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In this paper, a real-time online data-driven adaptive method is developed to deal with uncertainties such as high nonlinearity, strong coupling, parameter perturbation and external disturbances in attitude control of fixed-wing un manned aerial vehicles (UAVs). Firstly, a model-free adaptive control (MFAC) method requiring only input/output (I/O) data and no model information is adopted for control scheme design of angular velocity subsystem which contains all model information and up-mentioned uncertainties. Secondly, the internal model control (IMC) method featured with less tuning parameters and convenient tuning process is adopted for control scheme design of the certain Euler angle subsystem. Simulation results show that, the method developed is obviously superior to the cascade PID (CPID) method and the nonlinear dynamic inversion (NDI) method.

The performance of the attitude controller of a fixed-wing UAV determines the quality of its autonomous flight. Some accurate mathematical model-based methods were proposed for the attitude control, for example, PID and LQR methods (linearized model based) [

Though these methods have achieved their purpose, there are still some drawbacks. Firstly, as the UAV system is strongly coupled, highly nonlinear and time variant, etc., approximated models cannot fully represent all characteristics of plant model. In the meantime, accurate model is hard to obtain, too costly or even inapplicable; secondly, designing controller is complex and tuning is tedious due to too many parameters.

A data-driven control method for designing attitude control law of fixed-wing UAV is thus developed. The control system is divided into two cascade subsystems, including an inner loop system for angular velocity control and an outer loop system for Euler angle control. As the angular velocity control system contains all model information (certainties and uncertainties), a novel data-driven MFAC [

Consider the following general single-input-single-output (SISO) nonlinear system:

Σ : y ( k + 1 ) = f ( y ( k ) , ⋯ , y ( k − n y ) , u ( k ) , ⋯ , u ( k − n u ) ) (1)

which satisfies the following two assumptions:

A1: Function f ( * ) is smooth; and its continuous and bounded partial derivatives ∂ f ∂ y ( k − i ) , 0 ≤ i ≤ n y and ∂ f ∂ u ( k − j ) , 1 ≤ j ≤ n u exists;

A2: The system Σ satisfies generalized Lipschitz condition, that is, for two real numbers k 1 ≠ k 2 ( k 1 ≥ 0 , k 2 ≥ 0 ) , and a positive number b, the system Σ satisfies:

| y ( k 1 + 1 ) − y ( k 2 + 1 ) | ≤ b ‖ H ( k 1 ) − H ( k 2 ) ‖ (2)

In above, u ( k ) ∈ R and y ( k ) ∈ R represent input and output of the system, respectively. n y and n u are two integers. f ( * ) : R n y + n u + 2 ↦ R is a nonlinear mapping.

Denote

H ( k ) = [ y ( k ) , ⋯ , y ( k − L y + 1 ) , u ( k ) , ⋯ , u ( k − L u + 1 ) ] T (3)

and

△ y ( k ) = y ( k ) − y ( k − 1 ) △ u ( k ) = u ( k ) − u ( k − 1 ) △ H ( k ) = H ( k ) − H ( k − 1 ) = [ △ y ( k ) , ⋯ , △ y ( k − L y + 1 ) , △ u ( k ) , ⋯ , △ u ( k − L u + 1 ) ] T (4)

where, L y and L u ( 0 ≤ L y ≤ n y , 1 ≤ L u ≤ n u ) are integers called pseudo order (PO).

Theorem [

△ y ( k + 1 ) = Φ T ( k ) △ H ( k ) (5)

and for an arbitrary time k, Φ ( k ) = [ ϕ 1 ( k ) , ⋯ , ϕ L y ( k ) , ϕ L y + 1 ( k ) , ⋯ , ϕ L y + L u ( k ) ] T is bounded.

Proof: See [

Denote y d ( k ) as reference signal. The cost function of input signal is selected as:

J ( u ( k ) ) = | y d ( k + 1 ) − y ( k + 1 ) | 2 + λ | u ( k ) − u ( k − 1 ) | 2 (6)

where, λ > 0 is a weighting factor.

According to Equations (5) and (6) and let ∂ J ( u ( k ) ) ∂ u ( k ) = 0 , the control law can be derived as:

u ( k ) = u ( k − 1 ) + ϕ L y + 1 ( k ) { ρ L y + 1 [ y d ( k + 1 ) − y ( k ) ] − ∑ i = 1 L y ρ i ϕ i ( k ) △ y ( k − i + 1 ) − ∑ i = L y + 2 L y + L u ρ i ϕ i ( k ) △ u ( k − L y − i + 1 ) } λ + | ϕ L y + 1 ( k ) | 2 (7)

where, ρ i ∈ ( 0 , 1 ] , 1 ≤ i ≤ L y + L u .

In Equation (7), only PG is unknown and needs to be updated online.

To obtain PG, the following cost function is adopted:

J ( Φ ( k ) ) = | y ( k ) − y ( k − 1 ) − Φ T ( k ) △ H ( k − 1 ) | 2 + μ ‖ Φ ( k ) − Φ ( k − 1 ) ‖ 2 (8)

where, μ > 0 is a weighting factor.

Letting ∂ J ( Φ ( k ) ) ∂ Φ ( k ) = 0 yields

Φ ( k ) = Φ ( k − 1 ) + γ △ H ( k ) [ y ( k ) − y ( k − 1 ) − Φ T ( k − 1 ) △ H ( k − 1 ) ] μ + ‖ △ H ( k − 1 ) ‖ 2 (9)

where, γ ∈ ( 0 , 2 ] .

The reset conditions for PG are:

Φ ( k ) = Φ ( 1 ) , if ‖ Φ ( k ) ‖ ≤ ε or ‖ △ H ( k − 1 ) ‖ ≤ ε or

s i g n ( ϕ L y + 1 ( k ) ) ≠ s i g n ( ϕ L y + 1 ( 1 ) ) (10)

where, ε is a small positive number.

The model described here is only used to generate flight data for simulations. The nonlinear model of a UAV [

Rotational equations:

[ ϕ ˙ θ ˙ ψ ˙ ] = [ v 1 v 2 v 3 ] = R ⋅ [ p q r ] = [ 1 sin ϕ tan θ cos ϕ tan θ 0 cos ϕ − sin ϕ 0 sin ϕ sec θ cos ϕ sec θ ] [ p q r ] (11)

[ p ˙ , q ˙ , r ˙ ] T = [ f p , f q , f r ] T + [ w 1 , w 2 , w 3 ] T (12)

In which v 1 , v 2 and v 3 are virtual inputs which are used in the following of the design of Euler angle control law. And

[ w 1 w 2 w 3 ] = ρ V a 2 S 2 [ b I x I z − I x z 2 ⋅ ( I z C l δ a + I x z C n δ a ) 0 b I x I z − I x z 2 ⋅ ( I z C l δ r + I x z C n δ r ) 0 c I y ⋅ C m δ e 0 b I x I z − I x z 2 ⋅ ( I x z C l δ a + I x C n δ a ) 0 b I x I z − I x z 2 ⋅ ( I x z C l δ r + I x C n δ r ) ] [ δ a δ e δ r ] (14)

Translational equations:

and

V a = u 2 + v 2 + w 2 , α = tan − 1 ( w u ) , β = sin − 1 ( v V a ) (16)

In the above equations, m and g represent mass and gravitational constant, respectively; I x , I y , I z , and I x z represent moments of inertia; ϕ , θ , and ψ represent roll, pitch, and yaw angle, respectively; p, q, and r represent roll, pitch, and yaw angular velocity, respectively; δ a , δ e , and δ r are deflection angles of aileron, elevator, and rudder, respectively; δ t is throttle, ranging from 0 to 1; b represents wing span; c represents chord length; S represents wing area; V a represents air speed; ρ represents air density; α and β are attack and sideslip angle, respectively; S p r o p and C p r o p are coefficients with respect to propeller. C l 0 , C l β , C l p , C l r , C l δ a , and C l δ r are aerodynamic derivatives with respect to roll moment; C m 0 , C m α , C m q , and C m δ e are aerodynamic derivatives with respect to pitch moment; C n 0 , C n β , C n p , C n r , C n δ a , and C n δ r are aerodynamic derivatives with respect to yaw moment; C L 0 , C L α , C L q , and C L δ e are aerodynamic derivatives with respect to lift; C Y 0 , C Y β , C Y p , C Y r , C Y δ a , and C Y δ r are aerodynamic derivatives with respect to side force; C D 0 , C D α , C D q , and C D δ e are aerodynamic derivatives with respect to drag. Appendix E of Reference [

The general structure of an IMC-based control system [

G I M C ( s ) = G p − − 1 ( s ) f ( s ) (17)

where, G p − ( s ) includes the parts with minimum phase of G p ( s ) . f ( s ) is a low pass filter that has a formation like

f ( s ) = 1 ( λ f s + 1 ) n (18)

λ f is called filter coefficient. Rationality of G I M C ( s ) can be guaranteed by selecting appropriate value of n.

Thus, the two transfer functions of the above system can be obtained as:

Y ( s ) R ( s ) = G I M C ( s ) G p ( s ) 1 + G I M C ( s ) [ G p ( s ) − G m ( s ) ] (19)

Y ( s ) D ( s ) = [ 1 − G I M C ( s ) G m ( s ) ] G p ( s ) 1 + G I M C ( s ) [ G p ( s ) − G m ( s ) ] (20)

Hence, if G m ( s ) and G I M C ( s ) satisfy the following two conditions:

G p ( s ) = G m ( s ) (21)

G I M C ( s ) = G p − 1 ( s ) (22)

Equations (19) and (20) turn into:

Y ( s ) R ( s ) = 1 (23)

Y ( s ) D ( s ) = 0 (24)

Therefore, building on conditions (23) and (24), the system can track the reference signal and at the same time, would theoretically not be influenced by disturbances.

In this part, the attitude control laws are derived. In controlling a fixed-wing UAV, usually, deflecting aileron generates roll movement, deflecting elevator generates pitch movement, and deflecting rudder generates yaw movement. The overall control scheme is shown as:

Applying Equations ((7), (9), and (10)) yields the control laws of roll, pitch, and yaw angular rates, shown as:

Roll rate:

Φ p ( k ) = Φ p ( k − 1 ) + γ p △ H p ( k ) [ p ( k ) − p ( k − 1 ) − Φ p T ( k − 1 ) △ H p ( k − 1 ) ] μ p + ‖ △ H p ( k − 1 ) ‖ 2 (25)

Φ p ( k ) = Φ p ( 1 ) , if ‖ Φ p ( k ) ‖ ≤ ε p or ‖ △ H p ( k − 1 ) ‖ ≤ ε p or

s i g n ( ϕ L p y + 1 p ( k ) ) ≠ s i g n ( ϕ L p y + 1 p ( 1 ) ) (26)

δ a ( k ) = δ a ( k − 1 ) + ϕ L p y + 1 p ( k ) { ρ L p y + 1 p [ p d ( k + 1 ) − p ( k ) ] − ∑ i = 1 L p y ρ i p ϕ i p ( k ) △ p ( k − i + 1 ) − ∑ i = L p y + 2 L p y + L p u ρ i p ϕ i p ( k ) △ δ a ( k − L p y − i + 1 ) } λ p + | ϕ L p y + 1 p ( k ) | 2 (27)

Pitch rate:

Φ q ( k ) = Φ q ( k − 1 ) + γ q △ H q ( k ) [ q ( k ) − q ( k − 1 ) − Φ q T ( k − 1 ) △ H q ( k − 1 ) ] μ q + ‖ △ H q ( k − 1 ) ‖ 2 (28)

Φ q ( k ) = Φ q ( 1 ) , if ‖ Φ q ( k ) ‖ ≤ ε q or ‖ △ H q ( k − 1 ) ‖ ≤ ε q or

s i g n ( ϕ L q y + 1 q ( k ) ) ≠ s i g n ( ϕ L q y + 1 q ( 1 ) ) (29)

δ e ( k ) = δ e ( k − 1 ) + ϕ L q y + 1 q ( k ) { ρ L q y + 1 q [ q d ( k + 1 ) − q ( k ) ] − ∑ i = 1 L q y ρ i q ϕ i q ( k ) △ q ( k − i + 1 ) − ∑ i = L q y + 2 L q y + L q u ρ i q ϕ i q ( k ) △ δ e ( k − L q y − i + 1 ) } λ q + | ϕ L q y + 1 q ( k ) | 2 (30)

Yaw rate:

Φ r ( k ) = Φ r ( k − 1 ) + γ r △ H r ( k ) [ r ( k ) − r ( k − 1 ) − Φ r T ( k − 1 ) △ H r ( k − 1 ) ] μ r + ‖ △ H r ( k − 1 ) ‖ 2 (31)

Φ r ( k ) = Φ r ( 1 ) , if ‖ Φ r ( k ) ‖ ≤ ε r or ‖ △ H r ( k − 1 ) ‖ ≤ ε r or

s i g n ( ϕ L r y + 1 r ( k ) ) ≠ s i g n ( ϕ L r y + 1 r ( 1 ) ) (32)

δ r ( k ) = δ r ( k − 1 ) + ϕ L r y + 1 r ( k ) { ρ L r y + 1 r [ r d ( k + 1 ) − r ( k ) ] − ∑ i = 1 L r y ρ i r ϕ i r ( k ) △ r ( k − i + 1 ) − ∑ i = L r y + 2 L r y + L r u ρ i r ϕ i r ( k ) △ δ r ( k − L r y − i + 1 ) } λ r + | ϕ L r y + 1 r ( k ) | 2 (33)

The Euler angle control laws are included in Equation (11) and

Simulation was carried out in the MATLAB environment by writing .m file to demonstrate the feasibility and superiority of the method developed by making

comparison with CPID and NDI methods. In the simulation, the UAV was under external disturbances.

Parameter values of algorithms are shown as (

Reference signals:

{ ϕ d = 0.1 sin ( 0.5 π t ) rad θ d = 0.1 sin ( 0.5 π t ) rad ψ d = 0.1 sin ( 0.5 π t ) rad (34)

In this part, the control simulation of the small UAV was performed with the UAV under external low frequency disturbances. Simulation results are as follows:

From the figures the following conclusions can be drawn:

Figures 3-5 reveal that, without using any model information of the UAV, the developed data-driven method is obviously superior to the CPID method (also a data-driven method) and the NDI method (requires detailed model information) in control performance. The fundamental reason is that the method develop has a better control performance on body rate control, namely, realizing a better performance in dealing with uncertainties, which is revealed in Figures 6-11.

The essential difference between the MFAC method and the CPID method is the acquisition of dynamic model, see Equations (9) and (10). In the MFAC method, because of the identification of PG, a dynamic model can be obtained, which is foundation for optimal control law design, making deflection angles of control surfaces at each time point are optimum. This is also why the varying frequency of the control surfaces of the method developed is so high in Figures 6-8. The optimum control laws take into account of all uncertainties, thus even they may disturb the system, these control laws will make the output of UAV system track and approximate reference signal (see Figures 9-11). While for the

Channel | Roll | Pitch | Yaw |
---|---|---|---|

value | λ f ϕ = 0.07 | λ f θ = 0.07 | λ f ψ = 0.07 |

L p y = 3 , L p u = 1 | L q y = 3 , L q u = 1 | L q y = 3 , L q u = 1 | |

ρ i p = 0.5 ( i = 1 , 2 , 3 , 4 ) | ρ i q = 0.5 ( i = 1 , 2 , 3 , 4 ) | ρ i r = 0.05 ( i = 1 , 2 , 3 , 4 ) | |

η p = 1 | η q = 1 | η r = 1 | |

λ p = 10 − 4 | λ q = 10 − 3 | λ r = 10 − 3 | |

μ p = 0.01 | μ q = 0.01 | μ r = 0.01 | |

ε p = 10 − 5 | ε q = 10 − 5 | ε r = 10 − 5 |

most widely used CPID-based controller that lacks dynamic model obtaining capabilities, its deflection angles at each sampling point are not optimum and thus cannot cover the whole flight envelop with best performance (the control law may be optimal in a certain short time range, see 0 - 2.3 seconds in first and third enlarged figures in

The attitude control of fixed wing UAVs is studied based on the characteristics of inner loop and outer loop control systems, and realized by using MFAC to design the control law of inner loop angular velocity system and IMC to design that of Euler angle system. Firstly, the MFAC based controller has achieved inner loop angular velocity control using only I/O data without any model information. The IMC based controller has realized outer loop Euler angle control using a few tuning parameters (only one for each channel) and easy tuning process. Secondly, compared with conventional model-free CPID method and detailed model-based NDI method, it is discovered that the method developed is capable of dealing with strong coupling and highly nonlinear system such as that of fixed-wing UAVs; and it shows superiority in dealing with disturbances.

This publication was supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions and the Advanced Research Project of Army Equipment Development (No.301020803).

The authors declare no conflicts of interest regarding the publication of this paper.

Chen, M.L. and Wang, Y. (2019) A Data-Driven Adaptive Method for Attitude Control of Fixed-Wing Unmanned Aerial Vehicles. Advances in Aerospace Science and Technology, 4, 1-15. https://doi.org/10.4236/aast.2019.41001

D ( s ) Laplace transforms of disturbance signal

f p , f q , f r unknown nonlinear parts, rad/s^{2}

g gravitational constant, m/s^{2}

G m ( s ) transfer function of internal model

G p ( s ) transfer function of plant model

G I M C ( s ) Internal model controller

I x , I y , I z roll, pitch and yaw moments of inertial, kg∙m^{2}

I x z product of inertial, kg∙m^{2}

L y , L u pseudo order

L p y , L p u pseudo order for roll rate control law

L q y , L q u pseudo order for pitch control law

L r y , L r u pseudo order for yaw rate control law

m mass of UAV, kg

p , q , r fuselage roll, pitch and yaw angular rates, rad/s

p d , q d , r d reference signals for fuselage roll, pitch and yaw angular rates, rad/s

R ( s ) Laplace transforms of reference signal

u , v , w velocity components for fuselage, m/s

U ( s ) Laplace transforms of input signal

ν 1 , ν 2 , ν 3 virtual inputs for control systems of roll, pitch and yaw angular rates, rad/s^{2}

V a velocity of UAV in air coordinate, m/s

Y ( s ) Laplace transforms of output signal

α angle of attack, rad

r a d angle of sideslip, rad

δ a , δ e , δ r , δ t deflection angles for aileron, elevator, rudder and throttle, rad

η p , η q , η r parameters in MFAC algorithm

ε p , ε q , ε r lower limit for resetting PG

ϕ , θ , ψ roll, pitch and yaw angles of UAV, rad

ϕ d , θ d , ψ d reference signals for roll, pitch and yaw angles, rad

λ p , λ q , λ r parameters in MFAC algorithm

λ f ϕ , λ f θ , λ f ψ filter coefficient of internal model controller for Euler angle

μ p , μ q , μ r parameters in MFAC algorithm

ρ p , ρ q , ρ r parameters in MFAC algorithm

Φ p , Φ q , Φ r pseudo gradient in MFAC algorithm