Sense Of Lyapunov Research Articles

Human gait tracking for rehabilitation exoskeleton: adaptive fractional order sliding mode control approach

To improve the rehabilitation training effect of hemiplegic patients, in this paper, a discrete adaptive fractional order fast terminal sliding mode control approach is proposed for the lower limb exoskeleton system to implement high-precision human gait tracking tasks. Firstly, a discrete dynamic model is established based on the Lagrange system discretization criterion for the lower limb exoskeleton robot. Then, in order to design a discrete adaptive fractional order fast terminal sliding mode controller, the Grünwald–Letnikov fractional order operator is introduced to combine with fast terminal attractor to construct a fractional order fast terminal sliding surface. An adaptive parameter adjustment strategy is proposed for the reaching law of sliding mode control, which drives the sliding mode to the stable region dynamically. Moreover, the stability of the control system is proved in the sense of Lyapunov, and the guidelines for selecting the control parameters are given. Finally, the simulations are tested on the MATLAB-Opensim co-simulation platform. Compared with the conventional discrete sliding mode control and discrete fast terminal sliding mode control, the results verify the superiority of the proposed method in improving lower limb rehabilitation training.

Neural-Adaptive Switching Control of Task-Space Objectives on Strict-Feedback Robot

Switching between tasks in robot applications is necessary for autonomous behaviour, requiring a control framework for switching between objectives. Thus, this paper proposes a switching control framework that allows transitions in position-dependent objectives for robot systems in strict-feedback form with adaptive capabilities. The proposed controller integrates artificial neural networks in two ways; one approximates control transforms while another approximates additive terms. Furthermore, the proposed method includes a dynamic damped inverse to ensure the invertible control transform matrices. The proposed controller is derived using the backstepping method and is stable in a Lyapunov sense. The proposed task-switching controller is first demonstrated on a generic robot manipulator. Then, a simulated two-degree-of-freedom manipulator with a camera at the end-effector tasked with position tracking and visual servoing verifies the capabilities of the proposed controller. A baseline switching controller shows that the proposed controller has superior performance, specifically, root mean square (RMS) error and control effort. Also, executing the simulation for multiple cycles shows that the proposed controller's artificial neural network and adaptive laws are stable.

Improving transient performance of a class of nonlinear systems via indicator-based switching control scheme

This work presents an indicator-based switching control for a class of strict-feedback systems to exploit the potential performance improvement. The proposed approach is designed by injecting a switch mechanism into the backstepping control. An indicator is introduced to reflect the impact of the integral part on the system. The indicator-triggered control method is capable of wisely switching the control structure. Thus, the transient performance can be improved. In addition, the parameters of the proposed control scheme are optimized by Cuckoo Search algorithm to achieve the better performance. The stability is analyzed in the Lyapunov sense and the effectiveness is also verified by numerical simulations.

Mean-AVaR in credibilistic portfolio management via an artificial neural network scheme

ABSTRACT This paper focuses on the computation issue of portfolio optimisation with scenario-based mean-Average Value at Risk (AVaR) in credibilistic environment. The portfolio optimisation problem is designed in two cases: risk taker model and risk-averse model. The main idea is to replace the portfolio selection models with linear programming (LP) problems. Since the computing time required for solving LP greatly depends on the dimension and the structure of the problem, the conventional numerical methods are usually less effective in real-time applications. One promising approach to handle online applications is to employ recurrent neural networks based on circuit implementation. Hence, according to the convex optimisation theory and some concepts of ordinary differential equations, a neural network model for solving the LP problems related to portfolio selection problems is presented. The equilibrium point of the proposed model is proved to be equivalent to the optimal solution of the original problem. It is also shown that the proposed neural network model is stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the portfolio selection problem with fuzzy returns. Some illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper.

Lyapunov stability theorems for $$\psi $$-Caputo derivative systems

We introduce the stability concepts for \(\psi \)-Caputo derivative systems in the sense of Lyapunov’s idea. Two new Lyapunov theorems are formulated for the stability analysis of such systems. Many new inequalities that involve \(\psi \)-Caputo derivative are developed for the application of the proposed Lyapunov theorems. Examples are provided and it is shown that the proposed theorems are very useful for predicting the limiting behavior of the trajectory of such systems starting from any initial condition imposed at any initial time on the real axis.

An adaptive convex combination of CLMK and ACLMK algorithms for processing complex-valued signals

This paper introduces a novel adaptive convex combination (ACC) of the complex-valued least mean kurtosis (CLMK) and augmented CLMK (ACLMK) algorithms by inspiring their individual advantages to effectively process circular and noncircular complex-valued signals. For this purpose, the negated kurtosis-based cost function of the overall error signal in the complex domain is primarily defined and then it is minimized to obtain the update rule of the mixing parameter for the proposed ACC. The proposed ACC offers a significant opportunity for the CLMK and ACLMK algorithms to work collaboratively. By iteratively updating the mixing parameter based on the mentioned kurtosis cost function, the proposed ACC adaptively dominates the CLMK or ACLMK at its overall output according to the requirements of the filter operating environments to improve the overall filtering performance in terms of the convergence rate and steady-state error. The evolution of the mixing parameter in the proposed ACC also gives information about the characteristic behaviour of the considered signal or system. Moreover, for the first time in this paper, the stability of the mixing parameter update rule of the proposed ACC is theoretically analysed in the sense of Lyapunov to derive its step size bounds. On the other h and, since the analyses of the CLMK and ACLMK algorithms in the sense of Lyapunov are still missing in the literature, in the scope of this work, their Lyapunov analyses are also provided in detail. The comprehensive simulations on system identification and one-step ahead prediction scenarios in the complex domain support both the mentioned elegant properties of the proposed ACC and its superior performance over the existing ACC.

A neural network approach to solve geometric programs with joint probabilistic constraints

We study a dynamical neural network approach to solve nonconvex geometric programs with joint probabilistic constraints (GPPC for short) with normally distributed coefficients and independent matrix row vectors. The main feature of our framework is to solve the biconvex deterministic equivalent problem of GPPC without the use of any convex approximation unlike the state-of-the-art solving methods. The second feature of our approach is to solve the dynamical system without the use of any iterative method but only by solving an ordinary differential equation system. We prove the stability together with the convergence of our neural network in the sense of Lyapunov. We also prove the equivalence between the optimal solution of the deterministic equivalent problem of GPPC and the solution of the dynamical system. We provide numerical experiments to show the performances of our approach compared to the state-of-art.

Some remarks on the periodic motions of a bouncing ball

We consider the vertical motion of a free falling ball bouncing elastically on a racket moving in the vertical direction according to a regular 1-periodic function f. For fixed coprime p, q we study existence, stability in the sense of Lyapunov and multiplicity of p periodic motions making q bounces in a period. If f is real analytic we prove that one periodic motion is unstable and give some information on the set of these motions.

Fraction dynamic-surface-based adaptive neural finite-time control for stochastic nonlinear systems subject to unknown control directions, time-varying input delay and state delay

In this paper, an adaptive neural finite-time tracking control is studied for a category of stochastic nonlinearly parameterized systems with multiple unknown control directions, time-varying input delay, and time-varying state delay. To this end, a novel criterion of semi-globally finite-time stability in probability (SGFSP) is proposed, in the sense of Lyapunov, for stochastic nonlinear systems with multiple unknown control directions. Secondly, a novel auxiliary system with finite-time convergence is presented to cope with the time-varying input delay, the appropriate Lyapunov Krasovskii functionals are utilized to compensate for the time-varying state delay, Nussbaum functions are exploited to identify multiple unknown control directions, and the neural networks (NNs) are applied to approximate the unknown functions of nonlinear parameters. Thirdly, the fraction dynamic surface control (FDSC) technique is embedded in the process of designing the controller, which not only the “explosion of complexity” problems are successfully avoided in traditional backstepping methods but also the command filter convergence can be obtained within a finite time to lead greatly improved for the response speed of command filter. Meanwhile, the error compensation mechanism is established to eliminate the errors of the command filter. Then, based on the proposed novel criterion, all closed-loop signals of the considered systems are SGPFS under the designed controller, and the tracking error can drive to a small neighborhood of the origin in a finite time. In the end, three simulation examples are applied to demonstrate the validity of the control method.

Spherical Formation Tracking Control of Nonlinear Second-Order Agents With Adaptive Neural Flow Estimate.

This article addresses the spherical formation tracking control problem of nonlinear second-order vehicles moving in flowfields under both undirected networks and directed, strongly connected networks. Different from the previous adaptive estimate of the time-invariant parameters of flowfields, the flowfields under our consideration are spatial and absolutely unknown dynamics. Adaptive neural networks (ANNs) with the novel cooperative adaptive algorithms are proposed to approximate the flowfield acting on the channel of each vehicle's velocity (i.e., the mismatched flowfield) and the flowfield pushing the acceleration (i.e., the matched flowfield), respectively. For the purpose of avoiding the complex derivation derived from backstepping, the novel first-order filters are generated by dynamic surface based on barrier functions and relative positions of neighbors. The proposed control algorithms and adaptive upgrade law are fully distributed without using any global information of the graph. The uniform boundedness is analyzed in the Lyapunov sense. Simulation results are given to verify the theoretical analysis.

Suppressing UAV payload swing with time-varying cable length through nonlinear coupling

This paper introduces a method to navigate the UAV flight and to suppress the payload swing by a nonlinear coupling control augmented with time-varying cable length. Special error signals are introduced that bring further coupling of dynamics among various states of the UAV and enable the flight control to effectively suppress the payload swing. The control is designed with the help of the dynamic model of the UAV in lift and transport operation. The stability of the control is proven in the Lyapunov sense. Extensive simulations and experiments have been carried out to demonstrate the effectiveness of the proposed nonlinear coupling control. Two popular control approaches in the literature are chosen to compare with the current work. It is found that the proposed control is quite effective in suppressing the payload swing while tracking the pre-determined path at the same time.

Propagation dynamics of forced pulsating waves of a time periodic Lotka-Volterra competition system in a shifting habitat

Existence, uniqueness and stability of forced pulsating waves as well as gap formation of reaction diffusion models in a shifting habitat are challenging problems in the field of propagation dynamics. In this paper, we design to tackle these problems for a time periodic Lotka-Volterra competition system. By using alternatively-coupling upper-lower solution method, we establish the existence of forced pulsating traveling wave, as long as the shifting speed falls in a finite interval where the end points are obtained from KPP-Fisher speeds. The asymptotic behaviors of the forced pulsating traveling wave are derived and this helps us show that this wave is unique. Moreover in the Lyapunov sense, the stability of the forced pulsating wave is also studied. Finally, when the forced speed is beyond the finite interval, gap formation is investigated theoretically and illustrated further by numerical simulations.

General Equilibrium with Price Adjustments—A Dynamic Programming Approach

This research article develops a dynamic framework for the Walrasian pure exchange economy and thus extends the static Walrasian general equilibrium theory into a dynamic one with price adjustments. An evolution equation for the price vector is derived from dynamic programming considerations. The economy tries to move from disequilibrium to general equilibrium by minimizing certain cost functional. The cost functional measures transactions costs and the total expenditure of agents when they optimize individually. Price determination is directly related to a gradient search. The general equilibrium is shown to be stable in the sense of Lyapunov if price adjustments can be large, when needed. The conditional stability could be one reason for volatility clustering in financial time series.

Matrix-Form Neural Networks for Complex-Variable Basis Pursuit Problem With Application to Sparse Signal Reconstruction.

In this article, a continuous-time complex-valued projection neural network (CCPNN) in a matrix state space is first proposed for a general complex-variable basis pursuit problem. The proposed CCPNN is proved to be stable in the sense of Lyapunov and to be globally convergent to the optimal solution under the condition that the sensing matrix is not row full rank. Furthermore, an improved discrete-time complex projection neural network (IDCPNN) is proposed by discretizing the CCPNN model. The proposed IDCPNN consists of a two-step stop strategy to reduce the calculational cost. The proposed IDCPNN is theoretically guaranteed to be global convergent to the optimal solution. Finally, the proposed IDCPNN is applied to the reconstruction of sparse signals based on compressed sensing. Computed results show that the proposed IDCPNN is superior to related complex-valued neural networks and conventional basis pursuit algorithms in terms of solution quality and computation time.

Photogravitational magnetic-binary problem with oblateness and belt of material points

We study the motion of a charged particle in the framework of magnetic-binary problem where the bigger primary is the source of radiation and the smaller primary is the oblate body; and they are enclosed by a homogeneous circular truss of material points centered at the center of mass of the system. We have determined the equations of motion that govern the motion of a charged particle. The coordinates of collinear and non-collinear equilibrium points and their linear stability have been calculated. Numerical results reveal that the ratio of magnetic moment λ has a huge impact on the location, stability and orbital dynamics of the problem. We observed that there exists eight, eleven and thirteen equilibrium points for different values of mass parameter μ and the ratio of magnetic moment λ. Further, we observed that all non-collinear equilibrium points are unstable in the Lyapunov sense. But the collinear points L4 and L6 show a stable behavior for some values of μ and λ, while other collinear equilibrium points are unstable. The geometric configuration of zero velocity curves of the charged particle is numerically simulated and addressed. Moreover, first order approximations to a Lyapunov and Lissajous orbits are summarized near the collinear equilibrium points under the effect of λ.

Yaw stability control for steer-by-wire vehicle based on radial basis network and terminal sliding mode theory

To further improve the handling and steering performance of steer-by-wire vehicles, a layered control strategy is proposed for the steer-by-wire system, which includes the upper and lower controllers. In the upper control, considering the vehicle sideslip angle and yaw rate, an adaptive sliding mode control strategy is designed for the vehicle lateral stability. Due to the measurement difficulty of the sideslip angle, a robust sliding mode observer is designed to estimate the sideslip angle. In the lower control, a new adaptive global fast terminal sliding mode is proposed to ensure the tracking accuracy of the front wheel angle, considering the dynamic uncertainty characteristics of the steering-by-wire system. The adaptive part of the lower control adopts a radial basis function network, and the adaptive law is designed in Lyapunov sense to make the network approximate the unknown dynamics of the steering-by-wire system, thus the parameters of the steering-by-wire are not required to be completely known. Finally, the joint simulation of MATLAB/Simulink and Carsim verifies that the proposed layered controller has excellent control performance.

Sparse signal reconstruction via recurrent neural networks with hyperbolic tangent function

In this paper, several recurrent neural networks (RNNs) for solving the L1-minimization problem are proposed. First, a one-layer RNN based on the hyperbolic tangent function and the projection matrix is designed. In addition, the stability and global convergence of the previously presented RNN are proved by the Lyapunov method. Then, the sliding mode control technique is introduced into the former RNN to design finite-time RNN (FTRNN). Under the condition that the projection matrix satisfies the Restricted Isometry Property (RIP), a suitable Lyapunov function is constructed to prove that the FTRNN is stable in the Lyapunov sense and has the finite-time convergence property. Finally, we make a comparison of the proposed RNN and FTRNN with the existing RNNs. To achieve this, we implement experiments for sparse signal reconstruction and image reconstruction. The results further demonstrate the effectiveness and superior performance of the proposed RNN and FTRNN.

Robust backstepping sliding mode aircraft attitude and altitude control based on adaptive neural network using symmetric BLF

In this paper, asymptotic control of attitude and altitude of the aircraft is proposed by employing robust backstepping sliding mode control (BSMC) in conjunction with adaptive radial basis function neural network (RBFNN). Accurate knowledge of the non-linear aerodynamic forces and moments, particularly high-fidelity models, is of paramount importance to arrive at such a control strategy under a continuous dynamic environment. Adaptive RBFNN is used to approximate such an unknown non-linear function by continuously updating the network weights in rapidly varying conditions. Further, adaptation laws are used concurrently with neural networks to update the control power derivatives. These adaptive neural networks are used within the architecture of backstepping, integrated with the sliding surfaces where angular rates act as the virtual controller. The postulation of such law requires only minimal information about the aerodynamic model beyond well-known physical features. Moreover, Barrier Lyapunov Function (BLF) candidate is employed to constrain the state of the plant from transgressing a specific limit. Closed-loop signals are theoretically proved to be semi-globally uniformly ultimately bounded in the sense of Lyapunov. Finally, the robustness of the designed flight control law is explored by appending uncertainties and bounded exogenous disturbances in the plant. The results obtained in the present study signify good control performance where output tracks the reference signals by forcing the system states to remain in the designed sliding surfaces.

Guaranteed cost-based feedback control design for fractional-order neutral systems with input-delayed and nonlinear perturbations

Time delay in actuators is mainly caused by electrical and mechanical components. The effect is visible in the system response particularly when changing in the input command. Therefore, input delay is a problem in the control system design that must be taken into account. Besides, ignoring uncertainty in the dynamic models may compromise the controller design. Thus, how to mitigate the effect of this issue on the system stability and performance is a challenging topic. This article deals with the stabilization of fractional neutral systems considering input-delayed and nonlinear perturbations using the guaranteed cost-based feedback control technique. The main focus is to design the state- and output-feedback controllers to achieve a good performance. The stability criteria are formulated in the Lyapunov sense, which are described in terms of matrix inequalities. The proposed idea is validated using simulations.

Event-triggered controller design for LTI systems: A distributed interval observer-based approach

This paper investigates the distributed event-triggered control design problem for networked linear time-invariant (LTI) systems with partially measurable states, in the presence of bounded external disturbances. A distributed interval observer-based event-triggered control scheme has been proposed, the main features of this scheme lie in a) designing a novel distributed interval observer to deal with the unknown states and bound disturbances problem; b) proposing a distributed feedback control method using interval estimation information, the partially measurable limitation has been solved; c) introducing a decreased time-varying function with dead-zone modification to avoid the Zeno behavior. Moreover, sufficient conditions for the stability of closed-loop systems are given in the Lyapunov sense by using matrix inequalities and transmission strategy. Finally, the uniformly ultimately bounded stability and Zeno behavior avoidance of the proposed approach have been numerically demonstrated.