Time-varying Linear Inequality Research Articles

Robust Neural Dynamics Method for Redundant Robot Manipulator Control With Physical Constraints

Redundant robot manipulators play a significant role in modern industry. In this paper, we propose a solution scheme to the trajectory tracking problem of redundant robot manipulator with physical constraints through Zhang neural dynamics method. Such problem is integrated into a time-varying system consisting of time-varying nonlinear equation (TVNE) and time-varying linear inequality (TVLI) and solved online by the varying-parameter Zhang neural dynamics (VPZND) model. It is ensured that the redundant robot manipulator can still perform the tracking task perfectly under the coexistence of time-varying bounded noise and physical constraints. Theoretical analysis proves that this VPZND model also has an explicit fixed convergence time. Numerical experiments confirm the feasibility of our VPZND model for TVLI. The trajectory tracking problem of the redundant robot manipulator with six or three degrees of freedom under the dual influence of physical constraints and noise is perfectly solved by the VPZND model, which is enough to verify its practical value.

Two New Zhang Neural Networks for Solving Time-Varying Linear Equations and Inequalities Systems.

Recently, Xu et al. solved a class of time-varying linear equations and inequalities systems (LEIESs) by using a Zhang neural network (ZNN) model through introducing a nonnegative relaxation vector. However, the introduction of this unknown nonnegative slack vector will increase the size and complexity of the model, thereby increasing the cost of computation. In this article, we propose two new ZNN models (called traditional Zhang neural network (TZNN) and variant Zhang neural network (VZNN) models, respectively) in which no additional relaxation vector is needed. The convergence analysis of these two new models are performed, and two simulation experiments are given to illustrate their efficiency and effectiveness for solving the time-varying LEIESs, including the applicability of our proposed models to robot manipulator.

Solving Perturbed Time-Varying Linear Equation and Inequality Problem With Adaptive Enhanced and Noise Suppressing Zeroing Neural Network

Solving time-varying linear equation and inequality (TVLEI) problem has attracted extensive attention in numerous scientific and engineered fields. In this article, it is basically considered that the commonly used dynamics neural network in the virtual environment is inevitably interfered with by the variable measurement noises while dealing with the TVLEI problem. An adaptive enhanced and noise-suppressing zeroing neural network (AENSZNN) model is proposed as an improved algorithm for solving the TVLEI problem. An adaptive scale factor based on the residual error norm is designed to make the proposed AENSZNN model converge to the theoretical solution faster. Furthermore, the momentum enhancement terms added to the model enables the AENSZNN model to effectively solve the TVLEI problem in real-time under the obstruction of different measurement noises. Besides, theoretical results and numerical experiments indicate that the AENSZNN model has advantages in convergence accuracy and robustness to noises compared with the existing algorithms. Note that, the proposed AENSZNN model is successfully exploited for the estimation of mobile object localization.

Unified Model Solving Nine Types of Time-Varying Problems in the Frame of Zeroing Neural Network.

Many time-varying problems have been solved using the zeroing neural network proposed by Zhang et al. In this article, nine types of time-varying problems, namely time-varying nonlinear equation system, time-varying linear equation system, time-varying convex nonlinear optimization under linear equalities, unconstrained time-varying convex nonlinear optimization, time-varying convex quadratic programming under linear equalities, unconstrained time-varying convex quadratic programming, time-varying nonlinear inequality system, time-varying linear inequality system, and time-varying division, are investigated to better understand the essence of zeroing neutral network. Discrete-form time-varying problems are studied by considering the nature of unknown future and the requirement of real-time computation for time-varying problems. A unified model is proposed in the frame of zeroing neural network to uniformly solve these time-varying problems on the basis of their connections and a newly developed discretization formula. Theoretical analyses and numerical experiments, including the tracking control of PUMA560 robot manipulator, verify the effectiveness and precision of the proposed unified model.

Discrete-Time Zeroing Dynamics With Quadruplicate Error Pattern for Time-Varying Linear Inequality

Linear inequality (LI) plays an important role in many fields of science and engineering. Recently, a typical neural dynamics called zeroing dynamics (ZD) has been reported for online solution of time-varying LI (TVLI). On the basis of the previous work, the discrete-time form of the ZD with superior computational property is studied in this paper. Specifically, a Taylor-type difference rule is first presented for the first-order derivative approximation. By utilizing such a difference rule to discrete the previous ZD model, the new discrete-time ZD (DTZD) algorithm is thus established and proposed for TVLI solving. Such an algorithm performs better computational performance than the existing DTZD algorithm. Theoretical results show that the proposed DTZD algorithm has a quadruplicate error pattern on solving the TVLI. Comparative numerical results with two illustrative examples further substantiate the efficacy and superiority of the proposed DTZD algorithm over the existing DTZD algorithm.

Li-Function Activated Zhang Neural Network for Online Solution of Time-Varying Linear Matrix Inequality

In the previous work, a typical recurrent neural network termed Zhang neural network (ZNN) has been developed for various time-varying problems solving. Based on the previous work, by exploiting a special activation function (i.e., Li activation function), the resultant ZNN model is presented and investigated in this paper for online solution of time-varying linear matrix inequality (TVLMI). For such a Li-function activated ZNN (LFAZNN) model, theoretical results are provided to show its excellent computational performance on solving the TVLMI. That is, the presented LFAZNN model has the property of finite-time convergence. Comparative simulation results with two illustrative examples further substantiate the efficacy of the presented LFAZNN model for TVLMI solving.

Solving time-varying linear inequalities by finite-time convergent zeroing neural networks

In this paper, to solve time-varying linear inequalities much faster, on basis of zeroing neural network (ZNN), two finite-time convergent ZNN (FTCZNN) models are proposed by exploiting two novel nonlinear activation functions (AFs). The first FTCZNN model is established by using the sign-bi-power AF which is termed as FTCZNN-S for presentation convenience. The second one is established by amending the sign-bi-power AF through adding a linear term, and called FTCZNN-SL. Compared with existing ZNN models for time-varying linear inequalities, the proposed two FTCZNN models possess prominent finite-time convergence performance. In addition, theoretical analysis is given to estimate the finite-time convergence upper bounds of those two FTCZNN models. Numerical comparative results ulteriorly validate the effectiveness and dominance of two FTCZNN models for finding the solution of time-varying linear inequalities.

Design and analysis of three nonlinearly activated ZNN models for solving time-varying linear matrix inequalities in finite time

To obtain the superiority property of solving time-varying linear matrix inequalities (LMIs), three novel finite-time convergence zeroing neural network (FTCZNN) models are designed and analyzed in this paper. First, to make the Matlab toolbox calculation processing more conveniently, the matrix vectorization technique is used to transform matrix-valued FTCZNN models into vector-valued FTCZNN models. Then, considering the importance of nonlinear activation functions on the conventional zeroing neural network (ZNN), the sign-bi-power activation function (AF), the improved sign-bi-power AF and the tunable sign-bi-power AF are explored to establish the FTCZNN models. Theoretical analysis shows that the FTCZNN models not only can accelerate the convergence speed, but also can achieve finite-time convergence. Computer numerical results ulteriorly confirm the effectiveness and advantages of the FTCZNN models for finding the solution set of time-varying LMIs.

Zeroing Neural Network for Solving Time-Varying Linear Equation and Inequality Systems

A typical recurrent neural network called zeroing neural network (ZNN) was developed for time-varying problem-solving in a previous study. Many applications result in time-varying linear equation and inequality systems that should be solved in real time. This paper provides a ZNN model for determining the solution of time-varying linear equation and inequality systems. By introducing a nonnegative slack variable, the time-varying linear equation and inequality systems are transformed into a mixed nonlinear system. The ZNN model is established via the definition of an indefinite error function and the usage of an exponential decay formula. Theoretical results indicate the convergence property of the proposed ZNN model. Comparative simulation results prove the ZNN effectiveness and superiority for time-varying linear equation and inequality systems. Furthermore, the proposed ZNN model is employed to robot manipulators, thus showing the ZNN applicability.

Model predictive control for autonomous rendezvous and docking with a tumbling target

In this paper, we investigate the application of model predictive control (MPC) strategy for controlling a chaser spacecraft to dock with a tumbling target in space. The CW-based model is used to predict the proximity trajectory, while a receding prediction horizon is employed to complete the specific docking manipulation in finite time, and a 2-norm cost index about control input is used to minimize the fuel consumption of the chaser during docking. To ensure the implementation of the docking procedure, several engineering constraints such as control input saturation, collision avoidance, velocity constraint, and dock-enabling condition are considered simultaneously. A time-varying linear inequality constraint is used rather than the original quadratic constraint, to achieve the collision-avoidance requirement. Therefore, the constrained optimal rendezvous trajectory in the case of such a MPC framework can be obtained by solving a quadratic program subject to linear constraints. To verify the feasibility of the proposed MPC strategy, four AR&D scenarios are simulated. The simulation results show that MPC strategy can be an effective approach to achieve rendezvous and proximity maneuvering between a chaser spacecraft and a tumbling target while satisfying various constraints, and provide robustness to disturbances.

Zhang neural networks for a set of linear matrix inequalities with time-varying coefficient matrix

Zhang neural networks (ZNN) model is developed for solving a set of time-varying linear matrix inequalities, referred to as Stein matrix inequality, which exploits the time-derivative information of time-varying coefficients. Computer simulation results show that the proposed ZNN model is efficient and superior for such kind of linear matrix inequalities (LMIs).

ZNN for solving online time-varying linear matrix–vector inequality via equality conversion

In this paper, a special recurrent neural network termed Zhang neural network (ZNN) is proposed and investigated for solving online time-varying linear matrix–vector inequality (LMVI) via equality conversion. That is, by introducing a time-varying vector (of which each element is great than or equal to zero), such a time-varying linear inequality can be converted to a time-varying matrix–vector equation. Then, the ZNN model is developed and investigated for solving online the time-varying matrix–vector equation (as well as the time-varying LMVI) by employing the ZNN design formula. The resultant ZNN model exploits the time-derivative information of time-varying coefficients. Computer-simulation results further demonstrate the efficacy and superiority of the proposed ZNN model for solving online the time-varying LMVI (and the converted time-varying matrix–vector equation).

Zhang neural network for online solution of time-varying linear matrix inequality aided with an equality conversion.

In this paper, for online solution of time-varying linear matrix inequality (LMI), such an LMI is first converted to a time-varying matrix equation by introducing a time-varying matrix, of which each element is greater than or equal to zero. Then, by employing Zhang et al.'s neural dynamic method, a special recurrent neural network termed Zhang neural network (ZNN) is proposed and investigated for solving online the converted time-varying matrix equation as well as the time-varying LMI. Such a ZNN model showed in an explicit dynamics exploits the time-derivative information of time-varying coefficients. In addition, theoretical analysis and results of the proposed ZNN model are discussed and presented to show its excellent performance on solving the time-varying LMI. Computer simulation results further demonstrate the efficacy of the proposed ZNN model for online solution of the time-varying LMI and the converted time-varying matrix equation.

Distributed average filtering for sensor networks with sensor saturation

This study addresses the distributed average set-membership filtering of spatially varying processes using sensor networks. The system under consideration contains sensor saturation in the presence of unknown-but-bounded process and measurement noise in the sensor network. The so-called distributed average set-membership filtering is defined to quantify bounded consensus regarding the estimation error. A sufficient condition for distributed average set-membership filtering parameter design is established in terms of a set of time-varying linear matrix inequalities. A recursive algorithm is developed for computing the estimator, controller gains and the ellipsoid that guarantees to contain the true state. Simulation results are provided to demonstrate the effectiveness of the proposed method.

A new variant of the Zhang neural network for solving online time-varying linear inequalities

Since March 2001, a special class of recurrent neural networks termed the Zhang neural network (ZNN) has been proposed by Zhang and co-workers for solving online a rich repertoire of time-varying problems. By extending Zhang et al. 's design formula (or say, the ZNN design formula), a (new) variant of the ZNN design formula is proposed and investigated in this paper, which is also based on a matrix/vector-valued indefinite error function. In addition, by employing such a novel design formula, a new variant of the ZNN (NVZNN) is proposed, developed and investigated for online solution of time-varying linear inequalities (LIs). The resultant NVZNN models are depicted in implicit dynamics and methodologically exploit the time-derivative information of time-varying coefficients. Computer simulation results further demonstrate the novelty, efficacy and superiority of the proposed NVZNN models for solving online time-varying LIs.

Zhang Neural Network Versus Gradient Neural Network for Solving Time-Varying Linear Inequalities

By following Zhang design method, a new type of recurrent neural network [i.e., Zhang neural network (ZNN)] is presented, investigated, and analyzed for online solution of time-varying linear inequalities. Theoretical analysis is given on convergence properties of the proposed ZNN model. For comparative purposes, the conventional gradient neural network is developed and exploited for solving online time-varying linear inequalities as well. Computer simulation results further verify and demonstrate the efficacy, novelty, and superiority of such a ZNN model and its method for solving time-varying linear inequalities.

Set-Membership Filtering for Discrete-Time Systems With Nonlinear Equality Constraints

In this technical note, the problem of set-membership filtering is considered for discrete-time systems with nonlinear equality constraint between their state variables. The nonlinear equality constraint is first linearized and transformed into a state linear equality constraint with two uncertain quantities related to linearizing truncation error and base point error. S-procedure method is then applied to merge all inequalities into one inequality and the solution to the unconstrained set-membership filtering problem is provided. The set-membership filter with state constraint is finally derived from projecting the unconstrained set-membership filter onto the constrained surface by using Finsler's Lemma. A time-varying linear matrix inequality optimization based approach is proposed to design the set-membership filter with nonlinear equality constraint. A recursive algorithm is developed for computing the state estimate ellipsoid that guarantees to contain the true state. An illustrative example is provided to demonstrate the effectiveness of the proposed set-membership filtering with nonlinear equality constraint.

Set-membership filtering for systems with sensor saturation

This paper addresses the set-membership filtering problem for a class of discrete time-varying systems with sensor saturation in the presence of unknown-but-bounded process and measurement noises. A sufficient condition for the existence of set-membership filter is derived. A convex optimisation method is proposed to determine a state estimation ellipsoid that is a set of states compatible with sensor saturation and unknown-but-bounded process and measurement noises. A recursive algorithm is developed for computing the ellipsoid that guarantees to contain the true state by solving a time-varying linear matrix inequality. Simulation results are provided to demonstrate the effectiveness of the proposed method.

Robust Error Square Constrained Filter Design for Systems With Non-Gaussian Noises

In this letter, an error square constrained filtering problem is considered for systems with both non-Gaussian noises and polytopic uncertainty. A novel filter is developed to estimate the systems states based on the current observation and known deterministic input signals. A free parameter is introduced in the filter to handle the uncertain input matrix in the known deterministic input term. In addition, unlike the existing variance constrained filters, which are constructed by the previous observation, the filter is formed from the current observation. A time-varying linear matrix inequality (LMI) approach is used to derive an upper bound of the state estimation error square. The optimal bound is obtained by solving a convex optimization problem via semi-definite programming (SDP) approach. Simulation results are provided to demonstrate the effectiveness of the proposed method.