# Robust Decentralized Nonlinear Control for a Twin Rotor MIMO System

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## Abstract

**:**

## 1. Introduction

^{®}software environment. The dynamic behavior of the system is similar to a real helicopter, but with some differences due to the construction of the model that greatly hinder the modeling and design of control algorithms for this platform. As can be seen in Figure 1, the TRMS is formed by a base attached to a tower, at which end is a two-dimensional pivot that allows the mobile structure to rotate freely. The mobile part is composed of two metal beams: the horizontal beam in which ends the main and tail rotors with the corresponding propellers are positioned in perpendicular planes and the counterbalance beam affixed to the horizontal beam at the pivot to move the equilibrium point of the system.

^{®}, to the actual voltage value applied to each DC motor. Thus, a change in the control voltages produces a variation in the supply voltages of the motors, which results in a variation of the rotational speed of each propeller, measured by a tachometer. This way, a change in propulsive forces finally results in the movement of the platform. The movement, in the vertical and horizontal planes, is measured by two encoders that determine the pitch and yaw angles, respectively.

## 2. Dynamic Model

#### 2.1. Dynamics of the Electrical Part

- Main rotor:$$\begin{array}{ccc}\hfill {L}_{m}\frac{d{i}_{m}}{dt}& =& {v}_{m}-{k}_{{v}_{m}}{\omega}_{m}-{R}_{m}{i}_{m}\hfill \end{array}$$$$\begin{array}{ccc}\hfill {I}_{{m}_{1}}{\dot{\omega}}_{m}& =& {k}_{{t}_{m}}{i}_{m}-{f}_{{v}_{m}}{\omega}_{m}-{C}_{{Q}_{m}}{\omega}_{m}\left|{\omega}_{m}\right|\hfill \end{array}$$
- Tail rotor:$$\begin{array}{ccc}\hfill {L}_{t}\frac{d{i}_{t}}{dt}& =& {v}_{t}-{k}_{{v}_{t}}{\omega}_{t}-{R}_{t}{i}_{t}\hfill \end{array}$$$$\begin{array}{ccc}\hfill {I}_{{t}_{1}}{\dot{\omega}}_{t}& =& {k}_{{t}_{t}}{i}_{t}-{f}_{{v}_{t}}{\omega}_{t}-{C}_{{Q}_{t}}{\omega}_{t}\left|{\omega}_{t}\right|\hfill \end{array}$$

^{®}environment, defined as ${u}_{m}$ and ${u}_{t}$, respectively, and the motor terminal voltages, defined as ${v}_{m}$ and ${v}_{t}$, respectively, are nonlinear (the signals pass through a circuit interface), as was demonstrated in [10]. In our developments, it is assumed that the relationship between the control signals and the motor voltages is linear and that the differences will be canceled at the controller stage. Therefore, the relationships between the control signals and the MATLAB/Simulink

^{®}environment are the following:

#### 2.2. Dynamics of the Mechanical Part

#### 2.2.1. Evaluation of the Kinetic Energy

#### 2.2.2. Evaluation of the Potential Energy

#### 2.2.3. Lagrangian

#### 2.2.4. Generalized Forces

#### 2.2.5. Equations of Motion

## 3. Design of the Control System

#### 3.1. Inner Loop Control

^{®}environment, $\mathbf{u}\left(t\right)={[{u}_{m}\left(t\right),{u}_{t}\left(t\right)]}^{T}$, in order to reduce and eliminate the difference between the vector of angular velocities of the propellers of the TRMS, $\mathit{\omega}\left(t\right)={[{\omega}_{m}\left(t\right),{\omega}_{t}\left(t\right)]}^{T}$, and the reference vector of these angular velocities, ${\mathit{\omega}}^{*}\left(t\right)={[{\omega}_{m}^{*}\left(t\right),{\omega}_{t}^{*}\left(t\right)]}^{T}$, which is the output of the outer loop. In this sense, the feedback multivariate control input, $\mathbf{u}\left(t\right)={[{u}_{m}\left(t\right),{u}_{t}\left(t\right)]}^{T}$, is synthesized as a nonlinear input transformation and classical proportional controller with a nonlinear cancellation vector:

#### 3.2. Outer Loop Control

## 4. Experimental Section

#### 4.1. Experimental Setup

- A PC operating in a Windows
^{®}environment using software tools from MathWorks^{®}Inc (MATLAB^{®}, Simulink, Control Toolbox, Real Time Workshop^{®}(RTW), Real Time Windows Target^{®}(RTWT)) and Visual ${\mathrm{C}}^{++}$ Professional^{®}. - The real TRMS is connected to the computer by means of an Advantech
^{®}PCI1711 card, which is accessible in the MATLAB/Simulink^{®}environment through the Real-Time Toolbox^{®}. - The control signals in the MATLAB/Simulink
^{®}environment consist of two input voltages (in the range $[-2.5,2.5]$ V) for the two DC motors A-max 26 provided by Maxon Motor^{®}. - The vector of generalized coordinates, $\mathbf{q}\left(t\right)={[\psi \left(t\right),\varphi \left(t\right)]}^{T}$, are measured by using two HCTL 2016 digital encoders provided by Agilent Technologies
^{®}, and the angular velocity vector $\mathit{\omega}\left(t\right)={[{\omega}_{m}\left(t\right),{\omega}_{t}\left(t\right)]}^{T}$ is measured by using two DC-Tacho DCT 22 provided by Maxon Motor^{®}. - The sampling rate for the controlled system is $0.002$ s.

^{®}acts as the application host environment, in which the other MathWorks

^{®}products run, and Simulink

^{®}provides a well-structured graphical interface for the implementation of the proposed nonlinear control scheme. Real Time Workshop

^{®}automatically builds a ${\mathrm{C}}^{++}$ source program from the Simulink Model. The ${\mathrm{C}}^{++}$ Compiler

^{®}compiles and links the code created by Real Time Workshop

^{®}to produce an executable program. Real Time Windows Target

^{®}communicates with the executable program acting as the control program and interfaces with the TRMS through the PCI1711 card. Real Time Windows Target

^{®}controls the two-way data, or signal flow, to and from the model (which is now an executable program), and to and from the PCI1711 card. The advantage of this approach is that the designer only needs to model the process, using the graphical tools available in Simulink

^{®}, without having to worry about the mechanics of communication to and from the TRMS.

#### 4.2. Experimental Results

^{®}environment, which occur at $\pm 2.5$ V. The summary of the procedure carried out to tune the designer parameters is explained next. Firstly, the inner loop control has been tuned using the model of the electrical part of the TRMS by means of numerical simulations. In this first stage, the parameters of the proportional controller have been tuned in order to achieve the fast dynamics of the inner loop. In other words, the aim is to achieve a quick convergence of the closed loop tracking error vector, ${\mathbf{e}}_{\mathit{\omega}}\left(t\right)$, to a small vicinity around the origin of the tracking error phase space. Secondly, we have assumed the dynamics of the inner loop to be equal to ${\mathbf{I}}^{2\times 2}$, and then, we have tuned, again by means of numerical simulations, the parameters of the PID controller in the outer loop. Finally, the values obtained in the simulations have been slightly adjusted in the experimental trials with the laboratory platform. Thereby, for the inner loop controller, the values of the desired Hurtwitz $2\times 2$ complex diagonal matrix for the controller are ${\mathbf{p}}_{\mathbf{c}}^{\mathbf{e}}\left(s\right)=diag(12.0,9.0)$, and for the outer loop controller, the values of the matrices of the desired Hurtwitz polynomial vector for the feedback controller are ${\mathbf{p}}_{\mathbf{c}}^{\mathbf{m}}=diag(1.0,1.0)$, ${\mathit{\zeta}}_{\mathbf{c}}^{\mathbf{m}}=diag(1.5,1.5)$ and ${\mathit{\omega}}_{\mathbf{c}}^{\mathbf{m}}=diag(2.0,1.8)$. More details about how to tune controllers based on a cascade scheme can be consulted in some reference works [25,26,27,28].

^{®}environment, $\mathbf{u}\left(t\right)={[{u}_{m}\left(t\right),{u}_{t}\left(t\right)]}^{T}$, are shown in Figure 11. This graph illustrates that the smallest control input effort is provided by the proposed control scheme, which furthermore presents a smooth evolution of the input voltage vector without saturations unlike both PID controls, the standard PID and the PID with derivative filter coefficient. As you may observe at the top of this figure, both PID controls cause the saturation of the control signal of the main rotor, which occurs at $\pm 2.5$ V, for long periods of time during the trials. These saturations cause a worse performance of each one of these controllers in comparison with the proposed control scheme.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 7.**Real and desired evolution trajectories of the vector of the generalized coordinates of the TRMS, $\mathbf{q}\left(t\right)={[\psi \left(t\right),\varphi \left(t\right)]}^{T}$.

**Figure 8.**Evolution of the error vector of generalized coordinates of the TRMS, ${\mathbf{e}}_{\mathbf{q}}\left(\mathbf{t}\right)=\mathbf{q}\left(t\right)-{\mathbf{q}}^{*}\left(t\right)={[\psi \left(t\right)-{\psi}^{*}\left(t\right),\varphi \left(t\right)-{\varphi}^{*}\left(t\right)]}^{T}$.

**Figure 9.**Real and desired evolution trajectories of the angular velocity vector, ${\mathit{\omega}}^{*}\left(t\right)={[{\omega}_{m}^{*}\left(t\right),{\omega}_{t}^{*}\left(t\right)]}^{T}$ and $\mathit{\omega}\left(t\right)={[{\omega}_{m}\left(t\right),{\omega}_{t}\left(t\right)]}^{T}$.

**Figure 10.**Evolution of the angular velocity error vector, ${\mathbf{e}}_{\mathit{\omega}}\left(t\right)=\mathit{\omega}\left(t\right)-{\mathit{\omega}}^{*}\left(t\right)=[{\omega}_{m}\left(t\right)-{\omega}_{m}^{*}\left(t\right)$, ${\omega}_{t}\left(t\right)-{\omega}_{t}^{*}{\left(t\right)]}^{T}$.

**Figure 11.**Evolution of the input voltage vector, $\mathbf{u}\left(t\right)={[{u}_{m}\left(t\right),{u}_{t}\left(t\right)]}^{T}$, in the MATLAB/Simulink

^{®}environment.

Symbol | Parameter | Value | Units |
---|---|---|---|

Parameters of the Main Rotor | |||

${k}_{{v}_{m}}$ | Motor velocity constant | $0.0202$ | $V\xb7ra{d}^{-1}\xb7s$ |

${R}_{m}$ | Motor armature resistance | 8 | Ω |

${L}_{m}$ | Motor armature inductance | $0.86\times {10}^{-3}$ | H |

${k}_{{t}_{m}}$ | Electromagnetic constant torque motor | $0.0202$ | $N\xb7m\xb7{A}^{-1}$ |

${k}_{{u}_{m}}$ | Coefficient linear relationship interface circuit | $8.5$ | − |

${C}_{{Q}_{m}}^{+}$ | Load factor (${\omega}_{m}\ge 0$) | $2.695\times {10}^{-7}$ | $N\xb7m\xb7{s}^{2}\xb7ra{d}^{-2}$ |

${C}_{{Q}_{m}}^{-}$ | Load factor (${\omega}_{m}<0$) | $2.46\times {10}^{-7}$ | $N\xb7m\xb7{s}^{2}\xb7ra{d}^{-2}$ |

${f}_{{v}_{m}}$ | Viscous friction coefficient | $3.89\times {10}^{-6}$ | $N\xb7m\xb7ra{d}^{-1}\xb7s$ |

${I}_{m1}$ | Moment of inertia about the axis of rotation | $1.05\times {10}^{-4}$ | $kg\xb7{m}^{2}$ |

${c}_{{e}_{m}}$ | Electrical time constant (${L}_{m}/{R}_{m}$) | $1.075\times {10}^{-4}$ | s |

${c}_{{m}_{m}}$ | Mechanical time constant (${I}_{m1}{R}_{m}/{k}_{{t}_{m}}{k}_{{v}_{m}}$) | $2.058$ | s |

Parameters of the Tail Rotor | |||

${k}_{{v}_{t}}$ | Motor velocity constant | $0.0202$ | $V\xb7ra{d}^{-1}\phantom{\rule{0.166667em}{0ex}}s$ |

${R}_{t}$ | Motor armature resistance | 8 | Ω |

${L}_{t}$ | Motor armature inductance | $0.86\times {10}^{-3}$ | H |

${k}_{{t}_{t}}$ | Electromagnetic constant torque motor | $0.0202$ | $N\xb7m\xb7{A}^{-1}$ |

${k}_{{u}_{t}}$ | Coefficient linear relationship interface circuit | $6.5$ | − |

${C}_{{Q}_{t}}$ | Load factor | $1.164\times {10}^{-8}$ | $N\xb7m\xb7{s}^{2}\xb7ra{d}^{-2}$ |

${f}_{{v}_{t}}$ | Viscous friction coefficient | $1.715\times {10}^{-6}$ | $N\xb7m\xb7ra{d}^{-1}\xb7s$ |

${I}_{t1}$ | Moment of inertia about the axis of rotation | $2.1\times {10}^{-5}$ | $kg\xb7{m}^{2}$ |

${c}_{{e}_{t}}$ | Electrical time constant (${L}_{t}/{R}_{t}$) | $1.075\times {10}^{-4}$ | s |

${c}_{{m}_{t}}$ | Mechanical time constant (${I}_{t1}{R}_{t}/{k}_{{t}_{t}}{k}_{{v}_{t}}$) | $0.4117$ | s |

Symbol | Parameter | Value | Units |
---|---|---|---|

${l}_{t}$ | Length of the tail part of the free-free beam | $0.282$ | m |

${l}_{m}$ | Length of the main part of the free-free beam | $0.246$ | m |

${l}_{b}$ | Length of the counterbalance beam | $0.290$ | m |

${l}_{cb}$ | Distance between the counterweight and the joint | $0.276$ | m |

${r}_{ms}$ | Radius of the main shield | $0.155$ | m |

${r}_{ts}$ | Radius of the tail shield | $0.1$ | m |

h | Length of the pivoted beam | $0.06$ | m |

${m}_{tr}$ | Mass of the tail DC motor and tail rotor | $0.221$ | kg |

${m}_{mr}$ | Mass of the main DC motor and main rotor | $0.236$ | kg |

${m}_{cb}$ | Mass of the counterweight | $0.068$ | kg |

${m}_{t}$ | Mass of the tail part of the free-free beam | $0.015$ | kg |

${m}_{m}$ | Mass of the main part of the free-free beam | $0.014$ | kg |

${m}_{b}$ | Mass of the counterbalance beam | $0.022$ | kg |

${m}_{ts}$ | Mass of the tail shield | $0.119$ | kg |

${m}_{ms}$ | Mass of the main shield | $0.219$ | kg |

${m}_{h}$ | Mass of the pivoted beam | $0.01$ | kg |

Symbol | Parameter | Value | Units |
---|---|---|---|

Parameters of the Pitch movement | |||

${C}_{{T}_{m}}^{+}$ | Thrust torque coefficient of the main rotor (${\omega}_{m}\ge 0$) | $1.53\times {10}^{-5}$ | $N\xb7{s}^{2}\xb7ra{d}^{-2}$ |

${C}_{{T}_{m}}^{-}$ | Thrust torque coefficient of the main rotor (${\omega}_{m}<0$) | $8.8\times {10}^{-6}$ | $N\xb7{s}^{2}\xb7ra{d}^{-2}$ |

${C}_{{R}_{t}}$ | Load torque coefficient of the tail rotor | $9.7\times {10}^{-8}$ | $N\xb7m\xb7{s}^{2}\xb7ra{d}^{-2}$ |

${f}_{{v}_{\psi}}$ | Viscous friction coefficient | $0.0024$ | $N\xb7m\xb7s\xb7ra{d}^{-1}$ |

${f}_{{c}_{\psi}}$ | Coulomb friction coefficient | $5.69\times {10}^{-4}$ | $N\xb7m$ |

${k}_{t}$ | Coefficient of the inertial counter torque due to change in ${\omega}_{t}$ | $2.6\times {10}^{-5}$ | $N\xb7m\xb7{s}^{2}\xb7ra{d}^{-1}$ |

Parameters of the Yaw movement | |||

${C}_{{T}_{t}}^{+}$ | Thrust torque coefficient of the tail rotor (${\omega}_{t}\ge 0$) | $3.25\times {10}^{-6}$ | $N\xb7{s}^{2}\xb7ra{d}^{-2}$ |

${C}_{{T}_{t}}^{-}$ | Thrust torque coefficient of the tail rotor (${\omega}_{t}<0$) | $1.72\times {10}^{-6}$ | $N\xb7{s}^{2}\xb7ra{d}^{-2}$ |

${C}_{{R}_{m}}^{+}$ | Load torque coefficient of the main rotor (${\omega}_{m}\ge 0$) | $4.9\times {10}^{-7}$ | $N\xb7m\xb7{s}^{2}\xb7ra{d}^{-2}$ |

${C}_{{R}_{m}}^{-}$ | Load torque coefficient of the main rotor (${\omega}_{m}<0$) | $4.1\times {10}^{-7}$ | $N\xb7m\xb7{s}^{2}\xb7ra{d}^{-2}$ |

${f}_{{v}_{\varphi}}$ | Viscous friction coefficient | $0.03$ | $N\xb7m\xb7s\xb7ra{d}^{-1}$ |

${f}_{{c}_{\varphi}}$ | Coulomb friction coefficient | $3\times {10}^{-4}$ | $N\xb7m$ |

${c}_{c}$ | Coefficient of the elastic force torque created by the cable | $0.016$ | $N\xb7m\xb7ra{d}^{-1}$ |

${\varphi}_{0}$ | Constant for the calculation of the torque of the cable | 0 | $rad$ |

${k}_{m}$ | Coefficient of the inertial counter torque due to change in ${\omega}_{m}$ | $2\times {10}^{-4}$ | $N\xb7m\xb7{s}^{2}\xb7ra{d}^{-1}$ |

Control Method | ISE | IAE | ITAE |
---|---|---|---|

Robust Decentralized Nonlinear Control (DEC NON) | $0.3956$ | $6.6579$ | $435.7$ |

Standard PID control (PID CLASSIC) | $5.8275$ | $26.7591$ | $2002.4$ |

PID control with the derivative filter coefficient (PID DFC) | $5.0814$ | $24.8175$ | $1834.4$ |

© 2016 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**

Belmonte, L.M.; Morales, R.; Fernández-Caballero, A.; Somolinos, J.A.
Robust Decentralized Nonlinear Control for a Twin Rotor MIMO System. *Sensors* **2016**, *16*, 1160.
https://doi.org/10.3390/s16081160

**AMA Style**

Belmonte LM, Morales R, Fernández-Caballero A, Somolinos JA.
Robust Decentralized Nonlinear Control for a Twin Rotor MIMO System. *Sensors*. 2016; 16(8):1160.
https://doi.org/10.3390/s16081160

**Chicago/Turabian Style**

Belmonte, Lidia María, Rafael Morales, Antonio Fernández-Caballero, and José Andrés Somolinos.
2016. "Robust Decentralized Nonlinear Control for a Twin Rotor MIMO System" *Sensors* 16, no. 8: 1160.
https://doi.org/10.3390/s16081160