Adaptive Synergetic Controller for Stabilizing the Altitude and Angle of Mini Helicopter

This research proposes ASC (Adaptive Synergetic Controller) for the nonlinear model of MH (Mini Helicopter) to stabilize the desired altitude and angle. The model of MH is highly nonlinear, underactuated and multivariable in nature due to its dynamic uncertainties and restrictions of velocities during the flight. ASC can force the tracking errors of the system states converges to zero in a finite interval of time. The MH system requires smooth controller and fast precise transition response from initial state till the desired state, therefore the parametric calculations and simulations can be done by the proposed ASC algorithm. It is validated that the above simulated results of the proposed controller have a better convergence rate and smoother stability response in order to track the desired altitude and angle when compared with SMC (Sliding Mode Controller). Moreover, it does not need any linearization, transformation and variations in the system model.


INTRODUCTION
Formerly, many researchers, scientists pay attention on the linearization and stabilization of MH, because of its nonlinear dynamics [5].
For proper controlling, stabilizing altitude and heading angle of helicopter initially we require the stability of its nonlinear dynamics at different points commonly called as trim point conditions in control engineering [6][7][8].
Secondly, it may concern the referred position, velocity constraints and desired orientation to track given path. The breakup of this manuscript is structured as shown.
Section 2 defines the structure of MH. It is followed by the designing of the proposed controller in section 3. Section 4 discusses the convergence rate of the desired performance. The simulation result defines in section 5, which shows the robustness and desired performance of the system. Finally, section 6 outlines the conclusions.

SYSTEM MODEL
The basic helicopter model consists of four inputs called as lateral, longitudinal, pedal and collective. Two velocity components are angular, linear denoted as (p,q,r), (u,v,w), Euler angles () and its movements are (x,y,z) axis respectively, such that 'z' axis shows its altitude [25]. In this article we only consider two inputs of the MIMO model of MH and check their responses by tuning the system dynamics as per our desired altitude and collective pitch angle.
The following system model shown in Fig. 1 is taken from [18], Equations (1)(2) show the differential equations model in which vertical altitude and rotational speed of rotors are written as,

Adaptive Synergetic Controller for Stabilizing the Altitude and Angle of Mini Helicopter
Where altitude or height of helicopter is denoted by 'z' in meters that is above the ground level, the rotational speed of rotors is defined by '' (rad/s), gravitational force is 'g' (m/s 2 ), 'v c ' is the collective angle of pitch and 'C t ' is the coefficient of thrust. Where  1 ,  2 are the throttle and collective inputs which is utilized to control the altitude and angle of MH. The marginal values of the system constraints are taken from [17][18][19] are described in Table 1. with angle and at times the gravitational effects in the system in neglected [18].
Where the input control variables and the system states are defined by: There are two couplings in the dynamics of system states and the second is the control input (operational coupling).
In the following section, nonlinear model of the system is controlled by the ASC strategy to control the dynamics of the system and the results presented in this research.
Similarly Lie Bracket of (x), (x) and  2 (x) is given by:

SYNERGETIC CONTROLLER
Nonlinear systems are complex in nature, they can be controlled by adaptive, robust and synergetic control theory based on an analytical approach of aggregation [27]. The designing process of the controller begins by identifying the micro variable. In this research, an algorithm is proposed that follows an adaptive synergetic controlled which is the main contribution of this research [28]. In order to enhance the robust control design, including a continuous control law and finite time convergence of the errors in a fully non-linear system simulation [29]. The system dynamics of synergetic control along with attractors  i = 0 are shown by:

FIG. 2. THE MODEL OF MH
Where the convergence rate represented by T, is the time derivative of macro variable and reaching effect function (.) which is also equal to  Therefore the Equation (10) can be written as: The above differential equation is evaluated by: Where 't' is the time and 'T' is the convergence rate as shown above. Transient speed of the controller can be enhanced by varying 'T' inversely [30]. In the coming section, the algorithm can be made using the core parameters of synergetic controller for obtaining the roots of the expressions. The nonlinear equation of the system is already defined in Equation (3).

CONVERGENCE RATE
In accordance with dynamic Equation (10)  represented by  0 at time t 0 and the simplification of the above expression is given by: Here  t is inversely proportional to the time t i.e. as the time approaches to infinity it becomes zero, therefore the movement of the system state from the initial state to final state is possible. Furthermore, the T i in the above equation represent the convergence rate of the state variable to the final stage. If somehow the system variable is stable and the T i remains smaller, the convergence rate is faster to achieve its target. The desired output utilizes the shorter process and give better result. Where e 1 , e 2 are the error of altitude, heading angle and c ii are the constants of controller that has minor value less than (0.1).

SIMULATION RESULTS
In this section, the simulations of the proposed ASC is performed and the comparative analysis of ASC and SMC performance is shown by taking altitude=4m and angle=0.4rad as a reference signal.
In Fig. 3 the control inputs of the system are taken as  1 and  2 . Initially  1 starts from zero and converges to 200 after the delay of 1 second while  1 starts from 410 and converges to -100.
In Fig. 4, the reference altitude is set as 3m from the ground, proposed controller follows by ASC and SMC control techniques and it is found that ASC it achieves stability just after 0.6 seconds whereas SMC almost takes 1.5 seconds for the same task.