Discrete-time hypersonic flight control based on extreme learning machine
Introduction
Hypersonic flight vehicles may offer a reliable and more cost efficient way to access space by reducing flight time. Also quick response and global attack became possible. Hypersonic flight control is challenging since the longitudinal model of the dynamics is known to be unstable, non-minimum phase with respect to the regulated output, and affected by significant model uncertainty. The main difficulty of the control law design for the hypersonic aircraft is due to the high complexity of the motion equations and there is little knowledge of the aerodynamic parameters of the vehicle.
Recently adaptive control and robust control are popularly studied on hypersonic flight controller design [1]. Back-stepping design [2] is an explicit tool for systematic nonlinear design. The HFV dynamics are written in the linearly parameterized form [3] and then robust adaptive dynamic inversion with back-stepping arguments is conducted. Dynamic surface control with control inputs saturation design is studied in [4].
Intelligent control is one important aspect for hypersonic flight control since it is with the capability of uncertainty approximation [5], [6], [7], [22]. Since modern aircrafts are equipped with digital computers, the controller should be designed in discrete-time form [8]. Controller on the basis of continuous system is usually implemented by a digital computer with a certain sampling interval [9]. There are two methods for designing the digital controller. One method, called emulation, designs a controller with the continuous-time system, and then discretizes the controller. The other is to design the controllers directly based on the discrete system. In contrast to the emulation method, the discrete controller is designed in a discrete domain so that the performance of the controller may not depend on the sampling rate and the upper bounds of the neural network (NN) weight update rates guaranteeing the convergence can be estimated analytically while emulation method is otherwise [10].
Focused on discrete time design, the adaptive NN back-stepping HFV control [11] is studied to deal with the system uncertainty. The Kriging based adaptive controller is designed in [12] where the uncertainty is described as the realization of the Gaussian random functions [13]. The simulation shows the effectiveness of the controller design [11], [14]. In the above schemes, the structure of NN is determined according to some prior information regarding the system to be approximated. Then the stable adaptive laws can be generated in a linear fashion. However in practice, systems are time-varying and the prior information is difficult to obtain. In this case, the exact values for the NN are hard to determine.
In this paper, a new stable neural control scheme is presented. The SLFN with RBF nodes is used as the function approximator to estimate the unknown nonlinearity. Different from the existing methods, the parameters of SLFN are adjusted based on the ELM. ELM has attracted widespread concern in recent years [15], [16], [17] since it overcomes some challenges faced by other techniques [18] such as (1) slow learning speed, (2) trivial human intervene, and/or (3) poor computational scalability. ELM works for generalized SLFN. The essence of ELM is that the hidden layer of SLFN needs not to be tuned. Compared with those traditional computational intelligence techniques, ELM provides better generalization performance at a much faster learning speed and with least human intervene. In [19], an approach for performing regression on large data sets in reasonable time is proposed using an ensemble of ELMs and the experiments show that competitive performance is obtained on the regression tasks. In [20], the ELM is utilized to train the controller by randomly assigning the parameters of hidden nodes. The output weights are synthesized using a Lyapunov function for guaranteeing the stability of the closed-loop system. Also it is indicated that original ELM cannot follow its reference trajectory well since the original ELM lacks stability proof of the whole control system and thus the convergence of the tracking error cannot be satisfied.
In this paper, considering the use of digital computer, the back-stepping controller is designed with ELM by randomly assigning the parameters of hidden nodes. The updating law is designed with Lyapunov synthesis approach in discrete-time. Following the functional decomposition [11], we design the controller separately for the subsystems. Furthermore, the “minimal learning parameter” technique based on bound estimation of weight vector [14], [21] is incorporated to reduce the computation burden. In this paper, only the cruise trajectories are considered for the control problem in this paper and we does not consider the ascent or the reentry of the vehicle.
This paper is organized as follows. Section 2 describes the longitudinal dynamics of a generic hypersonic flight vehicle. The strict-feedback form is formulated and the discrete analysis model is obtained in Section 3. SLFN based on ELM is illustrated in Section 4. Section 5 presents the adaptive controller design based on ELM. The weight bound estimation based controller is designed in Section 6. The simulation result is included in Section 7. Section 8 presents several comments and final remarks.
Section snippets
Hypersonic air vehicle model
The model of the longitudinal dynamics of a generic hypersonic aircraft in [1] is considered. This model is composed of five state variables and two control inputs where V is the velocity, γ is the flight path angle, h is the altitude, α is the attack angle, q is the pitch rate, δe is the elevator deflection and is the throttle setting.
The dynamics of hypersonic aircraft are described by the following nonlinear equations:
Strict-feedback formulation
Referred to [11], [14], the formulation of the subsystems is presented in (6), (8). The related definition of the system is listed in Appendix B.
The velocity subsystem (1) can be rewritten as follows:
The tracking error of the altitude is defined as and the flight path command is chosen asif and are chosen and the flight path angle is controlled to follow γd, the altitude tracking error is regulated to zero exponentially [5]
SLFN based on ELM
For N arbitrary distinct samples , where and , standard SLFN with hidden neurons can be expressed as follows:where and are the learning parameters of hidden nodes, is the weight vector connecting the hidden neuron and the output neurons and is the output of the ith hidden node with respect to input .
Let , and
Adaptive control with ELM
When the SLFN is utilized to approximate the unknown nonlinear functions in the designed control scheme, according to the property of ELM, the hidden node parameters need not be tuned during training and may simply be assigned with random values.
The error definition is presented:where is derived from (7), , are the virtual control inputs to be designed.
For simplicity, we define , , ,
Weight vector bound estimation based controller design
As indicated in [14], the neural design could be further simplified by estimating the upper bound of . With this idea, we have the following definition: Assumption 3 For each on the compact set Ωi, satisfieswhere ϕi if and −1 otherwise. Remark 2 In [21], the algorithm is with restriction since the estimation can be positive only. But actually the NN approximation could be negative. Using the function and redefining the assumption, it is more reasonable that the ϕi
Simulations
The flight of the vehicle is at the condition M=15, V=15,060 ft/s, h=110,000 ft, , , q=0. In this simulation, step commands of 1000 ft and 100 ft/s are selected for altitude and velocity separately. Reference commands are generated by the filter:where , , , . The perturbation is set to be 3% for the parameter set .
To demonstrate the tracking performance, the control methods in 5 Adaptive control with
Conclusions
In this paper, discrete controller via back-stepping design is applied on hypersonic flight. In the design, the SLFN is used for uncertainty approximation. The hidden node parameters of SLFN are determined based on the ELM algorithm by assigning randomly the values. The robust updating law is provided to guarantee the closed-loop stability. Furthermore, the bound estimation based controller is presented to reduce the number of online adaptive parameters. Simulation results are presented to show
Acknowledgments
This work was supported by DSO National Laboratories of Singapore through a Strategic Project Grant (Project no. DSOCL10004), National Science Foundation of China (Grant nos. 61304098, 61134004, 61074185, 61005085), Fundamental Research Funds for the Central Universities (2012QNA4024) and NWPU Basic Research Funding (Grant no. JC20120236). The authors would like to thank the anonymous reviewers for constructive comments that helped to improve the quality and presentation of this paper.
Bin Xu received the B.S. degree in measuring and control instrument from Northwestern Polytechnical University in 2006 and the Ph.D. degree in computer science from Tsinghua University, Beijing, China in 2012. During 2012–2013, he was a research fellow with Nanyang Technological University. He is currently a lecturer in School of Automation, Northwestern Polytechnical University. His research interests include intelligent control and adaptive control with application on flight dynamics,
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Bin Xu received the B.S. degree in measuring and control instrument from Northwestern Polytechnical University in 2006 and the Ph.D. degree in computer science from Tsinghua University, Beijing, China in 2012. During 2012–2013, he was a research fellow with Nanyang Technological University. He is currently a lecturer in School of Automation, Northwestern Polytechnical University. His research interests include intelligent control and adaptive control with application on flight dynamics, multi-robot formation and transportation system.
Yongping Pan The Ph.D. degree in control theory and control engineering from the South China University of Technology (SCUT), Guangzhou, in 2011. From 2007 to 2008, he was a Control Software Engineer in Santak Electronic (Shenzhen) Co., Ltd., Eaton Co., and an R&D Engineer in Light Engineering (China) Co., Ltd. He is currently a Research Fellow of the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. His research interests include adaptive approximation-based control, fuzzy logic and neural networks, modeling and prediction, and embedded control system design.
Danwei Wang received his Ph.D. and MSE degrees from the University of Michigan, Ann Arbor in 1989 and 1984, respectively. He received his B.E degree from the South China University of Technology, China, in 1982. Since 1989, he has been with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. Currently, he is a professor in the Division of Control and Instrumentation. He has served as general chairman, technical chairman and various positions in international conferences, such as International Conference on Control, Automation, Robotics and Vision (ICARCVs) and IROS conferences. He is an Associate Editor for the International Journal of Humanoid Robotics and served as an Associate Editor of Conference Editorial Board, IEEE Control Systems Society from 1998 to 2005. He was a recipient of Alexander von Humboldt fellowship, Germany. His research interests include robotics, control theory and applications. He has published widely in technical areas of iterative learning control, repetitive control, robust control and adaptive control systems, manipulator/mobile robot dynamics, path planning, and control, as well as model-based fault diagnosis and satellite formation flying (Personal home page: http://www.ntu.edu.sg/home/edwwang).
Fuchun Sun received the B.S. and M.S. degrees from the Naval Aeronautical Engineering Academy, Yantai, China, in 1986 and 1989, respectively, and the Ph.D. degree from Tsinghua University, Beijing, China, in 1998. He was with the Department of Automatic Control, Naval Aeronautical Engineering Academy. From 1998 to 2000, he was a Postdoctoral Fellow with the Department of Automation, Tsinghua University. He is currently a Professor with the Department of Computer Science and Technology, Tsinghua University. His research interests include intelligent control, neural networks, fuzzy systems, variable structure control, nonlinear systems, and robotics. Dr. Sun is the recipient of the excellent Doctoral Dissertation Prize of China in 2000 and the Choon-Gang Academic Award by Korea in 2003, and was recognized as a Distinguished Young Scholar in 2006 by the Natural Science Foundation of China.