Robust Nonlinear Control Based on Disturbance Observer for a Small-Scale Unmanned Helicopter

A robust nonlinear controller based on disturbance observer for the trajectory tracking control of a smallscale unmanned helicopter with nonlinear structure under external disturbances and parameter uncertainties is designed. The control objective is to let the helicopter track a predefined trajectory. The proposed robust nonlinear controller is based on the backstepping sliding mode control technique which combines both the capabilities of backstepping control and sliding mode control. The control performance developed based on a time-varying disturbance observer. In order to obtain an efficient control law design, the nonlinear model of the helicopter is reformulated as an affine nonlinear system. The mathematical proof using Lyapunov stability theorem shows that the closed loop system is asymptotically stable in the presence of this controller. To verify the robustness and stability of the proposed controller, it is compared with conventional sliding mode controller. The chattering phenomenon is attenuated significantly and the tracking error is also alleviated. The simulation results confirm the desirable performance of proposed robust nonlinear controller.


Introduction
In the last years, small-scale unmanned helicopters with the capabilities of vertical taking-off and landing, hovering, low-altitude cruise, and low-velocity flight have attracted more attention.These features make them suitable in military and civilian applications.However control of small-scale unmanned helicopter is a challenging concept in both the theory and experimental implementation because of its strongly nonlinear structure, underactuated nature, strong coupling, and uncertainties caused by parameter uncertainties and external disturbances.Small-scale unmanned helicopters control methods consist of two categories of control methods: linear methods and nonlinear methods [1].Linear control methods are developed based on linear models, including PID [2] and  ∞ [3].Although these linear methods are uncomplicated and reliable, they are only effective http://www.ispacs.com/journals/jnaa/2017/jnaa-00390/International Scientific Publications and Consulting Services when the system states are near the equilibrium points.Therefore, in order to overcome this major defect, many nonlinear techniques have been developed, such as feedback linearization [4], adaptive control [5], backstepping technique [6], sliding mode control [7][8].Among these methods, backstepping as a recursive technique based on Lyapunov stability analysis is effective for underactuated systems (e.g.unmanned helicopters).Variable structure control with sliding mode or as it known, sliding mode control (SMC) is a nonlinear control method that is well known for its robust characteristics.However, the conventional SMC has some disadvantages such as chattering phenomenon which would affect helicopter control efforts.In this paper a novel robust nonlinear control method based on disturbance observer is proposed for the small-scale unmanned helicopter to make the helicopter position track the desired reference trajectory in the presence of disturbance.Firstly, the helicopter model is reformulated as an affine nonlinear equation system, and then the affine nonlinear model is applied to design the controller.Then a disturbance observer is designed to estimate bounded time-varying disturbance.Using a backstepping sliding mode control technique, a robust and stable controller which considers nonlinear structure of system and all uncertainties is designed, this aids in controlling the output and tracking a given trajectory.Proposed controller stability analysis based on Lyapunov's direct stability theorem is described and proved.Moreover, the undesirable chattering phenomenon which leads to high wear of mechanical parts removed in this method.The paper is organized as follows: In section 2, the dynamical model of the small-scale unmanned helicopter is derived.Backstepping sliding mode control with disturbance observer designed in section 3. Simulation results are given in section 4 to illustrate the robust performance of the proposed controller.Finally, conclusions are drawn in section 5.

Small-Scale Unmanned Helicopter Model
The helicopter model is determined in two reference frames: the earth reference frame ℇ = {  ,   ,   ,   } and the body fixed reference frame ℬ = {  ,   ,   ,   }, the definitions of which are in accordance with [1].  .Skew symmetric, rotation and inertia matrix of helicopter model are described below.In which  and  are shorts for (.) and (.), respectively.Without loss of generality, we omit disturbance, , in continue until the simulation.
External force and torque that are the results of the main and tail rotors are detailed below, with respect to the Table 1  Collective pitches of main and tail rotors

Controller Design
In this section the concept of proposed robust nonlinear controller based on disturbance observer will be described, and applied to the small-scale unmanned helicopter's position control.

Small-Scale Unmanned Helicopter Description
In this paper, we consider small-scale unmanned helicopter model as a second order affine nonlinear system [11]  And () is time-varying external disturbance.

Disturbance Observer Design
For the system (3.7), a time-varying disturbance observer designed as follow Where  ̂ is the estimation of ,  ̂ is the estimation of  ̇, and  1 and  2 are positive constants.Select the Lyapunov function as Where  ̃=  −  ̂, and  ̃=  ̇−  ̂.
Using the designed disturbance observer, we can estimate the disturbance , and the estimation of  will be applied in the feedback control law.

Backstepping Sliding Mode Control
Let the desired output be   and considering  as the tracking error  =  −   (3.15) Step Therefore, by defining  =  1 +  2 +  3 , the closed loop control system using the proposed control law is stable by the concept of Lyapunov stability theorem.In order to eliminate the chattering phenomenon, the saturated function sat() is adopted instead of sgn() in (3.22).
Where  is the boundary layer.The boundary layer near the sliding surface ensures that the system states remain on the sliding surface.

Simulation Results
The effectiveness of the proposed robust controller is confirmed by some simulations.The helicopter's parameters are introduced from [10].The helicopter is initially at {3, -2, 0, 0, 0, 0.5}.We choose  1 = 700,  2 = 400, the boundary layer  = 0.10, and set the controller with  3 = 45 and  4 = 30.In order to check the flight control performance, we use a desired 8-shaped flight path, which is described below: if  ≤ 10 sec Flight time length is 60 seconds, and the start point is considered to be an origin on the earth.For the ease of formulation, we omit the wind disturbances, but in simulation results, it is applied to system as below: In order to show the improvement due to the proposed robust nonlinear controller with disturbance observer, the simulation results of applying this method are compared with the related results of the conventional sliding mode controller.http://www.ispacs.com/journals/jnaa/2017/jnaa-00390/ International Scientific Publications and Consulting Services   As it is seen in Figure 2, after applying the proposed robust nonlinear controller with disturbance observer, the helicopter reaches to the desired path and tracks it very well.By comparing of Figure 3 and Figure 4, it can be seen that tracking error in case of proposed controller is much less than tracking error in case of conventional SMC.Numerical indices for conventional sliding mode controller (CSMC), adaptive fuzzy sliding mode controller (AFSMC) [11], and proposed robust nonlinear controller with disturbance observer are compared in Table 2, that are defined in Table 3.  Considering to Figure 6, it can be seen that after applying the conventional sliding mode controller to the helicopter, the intensive chattering appears in the control inputs while by considering Figure 5, the control inputs due to applying proposed robust nonlinear controller is free of chattering phenomenon.The existence of chattering phenomena can be an incitement for the dynamic modes of the helicopter.

Conclusion
A robust nonlinear backstepping sliding mode control approach with disturbance observer is proposed for trajectory tracking control of a small-scale unmanned helicopter in the presence of disturbance and by considering model uncertainties.First, the helicopter's dynamical model is rewritten as a second order affine nonlinear equation, and then it is used to design the controller.Then backstepping sliding mode control method as robust and stable controller based on disturbance observer presented.Stability of closed loop control system is proved using Lyapunov theory.The proposed controller's performance is compared with conventional sliding mode controller and simulation results verified the performance of the proposed disturbance observer based control method.An important property of the proposed controller is the chattering free control efforts.

Figure 1 :
Figure 1: Schematic of the small-scale unmanned helicopter

Figure 2 :
Figure 2: 3D trajectory tracking of proposed robust nonlinear controller with disturbance observer

Figure 3 :
Figure 3: Tracking errors of proposed robust nonlinear controller with disturbance observer

Figure 4 :
Figure 4: Tracking errors of conventional sliding mode controller

Figure 5 :Figure 6 :
Figure 5: Control inputs of proposed robust nonlinear controller with disturbance observer