Zhang Function Research Articles

Zhang Dynamics based Tracking Control of Knee Exoskeleton with Timedependent Inertial and Viscous Parameters

Knee exoskeleton plays an important role in robot-assisted rehabilitation for impaired pilots to restore their motor functionality of lower extremity through producing external movement compensation. Tracking control of knee exoskeleton often encounters time-dependent (time-varying) issues reflected in its dynamic behaviors. In many applications, inertial and viscous parameters of knee exoskeletons are measured to be time-dependent due to unexpected mechanical vibrations and contact interactions, which increases difficultly of accurate control of knee exoskeleton to follow desired joint angle trajectories. This paper proposes a novel control strategy for controlling knee exoskeleton with time-dependent (time-varying) inertial and viscous coefficients. Such controller is designed based on Zhang dynamics (ZD) method and utilizes twice Zhang function (ZF) so as to make the tracking error of joint angle exponentially converge to zero. Illustrative simulation examples and experimental validation are presented to show efficiency of this type of controller based on ZD method. Comparisons with gradient dynamic (GD) approach are also presented to demonstrate superiority of ZD-type control strategy for tracking joint angle of knee exoskeleton.

Generic type of Ahlswede–Zhang style identities

We introduce a generic identity based on the Ahlswede–Zhang (AZ) function, which can be considered as a template to establish AZ-style identities. Many AZ-style identities can be derived from the generic identity.

Infinitely many Zhang functions resulting in various ZNN models for time-varying matrix inversion with link to Drazin inverse

In this Letter, by generalizing the notion of Zhang functions (ZFs) from previous work, a novel general-form Zhang function (NGFZF) is proposed, developed and investigated. Specifically, based on the NGFZF, infinitely many ZFs (as error functions) can be readily generated by successively selecting the different values of its parameters. Besides, by employing the NGFZF, a novel general-form Zhang neural net (NGFZNN) is proposed and studied for real-time solution of a time-varying matrix inverse (also termed, Zhang matrix inverse, ZMI). Moreover, a link between ZMI and Drazin inverse is discovered and further generalized to solve for the time-varying Drazin inverse (TVDI).

Different Complex ZFs Leading to Different Complex ZNN Models for Time-Varying Complex Generalized Inverse Matrices

As a special class of recurrent neural network, Zhang neural network (ZNN) has been recently proposed since 2001 for solving various time-varying problems, and has shown high efficiency and excellent performance for solving the problems in the real domain. In this paper, to solve online the time-varying complex generalized inverse (in most cases, the pseudoinverse) problem in the complex domain, a new type of complex-valued ZNN is further proposed and investigated. The design of such a complex ZNN is based on a complex Zhang function (ZF) which is indefinite and quite different from the usual error function (specially, the scalar-valued energy function) in the studies of conventional algorithms. By introducing five different complex ZFs, five different complex ZNN models (termed complex ZNN-I, ZNN-II, ZNN-III, ZNN-IV, and ZNN-V models) are proposed, developed, and investigated for the online solution of the time-varying complex generalized inverse matrices. Theoretical results of convergence analysis are presented to show the desirable properties of complex ZNN models. In addition, we discover the link between the proposed complex ZNN models and the Getz-Marsden dynamic system in the complex domain. Computer-simulation results further demonstrate the effectiveness of complex ZNN models based on different complex ZFs for the time-varying complex generalized inverse matrices.

Different Zhang functions resulting in different ZNN models demonstrated via time-varying linear matrix–vector inequalities solving

Our previous work shows that Zhang neural network (ZNN) has the higher efficiency and better performance for solving online time-varying linear matrix–vector inequalities, as compared to the conventional gradient neural network. In this paper, introducing the concept of Zhang function, we further investigate the problem of time-varying linear matrix–vector inequalities solving. Specifically, by defining three different Zhang functions, three types of ZNN models are further elaborately constructed to solve time-varying linear matrix–vector inequalities. The first ZNN model is based on a vector-valued lower-bounded Zhang function and is termed ZNN-1 model. The second one is based on a vector-valued lower-unbounded Zhang function and is termed ZNN-2 model. The third one is based on a transformed lower-unbounded Zhang function and is termed ZNN-3 model. Compared with the ZNN-1 model for solving time-varying linear matrix–vector inequalities, it is surprisedly discovered that the ZNN-2 model incorporates the ZNN-1 model as its special case. Besides, we put research emphasis on the ZNN-3 model for solving time-varying linear matrix–vector inequalities (including its design process, theoretical analysis and simulation verification). When power-sum activation functions are exploited, the ZNN-3 model possesses the property of superior convergence and better accuracy. Computer-simulation results further verify and demonstrate the theoretical analysis and efficacy of the ZNN-3 model for solving time-varying linear matrix–vector inequalities.

From Different Zhang Functions to Various ZNN Models Accelerated to Finite-Time Convergence for Time-Varying Linear Matrix Equation

In addition to the parallel-distributed nature, recurrent neural networks can be implemented physically by designated hardware and thus have been found broad applications in many fields. In this paper, a special class of recurrent neural network named Zhang neural network (ZNN), together with its electronic realization, is investigated and exploited for online solution of time-varying linear matrix equations. By following the idea of Zhang function (i.e., error function), two ZNN models are proposed and studied, which allow us to choose plentiful activation functions (e.g., any monotonically-increasing odd activation function). It is theoretically proved that such two ZNN models globally and exponentially converge to the theoretical solution of time-varying linear matrix equations when using linear activation functions. Besides, the new activation function, named Li activation function, is exploited. It is theoretically proved that, when using Li activation function, such two ZNN models can be further accelerated to finite-time convergence to the time-varying theoretical solution. In addition, the upper bound of the convergence time is derived analytically via Lyapunov theory. Then, we conduct extensive simulations using such two ZNN models. The results substantiate the theoretical analysis and the efficacy of the proposed ZNN models for solving time-varying linear matrix equations.

Different Zhang functions leading to different ZNN models illustrated via time-varying matrix square roots finding

In view of the great potential in parallel processing and ready implementation via hardware, neural networks are now often employed to solve online nonlinear matrix equation problems. Recently, a novel class of neural networks, termed Zhang neural network (ZNN), has been formally proposed by Zhang et al. for solving online time-varying problems. Such a neural-dynamic system is elegantly designed by defining an indefinite matrix-valued error-monitoring function, which is called Zhang function (ZF). The dynamical system is then cast in the form of a first-order differential equation by using matrix notation. In this paper, different indefinite ZFs, which lead to different ZNN models, are proposed and developed as the error-monitoring functions for time-varying matrix square roots finding. Towards the final purpose of field programmable gate array (FPGA) and application-specific integrated circuit (ASIC) realization, the MATLAB Simulink modeling and verifications of such ZNN models are further investigated for online solution of time-varying matrix square roots. Both theoretical analysis and modeling results substantiate the efficacy of the proposed ZNN models for time-varying matrix square roots finding.

Different Zhang functions leading to different Zhang-dynamics models illustrated via time-varying reciprocal solving

Along with neural dynamics (based on analog solvers) widely arising in scientific computation and optimization fields in recent decades which attracts extensive interest and investigation of researchers, a novel type of neural dynamics, called Zhang dynamics (ZD), has been formally proposed by Zhang et al. for the online solution of time-varying problems. By following Zhang et al.’s neural-dynamics design method, the ZD model, which is based on an indefinite Zhang function (ZF), can guarantee the exponential convergence performance for the online time-varying problems solving. In this paper, different indefinite Zhang functions, which can lead to different ZD models, are proposed and developed as the error-monitoring functions for the time-varying reciprocal problem solving. Additionally, for the goal of developing the floating-point processors or coprocessors for the future generation of computers, the MATLAB Simulink modeling and simulative verifications of such different ZD models are further presented for online time-varying reciprocal solving. The modeling results substantiate the efficacy of such different ZD models for time-varying reciprocal solving.