# Adaptive Fixed-Time Sliding Mode Control for Uncertain Twin-Rotor System with Experimental Validation

Authors: Linwu Shen; Qiang Chen; Meiling Tao; Xiongxiong He
Journal: Complexity
 (2019)

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## Abstract

This paper proposes an adaptive fixed-time control scheme for twin-rotor systems subject to the inertia uncertainties and external disturbances. First of all, a fixed-time sliding mode surface is constructed and the corresponding controller is developed such that the fixed-time uniform ultimate boundedness of the sliding variable and tracking error could be guaranteed simultaneously, and the setting time is independent of the initial values. The adaptive update laws are developed to estimate the upper bounds of the lumped uncertainties and external disturbances such that no prior knowledge on the system uncertainties and disturbances is required. Finally, a twin-rotor platform is constructed to verify the effectiveness of proposed scheme. Comparative results show better position tracking performance of the proposed control scheme.


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## Body

## 1. Introduction

As a novel application of the small unmanned aerial vehicle, multirotor is a trending topic due to its wide advantages in practical systems [1–7]. So far, many research studies have been carried out on the control of multirotor systems such as robust adaptive control [8–12], backstepping control [13], and sliding mode control [14, 15]. However, only the asymptotic stability is achieved in the aforementioned control methods.Compared with the asymptotic stability of the system, the finite-time control is proposed to realize the better control performance, and it has been extensively employed in the multirotor attitude control [16–24]. In [21], a finite-time control was developed based on the first-order command filter and the prescribed performance boundary to realize the finite-time attitude stabilization of rigid spacecraft. In [22], a continuous multivariable attitude control law was constructed in a supertwisting-like algorithm, which drove the attitude tracking errors of quadrotor to origin in finite time. In [23], a model-free terminal sliding mode controller was constructed to control both the attitude and position of a quadrotor in the presence of inertia uncertainties and external disturbances. In [24], a finite-time integral sliding mode control scheme was developed for the quadrotor attitude tracking control with uncertainties and external disturbances.Based on the aforementioned literatures, the finite-time convergence of the system states is dependent on the initial values. However, when the initial system states are unknown, it is a challenge to require the exact estimation for the upper bound of the setting time [25–27]. In [28], a fixed-time convergence of the system was initially proposed, and the setting time was bounded by a designed constant with the unknown initial states, and the fixed-time technology was applied in some practical systems [29]. In [30], a nonsingular fixed-time sliding mode control law was presented to realize the fixed-time convergence of the rigid spacecraft, and the setting time is irrelevant of the system initial states. In [31], an inverse trigonometric function is used to construct a double power reaching law in a fixed-time fault-tolerant controller, which could speed up the state stabilization and reduce the chattering phenomenon simultaneously.Motivated by the aforementioned discussions, an adaptive fixed-time control law is proposed for twin-rotor systems subject to the inertia uncertainties and external disturbances. The two main contributions are summarized as follows:(1)
A fixed-time sliding mode controller is constructed to achieve the fixed-time uniform ultimate boundedness of the sliding variable and tracking error, and the setting time is independent of the initial values(2)
A twin-rotor platform is constructed to verify the effectiveness of the proposed control scheme, and the comparative results show better position tracking performance of the proposed schemeThe framework of this paper is shown as follows. Section2 describes the mathematical model and problem formulation. Section 3 introduces an adaptive fixed-time control scheme. The fixed-time convergence of the system states is analyzed in Section 4. Experiment results are shown in Section 5.


## 2. Mathematical Model and Problem Formulation

### 2.1. Dynamic Model

To describe system dynamics of twin-rotor rigid body, a twin-rotor equalizing bar with a single degree of freedom is considered. As shown in Figure1, the twin-rotor equalizing bar consists of mechanical and electrical parts. The mechanical part includes a base plate, a bracket, an equalizing bar, and two bearings, and the bearings’ axis is the mass center of the equalizing bar rotating around the bracket. The electrical part, including a controller unit (CU), an electron speed regulator (ESR), and two brushless direct current (BLDC) motors, is used to control the attitude and angular velocity. The blades of the two BLDC motors have contrast rotational directions in order to eliminate contrast axis torque effect of the BLDC motors.Figure 1
The twin-rotor equalizing bar.Since the twin-rotor equalizing bar is a single degree of freedom (i.e., only the pitch axis is used), the system model is given by(1)x˙1=x2,x˙2=bu+fx1,x2+dt,x1,x2,y=x1,where x1 and x2 are the outputs of the Gyro attitude angle and angular velocity, respectively, u∈R is the control input, b>0 is a known positive constant, fx1,x2 are the unknown smooth nonlinear uncertainties, and dt,x1,x2 represents the unknown external disturbances, satisfying the following assumptions.Assumption 1.
The unknown smooth nonlinear uncertaintiesfx1,x2 are assumed to be bounded. Therefore, for all u∈R, there exists a constant F>0 such that fx1,x2≤F.Assumption 2.
The lumped external disturbancesdt,x1,x2 are assumed to be bounded. Therefore, for all u∈R and t≥0, there exists a constant D>0 such that dt,x1,x2≤D.


### 2.2. Preliminaries

Before the controller design, a few useful lemmas are given as follows.Lemma 1 (see [32]).
Ifα>1, the following inequality holds:(2)∑i=1Nyiα+1≥N1−α/2∑i=1Nyi21+α/2,and if β<1, the following inequality holds:(3)∑i=1Nyiβ+1≥∑i=1Nyi21+β/2,where N is a positive integer and yi>0.Lemma 2 (see [33]).
For a scalar system,(4)y˙=−k1ya1−k2ya2,y0=y0,where k1>0, k2>0, a1>1, and 0<a2<1.
The convergence timeT of (4) is a fixed time and expressed as follows:(5)T<Tmax=1k1a1−1+1k21−a2.
Furthermore, ifδ=a1−1/1−a2≤1, a higher precise upper-bound estimation of the convergence time is obtained as follows:(6)T<Tmax′=11−a21k1k2tan−1k1k2+1k1δ.


## 2.1. Dynamic Model

To describe system dynamics of twin-rotor rigid body, a twin-rotor equalizing bar with a single degree of freedom is considered. As shown in Figure1, the twin-rotor equalizing bar consists of mechanical and electrical parts. The mechanical part includes a base plate, a bracket, an equalizing bar, and two bearings, and the bearings’ axis is the mass center of the equalizing bar rotating around the bracket. The electrical part, including a controller unit (CU), an electron speed regulator (ESR), and two brushless direct current (BLDC) motors, is used to control the attitude and angular velocity. The blades of the two BLDC motors have contrast rotational directions in order to eliminate contrast axis torque effect of the BLDC motors.Figure 1
The twin-rotor equalizing bar.Since the twin-rotor equalizing bar is a single degree of freedom (i.e., only the pitch axis is used), the system model is given by(1)x˙1=x2,x˙2=bu+fx1,x2+dt,x1,x2,y=x1,where x1 and x2 are the outputs of the Gyro attitude angle and angular velocity, respectively, u∈R is the control input, b>0 is a known positive constant, fx1,x2 are the unknown smooth nonlinear uncertainties, and dt,x1,x2 represents the unknown external disturbances, satisfying the following assumptions.Assumption 1.
The unknown smooth nonlinear uncertaintiesfx1,x2 are assumed to be bounded. Therefore, for all u∈R, there exists a constant F>0 such that fx1,x2≤F.Assumption 2.
The lumped external disturbancesdt,x1,x2 are assumed to be bounded. Therefore, for all u∈R and t≥0, there exists a constant D>0 such that dt,x1,x2≤D.


## 2.2. Preliminaries

Before the controller design, a few useful lemmas are given as follows.Lemma 1 (see [32]).
Ifα>1, the following inequality holds:(2)∑i=1Nyiα+1≥N1−α/2∑i=1Nyi21+α/2,and if β<1, the following inequality holds:(3)∑i=1Nyiβ+1≥∑i=1Nyi21+β/2,where N is a positive integer and yi>0.Lemma 2 (see [33]).
For a scalar system,(4)y˙=−k1ya1−k2ya2,y0=y0,where k1>0, k2>0, a1>1, and 0<a2<1.
The convergence timeT of (4) is a fixed time and expressed as follows:(5)T<Tmax=1k1a1−1+1k21−a2.
Furthermore, ifδ=a1−1/1−a2≤1, a higher precise upper-bound estimation of the convergence time is obtained as follows:(6)T<Tmax′=11−a21k1k2tan−1k1k2+1k1δ.


## 3. Adaptive Fixed-Time Controller Design

In this section, we construct a fixed-time sliding mode surface with the convergence time independent of the system initial states and design an adaptive fixed-time controller to ensure the shorter reaching time and sliding time than the general fast terminal sliding mode control and linear sliding mode control with the same parameters.


### 3.1. Fixed-Time Sliding Mode Surface

The fixed-time sliding mode surface of the system (1) is formulated by(7)s=e˙+k1siga1e+k2siga2e,where e=x1−xd, e˙=x2−x˙d, k1>0, k2>0, a1>1, 0<a2<1, and xd and x˙d are the setting inputs of the sliding mode controller.The time derivative of (7) is expressed as(8)s˙=e¨+k1a1ea1−1e˙+k2a2ea2−1e˙.In (8), owing to a2−1<0, ea2−1⟶∞, s˙⟶∞; when e˙=0 or e¨≠0, there exists a singularity problem. To solve the singularity and carry out in practical application, the following function is found to take place of the term siga2e.(9)ϕe=ea2sgne,s=0ors≠0,e>μ,l1e+l2e2sgne,s≠0,e≤μ,where μ is a small positive bounded constant, l1=2−a2μa2−1, and l2=a2−1μa2−1.Deriving (9) leads to(10)ϕ˙e=a2ea2−1e˙,s=0ors≠0,e>μ,l1e˙+2l2ee˙,s≠0,e≤μ.Based on the former analysis, (7) and (8) are rewritten as(11)s=e˙+k1siga1e+k2ϕe,(12)s˙=e¨+k1a1ea1−1e˙+k2ϕ˙e.


### 3.2. Controller Design

For the system (1), the sliding mode controller u is designed as(13)u=−1bk1a1ea1−1e˙+k2ϕ˙e−x¨d+αsigγ1s+βsigγ2s+F^sgns+D^sgns,where α>0, β>0, γ1>1, 0<γ2<1, α, β, γ1, and γ2 are the designed parameters and x¨d is the desirable angular acceleration.The adaptive update laws are given by(14)F^˙=1θ0c1s−p1F^,(15)D^˙=1θ0c2s−p2D^,where p1, p2, c1, c2, and θ0 satisfy p1>0, p2>0, c1>0, c2>0, and θ0>0.


## 3.1. Fixed-Time Sliding Mode Surface

The fixed-time sliding mode surface of the system (1) is formulated by(7)s=e˙+k1siga1e+k2siga2e,where e=x1−xd, e˙=x2−x˙d, k1>0, k2>0, a1>1, 0<a2<1, and xd and x˙d are the setting inputs of the sliding mode controller.The time derivative of (7) is expressed as(8)s˙=e¨+k1a1ea1−1e˙+k2a2ea2−1e˙.In (8), owing to a2−1<0, ea2−1⟶∞, s˙⟶∞; when e˙=0 or e¨≠0, there exists a singularity problem. To solve the singularity and carry out in practical application, the following function is found to take place of the term siga2e.(9)ϕe=ea2sgne,s=0ors≠0,e>μ,l1e+l2e2sgne,s≠0,e≤μ,where μ is a small positive bounded constant, l1=2−a2μa2−1, and l2=a2−1μa2−1.Deriving (9) leads to(10)ϕ˙e=a2ea2−1e˙,s=0ors≠0,e>μ,l1e˙+2l2ee˙,s≠0,e≤μ.Based on the former analysis, (7) and (8) are rewritten as(11)s=e˙+k1siga1e+k2ϕe,(12)s˙=e¨+k1a1ea1−1e˙+k2ϕ˙e.


## 3.2. Controller Design

For the system (1), the sliding mode controller u is designed as(13)u=−1bk1a1ea1−1e˙+k2ϕ˙e−x¨d+αsigγ1s+βsigγ2s+F^sgns+D^sgns,where α>0, β>0, γ1>1, 0<γ2<1, α, β, γ1, and γ2 are the designed parameters and x¨d is the desirable angular acceleration.The adaptive update laws are given by(14)F^˙=1θ0c1s−p1F^,(15)D^˙=1θ0c2s−p2D^,where p1, p2, c1, c2, and θ0 satisfy p1>0, p2>0, c1>0, c2>0, and θ0>0.


## 4. Stability Analysis

Lemma 3.
Considering the fixed-time sliding mode surface (7), once the sliding mode manifold s=0 is achieved, the attitude error e and angular velocity error e˙ could be guaranteed in fixed time, and the setting time Ts satisfies the following inequality:(16)Ts<12a1+1/22k1a1−1+12a2+1/22k21−a2.Proof.
Construct the following Lyapunov candidate function:(17)V0=12e2.
Once the sliding mode manifolds=0 is achieved, from (7), the time derivative of (17) is(18)V˙0=ee˙=e−k1siga1e−k2siga2e=−2a1+1/2k112e2a1+1/2−2a2+1/2k212e2a2+1/2=−2a1+1/2k1V0a1+1/2−2a2+1/2k2V0a2+1/2,where k1, k2>0, a1>1, and 0<a2<1.
From (18), it is concluded according to Lemma 2 that once the sliding mode manifold s=0 is achieved, the convergence of the attitude error e and angular velocity error e˙ could be guaranteed in fixed time, and the setting time Ts satisfies the following inequality:(19)Ts<12a1+1/22k1a1−1+12a2+1/22k21−a2.
This completes the proof.Theorem 1.
Considering the system (1) with the sliding mode surface (11), the control law (13), and the adaptive update laws (14) and (15), all signals of the closed-loop system are uniformly ultimately bounded within a fixed time.Proof.
Construct the following Lyapunov candidate function:(20)V=12s2+12cr1F˜2+12cr2D˜2,where cr1>0, cr2>0, F˜=F−F^, and F˜=F−F^.
By using (1) and (12), the derivative of (20) is obtained as(21)V˙=ss˙+cr1F˜F˜˙+cr2D˜D˜˙=sf+d+bu−x¨d+k1a1ea1−1e˙+k2ϕ˙e+cr1F˜F˜˙+cr2D˜D˜˙≤sF⋅sgns+D⋅sgns+bu−x¨d+k1a1ea1−1e˙+k2ϕ˙e+cr1F˜F˜˙+cr2D˜D˜˙.
Substituting (13) into (21) yields(22)V˙≤s−αsigγ1s−βsigγ2s−F^⋅sgns−D^⋅sgns+F⋅sgns+D⋅sgns+cr1F˜F˜˙+cr2D˜D˜˙=−αsγ1+1−βsγ2+1+F˜s+D˜s+cr1F˜F˜˙+cr2D˜D˜˙=−2γ1+1/2α12s2γ1+1/2−2γ2+1/2β12s2γ2+1/2−12cr1F˜2γ1+1/2−12cr1F˜2γ2+1/2−12cr2D˜2γ1+1/2−12cr2D˜2γ2+1/2+12cr1F˜2γ1+1/2+12cr1F˜2γ2+1/2+12cr2D˜2γ1+1/2+12cr2D˜2γ2+1/2+F˜s+D˜s−cr1F˜F^˙−cr2D˜D^˙.
Substituting (14) and (15) into (22) yields(23)V˙≤−η112s2γ1+1/2+12cr1F˜2γ1+1/2+12cr2D˜2γ1+1/2−η212s2γ2+1/2+12cr1F˜2γ2+1/2+12cr2D˜2γ2+1/2+12cr1F˜2γ1+1/2+12cr1F˜2γ2+1/2+12cr2D˜2γ1+1/2+12cr2D˜2γ2+1/2+p1F˜F^+p2D˜D^,where η1=min2γ1+1/2α,1, η2=min2γ2+1/2β,1, cr1=θ0c1, and cr2=θ0c2.
According to Lemma 1, (23) can be rewritten as(24)V˙≤−31−γ1/2η1Vγ1+1/2+η2Vγ2+1/2+12cr1F˜2γ1+1/2+12cr1F˜2γ2+1/2+12cr2D˜2γ1+1/2+12cr2D˜2γ2+1/2+p1F˜F^+p2D˜D^.
Due to Young’s inequalities, the following formulations hold:(25)p1F˜F^=p1F˜F−F˜=p1F˜F−F˜2≤p1F˜22θ1+θ1F22−F˜2=p11−2θ1F˜22θ1+p1θ1F22,(26)p2D˜D^=p2D˜D−D˜=p2D˜D−D˜2≤p2D˜22θ2+θ2D22−D˜2=p21−2θ2D˜22θ2+p2θ2D22,where θ1>1/2 and θ2>1/2.
Combining (25) and (26) into (24), (24) is transformed into(27)V˙≤−31−γ1/2η1Vγ1+1/2+η2Vγ2+1/2+p1θ1F22+p2θ2D22+12cr1F˜2γ1+1/2+12cr1F˜2γ2+1/2+12cr2D˜2γ2+1/2+12cr2D˜2γ2+1/2−c1F˜2−c2D˜2≤−31−γ1/2η1Vγ1+1/2+η2Vγ2+1/2+p1θ1F22+p2θ2D22+12cr1F˜2γ1+1/2+12cr1F˜2γ2+1/2+12cr2D˜2γ1+1/2+12cr2D˜2γ2+1/2−12θ0γ2+1/2c1F˜2−12θ0γ2+1/2c2D˜2−1−12θ0γ2+1/2c1F˜2−1−12θ0γ2+1/2c2D˜2,where c1=2θ1−1p1/2θ1, c2=2θ2−1p2/2θ2, and θ0 satisfying 1/2θ0γ2+1/2+1/2θ0−1=0.
Becausecr1=c1θ0 and cr2=c2θ0, (27) is rewritten as(28)V˙≤−31−γ1/2η1Vγ1+1/2+η2Vγ2+1/2+p1θ1F22+p2θ2D22+12c1θ0F˜2γ1+1/2+12c1θ0F˜2γ2+1/2+12c2θ0D˜2γ1+1/2+12c2θ0D˜2γ2+1/2−12θ0γ2+1/2c1F˜2−12θ0γ2+1/2c2D˜2−12θ0c1F˜2−12θ0c2D˜2.Case 1.
Whenc1F˜2≥1, we have c1F˜2γ2+1/2−c1F˜2≤0.Case 2.
When0<c1F˜2<1, c1F˜2=2/γ2+12/γ2−1, and υ0=2/γ2+1γ2+1/γ2−1−2/γ2+12/γ2−1, the term of c1F˜2γ2+1/2−c1F˜2 satisfies the following inequality:(29)0<c1F˜2γ2+1/2−c1F˜2<υ0.
Combining Cases 1 and 2, we can obtainc1F˜2γ2+1/2−c1F˜2<υ0. Similarly, c2D˜2γ2+1/2−c2D˜2<υ0. (28) is rewritten as(30)V˙≤−31−γ1/2η1Vγ1+1/2+η2Vγ2+1/2+12θ0c1F˜2γ1+1/2+12θ0c2D˜2γ1+1/2−12θ0c1F˜2−12θ0c2D˜2+υ1,where(31)υ1=212θ0γ2+1/2υ0+p1θ1F22+p2θ2D22.Case 3.
Whenξ≤min2/θ0c1,2/θ0c2, ξ is the value interval of F˜ and D˜; then, 1/2θ0c1F˜2γ1+1/2≤1/2θ0c1F˜2 and1/2θ0c2D˜2γ1+1/2≤1/2θ0c2D˜2, i.e.,(32)12θ0c1F˜2γ1+1/2−12θ0c1F˜2≤0,(33)12θ0c2D˜2γ1+1/2−12θ0c2D˜2≤0.
From (32) and (33), (30) is expressed as(34)V˙≤−31−γ1/2η1Vγ1+1/2+η2Vγ2+1/2+υ1.Case 4.
Whenξ>min2/θ0c1,2/θ0c2, then(35)12θ0c1F˜2γ1+1/2+12θ0c2D˜2γ1+1/2−12θ0c1F˜2−12θ0c2D˜2≤212θ0cmaxξ2γ1+1/2−212θ0cmaxξ2,where cmax=c1,c2.
(34) is rewritten as(36)V˙≤−31−γ1/2η1Vγ1+1/2+η2Vγ2+1/2+υ1+212θ0cmaxξ2γ1+1/2−212θ0cmaxξ2.
Based on the mentioned analysis, the time derivative of V is given by(37)V˙≤−31−γ1/2η1Vγ1+1/2+η2Vγ2+1/2+υ,where(38)υ=υ1,ξ≤min2θ0c1,2θ0c2,υ1+212θ0cmaxξ2γ1+1/2−212θ0cmaxξ2,ξ>min1θ0c1,1θ0c2.
The sliding variables converges to the region given by (39) in the fixed time:(39)Δs=min21/2υ31−γ1/2η11/γ1+1,21/2υη21/γ2+1.
Moreover, the sliding variables in (7) is rewritten as follows:(40)e˙+k1siga1e+k2siga2e=ε,where ε≤Δs.
According to Lemma 3, the attitude error convergence is fixed time, and the attitude error converges to the following region in the fixed time:(41)e≤minεk11/a1,εk21/a2.
Based on (40), the convergence region of angular velocity error is given by(42)e˙≤ε+k1ea1+k2ea2≤3Δs.
According to Lemma 2, the setting time upper boundary of the sliding variables in the reaching phase is given by(43)Tr<231−γ1/2η1γ1−1+2η21−γ2.
The setting time upper bound of the attitude and angular velocity errors in the sliding phase is given by(44)Ts<12a1+1/22k1a1−1+12a2+1/22k21−a2.
Then, the system setting timeT is depicted as(45)T<Tr+Ts=231−γ1/2η1γ1−1+2η21−γ2+12a1+1/22k1a1−1+12a2+1/22k21−a2.
In terms of former analysis, the sliding variables, attitude error e, and angular velocity e˙ are uniformly ultimately bounded in a fixed time. This completes the proof.Remark 1.
From (43) and (44), it is seen that the setting time is upper bounded and independent of initial conditions, such as the initial attitude, but depends on the designed parameters, such as k1, k2, a1, a2, γ1, γ2, η1, and η2.Remark 2.
From (43) and (44), large k1, k2, a1, γ1, η1, and η2 and less a2 and γ2 will reduce the setting time. However, too large parameter values of k1, k2, a1, γ1, η1, and η2 may lead to a high controller gain. Consequently, the parameter values should be chosen appropriately with a trade-off between the settling time and controller gain.


## 5. Experiment Results

### 5.1. Description of the Twin-Rotor Equalizing Bar System

To verify the applicability of the proposed control scheme, a twin-rotor equalizing bar platform is built based onSTM32F103C8T6 and employed as the test rig, as shown in Figure 2, which includes two BLDC motors X2212 , two ESRs SKYWALKER40A, a DC adapter, a Gyro module WT931, a CU STM32F103C8T6 performing controller, a host Lenovo W520 workstation operating for display and data analysis, and two universal serial bus (USB) communication circuits receiving and sending the data between the host and the CU. The proposed control scheme is implemented via a C program in the Keil IDE uVision V5.21 Evaluation in the CU.Figure 2
Diagram of the twin-rotor equalizing bar platform.In the experiments, the data transmission baud rate between the twin-rotor and the host PC is921600bps, and 38 variables from the twin-rotor to the host PC with 8 bytes per variable are transferred within 6ms of the sampling period. It spends about 3.6ms to complete a cycle data transmission. Furthermore, the direct memory access (DMA) in CU is adopted to realize data sending and receiving without affecting other software operation. Consequently, the data transmission is reliable and timely. The Gyro module W931T with an attitude dynamic precision 0.1∘ and an angular velocity range ±1000°/s meets the requirements of the practical application.This test rig is employed to implement the transient performance control for given attitude references. Only the pitch axis is used in the experiments, and a sampling rate of6ms is selected, which is significantly faster than the later considered closed-loop demand frequencies.The procedure of conducting an experiment is depicted in the following steps:(1)
Adjust a location screw to set the proper initial attitude(2)
Turn on the CU power and the ESR DC adapter power in sequence(3)
Compile the control schemes and download the compiling code to the CU via theSTlinkV2 debugger(4)
Send a standby message 0x00 0x01 to the CU via the USB1 interface(5)
Send a run operation code 0x00 0x02 to perform the control schemes(6)
The test data will be on display in the host and saved as a text file(7)
After finishing the experiment, transmit a terminal message 0x00 0x00, close all of powers, and restore to the initial states


### 5.2. Design of Comparative Controllers

Comparative the control scheme experiments are conducted in the CU. To validate the effectiveness of the control scheme transient performance and the fixed-time property, three different control schemes are arranged, includingM1 adaptive fixed-time sliding mode control, M2 adaptive finite-time sliding mode control [34], and M3 adaptive linear sliding mode control [35]. Adaptive items are adopted to approximate model uncertainties and external disturbances, and a switching scheme is taken to solve the singularity problem of sliding manifold and control laws. For fair comparison, the corresponding parameters in three different control schemes are defined as the same.


#### 5.2.1. Adaptive Fixed-Time Sliding Mode Control

In M1, the sliding variable is designed as (11), the control law is addressed by (13), and the adaptive update laws are depicted in (14) and (15), respectively, and the parameters are listed in Table 1, where F^0, D^0 are the initial values of F^, D^.Table 1
M1 parameters.
Parameter
Value
k
1
3.0
k
2
3.0
a
1
23.0
/
17.0
a
2
17.0
/
23.0
μ
0.1
α
0.02307
β
0.1725
γ
1
21.0
/
17.0
γ
2
11.0
/
17.0
θ
0
0.692
θ
1
0.6
θ
2
0.6
p
1
0.45
p
2
0.45
F
^
0
10.0
D
^
0
10.0


#### 5.2.2. Adaptive Finite-Time Sliding Mode Control

In M2, the fast terminal sliding variable is selected as(46)s=e˙+k1e+k2ϕe,where k1>0, k2>0, and ϕe is the same as (9).The control law is(47)u=−1bk1e˙+k2ϕ˙e−x¨d+αs+βsigγ2s+F^sgns+D^sgns,where α>0, β>0, 0<γ2<1, and ϕ˙e is expressed in (10).The adaptive update laws are defined as(48)F^˙=1θ0c1s−p1F^,(49)D^˙=1θ0c2s−p2D^,where p1>0, p2>0, θ1>1/2, θ2>1/2, c1=2θ1−1p1/2θ1, c2=2θ2−1p2/2θ2, and θ0 satisfying 1/2θ0γ2+1/2+1/2θ0−1=0. The parameters are listed in Table 2.Table 2
M2 parameters.
Parameter
Value
k
1
3.0
k
2
3.0
a
2
17.0
/
23.0
μ
0.1
α
0.02307
β
0.1725
γ
2
11.0
/
17.0
θ
0
0.692
θ
1
0.6
θ
2
0.6
p
1
0.45
p
2
0.45
F
^
0
10.0
D
^
0
10.0


#### 5.2.3. Adaptive Linear Sliding Mode Control

The sliding variable of M3 is given as(50)s=e˙+k1e.The control law is expressed as(51)u=−1bk1e˙−x¨d+αs+F^sgns+D^sgns.The adaptive update laws are acquired by(52)F^˙=1c1s−p1F^,(53)D^˙=1c2s−p2D^,where p1>0, p2>0, θ1>1/2, θ2>1/2, c1=2θ1−1p1/2θ1, and c2=2θ2−1p2/2θ2, and the design parameters are listed in Table 3.Table 3
M3 parameters.
Parameter
Value
k
1
3.0
α
0.02307
θ
1
0.6
θ
2
0.6
p
1
0.45
p
2
0.45
F
^
0
10.0
D
^
0
10.0


### 5.3. Comparative Results

For the system given by (1), the compared experiments are based on the different initial values of attitude x10, i.e.,(i)
x10=−30∘(ii)
x10=−50∘The experiment results are shown in Figures3–5. Figures 3–5 depict the time response of the sliding variable, attitude, and angular velocity for M1–M3 control schemes under the initial value i, respectively. Compared with M2 and M3, the proposed M1 could achieve less reaching time and faster convergence speed as shown in Figures 3–5.Figure 3
Time response of the sliding variables under the initial value i.Figure 4
Time response of the attitudex1 under the initial value i.Figure 5
Time response of the angular velocityx2 under the initial value i.The time response of the sliding variable, attitude, and angular velocity for M1–M3 control schemes under the initial valueii is shown in Figures 6–8, respectively. From Figures 6–8, the convergence time of M1 is irrelevant to the initial states, but the convergence time of M2 and M3 is related to the initial states of the twin-rotor system.Figure 6
Time response of the sliding variables under the initial value ii.Figure 7
Time response of the attitudex1 under the initial value ii.Figure 8
Time response of the angular velocityx2 under the initial value ii.To further illustrate the transient performance, based on six different initial attitude values, the time response of system states in the experiments is shown in Figures9–11. The attitude convergence time of M1 in Figure 9 is almost unchanged, the attitude convergence time of M2 in Figure 10 is about 6.5 s to 9.4 s, and the attitude of M3 is asymptotic convergence in Figure 11.Figure 9
Time response of attitude in M1 under different initial values.Figure 10
Time response of attitude in M2 under different initial values.Figure 11
Time response of attitude in M3 under different initial values.From Figures3–11, it is shown that less reaching time and faster convergence speed could be guaranteed with the proposed M1.


## 5.1. Description of the Twin-Rotor Equalizing Bar System

To verify the applicability of the proposed control scheme, a twin-rotor equalizing bar platform is built based onSTM32F103C8T6 and employed as the test rig, as shown in Figure 2, which includes two BLDC motors X2212 , two ESRs SKYWALKER40A, a DC adapter, a Gyro module WT931, a CU STM32F103C8T6 performing controller, a host Lenovo W520 workstation operating for display and data analysis, and two universal serial bus (USB) communication circuits receiving and sending the data between the host and the CU. The proposed control scheme is implemented via a C program in the Keil IDE uVision V5.21 Evaluation in the CU.Figure 2
Diagram of the twin-rotor equalizing bar platform.In the experiments, the data transmission baud rate between the twin-rotor and the host PC is921600bps, and 38 variables from the twin-rotor to the host PC with 8 bytes per variable are transferred within 6ms of the sampling period. It spends about 3.6ms to complete a cycle data transmission. Furthermore, the direct memory access (DMA) in CU is adopted to realize data sending and receiving without affecting other software operation. Consequently, the data transmission is reliable and timely. The Gyro module W931T with an attitude dynamic precision 0.1∘ and an angular velocity range ±1000°/s meets the requirements of the practical application.This test rig is employed to implement the transient performance control for given attitude references. Only the pitch axis is used in the experiments, and a sampling rate of6ms is selected, which is significantly faster than the later considered closed-loop demand frequencies.The procedure of conducting an experiment is depicted in the following steps:(1)
Adjust a location screw to set the proper initial attitude(2)
Turn on the CU power and the ESR DC adapter power in sequence(3)
Compile the control schemes and download the compiling code to the CU via theSTlinkV2 debugger(4)
Send a standby message 0x00 0x01 to the CU via the USB1 interface(5)
Send a run operation code 0x00 0x02 to perform the control schemes(6)
The test data will be on display in the host and saved as a text file(7)
After finishing the experiment, transmit a terminal message 0x00 0x00, close all of powers, and restore to the initial states


## 5.2. Design of Comparative Controllers

Comparative the control scheme experiments are conducted in the CU. To validate the effectiveness of the control scheme transient performance and the fixed-time property, three different control schemes are arranged, includingM1 adaptive fixed-time sliding mode control, M2 adaptive finite-time sliding mode control [34], and M3 adaptive linear sliding mode control [35]. Adaptive items are adopted to approximate model uncertainties and external disturbances, and a switching scheme is taken to solve the singularity problem of sliding manifold and control laws. For fair comparison, the corresponding parameters in three different control schemes are defined as the same.


### 5.2.1. Adaptive Fixed-Time Sliding Mode Control

In M1, the sliding variable is designed as (11), the control law is addressed by (13), and the adaptive update laws are depicted in (14) and (15), respectively, and the parameters are listed in Table 1, where F^0, D^0 are the initial values of F^, D^.Table 1
M1 parameters.
Parameter
Value
k
1
3.0
k
2
3.0
a
1
23.0
/
17.0
a
2
17.0
/
23.0
μ
0.1
α
0.02307
β
0.1725
γ
1
21.0
/
17.0
γ
2
11.0
/
17.0
θ
0
0.692
θ
1
0.6
θ
2
0.6
p
1
0.45
p
2
0.45
F
^
0
10.0
D
^
0
10.0


### 5.2.2. Adaptive Finite-Time Sliding Mode Control

In M2, the fast terminal sliding variable is selected as(46)s=e˙+k1e+k2ϕe,where k1>0, k2>0, and ϕe is the same as (9).The control law is(47)u=−1bk1e˙+k2ϕ˙e−x¨d+αs+βsigγ2s+F^sgns+D^sgns,where α>0, β>0, 0<γ2<1, and ϕ˙e is expressed in (10).The adaptive update laws are defined as(48)F^˙=1θ0c1s−p1F^,(49)D^˙=1θ0c2s−p2D^,where p1>0, p2>0, θ1>1/2, θ2>1/2, c1=2θ1−1p1/2θ1, c2=2θ2−1p2/2θ2, and θ0 satisfying 1/2θ0γ2+1/2+1/2θ0−1=0. The parameters are listed in Table 2.Table 2
M2 parameters.
Parameter
Value
k
1
3.0
k
2
3.0
a
2
17.0
/
23.0
μ
0.1
α
0.02307
β
0.1725
γ
2
11.0
/
17.0
θ
0
0.692
θ
1
0.6
θ
2
0.6
p
1
0.45
p
2
0.45
F
^
0
10.0
D
^
0
10.0


### 5.2.3. Adaptive Linear Sliding Mode Control

The sliding variable of M3 is given as(50)s=e˙+k1e.The control law is expressed as(51)u=−1bk1e˙−x¨d+αs+F^sgns+D^sgns.The adaptive update laws are acquired by(52)F^˙=1c1s−p1F^,(53)D^˙=1c2s−p2D^,where p1>0, p2>0, θ1>1/2, θ2>1/2, c1=2θ1−1p1/2θ1, and c2=2θ2−1p2/2θ2, and the design parameters are listed in Table 3.Table 3
M3 parameters.
Parameter
Value
k
1
3.0
α
0.02307
θ
1
0.6
θ
2
0.6
p
1
0.45
p
2
0.45
F
^
0
10.0
D
^
0
10.0


## 5.2.1. Adaptive Fixed-Time Sliding Mode Control

In M1, the sliding variable is designed as (11), the control law is addressed by (13), and the adaptive update laws are depicted in (14) and (15), respectively, and the parameters are listed in Table 1, where F^0, D^0 are the initial values of F^, D^.Table 1
M1 parameters.
Parameter
Value
k
1
3.0
k
2
3.0
a
1
23.0
/
17.0
a
2
17.0
/
23.0
μ
0.1
α
0.02307
β
0.1725
γ
1
21.0
/
17.0
γ
2
11.0
/
17.0
θ
0
0.692
θ
1
0.6
θ
2
0.6
p
1
0.45
p
2
0.45
F
^
0
10.0
D
^
0
10.0


## 5.2.2. Adaptive Finite-Time Sliding Mode Control

In M2, the fast terminal sliding variable is selected as(46)s=e˙+k1e+k2ϕe,where k1>0, k2>0, and ϕe is the same as (9).The control law is(47)u=−1bk1e˙+k2ϕ˙e−x¨d+αs+βsigγ2s+F^sgns+D^sgns,where α>0, β>0, 0<γ2<1, and ϕ˙e is expressed in (10).The adaptive update laws are defined as(48)F^˙=1θ0c1s−p1F^,(49)D^˙=1θ0c2s−p2D^,where p1>0, p2>0, θ1>1/2, θ2>1/2, c1=2θ1−1p1/2θ1, c2=2θ2−1p2/2θ2, and θ0 satisfying 1/2θ0γ2+1/2+1/2θ0−1=0. The parameters are listed in Table 2.Table 2
M2 parameters.
Parameter
Value
k
1
3.0
k
2
3.0
a
2
17.0
/
23.0
μ
0.1
α
0.02307
β
0.1725
γ
2
11.0
/
17.0
θ
0
0.692
θ
1
0.6
θ
2
0.6
p
1
0.45
p
2
0.45
F
^
0
10.0
D
^
0
10.0


## 5.2.3. Adaptive Linear Sliding Mode Control

The sliding variable of M3 is given as(50)s=e˙+k1e.The control law is expressed as(51)u=−1bk1e˙−x¨d+αs+F^sgns+D^sgns.The adaptive update laws are acquired by(52)F^˙=1c1s−p1F^,(53)D^˙=1c2s−p2D^,where p1>0, p2>0, θ1>1/2, θ2>1/2, c1=2θ1−1p1/2θ1, and c2=2θ2−1p2/2θ2, and the design parameters are listed in Table 3.Table 3
M3 parameters.
Parameter
Value
k
1
3.0
α
0.02307
θ
1
0.6
θ
2
0.6
p
1
0.45
p
2
0.45
F
^
0
10.0
D
^
0
10.0


## 5.3. Comparative Results

For the system given by (1), the compared experiments are based on the different initial values of attitude x10, i.e.,(i)
x10=−30∘(ii)
x10=−50∘The experiment results are shown in Figures3–5. Figures 3–5 depict the time response of the sliding variable, attitude, and angular velocity for M1–M3 control schemes under the initial value i, respectively. Compared with M2 and M3, the proposed M1 could achieve less reaching time and faster convergence speed as shown in Figures 3–5.Figure 3
Time response of the sliding variables under the initial value i.Figure 4
Time response of the attitudex1 under the initial value i.Figure 5
Time response of the angular velocityx2 under the initial value i.The time response of the sliding variable, attitude, and angular velocity for M1–M3 control schemes under the initial valueii is shown in Figures 6–8, respectively. From Figures 6–8, the convergence time of M1 is irrelevant to the initial states, but the convergence time of M2 and M3 is related to the initial states of the twin-rotor system.Figure 6
Time response of the sliding variables under the initial value ii.Figure 7
Time response of the attitudex1 under the initial value ii.Figure 8
Time response of the angular velocityx2 under the initial value ii.To further illustrate the transient performance, based on six different initial attitude values, the time response of system states in the experiments is shown in Figures9–11. The attitude convergence time of M1 in Figure 9 is almost unchanged, the attitude convergence time of M2 in Figure 10 is about 6.5 s to 9.4 s, and the attitude of M3 is asymptotic convergence in Figure 11.Figure 9
Time response of attitude in M1 under different initial values.Figure 10
Time response of attitude in M2 under different initial values.Figure 11
Time response of attitude in M3 under different initial values.From Figures3–11, it is shown that less reaching time and faster convergence speed could be guaranteed with the proposed M1.


## 6. Conclusion

This paper proposed an adaptive fixed-time control scheme for the uncertain twin-rotor systems. A fixed-time sliding mode controller is designed, and the sliding variable and tracking error are both guaranteed to be uniformly ultimately bounded within the fixed time, which is independent of the initial values. With the proposed control scheme, the prior knowledge on the system uncertainties and disturbances is not needed, and the upper bounds of the lumped uncertainties could be estimated by developing the adaptive update laws. The effectiveness of the proposed control scheme is verified on a twin-rotor platform, and comparative experimental results illustrate the superior performance of the presented scheme.



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*Source: 6582405-2019-10-29.xml*