Design of a PID-Lead Compensator for a Twin Rotor Aerodynamic System (TRAS)

This paper deals with a Twin Rotor Aerodynamic System (TRAS). It is a Multi-Input MultiOutput (MIMO) system with high crosscoupling between its two channels. It proposes a hybrid design procedure that combines frequency response and root locus approaches. The proposed controller is designated as PID-Lead Compensator (PIDLC); the PID controller was designed in previous work using frequency response design specifications, while the lead compensator is proposed in this paper and is designed using the root locus method. A general explicit formula for angle computations in any of the four quadrants is also given. The lead compensator is designed by shifting the dominant closed-loop poles slightly to the left in the s-plane. This has the effect of enhancing the relative stability of the closed-loop system by eliminating the oscillation in its transient part but at the expense of greater rise time. However, for some applications, long rise time may be an allowable price to get rid of undesired oscillation. To demonstrate the proposed hybrid controller's performance numerically, a new performance index, designated by Integral Reciprocal Time Absolute Error (IRTAE), is defined as a figure to measure the oscillation of the response in its transient part. The proposed controller enhances this performance index by 0.6771%. Although the relative enhancement of the performance index is small, it contributes to eliminating the oscillation of the response in its transient part. Simulation results are performed on the MATLAB/Simulink environment.


INTRODUCTION
The Twin Rotor Aerodynamic System (TRAS) is a lab machine that behaves like a helicopter; thus, it is an important issue to design a system controller. It consists of two propellers tied to the ends of a beam. These propellers are driven by two DC motors Fig. 1. The main rotor generates vertical pushes and rotation around the horizontal axis, and the tail rotor generates horizontal pushes and rotation around the vertical axis (Netto, Lakhani and Meenatchi Sundaram, 2019).
The TRAS is a nonlinear system with high nonlinearity and high coupling interaction between its two channels. It is a Multi-Input Multi-Output (MIMO) system where it has two inputs , The voltages to the DC motors are connected to the tail, and main rotors, respectively, and have two outputs and , which are the angles from the tail and main rotors, respectively (Haruna et al.,

2019).
Designing a controller for a nonlinear MIMO system with high coupling is a complex task. Therefore some simplification has to be done, such as linearizing the system. Many attempts have been made to control the TRAS. In , an SMC scheme assisted by Generalized Proportional Integral Observers (GPI) was proposed for TRAS. The proposed technique has shown to sustain the sliding system even in unfavorable operating environments, for example, parameter fluctuations, external disturbances, nonlinear effects, and actuator failures. This illustrates the robustness and practical efficiency of the proposed strategy. The most important features of the strategy used are (a) For each decoupling loop, only the order and function ( , ) are required, (b) A single GPI observer for each uncoupling loop can be used to estimate the conditions and disturbances associated with each sliding surface, (c) The architecture of each control law must be reduced to a single coefficient deciding the dynamics of the tracking error within the sliding surface. In (Castillo, Kutlu, and Atan, 2020), two intuitionistic fuzzy controllers were designed separately, one for the main rotors and the other is for the tail rotors. The output was obtained from combining them. Because of the MIMO twin-rotor system configuration, it has many uncertainties. The intuitionistic fuzzy control provides an alternative to the optimal PID method under system uncertainties. The Simulation results showed that intuitionistic fuzzy control increases the reference signal's efficiency according to the PID system. The control method is designed to provide good performance on highly nonlinear systems, in which uncertainties are often modeled. In (Hassan, Hossam and El-Badawy, 2020), robust H ∞ controller was designed. The nonlinear mathematical model of TRAS was obtained by using Euler's law and then linearized about an equilibrium point. The proposed control strategies for trajectory tracking can be shown to be successful. In (Goyal et al., 2020), a robust Linear Matrix Inequality (LMI) based PI controller was designed for tracking control of a TRAS. The system model was represented in a linear form from nonlinear that does not involve dropping any higherorder term. The proposed controller with a decentralized structure was contributed to removing the cross-coupling between two TRAS channels. Furthermore, it was provided a better setpoint tracking result in good robust performance. The rest of this paper is organized as follows. Section 2 describes the TRAS's mathematical model; Section 3 presents an overview of the PID controller design and the design of a Lead compensator; experimental results are presented and discussed in Section 4, and concluding remarks are given in Section 5.

TRAS MODEL
A Lagrange's equations are used to derive the equations that represent the dynamics of the TRAS. The mathematical model of the TRAS is given by: Journal of Engineering Volume 27 January 2021 Number 1

82
The parameters and represent the sums of moments of inertia relative to the vertical axis and the horizontal axis, respectively; they are: = m m 2 cos 2 + t t 2 cos 2 + 2 cw cw 2 sin 2 = cos 2 + = cw cw 2 = m m 2 + t t 2 − cw cw 2 = m m 2 + t t 2 + cw cw

DESIGN OF A LEAD COMPENSATOR
The proposed hybrid controller, the PIDLC, has two parts. The first part is a PID controller, and the second part is a Lead compensator that is proposed in this paper using the root locus method. The PID controller was designed in (Abdulwahhab and Abbas, 2017) and (Abdulwahhab, 2018) using frequency response specifications for a linearized decoupled TRAS model; its nonlinear differential equation was linearized about certain operating point, and the controller was tuned based on the linearized mode. (10) The PID controller equation is: ( ) = ( ) + ∫ ( ) 0 + ( ) where , , > 0.
The control system was designed as a two SISO system, as shown in Fig. 3.  To design a lead compensator using root locus method, the following procedure was adopted: Step 1: The values of poles and zeros of the open-loop transfer function are represented in Fig. 4, then a suitable desired point (Pd), which represents a dominant closed-loop pole, is suggested to shift the critical pole to the left. Let 1 = − + , 2 = − + , 3 = − + ℎ, and 4 = − + be open-loop poles or zero.
Step 2: Determining if the angle from a pole or zero to the desired dominant closed-loop pole. It is measured counterclockwise (Ogata, 2009), as shown in Fig. 4. The 1 , 2 , 3 and 4 are : Step 3 After that, the sum of compensator poles and zeroes must be determined. From the low of the sum of the triangle's angles: So, = + (20) Step 5: The location of the zero = − c can be chosen arbitrarily. The location of the pole = − can be determined as follows: Step 6: Finally, determining the value of the lead compensator gain from the magnitude condition: In this paper, has been chosen arbitrary equal to 0.1, since this value is small enough to make the rule of the lead compensator < . Utilizing equations Eq. (23) and Eq. (24) to design the remaining compensator parameters and , respectively, the resultant compensator parameters for the two channels are shown in Table 1. Thus, the two compensators are: Using compensator design, the close loop's imaginary poles were shifted to the left to make the TRAS more stable compared with TRAS without the compensator.

SIMULATION RESULTS AND DISCUSSION
A simulation was carried out using MATLAB/Simulink tool. The closed-loop system's responses with designed PIDLC obtained, as shown in Fig. 6, where the controller was applied to the nonlinear decoupled TRAS system. From Fig. 6, it is clear that the oscillation of response was eliminated, and the relative stability of the system was increased so that the PIDLC outperforms the PID controller. As a result, the poles in close loop pulls to the left, as shown in Table 1.
where ϵ is a sufficiently small positive number. In this paper, ϵ=0.001. IRTAE is a measure of how much the system oscillates in its transient part. For the PID based control system, its value is 5.612, while for the PIDLC based system, its value is 5.574; thus, the percentage improvement in this performance index is 0.6771%.
Step response of the TRAS with PID and PIDLC for (a) Azmith angle and input voltage to the tail rotor (b) Pitch angle and input voltage to the main rotor.

CONCLUSIONS
In this paper, a hybrid controller is proposed, which is a PID-Lead compensator (PIDLC) by combining frequency response and root locus design methods. The PIDLC enhances the relative stability of the control system compared to the original PID controller. This was demonstrated by enhancing the IRTAE performance index's value, which measures the oscillation of the system's transient response. However, the proposed controller increases the rise time, which is considered a drawback. The trade-off between relative stability and rise time can be further investigated in future work.