2.1 Mechanism design and system modelling

When starting the conceptual design of a wing kit, firstly, a designer should consider how ammunitions, particularly the ones with a wing kit, are carried in fighter aircrafts. In fact, ammunitions with a wing kit attachment can only be stored in fighter aircrafts when the wings are at their closed position, or in other words when the swept-back angle is 90° [Reference Teope29, Reference Rogers and Costello30]. Previous wing kits are designed in such a way that they could operate only one-time and one-direction to reach a fixed swept-back angle after the ammunition is launched from the aircraft as afore-mentioned in the introduction section [Reference Hoehn and Ryder5]. These systems are mostly driven by a pulley, an electromechanical actuator or piston member (i.e., pneumatic, pyrotechnic-based, hydraulic) to adjust and lock the wings [Reference Harris and Levy14–Reference O’Shea25, Reference Myers31–Reference Vainshtein and Bouhryakov33]. However, pulleys or pre-compressed tension springs need to be installed again for any reverse direction movements. For piston driven systems, an extra gas tank is necessary for the similar purpose. In other words, it is simply not possible to control the wings in two-directions with current wing kit designs, unless these improvements are implemented. On the other hand, these additions would inevitably increase the cost and complicate the mechanisms, as well as necessitating extra space. Instead of these, the proposed FWS mechanism could drive the wings in two directions by a BLDC motor. The electromechanical actuator (BLDC motor) could change the direction through changing the sign of current, and thus, the swept-back angle could easily be adjusted to reach any desired position during the flight without the necessity of any brake or fixing mechanisms.

In this study, working conditions were chosen as 40°–90° for swept-back angle, 1 s for time elapsed to open/close the wings, and two flight scenarios were considered with and without external loads on wings. In other words, the FWS considered here must open/close the wings from 40° to 90° (and vice versa) in 1 s and can operate under aerodynamic disturbances. A slider-crank mechanism was designed to fulfill these requirements, and the FWS contains a screw for slider movement, as well as twin connection rods to pull the wings as can be seen in Fig. 1. The FWS was designed with symmetrical connection rods and wings that rotate in reverse directions to balance the inertia and moments generated during the wing rotation process, and hence, minimising any adverse effects on flight stability. The motor assembly was put inside a casing and mounted through screw to the FWS. The connection rods were connected to the slider and wings by revolute joints. Link dimensions and eccentricity were optimised through the closed-loop equation to minimise control torque and mechanism stroke. The closed-loop equation Equation (1) and its derivative Equation (2) are:

(1) \begin{equation}s + ic = {L_2}{e^{i{\theta _2}}} + {L_3}{e^{i{\theta _3}}}\end{equation}

(2) \begin{equation}\dot s = {L_2}{\dot \theta _2}{e^{i{\theta _2}}} + {L_3}{\dot \theta _3}{e^{i{\theta _3}}}\end{equation}

where ${L_2},\;\;{L_3},\;\;c,\;\;{\theta _2},\;\;{\theta _3}$ , ${\dot \theta _2},\;{\dot \theta _3},\;s$ and $\dot s$ are the distance between wing hinge to link hinge, length of connection rod, eccentricity, wing angle, connection rod angle, angular velocity of wing ( ${L_2}$ ), angular velocity of connection rod ( ${L_3}$ ), slider position and velocity of slider, respectively (Fig. 1). When wing position changes from closed to open or vice versa, the relationship between the position and the velocity of each mechanism components and the swept-back angle is shown in Fig. 2.

Figure 1. CAD design of the FWS at 90° swept-back angle: components and mechanism illustration.

Figure 2. Changes in the swept-back angle versus position (a), and velocity (b) of each mechanism links.

In mechanism design, it is important to consider the mechanism transmission ratio, which defines the relation between input and output [Reference Rothenhofer, Walsh and Slocum34]. Regarding the transmission ratio of the FWS developed in this work, it represents the relation between angular velocity of motor and angular velocity of wings, and can be calculated by Equation (3):

(3) \begin{equation}{N_{velocity}} = \frac{{{{\dot \theta }_{motor}}}}{{{{\dot \theta }_2}}} = \frac{{2\pi i}}{p}{L_2}\frac{{sin\left( {{\theta _2} - {\theta _3}} \right)}}{{cos\left( {{\theta _3}} \right)}}\end{equation}

where ${N_{velocity}}$ is the transmission ratio, ${\dot \theta _{motor}}$ is the angular velocity of motor, $p$ is the screw pitch and i is the reducer ratio.

For convenience, the motor position (θ_motor) is divided by ${N_{velocity}}$ to calculate the actual rotation of wings. The mean of ${N_{velocity}}$ was found to be 671 by averaging it for different swept-back angle values as depicted in Fig. 3a. Briefly, it implies that when the motor completes about 1.86 turns (671° rotation), the wing swept-back angle is changed by 1 degree. Figure 3b illustrates the relation between the motor torque and swept-back angle to adjust opened position (40°) in 1 s under varying loads. The aerodynamic disturbance on the wings is increasing with swept-back angle, and thus higher motor torques are required to reach higher angles, as expected. Also, Fig. 3b shows normalised motor rpm values with respect to maximum motor rpm and as can be seen, the normalised values increase to some point, after that starting to decrease due to the behaviour of the slider-crank mechanism [Reference Uicker35, Reference Söylemez36].

Figure 3. Changing of transmission ratio (a), and motor torque under load (b) with swept-back angle.

Following the mechanical design, approximately 40 different components are produced to assemble the proposed FWS. As the proposed kit is designed to be used in guided ammunitions, structural components are manufactured from aluminum due to its optimum weight and mechanical properties [Reference Akgul, Erden and Ozbay37]. Transmission parts are manufactured from stainless steel due to its excellent wear resistance. The photographs of the as-manufactured FWS with 40° and 90° swept-back angle are given in Fig. 4 for better illustration.

Figure 4. Photographs of the FWS: the swept-back angle is 90° (a and c), and 40° (b and d) in the absence (a and b) and presence (c and d) of aerodynamic loads.

To evaluate the system dynamics, the mathematical model of the FWS is required. Since the current controlled BLDC motor driver is utilised in the present work, the system transfer function should be established between the applied current function and the measured motor positions. BLDC motor is modelled based on the assumptions that stator and rotor reluctance angles are not changing, power semiconductor devices are ideal, mechanical components are rigid, iron losses are negligible and the motor is unsaturated [Reference Krishnan38–Reference Shanmugasundram, Zakariah and Yadaiah41]. In this case, there should be a linear relationship between output torque $\tau $ and current $i$ through the windings, $\tau = {K_t}i$ [Reference Krishnan38–Reference Shanmugasundram, Zakariah and Yadaiah41]. The dynamic equation of the FWS, and its Laplace transform are given in Equation (4), (5) and (6):

(4) \begin{equation}{J_{eq}}.{\ddot \theta _{motor}}(t) + {B_{eq}}.{\dot \theta _{motor}}(t) = {\tau _{motor}} = {K_t}.{i_{motor}}(t)\end{equation}

(5) \begin{equation}{J_{eq}}.{\theta _{motor}}\left( s \right).{s^2} + {B_{eq}}.{\theta _{motor}}\left( s \right).s = .{K_t}.{I_{motor}}\left( s \right)\end{equation}

(6) \begin{equation}\frac{{{\theta _{motor}}\left( s \right)}}{{{I_{motor}}\left( s \right)}} = \frac{{{K_t}}}{{{J_{eq}}.{s^2} + {B_{eq}}.s}} = \frac{{\dfrac{{{K_t}}}{{{B_{eq}}}}}}{{\dfrac{{{J_{eq}}}}{{{B_{eq}}}}.{s^2} + s}}\end{equation}

where ${K_t}$ is torque constant of the motor, ${J_{eq}}$ is the equivalent moment of inertia of FWS reduced to motor, $i$ is motor current and ${B_{eq}}$ is equivalent viscous friction of the system reduced to motor. Following this, the relationship between motor current and swept-back angle is defined as a transfer function in Equation (7):

(7) \begin{equation}{G_{plant}}\left( s \right) = \frac{{\dfrac{{{\theta _{motor}}\left( s \right)}}{{{N_{velocity}}}}}}{{{I_{motor}}\left( s \right)}} = \frac{{{\theta _2}\left( s \right)}}{{{I_{motor}}\left( s \right)}} = \frac{{\dfrac{{{K_t}}}{{{N_{velocity}}}}}}{{{J_{eq}}.{s^2} + {B_{eq}}.s}} = \frac{{\dfrac{{{K_t}}}{{{N_{velocity}}{B_{eq}}}}}}{{\frac{{{J_{eq}}}}{{{B_{eq}}}}.{s^2} + s}}\end{equation}

2.2 System identification

System identification is a technique that converts physical systems or processes to mathematical models by obtaining experimental data through sending various signals. These mathematical models are used in control system design, fault detection, estimation of system characteristics and understanding of system dynamics [Reference Karnopp, Margolis and Rosenberg42]. In fact, a proper selection of the input signal increases the model prediction accuracy. There are various input signals that could be used to utilise for the identification of a system, including chirp, multi-sine, pseudo random binary sequence (PRBS) and impulse [Reference Soares and Serpa43, Reference Vuojolainen44]. Among them, PRBS is a popular input signal for system identification due to its simplicity [Reference Vermeulen, Strauss and Shikoana45, Reference Bnhamdoon, Hanif and Akmeliawati46]. It involves a sequence of step functions that could be utilised in a wide range of frequencies. Therefore, PRBS signal was selected in the present work to excite physical FWS system dynamics.

The PRBS signal excited the physical FWS for the identification process and the system was assumed as a grey box system. Besides, the model orders were known through the dynamic equation of the FWS. The amplitude of PRBS was arranged as 2A since our preliminary studies show that 2A amplitude allows the slider to go through the entire mechanism stroke.

Regarding estimation of the system models, Box-Jenkins (BJ) is one of the most widely used techniques to develop reliable models in system identification [Reference Bnhamdoon, Hanif and Akmeliawati46, Reference Guarin and Kearney47]. One of the advantages of using a BJ model is that it provides better predictions for closed-loop models [Reference Forssell and Ljung48]. During identification of the system, some problems might be encountered that make the model estimation worse. For example, if the noise model has the same poles as the system model, the sensor measurement noise might resemble the actual input signal, and thus, making it difficult to correctly estimate the plant states [Reference Schoukens49]. Using the BJ method helps to avoid such risks, because the noise and the process are modelled independently in the BJ model structure [Reference Schoukens49]. The BJ model is structured as a discrete-time transfer function with noise integration model and given in Equation (8) [Reference Ljung50–Reference Tangirala52]:

(8) \begin{equation}y(t) = \mathop \sum \limits_{i = 1}^{{n_u}} \frac{{{B_i}\left( q \right)}}{{{F_i}\left( q \right)}}\;u\left( {t - n{k_i}} \right) + \frac{{C\left( q \right)}}{{D\left( q \right)\left( {1 - {q^{ - 1}}} \right)}}e(t)\end{equation}

where y(t) is the measured noisy output, n_u is the number of input channel, ${n_k}$ is the input delay, u(t) is a controlled input, e(t) is white noise, $\frac{1}{{1 - {q^{ - 1}}}}$ is the integrator in noise channel, and B, C, D and F are polynomial variables expressed in time shift operator, ${q^{ - 1}}$ . Also, orders of the BJ model $\left( {{n_b},\;{n_c},\;{n_d},\;{n_f}} \right)\;$ are defined as [Reference Ljung50–Reference Tangirala52]:

(9) \begin{equation}{n_b}:\;B\left( q \right) = {b_1} + {b_2}{q^{ - 1}} + \ldots + {b_{{n_b}}}{q^{ - {n_b} + 1}}\;\end{equation}

\begin{equation*}{n_c}:\;C\left( q \right) = 1 + {c_1}{q^{ - 1}} + \ldots + {c_{{n_c}}}{q^{ - {n_c}}}\end{equation*}

\begin{equation*}{n_f}:\;F\left( q \right) = 1 + {f_1}{q^{ - 1}} + \ldots + {f_{{n_f}}}{q^{ - {n_f}}}\end{equation*}

\begin{equation*}{n_d}:\;D\left( q \right) = 1 + {d_1}{q^{ - 1}} + \ldots + {d_{{n_d}}}{q^{ - {n_d}}}\end{equation*}

(10) \begin{equation}\left[ {{n_b}\;{n_c}\;{n_d}\;{n_f}\;{n_k}\left] { = } \right[1\;4\;4\;2\;1} \right]\end{equation}

A coefficient vector is defined to represent Box–Jenkins model orders and delays [Reference Khalfi51]. This vector must consist of positive integers. We determined the coefficient vector of our BJ model iteratively through goodness of fit and mean squared error criteria as given in Equation (10). Therefore, the resultant vector suggests that we have a total of 11 coefficients (b1, c1, c2, c3, c4, f1, f2, f3, f4, d1, d2).

2.3 Controller design

The controller of the developed FWS was designed to control the swept-back angle and adjust the desired position in a short time. Simply, position error is calculated using an encoder and reference angle. Then, position error is used as an input of our controller. The controller provides a current signal as its output. The motor driver converts this signal to a proper current, which becomes the input of our BLDC motor. In this work, LQR, LQI, SMC and SOSMC are designed, and their real-time results are compared.

LQR is a widely used optima l control technique in practical engineering, and thus it is chosen for performing the precision position control of FWS. It seeks a controller that could minimise the cost function, J, as given in Equation (13). LQR takes the states of the dynamical system and control input into account to make optimal control decisions using two positive definite matrices, which are called Q and R. In fact, the elements of these matrices can be adjusted by designers to better satisfy the system requirements. Q matrix penalises the departure of system states, while R matrix penalises the control input [Reference Shehzad, Bilal and Ahmad53]. The state space model of the system is defined as below:

(11) \begin{equation}\dot x(t) = Ax(t) + Bu(t) = \left[ \begin{array}{l@{\quad}c} 0 & 1 \\[3pt] 0 & \dfrac{-{B_{eq}}}{{J_{eq}}}\end{array} \right]x(t) + \left[ \begin{array}{c} 0\\[3pt]\dfrac{K_t}{B_{eq}} \end{array}\right]i(t)\;\;\;\;\;\;\;\;\;t \ge 0,\;\;\;x\left( 0 \right) = {x_0}\end{equation}

(12) \begin{align}y(t) & = Cx(t) + Du(t) = \left[ {1\;\;\;\;0} \right]x(t) + Di(t)\nonumber\\u(t) & = - {K_{LQR}}\;x(t)\end{align}

where $x(t)$ is the state vector, $\dot x(t)$ is derivative of the state vector, $u(t)\;$ is the input vector and ${K_{LQR}}$ is LQR gain matrix that minimises the cost function J.

(13) \begin{equation}J = \mathop \smallint \nolimits_0^\infty \left( {{x^T}Qx + {u^{\rm{T}}}Ru} \right)dt\end{equation}

Considering LQI control, it contains a simple state feedback system used in LQR in addition there is output feedback by an integrator that allows servo reference tracking properties [Reference Kisszölgyémi, Beneda and Faltin54], and it can be a proper optimal controller for precision control of the swept-back angle. In LQI control, the error is defined as the state of the system and then, the state equation model of the system including the integrator is formed. Thus, steady state errors do not occur since the integral effect will be added to the output error. By this way, the error robustly tends to zero as quickly as possible, assuring robustness of tracking ability. Therefore, the controller has better performance especially against constant disturbances [Reference Kisszölgyémi, Beneda and Faltin54–Reference Altun56]. The matrices of dynamic system, A and B rearranged as $\hat A$ and $\hat B$ , and are given in Equation (14), respectively.

(14) \begin{equation}\hat A = \left[ \begin{array}{l@{\quad}l} A & 0 \\- C & 0 \end{array} \right],\;\hat B = \left[ \begin{array}{c} B\\ 0 \end{array} \right]\end{equation}

(15) \begin{equation}u(t) = - {K_{LQI}}\;z(t) = - {K_{LQI}}\;\left[ {\begin{array}{*{20}{c}}x\\{{x_i}}\end{array}} \right]\end{equation}

For sliding mode control (SMC), it can change the dynamics of a nonlinear system by applying a discontinuous control signal. The discontinuous control signal enforces the system to slide along sliding surface. Considering the SMC characteristics, it can be claimed that system performance has insensitivity to parameter variations, and thus, could result in complete rejection of disturbances [Reference Young, Utkin and Özgüner57, Reference Liu58]. Therefore, it is a suitable controller for FWS, which operates under uncertainty and external disturbance.

Although SMC is a robust controller, the sliding mode has disadvantages such as the chattering phenomenon [Reference Lee and Utkin59, Reference He and Lin60]. In fact, a number of analytical design methods were proposed to overcome the effects of the chattering [Reference Lee and Utkin59, Reference Utkin61–Reference Bartolini, Ferrara and Usai63]. For instance, one approach to reduce the chattering involves replacement of signum function with a high gain saturation function [Reference Lee and Utkin59]. Another approach is based on introducing observers for the unmodeled part of the system [Reference Utkin61]. The third approach is to design a higher order sliding mode controller to retain the robustness, and thus, eliminating the chattering [Reference Mondal and Mahanta62–Reference Liu64]. In this study, we, first, preferred to use the saturation function due primarily to its simplicity, and the plant is considered to be a single-input second-order linear model as given in Equation (16):

(16) \begin{equation}\ddot x(t) = - \left( {A + \Delta {\rm{A}}} \right)\dot x - \left( {B + \Delta {\rm{B}}} \right)x + \left( {C + \Delta {\rm{C}}} \right)u(t) + d(t)\end{equation}

where A, B and C are nominal system parameters, $\Delta {\rm{A}}$ , $\Delta {\rm{B}}$ and $\Delta {\rm{C}}$ are unknown model uncertainties introduced due to the plant parameters, nonlinear friction, and unmodeled dynamics, and d(t) is external disturbance of the system. In this case, $A = - \frac{{{B_{eq}}}}{{{J_{eq}}}},\;B = 0$ and $C = \frac{{{K_t}}}{{{J_{eq}}}}$ . The lumped uncertainty satisfies $\left| D \right| \le {D_{max}}$ condition, is bounded, unknown, denoted with $D\left( {t,u(t)} \right)$ , and is given below:

(17) \begin{equation}D\left( {t,u(t)} \right) = \pm \Delta A\dot x \pm \Delta Bx \pm \Delta Cu(t) + d(t)\end{equation}

Then, Equation (16) can be rewritten as

(18) \begin{equation}\ddot x(t) = - A\dot x - Bx + Cu(t) + D\left( {t,u(t)} \right)\;\end{equation}

The upper bound for the uncertainty ${D_{max}}$ is member of positive real constants ${R^ + }$ , and is given by

(19) \begin{equation}{D_{max}} = \Delta A\left| {\dot x} \right| + \Delta B\left| x \right| + \Delta C\bar u(t) + \left| {d(t)} \right|\end{equation}

where $\bar u$ is the input hard constraint | $u(t)| \le \bar u.$ In sliding mode, the purpose of the control is to specify a control input, u(t), such that the tracking error, e(t), converges to zero when the closed-loop system follows the desired trajectory. SMC enforces the error to approach the sliding manifold, and then error moves along the origin. The tracking error $e$ and its derivatives are defined in Equation (20), where, ${x_d}$ refers to the desired position.

(20) \begin{equation}e(t) = {x_d}(t) - x(t)\;,\;\;\dot e(t) = {\dot x_d}(t) - \dot x(t)\;,\;\;\ddot e(t) = {\ddot x_d}(t) - \ddot x(t)\end{equation}

When time goes to infinity, error should converge to zero, therefore, s must be a stable sliding surface; only then the error will die out asymptotically. This means that the desired reference trajectory can be followed asymptotically [Reference Eker65]. The sliding surface variable, s, is defined as [Reference Slotine and Li66]

(21) \begin{equation}s(t) = {\left( {\lambda + \frac{d}{{dt}}} \right)^2}\smallint e(t)dt = {\lambda ^2}\smallint e(t)dt + 2\lambda e(t) + \dot e(t),\;\;{\rm{\lambda }} \in {R^ + }\end{equation}

\begin{equation*}\;\dot s(t) = {\lambda ^2}e(t) + 2\lambda \dot e(t) + \ddot e(t)\end{equation*}

(22) \begin{equation}\dot s(t) = \left( {{{\ddot x}_d}(t) + A\dot x(t) + B{\rm{x}}\left( {\rm{t}} \right) - Cu(t) - D\left( {t,u(t)} \right) + {\lambda ^2}e(t) + 2\lambda \dot e(t)} \right)\end{equation}

The process of sliding mode control includes two separate phases, one of which is known as sliding phase, and is considered when “s(t) = 0” and “ $\dot s$ (t) = 0”. The second phase is the reaching phase, which is used when “s(t) ≠ 0”. The control laws (i.e., equivalent control, switching control) are derived for each of these phases [Reference Eker65]. The $equivalent\;control$ is described when the trajectory of states is near a sliding surface (s = 0), and is obtained from Equation (20). Also, it is represented by ${u_{eq}}$ , and is arranged to cancel out known terms as [Reference Eker65–Reference Eker67]:

(23) \begin{equation}{u_{eq}}(t) = \frac{1}{C}\left( {{{\ddot x}_d}(t) + A\dot x(t) + B{\rm{x}}\left( {\rm{t}} \right) + {\lambda ^2}e(t) + 2\lambda \dot e(t)} \right)\end{equation}

The switching control is introduced to deal with matching uncertainties and disturbances, and symbolised with ${u_{sw}}$ [Reference Eker65–Reference Fei68]:

(24) \begin{equation}{u_{sw}}(t) = - ksign\left( {s(t)} \right)\end{equation}

where, $k$ is a positive constant, $k \in {R^ + }$ , and to dominate uncertainty and disturbance, $k$ must be properly selected to satisfy $k \gt {D_{max}}.$ Reaching condition is known as fundamental inequality in sliding mode and can be given as $s(t)\dot s(t) \lt 0$ [Reference Slotine and Li66, Reference Eker67, Reference Saeed and Soltanpour69]. Considering the plant with uncertainties and disturbances given in Equation (16) and to satisfy reaching condition, the control input signal $u(t)$ could be designed as given below:

(25) \begin{equation}u(t) = {u_{eq}}(t) + {u_{sw}}(t)\end{equation}

If Equation (24) is substituted into Equation (22), one has:

(26) \begin{equation}\dot s(t) = \left( { - ksign\left( {s(t)} \right) - \;D\left( {t,u(t)} \right)} \right)\end{equation}

To avoid chattering, saturation function Equation (27) can also be used instead of the signum function Equation (24) [Reference Eker65].

(27) \begin{equation}sat\left( {\frac{s}{\alpha }} \right) = \left\{ {\begin{array}{*{20}{c}}{\dfrac{s}{\alpha }\;if\;\left| {\dfrac{s}{\alpha }} \right| \lt 1}\\[10pt]{sign\left( {\dfrac{s}{\alpha }} \right)\;if\;\left| {\dfrac{s}{\alpha }} \right| \geqslant 1}\end{array}} \right.\end{equation}

(28) \begin{equation}u = {u_{eq}} - k\frac{{s(t)}}{{\left| {s(t)} \right| + \alpha }}\qquad\end{equation}

where $\alpha $ and k are positive constants and $\alpha $ is defined as the thickness of the boundary layer.

Briefly, the value of continuous input signal changes between ±k. Thus, in this study, the switching control has the modified saturation function Equation (28).

Regarding second-order sliding mode control, it is implemented to overcome the chattering phenomenon, as also afore-mentioned as the third approach for this purpose in SMC. In fact, SOSMC provides smooth control and better performance in the practice with less chattering while preserving the robustness properties [Reference Eker67]. In the particular case of the SOSMC, the aim of the controller is to steer to zero both the sliding variable s and its first-order time derivative in a finite time (as $s(t) = \dot s(t)\; = 0$ ) [Reference Capisani, Ferrara and Magnani70]. In the literature, there are a number of sliding surfaces generated for SOSMC such as integral sliding surface [Reference Camacho, Rojas and García-Gabín71, Reference Yorgancıoğlu and Kömürcügil72], PID sliding surface with two independent gain parameters [Reference Young, Utkin and Özgüner57], and PID sliding surface with three gain parameters [Reference Eker67]. In the present case, sliding surface is defined as:

(29) \begin{equation}\dot s(t) + {\beta _c}s(t) = {\left( {\lambda + \frac{d}{{dt}}} \right)^2}\smallint e(t)dt,\;{\rm{\lambda }} \in {R^ + }\end{equation}

(30) \begin{equation}\dot s(t) + {\beta _c}s(t) = {\lambda ^2}\smallint e(t)dt + 2\lambda e(t) + \dot e(t)\;\end{equation}

(31) \begin{equation}\ddot s(t) + {\beta _c}\dot s(t) = {\lambda ^2}e(t) + 2\lambda \dot e(t) + \ddot e(t)\end{equation}

where $\lambda $ is an independent parameter gain which is related to error, and the gain parameter of sliding surface, ${\beta _c}$ , which is a positive constant, ${\beta _c} \in $ ${R^ + },\;$ contributes to the damping of s(t). The $\;D\left( {t,u(t)} \right)$ , which was also given in Equation (17), denotes the lumped uncertainty. When the second order derivative of the error, which was also given in Equation (20), substitutes into Equation (31), we can rewrite the equation as follows:

(32) \begin{equation}\ddot s(t) + {\beta _c}\dot s(t) = {\lambda ^2}e(t) + 2\lambda \dot e(t) + {\ddot x_d}(t) + A\dot x(t) + Bx(t) - Cu(t) - {\rm{D}}\left( {{\rm{t}},{\rm{u}}\left( {\rm{t}} \right)} \right)\end{equation}

When the system is on the sliding manifold, the error, e(t), will converge to zero exponentially if the coefficient $\lambda $ is chosen properly. If the limit of the error goes to zero in infinite time, $\mathop {\lim }\limits_{t \to \infty } e(t) = 0$ , it means that the system is asymptotic stable. Moreover, $\ddot s(t) = 0$ is recognised as an essential condition for the error to stay on the sliding manifold. With this assumption, the equivalent control is formed as given in Equation (33), and disturbances are not taken into account for the equivalent control since the nominal system model is assumed known.

(33) \begin{equation}{u_{eq}} = \frac{1}{C}\left( {{{\ddot x}_d}(t) + {\lambda ^2}e + 2\lambda \dot e + A\dot x(t) + Bx(t) - {\beta _c}\dot s} \right)\end{equation}

In our case, the switching control is introduced with saturation function as:

(34) \begin{equation}{u_{sw}} = {\lambda _2}s(t) + {k_s}sat\left( {\dot s(t)} \right)\end{equation}

where ${\lambda _2}$ and ${k_s}$ $ \in {R^ + }$ with ${\lambda _2} \gt \frac{1}{C}$ and ${k_s} \gt \frac{{{D_{max}}}}{C}$ with ${D_{max}} = su{p_{\forall t,s,\dot s = 0}}D\left( {t,u(t)} \right)$ . When Equations (33) and (36) are substituted into Equation (32), the final equation becomes:

(35) \begin{equation}\ddot s = - D\left( {t,u(t)} \right) - C{\lambda _2}s(t) - C{k_s}sign\left( {\dot s(t)} \right)\end{equation}

It is known that the stability is assured if the derivative of the Lyapunov function $\dot V\;$ is a negative definite. Thus, the Lyapunov function is chosen as given in Equation (36) to guarantee stability:

(36) \begin{equation} V = \frac{1}{2}s{(t)^2} + \frac{1}{2}\dot s{(t)^2}\,{\rm{with}}\,V\left( 0 \right) = 0,{\rm{\;\;}}s(t) \ne 0,{\rm{\;\;}}\dot s(t) \ne 0 \end{equation}

If taking the first-time derivative of Equation (36), and then Equation (35) substituted into it, one has:

(37) \begin{align} \dot V & = s(t)\dot s(t) + \dot s(t)\ddot s(t)\nonumber\\[3pt]& = s(t)\dot s(t) + \;\dot s(t)\left( { - D\left( {t,u(t)} \right) - C{\lambda _2}s(t) - C{k_s}sign(\dot s(t)} \right)\nonumber\\[3pt]& = s(t)\dot s(t) - \;\dot s(t)D\left( {t,u(t)} \right) - C{\lambda _2}s(t)\dot s(t) - C{k_s}\left| {\dot s(t)} \right|\nonumber\\[3pt]& \le \left| {\dot s(t)} \right|\;\left( {s(t) - C{\lambda _2}s(t) - D\left( {t,u(t)} \right) - C{k_s}} \right)\nonumber\\[3pt]& \le \left| {\dot s(t)} \right|\;\left( {\left| {s(t)} \right| - C{\lambda _2}\left| {s(t)} \right| - D\left( {t,u(t)} \right) - C{k_s}} \right)\nonumber\\[3pt]& \le \left| {\dot s(t)} \right|\;\left( {\left| {s(t)} \right| - C{\lambda _2}\left| {s(t)} \right| + {D_{max}}\left( {t,u(t)} \right) - C{k_s}} \right)\nonumber\\[3pt]& = - \left| {\dot s(t)} \right|\;\left( {\left| {s(t)} \right|\left( {C{\lambda _2} - 1} \right) - {D_{max}}\left( {t,u(t)} \right) + C{k_s}} \right) \lt 0.\end{align}

Hence, from the above analysis, the global asymptotic stability is assured since $s(t) \ne 0,\dot s(t) \ne 0,\;{k_s} \gt \dfrac{D_{max}}{C}$ , $\;{\lambda _2} \gt \dfrac{1}{C}$ , $\left| {s(t)} \right| \gt 0,\;\left| {\dot s(t)} \right| \gt 0,$ and thus, it can be proved that $\dot V \lt 0$ . The requirement of satisfying ${k_s}\; \gt \dfrac{{{D_{max}}}}{C}$ is obligatory to dominate the matching uncertainties.

2.4 System compositions and working principle

The FWS considered here consists of a slider-crank mechanism and motor assembly. Motor component assembly consists of encoder, gearbox, screw and BLDC motor. A gearbox is introduced to improve the output torque and BLDC motor position is measured by rotary encoder. Regarding control, LQR, LQI, sliding mode and second-order sliding mode controllers were tested to adjust the swept-back angle through BLDC motor position. The real-time controller responses were examined under aerodynamic disturbances that linearly vary with swept-back angle. In fact, these aerodynamic disturbances were implemented by extension springs which were attached to wings. The extension springs were carefully chosen and manufactured with specific properties to provide more moment than encountered in flight. They were placed inside the protection case owing to security reasons and were mounted to wings with these aluminum casing. The extension springs were manufactured properly with 17.4 active coils of 3.8 mm diameter wire to provide a spring coefficient of 3.73. The test system was designed as while decreasing of the swept-back angle, the value of aerodynamic disturbance torque increases as shown in Fig. 5.

Figure 5. Variation of disturbance torque versus swept-back angle.

In selection of the system components, kinematic and dynamic analyses were taken into account. The maximum value of the net force on the slider might approach about 10³ N, and thus, safety should be provided with proper screw selection. Maxon spindle drive component, consisting of a ball-screw, 1:3.7 reducer, and a rotary incremental encoder was chosen to meet the system requirements. Regarding the motor and encoder, it can be seen from Fig. 3b that the motor must accomplish a minimum of 780 mNm torque.

Figure 6. The diagram of xPC Target and data acquisition system.

The kinematic and dynamic analysis, as well as the system identification and controller design were all conducted with MATLAB. Following the assembly, the system identification process was performed, and then the developed controllers were applied to the presented system. Data collection and transfer processes between host and target were conducted using a NI PCI-6221 data acquisition card working with 1 kHz sample time via MATLAB xPC Target toolbox (Fig. 6). The BLDC motor was driven by an AMC motor driver. Briefly, the system is controlled by adjusting the current output and the current direction through the motor driver.