1 Introduction

The exceptional structural design of the tilt rotor aids in the vertical take-off and landing capacity of the helicopter, and also facilitates characteristics such high cruise speed, long operational range, and heavy load of fixed wing aircraft^[1]. Quad tilt rotor (QTR) is a class of controlled unmanned aerial vehicles with complex structure, the QTR's attitude control system exhibits characteristics such as strong coupling, multi input, multi output, and nonlinearity and the system possesses modeling error and external disturbance, which causes the design of the flight control system to be exceedingly demanding^[2]. To solve these problems, many scholars have carried out extensive research, for instance, designing flight control system of the tilting rotor using nonlinear control strategy^[3-5], design of a tilting rotor flight control system based on full mode flight strategy^[6-7], unified velocity control of unmanned tilt-wing aircraft^[8], the flight control system in a large attitude change ^[9], the optimal control theory ^[10-11], tilt rotorcraft flight control method based on switch controller initialization^[12], four tilt rotor unmanned aerial vehicle control system with adaptive control ^[13]. Especially in the stage of vertical take-off and landing, it is exceedingly important that the attitude control of the aircraft, throughout the tilting of the Rotorcraft V-22 four major accidents occurred, these were all caused by out of control attitude during the vertical take-off and landing stages, therefore, this study focuses on the QTR's attitude control during the vertical takeoff and landing modes. As a nonlinear control method, because of its characteristic of significant robustness, the sliding mode variable structure control has been given considerable attention, it has been widely used in the design of motor control and aircraft control systems^[14-15]. In the general sliding mode variable structure control, usually select a linear sliding surface, the state of the system can only asymptotically converging to the equilibrium point, and the convergence rate is slow, thus when the system state error is large, the control effect of linear sliding mode is not ideal. To solve this problem, Zak et al.^[16] proposed terminal sliding mode (TSM), it is a finite time convergent sliding mode control strategy. Introducing nonlinear term in the sliding mode, the convergence characteristics of the system are improved, so that the system state is exhibits finite time convergence with respect to the desired trajectory. In Ref.[17], the final attractor in the neural network is introduced so that the state error converges to zero within a limited time. However, there is a problem of singularity and slow convergence rate in the traditional terminal sliding mode control when situated far from the sliding mode surface, this is owing to the introduction of nonlinear components the convergence speed with respect to the equilibrium state is improved, and the closer it is to the equilibrium state, the faster is the convergence rate, but, when the state of the system is away from the equilibrium point, the convergence rate of the nonlinear sliding mode is slower than that of the linear. To solve the singularity problem, scholars have studied several methods, such as switching between nonlinear and linear sliding mode surfaces^[18]; the trajectory is transformed to the prescribed interval and the terminal sliding mode in this interval is non-singular^[19]; however, these methods are incorporated to avoid the singularity problem indirectly. In Ref.[20], the redesigned terminal sliding mode directly avoids the singularity problems. In order to solve the shortcomings of convergence rate, the characteristics of a linear sliding surface is used in Ref.[21] to improve the convergence speed of the terminal sliding mode far away from the equilibrium point. In Refs.[22-23], an exponential and logarithmic type of sliding mode surface is designed, which improves the convergence speed away from the equilibrium point. However, the previous studies only consider one of the two problems, but do not consider the singular problem and the convergence speed problem simultaneously, therefore, this study is based on the aforementioned research a logarithmic form of the global fast nonsingular terminal sliding surface is designed, which avoids the singularity directly and improves the convergence speed of the sliding surface of the terminal away from the equilibrium point.

To realize the sliding mode, Gao^[24]proposed the concept of reaching law, but to reduce the chattering and convergence time, the traditional reaching law still possesses a scope for improvement. In Refs.[25-27], "double power reaching law", "variable exponent reaching law", and "an improved the power reaching law" were proposed, thereby improving the system state of convergence and suppressing the chattering of variable structure control. Based on the aforementioned research, a fast double power reaching law with variable coefficients is proposed in this study, which renders the motion quality in the process of system state approaching the sliding mode outstanding.

Because of the existence of the modeling error and external disturbance term in the QTR attitude model, it is difficult to eliminate the chattering of the controller, managing the complex disturbance becomes the problem that should be considered when designing the controller. The appearance of the extended state observer(ESO) theory^[28] provides a new method to solve the complex disturbance. The ESO is the principal component of the active disturbance rejection control (ADRC) technique. It can be used to estimate the complex disturbance and compensate it in the control system. Based on the comprehensive analysis of the above, in this study, a new logarithmic fast nonsingular terminal sliding mode controller with ESO is designed and the relevant stability proof is provided. The simulation results show that the scheme has good tracking control performance.

2 Attitude Model of QTR Vertical Take-Off and Landing Model

The attitude control of the QTR vertical take-off and landing mode is realized via differential adjustment of the lift force of the propellers or nacelle angle. The control of the pitching channel is achieved via differential adjustment of the two sets of front and rear propellers, the rolling channel is realized via differential adjustment of the rotation rate of the rotor on both sides of the fuselage, the yaw channel is achieved via the differential adjustment of the two groups of nacelle angles on both sides of the fuselage. The system structure is shown in Fig. 1.

Neglecting the impact of the QTR's elastic vibration, a new state variable x is defined as follows:

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{x}} = {{\left[ {\begin{array}{*{20}{c}} {{x_1}}&{{x_2}}&{{x_3}}&{{x_4}}&{{x_5}}&{{x_6}} \end{array}} \right]}^{\rm{T}}} = }\\ {{{\left[ {\begin{array}{*{20}{c}} \phi &{\dot \phi }&\theta &{\dot \theta }&\varphi &{\dot \varphi } \end{array}} \right]}^{\rm{T}}}} \end{array} $

The attitude model of the QTR vertical take-off and landing model can be expressed as follows with respect to the nonlinear system with uncertainties:

$ \left\{ \begin{array}{l} {{\dot x}_{\left( {2k - 1} \right)}}\left( t \right) = {x_{\left( {2k} \right)}}\left( t \right)\\ {{\dot x}_{\left( {2k} \right)}}\left( t \right) = {f_k}\left( {x,t} \right) + {g_k}{u_k} + {\sigma _{dk}} \end{array} \right. $

(1)

where k=1, 2, 3, f_k is

$ {\mathit{\boldsymbol{f}}_k} = \left[ {\begin{array}{*{20}{c}} {\frac{{{I_{xz}}\left( {{I_{xx}} - {I_{yy}} + {I_{zz}}} \right){x_4}{x_6} + \left[ {{I_{zz}}\left( {{I_{yy}} - {I_{xx}}} \right) - I_{xz}^2} \right]{x_2}{x_6}}}{{{I_{xx}}{I_{zz}} - I_{xz}^2}}}\\ {\frac{{\left( {{I_{zz}} - {I_{xx}}} \right){x_2}{x_6} + {I_{xz}}\left( {x_6^2 - x_2^2} \right)}}{{{I_{yy}}}}}\\ {\frac{{{I_{xx}}\left( {{I_{xx}} - {I_{yy}}} \right){x_4}{x_6} + {I_{xz}}\left( {{I_{xx}} - {I_{yy}} + {I_{zz}}} \right){x_2}{x_6}}}{{{I_{xx}}{I_{zz}} - I_{xz}^2}}} \end{array}} \right] $

and g_k is

$ {\mathit{\boldsymbol{g}}_k} = \left[ {\begin{array}{*{20}{c}} {\frac{{{I_{zz}}}}{{{I_{xx}}{I_{zz}} - I_{xz}^2}}}&0&{\frac{{{I_{xz}}}}{{{I_{xx}}{I_{zz}} - I_{xz}^2}}}\\ 0&{\frac{1}{{{I_{yy}}}}}&0\\ {\frac{{{I_{xz}}}}{{{I_{xx}}{I_{zz}} - I_{xz}^2}}}&0&{\frac{{{I_{xz}}}}{{{I_{xx}}{I_{zz}} - I_{xz}^2}}} \end{array}} \right] $

$ {\mathit{\boldsymbol{u}}_k} = {\left[ {\begin{array}{*{20}{l}} {{u_1}}&{{u_2}}&{{u_3}} \end{array}} \right]^{\rm{T}}} $

$ {\sigma _{dk}} = \Delta {f_k}\left( {x,t} \right) + \Delta {g_k}{u_k} + {d_k} $

where, Δf_k, Δg_k are the modeling error terms; σ_dkis composite disturbance; φ, θ, ϕ are the roll angle, pitch angle, and yaw angle; I_xx, I_yy, I_zz are the moment of inertia around the x, y, z axis of the fuselage; I_xz is the inertial product; d_k refers to the external disturbances of each channel; u₁, u₂, u₃ are the control input of each channel, the specific expression is

$ {u_1} = {L_\varphi }{k_T}\left( {w_1^2 - w_2^2 + w_3^2 - w_4^2} \right) $

$ {u_2} = {L_\theta }{k_T}\left( {w_1^2 + w_2^2 - w_3^2 - w_4^2} \right) $

$ {u_3} = {L_\phi }mgv $

where k_T is the propeller thrust coefficient, w_i is the corresponding propeller speed, m is the aircraft mass, υ is the nacelle angle, L_φ, L_θ, L_ϕ are the arms of each channel.

3 Terminal Sliding Mode Control

The common terminal sliding mode surface (TSM) is designed as ^[17]

$ {s_T} = \dot x + {\beta _T}{\rm{sig}}{(x)^{{q_T}}} $

(2)

where, sliding surface parameters β_T>0, and 0 < p_T < 1.

The state of the system based on the sliding mode surface of Eq.(2) can be stable for a limited time, however, when the state is far from the equilibrium, the convergence rate is slow, thus the design described in Ref.[18] is as follows, fast terminal sliding mode surface (FTSM):

$ {s_F} = \dot x + {\alpha _F}x + {\beta _F}{\rm{sig}} {(x)^{{p_F}}} $

(3)

where, α_F, β_F>0;p_F∈(0, 1). The convergence time of the system is T_s=max{T_si} (i=1, 2, …, N), where:

$ {T_{si}} = \frac{1}{{{\alpha _F}\left( {1 - {P_F}} \right)}}\ln \frac{{{\alpha _F}{{\left| {{x_{si}}} \right|}^{1 - {p_F}}} + {\beta _F}}}{{{\beta _F}}} $

(4)

where, x_si∈ R is the state of the system when it reaches the sliding surface.

However, the above two types of sliding surface designs will inevitably exhibit the singularity phenomenon, thus, in Ref.[18] a nonsingular terminal sliding mode surface (NTSM) is designed, in Refs. [29] and [30] a fast nonsingular terminal sliding mode surface (FNTSM) is designed; the FNTSM surface described in Ref.[30] is as follows:

$ {s_N} = x + \frac{1}{{{\alpha _N}}}{\rm{sig}}{\left( x \right)^{{q_N}}} + \frac{1}{{{\beta _N}}}{\rm{sig}}{\left( {\dot x} \right)^{{p_N}}} $

(5)

where, α_N, β_N>0;p_N∈(1, 2);q_N>p_N. The convergence time of the system is

$ \begin{array}{l} {T_{si}} = \frac{{{p_N}}}{{{p_N} - 1}}\beta _N^{ - 1/{P_N}}{\left| {{x_{si}}} \right|^{1 - 1/{p_N}}} \cdot \bar F\left( {\frac{1}{{{P_N}}},} \right.\\ \;\;\;\;\left. {\frac{{{P_N} - 1}}{{{P_N}\left( {{q_N} - 1} \right)}},1 + \frac{{{P_N} - 1}}{{{P_N}\left( {{q_N} - 1} \right)}}, - \frac{1}{{{\alpha _N}}}{{\left| {{x_{si}}} \right|}^{{q_N} - 1}}} \right) \end{array} $

(6)

where F is Gauss hypergeometric function^[31].

4 New Logarithmic Nonsingular Fast Terminal Sliding Mode

Based on the aforementioned research, a new logarithmic fast nonsingular terminal sliding mode (LFNTSM) surface is designed in this study:

$ \begin{array}{l} {s_k} = {\left( {{I_k} + \left| x \right|} \right)^p}{\rm{diag}}\left( {{{\ln }^{\left( {1 + \left| x \right|} \right)}}} \right){\rm{sgn}}(x) + \\ \;\;\;\;\;\;\frac{1}{\alpha }{\left( {{I_k} + \left| x \right|} \right)^p}{\rm{diag}}{\left( {{{\ln }^{\left( {1 + \left| x \right|} \right)}}} \right)^q}{\rm{sgn}}(x) + \\ \;\;\;\;\;\;\frac{1}{\beta }{\rm{sig}}{\left( {\dot x} \right)^p} \end{array} $

(7)

where |x|=diag(x_k)sgn(x).

Lemma 1   For Gaussian hypergeometric functions^[20]

$ \bar F\left( {A,B,C,z} \right) = \sum\limits_{k = 0}^{ + \infty } {\frac{{{{\left( A \right)}_k}{{\left( B \right)}_k}}}{{{{\left( C \right)}_k}k!}}{z^k}} $

(8)

If A, B, and C are positive real numbers, and C-B-A>0, then F(A, B, C, z) is convergent on the domain z < 0.

Theorem 1  For the nonlinear system described in Eq.(1), if the sliding mode surface of Eq.(7) is used to design the controller, and the parameters satisfy the condition α, β>0, p∈(1, 2), q>p, then the state of the system will slide to x≡0 in the finite time T_sk after reaching the sliding surface, where:

$ \begin{array}{l} {T_{sk}} = \frac{{{p_L}}}{{{p_L} - 1}}\beta _L^{ - 1/{P_L}}{\left| {{{\ln }^{\left( {1 + \left| {{x_k}} \right|} \right)}}} \right|^{1 - 1/{p_L}}} \cdot \\ \;\;\;\;\bar F\left( {\frac{1}{{{P_L}}},\frac{{{P_L} - 1}}{{{P_L}\left( {{q_L} - 1} \right)}},1 + \frac{{{P_L} - 1}}{{{P_L}\left( {{q_L} - 1} \right)}},} \right.\\ \;\;\;\;\left. { - \frac{1}{{{\alpha _L}}}{{\left| {{{\ln }^{\left( {1 + \left| {{x_k}} \right|} \right)}}} \right|}^{{q_N} - 1}}} \right) \end{array} $

(9)

Proof   When s=0, Eq. (7) is transformed as follows:

$ \begin{array}{l} \dot x = - {\left[ {\beta \left[ {{{\ln }^{(1 + |x|)}}} \right] + \frac{\beta }{\alpha }{{\left[ {{{\ln }^{(1 + |x|)}}} \right]}^q}} \right]^{1/p}}(1 + \\ \;\;\;|x|) {\rm{sgn}} (x) \end{array} $

which can be obtained using ${\mathit{\boldsymbol{\dot x}}^{\rm{T}}}\mathit{\boldsymbol{x}} < 0$, thus x will approach zero from any initial value. To transpose and integral on both sides of equation can be obtained using Eq.(9). Using Lemma 1, we can understand that when the parameters satisfy α, β>0, p∈(1, 2), q>p, F(A, B, C, z) is convergent, thus T_s will converge to a constant, i.e., the time required for the state of the system to slide to zero is T_s and is a constant, x≡0 exists.

Remark 1  Comparing the convergence time of FNTSM and LFNTSM, the term ln^(1+|x|) is observed to be the only difference between Eq.(6) and Eq.(9), and when x_i≠0, the term ln^(1+|x|) < |x| is always tenable. Therefore, under the same parameters, the terminal sliding mode based on Theorem 1 is faster than the existing fast nonsingular terminal sliding mode technique with regard to avoiding singularity.

5 New Fast Reaching Law

To realize the system state fast reaching sliding surface, Professor Gao ^[24] proposed the following exponential reaching law:

$ {{\dot s}_k} = - {\varepsilon _{k1}} sgn \left( {{s_k}} \right) - {\varepsilon _{k2}}{s_k},{\varepsilon _{k1}} > 0,{\varepsilon _{k2}} > 0 $

(10)

Based on the form of the exponential reaching law, it can be observed that the derivative of the sliding mode surface is not zero when the state of system converges to s=0, it slides the system state out of the sliding surface, and the reaching law does not possess the two order sliding mode characteristics, the chattering is relatively large, if the saturation function is used instead of the switching function, the robustness of the system will be eliminated. In this study, to realize the fast chattering free approach to sliding surface, a double power reaching law of variable coefficients with two order sliding mode characteristics is proposed:

$ \begin{array}{l} {{\dot s}_k} = - \left( {{I_k} + \left| {{{\dot \lambda }_k}} \right|} \right)\left[ {{\varepsilon _{k1}}{\rm{diag}}\left\{ {\left| {{s_k}} \right|} \right\}{\rm{sgn}}\left( {{s_k}} \right) + } \right.\\ \;\;\;\left. {{\varepsilon _{k2}}{\rm{diag}}\left\{ {{{\left| {{s_k}} \right|}^{m/n}}} \right\}{\rm{sgn}}\left( {{s_k}} \right)} \right]\\ \;\;\;{\varepsilon _{k1}} > 0,{\varepsilon _{k2}} > 0,n > m > 0,k = 1,2,3\\ \;\;\;\left| {{{\dot \lambda }_k}} \right| = {\rm{diag}}\left( {\left| {{{\dot \lambda }_1}} \right|,\left| {{{\dot \lambda }_2}} \right|,\left| {{{\dot \lambda }_3}} \right|} \right) \end{array} $

(11)

When the state of system is far away from the sliding surface, the convergence rate of the new reaching law is primarily determined via the term $-\left(I_{k}+\left|\dot{\lambda}_{k}\right|\right) \varepsilon_{k 1} \operatorname{diag}\left\{\left|s_{k}\right|\right\}$ sgn (s_k), and the rate of convergence is faster than the traditional approach law; when the system state is near the sliding surface, the term $-\left(I_{k}+\left|\dot{\lambda}_{k}\right|\right) \varepsilon_{k 2} \operatorname{diag}\left\{\left|s_{k}\right|^{m / n}\right\} \operatorname{sgn}\left(s_{k}\right)$ in the reaching law plays a decisive role, so that the system can smoothly enter the sliding mode; with the combination of the two terms, the sliding quality in the process of approaching the sliding mode is guaranteed. Using Theorem 1 of Ref.[14], it can be deduced that the time when the state of system reaches the sliding mode surface is:

$ {t_{rk}} < \left\{ {\begin{array}{*{20}{l}} {\frac{1}{{1 + \left| {{{\dot \lambda }_k}} \right|}} \cdot \frac{1}{{{\varepsilon _{k2}}(1 - m/n)}},\left| {s\left( 0 \right)} \right| > 1}\\ {\frac{1}{{1 + \left| {{{\dot \lambda }_k}} \right|}} \cdot \frac{{{{\left| {s\left( 0 \right)} \right|}^{1 - m/n}}}}{{{\varepsilon _{k2}}(1 - m/n)}},0 < \left| {s\left( 0 \right)} \right| < 1} \end{array}} \right. $

(12)

It can be observed that the system state converges to the sliding mode surface in a finite time, and it is $\dot{s}_{k}$=0 when s_k=0, the convergence rate decreases to zero when the state reaches the sliding mode, which significantly reduces the chattering of the system.

6 Design of New Logarithmic Nonsingular Terminal Sliding Mode Control with ESO 6.1 New Logarithmic Nonsingular Terminal Sliding Mode Controller

Introducing the new logarithmic fast nonsingular fast terminal sliding surface:

$ \begin{array}{l} {s_k} = {\left( {{I_k} + \left| {{\lambda _k}} \right|} \right)^p}{\rm{diag}}\left( {{{\ln }^{\left( {1 + \left| {{\lambda _k}} \right|} \right)}}} \right){\rm{sgn}}\left( {{\lambda _k}} \right) + \\ \;\;\;\;\;\;\frac{1}{\alpha }{\left( {{I_k} + \left| {{\lambda _k}} \right|} \right)^p}{\rm{diag}}{\left[ {{{\ln }^{\left( {1 + \left| {{\lambda _k}} \right|} \right)}}} \right]^q}{\rm{sgn}}\left( {{\lambda _k}} \right) + \\ \;\;\;\;\;\;\frac{1}{\beta }{\rm{sig}}{\left( {{{\dot \lambda }_k}} \right)^p} \end{array} $

(13)

where,

$ \left| {{\lambda _k}} \right| = {\rm{diag}}\left( {{\lambda _k}} \right){\rm{sgn}}\left( {{\lambda _k}} \right) $

$ {\lambda _k} = {x_{(2k - 1)}} - {x_{d(2k - 1)}},k = 1,2,3 $

The following is obtained via the derivation of Eq.(13):

$ {{\dot s}_k} = h\left( {{\lambda _k},{{\dot \lambda }_k}} \right) + \frac{p}{\beta }{\left( {{{\dot \lambda }_k}} \right)^{p - 1}}{{\ddot \lambda }_k} $

(14)

where:

$ \begin{array}{l} h\left( {{\lambda _k},} \right.\left. {{{\dot \lambda }_k}} \right) = \left[ {p{{\left( {1 + \left| {{\lambda _k}} \right|} \right)}^{p - 1}} \cdot {{\ln }^{\left( {1 + \left| {{\lambda _k}} \right|} \right)}}} \right] \cdot \\ \;\;\;\;\;\;\;{{\dot \lambda }_k} + {\left( {1 + \left| {{\lambda _k}} \right|} \right)^{p - 1}}{{\dot \lambda }_k} + \\ \;\;\;\;\;\;\frac{p}{\alpha }{\left( {1 + \left| {{\lambda _k}} \right|} \right)^{p - 1}}{\ln ^{{{\left( {1 + \left| {{\lambda _k}} \right|} \right)}^q}}}{{\dot \lambda }_k} + \\ \;\;\;\;\;\;\frac{q}{\alpha } \cdot {\left( {1 + \left| {{\lambda _k}} \right|} \right)^{p - 1}} \cdot {\ln ^{\left( {1 + {{\left| {{\lambda _k}} \right|}^{q - 1}}} \right)}}{{\dot \lambda }_k} \end{array} $

Combined with the new fast approaching law proposed in the previous section of this paper, the corresponding control law is designed as follows:

$ \begin{array}{l} {u_k} = - g_k^{ - 1}\left\{ {\frac{\beta }{p}{{\left( {{{\dot \lambda }_k}} \right)}^{1 - p}}h\left( {{\lambda _k},{{\dot \lambda }_k}} \right) + {f_k} - {{\ddot x}_{d(2k - 1)}} + } \right.\\ \;\;\;\;\;\;\;{\sigma _{dk}} + \left[ {{\varepsilon _{k1}}{\rm{diag}}\left\{ {\left| {{s_k}} \right|} \right\}{\rm{sgn}}\left( {{s_k}} \right) + } \right.\\ \;\;\;\;\;\;\;\left. {\left. {{\varepsilon _{k2}}{\rm{diag}}\left\{ {{{\left| {{s_k}} \right|}^{m/n}}} \right\}{\rm{sgn}}\left( {{s_k}} \right)} \right]\left( {1 + \left| {{{\dot \lambda }_k}} \right|} \right)} \right\} \end{array} $

(15)

where, ε_kj∈R⁺(k=1, 2, 3, j=1, 2), and σ_dk is the compound disturbance to be estimated.

6.2 Disturbance Observer Based on ESO

For system (1), the effect of the modeling error and external disturbance is exceedingly serious. To ensure the control precision and robustness and to minimize the chattering as much as possible, this study incorporates the extended state observer (ESO) to conduct the estimation and compensation of the online real-time disturbance. In this study, a disturbance observer is designed using the ESO model proposed in Ref.[17],

$ \left\{ {\begin{array}{*{20}{l}} {{{\dot {\hat x}}_{(2k - 1)}} = {{\hat x}_{(2k)}} + \frac{{{\rho _{k1}}}}{{{\xi _k}}}\left( {{x_{(2k - 1)}} - {{\hat x}_{(2k - 1)}}} \right)}\\ {{{\dot {\hat x}}_{(2k)}} = {{\hat f}_{(k)}} + {{\hat g}_{(k)}}{u_k} + {{\hat \sigma }_{(k)}} + \frac{{{\rho _{k2}}}}{{\xi _k^2}}\left( {{x_{(2k - 1)}} - {{\hat x}_{(2k - 1)}}} \right)}\\ {{{\dot {\hat \sigma} }_{(k)}} = \frac{{{\rho _{k3}}}}{{\xi _k^3}}\left( {{x_{(2k - 1)}} - {{\hat x}_{(2k - 1)}}} \right)} \end{array}} \right. $

(16)

where k=1, 2, 3. $\hat{x}_{(2 k-1)}, \hat{x}_{(2 k)}, \hat{\sigma}_{(d k)}$ are the observer states, ξ_k>0, ρ_k1, ρ_k2, ρ_k3 are positive real numbers. In Ref.[17], the self-stabilization domain method is used to prove that the observation error of the observer in Eq. (16) is gradually converged to zero. Based on the above condition the system loop control law with ESO is as follows:

$ \begin{array}{l} {u_k} = - g_k^{ - 1}\left\{ {\frac{\beta }{p}{{\left( {{{\dot {\hat \lambda} }_k}} \right)}^{1 - p}}h\left( {{{\hat \lambda }_k},{{\dot {\hat \lambda} }_k}} \right) + {{\hat f}_k} - {{\ddot x}_{d(2k - 1)}} + } \right.\\ \;\;\;\;{{\hat \sigma }_{dk}} + \left[ {{\varepsilon _{k1}}{\rm{diag}}\left\{ {\left| {{s_k}} \right|} \right\}{\rm{sgn}}\left( {{s_k}} \right) + } \right.\\ \left. {\left. {\;\;\;\;{\varepsilon _{k2}}{\rm{diag}}\left\{ {{{\left| {{s_k}} \right|}^{m/n}}} \right\}{\rm{sgn}}\left( {{s_k}} \right)} \right]\left( {1 + \left| {{{\hat \lambda }_k}} \right|} \right)} \right\} \end{array} $

(17)

Assumption 1   Composite disturbance σ_dk and composite disturbance estimate ${\widehat \sigma _{dk}}$, there is σ_dk^*∈R⁺, satisfying

$ \left| {{\sigma _{dk}} - {{\hat \sigma }_{dk}}} \right| \le \sigma _{dk}^* $

(18)

Theorem 2  For the attitude control system of QTR, if the composite disturbance σ_dk satisfies the condition of Assumption 1, using the new type of nonsingular terminal sliding mode controller with ESO disturbance observer as shown in Eq.(15), the system state trajectory will reach a neighborhood Ω_k of the sliding surface in a finite time, and

$ \left\{ {\begin{array}{*{20}{l}} {\left| {{\mathit{\Omega }_k}} \right| \le \Delta = \min \left( {{\Delta _1},{\Delta _2}} \right)}\\ {{\Delta _1} = \frac{{\sigma _{dk}^*}}{{{\varepsilon _{k1}}\left( {1 + \left| {{{\dot \lambda }_k}} \right|} \right)}}}\\ {{\Delta _2} = {{\left( {\frac{{\sigma _{dk}^*}}{{{\varepsilon _{k2}}\left( {1 + \left| {{{\dot \lambda }_k}} \right|} \right)}}} \right)}^{n/m}}} \end{array}} \right. $

(19)

λ_k and $\dot{\lambda}_{k}$ converge respectively to the following areas in finite time:

$ \left\{ {\begin{array}{*{20}{l}} {\left| {{\lambda _k}} \right| \le {\Delta _{{\lambda _k}}}}\\ {\left| {{{\dot \lambda }_k}} \right| \le {\Delta _{{{\dot \lambda }_k}}}} \end{array}} \right. $

(20)

where,

$ {\Delta _{{\lambda _k}}} = \min \left[ {{{\left( {\frac{{a\sigma _{dk}^*}}{{{\varepsilon _{k1}}}}} \right)}^{1/q}},{{\left( {\frac{{{a^{m/n}}\sigma _{dk}^*}}{{{\varepsilon _{k2}}}}} \right)}^{n/qm}}} \right] $

$ {\Delta _{{{\dot \lambda }_k}}} = \min \left[ {{{\left( {\frac{{\beta \sigma _{dk}^*}}{{{\varepsilon _{k1}}}}} \right)}^{1/p}},{{\left( {\frac{{{\beta ^{m/n}}\sigma _{dk}^*}}{{{\varepsilon _{k2}}}}} \right)}^{n/pm}}} \right] $

Proof  The selected Lyapunov function is

$ {\mathit{\boldsymbol{V}}_k} = \frac{1}{2}\mathit{\boldsymbol{S}}_k^{\rm{T}}{\mathit{\boldsymbol{S}}_k} $

(21)

$ \begin{array}{l} {{\mathit{\boldsymbol{\dot V}}}_k} = {\mathit{\boldsymbol{S}}^{\rm{T}}}\mathit{\boldsymbol{S}} = {\mathit{\boldsymbol{S}}^{\rm{T}}}\left( {h + p{{(\dot \lambda )}^{p - 1}}\ddot \lambda /\beta } \right) = \\ \;\;\;\;\;{\mathit{\boldsymbol{S}}^{\rm{T}}}\left( {h + p{{(\dot \lambda )}^{p - 1}}\left( {{f_k}(x,t) + {g_k}{u_k} + {\sigma _{dk}}} \right)/\beta } \right) \end{array} $

(22)

where k=1, 2, 3.

The following formula can be obtained by substituting Eq.(17) into Eq.(22):

$ \begin{array}{l} {{\mathit{\boldsymbol{\dot V}}}_k} = {\mathit{\boldsymbol{S}}^{\rm{T}}}\frac{p}{\beta }{\left( {\mathit{{\dot \lambda }}} \right)^{p - 1}}\left\{ {\left( {{\sigma _{dk}} - {{\hat \sigma }_{dk}}} \right) - } \right.\\ \;\;\;\;\left[ {{\varepsilon _{k1}}{\rm{diag}}\left\{ {\left| {{s_k}} \right|} \right\}{\rm{sgn}}\left( {{s_k}} \right) + } \right.\\ \;\;\;\;\left. {{\varepsilon _{k2}}{\rm{diag}}\left\{ {{{\left| {{s_k}} \right|}^{m/n}}} \right\}{\rm{sgn}}\left( {{s_k}} \right)\left( {1 + \left| {{{\dot \lambda }_k}} \right|} \right)} \right\} = \\ \;\;\;\; - \eta \left( {{\delta _{k1}}{{\left| {{s_k}} \right|}^2} + {\varepsilon _{k2}}{{\left| {{s_k}} \right|}^{1 + \frac{m}{n}}}} \right)\left( {1 + \left| {{{\dot \lambda }_k}} \right|} \right) = \\ \;\;\;\; - 2\eta \left( {{\delta _{k1}}{V_k} + {\varepsilon _{k2}}V_k^{(n + m)/2n}} \right)\left( {1 + \left| {{{\dot \lambda }_k}} \right|} \right)\\ \;\;\;\;k = 1,2,3 \end{array} $

(23)

where,

$ \eta = \frac{p}{\beta }{\left( {{{\dot \lambda }_k}} \right)^{p - 1}} $

$ {\delta _{k1}} = \frac{{ - {s_k}\left( {{\sigma _{dk}} - {{\hat \sigma }_{dk}}} \right)}}{{\left( {1 + \left| {{{\dot \lambda }_k}} \right|} \right){{\left| {{s_k}} \right|}^2}}} + {\varepsilon _{k1}} $

$ k = 1,2,3 $

when, δ_k1>0, $\dot{\lambda}_{k}$≠0, the format of Eq.(23) and the proposed approach law (11) is the same, it can ensure that the system state converges within a limited time t_k, t_k satisfies the Eq.(12), because ε_k1>0 and combining the assumption 1, we can obtain the outside the area as

$ \left| {{s_k}} \right| \le \frac{{\sigma _{dk}^*}}{{{\varepsilon _{k1}}\left( {1 + \left| {{{\dot \lambda }_k}} \right|} \right)}} $

(24)

$\dot{V}$≤0, thus, in this case the convergence domain is Eq.(24), similarly, the Eq.(22) can be transformed into the following equation:

$ {{\dot V}_k} = - \eta \left( {{\varepsilon _{k1}}{V_k} + {\delta _{k2}}V_k^{(n + m)/2n}} \right)\left( {1 + \left| {{{\dot \lambda }_k}} \right|} \right) $

(25)

where, k=1, 2, 3

$ {\delta _{k2}} = \frac{{ - {s_k}\left( {{\sigma _{dk}} - {{\hat \sigma }_{dk}}} \right)}}{{\left( {1 + \left| {{{\dot \lambda }_k}} \right|} \right){{\left| {{s_k}} \right|}^{1 + m/n}}}} + {\varepsilon _{k2}},{\delta _{k2}} > 0 $

Using the same analysis method, we can obtain another description of the convergence domain:

$ \left| {{s_k}} \right| \le {\left( {\frac{{\sigma _k^*}}{{{\varepsilon _{k2}}\left( {1 + \left| {{{\dot \lambda }_k}} \right|} \right)}}} \right)^{n/m}} $

(26)

It can be derived from Eqs. (24) and (26) that the region where the system state converges within a finite time is as described in Eq. (19).

When $\dot{\lambda}_{k}$=0, the control law Eq.(17) is substituted into the system (1) to obtain

$ \begin{array}{l} {{\dot x}_{(2k)}}(t) = - \left[ {{\varepsilon _{k1}}{\rm{diag}}\left\{ {\left| {{s_k}} \right|} \right\}{\rm{sgn}}\left( {{s_k}} \right) + {\varepsilon _{k2}}{\rm{diag}}\left\{ {{{\left| {{s_k}} \right|}^{m/n}}} \right\}} \right.\\ \left. {\;\;\;\;\;\;\;\;\;\;\;{\rm{sgn}}\left( {{s_k}} \right)} \right] - \left( {{\sigma _{dk}} - {{\hat \sigma }_{dk}}} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\left\{ {\begin{array}{*{20}{l}} { - \left( {{\delta _{k1}}\left| {{s_k}} \right| + {\varepsilon _{k2}}{{\left| {{s_k}} \right|}^{m/n}}} \right)}\\ { - \left( {{\varepsilon _{k1}}\left| {{s_k}} \right| + {\delta _{k2}}{{\left| {{s_k}} \right|}^{m/n}}} \right)} \end{array}} \right. \end{array} $

(27)

for the sliding surface s_k outside the region Δ₁, Δ₂, there exists δ_k1, δ_k2>0, s_k≠0, thus we can obtain $\dot{x}_{(2 k)}$≠0 using the Eq.(23), i.e., when the state of the system reaches the neighborhood range of the sliding surface Ω_k, $\dot{\lambda}_{k}$ is not an attractor, therefore, the conclusion that the system state reaches the neighborhood of the sliding surface within a limited time is obtained.

When the state of the system enters the |s_k| ≤Δ region, there is the |τ| ≤Ω_k

$ \begin{array}{l} {\left( {1 + \left| {{\lambda _k}} \right|} \right)^p}{\ln ^{\left( {1 + \left| {{\lambda _k}} \right|} \right)}}{\rm{sgn}}\left( {{\lambda _k}} \right) + \\ \;\;\;\;\;\frac{1}{\alpha }{\left( {1 + \left| {{\lambda _k}} \right|} \right)^p}{\left[ {{{\ln }^{\left( {1 + \left| {{\lambda _k}} \right|} \right)}}} \right]^q}{\rm{sgn}}\left( {{\lambda _k}} \right) + \\ \;\;\;\;\;\frac{1}{\beta } {\rm{sig}} {\left( {{{\dot \lambda }_k}} \right)^p} = \tau \end{array} $

(28)

Eq. (28) can be converted to

$ \begin{array}{l} {\left( {1 + \left| {{\lambda _k}} \right|} \right)^p}{\ln ^{\left( {1 + \left| {{\lambda _k}} \right|} \right)}}{\rm{sgn}}\left( {{\lambda _k}} \right) + \\ \;\;\;\;\;\frac{1}{\alpha }{\left( {1 + \left| {{\lambda _k}} \right|} \right)^p}{\left[ {{{\ln }^{\left( {1 + \left| {{\lambda _k}} \right|} \right)}}} \right]^q}{\rm{sgn}}\left( {{\lambda _k}} \right) + \left( {\frac{1}{\beta } - } \right.\\ \;\;\;\;\;\;\left. {\frac{\tau }{{ {\rm{sig}} {{\left( {{{\dot \lambda }_k}} \right)}^p}}}} \right) {\rm{sig}} {\left( {{{\dot \lambda }_k}} \right)^p} = 0 \end{array} $

(29)

When β^-1-τsig(${\dot \lambda _k}$)^-p>0, the Eq.(29) satisfies the LFTSM form as described in Eq.(7), therefore, the state ${\dot \lambda _k}$ can converge rapidly to the region $\left|\dot{\lambda}_{k}\right| \leqslant(\tau \beta)^{1 / p} \leqslant(\Delta \beta)^{1 / p} \leqslant \Delta_{\dot{\lambda}_{k}}$ in finite time. The same method is used to transform the Eq.(28), we can conclude that the convergence region of the state λ_k is |λ_k|≤Δ_λk, where Δ_λk and ${\Delta _{{{\dot \lambda }_k}}}$ satisfy the form of the Eq.(20).

7 Simulation and Result Analysis 7.1 Simulation Parameters

The QTR's quality is 6.65 kg, the moment of inertia are

$ {I_{xx}} = 4.782{\rm{kg}} \cdot {{\rm{m}}^2} $

$ {I_{yy}} = 3.258{\rm{kg}} \cdot {{\rm{m}}^2} $

$ {I_{zz}} = 2.837{\rm{kg}} \cdot {{\rm{m}}^2} $

The product of inertia is I_xz=0.032 kg·m², the initial state is ϕ₀=θ₀=φ₀=0°, the target instructions are φ_d=3°, θ_d=5°, ϕ_d=2°, the moment of inertia is applied to 12% uncertainty, assumptions from the beginning of the simulation, the attitude angle measurement exhibits a Gauss white noise with a magnitude of 0.05, and filters 50/(s+50) are used, the time-varying disturbance torque applied to the system is

$ \Delta \mathit{\boldsymbol{M}} = \left[ {\begin{array}{*{20}{l}} {3\sin (0.9t) + 6\cos (0.2t)}\\ {2\sin (0.3t) + 7\cos (0.6t)}\\ {6\sin (0.7t) + 2\cos (0.5t)} \end{array}} \right] $

In order to verify the validity of the method proposed in this paper, the following two control schemes are compared.

Control scheme 1   Applying the new logarithmic fast nonsingular terminal sliding mode (LFNTSM) controller proposed in this study, the reaching law uses a new fast reaching law with two order sliding mode characteristics.

Control scheme 2   Using the traditional fast nonsingular terminal sliding mode (FNTSM) controller, the reaching law is selected based on the traditional exponential reaching law, the corresponding control law is as follows:

$ \begin{array}{l} {u_k} = - g_k^{ - 1}\left\{ {\frac{\beta }{p}{{\left( {{{\dot {\hat \lambda} }_k}} \right)}^{2 - p}}{\rm{sgn}}\left( {{{\dot {\hat \lambda} }_k}} \right)\left( {1 + \frac{q}{\alpha }{{\left| {{{\hat \lambda }_k}} \right|}^{q - 1}}} \right) + } \right.\\ \;\;\;\;\;\;\left. {{{\hat f}_k} - {{\ddot x}_{d(2k - 1)}} + {{\hat \sigma }_{dk}} + {\varepsilon _{k1}}{\rm{sgn}}\left( {{s_k}} \right) + {\varepsilon _{k2}}{s_k}} \right\} \end{array} $

In control schemes 1 and 2, the extended state disturbance observer is used to estimate and compensate the complex disturbances. For the two sliding mode control schemes, the selected controller parameters are as follows:

In control scheme 1:

$ \begin{array}{*{20}{l}} {p = {\rm{diag}}\left[ {1.7,1.7,1.7} \right],q = {\rm{diag}}\left[ {3,3,3} \right]}\\ {\alpha = {\rm{diag}}\left[ {10,10,10} \right],\beta = {\rm{diag}}\left[ {40,40,40} \right]}\\ {{\varepsilon _{k1}} = {\rm{diag}}\left[ {0.6,0.6,0.6} \right],{\varepsilon _{k2}} = {\rm{diag}}\left[ {1.8,1.8,1.8} \right]}\\ {n = 4,m = 3} \end{array} $

In control scheme 2:

$ \begin{array}{*{20}{l}} {p = {\rm{diag}}\left[ {1.7,1.7,1.7} \right],q = {\rm{diag}}\left[ {3,3,3} \right]}\\ {\alpha = {\rm{diag}}\left[ {10,10,10} \right],\beta = {\rm{diag}}\left[ {40,40,40} \right]}\\ {{\varepsilon _{k1}} = {\rm{diag}}\left[ {0.8,0.8,0.8} \right],{\varepsilon _{k2}} = {\rm{diag}}\left[ {2,2,2} \right]} \end{array} $

In the two control schemes, the extended state disturbance observer has the same parameter setting, ρ_k1=ρ_k2=ρ_k3=10, ξ_k=3.

7.2 Simulation Result Analysis

It can be observed from Fig. 2 that although the disturbance estimation curve with the extended disturbance observer exhibits a brief oscillation at the initial stage, the time-varying complex disturbance is tracked in a short time and it coincides with the actual time-varying perturbation curve. Thus, on-line continuous disturbance compensation can be provided in the execution of the controller, the effect of the complex disturbance on the system is effectively suppressed, and the control accuracy and robustness of the two kinds of control methods are improved.

Fig. 3 is the attitude angle response curve of the QTR under two control schemes. It can be observed from the diagram that the control scheme 1 tracks the control instructions, and exhibits no overshoot, with steady state accuracy, and a fast convergence speed, control scheme 2 does not track the instructions satisfactorily, and the UAV attitude angle around the control command fluctuates and the steady-state performance is poor. This is because the logarithmic nonsingular terminal sliding mode presented in this study compared with the traditional fast non-singular terminal sliding surface, the convergence rate becomes faster and in the control scheme 1, a double power reaching law of variable coefficients with two order sliding mode characteristics is applied, which softens the motion trajectory of the controlled system.

As shown in Fig. 4, in the process of arrival of the system from the initial state to the sliding surface, the convergence rate of the sliding surface in control scheme 1 is higher than that in control scheme 2, reducing the movement time of the system in the approaching stage. When the state of system is far away from the sliding surface, the system state in control scheme 1 can move to the sliding surface rapidly. But, when the motion is close to the sliding surface, the approaching speed is significantly reduced and the smooth transition of sliding mode surface is realized; the chattering is also effectively reduced.

Fig. 5 shows a change in the curve of each channel control quantity using the control scheme 1, it can be observed from the graph that the curve of each channel control moment is smooth and chattering free. This is because the new fast reaching law used in the control scheme 1 possesses two order sliding mode characteristics, thus when the system enters the sliding mode, no tendency exists to leave the sliding surface, which effectively suppresses the chattering of the system. In summary, the control strategy proposed in this study exhibits satisfactory control effect, and achieves the design purpose.

8 Conclusions

In this study, a new type of logarithmic fast non-singular terminal sliding mode surface is designed regarding the QTR attitude control problem, which significantly improves the convergence speed of the state of system, and a new fast reaching law with two order sliding mode characteristics is proposed, smooth and chattering free to increase the system convergence speed, the extended state observer is adopted to compensate the controller on-line, which effectively suppresses the effect of complex disturbance on the system. The simulation analysis indicates that the proposed method is effective and exhibits significant practical value.