Attitude tracking control design of fixed-wing UAVs having uncertain dynamics and corrupted gyro sensor outputs

Abstract

Flight performance of unmanned aerial vehicles (UAVs) strongly depends on implemented attitude tracking control. For designing better controllers, nonlinear control design techniques are often opted instead of control design based on linearized models. Uncertainty in nonlinear dynamics estimation may arise due to inaccuracies in aerodynamic derivatives and simplifications/assumptions made during the derivation of nonlinear models. This paper considers attitude tracking control of fixed-wing UAVs having uncertain dynamics and corrupted gyro sensor outputs. An integral chain differentiator (ICD) is used to provide the analytical redundancy to the gyros used to measure the angular rates. Two control design schemes are proposed, a neuro-fuzzy adaptive sliding mode control (NFASMC) and an ICD approximation-based fuzzy adaptive sliding mode control (ICD-FASMC). In NFASMC, the uncertain part of the dynamics is estimated using an adaptive radial basis function neural network. Gyro sensor output errors are estimated in real-time, using ICD based error estimation scheme and used in the control law along with the sensor’s corrupted outputs. In ICD-FASMC, the uncertain dynamics and angular rates of UAV are estimated using the ICD such that the requirement of the gyro sensor outputs for control design is bypassed. The switching gain of the designed controllers is made adaptive using fuzzy logic to mitigate the chattering effect. The stability of the proposed controllers is proved using the Lyapunov approach. The proposed schemes are implemented using a nonlinear simulation of a fixed-wing UAV. Simulation results are presented to show the effectiveness of the proposed techniques.

Keywords

Neural network tracking control sliding mode control fuzzy logic UAV sensors

1 Introduction

Attitude tracking control is a critical part of the control system design for aircraft and unmanned aerial vehicles (UAVs). UAVs have to follow the commands generated by the guidance system for the desired trajectory tracking. For this, attitude reference command tracking control design must be efficient in the presence of modelling/parametric uncertainties and external disturbances. In addition, any false/corrupted state feedback from the sensors to the control system must be accounted for as it may degrade the control performance. Dynamical models of UAVs are simplified mathematical equations representing the approximation of complex and highly coupled nonlinear dynamics. The aerodynamic derivatives obtained from wind tunnel tests or computational techniques may also contain uncertainties. In UAVs, the available aerodynamic derivatives may not be accurate enough to design a reliable model-based controller. In such cases, we can incorporate real-time approximation strategies into the control design. In this work, we have considered the attitude control design task of fixed-wing UAVs having uncertain/unavailable dynamics and inaccurate/corrupted gyro sensor outputs.

Sliding mode control (SMC) is a well-known control technique having the advantage of robustness against modelling uncertainties, parametric variations and external disturbances [1, 2]. SMC has been applied extensively to control uncertain nonlinear systems, including aerospace systems. In contrast to other nonlinear control schemes, such as dynamic inversion, feedback linearization and backstepping, SMC does not require a precise mathematical model of the system [3]. However, the so-called switching control term added to the SMC for robustness, against external disturbances, modelling and parametric uncertainties, is discontinuous. The discontinuous switching control term of the SMC may cause undesirable oscillations/chattering of the control surfaces. An appropriate value of the gain of the switching control term (switching gain) is required to minimize the chattering. In the case of an uncertain dynamic system, the switching gain value can be low if we estimate the system dynamics accurately enough and use it in the control law. However, to cancel the effect of the bounded external disturbances, the switching gain must be increased according to the upper bound of the external disturbance. Instead of selecting a conservative high switching gain value, adaptive techniques can be incorporated in the SMC to adaptively tune the switching gain to maintain the control performance and minimize the chattering effect.

Several techniques exist for the online estimation of uncertain dynamics of nonlinear systems. Among them, artificial neural networks (ANNs) [4 –8], fuzzy logic [9 –11] and differentiator-based approximation techniques [12 –15] provide good results. These techniques can be incorporated into the SMC to reduce/eliminate the dependency of the control design on the analytical calculations of the system dynamics. ANNs have excellent nonlinear function approximation capabilities due to their quick online learning and network parameters self-adaptation [5, 16]. A type of ANN, radial basis functions neural network (RBFNN), is good at avoiding local minima and has a simple structure with quick learning abilities [17, 18]. Learning refers to the auto-adjustment of the network parameters through some adaptation laws such that the network output converges to the desired value.

Using the system outputs, differentiators can estimate the higher-order system states for use in the control design [13]. These estimations of the higher-order system states can also be used to estimate the uncertain dynamics of the system [15]. In case of a high degree of uncertainty or unavailability of the system dynamics, higher-order differentiator-based approximations of the system dynamics can be incorporated into the SMC [14, 19]. Moreover, the angular rates of the dynamic system which are usually measured using the gyro sensors can be accurately estimated using the differentiator technique. Thus, differentiators can provide analytical redundancy to the gyros such that any gyro output errors can be approximated and isolated [20, 21].

Fuzzy logic is an effective method to mitigate the chattering associated with the SMC [22, 23]. Fuzzy models can be developed using the process inputs and outputs without the requirement of the exact dynamic model [24, 25]. A Fuzzy logic system (FLS) can be designed for nonlinear control when the exact mathematical relationship between inputs and outputs of the process is unknown. Fuzzy logic can be used to approximate the uncertain dynamics or/and adaptively tune the switching control gain to mitigate the chattering associated with the SMC [22, 26]. In [26], based on FLS, an adaptive compensation FTC is designed for nonlinear Markov jump systems subject to mismatched uncertainty and simultaneous multiplicative and additive actuator faults. The FLS is used to approximate the nonlinear functions. Whereas in [22], FLS is used to tune the switching gain of the SMC.

This paper considers the attitude tracking control of fixed-wing UAVs under uncertainty (including modelling and parametric)/unavailability of the system dynamics and corrupted gyro sensor outputs. Two control schemes are proposed and their control performance is presented. In the first scheme, a neuro-fuzzy adaptive SMC (NFASMC) is designed. ICD is used to estimate and isolate the sensor output errors. A new sliding surface based on erroneous gyro sensor output and the estimated values of the sensor output error from the ICD-based error estimation scheme is proposed. The control law is designed using both the corrupt sensor output and the error estimate from the ICD-based error estimation scheme. The uncertain part of the system dynamics is estimated using the RBFNN, where the adaptive law for the RBFNN is derived using the Lyapunov theory. The switching gain of the controller is made adaptive using a fuzzy logic system (FLS) to mitigate the chattering effect of the SMC. The innovative combination of the ANN, SMC, fuzzy logic and the ICD-based gyro sensor output error estimation in NFASMC provides excellent tracking performance in the presence of gyro sensor output errors. The uncertain dynamics does not affect the controller performance as it is estimated using the adaptive RBFNN and the chattering effect of the SMC is mitigated by the FLS. The second control scheme, ICD approximation-based fuzzy adaptive SMC (ICD-FASMC), uses ICD to approximate the unknown system dynamics and the higher-order system states required for the control design. The switching gain of the proposed ICD-FASMC is also made adaptive using the fuzzy logic system (FLS). Chattering associated with the conservative high values of switching gains is eliminated using the adaptive gain strategy. The ICD-FASMC can be used when the UAV is susceptible to gyro sensor output errors. It does not require the gyro sensor outputs, as the angular rates are computed by the ICD. The exact dynamics of the system is also not required as it is estimated using the ICD output. For indirect use of the ICD-FASMC, the system can be switched from any nominal controller to the ICD-FASMC when the sensor output error from the ICD-based error estimation scheme crosses some predefined threshold. The proposed control schemes are implemented using a nonlinear simulation of a fixed-wing UAV.

The main features of this work are summarized as

Efficient and robust attitude control schemes, applicable to nonlinear systems in general and fixed-wing UAVs in particular, are proposed that provide excellent tracking performance.

The proposed control schemes do not require a precise mathematical model and dynamic parameters as the uncertain dynamics is approximated in real-time and used in the control laws.

In both the proposed control schemes, gyro sensor output errors are taken care of. In NFASMC, ICD is used to estimate the sensor output error and the control is designed using the values of sensor error estimates and the erroneous sensor output. In ICD-FASMC, the controller is designed using the estimated values of the gyro sensor output using the ICD so that the erroneous output from the gyro sensor is not required for the control design.

The gain of the switching part of the controllers is made adaptive to mitigate the chattering effects of the SMC. A FLS is designed which takes the sliding surface as the input and gives the required value of the switching gain as the output to be used in the proposed control laws.

The remaining paper is organized as

Section 2 of the paper includes UAV nonlinear dynamics and the problem formulation. Section 3 describes the design and stability analysis of the two proposed control schemes. Simulation results are presented and discussed in section 4. Finally, section 5 concludes the paper.

2 UAV nonlinear dynamics and problem formulation

In this section, first a general formulation of the nonlinear dynamics of fixed-wing UAVs is presented and then the control problem addressed in this work is formulated.

2.1 UAV nonlinear dynamics

Taking the assumption of the flat earth, six degrees of freedom (6-DOF) general nonlinear equations for UAV dynamics in body-axis are listed in (1) [27]. $\begin{array}{l} \dot{u} = r v - q w - {g^{'}}_{0} \sin θ + F_{x} / m \\ \dot{v} = - r u + p w + {g^{'}}_{0} \sin φ \cos θ + F_{y} / m \\ \dot{w} = q u - p v + {g^{'}}_{0} \cos φ \cos θ + F_{z} / m \\ \dot{\bar{h}} = - u \sin θ + v \sin φ \cos θ + w \cos φ \cos θ \\ \dot{p} = (a_{1} p + a_{2} r) q + a_{3} \bar{L} + a_{4} N \\ \dot{q} = a_{5} p r - a_{6} (p^{2} - r^{2}) + a_{7} M \\ \dot{r} = (a_{8} p + a_{1} r) q + a_{4} \bar{L} + a_{9} N \\ \dot{φ} = p + \tan θ (q \sin φ + r \cos φ) \\ \dot{θ} = q \cos φ + r \sin φ \\ \dot{ψ} = (q \sin φ + r \cos φ) / \cos θ \end{array}}$ (1)

In (1), u, v and w represent the UAV velocities along x, y and z axis respectively. $\bar{h}$ is the height of the aircraft from ground. p, q and r are the angular rates about x, y and z axis respectively. φ, θ and ψ, termed as Euler angles, correspond to roll, pitch and yaw angular positions respectively. $g_{o}^{'}$ is acceleration due to gravity. Further details of these equations (1) can be seen in the Appendix.

2.2 Problem formulation

Attitude tracking control of a UAV refers to the control logic implemented to regulate the attitude angles (φ, θ and ψ). The guidance system generates the desired attitude angles (φ_d, θ_d and ψ_d) required to keep the UAV on the desired path. The attitude tracking control generates the required control surface deflections such that the attitude angles follow the desired angles. In this work, the problem of attitude tracking control is addressed for a UAV having uncertain/unavailable dynamics and corrupted gyro sensor output. The attitude control can be designed by using the relations for $\dot{p}, \dot{q}$ and $\dot{r}$ (from (1)) which can be written as a general nonlinear 2^nd order system as $\begin{matrix} {\dot{x}}_{2} = f (x) + g (x) u_{o} + d (t) \\ {\dot{x}}_{1} = x_{2} + E_{s} (t) \\ y = x_{1} \end{matrix}$ (2)

Where, ${\dot{x}}_{2} = {[\dot{p} \dot{q} \dot{r}]}^{T}, f (x) = {[f_{p} f_{q} f_{r}]}^{T}$ , ${\dot{x}}_{1} = {[p q r]}^{T}$ , u_o = [δa δe δr] ^T is the input vector such that δa, δe and δr represent aileron, elevator and rudder deflections respectively and y = [φ θ ψ] ^T is the output vector. d(t) represents the time varying external disturbance. E_s(t) represents the sensor output error such that E_s = [E_p E_q E_r] ^T. Where, E_p, E_q and E_r represent the error in the outputs corresponding to the roll, pitch and yaw rate sensors respectively. Equations for f_p, f_q, f_r and g(x) are derived in the Appendix. The function f(x) is uncertain due to the uncertainties in modelling and aerodynamic derivatives and g(x) is assumed to be invertible. Following additional assumptions are made for the development of the FTC.

Sensor output errors E_s(t) are assumed to be unknown but bounded.

The external disturbances d(t) are bounded such that ∥d(t) ∥ ⩽ D .

Remark 1: Gyro sensors are assumed inherently faultless such that the required attitude angles are obtained using the implemented integration algorithms within the onboard attitude-heading reference system (AHRS). Only the outputs of the gyro sensors are assumed to be potentially corrupted/erroneous. These types of errors in the output signals can occur due to improper/loose connections at the output port of sensors, transmission problems, external magnetic fields etc.

3 Controller design

For the uncertain non-linear system (2), we present the design and stability analysis of two different control schemes in this section. A neuro-fuzzy adaptive SMC (NFASMC) and an ICD approximation based fuzzy adaptive SMC (ICD-FASMC) are presented. Following two paragraphs describe the work flow for the design of the proposed control schemes.

ICD is used in the design process of NFASMC to provide the analytical redundancy to the gyro sensors. The error in the sensor output is estimated by using the ICD and the gyro sensor output. The NFASMC is designed using erroneous output of the gyro sensors and the real-time error estimate. Further, the switching gain of the control is made adaptive using a fuzzy logic system (FLS). For the stated procedure, design of the ICD is described in the Section 3.1, Section 3.2 describes the design and stability analysis of the NFASMC and Section 3.3 describes the FLS used for switching gain adaptation.

For the design of the ICD-FASMC, the requirement of gyro sensor output for the control design are bypassed by using the ICD based approximation of the attitude rates in the development of the controller. For the proposed ICD-FASMC, ICD described in Section 3.1 and the switching gain adaptation procedure described in Section 3.3 are used. The design and stability analysis of the ICD-FASMC is described in Section 3.4.

3.1 Integral chain differentiator

ICD can be used to estimate the unknown system states. Using the system’s outputs (y = x₁), the higher order unknown states (x₂, x₃, . . . , x_n) can be acquired by using the ICD. Following lemma provides the illustration of estimation procedure for the unknown higher order system states using the ICD.

Lemma 1 [15]: For any nth (n = 1, 2, . . . , n) order nonlinear system with unknown parameters and unknown higher order system states (x₂, x₃, . . . , x_n), using the output (y = x₁), the ICD is designed as $\begin{array}{l} x_{1} (t) = {\hat{x}}_{2} (t) \\ ⋮ \\ x_{n - 1} (t) = {\hat{x}}_{n} (t) \\ x_{n} (t) = {\hat{x}}_{n + 1} (t) \\ x_{n + 1} (t) = - \frac{λ_{1}}{ε^{n + 1}} ({\hat{x}}_{1} (t) - x_{1} (t)) \\ - \frac{λ_{2}}{ε^{n}} {\hat{x}}_{2} (t) - \dots - \frac{λ_{n + 1}}{ε} {\hat{x}}_{n + 1} (t) \end{array}$ (3)

Where, ɛ > 0 is the perturbation parameter of sufficiently small value. The parameters λ₁, λ₂, . . . , λ_n+1 are positive coefficients of Hurwitz polynomial such that sⁿ⁺¹ + λ_n+1sⁿ + … λ₂s + λ₁ = 0. Therefore, $lim_{ɛ \to 0} {\hat{x}}_{i} (t) = x_{i} (t), i = 1, 2, . . ., n$ $and lim_{ɛ \to 0} {\hat{x}}_{n + 1} (t) = {\dot{x}}_{n} (t)$ where, $\hat{x} (t) = {[\begin{matrix} {\hat{x}}_{1} (t) & {\hat{x}}_{2} (t) & . . . & {\hat{x}}_{n} (t) \end{matrix}]}^{T}$ taking the Laplace transform of the ICD we have $\begin{matrix} {\hat{X}}_{i - k} (s) = \frac{{\hat{X}}_{i} (s)}{s^{k}}, k = 0, 1, \dots, i - 1; i = 1, 2, \dots, n + 1 \\ s {\hat{X}}_{n + 1} (s) + \frac{λ_{n + 1}}{ɛ} {\hat{X}}_{n + 1} (s) + \frac{λ_{n}}{ɛ^{2}} {\hat{X}}_{n} (s) + \dots \\ + \frac{λ_{2}}{ɛ^{n}} {\hat{X}}_{2} (s) + \frac{λ_{1}}{ɛ^{n + 1}} {\hat{X}}_{1} (s) = \frac{λ_{1}}{ɛ^{n + 1}} {\hat{X}}_{1} (s) \end{matrix}$ (4) From (4) we get $\frac{{\hat{X}}_{1} (s)}{X_{1} (s)} = \frac{\frac{λ_{1}}{ɛ^{n + 1}}}{s^{n + 1} + \frac{λ_{n + 1}}{ɛ} s^{n} + \frac{λ_{n}}{ɛ^{2}} s^{n - 1} + \dots + \frac{λ_{1}}{ɛ^{n + 1}}}$

For suitable values of λ₁, λ₂, . . . , λ_n+1 filtering can be obtained by ICD.

Corollary 1: For the 2^nd order uncertain nonlinear system (2), the integral-chain differentiator can be designed as $\begin{array}{l} {\dot{\hat{x}}}_{1} = {\hat{x}}_{2} \\ {\dot{\hat{x}}}_{2} = {\hat{x}}_{3} \\ {\dot{\hat{x}}}_{3} = - \frac{λ_{1}}{ε^{3}} ({\hat{x}}_{1} - y) - \frac{λ_{2}}{ε^{2}} {\hat{x}}_{2} - \frac{λ_{3}}{ε} {\dot{\hat{x}}}_{3} \end{array}$ (5) for the approximation of f(x) and to estimate x₂ and x₃ using the input u_o and output y = x₁ of the uncertain nonlinear system (2). Where ɛ > 0 and is sufficiently small and s³ + λ₃s² + λ₂s + λ₁ = 0 is Hurwitz. Therefore, $lim_{ɛ \to 0} {\hat{x}}_{2} = x_{2}$ and $lim_{ɛ \to 0} {\hat{x}}_{3} = f (x) + g (x) u$ or $f (x) = lim_{ɛ \to 0} ({\hat{x}}_{3} - g (\hat{x}) u)$ where $\hat{x} = [\begin{matrix} {\hat{x}}_{1} & {\hat{x}}_{2} \end{matrix}]^{T}$

3.2 Neuro-fuzzy adaptive SMC

For the nonlinear system (2), f(x) is assumed uncertain/unavailable. The attitude control must ensure the tracking of the desired trajectory x_1d = [φ_d θ_d ψ_d] in the presence of external disturbances d(t) and gyro sensor output errors. The uncertain part of the dynamics (f(x)) is estimated using adaptive RBFNN. The adaptive law for RBFNN is derived using the Lyapunov theory. For the estimation of gyro sensor (measuring ${\dot{x}}_{1}$ ) error we use the ICD (5) as an observer. The ICD gives the estimated value of ${\dot{x}}_{1}$ as ${\dot{\hat{x}}}_{1}$ . Where, ${\dot{\hat{x}}}_{1} = [\hat{p} \hat{q} \hat{r}]^{T}$ The sensor output error is estimated as ${\hat{E}}_{s} = {\dot{\hat{x}}}_{1} - {\dot{x}}_{1}$ . Where, $E_{s} = {[{\hat{E}}_{p} {\hat{E}}_{q} {\hat{E}}_{r}]}^{T}$ . These output error estimation can be used to correct the gyro sensor output as ${\dot{x}}_{1 c} = x_{2} + E_{s} - {\hat{E}}_{s}$ . Defining ${\tilde{E}}_{s} = E_{s} - {\hat{E}}_{s}$ as the estimation error of the ICD, where ${\tilde{E}}_{s} = {[{\tilde{E}}_{p} {\tilde{E}}_{q} {\tilde{E}}_{r}]}^{T}$ , ${\dot{x}}_{1 c}$ can be written in corrected form as ${\dot{x}}_{1 c} = x_{2} + {\tilde{E}}_{s}$ . It is clear that as ${\dot{\hat{x}}}_{1} \to {\dot{x}}_{1}$ (Sec. 3.1), ${\hat{E}}_{s} \to E_{s}$ and hence, ${\tilde{E}}_{s} \to 0$ . This proves that the ICD observer’s estimation error for the gyro sensor output converges to zero.

The tracking error can be defined as e = x₁ - x_1d, where e = [e_φ e_θ e_ψ] ^T, and the derivative of the tracking error is $\dot{e} = {\dot{x}}_{1} - {\dot{x}}_{1 d}$ , where $\dot{e} = {[e_{p} e_{q} e_{r}]}^{T}$ . The sliding surface is defined as $S = \dot{e} + ce$ where S = [S₁ S₂ S₃] ^T and c = [c₁ c₂ c₃] ^T, c₁, c₂ and c₃ are positive constants to be tuned during the simulation. The sliding surface based on the corrected value $({\dot{x}}_{1 c})$ of sensors output can be rewritten as

$\begin{matrix} S = ({\dot{x}}_{1 c} - {\dot{x}}_{1 d}) + c (x_{1} - x_{1 d}) = (x_{2} - {\dot{x}}_{1 d}) \\ + c (x_{1} - x_{1 d}) + {\tilde{E}}_{s} \end{matrix}$ (6) such that x₁ → x_1d and $x_{2} \to {\dot{x}}_{1 d}$ when S → 0 as ${\tilde{E}}_{s} \to 0$ .

For actual implementation, we propose the use of both the erroneous sensor output ${\dot{x}}_{1}$ and the sensor error estimate ${\hat{E}}_{s}$ obtained using the ICD observer. Equation (7), obtained from (6), depicts the actual implementation form of the sliding surface. $S = (\underset{{\dot{x}}_{1}}{\underset{︸}{x_{2} + E_{s}}} - {\dot{x}}_{1 d}) + c (x_{1} - x_{1 d}) - {\hat{E}}_{s}$ (7)

Using the sliding surface (7), the gyro sensor output ${\dot{x}}_{1}$ and the sensor error estimate ${\hat{E}}_{s}$ , the sliding mode based attitude tracking control for the nonlinear system (2) is proposed as $u_{o} = - g (x)^{- 1} [\begin{matrix} \hat{f} (x) - {\ddot{x}}_{1 d} + c ({\dot{x}}_{1} - {\dot{x}}_{1 d}) \\ - c {\hat{E}}_{s} + k sign (S) \end{matrix}]$ (8)

For nonlinear system (2), the parameter vectors of the controller (8) can be defined as

u_o = [u₁ u₂ u₃] ^T = [δa δe δr] ^T and $\hat{f} (x) = {[{\hat{f}}_{p} {\hat{f}}_{q} {\hat{f}}_{r}]}^{T}$ represets the estimate of f(x), the uncertain dynamics part, using adaptive RBFNN. k = [k₁ k₂ k₃] ^T, where k₁ > 0, k₂ > 0 and k₃ > 0 are the switching control gains to make the controller robust against external disturbances, RBFNN and ICD approximation errors. Figure 1 shows the block diagram representing the architecture of the NFASMC.

Fig. 1

Architecture of NFASMC.

Approximation of f ( x )

To approximate the unknown/uncertain f(x), RBFNN is used. RBFNN is a three layered feed forward neural network. Number of neurons in each of the three layers are chosen according to the suitability to the problem. The middle layer neurons use radial basis functions (h_j) as activation functions. The output layer linearly combines the signals from the hidden layer. The RBFNN algorithm is [11] $h_{j} = exp (- {∥ ς_{o} - {\bar{c}}_{j} ∥}^{2} / 2 {b_{j}}^{2})$ (9) $f (ς) = W^{* T} h (ς_{o}) + ɛ_{o}$ (10)

Where ς_o is the input to the network. $\bar{c} = [{\bar{c}}_{ij}]$ , i is the input number and j is the number of nodes in the hidden layer. Output of the Gaussian function is h = [h_j] ^T. b_j represents the bias value for jth neuron of the hidden layer. W^* is the ideal weight matrix of the network for small approximation error ɛ_o, (ɛ_o ⩽ ɛ_N), and f(ς_o) represents the output of the RBFNN.

For the approximation of unknown f(x), the input to the RBFNN is $ς_{o} = {[e \dot{e}]}^{T}$ , where e = x₁ - x_1d and $\dot{e} = {\dot{x}}_{1 c} - {\dot{x}}_{1 d}$ . The RBFNN estimate of f(x) is $\hat{f} (x) = {\hat{W}}^{T} h (ς_{o})$ (11)

$\hat{W}$ represent the weight values from hidden to output layer, W = [W₁ W₂ ⋯ W_j] ^T. This estimation $\hat{f} (x)$ is used to design the control law (8). For the RBFNN learning an adaptive law is derived using the Lyapunov theory. We put forward the following theorem.

Theorem 1: The uncertain nonlinear system (2), subject to external disturbances d(t) and gyro sensor output errors E_s, under the action of the designed controller (8) using RBFNN adaptive law $\dot{\hat{W}} = γ^{- 1} Sh (ς_{o})$ is stable provided the switching gain k is chosen suitable.

Proof: Taking the derivative of the sliding surface (6), using (the corrected sensor outputs) instead of ${\dot{x}}_{1}$ and using (2) we get $\begin{matrix} \dot{S} = ({\dot{x}}_{2} - {\ddot{x}}_{1 d}) + c ({\hat{\dot{x}}}_{1 c} - {\dot{x}}_{1 d}) + {\dot{\tilde{E}}}_{s} \\ = (f (x) + g (x) u + d (t) - {\ddot{x}}_{1 d}) + c (x_{2} + E_{s} - {\hat{E}}_{s} - {\dot{x}}_{1 d}) + {\dot{\tilde{E}}}_{s} \\ = (f (x) + g (x) u + d (t) - {\ddot{x}}_{1 d}) + c ({\dot{x}}_{1} - {\dot{x}}_{1 d}) - c {\hat{E}}_{s} + {\dot{\tilde{E}}}_{s} \end{matrix}$ (12)

Submitting (8) into (12) we get $\begin{array}{l} \dot{S} = f (x) - \hat{f} (x) + d (t) + \dot{\tilde{E}} - k sign (S) \\ = \tilde{f} (x) + d (t) + \dot{\tilde{E}} - k sign (S) \end{array}$ (13)

Where, $\tilde{f} (x) = f (x) - \hat{f} (x)$

Using (10) and (11), we can write $\tilde{f} (x) = W^{* T} h (ς_{o}) + ɛ_{o} - {\hat{W}}^{T} h (ς_{o}) = {\tilde{W}}^{T} h (ς_{o}) + ɛ_{o}$ (14) for $\tilde{W} = W^{*} - \hat{W}$

We define a suitable Lyapunov function as $L = \frac{1}{2} S^{2} + \frac{1}{2} γ {\tilde{W}}^{T} \tilde{W} (γ > 0)$ (15)

Taking the derivative of (15) and using (13) and (14) $\begin{array}{l} \dot{L} = S \dot{S} + γ {\tilde{W}}^{T} \dot{\tilde{W}} = S (\tilde{f} (x) + d (t) + {\dot{\tilde{E}}}_{s} - k sign (S)) - γ {\tilde{W}}^{T} \dot{\hat{W}} \\ = S ({\tilde{W}}^{T} h (ς_{o}) + ε_{o} + d (t) + {\dot{\tilde{E}}}_{s} - k sign (S)) - γ {\tilde{W}}^{T} \dot{\hat{W}} \\ = {\tilde{W}}^{T} (S h (ς_{o}) - γ \dot{\hat{W}}) + S (ε_{o} + d (t) + {\dot{\tilde{E}}}_{s} - k sign (S)) \end{array}$ (16)

We define the adaptive law for the RBFNN as $\dot{\hat{W}} = γ^{- 1} Sh (ς_{o})$ . Now (16) becomes $\begin{array}{l} \dot{L} = S ((ε_{o} + d (t) + {\dot{\tilde{E}}}_{s}) - k sign (S)) \\ = S (ε_{o} + d (t) + {\dot{\tilde{E}}}_{s}) - k | S | \end{array}$ (17)

The RBFNN approximation error ɛ_o is sufficiently small and limited (ɛ_o ⩽ ɛ_N), d(t) is bounded with known upper bound (d(t) ⩽ D) and also ${\tilde{\dot{E}}}_{s}$ is bounded as ${\tilde{E}}_{s}$ is bounded let $| {\tilde{\dot{E}}}_{s} | \leq Λ$ , where Λ is a positive number. We can set k >(ɛ_N + D + Λ) to ensure that $\dot{L} < 0 .$ Thus, with proper selection of the gain k, $\dot{L}$ becomes negative definite. This proves the asymptotic stability of the system.

3.3 Switching gain adaptation using fuzzy logic

The proposed controller (8) for nonlinear system (2) is effectively robust against external disturbances. However, during the sliding phase, chattering phenomena (high frequency oscillations of the control output) appears near the sliding surface. Chattering is a major drawback of the SMC which can damage the control surfaces and can excite the unmodelled dynamics of the system. Mathematically, the sign function in the control law (8), induces chattering due to the frequent switching, while the chattering intensity depends on the gain k of the sign function. There are various methods to address the chattering of the SMC. The most common method is the design of boundary layer or in other words using a saturation function instead of the sign function. However, there is a trade-off between chattering mitigation and the control performances using the saturation function. In this work, we used an adaptive method to mitigate the chattering effect. Using a fuzzy logic system (FLS), the gain of the switching term of the SMC is made adaptive with the sliding surface. The sliding surface is used as the input of the FLS to get the required value of the gain as an output. The FLS architecture is shown in Fig. 2. MATLAB inbuilt fuzzy logic tool is used to create the FLS system. The design steps of the FLS are described as under.

(1) Inputs and outputs selection

Fig. 2

The FLS architecture.

The input of the FLS is selected as the sliding surface S_i ∈ S (i = 1, 2, 3). The gain k_i ∈ k (i = 1, 2, 3) of the switching control term of the control law (8) is the output of the FLS.

(2) Defining the fuzzy sets

The input and output fuzzy sets of the FLS are defined similarly as [NB, NS, Z, PS, PB]. Where NB, NS, Z, PS and PB are used to denote negative big, negative small, zero, positive small, and positive big respectively. FLS membership functions are chosen as gauss2mf within the MATLAB inbuilt fuzzy logic tool. Gauss2mf is a formed by combining two Gaussian functions such that each function defines the shape of one side of the membership function. The Gaussian function can be defined as $f (\bar{x}; ς, z) = exp (- (\bar{x} - z)^{2} / 2 ς^{2})$ where $\bar{x}$ is the input which is sliding surface in our case. z and ς represent the standard deviation and mean of the Gaussian function respectively. For creating gauss2mf, we have to set four parameters for the two Gaussian functions forming the membership function. In MATLAB fuzzy logic tool these are set as a parameter vector $[\begin{matrix} ς_{1} & z_{1} & ς_{2} & z_{2} \end{matrix}]$ , where ς₁ and z₁ are respectively the standard deviation and mean of the left Gaussian function, ς₂ and z₂ are respectively the standard deviation and mean of the right Gaussian function of any membership function, the parameter vectors of the input membership functions NB, NS, Z, PS and PB are respectively set as $\begin{matrix} [1.425 - 12.07 1.425 - 9.928] \\ [1.247 - 7.757 2.49 - 6.657] \\ [2.49 - 0.733 2.49 0.733] \\ [2.49 6.596 1.247 7.696] \\ [1.425 9.928 1.425 12.07] \end{matrix}$

Similarly, the parameter vectors of the output membership functions NB, NS, Z, PS and PB are respectively set as $\begin{matrix} [0.1296 - 1.098 0.1296 - 0.9025] \\ [0.1133 - 0.3666 0.1132 - 0.3] \\ [0.1133 - 0.03 0.1133 0.03] \\ [0.1132 0.3 0.1133 0.366] \\ [0.1295 0.9025 0.1295 1.097] \end{matrix}$

The input and output membership functions are plotted in Figs. 3 and 4 respectively. Note that the range of input in Fig. 3 corresponds to the designed controller for pitch tracking using NFASMC. The input range has to be adjusted for each controller according to the expected value of the sliding surface variable. The fuzzy logic tool automatically adjusts the membership parameter vectors according to the new input range such that the shape of the membership functions is preserved

Fig. 3

FLS input membership functions.

Fig.4

FLS output membership functions.

(3) Fuzzy rules selection

For the designed controller (8), during the sliding motion, whenever the state trajectories deviate from the sliding surface the switching term ksign(S) comes into action and forces the states towards the sliding surface. The purpose here is to make gain k adaptive to minimize the chattering. The FLS is designed based on the fact that if the value of the gain k is adjusted based on the value of S, we can minimize the chattering. For this, the value of gain should be decreased when the states are approaching the sliding surface and it should be increased when the states are moving away from the sliding surface. FLS rules based on the knowledge and experience of experts are formed as “IF-THEN” rules such as if one condition is satisfied then one predefined action has to be taken. The designed fuzzy rules are given as $\begin{matrix} IF Input is PB THEN Output is PB \\ IF Input is PS THEN Output is PS \\ IF Input is Z THEN Output is Z \\ IF Input is NS THEN Output is NS \\ IF Input is NB THEN Output is NB \end{matrix}$

(4) Defuzzification of the outputs

Defuzzification of the outputs is made by using centroid method within the MATLAB fuzzy logic tool. The FLS thus created is used to adaptively tune the switching gain of the designed sliding mode controllers. The input (sliding surface value) output (adaptive gain) relationship curve is plotted in Fig. 5. It shows that using the proposed FLS the switching gain can be adjusted adaptively according to the value of the sliding surface.

Fig. 5

Input output variation of the designed FLS.

Remark 2. The designed FLS is used to adaptively tune the gain k according to the fuzzy rules using the sliding surface. Therefore the NFASMC control law (8) can now be written as $u_{o} = - g (x)^{- 1} [\begin{matrix} \hat{f} (x) - {\ddot{x}}_{1 d} + c ({\dot{x}}_{1} - {\dot{x}}_{1 d}) \\ - c {\hat{E}}_{s} + \hat{k} sign (S) \end{matrix}]$ (18)

Where $\hat{k}$ represents FLS output as the adaptive switching control gain. The value of $\hat{k}$ is obtained by taking the absolute value of the FLS output.

3.4 Fuzzy adaptive SMC based on ICD approximation

In this section we propose an attitude tracking control based on the ICD described in Sec.3.1 for the nonlinear system (2) without the knowledge of the system dynamics and higher order system states. Using the output of the system, the dynamics of the system is approximated using the ICD during the process of the control design. The designed controller does not require the higher order system states from the onboard sensors as they are approximated using the ICD. Hence, any gyro sensor output errors will not impact the performance of the controller. For the nonlinear system (2), let the desired trajectory y_d = [φ_d θ_d ψ_d] ^T is differentiable up to 2^nd order. Then the system (2) based on tracking error dynamics can be written as $\begin{matrix} {\dot{e}}_{1} = e_{2} \\ {\dot{e}}_{2} = f (x) + g (x) u_{o} - {\ddot{y}}_{d} \end{matrix}$ (19)

Where, e₁ = x₁ - y_d and $e_{2} = x_{2} - {\dot{y}}_{d}$ . Let ${\hat{e}}_{1} = {\hat{x}}_{1} - y_{d}$ and ${\hat{e}}_{2} = {\hat{x}}_{2} - {\dot{y}}_{d}$

We choose a bounded controller u_o such that |u_o| ⩽ l_u

Using (5) we can write $\hat{f} (x) = {\hat{x}}_{3} - g (\hat{x}) u_{o}$ . From (2), we have $f (x) = {\dot{x}}_{2} - g (x) u_{o}$ . Therefore $f (x) - \hat{f} (x) = {\dot{x}}_{2} - {\hat{x}}_{3} + (g (\hat{x}) - g (x)) u_{o}$ (20) therefore, $| f (x) - \hat{f} (x) | ⩽ | {\dot{x}}_{2} - {\hat{x}}_{3} | + | g (\hat{x}) - g (x) | | u_{o} |$ (21) For u_o, let us suppose that a constant l_g exists such that $| g (\hat{x}) - g (x) | ⩽ l_{g} ∥ \hat{x} - x ∥$ therefore from (21) we get $| f (x) - \hat{f} (x) | ⩽ | {\dot{x}}_{2} - {\hat{x}}_{3} | + l_{u} l_{g} ∥ \hat{x} - x ∥$ (22)

To design a trajectory tracking controller based on ICD, we assume that $| - {\hat{x}}_{3} + {\ddot{y}}_{d} - c {\hat{e}}_{2} |$ is bounded, i. e. $| - \hat{f} (x) + {\ddot{y}}_{d} + c \hat{e} | ⩽ l_{1}$ where $\hat{e} = {\hat{e}}_{2}$ and l_inf ⩽ |g(x) | ⩽ l_sup

We define a sliding surface as $S_{o} = e_{2} + c e_{1}$ (23) where S_o = [S₁ S₂ S₃] ^T and c = [c₁ c₂ c₃] ^T, c₁, c₂ and c₃ are positive constants to be tuned during the simulation. The sliding surface based on the ICD observer is ${\hat{S}}_{o} = {\hat{e}}_{2} + c {\hat{e}}_{1}$ (24)

Theorem 2: For system (2) with unknown f(x) and x₂ we design an SMC controller based on ICD approximation as $u_{o} = \frac{1}{g (\hat{x})} (- \hat{f} (x) + {\ddot{y}}_{d} - c {\hat{e}}_{2} - l sign ({\hat{S}}_{o}))$ (25)

Using ICD approximation (5), we have

$x_{1} \to y_{d}, x_{2} \to {\dot{y}}_{d}$ as t→ ∞ where

Proof: For Stability proof we select the Lyapunov function as $V = \frac{1}{2} {S_{o}}^{2}$ (26)

Taking the derivative of (26), we get $\dot{V} = S_{o} {\hat{S}}_{o}$ (27) using (23) in (27), we obtain $\dot{V} = S_{o} ({\dot{e}}_{2} + c e_{2})$ (28) using (19) in (28), we get $\dot{V} = S_{o} (f (x) + g (x) u_{o} - {\ddot{y}}_{d} + c e_{2})$ (29) Now using (25) in (29), we get $\dot{V} = S_{o} (\begin{matrix} f (x) + g (x) \frac{1}{g (\hat{x})} (- \hat{f} (x) + {\ddot{y}}_{d} - c {\hat{e}}_{2} \\ - l sign ({\hat{S}}_{o})) - {\ddot{y}}_{d} + c e_{2} \end{matrix})$ $\begin{matrix} = S_{o} (\begin{matrix} f (x) + \frac{g (\hat{x}) + g (x) - g (\hat{x})}{g (\hat{x})} (- \hat{f} (x) \\ + {\ddot{y}}_{d} - c {\hat{e}}_{2} - l sign ({\hat{S}}_{o})) - {\ddot{y}}_{d} + c e_{2} \end{matrix}) \\ = S_{o} (\begin{matrix} f (x) - \hat{f} (x) - c ({\hat{x}}_{2} - x_{2}) - l sign ({\hat{S}}_{o}) + \\ \frac{g (x) - g (\hat{x})}{g (\hat{x})} (- \hat{f} (x) + {\ddot{y}}_{d} - c {\hat{e}}_{2} - l sign ({\hat{S}}_{o})) \end{matrix}) \end{matrix}$ Therefore, $\begin{matrix} \dot{V} = S_{o} (f (x) - \hat{f} (x)) - S_{o} c ({\hat{x}}_{2} - x_{2}) - {\hat{S}}_{o} l sign ({\hat{S}}_{o}) - \\ (S_{o} - {\hat{S}}_{o}) l sign ({\hat{S}}_{o}) + \frac{g (x) - g (\hat{x})}{g (\hat{x})} S_{o} (- \hat{f} (x) + \\ {\ddot{y}}_{d} - c {\hat{e}}_{2} - l sign ({\hat{S}}_{o})) \end{matrix}$ $\begin{matrix} = {\hat{S}}_{o} (f (x) - \hat{f} (x)) + (S_{o} - {\hat{S}}_{o}) (f (x) - \hat{f} (x)) - \\ {\hat{S}}_{o} c ({\hat{x}}_{2} - x_{2}) - (S_{o} - {\hat{S}}_{o}) c ({\hat{x}}_{2} - x_{2}) + \\ \frac{g (x) - g (\hat{x})}{g (\hat{x})} {\hat{S}}_{o} (- \hat{f} (x) + {\ddot{y}}_{d} - c {\hat{e}}_{2} - l sign ({\hat{S}}_{o})) + \\ \frac{g (x) - g (\hat{x})}{g (\hat{x})} (S_{o} - {\hat{S}}_{o}) (- \hat{f} (x) + {\ddot{y}}_{d} - c {\hat{e}}_{2} - \\ l sign ({\hat{S}}_{o})) - {\hat{S}}_{o} l sign ({\hat{S}}_{o}) - (S_{o} - {\hat{S}}_{o}) l sign ({\hat{S}}_{o}) \end{matrix}$ (30) This implies $\begin{matrix} \dot{V} ⩽ | f (x) - \hat{f} (x) | | {\hat{S}}_{o} | + | S_{o} - {\hat{S}}_{o} | | f (x) - \hat{f} (x) | + \\ | {\hat{S}}_{o} | | c | | ({\hat{x}}_{2} - x_{2}) | + | S_{o} - {\hat{S}}_{o} | | c | | ({\hat{x}}_{2} - x_{2}) | + \\ \frac{| g (x) - g (\hat{x}) |}{| g (\hat{x}) |} | {\hat{S}}_{o} | | - \hat{f} (x) + {\ddot{y}}_{d} - c {\hat{e}}_{2} - l sign ({\hat{S}}_{o}) | + \\ \frac{| g (x) - g (\hat{x}) |}{| g (\hat{x}) |} | S_{o} - {\hat{S}}_{o} | | - \hat{f} (x) + {\ddot{y}}_{d} - c {\hat{e}}_{2} - \\ l sign ({\hat{S}}_{o}) | + l | S_{o} - {\hat{S}}_{o} | - l | {\hat{S}}_{o} | \end{matrix}$ $\begin{matrix} ⩽ | f (x) - \hat{f} (x) | | {\hat{S}}_{o} | + | S_{o} - {\hat{S}}_{o} | | f (x) - \hat{f} (x) | + \\ | {\hat{S}}_{o} | | c | | ({\hat{x}}_{2} - x_{2}) | + | S_{o} - {\hat{S}}_{o} | | c | | ({\hat{x}}_{2} - x_{2}) | + \end{matrix}$ $\begin{matrix} \frac{l_{g} | \hat{x} - x |}{l_{inf}} | {\hat{S}}_{o} | | - \hat{f} (x) + {\ddot{y}}_{d} - c {\hat{e}}_{2} - l sign ({\hat{S}}_{o}) | + \\ \frac{l_{g} | \hat{x} - x |}{l_{inf}} | S_{o} - {\hat{S}}_{o} | | - \hat{f} (x) + {\ddot{y}}_{d} - c {\hat{e}}_{2} - \end{matrix}$ $l sign ({\hat{S}}_{o}) | + l | S_{o} - {\hat{S}}_{o} | - l | {\hat{S}}_{o} |$ (31) When ɛ is sufficiently small, we have $lim_{ɛ \to 0} | S_{o} - {\hat{S}}_{o} | = 0, lim_{ɛ \to 0} | {\hat{x}}_{2} - x_{2} | = 0$ and $lim_{ɛ \to 0} | {\dot{x}}_{2} - {\hat{x}}_{3} | = 0$ and from (22), we have $lim_{ɛ \to 0} | f (x) - \hat{f} (x) | = 0$

Finally, we have $\dot{V} ⩽ 0$ . Therefore, x₁ → y_d as t→ ∞. This concludes the proof. The architecture of ICD-FASMC is depicted in Fig. 6.

Fig. 6

Architecture of ICD-FASMC.

Remark 3: To mitigate the chattering effect of the SMC, the switching gain l of the ICD-FASMC (25) is made adaptive using the FLS described in section 3.3. The final form of ICD-FASMC (25) becomes $u_{o} = \frac{1}{g (\hat{x})} (- \hat{f} (x) + {\ddot{y}}_{d} - c {\hat{e}}_{2} - \hat{l} sign ({\hat{S}}_{o}))$ (32)

Where, $\hat{l}$ is the adaptive switching gain obtained from the FLS.

Where for nonlinear system (2), the parameter vectors of the controller (32) can be defined as u_o = [u₁ u₂ u₃] ^T = [δa δe δr] ^T. $\hat{f} (x) = {[{\hat{f}}_{p} {\hat{f}}_{q} {\hat{f}}_{r}]}^{T}$ represets the estimate of f(x), the uncertain dynamics part, using the ICD. $\hat{l} = {[{\hat{l}}_{1} {\hat{l}}_{2} {\hat{l}}_{3}]}^{T}$ , where ${\hat{l}}_{1}, {\hat{l}}_{2}$ and ${\hat{l}}_{3}$ are the switching control gains obtained using the FLS.

4 Simulation, results and discussion

The proposed control strategies are implemented in MATLAB/SIMULINK environment using a high-fidelity nonlinear simulation of scaled YAK-54 UAV. The UAV simulation incorporates the UAV dynamics, environmental model and proper actuator models to imitate the actual flight performance. The UAV has been used in several research works [28 –30]. Important parameters of the UAV are listed in Table 1 [31]. Small fixed-wing UAVs are normally controlled as bank (roll) to turn. Hence, longitudinal pitch control and lateral roll control are designed and simulated using the proposed control methods in this paper. The aerodynamic and control derivatives of the UAV can be found in [31]. In both the proposed control schemes, ICD is used. The parameters of the ICD used in all the simulations are set as the same with λ₁ = 1, λ₂ = 3, λ₃ = 3, and ɛ = 0.005. For pitch control design using NFASMC, RBFNN consists of two inputs, pitch angle tracking error and the corrected pitch rate as described in Sec. 3.2. Similarly, for roll control design the inputs of the RBFNN are roll angle tracking error and the corrected roll rate. The hidden layer of the RBFNN in both pitch and roll control consists of nine neurons. All the weights of the RBFNN in both cases are initialized with the value of 0.1, bias values as 9 and center values as $[\begin{matrix} - 2 & - 1.5 & - 1 & - 0.5 & 0 & 0.5 & 1 & 1.5 & 2 \\ - 2 & - 1.5 & - 1 & - 0.5 & 0 & 0.5 & 1 & 1.5 & 2 \end{matrix}]$ . The learning rate γ of the RBFNN in NFASMC pitch control is set as 0.09 and in roll control as 0.06. c₂ in pitch control is set as 2 and for roll control c₁ is set as 2.4. All the simulations are run using a fixed step size of 5 milliseconds and simulation results are plotted. Figures 7–9 show the error estimation and pitch control results for NFASMC. Figure 7 shows the error estimation results of the ICD-based error estimation in the case of injected saw tooth type error signal in the output of the pitch rate gyro sensor. It can be seen that the error is estimated with good accuracy with a small difference in the range of –0.1 to+0.1 rad/sec. This difference arises due to the small time lag in the convergence of the ICD to time-varying input θ which is shown in the Fig. 8. In Fig. 8, pitch angle tracking results in the presence of the gyro sensor output errors are plotted including the pitch angle θ, the pitch rate q and the elevator deflection δe. It can be seen that the controller provides good convergence and tracking performance with smooth control action without any chattering (as is clear from the δe plot). Figure 9 shows, the variation of the adaptive gain using the FLS, the sliding surface convergence and the estimation of f_q(x) using the adaptive RBFNN within the NFASMC scheme. It is clear from Fig. 9 that during the pitch control the switching gain of the SMC varies adaptively using the designed FLS. The sliding surface converges to zero as the tracking error becomes zero. The RBFNN approximation of the nonlinear function f_q(x) is also plotted along with the calculated f_q(x) from the available dynamics of the UAV, which shows that the RBFNN’s good approximation performance during the simulation. The roll control design using the proposed NFASMC is validated similarly as the pitch control. Figures 10–12 show the roll rate gyro error estimation results and roll angle command tracking results in the presence of roll rate gyro output errors using the proposed NFASMC. It can be seen from Fig. 10 that roll rate gyro sensor output errors are estimated with good accuracy in real-time. These estimates are used in the NFASMC (18) and the roll angle command tracking control performance is plotted in Figs. 11 and 12. The roll command tracking control also shows good performance as shown in the plots of the roll angle φ and the roll rate p in Fig. 11. Also the aileron deflection δa is smooth without chattering. It is also clear from Fig. 12 that during the roll control the switching gain of the SMC varies adaptively using the designed FLS. The sliding surface converges to zero as the tracking error becomes zero. The RBFNN approximation of the nonlinear function f_p (x) is also plotted along with the calculated f_p (x) from the available dynamics of the UAV, which shows that the RBFNN’s good approximation performance during the simulation.

Table 1
UAV Parameters [31]

No. Parameter Value

1 Length 2080 mm

2 Take-off mass 11 kg

3 Wing area 943869mm²

4 Wing span 2235 mm

5 Vertical tail root/tip airfoil NACA 0009 / 0010

6 Horizontal tail root/tip airfoil NACA 0015 / 0012

7 Wing root/tip airfoil NACA 0016 / 0017

8 Moments of inertia: I_x, I_y, I_z 1.36, 2.848, 4.07 kgm²

No.	Parameter	Value
1	Length	2080 mm
2	Take-off mass	11 kg
3	Wing area	943869mm²
4	Wing span	2235 mm
5	Vertical tail root/tip airfoil	NACA 0009 / 0010
6	Horizontal tail root/tip airfoil	NACA 0015 / 0012
7	Wing root/tip airfoil	NACA 0016 / 0017
8	Moments of inertia: I_x, I_y, I_z	1.36, 2.848, 4.07 kgm²

Fig. 7

Pitch rate gyro sensor error estimation.

Fig. 8

Pitch tracking control performance of NFASMC (θ, q and δe variation).

Fig. 9

Pitch tracking control performance of NFASMC (Adaptive gain, S₂ and f_q(x) variation).

Fig. 10

Roll rate gyro sensor error estimation.

Fig. 11

Roll tracking control performance of NFASMC (φ, p and δa variation).

Fig. 12

Roll tracking control performance of NFASMC (Adaptive gain, S₁ and f_p(x) variation).

The ICD-FASMC (32) as described in Sec. 3.4 is also implemented and validated as the NFASMC. The unknown/uncertain dynamics of the system in the case of ICD-FASMC is estimated using the ICD described in Section 3.1. ICD is also used to estimate the expected gyro sensor outputs, pitch rate q and roll rate p from the attitude angles θ and φ respectively. Therefore, in the case of the proposed ICD-FASMC, gyro sensor outputs are not required for the control implementation as they are being estimated using the ICD. Moreover, the switching gain of the designed controller (32) is made adaptive using the FLS described in Sec. 3.3 to address the chattering effect of the SMC. The parameters of the ICD are kept the same as described earlier. The gain c₂ in pitch control is set as 2 and for roll control c₁ is set as 4.5. ICD-FASMC based pitch command tracking control results are shown in Figs. 13 and 14. It is clear from Fig. 13 that the controller provides good convergence and tracking performance without any chattering of the control surface. Figure 14 shows that the switching gain varies adaptively with the sliding surface value and the designed strategy approximates the uncertain f_q(x) with good accuracy. Similarly, the ICD-FASMC is implemented and validated for roll command tracking control and the simulation results are shown in Figs. 15 and 16. The roll command tracking control also shows good convergence and tracking performance without chattering owing to the adaptive gain using the FLS.

Fig. 13

Pitch tracking control performance of ICD-FASMC (θ, q and δe variation).

Fig. 14

Pitch tracking control performance of ICD-FASMC (Adaptive gain, S₂ and f_q(x) variation).

Fig. 15

Roll tracking control performance of ICD-FASMC (φ, p and δa variation).

Fig. 16

Roll tracking control performance of ICD-FASMC (Adaptive gain, S₁ and f_p(x) variation).

5 Conclusion

Two adaptive control schemes for fixed-wing UAVs with uncertain dynamics and corrupted/erroneous gyro sensor outputs are proposed and implemented in this work. A NFASMC where the sensor output errors are estimated using an ICD as an observer. Uncertain dynamics of the UAV in NFASMC is estimated using an adaptive RBFNN. The control is designed using both the erroneous sensor outputs and the estimate of the sensor output errors provided by the ICD-based error estimation. The gain of the switching part of the NFASMC is made adaptive using a FLS to address the chattering of the SMC. The simulation results for attitude tracking of the UAV in the presence of sensor output errors show the effectiveness of the proposed NFASMC. Another control scheme ICD-FASMC is also proposed, where the requirement of the erroneous gyro sensor outputs is bypassed using ICD to estimate the angular rates of the UAV. The uncertain dynamics of the UAV in ICD-FASMC is estimated within the ICD-based proposed control scheme. Additionally, the gain of the switching term of the ICD-FASMC is also made adaptive using the FLS. The simulation results of the ICD-FASMC also show good convergence and tracking performance with smooth control action. The control problem can be efficiently handled by adopting one of the proposed control strategies. The proposed NFASMC can work well both in the presence and absence of sensor output errors. Moreover, there is no requirement for controller reconfiguration at the onset of the sensor errors. Alternatively, we can use the ICD-FASMC which does not require the gyro sensor outputs. Another scheme can be developed where we can use any good controller in the sensor error-free condition and switch to ICD-FASMC when errors in the gyro sensors are detected using the ICD-based error estimation scheme.

Appendix

I_xx, I_yy and I_zz respectively represent the inertia moments of the UAV about the x, y and z axis. I_xz is the inertia moment about xz plane. If we define, $τ = {(I_{xx} I_{zz} - I_{xz}^{2})}^{- 1}$ , then a₁-a₉ can be defined as; $a_{1} = (I_{xx} - I_{yy} + I_{zz}) I_{xz} τ, a_{2} = ((I_{yy} - I_{zz}) I_{zz} - I_{xz}^{2}) τ$ $a_{3} = I_{zz} τ, a_{4} = I_{xz} τ, a_{5} = (I_{zz} - I_{xx}) I_{yy}^{- 1}$ $a_{6} = I_{xz} I_{yy}^{- 1}, a_{7} = I_{yy}^{- 1}, a_{8} = ((I_{xx} - I_{yy}) I_{xx} + I_{xz}^{2}) τ, a_{9} = I_{xx} τ$

The moments about x, y and z body axis are denoted by L M and N respectively.

\begin{array}{l} \bar{L} = \bar{q} \bar{S} b [C_{l_{β}} β + C_{l_{δ a}} δ a + C_{l_{δ r}} δ r + (C_{l_{p}} + C_{l_{r}} r) b / 2 V_{\infty}] \\ M = \bar{q} \bar{S} \bar{c} [C_{m_{0}} + C_{m_{α}} α + C_{m_{δ e}} δ e + (C_{m_{q}} q + C_{m_{\dot{α}}} \dot{α}) \bar{c} / 2 V_{\infty}] \end{array}

$N = \bar{q} \bar{S} b [C_{n_{β}} β + C_{n_{δ a}} δ a + C_{n_{δ r}} δ r + (C_{n_{p}} p + C_{n_{r}} r) b / 2 V_{\infty}] \bar{c}$ represents the mean aerodynamic chord, b the wing span, S represents the surface area of the wing and q the dynamic pressure. The UAV’s total velocity is $V_{\infty} = \sqrt{u^{2} + v^{2} + w^{2}}$

The aerodynamic forces F_x, F_y and F_z projected into the body x, y and z axis respectively are defined as $\begin{array}{l} F_{x} = \bar{q} \bar{S} [C_{L} \sin α - C_{D} \cos α] \\ F_{y} = \bar{q} \bar{S} [C_{Y}_{_{β}} β + C_{Y_{u}} (u / V_{\infty}) + C_{Y_{δ a}} δ a + C_{Y_{δ r}} δ r + (C_{y_{p}} p + C_{Y_{r}} r) b / 2 V_{\infty}] \\ F_{z} = - \bar{q} \bar{S} [C_{D} \sin α + C_{L} \cos α] \end{array}$

The angle of attack is $α = \tan^{- 1} (w / u)$ and the side-slip angle $β = \sin^{- 1} (v / V_{\infty})$

Lift and drag coefficients, C_L and C_D respectively, are defined as $\begin{array}{l} C_{L} = C_{L_{0}} + C_{L_{α}} α + C_{L_{u}} (u / V_{\infty}) + C_{L_{δ e}} δ e + (C_{L_{\dot{α}}} \dot{α} + C_{L_{q}} q) \bar{c} / 2 V_{\infty} \\ C_{D} = C_{D_{0}} + C_{D_{α}} α + C_{D_{u}} u / V_{\infty} + C_{D_{δ e}} δ e + C_{L}^{2} / π A R \end{array}$

Where AR is the aspect ratio. Now, f_p, f_q, f_r and g₁, g₂, g₃, g₄, g₅ can be calculated as $f_{p} = (a_{1} p + a_{2} r) q + \bar{q} \bar{S} b (a_{3} (C_{l_{β}} β + (C_{l}_{p} p + C_{l}_{r} r) b / 2 V_{\infty}) + a_{4} (C_{n_{β}} β + (C_{n_{p}} p + C_{n_{r}} r) b / 2 V_{\infty}))$ $f_{q} = a_{5} p r - a_{6} (p^{2} - r^{2}) + a_{7} \bar{q} \bar{S} \bar{c} (C_{m_{0}} + C_{m_{α}} α + (C_{m_{q}} q + C_{m_{\dot{α}}} \dot{α}) \bar{c} / 2 V_{\infty})$ $f_{r} = (a_{8} p + a_{1} r) q + \bar{q} \bar{S} b (a_{4} (C_{l_{β}} β + (C_{l}_{p} p + C_{l}_{r} r b / 2 V_{\infty}) + a_{9} (C_{n_{β}} β + (C_{n_{p}} p + C_{n_{r}} r) b / 2 V_{\infty}))$ $\begin{array}{l} g_{1} = a_{3} \bar{q} \bar{S} b (a_{3} C_{l_{δ a}} + a_{4} C_{n_{δ a}}), g_{2} = \bar{q} \bar{S} b (a_{3} C_{l_{δ r}} + a_{4} C_{n_{δ r}}) \\ g_{3} = a_{7} \bar{q} \bar{S} \bar{c} C_{m_{δ e}}, g_{4} = \bar{q} \bar{S} b (a_{4} C_{l_{δ a}} + a_{9} C_{n_{δ a}} (C_{n_{p}} p + C_{n_{r}} r)) \end{array}$ and $g_{5} = \bar{q} \bar{S} b (a_{4} C_{l_{δ r}} + a_{9} C_{n_{δ r}})$ Thus g (x) can be defined as $g (x) = [\begin{matrix} g_{1} & 0 & g_{2} \\ 0 & g_{3} & 0 \\ g_{4} & 0 & g_{5} \end{matrix}]$

The UAV parameters, aerodynamic and control derivatives of the UAV (C_{•_•}) can be found in [31].

Footnotes

Acknowledgment

We are thankful to the editors and reviewers for processing and reviewing this paper.

This work is supported by the Aeronautical Science Foundation of China under Grant No. 20180753005 and Grant No. 201958053003.

References

Utkin

, Guldner

, Shi

Sliding mode control in electro-mechanical systems. CRC Press (2017).

Riaz

, Tayyeb

and Amin

A.A.

, A review of sliding mode control with the perspective of utilization in fault tolerant control, Recent Advances in Electrical & Electronic Engineering (Formerly Recent Patents on Electrical & Electronic Engineering) 14(3) (2021), 312–324.

Alyoussef

, Kaya

A review on nonlinear control approaches: sliding mode control back-stepping control and feedback linearization control, In International Engineering and Natural Sciences Conference (IENSC 2019) (2019).

Schilling

R.J.

, Carroll

J.J.

and Al-Ajlouni

A.F.

, Approximation of nonlinear systems with radial basis function neural networks, IEEE Transactions on neural networks 12(1) (2001), 1–15.

et al., Using radial basis function networks for function approximation and classification, International Scholarly Research Notices 2012 (2012).

Yang

et al., Investigation of neural networks for function approximation, Procedia Computer Science 17 (2013), 586–594.

Cheng

et al., Adaptive neural network control of nonlinear systems with unknown dynamics, Advances in Space Research 67(3) (2021), 1114–1123.

Taimoor

et al., Novel neural observer based fault estimation, reconstruction and fault-tolerant control scheme for nonlinear systems, Journal of Intelligent & Fuzzy Systems 41(1) (2021), 355–386.

Xiang

et al., Survey on fuzzy-logic-based guidance and control of marine surface vehicles and underwater vehicles, International Journal of Fuzzy Systems 20(2) (2018), 572–586.

10.

Abro

G.E.M.

et al., Model-free-based single-dimension fuzzy SMC design for underactuated quadrotor UAV. in Actuators. 2021. Multidisciplinary Digital Publishing Institute.

11.

Liu

Intelligent control design and Matlab simulation. Springer (2018).

12.

Wang

Universal approximation using differentiators and application to feedback control. arXiv preprint arXiv:1102.2794, 2011.

13.

Wang

and Shirinzadeh

, High-order nonlinear differentiator and application to aircraft control, Mechanical Systems and Signal Processing 46(2) (2014), 227–252.

14.

and Zhang

, Sliding mode tracking control with differential evolution optimisation algorithm using integral-chain differentiator in uncertain nonlinear systems, International Journal of Systems Science 49(6) (2018), 1345–1352.

15.

Liu

, Wang

Advanced sliding mode control for mechanical systems. Springer (2012).

16.

DeVore

, Hanin

and Petrova

, Neural network approximation. Acta Numerica, 30 (2021), 327–444.

17.

Montazer

G.A.

et al., Radial basis function neural networks: A review, Comput. Rev. J 1(1) (2018), 52–74.

18.

Shabbir

et al., Global fast terminal sliding mode based radial basis function neural network for accurate fault estimation in nonlinear systems, Journal of Intelligent & Fuzzy Systems 42(3) (2022), 2229–2245.

19.

et al., Sliding mode differentiator based tracking control of uncertain nonlinear systems with application to hypersonic flight, Asian Journal of Control 21(1) (2019), 143–155.

20.

Taimoor

, Aijun

Neural-sliding mode approachbased adaptive estimation, isolation and tolerance of aircraft sensor fault, Aircraft Engineering and Aerospace Technology (2019).

21.

Ryadchikov

et al., Differentiator-based velocity observer with sensor bias estimation: an inverted pendulum case study, IFAC-PapersOnLine 52(16) (2019), 436–441.

22.

Yang

and Yan

, Attitude regulation for unmanned quadrotors using adaptive fuzzy gain-scheduling sliding mode control, Aerospace Science and Technology 54 (2016), 208–217.

23.

Hamza

M.F.

et al., A survey on advancement of hybrid type 2 fuzzy sliding mode control, Neural Computing and Applications 30(2) (2018), 331–353.

24.

Fuzzy Approximation-based task-space control of robot manipulators with remote center of motion constraint, IEEE Transactions on Fuzzy Systems (2022).

25.

Liu

et al., Fuzzy approximation-based adaptive finite-time control for nonstrict feedback nonlinear systems with state constraints, Information Sciences 548 (2021), 101–117.

26.

Yang

, Jiang

and Yin

, Adaptive fuzzy fault-tolerant control for Markov jump systems with additive and multiplicative actuator faults, IEEE Transactions on Fuzzy Systems 29(4) (2020), 772–785.

27.

Stevens

B.L.

, Lewis

F.L.

, Johnson

E.N.

Aircraft control and simulation: dynamics, controls design, and autonomous systems. John Wiley & Sons (2015).

28.

Shah

M.Z.

, Samar

and Bhatti

A.I.

, Lateral track control of UAVs using the sliding mode approach: from design to flight testing, Transactions of the Institute of Measurement and Control 37(4) (2015), 457–474.

29.

Shabbir

et al., Active Fault Tolerant Control for a UAV Subject to Sensor Faults Using Sliding Mode Control, In 2022 8th International Conference on Control, Automation and Robotics (ICCAR). IEEE (2022).

30.

Ali

S.U.

et al., Higher-order sliding mode based lateral guidance for unmanned aerial vehicles, Transactions of the Institute of Measurement and Control 39(5) (2017), 715–727.

31.

Shah

M.Z.

Guidance of Aerospace Vehicles: A Sliding Mode Approach. Mohammad Ali Jinnah University Islamabad (2015).