sardella, M.F.; Serrano, M.E.; Camacho, O.; Scaglia, G.J. Design and application of a linear algebra based controller from a reduced-order model for regulation and tracking of chemical processes under uncertainties. Ind. Eng. Chem. Res.
**2019**, 58, 15222–15231. [Google Scholar] [CrossRef] - Utkin, V.; Poznyak, A.; Orlov, Y.V.; Polyakov, A. Road Map for Sliding Mode Control Design; Springer: Berlin/Heidelberg, Germany, 2020. [Google Scholar]
- Sira-Ramírez, H. Dynamical sliding mode control strategies in the regulation of nonlinear chemical processes. Int. J. Control
**1992**, 56, 1–21. [Google Scholar] [CrossRef] - Utkin, V.; Lee, H. Chattering problem in sliding mode control systems. In Proceedings of the International Workshop on Variable Structure Systems, VSS’06, Alghero, Italy, 5–7 June 2006; pp. 346–350. [Google Scholar]
- Cortes, D.; Vázquez, N.; Alvarez-Gallegos, J. Dynamical sliding-mode control of the boost inverter. IEEE Trans. Ind. Electron.
**2008**, 56, 3467–3476. [Google Scholar] [CrossRef] - Koshkouei, A.J.; Burnham, K.J.; Zinober, A.S. Dynamic sliding mode control design. IEEE Proc. Control Theory Appl.
**2005**, 152, 392–396. [Google Scholar] [CrossRef] [Green Version] - Shtessel, Y.; Edwards, C.; Fridman, L.; Levant, A. Sliding Mode Control and Observation; Springer: Berlin/Heidelberg, Germany, 2014; Volume 10. [Google Scholar]
- Liu, J.; Wang, X.; Liu, J.; Wang, X. Advanced Sliding Mode Control; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Yu, X.; Feng, Y.; Man, Z. Terminal sliding mode control–an overview. IEEE Open J. Ind. Electron. Soc.
**2020**, 2, 36–52. [Google Scholar] [CrossRef] - Li, C.; Kim, J.; Lee, M.C. Fast Terminal SMC with SPO for Trajectory Tracking of Robot Manipulator for Nuclear Reactor Dismantlement. In Proceedings of the 2022 22nd International Conference on Control, Automation and Systems (ICCAS), Busan, Republic of Korea, 27 November–1 December 2022; pp. 196–199. [Google Scholar]
- Báez, E.; Bravo, Y.; Leica, P.; Chávez, D.; Camacho, O. Dynamical sliding mode control for nonlinear systems with variable delay. In Proceedings of the 2017 IEEE 3rd Colombian Conference on Automatic Control (CCAC), Cartagena, Colombia, 18–20 October 2017; pp. 1–6. [Google Scholar]
- Herrera, M.; Camacho, O.; Leiva, H.; Smith, C. An approach of dynamic sliding mode control for chemical processes. J. Process Control
**2020**, 85, 112–120. [Google Scholar] [CrossRef] - Proaño, P.; Capito, L.; Rosales, A.; Camacho, O. A dynamical sliding mode control approach for long deadtime systems. In Proceedings of the 2017 4th International Conference on Control, Decision and Information Technologies (CoDIT), Barcelona, Spain, 5–7 April 2017; pp. 0108–0113. [Google Scholar]
- Coronel, W.; Camacho, O. A Dynamic Sliding Mode Controller using a Rotating Type Moving Sliding Surface for Chemical Processes with Variable Delay. In Proceedings of the 2021 IEEE CHILEAN Conference on Electrical, Electronics Engineering, Information and Communication Technologies (CHILECON), Online, 6–9 December 2021; pp. 1–6. [Google Scholar]
- Asimbaya, E.; Cabrera, H.; Camacho, O.; Chávez, D.; Leica, P. A dynamical discontinuous control approach for inverse response chemical processes. In Proceedings of the 2017 IEEE 3rd Colombian Conference on Automatic Control (CCAC), Cartagena, Colombia, 18–20 October 2017; pp. 1–6. [Google Scholar]
- Rojas, R.; Camacho, O.; González, L. A sliding mode control proposal for open-loop unstable processes. ISA Trans.
**2004**, 43, 243–255. [Google Scholar] [CrossRef] - Galluzzo, M.; Cirino, C. Sliding mode fuzzy logic control of an unstable bioreactor. Chem. Eng. Trans.
**2013**, 32, 1213–1218. [Google Scholar] - Mehta, U.; Rojas, R. Smith predictor based sliding mode control for a class of unstable processes. Trans. Inst. Meas. Control
**2017**, 39, 706–714. [Google Scholar] [CrossRef] [Green Version] - Pandey, S.; Dourla, V.; Dwivedi, P.; Junghare, A. Introduction and realization of four fractional-order sliding mode controllers for nonlinear open-loop unstable system: A magnetic levitation study case. Nonlinear Dyn.
**2019**, 98, 601–621. [Google Scholar] [CrossRef] - Siddiqui, M.A.; Anwar, M.N.; Laskar, S.H. Sliding mode controller design for second-order unstable processes with dead-time. J. Electr. Eng.
**2020**, 71, 237–245. [Google Scholar] [CrossRef] - Kumar, S.; Ajmeri, M. Optimal variable structure control with sliding modes for unstable processes. J. Cent. South Univ.
**2021**, 28, 3147–3158. [Google Scholar] [CrossRef] - Kumar, S.; Ajmeri, M. Smith predictor–based sliding mode control with hyperbolic tangent function for unstable processes. Trans. Inst. Meas. Control
**2023**, 01423312221146338. [Google Scholar] [CrossRef] - Espín, J.; Castrillon, F.; Leiva, H.; Camacho, O. A modified Smith predictor based–Sliding mode control approach for integrating processes with dead time. Alex. Eng. J.
**2022**, 61, 10119–10137. [Google Scholar] [CrossRef] - Camacho, C.; Camacho, O. A Dynamic Sliding Mode Controller Approach for Open-Loop Unstable Systems. In Proceedings of the 2022 IEEE International Autumn Meeting on Power, Electronics and Computing (ROPEC), Ixtapa, Mexico, 9–11 November 2022; Volume 6, pp. 1–6. [Google Scholar]
- Sree, R.P.; Chidambaram, M. Identification and Controller Design for Unstable System in Control of Unstable Systems; Alpha Science Int’l Ltd.: London, UK, 2006. [Google Scholar]
- Vozäk, D.; Ilka, A. Application of Unstable System in Education of Modern Control Methods. IFAC Proc. Vol.
**2013**, 46, 114–119. [Google Scholar] [CrossRef] - Ananth, I.; Chidambaram, M. Closed-loop identification of transfer function model for unstable systems. J. Frankl. Inst.
**1999**, 336, 1055–1061. [Google Scholar] [CrossRef] - Seer, Q.H.; Nandong, J. Stabilization and PID tuning algorithms for second-order unstable processes with time-delays. ISA Trans.
**2017**, 67, 233–245. [Google Scholar] [CrossRef] - Saat, M.S.M.; Nguang, S.K.; Nasiri, A. Analysis and Synthesis of Polynomial Discrete-Time Systems: An SOS Approach; Butterworth-Heinemann: Oxford, UK, 2017. [Google Scholar]
- Irshad, M.; Ali, A. Robust PI-PD controller design for integrating and unstable processes. IFAC-PapersOnLine
**2020**, 53, 135–140. [Google Scholar] [CrossRef] - Rivera, D.E.; Morari, M.; Skogestad, S. Internal model control: PID controller design. Ind. Eng. Chem. Process Des. Dev.
**1986**, 25, 252–265. [Google Scholar] [CrossRef] - Yuwana, M.; Seborg, D.E. A new method for on-line controller tuning. AIChE J.
**1982**, 28, 434–440. [Google Scholar] [CrossRef] - Kavdia, M.; Chidambaram, M. On-line controller tuning for unstable systems. Comput. Chem. Eng.
**1996**, 20, 301–305. [Google Scholar] [CrossRef] - Marlin, T.E. Process Control: Designing Processes and Control Systems for Dynamic Performance; McGraw-Hill Science, Engineering & Mathematics: New York, NY, USA, 2000. [Google Scholar]
- Vásquez, M.; Yanascual, J.; Herrera, M.; Prado, A.; Camacho, O. A hybrid sliding mode control based on a nonlinear PID surface for nonlinear chemical processes. Eng. Sci. Technol. Int. J.
**2023**, 40, 101361. [Google Scholar] [CrossRef] - Obando, C.; Rojas, R.; Ulloa, F.; Camacho, O. Dual-Mode Based Sliding Mode Control Approach for Nonlinear Chemical Processes. ACS Omega
**2023**, 8, 9511–9525. [Google Scholar] [CrossRef] - Camacho, O.; Smith, C.A. Sliding mode control: An approach to regulate nonlinear chemical processes. ISA Trans.
**2000**, 39, 205–218. [Google Scholar] [CrossRef] [PubMed] - De Paor, A.M.; O’Malley, M. Controllers of Ziegler-Nichols type for unstable process with time delay. Int. J. Control
**1989**, 49, 1273–1284. [Google Scholar] [CrossRef] - Agrawal, P.; Lim, H.C. Analyses of various control schemes for continuous bioreactors. In Bioprocess Parameter Control; Springer: Berlin/Heidelberg, Germany, 1984; pp. 61–90. [Google Scholar]
- Moliner, R.; Tanda, R. Herramienta para la sintonía robusta de controladores PI/PID de dos grados de libertad. Rev. Iberoam. Autom. Inform. Ind.
**2016**, 13, 22–31. [Google Scholar] [CrossRef] [Green Version] - Torralba-Morales, L.; Reynoso-Meza, G.; Carrillo-Ahumada, J. Sintonización y comparación de conceptos de diseño aplicando la optimalidad de Pareto. Un caso de estudio del biorreactor de Cholette. Rev. Iberoam. Autom. Inform. Ind.
**2020**, 17, 190–201. [Google Scholar] [CrossRef] - Liptak, B.G. Process control and optimization. In Instrument Engineers’ Handbook; CRC Press: Boca Raton, FL, USA, 2018; Volume 2. [Google Scholar]
- Smith, C.A.; Corripio, A.B. Principles and Practices of Automatic Process Control; John Wiley & Sons: Hoboken, NJ, USA, 2005. [Google Scholar]

**Figure 5.**Closed-loop response of the nonlinear model and identified model using the PID controller.

**Figure 6.**Proposed control scheme. where: $R\left(s\right)$: Reference; ${U}_{1}\left(s\right)$: DSMC output; ${U}_{2}\left(s\right)$: P/PD controller output; ${U}_{t}\left(s\right)$: Total controller output; ${X}_{m}^{-}\left(s\right)$: Invertible part output; ${X}_{m}^{+}\left(s\right)$: Non-invertible part output; ${X}_{r}\left(s\right)$: Process output; $e\left(s\right)$: Overall error; ${e}_{m}\left(s\right)$: Modeling error; $X\left(s\right)$: Total output of the new system.

**Figure 10.**Radial graph of the performance indices and transient parameters for the set-point changes.

**Figure 14.**Radial graph of performance indices and transient parameters for substrate feed variations.

**Figure 18.**Radial graph of performance indices and transient parameters with substrate feed variations and set-point changes.

Parameter | Nominal Value |
---|---|

$X{p}_{1}$ | 1.391 [g/L] |

$X{m}_{1}$ | 1.143 [g/L] |

$X{p}_{2}$ | 1.208 [g/L] |

${X}_{\infty}$ | 1.194 [g/L] |

$\Delta t$ | 9.200 [h] |

${k}_{p}$ | −0.740 |

${\tau}_{i}$ | 3.300 |

${\tau}_{d}$ | 0.200 |

Controllers | |||
---|---|---|---|

Parameter | DSMC | SMC | PID |

K | −6.057 | −6.057 | - |

$\tau $ | 5.959 | 5.959 | - |

${t}_{0}$ | 1 | 1 | - |

${t}_{f}$ | 4 | - | - |

${k}_{p}$ | 0.200 | - | - |

${l}_{d}$ | 5.200 | - | - |

${l}_{d0}$ | - | 0.062 | - |

${l}_{d1}$ | - | 0.500 | - |

${K}_{c}$ | −0.637 | - | −0.347 |

${\tau}_{i}$ | - | - | 20.12 |

${\tau}_{d}$ | 0.502 | - | 1.666 |

$\u03f5$ | - | - | 4.5 |

Parameter | Nominal Value | Parameter | Nominal Value |
---|---|---|---|

$\gamma $ | 0.40 [g/g] | ${k}_{1}$ | 0.455 [g/L] |

${S}_{f}$ | 4.00 [g/L] | D | 0.300 [h${}^{-1}$] |

${\mu}_{max}$ | 0.53 [L/h] | $B\left(0\right)$ | 0.995 [g/L] |

${k}_{m}$ | 0.12 [g/L] | $S\left(0\right)$ | 1.512 [g/L] |

Controller | Indices | |||
---|---|---|---|---|

ISE | TVu | Mp [%] | Ts [h] | |

DSMC | 0.339 | 279.1 | 0.360 | 26.78 |

SMC | 0.415 | 279.2 | 3.001 | 67.28 |

PID | 0.807 | 279.2 | 1.858 | 93.03 |

Controller | Indices | |||
---|---|---|---|---|

ISE | TVu | Mp[%] | Ts [h] | |

DSMC | 0.0023 | 479.4 | 0.670 | 25.75 |

SMC | 0.0773 | 479.8 | 3.196 | 77.39 |

PID | 0.0834 | 479.8 | 2.814 | 110.06 |

**Table 6.**Performance indices and transient parameters with substrate feed variations and set-point changes.

Controller | Indices | |||
---|---|---|---|---|

ISE | TVu | Mp [%] | Ts [h] | |

DSMC | 0.347 | 294.0 | 0.343 | 26.96 |

SMC | 0.470 | 294.2 | 2.198 | 74.70 |

PID | 0.873 | 294.2 | 1.714 | 92.56 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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**

Camacho, C.; Alvarez, H.; Espin, J.; Camacho, O.
An Internal Model Based—Sliding Mode Control for Open-Loop Unstable Chemical Processes with Time Delay. *ChemEngineering* **2023**, *7*, 53.
https://doi.org/10.3390/chemengineering7030053

**AMA Style**

Camacho C, Alvarez H, Espin J, Camacho O.
An Internal Model Based—Sliding Mode Control for Open-Loop Unstable Chemical Processes with Time Delay. *ChemEngineering*. 2023; 7(3):53.
https://doi.org/10.3390/chemengineering7030053

**Chicago/Turabian Style**

Camacho, Christian, Hernan Alvarez, Jorge Espin, and Oscar Camacho.
2023. "An Internal Model Based—Sliding Mode Control for Open-Loop Unstable Chemical Processes with Time Delay" *ChemEngineering* 7, no. 3: 53.
https://doi.org/10.3390/chemengineering7030053