Design of Preset Performance Reverse Step Attitude Controller for Four ‐ rotor UAV

: Aiming at the attitude control problem in flight control of four-rotor UAV, a control scheme based on the preset performance backstepping controller (ESO-NPPCBSC) for attitude Angle of four-rotor UAV with extended state observer is proposed. A preset performance function with specified time convergence is designed to constrain the transient performance and steady-state performance of the tracking error. Compared with the traditional scheme, the preset performance function can make the tracking error of the controlled system converge to the preset precision range within the specified time, and the convergence rate can be adjusted flexibly and the error conversion function can be used to transform the tracking error without constraints. The attitude Angle tracking error satisfies the preset performance condition by controlling the conversion error. The control law of attitude Angle of four-rotor UAV is designed based on reverse step method, which solves the problem of low control accuracy of four-rotor UAV under the condition of uncertain interference.


Introduction
In recent years, the design, development and application of four-rotor Unmanned Aerial Vehicles (UAVs) have attracted the attention of scholars around the world [1]. Compared with fixed wing UAVs, four-rotor uAVs have advantages such as simple structure, low manufacturing cost, easy maintenance, light fuselage and easy manipulation. In addition, four-rotor UAVs also have excellent maneuverability, environmental durability, hover, vertical takeoff and landing [2], which makes four-rotor uAVs widely used in military, agricultural, commercial, civil and scientific research and other related fields [3].
Four-rotor UAV is a control system with multi-input multioutput (MIMO), high nonlinear and underdriven internal parameter coupling degree [4]. At the same time, the dynamics modeling of quadrotor UAV contains many unknown interference factors, such as model parameter change, model mismatch, external environmental interference, etc. [5]. Therefore, it is of high practical value and practical significance to design a reliable flight control algorithm to solve flight control problems related to poor anti-interference ability and low control accuracy of quadrotor UAV in actual operation [6].
Sun Zidong [7] et al. designed an improved backstepping control algorithm combining fuzzy control and dynamic surface control, which not only approached the unknown function appearing in the system, but solved the complexity explosion problem existing in the backstepping design process. By introducing a performance function, the tracking error of the controlled system converges in a finite time. Dalwadi Niha l [8] et al. designed a backstepping controller based on nonlinear observer. The backstepping controller can achieve good trajectory tracking when the nominal altitude drops, and improve the control quality of the quadrotor UAV during the transition from hovering state to horizontal flight state. Ma TieNan [9] et al. designed a single nonlinear extended state observer by using the feedback linearization technique to reduce the complexity of observer parameter adjustment. A preset performance control method of the four-rotor UAV under the attitude and input saturation constraints is designed to improve the stability of the controlled system. Davood Allahverd [10] et al. designed an inverse integral sliding mode control (BISMC) based on iterative learning control, which improved the problems existing in the nonlinear translational and rotational dynamics of the fourrotor UAV and enabled the controlled system to have high transient and steady-state performance. RodriguezAbreo Omar [11] and others through a based on genetic algorithm is designed to step of parameters self-tuning control algorithm, the algorithm is based on four rotor uav autonomous trajectory tracking to backstepping controller gain, implements the gain of different magnitude error self-tuning. Chen Yanjie [12] et al. designed an adaptive sliding mode interference observer (ASMDO) and designed a finite time preset performance control method on the basis of ASMDO to realize effective control of UAV manipulator with uncertainty and external interference. The system converges in finite time and has specified transient and steady-state properties.
Based on the inspiration of the above literature, this paper aims to design a preset performance control scheme whose convergence time can be artificially set for small quadrotor UAV as the research object, and verifies the control performance of this scheme in the presence of incident interference through numerical simulation. The main contributions are as follows: An extended observer is designed to dynamically compensate the complex interference. A preset performance function whose convergence time can be artificially specified is proposed. Based on this, a preset performance control law based on backward step method is designed, which can make the tracking error of the controlled system converge within the specified time and the convergence speed can be adjusted according to the requirements.

System Modeling and Problem Description
Driven by four propellers, four-rotor UAV can realize six degrees of freedom movement in three-dimensional space. Therefore, the mathematical model of four-rotor UAV is a typical multi-input, multi-output and strongly coupled underactuated controlled system. In this paper, Newton-Euler formula is adopted to deduce the attitude dynamics of the four-rotor UAV [13]. The following hypothesis is given: (1) Quadrotor UAV is a uniform and symmetrical rigid body; (2) The mass and moment of inertia of the four-rotor UAV do not change; (3) The geometric center of the quadrotor UAV coincides with its own center; (4) Quadrotor UAV only accepts gravity and propeller pull. According to Newton-Euler formula, the attitude dynamics equation of the four-rotor UAV is established as follows: Where,  ,  and  represent the roll Angle, pitch Angle and yaw Angle of the attitude Angle system of the fourrotor UAV respectively. d  , d  and d  respectively represents the interference of roll Angle system, the interference of pitch Angle system and Interference of yaw Angle system. x I , y I and z I respectively represent the moment of inertia of the four-rotor UAV with respect to the x axis, the moment of inertia with respect to the y axis and the moment of inertia with respect to the z axis. U  , U  and U  respectively represent the control input of the three attitude angles of the four-rotor UAV.
In order to facilitate the subsequent controller design, the original four-rotor UAV attitude Angle system dynamics model needs to be converted into n order integral series dynamics model with strict feedback form. In order to simplify the design process, a symbol representation method as shown in Equation (2) is proposed before model transformation: where, 1 X , 2 X and 2 2 K U represents the control input of attitude Angle, attitude angular velocity and yaw Angle of the four-rotor UAV respectively. Based on Formula (1) and combined with Formula (2), the state space equation of the integral series attitude Angle system of the four-rotor UAV can be described as: Where, 2 d represents the sum of external airflow disturbance, unmodeled system dynamics and model uncertainty influence disturbances received by the four-rotor UAV.
Due to the under-actuation characteristic of the four-rotor UAV, the designed controller cannot track the 6 degrees of freedom at the same time. The controller designed in this paper can ensure no attitude control of UAV. The humanmachine attitude Angle stably tracks the expected input

Default Performance Function and Error Transform
Presetting performance means that the transient and steadystate performance of the system is limited within the range of the pre-designed performance function by designing a conversion function, so as to improve the transient and steady-state performance [14]. The default performance method can be expressed as the following inequality: that can converge in a finite time is selected as follows: Where, (0,1) r  ,   By inverting equation (7), the expression of conversion error ( ) t  can be obtained as follows: According to different control requirements, reasonable error conversion function ( ( )) s z t and error conversion function ( ) z t are designed, the transient performance and steady-state performance of the UAV attitude Angle system can be set in advance so that the tracking error of the system is always limited within the error ( For equation (9), make   ,Then formula (9) can be rewritten as:

Preset Performance Backstepping Controller Design
On the basis of given virtual control quantity, attitude closed-loop control can be regarded as a linear control. By using the idea of error conversion, the tracking error is guaranteed to be in the preset boundary function, and the extended state observer is designed to estimate the sum of the uncertain influence interference, and then the adaptive backstepping controller is designed.
Step 1: Conversion error 1 e is introduced into the attitude Angle tracking error 1 ( ) z t of the four-rotor UAV, then the tracking error equation of the system can be described as: 1 1 ( ) ( ) r e t z t p p    (11) Where, p and r p represents the actual attitude Angle and the expected attitude Angle respectively. Based on the rolling Angle channel tracking error equation of Equation (11), the Lyapunov function is constructed as follows: The derivative of equation (12) is obtained: Where, 2 U represents the virtual control input of the attitude Angle system of the four-rotor UAV. In order to ensure the tracking error of attitude Angle is stable, A must be satisfied 1 1 ( ) 0 V e   . Based on Equation (13), the virtual control law of the stable attitude Angle system satisfying Lyapunov's sense is designed as follows:

r t e t p t e t v t
If equation (14) is substituted into Equation (13), equation (13) can be rewritten as: Step 2: The error equation for defining the tracking speed of the attitude Angle system of the four-rotor UAV is as follows: 2 2 r e p p p U        (16) Where, p  , r p  Respectively represent the actual attitude Angle tracking speed. The degree and desired attitude Angle track the velocity. Aiming at the error equation of attitude Angle tracking velocity, the Lyapunov function is constructed as follows: The derivative of Equation (17)  , where 0 M  . Finally, the augmented statespace equation of the attitude Angle system of a four-rotor UAV can be described as: is used to represent formula (4), then formula (4) can be rewritten as:  . The comparison of attitude Angle tracking response curves of ESO-NPPCBSC controller and PID controller is shown in Figure 1 to Figure 3:    The experimental results show that, compared with PID controller, the attitude Angle tracking response curve of the four-rotor UAV based on ESO-NPPCBSC controller has faster response time, smaller overshoot and shorter adjustment time, and the convergence time can be manually specified by adjusting preset performance parameters. Compared with the ESO-BSC without preset performance, the attitude Angle tracking response curve of the four-rotor UAV using the ESO-NPPCBSC controller can be limited within the envelope of preset performance, and the static difference after reaching the steady state is relatively small.