# Improved Active Disturbance Rejection-Based Decentralized Control for MIMO Nonlinear Systems: Comparison with The Decoupled Control Scheme

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Active Disturbance Rejection Control

#### 1.2. Related Works

#### 1.3. Paper Contribution

#### 1.4. Paper Organization

## 2. Problem Statement

- Disassociation of the state couplings.
- Disassociation of the input couplings.
- Rejects the effect of the generalized disturbance, ${F}_{i}$, $i\in \left\{1,2,\mathrm{..},p\right\},$ on the outputs.
- Maintaining an acceptable performance during both transient and steady-state responses.

## 3. Decoupled Control [23]

## 4. Main Results

#### 4.1. The Proposed Decentralized Scheme Based on ADRC

#### 4.2. The Improved ADRC (IADRC)

#### 4.3. Closed-Loop Stability Analysis

**Assumption**

**1.**

**Assumption**

**2.**

**Theorem**

**1.**

**Closed–Loop Stability.**Given the MIMO nonlinear uncertain system of Equation (10), which has a linearization control law (LCL)${\mathrm{u}}_{\mathrm{i}}^{*}$of the form,

**Proof.**

## 5. Numerical Simulations

**1. First configuration: The Conventional ADRC (CADRC)**

- Conventional TD, described by Reference [62]:$$\{\begin{array}{c}{\dot{r}}_{i,1}={r}_{i,2},\\ {\dot{r}}_{i,2}=-{R}_{i}sign\left({r}_{i,1}-{r}_{i}+\frac{{r}_{i,2}\left|{r}_{i,2}\right|}{2{R}_{i}}\right),i\in \left\{1,2\right\}\end{array}$$
- fal-based control law, given as:$$\{\begin{array}{l}{u}_{1}^{*}={k}_{1,1}fal\left({\tilde{e}}_{1,1},{\alpha}_{1,1},{\delta}_{1,1}\right)+{k}_{1,2}fal\left({\tilde{e}}_{1,2},{\alpha}_{1,2},{\delta}_{1,2}\right)-{\widehat{\xi}}_{1,3},\\ {u}_{2}^{*}={k}_{2,1}fal\left({\tilde{e}}_{2,1},{\alpha}_{2,1},{\delta}_{2,1}\right)+{k}_{2,2}fal\left({\tilde{e}}_{2,2},{\alpha}_{2,2},{\delta}_{2,2}\right)-{\widehat{\xi}}_{2,3}.\end{array}$$
- The LESO, given as follows [62]:$$\{\begin{array}{l}{\dot{\widehat{\xi}}}_{i,1}={\widehat{\xi}}_{i,2}+3{\omega}_{o,i}\left({y}_{i}-{\widehat{\xi}}_{i,1}\right),\\ {\dot{\widehat{\xi}}}_{i,2}={\widehat{\xi}}_{i,3}+{u}_{i}^{*}+3{\omega}_{o,i}^{2}\left({y}_{i}-{\widehat{\xi}}_{i,1}\right),\\ {\dot{\widehat{\xi}}}_{i,3}={\omega}_{o,i}^{3}\left({y}_{i}-{\widehat{\xi}}_{i,1}\right),i\in \left\{1,2\right\}\end{array}$$

**2. Second configuration: The Improved ADRC (IADRC)**

- Conventional TD is given by Equation (43).
- fal-based control law given by Equation (44).
- A novel NHOESO, proposed as:$$\{\begin{array}{l}{\dot{\widehat{\xi}}}_{i,1}={\widehat{\xi}}_{i,2}+{a}_{i,1}{\omega}_{o,i}^{1}{\u210a}_{i}\left({y}_{i}-{\widehat{\xi}}_{i,1}\right),\\ {\dot{\widehat{\xi}}}_{i,2}={\widehat{\xi}}_{i,3}+{u}_{i}^{*}+{a}_{i,2}{\omega}_{o,i}^{2}{\u210a}_{i}\left({y}_{i}-{\widehat{\xi}}_{i,1}\right),\\ {\dot{\widehat{\xi}}}_{i,3}={\widehat{\xi}}_{i,4}+{a}_{i,3}{\omega}_{o,i}^{3}{\u210a}_{i}\left({y}_{i}-{\widehat{\xi}}_{i,1}\right),\\ {\dot{\widehat{\xi}}}_{i,4}={a}_{i,4}{\omega}_{o,i}^{4}{\u210a}_{i}\left({y}_{i}-{\widehat{\xi}}_{i,1}\right),i\in \left\{1,2\right\}\end{array}$$$${\u210a}_{i}\left({e}_{i}\right)={K}_{i,\alpha}{\left|{e}_{i}\right|}^{{\alpha}_{i}}sign\left({e}_{i}\right)+{K}_{i,\beta}{\left|{e}_{i}\right|}^{{\beta}_{i}}{e}_{i}$$

#### 5.1. Results of the Decoupled ADRC Control Scheme [23]

**Case (1): Output Tracking**

_{1,}where it slightly reduced from its value in the CADRC. This has been reflected in the control efforts ${u}_{1}$ and ${u}_{2},$shown in Figure 6c and Figure 7c, where ${u}_{1}$ and ${u}_{2}$ for the IADRC witnessed less activity than the CADRC. The tracking output response for the IADRC was better than in the CADRC, specifically during the transient period, where both configurations entirely attenuated the effect of the exogenous disturbances ${w}_{1}$ and ${w}_{2}$, the state couplings for each subsystem, and the time-varying input gains ${g}_{1,1}$, ${g}_{1,2}$, ${g}_{2,1}$, and ${g}_{2,2}$ on the output response of the two channels.

**Case (2): Input and State Decoupling**

#### 5.2. Results of the Decentralized ADRC Control Scheme

**Case (1): Output tracking**

**Case (2): Input and State Decoupling**

#### 5.3. Comparison between the Proposed Decentralized Scheme and the Decoupled Control [23]

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Han, J. A Class of Extended State Observers for Uncertain Systems. Control Decis.
**1995**, 10, 85–88. [Google Scholar] - Gao, Z. Scaling and bandwidth-parameterization based controller tuning. In Proceedings of the American Control Conference, Denver, CO, USA, 4–6 June 2003; pp. 4989–4996. [Google Scholar] [CrossRef]
- Humaidi, A.J.; Ibraheem, I.K. Speed Control of Permanent Magnet DC Motor with Friction and Measurement Noise Using Novel Nonlinear Extended State Observer-Based Anti-Disturbance Control. Energies
**2019**, 12, 1651. [Google Scholar] [CrossRef] [Green Version] - Ibraheem, I.K.; Abdul-Adheem, W.R. A Novel Second-Order Nonlinear Differentiator with Application to Active Disturbance Rejection Control. In Proceedings of the 1st International Scientific Conference of Engineering Sciences—3rd Scientific Conference of Engineering Science (ISCES), Diyalah, Iraq, 10–11 January 2018; pp. 68–73. [Google Scholar] [CrossRef] [Green Version]
- Ibraheem, I.K.; Abdul-Adheem, W.R. An Improved Active Disturbance Rejection Control for a Differential Drive Mobile Robot with Mismatched Disturbances and Uncertainties. arXiv
**2018**, arXiv:1805.12170. [Google Scholar] - Chen, Z.; Wang, Y.; Sun, M.; Sun, Q. Convergence and stability analysis of active disturbance rejection control for first-order nonlinear dynamic systems. Trans. Inst. Meas. Control
**2019**, 41, 2064–2076. [Google Scholar] [CrossRef] - Cheng, Y.; Chen, Z.; Sun, M.; Sun, Q. Cascade Active Disturbance Rejection Control of a High-Purity Distillation Column with Measurement Noise. Ind. Eng. Chem. Res.
**2018**, 57, 4623–4631. [Google Scholar] [CrossRef] - Hou, Y.; Gao, Z.; Jiang, F.; Boulter, B.T. Active Disturbance Rejection Control for Web Tension Regulation. In Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, USA, 4–7 December 2001; pp. 4974–4979. [Google Scholar] [CrossRef] [Green Version]
- Su, Y.X.; Zheng, C.H.; Sun, D.; Duan, B.Y. A simple nonlinear velocity estimator for high-performance motion control. IEEE Trans. Ind. Electron.
**2005**, 52, 1161–1169. [Google Scholar] [CrossRef] - Sun, B.; Gao, Z. A DSP-Based Active Disturbance Rejection Control Design for a 1-kW H-Bridge DC-DC Power Converter. IEEE Trans. Ind. Electron.
**2005**, 52, 1271–1277. [Google Scholar] [CrossRef] [Green Version] - Zheng, Q.; Chen, Z.; Gao, Z. A Dynamic Decoupling Control Approach and Its Applications to Chemical Processes. In Proceedings of the American Control Conference, New York, NY, USA, 11–13 July 2007; pp. 5176–5181. [Google Scholar] [CrossRef]
- Zheng, Q.; Dong, L.; Gao, Z. Control and rotation rate estimation of vibrational MEMS gyroscopes. In Proceedings of the International Conference on Control Applications, Singapore, 1–3 October 2007; pp. 118–123. [Google Scholar] [CrossRef]
- Goforth, F.J.; Gao, Z. An Active Disturbance Rejection Control Solution for Hysteresis Compensation. In Proceedings of the American Control Conference, Seattle, WA, USA, 11–13 June 2008; pp. 2202–2208. [Google Scholar] [CrossRef]
- Yan, B.; Tian, Z.; Shi, S.; Weng, Z. Fault diagnosis for a class of nonlinear systems via ESO. ISA Trans.
**2008**, 47, 386–394. [Google Scholar] [CrossRef] - Wu, D.; Chen, K. Design and analysis of precision active disturbance rejection control for noncircular turning process. IEEE Trans. Ind. Electron.
**2009**, 56, 2746–2753. [Google Scholar] [CrossRef] - Li, S.; Yang, X.; Yang, D. Active disturbance rejection control for high pointing accuracy and rotation speed. Automatica
**2009**, 45, 1854–1860. [Google Scholar] [CrossRef] - Sira-Ramirez, H.; Lopez-Uribe, C.; Velasco-Villa, M. Linear observer-based active disturbance rejection control of the omnidirectional mobile robot. Asian J. Control
**2013**, 15, 51–63. [Google Scholar] [CrossRef] - Leonard, F.; Martini, A.; Abba, G. Robust nonlinear controls of model-scale helicopters under lateral and vertical wind gusts. IEEE Trans. Control Syst. Technol.
**2012**, 20, 154–163. [Google Scholar] [CrossRef] - Madoński, R.; Kordasz, M.; Sauer, P. Application of a disturbance-rejection controller for robotic-enhanced limb rehabilitation trainings. ISA Trans.
**2014**, 53, 899–908. [Google Scholar] [CrossRef] [PubMed] - Texas Instruments. Technical Reference Manual for TMS320F28069M, TMS320F28068M InstaSPIN-MOTION Software; Texas Instruments: Dallas, TX, USA, 2014; pp. 1–57. [Google Scholar]
- Zheng, Q.; Gao, Z. On Practical Applications of Active Disturbance Rejection Control. In Proceedings of the 29th Chinese Control Conference, Beijing, China, 29–31 July 2010; pp. 6095–6100. [Google Scholar]
- Yi, H.; Wenchao, X.U.E.; Gao, Z.; Sira-ramirez, H.; Dan, W.; Mingwei, S. Active Disturbance Rejection Control: Methodology, Practice and Analysis. In Proceedings of the 33rd Chinese Control Conference, Nanjing, China, 28–30 July 2014; pp. 1–5. [Google Scholar] [CrossRef]
- Abdul-adheem, W.R.; Ibraheem, I.K. Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme. Alex. Eng. J.
**2019**, 58, 1145–1156. [Google Scholar] [CrossRef] - Elmali, H.; Olgac, N. Robust output tracking control of nonlinear MIMO systems via sliding mode technique. Automatica
**1992**, 28, 145–151. [Google Scholar] [CrossRef] - Costa, R.R.; Hsu, L.; Imai, A.K.; Kokotovic, P. Lyapunov-based adaptive control of MIMO systems. Automatica
**2003**, 39, 1251–1257. [Google Scholar] [CrossRef] - Liu, X.; Jutan, A.; Rohani, S. Almost disturbance decoupling of MIMO nonlinear systems and application to chemical processes. Automatica
**2004**, 40, 465–471. [Google Scholar] [CrossRef] - Bagni, G.; Basso, M.; Genesio, R.; Tesi, A. Synthesis of MIMO controllers for extending the stability range of periodic solutions in forced nonlinear systems. Automatica
**2005**, 41, 645–654. [Google Scholar] [CrossRef] - Mizumoto, I.; Chen, T.; Ohdaira, S.; Kumon, M.; Iwai, Z. Adaptive output feedback control of general MIMO systems using multirate sampling and its application to a cart—Crane system. Automatica
**2007**, 43, 2077–2085. [Google Scholar] [CrossRef] - Gündes, A.N.; Mete, A.N.; Palazoglu, A. Reliable decentralized PID controller synthesis for two-channel MIMO processes. Automatica
**2009**, 45, 353–363. [Google Scholar] [CrossRef] - Chen, M.; Ge, S.S.; Ren, B. Adaptive tracking control of uncertain MIMO nonlinear systems with input constraints. Automatica
**2011**, 47, 452–465. [Google Scholar] [CrossRef] - Chiu, C.S. Derivative and integral terminal sliding mode control for a class of MIMO nonlinear systems. Automatica
**2012**, 48, 316–326. [Google Scholar] [CrossRef] - Lu, L.; Yao, B. Online constrained optimization based adaptive robust control of a class of MIMO nonlinear systems with matched uncertainties and input/state constraints. Automatica
**2014**, 50, 864–873. [Google Scholar] [CrossRef] - Hashemi, M.; Ghaisari, J.; Askari, J. Adaptive control for a class of MIMO nonlinear time delay systems against time varying actuator failures. ISA Trans.
**2015**, 57, 23–42. [Google Scholar] [CrossRef] [PubMed] - Zhu, J.; Yu, X.; Zhang, T.; Cao, Z.; Yang, Y.; Yi, Y. Sliding mode control of MIMO Markovian jump systems. Automatica
**2016**, 68, 286–293. [Google Scholar] [CrossRef] - Pandey, V.K.; Kar, I.; Mahanta, C. Controller design for a class of nonlinear MIMO coupled system using multiple models and second level adaptation. ISA Trans.
**2017**, 69, 256–272. [Google Scholar] [CrossRef] - Wang, L.; Isidori, A.; Liu, Z.; Su, H. Robust output regulation for invertible nonlinear MIMO systems. Automatica
**2017**, 82, 278–286. [Google Scholar] [CrossRef] - Feng, Y.; Zhou, M.; Zheng, X.; Han, F.; Yu, X. Full-order terminal sliding-mode control of MIMO systems with unmatched uncertainties. J. Frankl. Inst.
**2018**, 355, 653–674. [Google Scholar] [CrossRef] - Zhao, K.; Song, Y.; Zhang, Z. Tracking control of MIMO nonlinear systems under full state constraints: A Single-parameter adaptation approach free from feasibility conditions. Automatica
**2019**, 107, 52–60. [Google Scholar] [CrossRef] - Li, J.; Du, J. Adaptive disturbance cancellation for a class of MIMO nonlinear Euler–Lagrange systems under input saturation. ISA Trans.
**2020**, 96, 14–23. [Google Scholar] [CrossRef] - Chen, M.; Mei, R.; Jiang, B. Sliding mode control for a class of uncertain MIMO nonlinear systems with application to near-space vehicles. Math. Probl. Eng.
**2013**, 2013, 180589. [Google Scholar] [CrossRef] - Chen, M.; Shi, P.; Lim, C.C. Robust Constrained Control for MIMO Nonlinear Systems Based on Disturbance Observer. IEEE Trans. Automat. Contr.
**2015**, 60, 3281–3286. [Google Scholar] [CrossRef] - Floquet, T.; Spurgeon, S.K.; Edwards, C. An output feedback sliding mode control strategy for MIMO systems of arbitrary relative degree. Int. J. Robust Nonlinear Control
**2011**, 21, 119–133. [Google Scholar] [CrossRef] - Polyakov, A.; Efimov, D.; Perruquetti, W. Sliding Mode Control Design for MIMO Systems: Implicit Lyapunov Function Approach. In Proceedings of the 2014 European Control Conference (ECC), Strasbourg, France, 24–27 June 2014; pp. 2612–2617. [Google Scholar] [CrossRef] [Green Version]
- Ibraheem, I.K. Anti-Disturbance Compensator Design for Unmanned Aerial Vehicle. J. Eng.
**2020**, 26, 86–103. [Google Scholar] [CrossRef] [Green Version] - Khalil, H.K. Extended High-Gain Observers as Disturbance Estimators. SICE J. Control. Meas. Syst. Integr.
**2017**, 10, 125–134. [Google Scholar] [CrossRef] - Tong Shaocheng, T.; Bin, C.; Yongfu, W. Fuzzy adaptive output feedback control for MIMO nonlinear systems. Fuzzy Sets Syst.
**2005**, 156, 285–299. [Google Scholar] [CrossRef] - Kim, E.; Lee, S. Output feedback tracking control of MIMO systems using a fuzzy disturbance observer and its application to the speed control of a PM synchronous motor. IEEE Trans. Fuzzy Syst.
**2005**, 13, 725–741. [Google Scholar] - Zhang, T.P.; Yang, Y.I. Adaptive Fuzzy Control for a Class of MIMO Nonlinear Systems with Unknown Dead-zones. Acta Autom. Sin.
**2007**, 33, 96–99. [Google Scholar] [CrossRef] - Chen, C.H.; Lin, C.M.; Chen, T.Y. Intelligent adaptive control for MIMO uncertain nonlinear systems. Expert Syst. Appl.
**2008**, 35, 865–877. [Google Scholar] [CrossRef] - Yousef, H.; Hamdy, M.; El-Madbouly, E.; Eteim, D. Adaptive fuzzy decentralized control for interconnected MIMO nonlinear subsystems. Automatica
**2009**, 45, 456–462. [Google Scholar] [CrossRef] - Liu, Y.J.; Tong, S.C.; Li, T.S. Observer-based adaptive fuzzy tracking control for a class of uncertain nonlinear MIMO systems. Fuzzy Sets Syst.
**2011**, 164, 25–44. [Google Scholar] [CrossRef] - Li, T.; Li, R.; Wang, D. Adaptive neural control of nonlinear MIMO systems with unknown time delays. Neurocomputing
**2012**, 78, 83–88. [Google Scholar] [CrossRef] - Li, C.; Wang, W. Fuzzy almost disturbance decoupling for MIMO nonlinear uncertain systems based on high-gain observer. Neurocomputing
**2013**, 111, 104–114. [Google Scholar] [CrossRef] - Zhen, H.T.; Qi, X.H.; Li, J.; Tian, Q.M. Neural network L1 Adaptive control of MIMO systems with nonlinear uncertainty. Sci. World J.
**2014**, 2014, 1–8. [Google Scholar] [CrossRef] [Green Version] - Rong, H.J.; Wei, J.T.; Bai, J.M.; Zhao, G.S.; Liang, Y.Q. Adaptive neural control for a class of MIMO nonlinear systems with extreme learning machine. Neurocomputing
**2015**, 149, 405–414. [Google Scholar] [CrossRef] - Yan, P.; Liu, D.; Wang, D.; Ma, H. Data-driven controller design for general MIMO nonlinear systems via virtual reference feedback tuning and neural networks. Neurocomputing
**2016**, 171, 815–825. [Google Scholar] [CrossRef] - Shi, W. Observer-based adaptive fuzzy prescribed performance control for feedback linearizable MIMO nonlinear systems with unknown control direction. Neurocomputing
**2019**, 368, 99–113. [Google Scholar] [CrossRef] - Huo, X.; Ma, L.; Zhao, X.; Zong, G. Event-triggered adaptive fuzzy output feedback control of MIMO switched nonlinear systems with average dwell time. Appl. Math. Comput.
**2020**, 365, 124665. [Google Scholar] [CrossRef] - Wang, L.Y.; Zhang, J.F. Fundamental Limitations and Differences of Robust and Adaptive Control. In Proceedings of the 2001 American Control Conference, Arlington, VA, USA, 25–27 June 2001; pp. 4802–4807. [Google Scholar]
- Hespanha, J.P.; Liberzon, D.; Morse, A.S. Overcoming the limitations of adaptive control by means of logic-based switching. Syst. Control Lett.
**2003**, 49, 49–65. [Google Scholar] [CrossRef] [Green Version] - Mandoski, R. On Active Disturbance Rejection in Robotic Motion Control. Ph.D. Thesis, University of Poznan, Poznań, Poland, 2016. [Google Scholar] [CrossRef]
- Han, J. From PID to active disturbance rejection control. IEEE Trans. Ind. Electron.
**2009**, 56, 900–906. [Google Scholar] [CrossRef] - Abdul-adheem, W.R.; Ibraheem, I.K. Improved Sliding Mode Nonlinear Extended State Observer based Active Disturbance Rejection Control for Uncertain Systems with Unknown Total Disturbance. Int. J. Adv. Comput. Sci. Appl.
**2016**, 7, 80–93. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Structure of the conventional SISO active disturbance rejection control (ADRC) configuration, $\rho $ is the nonlinear system’s relative degree.

**Figure 5.**The proposed decentralized ADRC control scheme, where each ADRC block involves a SISO connection of TD, NLSEF, and ESO, as in Figure 1.

**Figure 6.**The output response of Equation (42) using CADRC configuration, (

**a**) output ${y}_{1}$, (

**b**) output ${y}_{2}$, (

**c**) control signals ${u}_{1}$ and ${u}_{2}$, and (

**d**) estimated generalized disturbances ${\widehat{\xi}}_{1,3}$ and ${\widehat{\xi}}_{2,3}$ [23].

**Figure 7.**The output response of Equation (42) using IADRC, (

**a**) output ${y}_{1}$, (

**b**) output ${y}_{2}$, (

**c**) control signals ${u}_{1}$ and ${u}_{2}$, and (

**d**) estimated generalized disturbances ${\widehat{\xi}}_{1,3}$ and ${\widehat{\xi}}_{2,3}$ [23].

**Figure 8.**The output tracking of Equation (42) due to reference inputs ${\mathit{r}}_{1}$ and ${\mathit{r}}_{2}$ using a decoupled control scheme with CADRC, (

**a**) $\left({\mathit{r}}_{1},{\mathit{r}}_{2}\right)=\left(\mathbf{sin}\left(\mathit{t}\right),0\right)$; (

**b**) $\left({\mathit{r}}_{1},{\mathit{r}}_{2}\right)=\left(0,\mathbf{cos}\left(t\right)\right)$ [23].

**Figure 9.**The output tracking of Equation (42) due to reference inputs ${\mathit{r}}_{1}$ and ${\mathit{r}}_{2}$ using a decoupled control scheme with IADRC, (

**a**) $\left({\mathit{r}}_{1},{\mathit{r}}_{2}\right)=\left(\mathbf{sin}\left(\mathit{t}\right),0\right)$; (

**b**) $\left({\mathit{r}}_{1},{\mathit{r}}_{2}\right)=\left(0,\mathbf{cos}\left(\mathit{t}\right)\right)$ [23].

**Figure 10.**The output response of Equation (42) using CADRC, (

**a**) output ${y}_{1}$, (

**b**) output ${y}_{2}$, (

**c**) control signals ${u}_{1}$ and ${u}_{2}$, and (

**d**) estimated generalized disturbances ${\widehat{\xi}}_{1,3}$ and ${\widehat{\xi}}_{2,3}$.

**Figure 11.**The output response of (42) using IADRC, (

**a**) output curve ${y}_{1}$, (

**b**) output curve ${y}_{2}$, (

**c**) control signals ${u}_{1}$ and ${u}_{2}$, and (

**d**) estimated generalized disturbances ${\widehat{\xi}}_{1,3}$ and ${\widehat{\xi}}_{2,3}$.

**Figure 12.**The output tracking of Equation (42) due to reference inputs ${r}_{1}$ and ${r}_{2}$ using decentralized control scheme with CADRC, (

**a**) $\left({r}_{1},{r}_{2}\right)=\left(\mathrm{sin}\left(t\right),0\right)$, (

**b**) $\left({r}_{1},{r}_{2}\right)=\left(0,\mathrm{cos}\left(t\right)\right)$.

**Figure 13.**The output tracking of (42) due to reference inputs ${r}_{1}$ and ${r}_{2}$, using a decentralized control scheme with IADRC, (

**a**) $\left({r}_{1},{r}_{2}\right)=\left(\mathrm{sin}\left(t\right),0\right)$; (

**b**) $\left({r}_{1},{r}_{2}\right)=\left(0,\mathrm{cos}\left(t\right)\right)$.

**Table 1.**The parameters of the first configuration (conventional ADRC (CADRC)) [23].

Unit | First Channel Parameters | Second Channel Parameters | ||
---|---|---|---|---|

Parameter | Value | Parameter | Value | |

TD | ${R}_{1}$ | 92.2713 | ${R}_{2}$ | 88.4424 |

LESO | ${\omega}_{o,1}$ | 68.3308 | ${\omega}_{o,2}$ | 53.1690 |

fal-based control law | ${\delta}_{1,1}$ | 0.0010 | ${\delta}_{2,1}$ | 0.14456 |

${\delta}_{1,2}$ | 0.2834 | ${\delta}_{2,2}$ | 0.73456 | |

${\alpha}_{1,1}$ | 0.1629 | ${\alpha}_{2,1}$ | 0.02730 | |

${\alpha}_{1,2}$ | 0.7946 | ${\alpha}_{2,2}$ | 0.93745 | |

${k}_{1,1}$ | 12.8015 | ${k}_{2,1}$ | 18.3095 | |

${k}_{1,2}$ | 11.2999 | ${k}_{2,2}$ | 19.52670 | |

${\delta}_{1}$ | 40 | ${\delta}_{2}$ | 40 |

**Table 2.**The parameters of the second configuration (improved active disturbance rejection control (IADRC)) [23].

Unit | First Channel Parameters | Second Channel Parameters | ||
---|---|---|---|---|

Parameter | Value | Parameter | Value | |

TD | ${R}_{1}$ | 192.7715 | ${R}_{2}$ | 148.9279 |

NHOESO | ${\omega}_{o,1}$ | 135.6086 | ${\omega}_{o,2}$ | 22.8802 |

${a}_{1,1}$ | 2.31423 | ${a}_{2,1}$ | 3.3264 | |

${a}_{1,2}$ | 4.5361 | ${a}_{2,2}$ | 4.66885 | |

${a}_{1,3}$ | 2.0465 | ${a}_{2,3}$ | 1.48218 | |

${a}_{1,4}$ | 0.1658 | ${a}_{2,4}$ | 0.04076 | |

${K}_{1,\alpha}$ | 0.9000 | ${K}_{2,\alpha}$ | 0.9000 | |

${\alpha}_{1}$ | 0.9000 | ${\alpha}_{2}$ | 0.9000 | |

${K}_{1,\beta}$ | 0.1000 | ${K}_{2,\beta}$ | 0.1000 | |

${\beta}_{1}$ | 0.0100 | ${\beta}_{2}$ | 0.0100 | |

fal-based control law | ${\delta}_{1,1}$ | 0.0341 | ${\delta}_{2,1}$ | 0.0082 |

${\delta}_{1,2}$ | 0.6008 | ${\delta}_{2,2}$ | 0.8162 | |

${\alpha}_{1,1}$ | 0.0207 | ${\alpha}_{2,1}$ | 0.0120 | |

${\alpha}_{1,2}$ | 0.3372 | ${\alpha}_{2,2}$ | 0.7222 | |

${k}_{1,1}$ | 18.3186 | ${k}_{2,1}$ | 6.84822 | |

${k}_{1,2}$ | 8.9993 | ${k}_{2,2}$ | 6.5260 | |

${\delta}_{1}$ | 40 | ${\delta}_{2}$ | 40 |

**Table 3.**Performance of the decoupled ADRC scheme [23].

Performance Index | CADRC | IADRC | %Reduction |
---|---|---|---|

ITAE_{1} | 0.1628 | 0.1210 | 25.7% |

ITAE_{2} | 0.3536 | 0.0937 | 73.5% |

ISU_{1} | 314.1064 | 308.4248 | 1.8% |

ISU_{2} | 296.8865 | 225.7019 | 24% |

Unit | First Channel Parameters | Second Channel Parameters | ||
---|---|---|---|---|

Parameter | Value | Parameter | Value | |

TD | ${R}_{1}$ | 92.2713 | ${R}_{2}$ | 88.4423 |

LESO | ${\omega}_{\mathrm{o},1}$ | 68.3308 | ${\omega}_{\mathrm{o},2}$ | 53.1690 |

${b}_{1,1}$ | 1.0000 | ${b}_{2,2}$ | −1.0000 | |

fal-based Control law | ${\delta}_{1,1}$ | 0.0010 | ${\delta}_{2,1}$ | 0.1445 |

${\delta}_{1,2}$ | 0.2834 | ${\delta}_{2,2}$ | 0.7346 | |

${\alpha}_{1,1}$ | 0.1629 | ${\alpha}_{2,1}$ | 0.0273 | |

${\alpha}_{1,2}$ | 0.7946 | ${\alpha}_{2,2}$ | 0.9375 | |

${k}_{1,1}$ | 12.8015 | ${k}_{2,1}$ | 18.3095 | |

${k}_{1,2}$ | 11.2999 | ${k}_{2,2}$ | 19.5267 | |

${\delta}_{1}$ | 40 | ${\delta}_{2}$ | 40 |

Unit | First Channel | Second Channel | ||
---|---|---|---|---|

Parameter | Value | Parameter | Value | |

TD | ${R}_{1}$ | 155.2564 | ${R}_{2}$ | 107.6494 |

NHOESO | ${\omega}_{o,1}$ | 94.9942 | ${\omega}_{\mathrm{o},2}$ | 123.7601 |

${b}_{1,1}$ | 1.0000 | ${b}_{2,2}$ | −1.0000 | |

${a}_{1,1}$ | 1.7315 | ${a}_{2,1}$ | 3.6546 | |

${a}_{1,2}$ | 5.0845 | ${a}_{2,2}$ | 3.8128 | |

${a}_{1,3}$ | 1.5151 | ${a}_{2,3}$ | 2.0353 | |

${a}_{1,4}$ | 1.1444 × 10^{−6} | ${a}_{2,4}$ | 1.1230 × 10^{−6} | |

${K}_{1,\alpha}$ | 0.8028 | ${K}_{2,\alpha}$ | 0.5043 | |

${\alpha}_{1}$ | 0.9300 | ${\alpha}_{2}$ | 0.6982 | |

${K}_{1,\beta}$ | 0.2381 | ${K}_{2,\beta}$ | 0.8338 | |

${\beta}_{1}$ | 0.6221 | ${\beta}_{2}$ | 0.9534 | |

fal-based Control law | ${\delta}_{1,1}$ | 0.1250 | ${\delta}_{2,1}$ | 0.2510 |

${\delta}_{1,2}$ | 0.4163 | ${\delta}_{2,2}$ | 0.4531 | |

${\alpha}_{1,1}$ | 0.2750 | ${\alpha}_{2,1}$ | 0.3312 | |

${\alpha}_{1,2}$ | 0.7658 | ${\alpha}_{2,2}$ | 0.2783 | |

${k}_{1,1}$ | 25.6305 | ${k}_{2,1}$ | 30.3227 | |

${k}_{1,2}$ | 10.6899 | ${k}_{2,2}$ | 20.2694 | |

${\delta}_{1}$ | 40 | ${\delta}_{2}$ | 40 |

Performance Index | CADRC | IADRC | %Reduction |
---|---|---|---|

ITAE_{1} | 0.3890 | 0.3081 | 20.8% |

ITAE_{2} | 0.6434 | 0.4600 | 28.5% |

ISU_{1} | 181.5489 | 123.6903 | 31.9% |

ISU_{2} | 302.3266 | 265.2197 | 12.3% |

CARDC | IADRC | ||||
---|---|---|---|---|---|

Decoupled | Decentralized | %Reduction | Decoupled | Decentralized | %Reduction |

610.9921 | 483.8755 | 20.8 | 534.1267 | 388.91 | 27.18 |

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Abdul-Adheem, W.R.; Ibraheem, I.K.; Azar, A.T.; Humaidi, A.J.
Improved Active Disturbance Rejection-Based Decentralized Control for MIMO Nonlinear Systems: Comparison with The Decoupled Control Scheme. *Appl. Sci.* **2020**, *10*, 2515.
https://doi.org/10.3390/app10072515

**AMA Style**

Abdul-Adheem WR, Ibraheem IK, Azar AT, Humaidi AJ.
Improved Active Disturbance Rejection-Based Decentralized Control for MIMO Nonlinear Systems: Comparison with The Decoupled Control Scheme. *Applied Sciences*. 2020; 10(7):2515.
https://doi.org/10.3390/app10072515

**Chicago/Turabian Style**

Abdul-Adheem, Wameedh Riyadh, Ibraheem Kasim Ibraheem, Ahmad Taher Azar, and Amjad J. Humaidi.
2020. "Improved Active Disturbance Rejection-Based Decentralized Control for MIMO Nonlinear Systems: Comparison with The Decoupled Control Scheme" *Applied Sciences* 10, no. 7: 2515.
https://doi.org/10.3390/app10072515