Zeroing neural network with comprehensive performance and its applications to time-varying Lyapunov equation and perturbed robotic tracking
Introduction
The Lyapunov equation plays a crucial part in many scientific and engineering fields. For instance, it can be applied in communication [1], control theory [2], [3] and automatic control [4], [5]. Furthermore, Lyapunov equation is indispensable in the domain of optimal control. Thus, the solution of Lyapunov equation has earned a large amount of efforts on account of its extensive applications. For solving Lyapunov equation, approaches that have the longest history should be traditional numerical algorithms. In [6], an iterative algorithm was proposed to tackle Lyapunov equation with Markov jump. Other types of numerical algorithms for Lyapunov equation have also been investigated. [7] proposed an iterative algorithm based on the gradient to solve this problem, while [8] presented a method based on minimal residual error. However, the minimum time complexity of these numerical algorithms is about the cube of dimensions of input matrix. Thus, solving large scale Lyapunov equations using these numerical algorithms becomes a time consuming task. The large time cost of numerical algorithms has considerably limited their application ranges, especially for online Lyapunov equation problem.
Nowadays, we can accelerate many algorithms by taking advantage of hardware or software level parallelism, but this approach is hard for above mentioned numerical algorithms because of their serial processing nature. Therefore, Artificial Neural Networks (ANNs) including the Recurrent Neural Networks (RNNs) as other types of approaches, have been found efficient in solving various numerical computation, optimization and robot control problems [9], [10], [11], [12], [13] due to their parallel and distributed computing properties. Gradient-based Neural Network (GNN) is a variation of RNN and was investigated for the online stationary Lyapunov equation [14], [15], [16], [17]. GNN uses the norm of error matrix as its performance index and the neural network will evolve along the gradient-descent direction until its performance index converges to zero. GNN performs well in stationary Lyapunov equation problems, but it suffers from large residual error on solving time-varying Lyapunov equations and such error exists even after infinite time of evolving, which promotes researchers to overcome this drawback by discovering new neural network models.
Under such background, Zeroing Neural Network (ZNN) as a special type of RNN was proposed and utilized in Refs. [18], [19], [20], [21]. This kind of RNN model utilizes the velocity information of time-varying coefficients and uses the error matrix instead of its norm as performance index, successfully surpassing GNN model on solving both stationary and nonstationary Lyapunov equations. Nevertheless, the ordinary ZNN model is far from being perfect because it uses linear function as activation function and can only converge to the solution exponentially. That is, its error cannot converge to zero in finite time. Recently, many research works about speeding up the convergence speed of ANNs are springing up [22], [23], [24]. Therefore, to improve the convergence speed of the ordinary ZNN model, a finite-time convergent ZNN model which exploited the Sign-Bi-Power (SBP) activation function is presented in Ref. [25]. Another practical problem is that ZNN usually works in an environment that exists various noise and this can potentially decrease the accuracy of the neural network, but the general ZNN design formula didn’t take this factor into account. Thus, [26], [27], [28] designed a Noise-Tolerance ZNN (NTZNN) model to enhance ZNN model’s robustness against additive noise. However, NTZNN’s activation function is linear function and cannot be changed, so that it cannot converge in finite time. In this paper, we present a Noise-Tolerance Finite-time convergent ZNN (NTFZNN) model for the time-varying Lyapunov equation problem following the motivation that both finite-time convergence and noise suppression ability are highly demanded in ZNN applications [29]. The NTFZNN model not only converges to the accurate solution of time-varying Lyapunov equation in finite time in noise-free environments, but also have less stable residual error than the NTZNN model within additive noise polluted environment.
The main contributions of this paper are summarized as below:
- 1)
A novel NTFZNN model is developed for solving time-varying Lyapunov equation problem, which possesses finite-time convergence and powerful additive noise suppression ability at the same time.
- 2)
Using the SFAF (Simplified Finite-time convergent Activation Function), the (NTFZNN using SFAF) model is proposed, which has simpler structure and lower calculation complexity than the (NTFZNN using SBP) model but still being finite-time convergent.
- 3)
The convergence time upper bound and stable residual error upper bound of the NTFZNN model have both been quantitatively analyzed and then validated by simulation experiments. Besides, in the robustness analysis, the more practical time-varying unknown noises are investigated rather than the usual constant or limited known noises.
The rest of this paper is organized into 6 sections. In Section 2, the problem description and some preliminaries of the time-varying Lyapunov equation are provided for the following discussion and analyses. In Section 3, the NTFZNN model is designed for solving the time-varying Lyapunov equation, and two finite-time convergent activation functions have been introduced. In Section 4, the NTFZNN model’s convergence performance in noise-free environment and robustness performance when perturbed by additive noises have been analyzed in detail. In Section 5, numerical simulations and comparisons are presented to verify the previous theoretical conclusions. In Section 6, the NTFZNN model is successfully applied in controlling a mobile robot manipulator to track the desired path with additive noise perturbation, which has further validated the NTFZNN model’s applicability and superiority. Section 7 concludes this paper briefly.
Section snippets
Problem description and preliminaries
In this paper, the problem we mainly concerned about is solving the time-varying Lyapunov equation. Let be a nonstationary coefficient matrix with being a nonstationary symmetric positive-definite matrix. We have following equation:
Then, the above equation is widely known as the Lyapunov equation, where is a unique time-varying matrix that we should obtain given that and both satisfied the unique solution condition [17]. In the
Design of noise-tolerance finite-time convergent Zeroing Neural Network
The ordinary ZNN model introduced in Section 2 is able to handle the nonstationary Lyapunov equation well in the ideal noise-free environment. However, because continuous ZNN models are mainly implemented in analog circuits, there are many factors that may create noises. These factors include round-off error, circuit implementation errors and so on, which can result in the great loss for ZNN model (4) in accuracy. Therefore, we propose a novel NTFZNN model with better noise suppression ability
Theoretical analysis of noise-tolerance finite-time convergent Zeroing Neural Network
In this section, we focus on proving NTFZNN to be globally stable, computing the convergence time upper bound, and analyzing inherent noise suppression ability of the NTFZNN model (7). All of these theoretical analyses eventually illustrate the superiority of our NTFZNN model. As it has been shown in Section 3, the NTFZNN model formula (7) is equivalent to the formula (8) and the latter one is mainly used in the following analyses and experiments. In addition, the error matrix in (7)
Simulative verification and comparison
Theoretical analyses in previous sections have laid a solid theoretical foundation for the NTFZNN model. In this section, we mainly focus on validating the convergence and noise suppress abilities of the NTFZNN model when applied to time-varying Lyapunov equation problem solving. Without loss of generality, the coefficients in the time-varying Lyapunov problem (1) are selected as the following form
Because the
Application to perturbed mobile manipulator control
In the previous section, we have used NTFZNN model (7) to solve the nonstationary Lyapunov equation problem and we only solve a slightly simple Lyapunov equation for concise demonstration. However, to validate the efficacy of our proposed NTFZNN model on applying to the real-world task, we introduce a mobile manipulator to test it as in [13], [38], [39]. The robot manipulator is a wheeled mobile manipulator, whose model and mechanical structure can be seen in [40]. Furthermore, one can learn
Conclusion
In order to accelerate the convergence speed to finite time as well as to perform reliably even when there exists various types of internal and external noise, a novel Noise-Tolerance Finite-time convergent ZNN (NTFZNN) model is established to deal with the time-varying Lyapunov equation. Equipped with two finite-time convergent activation functions, the advanced properties of the NTFZNN model are firstly proved theoretically. Numerical simulative experiments have validated that NTFZNN is able
CRediT authorship contribution statement
Zeshan Hu: Data curation, Writing - original draft, Visualization. Kenli Li: Supervision, Validation. Keqin Li: Supervision, Validation. Jichun Li: Writing - review & editing. Lin Xiao: Conceptualization, Methodology, Supervision, Investigation, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by the National Key Research and Development Program of China (Grant No. 2016YFB0201800); the National Natural Science Foundation of China under grants 61866013, 61503152, 61976089, 61473259, and 61563017; the Natural Science Foundation of Hunan Province of China under grants 2019JJ50478, 18A289, JGY201926, 2016JJ2101, 2018TP1018, and 2018RS3065.
Zeshan Hu received the B.S. degree in Computer Science and Technology from Fuzhou University, Fuzhou, China, in 2018. He is currently a postgraduate with the College of Computer Science and Electronic Engineering, Hunan University, Changsha, China. His main research interests include neural networks, and robotics.
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Zeshan Hu received the B.S. degree in Computer Science and Technology from Fuzhou University, Fuzhou, China, in 2018. He is currently a postgraduate with the College of Computer Science and Electronic Engineering, Hunan University, Changsha, China. His main research interests include neural networks, and robotics.
Lin Xiao received the Ph.D. degree in Communication and Information Systems from Sun Yat-sen University, Guangzhou, China, in 2014. He is currently a Professor with the College of Information Science and Engineering, Hunan Normal University, Changsha, China. He has authored over 100 papers in international conferences and journals, such as the IEEE-TNNLS, the IEEE-TCYB, the IEEE-TII and the IEEE-TSMCA. His main research interests include neural networks, robotics, and intelligent information processing.
Kenli Li received the Ph.D. degree in computer science from the Huazhong University of Science and Technology, Wuhan, China, in 2003. He was a Visiting Scholar with the University of Illinois at Urbana-Champaign, Champaign, IL, USA, from 2004 to 2005. He is currently a Full Professor of Computer Science and Technology with Hunan University, Changsha, China, and also the Deputy Director of the National Supercomputing Center, Changsha. His current research interests include parallel computing, cloud computing, big data computing, and neural computing.
Jichun Li received his Ph.D. degree in mechanical engineering from King’s College London, University of London, U.K. in 2013. He is currently a Lecturer with the School of Computer Science and Electronic Engineering, University of Essex, U.K. His main research interests include neural networks, robotics, intelligent transportation and communication, control, IoT and AI solutions for bespoke automated systems for chemical, medical, energy, agri-food and biosciences industries. He is a member of the IEEE, IET, and IMeChE.
Keqin Li is a SUNY Distinguished Professor of computer science with the State University of New York. He is also a Distinguished Professor at Hunan University, China. His current research interests include cloud computing, fog computing and mobile edge computing, energy-efficient computing and communication, embedded systems and cyberphysical systems, heterogeneous computing systems, big data computing, high-performance computing, CPU-GPU hybrid and cooperative computing, computer architectures and systems, computer networking, machine learning, intelligent and soft computing.