Robust control design for path tracking of non-affine UAV

ABSTRACT Path tracking of Unmanned Aerial Vehicles (UAVs) with three degrees of freedom is studied in this paper with approach of dynamic sliding mode control. For this purpose, the equations of UAV are written. The difficulty and complexity of these equations is that they are non-affine with respect to control inputs. Moreover, they are not directly in the inertial coordinate system while the desired flight path is given in the inertial coordinate system. These two major problems add complexity to the design procedure. Therefore, it is necessary that equations be rewritten in the inertial coordinate system. By definition of virtual inputs; the equations convert to affine structure with respect to virtual inputs and the transformation between the real and virtual inputs has been obtained. After that, the Input/output (I-O) equations of the system are written and converted into controller canonical form. The dynamic sliding mode control law is then designed based on (I-O) equations. Optimal coefficients are also achieved numerically by considering an appropriate cost function. Finally, computer simulation is utilized to illustrate the performance of the designed controller.

Among the nonlinear robust control methods, the dynamic sliding mode technique has especial characteristics. This technique is a robust technique in stabilization of nonlinear systems with uncertainty and has greater benefits than the classical sliding mode technique (Wang, Bao, & Li, 2017Yan, Spurgeon, & Edwards, 2005. Dynamic sliding mode controller is designed based on output information and requires no observer designing. This CONTACT Tahereh Binazadeh binazadeh@sutech.ac.ir characteristics makes the dynamic sliding mode method as an efficient robust controller for practical implementations. However; designing the autopilot for path tracking, based on dynamic sliding mode technique for UAV with three degrees of freedom is not simply possible because of the two main problems. First, UAV equations are nonaffine with respect to their inputs. Moreover, the dynamical equations are not directly in the inertial coordinate system while the desired flight path is given in the inertial coordinate system. Robust autopilot designing for path tracking of UAV with three degrees of freedom is investigated in this paper. The main contributions of this paper are: new autopilot design for putting UAV on the predetermined desired flight path based on dynamic sliding mode method and tuning the coefficients with the optimality approach. For this purpose, after rewriting the dynamical equations in the inertial coordinate system, the virtual input vector is defined such that the state space equations have affine structure with respect to virtual inputs. Also, the transformation between the real and virtual inputs are obtained. Then considering the error vector of position as the output vector, the (I-O) equations of the system have been written and converted into controller canonical form. The obtained canonical subsystems have suitable structures which are adequate for design of sliding surfaces. These surfaces contain coefficients which affect the transient response and performance of the system. The optimal values of these coefficients are obtained by definition of an appropriate cost function and using a numerical algorithm. Finally, computer simulations are performed to show the performance of the designed control law.

UAV equations in three-dimensional space
UAV point mass model with three degrees of freedom is considered as follows which is a nonlinear and multivariable model (Boškovic, Chen, & Mehra, 2004).
where airspeed (V), flight path (γ ) and flight path-heading (θ) are system state variables and thrust force (T), load factor (n) and bank angle (μ) represent control inputs. D is the representation of drag force that can be described as follows: where, W, C D0 , k, ρ, S 0 , and g 0 are UAV weight, parasite drag coefficient, induced drag coefficient, density, reference area and acceleration due to the Earth gravity constant, respectively.

Obtain appropriate equations for tracking
Equation (1) shows that motion components are not directly in the inertial coordinate system. In order to achieve the control target in path tracking, it is necessary to define the state variables based on the inertial coordinate system. On the other hand, the Equation (1) has non-affine structure with respect to their inputs (T,n, and μ). These are two main difficulties that will be solved in the following.

4-1-motion equations in inertial coordinate system
The position vector of UAV is shown with x = [ x y z ] T where x, y and z are longitudinal, transverse and vertical positions of UAV in the inertial coordinate system, respectively. Derivative of the position vector is the velocity vector ( ẋ). Velocity vector in the space of inertial coordinate system has three components of longitudinal speed, transverse speed and vertical speed. Therefore, the components of velocity vector are obtained as Equation (3) and are shown in Figure 1.
where h is a nonlinear vector function of state vector q. Now by applying the chain rule in differentiation of Equation (3), one has: Substituting Equation (1) in Equation (4), yields: Thus, one has: The above phrase is composed of two sentences. After multiplying, the result of the first sentence is as follows: Thus the acceleration equations are obtained as follows: Therefore Equation (1) has been converted into Equation (6) which show the UAV motion equations in the inertial coordinate system. Equation (6) is still non-affine with respect to control inputs (i.e. T,n, μ). This problem will be solved in the following.

4-2-convert the equation to affine structure
The structure of Equation (6) is non-affine with respect to their control inputs. To obtain the affine structure for UAV model, the virtual input vector is defined as follows.
where a g is as follows: Although, Equation (7) has the affine structure with respect to the virtual control vector u * , the real inputs are T,nand μ and the goal is designing the real inputs. In order to solve this difficulty, after designing u * , the real inputs have been obtained from the following relations: In which, a = 0.5ρS C D0 and b = 2kW 2 /ρS.

Designing optimal dynamic sliding mode control for UAV
As stated before, the task is tracking the desired flight path by UAV. The position vector of the desired flight path is stated as The task is designing the virtual control law u * in such a way that After that the real inputs will be achieved according to Equations (8). Consequently, the error vector of position (i.e. e(t) = x(t) − x d (t)) is considered as the output vector of the system. The control objective is asymptotic convergence of output vector toward the zero.

I-O equations for UAV motion
Conversion of equations into the controllable canonical form is the next step for implementation of dynamic sliding mode method. For this purpose, input/output model of system should be obtained. Define output vector as follows; By differentiation from each output and obtaining n i (where n i is the relative degree related to ith output for i = 1, 2, 3), one has:ẏ 1 =ẋ −ẋ d y 1 =ẍ −ẍ d In Equation (6),ẍ,ÿ,z are elements of ẍ vector. Thus, u 1 * appears inÿ 1 and therefore, n 1 = 2 is the relative degree of the first output. By differentiating the second and third output, one has: Considering Equation (6), u 2 * appears inÿ 2 and u * 3 appears inÿ 3 , hence, n 2 = n 3 = 2.

Controller canonical equations
To transfer I-O equations to controllable canonical form, the following state variables are define: Now, the I-O equations may be rewritten in the following three subsystems which have the controllable canonical structure: First subsystem: Second subsystem: Third subsystem: and i s represent uncertain terms due to parametric uncertainties or external disturbances. It is assumed that the upper bound of | i | is known and where ξ (i) = [ξ (i) 1 , ξ (i) 2 ] T . Since system (10) has three inputs (u * 1 , u * 2 and u * 3 ), and three subsystem, thus three sliding surfaces should be designed. The sliding surfaces are designed such that the reduced-order model of each subsystem (which is the motion equations on the related sliding surface) be asymptotically stable.
Without loss of generally, considering a The positive constants a (i) 1 s results in asymptotic stability of motion equations on the sliding surfaces (i.e. 1 is the reduced order model of the ith subsystem and shows the motion equation on ith surface). By differentiating the sliding surfaces, one has: Now, the following equation which is the reaching condition will be solved to get the control law u * (see more details about the reaching condition in (Edwards & Spurgeon, 2002)).
This results in where where the free parameters k ij , k 0i and a (i) 1 (for i = 1, 2, 3 and j = 1, 2, 3) will be chosen with an optimal numerical solver (like PSO or genetic toolbox of Matlab) such that the following cost function be minimized.

Simulations
In    Also the desired position vector is considered as follows: Considering the cost function J = t f 0 ( y T Q y + u T R u u) dt, with Q = diag(1, 3, 1) and R u = diag(10, 10, 10)the optimal parameters are obtained (using Matlab genetic toolbox) as follows.  Figure 2. As can be seen, x(t) → x d (t), y(t) → y d (t) and z(t) → z d (t) and therefore tracking of reference signals is well performed despite presence of external disturbances and uncertainties of the model. Moreover, the desired and actual flight paths are shown in Figure 3. This figure illustrates the ability of the designed control law in tracking of the desired flight path. The time-responses of sliding surfaces are illustrated in Figure 4. As seen the system's trajectories reach the sliding surfaces in finite-time.

Conclusion
This paper has focused on dynamic sliding mode controller design for UAV with three-degree-of-freedom.
Firstly, the motion equations of UAV were written. Then the dynamical model was rewritten in the inertial coordinate system. Moreover, by define of virtual input vector, the equations was converted to the affine structure with respect to virtual control inputs and the transformation between the real inputs and virtual inputs was given. By definition of output vectors as the difference between the current position in the inertial system and the desired one, the I-O equations were achieved and rewritten in canonical controller form which was adequate for designing sliding surfaces. Furthermore, dynamic sliding mode control law has been calculated based on the designed sliding surfaces and the desired reaching law. Additionally, the appropriate cost function was given and free coefficients in control law were obtained by optimality approach. Computer simulations shown the performance of the proposed control law.

Disclosure statement
No potential conflict of interest was reported by the authors.