# Robust Controller Design for Multi-Input Multi-Output Systems Using Coefficient Diagram Method

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- Converts the compensator design problem used for decoupling into parameter optimization problems to reduce the difficulty of decoupling.
- (2)
- Gives an interaction measurement to quantify the interaction of coupled systems.
- (3)
- Analyzes the controller’s suppression effect on measurement noise based on the CDM.
- (4)
- Research on the application of CDM methods to MIMO systems needs to be promoted. In order to make up for the shortcomings of existing research, this paper presents a design strategy of a robust controller based on CDM, which provides a reference for the design of the MIMO system controller in other articles.

## 2. Decoupling Design

#### 2.1. Compensator Design

**Example**

**1.**

#### 2.2. Interaction Measurement

**Remark**

**1.**

#### 2.3. Parameter Tuning of Compensator

## 3. CDM Controller Design and Measurement Noise Rejection

#### 3.1. CDM Controller Design

#### 3.2. Measurement Noise Rejection

## 4. Overall Design Ideas

- 1.
- A characteristic polynomial and controller are simultaneously designed. The characteristic polynomial specifies stability and response. The structure of the controller guarantees robustness. Thus, a simple controller, which satisfies the stability, response, and robustness requirements, can be designed with ease.
- 2.
- Compared with PID control that needed to develop different tuning methods for the process with various properties, it is sufficient to use a single design procedure in the CDM technique. This is an outstanding advantage.

- 1.
- Set the SISO controller parameters $A\left(s\right)$ and $B\left(s\right)$. ${k}_{0}$ = 0 is a good choice for measurement noise suppression.
- 2.
- Select CDM design parameters. The stability index ${\gamma}_{i}$ in this paper is in the Manabe standard form of Equation (20). As long as the value of the equivalent time constant $\tau $ is determined, the controller parameters can be obtained. The $\tau $ value mainly determines the response time of the system. Generally, the $\tau $ value is determined according to the design requirements of the system setting time and bandwidth.
- 3.
- Solve the SISO controller parameters. $A\left(s\right)$, $B\left(s\right)$ can be obtained by Equations (17) and (21). $F\left(s\right)$ can be obtained by Equation (16).

**Remark**

**2.**

## 5. Simulation Experiment

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Liu, L.; Yin, S.; Zhang, L.; Yin, X.; Yan, H. Improved results on asymptotic stabilization for stochastic nonlinear time-delay systems with application to a chemical reactor system. IEEE Trans. Syst. Man Cybern. Syst.
**2016**, 47, 195–204. [Google Scholar] [CrossRef] - Kumar, S.D.; Chandramohan, D.; Purushothaman, K.; Sathish, T. Optimal hydraulic and thermal constrain for plate heat exchanger using multi objective wale optimization. Mater. Today Proc.
**2020**, 21, 876–881. [Google Scholar] [CrossRef] - Meng, F.; Song, P.; Liu, K. PID–P compound control of flexible transmission system with sandwich structure. Control Theory Appl.
**2020**, 37, 2432–2440. [Google Scholar] - Komareji, M.; Shang, Y.; Bouffanais, R. Consensus in topologically interacting swarms under communication constraints and time-delays. Nonlinear Dyn.
**2018**, 93, 1287–1300. [Google Scholar] [CrossRef] - Meng, F.; Wang, D.; Yang, P.; Xie, G. Application of Sum of Squares Method in Nonlinear H infinity Control for Satellite Attitude Maneuvers. Complexity
**2019**, 2019, 5124108. [Google Scholar] [CrossRef] [Green Version] - Lee, J.; Hyun Kim, D.; Edgar, T.F. Static decouplers for control of multivariable processes. AIChE J.
**2005**, 51, 2712–2720. [Google Scholar] [CrossRef] - Hagglund, T. The one-third rule for PI controller tuning. Comput. Chem. Eng.
**2019**, 127, 25–30. [Google Scholar] [CrossRef] - Diaz-Rodriguez, I.D.; Han, S.; Bhattacharyya, S.P. PID Control of Multivariable Systems. Anal. Des. PID Controll.
**2019**, 217–231. [Google Scholar] [CrossRef] - Coelho, M.S.; da Silva Filho, J.I.; Côrtes, H.M.; de Carvalho, A., Jr.; Blos, M.F.; Mario, M.C.; Rocco, A. Hybrid PI controller constructed with paraconsistent annotated logic. Control Eng. Pract.
**2019**, 84, 112–124. [Google Scholar] [CrossRef] - Liao, Q.; Sun, D. Sparse and decoupling control strategies based on takagi-sugeno fuzzy models. IEEE Trans. Cybern.
**2019**, 51, 947–960. [Google Scholar] [CrossRef] - Luan, X.L.; Wang, Z.Q.; Liu, F. Centralized PI control for multivariable non-square systems. Control Decis.
**2016**, 31, 811–816. [Google Scholar] - Meng, F.; Pang, A.; Dong, X.; Han, C.; Sha, X. H infinity optimal performance design of an unstable plant under bode integral constraint. Complexity
**2018**, 2018, 4942906. [Google Scholar] [CrossRef] [Green Version] - Manabe, S. Coefficient diagram method. IFAC Proc. Vol.
**1998**, 31, 211–222. [Google Scholar] [CrossRef] - Coelho, J.P.; Pinho, T.M.; Boaventura-Cunha, J. Controller system design using the coefficient diagram method. Arab. J. Sci. Eng.
**2016**, 41, 3663–3681. [Google Scholar] [CrossRef] - Hariz, M.B.; Bouani, F.; Ksouri, M. Robust controller for uncertain parameters systems. ISA Trans.
**2012**, 51, 632–640. [Google Scholar] [CrossRef] [PubMed] - Abtahi, S.F.; Yazdi, E.A. Robust control synthesis using coefficient diagram method and m-analysis: An aerospace example. Int. J. Dyn. Control
**2019**, 7, 595–606. [Google Scholar] [CrossRef] - Kumar, M.; Hote, Y.V. Maximum sensitivity-constrained coefficient diagram method-based PIDA controller design: Application for load frequency control of an isolated microgrid. Electr. Eng.
**2021**. [Google Scholar] [CrossRef] - Mohamed, T.H.; Shabib, G.; Ali, H. Distributed load frequency control in an interconnected power system using ecological technique and coefficient diagram method. Int. J. Electr. Power Energy Syst.
**2016**, 82, 496–507. [Google Scholar] [CrossRef] - Ma, C.; Cao, J.; Qiao, Y. Polynomial-method-based design of low-order controllers for two-mass systems. IEEE Trans. Ind. Electron.
**2012**, 60, 969–978. [Google Scholar] [CrossRef] - Banu, U.S.; Aparna, V.; Hussain, M. Coefficient diagram method based control for two interacting conical tank process. In Proceedings of the 2017 Trends in Industrial Measurement and Automation (TIMA), Chennai, India, 6–8 January 2017; pp. 1–5. [Google Scholar]
- Mitsantisuk, C.; Nandayapa, M.; Ohishi, K.; Katsura, S. Design for sensorless force control of flexible robot by using resonance ratio control based on coefficient diagram method. Automatika
**2013**, 54, 62–73. [Google Scholar] [CrossRef] - Derrac, J.; García, S.; Molina, D.; Herrera, F. A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evol. Comput.
**2011**, 1, 3–18. [Google Scholar] [CrossRef] - Manabe, S. Sufficient condition for stability and instability by Lipatov and its application to the coefficient diagram method. In Proceedings of the 9th Workshop on Astrodynamics and Flight Mechanics, Sagamihara, Japan, 22–23 July 1999; Volume 440, p. 449. [Google Scholar]
- Ahrens, J.H.; Khalil, H.K. High-gain observers in the presence of measurement noise: A switched-gain approach. Automatica
**2009**, 45, 936–943. [Google Scholar] [CrossRef] - Oppenheim, A.V.; Willsky, A.S.; Hamid Nawab, S. Signals & Systems; Prentice-Hall: Upper Saddle River, NJ, USA, 1997. [Google Scholar]
- Vajta, M. Some remarks on Pade-approximations. In Proceedings of the 3rd TEMPUS-INTCOM Symposium, Veszprém, Hungary, 9–14 September 2000; Volume 242. [Google Scholar]
- Sasaki, M.; Mori, K.; Li, X. PID Controller Design of MIMO Systems by using Coefficient Diagram Method. Essays Soc. Electron.
**2012**, 132, 1465–1472. [Google Scholar] - Vadigepalli, R.; Gatzke, E.P.; Doyle, F.J., III. Robust control of a multivariable experimental four-tank system. Ind. Eng. Chem. Res.
**2001**, 40, 1916–1927. [Google Scholar] [CrossRef] - Viknesh, R.; Sivakumaran, N.; Sarat Chandra, J.; Radhakrishnan, T.K. A critical study of decentralized controllers for a multivariable system. Chem. Eng. Technol.
**2004**, 27, 880–889. [Google Scholar] [CrossRef]

**Figure 8.**Step response curve of the original system (29) without decoupling design. (

**a**) ${x}_{1}\ne 0$; (

**b**) ${x}_{2}\ne 0$.

**Figure 9.**Step response curve of the original system (29) with decoupling design. (

**a**) ${A}_{1}$; (

**b**) ${A}_{2}$.

**Figure 12.**Use the PID controller in [27] and CDM controller to control the decoupling system $Q\left(s\right)$ step response. (

**a**) controlled variable ${x}_{1}$; (

**b**) controlled variable ${x}_{2}$.

**Figure 13.**Use the method in this paper to control the step response of the original system (29) with the step disturbance.

Evolutionary Algorithms | Parameter Settings |
---|---|

Genetic Algorithm (GA) | Population size = 50 |

The times of iterations = 100 | |

Crossover probability = 0.9 | |

Mutation probability = 0.1 | |

Shuffled Frog Leaping Algorithm (SFLA) | Population size = 50 |

The times of iterations = 100 | |

Moving maximum distance = 0.02 | |

Cuck Search (CS) | Population size = 50 |

The times of iterations = 100 | |

Maximum discovery probability = 0.05 | |

Particle Swarm Optimization (PSO) | Population size = 50 |

The times of iterations = 100 | |

The weight of inertia = 0.35 | |

The self-learning factor = 1.5 | |

The population-learning factor=2.5 |

Fmax | Fmin | Fave | Fstd | Time (s) | |
---|---|---|---|---|---|

GA | 0.8885 | −0.2030 | 0.0859 | 0.2911 | 0.2099 |

SFLA | 3.9953 | −0.2095 | 0.5383 | 1.1147 | 0.2389 |

CS | 0.1941 | −0.2124 | −0.1844 | 0.0845 | 0.1347 |

PSO | −0.1908 | −0.2128 | −0.2117 | 0.0039 | 0.0756 |

Friedman Ranks | Friedman Aligned Ranks | Quade Ranks | |
---|---|---|---|

GA | 1.9 | 7.32 | 1.81 |

SFLA | 2.5 | 10.7 | 2.41 |

CS | 1.2 | 5.6 | 1.38 |

PSO | 1.1 | 4.5 | 1.10 |

System A_{1} | System A_{2} | |
---|---|---|

$\tau $ | 11.2 | 16 |

$F\left(s\right)$ | 1.6811 | 0.8358 |

$A\left(s\right)$ | $0.0313{s}^{2}+0.0855s$ | $0.0921{s}^{2}+0.0928s$ |

$B\left(s\right)$ | $\begin{array}{c}6.9322{s}^{2}+6.1273s+1.681\hfill \end{array}$ | $\begin{array}{c}25.3384{s}^{2}+7.9861s+0.83576\hfill \end{array}$ |

Ref. [27] | Proposed Method | |
---|---|---|

evaluating value | 0.048587 | 0.0000067 |

**Table 6.**Performance values of the time response curves shown in Figure 12.

Settling Time | Max Overshoot % | |
---|---|---|

Masaya-${y}_{1}$ | 43 | 15 |

CDM-${y}_{1}$ | 41 | 13 |

Masaya-${y}_{2}$ | 172 | 1.5 |

CDM-${y}_{2}$ | 26 | 0 |

System A_{1} | System A_{2} | |
---|---|---|

$\tau $ | 120 | 100 |

$F\left(s\right)$ | 3.0833 | 2.8187 |

$A\left(s\right)$ | $69.2884{s}^{2}+4.1572s+0.9019$ | $75.3466{s}^{2}+3.8905s+0.9513$ |

$B\left(s\right)$ | $-3231.1{s}^{2}-109.6204s$ | $-3056.7{s}^{2}-88.0734s$ |

System A_{1} | System A_{2} | |
---|---|---|

$\tau $ | 76 | 64 |

$F\left(s\right)$ | 0.0017 | 0.00028 |

$A\left(s\right)$ | $0.3665{s}^{2}-0.0016s$ | $0.0015{s}^{2}+0.000555s$ |

$B\left(s\right)$ | $\begin{array}{c}0.2464{s}^{2}+0.1321s+0.0017\hfill \end{array}$ | $\begin{array}{c}0.0728{s}^{2}+0.0102s+0.0002773\hfill \end{array}$ |

System A_{1} | System A_{2} | System A_{3} | |
---|---|---|---|

$\tau $ | 38.67 | 100 | 68 |

$F\left(s\right)$ | −5.4186 | 3.6895 | 0.5582 |

$A\left(s\right)$ | $0.0011{s}^{2}+0.0169s$ | $\begin{array}{c}0.0253{s}^{3}+0.08397{s}^{2}+12.9567s\hfill \end{array}$ | $0.0027{s}^{2}+0.0055s$ |

$B\left(s\right)$ | $\begin{array}{c}-25.92{s}^{2}-19.0082s\hfill \\ -5.4186\hfill \end{array}$ | $\begin{array}{c}41568.98{s}^{3}+8205.15{s}^{2}\hfill \\ +447.8735s+3.6895\hfill \end{array}$ | $\begin{array}{c}9.8979{s}^{2}+3.4319s\hfill \\ +0.5582\hfill \end{array}$ |

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

© 2021 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**

Liu, K.; Meng, F.; Meng, S.; Wang, C.
Robust Controller Design for Multi-Input Multi-Output Systems Using Coefficient Diagram Method. *Entropy* **2021**, *23*, 1180.
https://doi.org/10.3390/e23091180

**AMA Style**

Liu K, Meng F, Meng S, Wang C.
Robust Controller Design for Multi-Input Multi-Output Systems Using Coefficient Diagram Method. *Entropy*. 2021; 23(9):1180.
https://doi.org/10.3390/e23091180

**Chicago/Turabian Style**

Liu, Kai, Fanwei Meng, Shengya Meng, and Chonghui Wang.
2021. "Robust Controller Design for Multi-Input Multi-Output Systems Using Coefficient Diagram Method" *Entropy* 23, no. 9: 1180.
https://doi.org/10.3390/e23091180