Article Contents
Citation: Y. Mousavi, A. Zarei, A. Mousavi, M. Biari. Robust optimal higher-order-observer-based dynamic sliding mode control for VTOL unmanned aerial vehicles. International Journal of Automation and Computing. http://doi.org/10.1007/s11633-021-1282-3 doi:  10.1007/s11633-021-1282-3
Cite as: Citation: Y. Mousavi, A. Zarei, A. Mousavi, M. Biari. Robust optimal higher-order-observer-based dynamic sliding mode control for VTOL unmanned aerial vehicles. International Journal of Automation and Computing . http://doi.org/10.1007/s11633-021-1282-3 doi:  10.1007/s11633-021-1282-3

Robust Optimal Higher-order-observer-based Dynamic Sliding Mode Control for VTOL Unmanned Aerial Vehicles

Author Biography:
  • Yashar Mousavi received the M. Sc. degree in control engineering from Shahrood University of Technology, Iran in 2014. He is currently a Ph. D. degree candidate as a member of the Power and Renewable Energy Systems (PRES) Research Division with Department of Applied Science, School of Computing, Engineering and Built Environment, Glasgow Caledonian University, UK.His research interests include evolutionary optimization, renewable energy, robotic systems and control, robust nonlinear control, fractional-order control, and fault-tolerant control. E-mail: seyedyashar.mousavi@gcu.ac.uk (Corresponding author) ORCID ID: 0000-0002-6718-3599

    Amin Zarei received the B. Sc. degree in electrical engineering from University of Sistan and Baluchestan, Iran in 2011 and the M. Sc. degree in control engineering from Shahrood University of Technology, Iran in 2014. He is currently a Ph. D. degree candidate in control engineering at University of Sistan and Baluchestan, Iran.His research interests include complex systems, networked control systems, nonlinear control, chaos theory, and time series prediction. E-mail: amin.zarei@pgs.usb.ac.ir ORCID ID: 0000-0003-2326-4431

    Arash Mousavi received the B. Sc. degree in control engineering from Payam University, Iran in 2013, and the M. Sc. degree in electrical engineering from Islamic Azad University of Jahrom, Iran in 2016. He is currently the head of Electrical Engineering Research Lab (EERL) with Department Electrical Engineering and Applied Sciences, Paradise Research Center, Iran.His research interests include renewable energy, robotic systems and control, power systems, impacts of distributed generations on power systems, and reliability of power systems. E-mail: mousavii.arash@gmail.com

    Mohsen Biari received the B. Sc. and the M. Sc. degrees in control engineering from Shahrood University of Technology, Iran in 2010 and 2013, respectively. He is currently the head of Robotics and Automation research Lab with Science and Technology Center, Iran.His research interests include robotic systems and control, nonlinear control, fault-tolerant control, autonomous control, and computer vision. E-mail: mohsen.biari@gmail.com

  • Received: 2020-09-27
  • Accepted: 2021-01-22
  • Published Online: 2021-03-22
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Figures (8)  / Tables (2)

Metrics

Abstract Views (23) PDF downloads (34) Citations (0)

Robust Optimal Higher-order-observer-based Dynamic Sliding Mode Control for VTOL Unmanned Aerial Vehicles

Abstract: This paper investigates the precise trajectory tracking of unmanned aerial vehicles (UAV) capable of vertical take-off and landing (VTOL) subjected to external disturbances. For this reason, a robust higher-order-observer-based dynamic sliding mode controller (HOB-DSMC) is developed and optimized using the fractional-order firefly algorithm (FOFA). In the proposed scheme, the sliding surface is defined as a function of output variables, and the higher-order observer is utilized to estimate the unmeasured variables, which effectively alleviate the undesirable effects of the chattering phenomenon. A neighboring point close to the sliding surface is considered, and as the tracking error approaches this point, the second control is activated to reduce the control input. The stability analysis of the closed-loop system is studied based on Lyapunov stability theorem. For a better study of the proposed scheme, various trajectory tracking tests are provided, where accurate tracking and strong robustness can be simultaneously ensured. Comparative simulation results validate the proposed control strategy′s effectiveness and its superiorities over conventional sliding mode controller (SMC) and integral SMC approaches.

Citation: Y. Mousavi, A. Zarei, A. Mousavi, M. Biari. Robust optimal higher-order-observer-based dynamic sliding mode control for VTOL unmanned aerial vehicles. International Journal of Automation and Computing. http://doi.org/10.1007/s11633-021-1282-3 doi:  10.1007/s11633-021-1282-3
Citation: Citation: Y. Mousavi, A. Zarei, A. Mousavi, M. Biari. Robust optimal higher-order-observer-based dynamic sliding mode control for VTOL unmanned aerial vehicles. International Journal of Automation and Computing . http://doi.org/10.1007/s11633-021-1282-3 doi:  10.1007/s11633-021-1282-3
    • In recent years, unmanned aerial vehicles (UAVs) with vertical take-off and landing (VTOL) maneuverability have received considerable attention due to their capacities of hovering and low-speed/low-altitude flight[1], and potential applications in agriculture, structures or oil pipelines inspection, law enforcement, sports, surveillance, mining, fire and traffic monitoring, and aerial imaging. These underactuated vehicles are canonical nonlinear systems with a three degrees of freedom mechanism and only two actuators, and the translational motion is controlled by a thrust force along a single body-fixed axis[2]. Moreover, VTOLs are members of a class of systems that cannot be fully controlled, i.e., some degrees of freedom cannot be directly controlled. Consequently, many strategies developed for fully controlled systems are not suitable and applicable to these systems. The existence of different characteristics in UAVs has generated a number of technical problems for control engineers. Accordingly, they can be considered as a benchmark for developing mathematical techniques to control actual unmanned flying devices.

      Control of non-minimum phase nonlinear VTOL UAVs is a challenging problem in aerospace engineering, and several methodologies, including position stabilization and trajectory tracking for UAVs in the presence of external disturbances, have been investigated in the literature to deal with this problem[3-10]. However, many of these studies fail to deal with the control performance and robustness issues simultaneously. Although some controllers allow acceptable set-point tracking performance, the disturbance rejection issue has been found to be more significant for many others. Thus, designing a control strategy that emphasizes disturbance rejection with good set-point tracking is a real design problem that this work will focus on.

      In [3], a new control strategy was proposed and formulated using an approximated solution for the tracking problem in the basin of the linear algebra. In addition, some control strategies were successfully developed to guarantee global asymptotic stability[4-6]. A number of feedback control designs and various operating modes encountered in practice are studied for VTOL UAVs[7]. Based on the dynamic communication network and distributed robust feedback control approach, formation and reconfiguration control and collision avoidance of a team of VTOL UAVs can be analyzed. An inner-outer loop trajectory tracking feedback controller for a class of under-actuated VTOL UAVs was developed in [8]. As the authors reported, the proposed paradigm demonstrated a remarkable performance through position and attitude tracking in the presence of uncertainties and exogenous disturbances. A centralized predictive interaction control scheme was investigated in [9] to deal with the ceiling effort problem of VTOL UAVs. According to the authors, the system's aerodynamic parameters were identified in real-time, where experimental investigations demonstrated the superior performance of the proposed strategy compared with the proportional-integral-derivative controller. Bouzid et al.[10] proposed a novel nonlinear internal model control scheme for trajectory tracking of a VTOL multi-rotors UAV under structured and unstructured disturbances. The coordinated trajectory tracking of multiple VTOL UAVs to drive all follower VTOL UAVs and to track the desired trajectory associated with a leader accurately based on defining novel distributed estimators is investigated in [11, 12]. Image-based tracking control algorithms were studied for VTOL UAVs to track a moving target[13, 14], which incorporated estimations of the target's acceleration without linear velocity measurements. Nevertheless, the position and orientation dynamics control problem was carried out by linear velocity estimations and the gravitational inertial direction extracted from image features[15]. Sliding modes-based methodologies for the planar VTOL (PVTOL) system were presented in [16, 17]. In addition, a two-step output-feedback control strategy was presented in [18] for regulation of a planar VTOL aircraft, such that the first controller stabilized the vertical variable based on an augmentation of a terminal sliding mode controller and a simple feedback linearization procedure, while the second controller stabilized the horizontal and angular variables using an energy-control method.

      The firefly algorithm (FA) is a metaheuristic algorithm inspired by nature and has been successfully utilized for solving optimization problems[19]. Although FA has proved its exceptional advantages in comparison with other evolutionary algorithms, it has some shortcomings, such as inability to make a reasonable trade-off between exploration and exploitation behaviors, and suffering from getting trapped into local minima, significantly when solving complex multimodal problems[20, 21]. To overcome these drawbacks, various modifications have been investigated[22-25]. One of the most recent well-performed modifications of FA introduced in the literature is the fractional-order FA (FOFA), which incorporates the fractional calculus concepts into FA's search process. Since the performance of fractional-based optimization algorithms has been well-proven in [24, 26], FOFA will be utilized in this study for performance enhancement of the proposed control strategy.

      Sliding mode control (SMC) has been utilized as one of the most efficient control approaches to deal with nonlinear systems subjected to external disturbances[27-30], however, it suffers from the chattering phenomenon[31-33]. To alleviate the chattering problem arising from the switching control action, a dynamic sliding mode control (DSMC) technique as a member of the variable structure controllers' family was investigated for nonlinear systems[34]. One contribution of this paper is to develop an optimal dynamical sliding mode control to deal with the VTOL position trajectory tracking control problem subjected to parametric uncertainties and external disturbances. A higher-order sliding mode observer (HSMO) is utilized to estimate the unmeasured variables in order to enhance the controllers′ performance with respect to chattering effects. Besides, the sliding surface is defined as a function of output variables, as a virtual output, which effectively reduces the undesirable properties of classical sliding mode such as the chattering phenomenon. To this end, a robust variable-structure control law is applied so that a neighboring point near the sliding surface is considered; and when the tracking error reaches to this point, the second control is activated and exponentially reduces the control signal. An output regulation scheme is employed on the state-space model of nonlinear time-variable systems in the present work. As a result, the proposed control strategy sets the output regulation error to the least possible value, while the states and input remain limited. To better illustrate the proposed control strategy's effectiveness and performance, it is applied to various reference trajectories.

      The rest of this paper is organized as follows. In Section 2, the problem and some useful preliminaries are provided. In Section 3, the proposed control scheme and stability analysis are stated. Simulation results are provided in Section 4 to verify the theoretical results. Finally, conclusions are presented in Section 5.

    • In FA, the swarm of fireflies moves toward more attractive and brighter fireflies. The attractiveness directly relates to the flashing light intensity associated with the objective function to be optimized. The relation between light intensity ${I_l}$ and the distance $r$ is expressed as

      ${I_l} = {I_{l,0}}{{\rm e}^{ - \gamma r}}$

      (1)

      where ${I_0}$ and $\gamma $ denote the initial light intensity and the light absorption coefficient, respectively. The firefly's attractiveness $\beta $ of firefly can be expressed as

      ${\beta _r} = {\beta _0}{{\rm e}^{ - \gamma r}}$

      (2)

      where ${\beta _0}$ is the attractiveness at $r = 0$. According to [24], the movement of a firefly $i$ attracted to another more attractive firefly $j$ is calculated by

      ${\mathfrak{D}^\Omega }\left[ {{x_{t + 1}}} \right] = {\beta _r}\left( {{x_j} - {x_i}} \right) + \eta (\psi - 0.5)$

      (3)

      where ${\mathfrak{D}^\Omega }[x(t)]$ denotes the Grünwald–Letnikov discrete-time fractional differential operator expressed as

      ${\mathfrak{D}^\Omega }[x(t)] = \frac{1}{{{T^\varOmega }}}\mathop \sum \limits_{q = 0}^z \frac{{{{\left( { - 1} \right)}^q}\varGamma (\varOmega + 1)x(t - qT)}}{{\varGamma (q + 1)\varGamma (\varOmega - q + 1)}}$

      (4)

      where $T = \varOmega = 1$, and $z$ is the truncation order. According to (3) and the first $z = 4$ terms of differential derivative, the following result obtains[24]:

      $\begin{split} &x(t + 1) = \varOmega x(t) + {\beta _r}\left( {{x_j} - {x_i}} \right) + \eta (\psi - 0.5) +\\ &\,\,\,\,\,\,\,\,\, \frac{1}{{2!}}\varOmega (1 - \varOmega )x(t - 1) + \frac{1}{{3!}}\varOmega (1 - \varOmega )\,(2 - \varOmega )x(t - 2) +\\ &\,\,\,\,\,\,\,\,\, \frac{1}{{4!}}\varOmega (1 - \varOmega )(2 - \varOmega )\,(3 - \varOmega )x(t - 3). \\[-10pt]\end{split} $

      (5)
    • The dynamic equations of motion have been described as (6)[35]. Also, Fig. 1 shows the normalized equations of the VTOL.

      Figure 1.  VTOL model

      $ \begin{split}&\ddot{x}(t)=-{u}_{1}\left(t\right)\mathrm{sin}\theta (t)+\tau {u}_{2}\left(t\right)\mathrm{cos}\theta (t)+{\psi }_{1}\left(t\right)\\ &\ddot{y}(t)={u}_{1}\left(t\right)\mathrm{cos}\theta (t)+\tau {u}_{2}\left(t\right)\mathrm{sin}\theta (t)-1+{\psi }_{2}\left(t\right)\\ &\ddot{\theta }(t)= {u}_{2}(t)\end{split}$

      (6)

      where $x(t)$ and $y(t)$ are the center of mass horizontal and vertical positions, respectively. $\theta $ represents the angle concerning the imaginary horizontal line, ${u_1}(t)$ and ${u_2}(t)$ respectively represent the thrust force that causes the elevation of the VTOL and the rolling moment of its corresponding center of mass, $\tau $ is the coupling between the rolling moment and the lateral acceleration; a minimal parameter which is usually neglected in the literature. To consider real-life applications, the system is subjected to parametric uncertainties and additive environmental external disturbances expressed by unknown terms ${\psi _1}(t)$ and ${\psi _2}(t)$. Although the values of the terms mentioned above are inherently uncertain, the upper bounds are assumed to be known as Assumption 1.

      Assumption 1. The additive uncertainties and external disturbances ${\psi _1}(t)$ and ${\psi _2}(t)$ satisfy the following conditions:

      $ \left|{\psi }_{i}\left(t\right)\right|\le {\rho }_{i}\left(x,\dot{x},y,\dot{y}\right)+{l}_{i},\;\;\;i=1,2$

      (7)

      where ${\rho _i}$ is a positive term and ${l_i}$ is a non-negative number. The VTOL UAV regulation problem consists of designing a robust controller such that the control laws ${u_1}$ and ${u_2}$ guarantee asymptotic convergence of $x(t) \to {x_d}(t)$ and $y(t) \to {y_d}(t)$ such that

      $\mathop {\lim }\limits_{x \to \,\,\,\infty } \left\| {\left( {\begin{array}{*{20}{c}} {y - {y_d}}&{x - {x_d}}&\theta \end{array}} \right)} \right\| = 0.$

      (8)

      Accordingly, new state variables can be expressed as follows:

      $ J=\left({J}_{1},{J}_{2},{J}_{3},{J}_{4}\right)=\left(x-{x}_{d}, \dot{x}-{\dot{x}}_{d}, y-{y}_{d}, \dot{y}-{\dot{y}}_{d}\right).$

      (9)

      Rewriting (6) in the new coordinate system yields

      $ \begin{split}&{\dot{J}}_{1}= {J}_{2}\\ &{\dot{J}}_{2}= -{u}_{1}\mathrm{sin}\theta +\tau {u}_{2}\mathrm{cos}\theta -{\ddot{x}}_{d}+{\psi }_{1}\\ &{\dot{J}}_{3}={J}_{4}\\ &{\dot{J}}_{4}={u}_{1}\mathrm{cos}\theta +\tau {u}_{2}\mathrm{sin}\theta -{\ddot{y}}_{d}-1+{\psi }_{2}.\end{split}$

      (10)

      Assumption 2. The angle $\theta (t) \in {I_0}$ holds for all the time, meaning that the acrobatic behavior is not possible.

    • Consider the following nonlinear system:

      $\begin{split} & \dot x = f\left( {x,u} \right) \\ & y = h\left( {x,u} \right) \\ \end{split} $

      (11)

      where $u \in {{\bf R}^m}$, $x \in {{\bf R}^n}$ and $y \in {{\bf R}^m}$ denote the system inputs, states and outputs vectors, respectively. $f\left( {x,u} \right)$ and $h\left( {x,u} \right)$ are nonlinear differentiable functions. The input-output model can be expressed as

      $ {y}_{i}^{\left({n}_{i}\right)}={\phi }_{i}\left(\widehat{{{y}}},\widehat{{{u}}},t\right)+{\psi }_{i}, i=1,2,\mathrm{\cdots},m$

      (12)

      where ${\widehat{ u}} = \left( {{u_1}, \cdots ,u_1^{\left( {{\beta _1}} \right)}, \cdots ,{u_m}, \cdots ,u_m^{\left( {{\beta _m}} \right)}} \right)$, ni is the integer derivative order, ${\widehat{ y}} = \left( {y_1}, \cdots ,y_1^{\left( {{n_1} - 1} \right)}, \cdots ,{y_m}, \cdots, $$ y_m^{({n_m} - 1)} \right)$, while ${\beta _j},j = 1,\cdots,m$ is the derivative order of input vector components. Considering known values ${\rho _i} > 0$ and $ {l}_{i} \ge 0$, the upper bond of additive uncertainties ${\psi _i}$ can be expressed as follows:

      $ \left|{\psi }_{i}\right|\le {\rho }_{i}\Vert \widehat{{{y}}}\Vert +{l}_{i}, i=1,\cdots ,m$

      (13)

      where

      $\begin{split} &{\rho ^{\left( 0 \right)}} = {\left[ {\mathop \sum \limits_{i = 1}^m \rho _i^2} \right]^\tfrac{1}{2}} \\ &{\rho ^{\left( 1 \right)}} = {\left[ {\mathop \sum \limits_{i = 1}^m \rho _i^2} \right]^\tfrac{1}{2}}(1 + \mathop {\max }\limits_{} \left\{ {a_j^{\left( i \right)}} \right\}\mathop {\max }\limits_{} \left\{ {\sqrt {{n_i} - 1} } \right\}) \\ &\rho = {\rho ^{\left( 0 \right)}} + {\rho ^{\left( 1 \right)}}/\left( {4{\theta _1}} \right) \end{split} $

      (14)

      where ${\theta _1} \in \left( {0,1} \right)$ is a constant parameter and ${\theta _0}$ is selected so that ${\theta _0} + {\theta _1} = 1$.

      Definition 1. The input-output model (12) with ${\psi _i} = 0$ is proper if both input and output vectors have the same dimensions[32], and for ${\widehat{ y}} \in {N_\delta }(0) = \{ J{\rm{|}}J \in {{\bf R}^n}, $$ J < \delta \}$ and $t \ge 0$, the following inequality is satisfied:

      $\det \left[ {\frac{{\partial \left( {{\varphi _1},\cdots,{\varphi _m}} \right)}}{{\partial \left( {{u^{\left( {{\beta _1}} \right)}},\cdots,{u^{\left( {{\beta _m}} \right)}}} \right)}}} \right] \ne 0.$

      (15)

      Definition 2. The related zero dynamics for ${\psi }_{i}=0, i=1,\cdots ,m$ to the input-output model (12) are defined as follows[36]:

      $ {\phi }_{i}\left(0,\widehat{{{u}}},t\right)=0, i=1,2,\mathrm{\cdots},m.$

      (16)

      System (12) is minimum phase if and only if ${{\widehat{ u}}_{0}} \in {{\bf R}^\beta }$ and $\delta > 0$, where $\beta = {\beta _1} + \cdots + {\beta _m}$. As a result, (16) is globally asymptotically stable with the initial condition ${\widehat{ u}}\left( 0 \right) \in {N_\delta }\left( {{{{\widehat{ u}}}_{\bf{0}}}} \right)$, where ${\widehat{ u}} = \left( {u_1}, \cdots ,u_1^{\left( {{\beta _1} - 1} \right)}, \cdots ,{u_m}, \cdots , $$ u_m^{\left( {{\beta _m} - 1} \right)} \right)$. In order to simplify the design procedure, the following generalized canonical representation of (12) can be expressed.

      $\left\{ \begin{array}{*{20}{c}} {\dot J_1^{\left( 1 \right)} = J_2^{\left( 1 \right)}} \\ \vdots \\ {\dot J_{{n_1} - 1}^{\left( 1 \right)} = J_{{n_1}}^{\left( 1 \right)}} \\ {\dot J_{{n_1}}^{\left( 1 \right)} = {\varphi _1}\left( {J,{\widehat{ u}},t} \right) + {\psi _1}} \\ \vdots \\ {\dot J_1^{\left( m \right)} = J_2^{\left( m \right)}} \\ \vdots \\ {\dot J_{{n_{m - 1}}}^{\left( m \right)} = J_{{n_m}}^{\left( m \right)}} \\ {\dot J_{{n_m}}^{\left( m \right)} = {\varphi _m}\left( {J,{\widehat{ u}},t} \right) + {\psi _m}} \end{array} \right.$

      (17)

      where ${J}^{\left(i\right)}=({J}_{1}^{\left(i\right)},\cdots ,{J}_{{n}_{1}}^{\left(i\right)})=({y}_{i},\cdots ,{y}_{i}^{\left({n}_{i}-1\right)}), i=1,\cdots , $$ m$ and $J = {\left( {{J^{\left( 1 \right)}},\cdots,{J^{\left( m \right)}}} \right)^{\rm{T}}}$.

      Unlike classical SMC, for which the sliding surface is defined based on state variables, in DSMC, it is defined according to the output and its derivatives. Matrices ${A_i}$ and ${D_i}$ are defined as follows:

      $\begin{split} & {A_i} = \left[ {\begin{array}{*{20}{c}} 0&1&0&0& \cdots &0 \\ 0&0&1&0& \cdots &0 \\ 0&0&0&1& \cdots & \vdots \\ \vdots & \vdots & \vdots & \ddots & \ddots &0 \\ 0&0&0&0&0&1 \\ { - a_1^{\left( i \right)}}&{ - a_2^{\left( i \right)}}&{ - a_3^{\left( i \right)}}&{ - a_4^{\left( i \right)}}& \cdots &{ - a_{n - 1}^{\left( i \right)}} \end{array}} \right] \\ & {D_i} = {\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 0&0 \end{array}}& \cdots &0&1 \end{array}} \right]^{\rm{T}}} . \\[-10pt]\end{split} $

      (18)

      Since ${A_i}$ is Hurwitz, $A$ is Hurwitz; thus $A$ and $P$ satisfy the Lyapunov equation ${A^{\rm T}}P + AP = - {I_2}$, where P is a positive definite matrix. By determining P and considering (12), $K$ is obtained such that satisfies the following condition:

      ${\lambda _{\rm min}}\left( K \right) - \left[ {\frac{1}{{{\theta _0}}}{{\left( {PD} \right)}^{\rm T}}\left( {PD} \right) + \rho {I_2}} \right] > 0$

      (19)

      where ${\lambda _{\rm min}}(K)$ represents the smallest eigenvalue of K, and ${I_2} \in {{\bf R}^{m \times m}}$ is a unit vector. The sliding surface is then considered as

      $ {s}_{i}=\sum \limits_{j=1}^{{n}_{i}+1}{a}_{j}^{\left(i\right)}{J}_{j}^{\left(i\right)}+{\phi }_{i}\left(J,\widehat{ u},t\right),\;\;\; i=1,\cdots ,m$

      (20)

      where $a_j^{(i)}$ is selected so that the polynomial ${ {\displaystyle\sum }_{j=1}^{{n}_{i}+1}{a}_{j}^{(i)}\lambda { }^{(j-1)},i=1,\mathrm{\cdots},m}$ is Hurwitz. In this work, the DSMC is designed considering the following reachability condition.

      $ \dot{s}= -\gamma \left(\kappa ,s\right)$

      (21)

      where $s = [{s_1}, \cdots ,{s_m}]$, and $\kappa = [{\kappa _1}, \cdots ,{\kappa _l}]$ denotes the constant design parameters. Furthermore, $\gamma (\kappa ,s) = $$ [{\gamma _1}(\kappa ,s),\cdots,{\gamma _m}(\kappa ,s)]$ satisfies the following conditions: (a) $\gamma (\kappa ,0) = 0$, (b) ${\gamma _i}(\kappa ,s){|_{i = 1,\cdots,m}}$ for ${s_i} \ne 0$ is a continuous function, and there exists a positive definite matrix K such that ${s^{\rm T}}$$\gamma $$ (\kappa ,s) > {s^{\rm T}}Ks$ for ${s_i} \ne 0$. Thus, the realization condition can be expressed as

      $ {\gamma }_{i}(\kappa ,s)={(K,s)}_{i}+{k}_{0i}sa{t}_{\epsilon }({s}_{i}) , \;\;\; i=1,\mathrm{\cdots},m$

      (22)

      where ${(Ks)_i}$ is the $i$-th component of $Ks$ vector, and ${K_0} = {\rm diag}\left[ {{k_{01}},{k_{0m}}} \right] \in {{\bf R}^{m \times m}}$. Considering the establishment of (21), (22) will satisfy the reaching law for all sliding surfaces. The derivative of (20) with respect to time is given by:

      $ {\dot{s}}_{i}=\displaystyle\sum \limits_{j=1}^{{n}_{i}-1}{a}_{j}^{\left(i\right)}{J}_{j+1}^{\left(i\right)}+{\phi }_{i}\left(J,\widehat{{ u}},t\right)+{\psi }_{i}\left(t\right), \;\;\; i=1,\cdots ,m.$

      (23)

      Thus, the control law must be designed such that it satisfies the following equation:

      $ \begin{split}&\displaystyle\sum \limits_{j=1}^{{n}_{i}-1}{a}_{j}^{\left(i\right)}{J}_{j+1}^{\left(i\right)}+{\phi }_{i}\left(J,\widehat{{ u}},t\right)+{\psi }_{i}\left(t\right)=\\ &\quad\quad-{\left(Ks\right)}_{i}-{k}_{0i}sa{t}_{\epsilon }\left({s}_{i}\right),\;\;\; i=1,\cdots ,m.\end{split}$

      (24)

      Also, $sa{t_\varepsilon }\left( {{s_i}} \right) = {[sa{t_\varepsilon }\left( {{s_1}} \right), \cdots ,sa{t_\varepsilon }\left( {{s_m}} \right)]^{\rm{T}}}$ and ${k_{0i}} > $$ {l_0} = {\left(\displaystyle\sum\nolimits_{i = 1}^m {l_i^2} \right)^\tfrac{1}{2}}$, where $sa{t_\varepsilon }({s_i})$ is defined as follows:

      $sa{t_\varepsilon }\left( {{s_i}} \right) = \varepsilon sat\left( {\frac{{{s_i}}}{\varepsilon }} \right) = \left\{ {\begin{array}{*{20}{c}} {\begin{aligned} &1,&\;{\rm if}\;{{s_i} > \varepsilon } \end{aligned}}\\ {\begin{aligned} &{\frac{{{s_i}}}{\varepsilon }},&\;{\rm if}\;{\left| {{s_i}} \right| \le \varepsilon } \end{aligned}}\\ {\begin{aligned} &{ - 1},&\;{\rm if}\;{{s_i} < - \varepsilon }. \end{aligned}} \end{array}} \right.$

      (25)

      Considering (24), (21) can be rewritten as

      $\dot s = - {\left( {Ks} \right)_i} - {K_0}sa{t_\varepsilon }\left( {{s_i}} \right) + \psi \left( {J,t} \right).$

      (26)
    • In this section, assuming that the state variables $({J_2},{J_4})$ are unmeasured, the proposed optimal HOB-DSMC approach is presented to deal with the trajectory tracking problem of the VTOL UAV system (12) subjected to additive uncertainties and external disturbances. The HSMO utilized to estimate the unmeasured variables $({J_2},{J_4})$ as well as disturbances $({\psi _1},{\psi _2})$ is given as

      $\begin{split} & {\widehat{ \dot J}_1} = {{\widehat J}_2} + {\varOmega _1}{\left| {{{\widetilde J}_1}} \right|^\tfrac{2}{3}}{\rm sgn}\left( {{{\widetilde J}_1}} \right) \\ &{{\widehat{ \dot J}}_2} = {\varOmega _2}{\left| {{{\widetilde J}_1}} \right|^\tfrac{1}{3}}{\rm sgn}\left( {{{\widetilde J}_1}} \right) - {u_1}\sin \theta +\\ &\,\,\,\,\, \tau {u_2}\cos \theta + {\psi _1}\left( {J,t} \right) \\ &{{\widehat{ \dot J}}_3} = {{\widehat J}_4} + {\varOmega _3}{\left| {{{\widetilde J}_3}} \right|^\tfrac{2}{3}}{\rm sgn}\left( {{{\widetilde J}_3}} \right) \\ &{{\widehat{ \dot J}}_4} = {\varOmega _4}{\left| {{{\widetilde J}_3}} \right|^{\tfrac{1}{3}}}{\rm sgn}\left( {{{\widetilde J}_3}} \right) + {u_1}\cos \theta +\\ &\,\,\,\,\, \tau {u_2}\sin \theta - 1 + {\psi _2}\left( {J,t} \right) \\ &{{\widehat {\dot J}}_5} = {\varOmega _5}{\rm sgn}\left( {{{\widetilde J}_1}} \right) \\ &{{\widehat {\dot J}}_6} = {\varOmega _6}{\rm sgn}\left( {{{\widetilde J}_3}} \right) \end{split} $

      (27)

      where ${\widehat J_i},i = 1,2,\cdots,6$ denote the estimated variables. Accordingly, the estimation errors can be expressed as ${\widetilde J_i} = {J_i} - {\widehat J_i},i = 1,\cdots,4$, ${\widetilde J_5} = - {\widehat J_5} + {\psi _1}$, and ${\widetilde J_6} = - {\widehat J_6} + $$ {\psi _2}$. Similar definitions have been used in [37, 38]. Considering (10) and (27), the estimation error dynamics yield

      $\begin{split} &{{\dot {\widetilde J}}_1} = - {\varOmega _1}{\left| {{{\widetilde J}_1}} \right|^\tfrac{2}{3}}{\rm sgn}\left( {{{\widetilde J}_1}} \right) + {{\widetilde J}_2} \\ &{{\dot {\widetilde J}}_2} = - {\varOmega _2}{\left| {{{\widetilde J}_1}} \right|^\tfrac{1}{3}}{\rm sgn}\left( {{{\widetilde J}_1}} \right) + {{\widetilde J}_5} \\ &{{\dot {\widetilde J}}_3} = - {\varOmega _3}{\left| {{{\widetilde J}_3}} \right|^\tfrac{1}{3}}{\rm sgn}\left( {{{\widetilde J}_3}} \right) + {{\widetilde J}_4} \\ &{{\dot {\widetilde J}}_4} = - {\varOmega _4}{\left| {{{\widetilde J}_3}} \right|^\tfrac{1}{3}}{\rm sgn}\left( {{{\widetilde J}_3}} \right) + {{\widetilde J}_6} \\ &{{\dot {\widetilde J}}_5} = - {\varOmega _5}{\rm sgn}\left( {{{\widetilde J}_1}} \right) + {{\dot \psi }_1} \\ &{{\dot {\widetilde J}}_6} = {\varOmega _6}{\rm sgn}\left( {{{\widetilde J}_3}} \right) + {{\dot \psi }_2}. \end{split} $

      (28)

      The above-mentioned estimation error dynamics (28) are presented in the form of the non-recursive exact robust differentiator presented in [39], where the convergence proofs of (28) are well-studied utilizing quadratic and strict Lyapunov function[40], geometric approaches[40], and homogeneity properties[41]. Accordingly, an appropriate definition of the gains $\left\{ {{\varOmega _1},{\varOmega _2},\cdots,{\varOmega _6}} \right\}$ will lead to finite-time convergence of estimation error dynamics to zero[39, 40].

      The sliding surface $s = \left[ {{s_1},{s_2}} \right]$ is proposed as follows:

      $ \begin{split} &{s}_{1}(t)= {a}_{1}{J}_{1}(t)+{\widehat{J}}_{2}(t)\\ &{s}_{2}(t)={a}_{2}{J}_{3}(t)+{\widehat{J}}_{4}(t) \end{split}$

      (29)

      where $ \left\{ {a}_{1}, {a}_{2}\right\}>0$. The derivative of (29) with respect to time yields

      $\begin{split} {{\dot s}_1} = &{a_1}{J_2} + {\varOmega _2}{\left| {{{\widetilde J}_1}} \right|^\tfrac{1}{3}}{\rm sgn}\left( {{{\widetilde J}_1}} \right) - {u_1}\sin \theta +\\ & \tau {u_2}\cos \theta + {\psi _1}\left( {J,t} \right) \\ {{\dot s}_2} =& {a_2}{J_4} + {\varOmega _4}{\left| {{{\widetilde J}_3}} \right|^\tfrac{1}{3}}{\rm sgn}\left( {{{\widetilde J}_3}} \right) + {u_1}\cos \theta +\\ & \tau {u_2}\sin \theta - 1 + {\psi _2}\left( {J,t} \right). \end{split} $

      (30)

      Applying the reaching law conditions yields

      $\begin{split} & {a_1}{J_2} + {\varOmega _2}{\left| {{{\widetilde J}_1}} \right|^\tfrac{1}{3}}{\rm sgn}\left( {{{\widetilde J}_1}} \right) - {u_1}\sin \theta + \tau {u_2}\cos \theta + \\ &\quad\quad{\psi _1}\left( {J,t} \right) = - {\omega _{11}}{s_1} - {\omega _{12}}{s_2} - {k_{01}}sa{t_\varepsilon }\left( {{s_1}} \right) \\ &{a_2}{J_4} + {\varOmega _4}{\left| {{{\widetilde J}_3}} \right|^\tfrac{1}{3}}{\rm sgn}\left( {{{\widetilde J}_3}} \right) + {u_1}\cos \theta + \tau {u_2}\sin \theta - \\ & \quad\quad1 + {\psi _2}\left( {J,t} \right) = - {\omega _{21}}{s_1} - {\omega _{22}}{s_2} - {k_{02}}sa{t_\varepsilon }\left( {{s_2}} \right) \\ \end{split} $

      (31)

      where ${\omega _{ij}}$ is positive definite, ${k_{0i}} > {I_0}$, and in the sliding reachability condition ${\gamma _0}\left( {\kappa ,s} \right) = Ks$. The control laws are obtained as

      $ \begin{split} &{u_1} = - \frac{1}{{\sin \theta \left( {1 + {{\cot }^2}\theta } \right)}}\times\\ & \begin{array}{l} \Bigg[ - {\omega _{11}}{s_1} - {\omega _{12}}{s_2} - {k_{01}}sa{t_\varepsilon }\left( {{s_1}} \right) - {a_1}{J_2}-\\ {\varOmega _2}{\left| {{{\widetilde J}_1}} \right|^{1/3}}{\rm sgn}\left( {{{\widetilde J}_1}} \right)+\\ \cot \theta \Bigg( \begin{array}{l} + {\omega _{21}}{s_1} + {\omega _{22}}{s_2} + {k_{02}}sa{t_\varepsilon }\left( {{s_2}} \right)+\\ {a_2}{J_4} - {\varOmega _4}{\left| {{{\widetilde J}_3}} \right|^{1/3}}{\rm sgn}\left( {{{\widetilde J}_3}} \right) - 1 \end{array} \Bigg)\Bigg] \end{array} \\ &{u_2} = - \frac{1}{{\sin \theta \left( {1 - {{\cot }^2}\theta } \right)}}\times\\ & \begin{array}{l} \Bigg[- {\omega _{21}}{s_1} - {\omega _{22}}{s_2} - {k_{02}}sa{t_\varepsilon }\left( {{s_2}} \right) - {a_2}{J_4}-\\ {\varOmega _4}{\left| {{{\widetilde J}_3}} \right|^{1/3}}{\rm sgn}\left( {{{\widetilde J}_3}} \right)+\\ \cot \theta \Bigg( \begin{array}{l} + {\omega _{11}}{s_1} + {\omega _{12}}{s_2} + {k_{01}}sa{t_\varepsilon }\left( {{s_1}} \right)\\ + {a_1}{J_2} - {\varOmega _2}{\left| {{{\widetilde J}_1}} \right|^{1/3}}{\rm sgn}\left( {{{\widetilde J}_1}} \right) \end{array} \Bigg) + 1\Bigg]. \end{array} \end{split}$

      (32)

      The control laws stated in (32) along with (12) yield a closed-loop system with arbitrarily initial conditions. To overcome the difficulty caused by the non-minimum phase behaviour, a robust dynamic control law is applied to minimize the chattering phenomenon. Considering (10) and (29), the error value $e = |{J_1}| + |{J_2}| + |{J_3}| + \left| {{J_4}} \right|=$ $\left\| {{J_1} + {J_2} + {J_3} + {J_4}} \right\|$ can be used. If $e > {e_0}$, where $0 < {e_0} < 1$ is a small positive user-defined value, the control laws (32) are applied. As the error value reaches $e \le {e_0}$, the second control is activated and exponentially reduces the control effort. Let $\sigma > 0$,

      $\left( {{{\dot \eta }_1},{{\dot \eta }_2}} \right) = \left\{ {\begin{array}{*{20}{c}} {\left\{ {\begin{array}{*{20}{c}} {\begin{aligned} &{{{\dot \eta }_1} = - {b_1}{\rm sgn}\left( {{\eta _1}} \right)}\\ &{{{\dot \eta }_2} = 0} \end{aligned}}&,{{\rm if}\left| {{\eta _1}} \right| > {\sigma _1}} \end{array}} \right.}\\ {\left\{ {\begin{array}{*{20}{c}} {\begin{aligned} &{{{\dot \eta }_1} = {\eta _2}}\\ &{{{\dot \eta }_2} = - {b_2}{\eta _1} - {b_3}{\eta _2}} \end{aligned}}&,{{\rm if}\left| {{\eta _1}} \right| \le {\sigma _1}} \end{array}} \right.} \end{array}} \right.$

      (33)

      where $({\eta _1},{\eta _2}) = (\theta ,\dot \theta )$ and $\left\{ {{b_1},{b_2},{b_3}} \right\} > 0$ are optimized using FOFA in order to achieve the minimum control effort.

    • Consider (29) as a coordinate transformation $ J= $$ \left({J}_{1},{J}_{2},{J}_{3},{J}_{4}\right) \leftrightarrow \left({J}_{1},{s}_{1},{J}_{2},{s}_{2}\right)$. In order to analyze the closed-loop system stability, it is essential to have an estimation of the uncertainty bounds in the $\left( {\widetilde J,s} \right)$ coordinate. The additive uncertainties and external disturbances expressed by (7) can be written in the $\left( {\widetilde J,s} \right)$ coordinate as follows:

      $ \Vert {\psi }_{i}\left(J,t\right)\Vert \le {\rho }_{i}\Vert J\Vert +{l}_{i}, \;\; i=1,2.$

      (34)

      This leads to the estimate

      $\left\| {\psi \left( {J,t} \right)} \right\| \le {\rho ^{\left( 0 \right)}}\left\| J \right\| + {l_0}.$

      (35)

      where ${\rho ^{\left( 0 \right)}} = {\left( {\rho _1^2 + \rho _2^2} \right)^\tfrac{1}{2}}$, ${l_0} = {\left( {l_1^2 + l_2^2} \right)^\tfrac{1}{2}}$. Also

      $\begin{split} \left\| J \right\| &\le \left\| {\widetilde J} \right\| + {\left( {{{\left( {{s_1} + {a_1}{J_1}} \right)}^2} + {{\left( {{s_2} + {a_2}{J_3}} \right)}^2}} \right)^\tfrac{1}{2}} \le \\ &\left\| {\widetilde J} \right\| + \left\| s \right\| + {\left( {{{\left( {{a_1}{J_1}} \right)}^2} + {{\left( {{a_2}{J_3}} \right)}^2}} \right)^\tfrac{1}{2}} \le \\ & \left\| {\widetilde J} \right\| + \left\| s \right\| + \max \left\{ {{a_1},{a_2}} \right\}\left\| {\widetilde J} \right\| . \\ \end{split} $

      (36)

      Thus,

      $\left\| {\psi \left( {J,t} \right)} \right\| = \left\| {\psi \left( {\widetilde J,s,t} \right)} \right\| \le {\rho ^{\left( 1 \right)}}\left\| {\widetilde J} \right\| + {\rho ^{\left( 0 \right)}}\left\| s \right\| + {l_0}$

      (37)

      where ${\rho ^{\left( 1 \right)}} = {\rho ^{\left( 0 \right)}}\left( {1 + \max \left\{ {{a_1},{a_2}} \right\}} \right)$. This yields the parameters stated in (14). It is also worth mentioning that the control law ${u_1}$ in (32) is static and bounded as long as $\left( {\widetilde J,s} \right)$ is bounded. In order to investigate the stability of the $\left( {\widetilde J,s} \right)$ dynamics, consider the following system:

      $\begin{split} & {\dot {\widetilde J }}= A\widetilde J + Ds \\ & \dot s = - {\gamma _0}\left( {\kappa ,s} \right) - {K_0}sa{t_\varepsilon }\left( s \right) + \psi \left( {J,t} \right) . \end{split} $

      (38)

      Suppose ${s^{\rm{T}}}{\gamma _0}\left( {\kappa ,s} \right) \ge {s^{\rm{T}}}Ks$, where $K$ satisfies

      $ {\lambda }_{{\rm{min}}}\left(K\right){I}_{2}-\left[\frac{1}{{\theta }_{0}}{\left(PD\right)}^{{\rm{T}}}\left(PD\right)+\rho {I}_{2}\right]>0 $

      (39)

      where ${K_0} > {l_0}{I_2}$. For arbitrary ${\varepsilon _0} > 0$, the parameter $\varepsilon $ in (25) can be chosen such that (38) is uniformly ultimately bounded by ${\varepsilon _0}$. The following Lyapunov function candidate is suggested:

      $V = P\widetilde J + \frac{1}{2}{s^{\rm{T}}}s.$

      (40)

      Differentiating (40) along with the trajectories of (38) and using (37) yields

      $\begin{split} \dot V = & - {\left\| {\widetilde J} \right\|^2} + 2{{\widetilde J}^{\rm{T}}}PDs - {s^{\rm{T}}}{\gamma _0}\left( {\kappa ,s} \right) -\\ & {s^{\rm{T}}}{k_0}sa{t_\varepsilon }\left( s \right) + {s^{\rm{T}}}\psi \left( {\widetilde J,s,t} \right) \le\\ & - {\left\| {\widetilde J} \right\|^2} + 2{{\widetilde J}^{\rm{T}}}PDs - {s^{\rm{T}}}Ks - {s^{\rm{T}}}{k_0}sa{t_\varepsilon }\left( s \right) + \\ & \left\| s \right\|\left( {{\rho ^{\left( 1 \right)}}\left\| {\widetilde J} \right\| + {\rho ^{\left( 0 \right)}}\left\| s \right\| + {l_0}} \right) . \end{split} $

      (41)

      Considering ${\rm sgn}\left( s \right) = {\left[ {{\rm sgn}\left( {{s_1}} \right),{\rm sgn}\left( {{s_2}} \right)} \right]^{\rm T}}$ and using (42) and (43), (44) is obtained.

      ${\rho ^{\left( 1 \right)}}\left\| {\widetilde J} \right\|\left\| s \right\| \le \theta {\left\| {\widetilde J} \right\|^2} + \frac{{{{\left( {{\rho ^{\left( 1 \right)}}} \right)}^2}}}{{4\theta }}{\left\| s \right\|^2}$

      (42)

      $\left\| s \right\| \le \left| {{s_1}} \right| + \left| {{s_2}} \right| = {s^{\rm{T}}}{\rm{sgn}}\left( s \right)$

      (43)

      $\begin{split} \dot V=& -{\left\| {\widetilde J} \right\|^2}+2{{\widetilde J}^{\rm{T}}}PDs - {s^{\rm{T}}}\left( {{\lambda _{\min }}\left( K \right)} \right){I_2}s-{s^{\rm{T}}}{k_0}sa{t_\varepsilon }\left( s \right)+\\ & \left[ {\theta {{\left\| {\widetilde J} \right\|}^2} + \left( {\frac{{{{\left( {{\rho ^{\left( 1 \right)}}} \right)}^2}}}{{4\theta }} + {\rho ^{\left( 0 \right)}}} \right){{\left\| s \right\|}^2}+{s^{\rm{T}}}\left( {{l_0}I} \right){\rm sgn}\left( s \right)} \right]=\\ & \left[ {{{\widetilde J}^{\rm{T}}},{s^{\rm{T}}}} \right]\left[ {\begin{array}{*{20}{c}} { - {\theta _0}{I_2}}&{\left[ {PD} \right]}\\ {{{\left[ {PD} \right]}^{\rm{T}}}}&{ - \left( {\left( {{\lambda _{\min }}\left( K \right) - \rho } \right){I_2}} \right)} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\widetilde J}\\ s \end{array}} \right]-\\ & \left( {\sum\limits_\mu {\left( {{k_{{0_\mu }}} - {l_0}} \right)\left| {{s_\mu }} \right| + \sum\limits_\chi {\left( {\frac{{\left| {{s_\chi }} \right|}}{\varepsilon }{k_{{0_\chi }}} - {l_0}} \right)\left| {{s_\chi }} \right|} } } \right) \end{split}$

      (44)

      where ${\theta _0} = 1 - \theta $, $\mu $ and $\chi $ are subscripts where $\left| {{s_\mu }} \right| \ge \varepsilon$ and $\left| {{s_\chi }} \right| < \varepsilon $, respectively. Thus, suppose $K$ satisfing:

      $ \begin{split}\Gamma =&\left[\begin{array}{cc}-{\theta }_{0}{I}_{n-m}& -\left[PD\right]\\ -{\left[PD\right]}^{{\rm{T}}}& \left(\left({\lambda }_{\mathrm{min}}\left(K\right)-\rho \right){I}_{2}\right)\end{array}\right]>0\\ & {\lambda }_{\mathrm{min}}\left(K\right)-\rho >0.\end{split}$

      (45)

      If all the ${s_i}$ satisfy $\left| {{s_i}} \right| > \varepsilon $, then

      $ \dot{V}\le -{\varphi }_{0}\left({\Vert \widetilde{J}\Vert }^{2}+{\Vert s\Vert }^{2}\right)<0, \left(\widetilde{J},s\right)\ne 0$

      (46)

      where

      ${\phi _0} = \min \left\{ {{\lambda _{\min }}\left( \Gamma \right),\left( {{\lambda _{\min }}\left( K \right),\rho } \right),{\lambda _{\min }}\left( {{K_0} - {l_0}{I_2}} \right)} \right\}.$

      (47)

      Otherwise,

      $ \begin{split}\dot{V}\le &-{\lambda }_{\mathrm{min}}\left(K\right)\left({\Vert \widetilde{J}\Vert }^{2}+{\Vert s\Vert }^{2}\right)-{ \displaystyle\sum\limits _{\mu }\left({k}_{{0}_{\mu }}-{l}_{0}\right)\left|{s}_{\mu }\right|}-\\& { \displaystyle\sum\limits_{\chi }\left(\frac{\left|{s}_{\chi }\right|}{\epsilon }{k}_{{0}_{\chi }}-{l}_{0}\right)\left|{s}_{\chi }\right|}\le\\ & -{\varphi }_{0}{\Vert \widetilde{J},s\Vert }^{2}+{ \displaystyle\sum _{\tau }\left|{s}_{\tau }\right|{l}_{0}}\le \\& -{\varphi }_{0}{\Vert \widetilde{J},s\Vert }^{2}+2{l}_{0}\epsilon <0 , \Vert \widetilde{J},s\Vert >\sqrt{\dfrac{2{l}_{0}\epsilon }{{\varphi }_{0}}}.\\[-10pt]\end{split}$

      (48)

      Thus, choosing $\varepsilon = \varepsilon _0^2{\phi _0}/2{l_0}$, system (38) is ultimately bounded by ${\varepsilon _0} = {\left( {2{l_0}\varepsilon /{\phi _0}} \right)^\frac{1}{2}}$ if (45) is satisfied. Since the Lyapunov function candidate $V$ is radially bounded, (38) is globally uniformly ultimately bounded by ${\varepsilon _0}$ if (45) is satisfied.

    • In this section, simulation results for two maneuvers are illustrated to testify the proposed optimal HOB-DSMC control scheme's performance for VTOL UAV trajectory tracking. The first maneuver is dedicated to performance verification of the proposed control scheme, with different reference trajectories, while the second maneuver validates the performance of the proposed control scheme in comparison with conventional SMC and integral SMC (ISMC)[42] approaches. It is worth mentioning that the VTOL is assumed to move in the XY plane. Thus, the height and the yaw angle of the VTOL are not involved in the trajectory tracking and specified to be constants; however, they can be controlled separately.

    • In this maneuver, the closed-loop system consisting of the VTOL model presented in (6) and the proposed optimal HOB-DSMC control scheme is required to track various reference trajectories in the presence of external disturbances ${\psi _1}(t)$ and ${\psi _2}(t)$. Three reference trajectories are considered in this maneuver: (a) circle, (b) reverse spiral circle, and (c) 8-shape. The parameters settings for the controller and optimization algorithm are listed in Tables 1 and 2, respectively.

      ParameterValueParameterValue
      ${\omega _{11}}$6.5$\tau $0.01
      ${\omega _{12}}$0.5${a_1}$3
      ${\omega _{21}}$0.5${a_2}$4
      ${\omega _{22}}$6.5${b_1}$9
      ${k_{01}}$1.5${b_2}$6.8
      ${k_{02}}$1.5${b_3}$7.3

      Table 1.  Proposed controller parameters

      ParameterValue
      Fractional order$\Omega = 0.7$
      Number of population$50$
      Max of iteration$50$
      Light absorption$\gamma = 0.9$
      Attractiveness at $r = 0$${\beta _0} = 1.8$
      Mutation coefficient$\alpha = 0.25$

      Table 2.  Fractional-order firefly algorithm parameter settings

      The achieved results are illustrated in Fig. 2, where it can be considered as a well-suited performance index to investigate the performance of the proposed control strategy in the presence of external disturbances. According to Fig. 2, three following prescribed trajectories are considered: round, reverse spiral round, and 8-shaped trajectories. As it is observed, with respect to the reference trajectory, each position tracking error component asymptotically converges to almost zero in less than 5 s. Consequently, the simulation results demonstrated in Fig. 2 show that the proposed HOB-DSMC can effectively compensate for the effects of external disturbances, delivering asymptotically stable position errors, which demonstrates the accomplishment of the tracking mission.

      Figure 2.  Closed-loop system trajectory tracking responses using optimal HOB-DSMC

    • Comparative simulations are carried out in this maneuver to testify and highlight the performance of the proposed optimal DOB-DSMC controller. To this end, a reverse spiral 8-shape trajectory is considered as the reference, and the tracking performance of the proposed DOB-DSMC is evaluated compared to the conventional SMC and ISMC approaches. Simulation results are depicted in Figs. 3-8. Fig. 3 illustrates the reference trajectory tracking performance comparison of the control approaches under investigation. According to Fig. 3, all three approaches can track the reference trajectory with different performance levels. Compared to SMC and ISMC, it is readily observed that the optimized dynamic SMC augmented with the high-order observer accomplishes the trajectory tracking mission more precisely with less position tracking error, as shown in Fig. 4.

      Figure 3.  Performance comparison of SMC, ISMC, and HOB-DSMC for reverse spiral 8-shape reference trajectory tracking

      Figure 8.  Comparison of sliding variables for SMC, ISMC and HOB-DSMC

      Figure 4.  Comparison of closed-loop position tracking errors

      The comparative closed-loop horizontal and vertical position and velocity tracking responses are demonstrated in Figs. 5 and 6, respectively, where accordingly, the superior performance of optimal HOB-DSMC with respect to SMS and ISMC approaches is apparent. Figs. 7 and 8 illustrate comparisons of control inputs and sliding variables of SMC, ISMC, and HOB-DSMC approaches, respectively. As shown in Fig. 7, compared with SMC and ISMC, the proposed controller's oscillations are much lower. Furthermore, the markedly better convergence speed of sliding variables in HOB-DSMC is observed in Fig. 8, which demonstrates the superior performance of HOB-DSMC in comparison with SMC and ISMC.

      Figure 5.  Comparison of closed-loop horizontal position and velocity tracking responses

      Figure 6.  Comparison of closed-loop vertical position and velocity tracking responses

      Figure 7.  Comparison of control inputs for SMC, ISMC and HOB-DSMC

      According to the results achieved, it is observed that utilizing the higher-order sliding mode observer to estimate the unmeasured variables, taking advantage of the robust variable-structure control law and the optimization procedure has noticeably enhanced the controllers' performance.

    • This paper proposed a dynamic SMC scheme to deal with the unmanned aerial vehicles' trajectory tracking problem subjected to external disturbances. The proposed control scheme consists of a higher-order observer to estimate the unmeasured variables, alleviate the chattering phenomenon's effects, and a second control strategy to reduce the control input. In this regard, a neighboring point close to the sliding surface was considered as the error threshold for activating the second control, augmented with the fractional-order firefly algorithm to maintain the optimal controller parameters. In order to validate the performance of the proposed scheme, three reference trajectories were considered: circle, reverse spiral circle, and 8-shape. Furthermore, the efficiency of the proposed HOB-DSMC on the VTOL UAV model in the presence of external disturbances has been investigated compared to conventional SMC and integral SMC approaches, where the controller parameters were tuned using FOFA. Simulation results indicated the effectiveness of the proposed optimal HOB-DSMC scheme. Several future works can be explored according to the present study; for instance, the control procedure can be used for the image-based control used for obstacle avoidance, proposing new control schemes to stabilize a wide range of underactuated systems and experimental validations of the proposed approach.

Reference (42)

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return