Advanced Control and Optimization for Complex Energy SystemsView this Special Issue
Research Article | Open Access
Linwu Shen, Qiang Chen, Meiling Tao, Xiongxiong He, "Adaptive Fixed-Time Sliding Mode Control for Uncertain Twin-Rotor System with Experimental Validation", Complexity, vol. 2019, Article ID 6582405, 11 pages, 2019. https://doi.org/10.1155/2019/6582405
Adaptive Fixed-Time Sliding Mode Control for Uncertain Twin-Rotor System with Experimental Validation
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.
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 , 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 , 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 , 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 , 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 , 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 , 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 . In , 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 , 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 scheme
The framework of this paper is shown as follows. Section 2 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 Figure 1, 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.
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 bywhere and are the outputs of the Gyro attitude angle and angular velocity, respectively, is the control input, is a known positive constant, are the unknown smooth nonlinear uncertainties, and represents the unknown external disturbances, satisfying the following assumptions.
Assumption 1. The unknown smooth nonlinear uncertainties are assumed to be bounded. Therefore, for all , there exists a constant such that .
Assumption 2. The lumped external disturbances are assumed to be bounded. Therefore, for all and , there exists a constant such that .
Before the controller design, a few useful lemmas are given as follows.
Lemma 1 (see ). If , the following inequality holds:and if , the following inequality holds:where N is a positive integer and .
Lemma 2 (see ). For a scalar system,where , , , and .
The convergence time T of (4) is a fixed time and expressed as follows:Furthermore, if , a higher precise upper-bound estimation of the convergence time is obtained as follows:
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 bywhere , , , , , , and and are the setting inputs of the sliding mode controller.
The time derivative of (7) is expressed as
In (8), owing to , , ; when or , 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 .where μ is a small positive bounded constant, , and .
Deriving (9) leads to
3.2. Controller Design
For the system (1), the sliding mode controller u is designed aswhere , , , , α, β, , and are the designed parameters and is the desirable angular acceleration.
The adaptive update laws are given bywhere , , , , and satisfy , , , , and .
4. Stability Analysis
Lemma 3. Considering the fixed-time sliding mode surface (7), once the sliding mode manifold is achieved, the attitude error e and angular velocity error could be guaranteed in fixed time, and the setting time satisfies the following inequality:
Proof. Construct the following Lyapunov candidate function:Once the sliding mode manifold is achieved, from (7), the time derivative of (17) iswhere , , , and .
From (18), it is concluded according to Lemma 2 that once the sliding mode manifold is achieved, the convergence of the attitude error e and angular velocity error could be guaranteed in fixed time, and the setting time satisfies the following inequality: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:where , , , and .
By using (1) and (12), the derivative of (20) is obtained asSubstituting (13) into (21) yieldsSubstituting (14) and (15) into (22) yieldswhere , , , and .
According to Lemma 1, (23) can be rewritten asDue to Young’s inequalities, the following formulations hold:where and .
Combining (25) and (26) into (24), (24) is transformed intowhere , , and satisfying .
Because and , (27) is rewritten as
Case 1. When , we have
Case 2. When , , and , the term of satisfies the following inequality:Combining Cases 1 and 2, we can obtain . Similarly, . (28) is rewritten aswhere
Case 4. When , thenwhere .
(34) is rewritten asBased on the mentioned analysis, the time derivative of V is given bywhereThe sliding variable s converges to the region given by (39) in the fixed time:Moreover, the sliding variable s in (7) is rewritten as follows:where .
According to Lemma 3, the attitude error convergence is fixed time, and the attitude error converges to the following region in the fixed time:Based on (40), the convergence region of angular velocity error is given byAccording to Lemma 2, the setting time upper boundary of the sliding variable s in the reaching phase is given byThe setting time upper bound of the attitude and angular velocity errors in the sliding phase is given byThen, the system setting time T is depicted asIn terms of former analysis, the sliding variable s, attitude error e, and angular velocity 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 , , , , , , , and .
Remark 2. From (43) and (44), large , , , , , and and less and will reduce the setting time. However, too large parameter values of , , , , , and 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 on and employed as the test rig, as shown in Figure 2, which includes two BLDC motors , two ESRs , a DC adapter, a Gyro module , a CU performing controller, a host Lenovo 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.
In the experiments, the data transmission baud rate between the twin-rotor and the host PC is , and 38 variables from the twin-rotor to the host PC with 8 bytes per variable are transferred within of the sampling period. It spends about 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 with an attitude dynamic precision and an angular velocity range 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 of 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 the 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, including adaptive fixed-time sliding mode control, adaptive finite-time sliding mode control , and adaptive linear sliding mode control . 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 , are the initial values of , .
5.2.2. Adaptive Finite-Time Sliding Mode Control
In M2, the fast terminal sliding variable is selected aswhere , , and is the same as (9).
The control law iswhere , , , and is expressed in (10).
The adaptive update laws are defined aswhere , , , , , , and satisfying . The parameters are listed in Table 2.
5.2.3. Adaptive Linear Sliding Mode Control
The sliding variable of M3 is given as
The control law is expressed as
The adaptive update laws are acquired bywhere , , , , , and , and the design parameters are listed in Table 3.
5.3. Comparative Results
For the system given by (1), the compared experiments are based on the different initial values of attitude , i.e.,(i)(ii)
The experiment results are shown in Figures 3–5. Figures 3–5 depict the time response of the sliding variable, attitude, and angular velocity for M1–M3 control schemes under the initial value , respectively. Compared with M2 and M3, the proposed M1 could achieve less reaching time and faster convergence speed as shown in Figures 3–5.
The time response of the sliding variable, attitude, and angular velocity for M1–M3 control schemes under the initial value 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.
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 Figures 9–11. 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.
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.
The data used to support the findings of the study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This study was supported by the National Natural Science Foundation of China (Project nos. 61873239, 61473262, and 61973274) and Zhejiang Provincial Natural Science Foundation (no. LY17F030018).
- A. Tayebi and S. McGilvray, “Attitude stabilization of a VTOL quadrotor aircraft,” IEEE Transactions on Control Systems Technology, vol. 14, no. 3, pp. 562–571, 2006.
- Q. Chen, L. Shi, J. Na, X. Ren, and Y. Nan, “Adaptive echo state network control for a class of pure-feedback systems with input and output constraints,” Neurocomputing, vol. 275, no. 1, pp. 1370–1382, 2018.
- M. Fanni and A. Khalifa, “A new 6-DOF quadrotor manipulation system: design, kinematics, dynamics, and control,” IEEE/ASME Transactions on Mechatronics, vol. 22, no. 3, pp. 1315–1326, 2017.
- F. Cheng, L. Qu, W. Qiao, C. Wei, and L. Hao, “Fault diagnosis of wind turbine gearboxes based on dfig stator current envelope analysis,” IEEE Transactions on Sustainable Energy, vol. 10, no. 3, pp. 1044–1053, 2019.
- C. Wei, M. Benosman, and T. Kim, “Online parameter identification for state of power prediction of lithium- ion batteries in electric vehicles using extremum seeking,” International Journal of Control, Automation and System, pp. 1–11, 2019.
- B. Xiao and S. Yin, “A new disturbance attenuation control scheme for quadrotor unmanned aerial vehicles,” IEEE Transactions on Industrial Informatics, vol. 13, no. 6, pp. 2922–2932, 2017.
- J. Na, Y. Li, Y. Huang, G. Gao, and Q. Chen, “Output feedback control of uncertain hydraulic servo systems,” IEEE Transactions on Industrial Electronics, vol. 67, no. 1, pp. 490–500, 2019.
- B. Zhao, B. Xian, Y. Zhang, and X. Zhang, “Nonlinear robust sliding mode control of a quadrotor unmanned aerial vehicle based on immersion and invariance method,” International Journal of Robust and Nonlinear Control, vol. 25, no. 18, pp. 3714–3731, 2015.
- B. Zhao, B. Xian, Y. Zhang, and X. Zhang, “Nonlinear robust adaptive tracking control of a quadrotor UAV via immersion and invariance methodology,” IEEE Transactions on Industrial Electronics, vol. 62, no. 5, pp. 2891–2902, 2015.
- R. Sun, J. Na, and B. Zhu, “Robust approximation-free prescribed performance control for nonlinear systems and its application,” International Journal of Systems Science, vol. 49, no. 3, pp. 511–522, 2018.
- M. Sun, T. Wu, L. Chen, and G. Zhang, “Neural AILC for error tracking against arbitrary initial shifts,” IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 7, pp. 2705–2716, 2018.
- J. Na, A. S. Chen, G. Herrmann, R. Burke, and C. Brace, “Vehicle engine torque estimation via unknown input observer and adaptive parameter estimation,” IEEE Transactions on Vehicular Technology, vol. 67, no. 1, pp. 409–422, 2018.
- Q. Chen, H. Shi, and M. Sun, “Echo state network-based backstepping adaptive iterative learning control for strict-feedback systems: an error-tracking approach,” IEEE Transactions on Cybernetics, pp. 1–14, 2019.
- S. Ding, K. Mei, and S. Li, “A new second-order sliding mode and its application to nonlinear constrained systems,” IEEE Transactions on Automatic Control, vol. 64, no. 6, pp. 2545–2552, 2019.
- Q. Chen, L. Tao, Y. Nan, and X. Ren, “Adaptive nonlinear sliding mode control of mechanical servo system with lugre friction compensation,” Journal of Dynamic Systems, Measurement and Control, vol. 138, no. 2, Article ID 021003–1–9, 2016.
- Q. Chen, X. Ren, and J. Na, “Robust finite-time chaos synchronization of uncertain permanent magnet synchronous motors,” ISA Transactions, vol. 58, no. 9, pp. 262–269, 2015.
- Q. Chen, X. Ren, J. Na, and D. Zheng, “Adaptive robust finite-time neural control of uncertain PMSM servo system with nonlinear dead zone,” Neural Computing and Applications, vol. 28, no. 12, pp. 3725–3736, 2017.
- J. Yu, P. Shi, and L. Zhao, “Finite-time command filtered backstepping control for a class of nonlinear systems,” Automatica, vol. 92, pp. 173–180, 2018.
- L. Fang, L. Ma, S. Ding, and D. Zhao, “Finite-time stabilization for a class of high-order stochastic nonlinear systems with an output constraint,” Applied Mathematics and Computation, vol. 358, pp. 63–79, 2019.
- H. Gui, L. Jin, and S. Xu, “Simple finite-time attitude stabilization laws for rigid spacecraft with bounded inputs,” Aerospace Science and Technology, vol. 42, pp. 176–186, 2015.
- Z. Chen, Q. Chen, X. He, and M. Sun, “Adaptive finite-time command filtered fault-tolerant control for uncertain Spacecraft with Prescribed Performance,” Complexity, vol. 2018, Article ID 4912483, 12 pages, 2018.
- B. Tian, L. Liu, H. Lu, Z. Zuo, Q. Zong, and Y. Zhang, “Multivariable finite time attitude control for quadrotor UAV: theory and experimentation,” IEEE Transactions on Industrial Electronics, vol. 65, no. 3, pp. 2567–2577, 2018.
- H. Wang, X. Ye, Y. Tian, G. Zheng, and N. Christov, “Model-free-based terminal SMC of quadrotor attitude and position,” IEEE Transactions on Aerospace and Electronic Systems, vol. 52, no. 5, pp. 2519–2528, 2016.
- B. Mu, K. Zhang, and Y. Shi, “Integral sliding mode flight controller design for a quadrotor and the application in a heterogeneous multi-agent system,” IEEE Transactions on Industrial Electronics, vol. 64, no. 12, pp. 9389–9398, 2017.
- S. Wang, H. Yu, J. Yu, J. Na, and X. Ren, “Neural-network-based adaptive funnel control for servo mechanisms with unknown dead-zone,” IEEE Transactions on Cybernetics, pp. 1–12, 2018.
- J. Na, S. Wang, Y.-J. Liu, Y. Huang, and X. Ren, “Finite-time convergence adaptive neural network control for nonlinear servo systems,” IEEE Transactions on Cybernetics, pp. 1–12, 2019.
- J. Na, Y. Huang, X. Wu, G. Gao, G. Herrmann, and J. Z. Jiang, “Active adaptive estimation and control for vehicle suspensions with prescribed performance,” IEEE Transactions on Control Systems Technology, vol. 26, no. 6, pp. 2063–2077, 2018.
- A. Polyakov, “Nonlinear feedback design for fixed-time stabilization of linear control systems,” IEEE Transactions on Automatic Control, vol. 57, no. 8, pp. 2106–2110, 2012.
- Q. Chen, M. Gao, L. Tao, and Y. Nan, “Adaptive fixed time parameter estimation and synchronization control for multiple robotic manipulators,” International Journal of Control, Automation and Systems, vol. 17, no. 9, pp. 2375–2387, 2019.
- Q. Chen, S. Xie, M. Sun, and X. He, “Adaptive nonsingular fixed-time attitude stabilization of uncertain spacecraft,” IEEE Transactions on Aerospace and Electronic Systems, vol. 54, no. 6, pp. 2937–2950, 2018.
- M. Tao, Q. Chen, X. He, and M. Sun, “Adaptive fixed-time fault-tolerant control for rigid spacecraft using a double power reaching law,” International Journal of Robust and Nonlinear Control, vol. 29, no. 12, pp. 4022–4040, 2019.
- M. Sun, “Two-phase attractors for finite-duration consensus of multiagent systems,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, pp. 1–9, 2018.
- Z. Zuo, “Non-singular fixed-time terminal sliding mode control of non-linear systems,” IET Control Theory & Applications, vol. 9, no. 4, pp. 545–552, 2015.
- L. Yang and J. Yang, “Nonsingular fast terminal sliding-mode control for nonlinear dynamical systems,” International Journal of Robust and Nonlinear Control, vol. 21, no. 16, pp. 1865–1879, 2011.
- S. Yu, X. Yu, B. Shirinzadeh, and Z. Man, “Continuous finite-time control for robotic manipulators with terminal sliding mode,” Automatica, vol. 41, no. 11, pp. 1957–1964, 2005.
Copyright © 2019 Linwu Shen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.