Robust neural network tracking controller using simultaneous perturbation stochastic approximation

This paper considers the design of robust neural network tracking controllers for nonlinear systems. The neural network is used in the closed-loop system to estimate the nonlinear system function. We introduce the conic sector theory to establish a robust neural control system, with guaranteed boundedness for both the input/output (I/O) signals and the weights of the neural network. The neural network is trained by the simultaneous perturbation stochastic approximation (SPSA) method instead of the standard backpropagation (BP) algorithm. The proposed neural control system guarantees closed-loop stability of the estimation system, and a good tracking performance. The performance improvement of the proposed system over existing systems can be quantified in terms of preventing weight shifts, fast convergence, and robustness against system disturbance.     

Adaptive stochastic approximation by the simultaneous perturbation method

Stochastic approximation (SA) has long been applied for problems of minimizing loss functions or root finding with noisy input information. As with all stochastic search algorithms, there are adjustable algorithm coefficients that must be specified, and that can have a profound effect on algorithm performance. It is known that choosing these coefficients according to an SA analog of the deterministic Newton-Raphson algorithm provides an optimal or near-optimal form of the algorithm. However, directly determining the required Hessian matrix (or Jacobian matrix for root finding) to achieve this algorithm form has often been difficult or impossible in practice. The paper presents a general adaptive SA algorithm that is based on a simple method for estimating the Hessian matrix, while concurrently estimating the primary parameters of interest. The approach applies in both the gradient-free optimization (Kiefer-Wolfowitz) and root-finding/stochastic gradient-based (Robbins-Monro) settings, and is based on the "simultaneous perturbation (SP)" idea introduced previously. The algorithm requires only a small number of loss function or gradient measurements per iteration-independent of the problem dimension-to adaptively estimate the Hessian and parameters of primary interest. Aside from introducing the adaptive SP approach, the paper presents practical implementation guidance, asymptotic theory, and a nontrivial numerical evaluation. Also included is a discussion and numerical analysis comparing the adaptive SP approach with the iterate-averaging approach to accelerated SA.     

Convergence of simultaneous perturbation stochastic approximation for nondifferentiable optimization

We consider simultaneous perturbation stochastic approximation for function minimization. The standard assumption for convergence is that the function be three times differentiable, although weaker assumptions have been used for special cases. However, all work that we are aware of at least requires differentiability. We relax the differentiability requirement and prove convergence using convex analysis.     

Vibration control of a two-link flexible robot arm

Analysis and experimentation is described for a two-link apparatus in which both members are very flexible. Attention is focused on endpoint position control for point-to-point movements, assuming a fixed reference frame for the base, with two rotary joints. Each link is instrumented with acceleration sensing and is driven by a separate motor equipped with velocity and position sensing. The control perspective adopted is to implement a two-stage control strategy in which the vibration control problem for fine-motion endpoint positioning is considered separate from the gross-motion, large-angle slew problem. In the first stage the control law shapes the actuator inputsfor the large-angle movement in such a way that minimal energy is injected into the flexible modes, while in the second phase an endpoint acceleration feedback scheme is employed in independent joint controlfor vibration suppression at the link endpoints.     

Identification of continuous-time Hammerstein systems by simultaneous perturbation stochastic approximation

This paper proposes an identification method for Hammerstein systems using simultaneous perturbation stochastic approximation (SPSA). Here, the structure of nonlinear subsystem is assumed to be unknown, while the structure of linear subsystem, such as the system order, is assumed to be available. The main advantage of the SPSA-based method is that it can be applied to identification of Hammerstein systems with less restrictive assumptions. In order to clarify this point, piecewise affine functions with a large number of parameters are adopted to approximate the unknown nonlinear subsystems. Furthermore, the linear subsystems are supposed to be described in continuous-time. Though this class of systems closely reflects the actual systems, there are few methods to identify such models. Hence, the SPSA-based method is utilized to identify the parameters in both linear and nonlinear subsystems simultaneously. The effectiveness of the proposed method is evaluated through several numerical examples. The results demonstrate that the proposed algorithm is useful to obtain accurate models, even for high-dimensional parameter identification.     

Vibration suppression control of 3D flexible robots using velocity inputs

Research on vibration suppression control of flexible robots has concentrated mainly on the one-link and two-link planar manipulators. Most of the techniques that have been presented cannot be easily extended to the case of a general 3D flexible robot. In this article we present a general control scheme based on hardware velocity servo cards. The velocity commands to move the robot are calculated by adding a vibration suppression term to the joint position feedback employed in “rigid” robots. Two different methods are proposed to calculate this term, one based on optimum quadratic control and the other based on pseudo-inverse nonlinear decoupling. These techniques are studied numerically in the case of a real two-link three-joint flexible robot, by computing the values of the closed-loop poles at different configurations. Experiments on position stabilization of the robot prove the validity of our methods.© 1997 John Wiley & Sons, Inc.     

Fluctuations of a stochastic approximation procedure with diffusion perturbation

A stochastic approximation procedure with a singularly perturbed regression function is considered. The form of the limit process is obtained that corresponds to the fluctuation of the stochastic approximation procedure in the neighborhood of an equilibrium point. A generator for the limit process is also constructed. The solution of the singular perturbation problem is given for the asymptotic representation of the generator of a Markov renewal process. The results obtained allow one to extend the possibilities of investigation of the asymptotic behavior of the procedure itself.     

Experimental two-axis vibration suppression and control of a flexible robot arm

This article focuses on the implementation of a dual-mode controller for the maneuver of a two-axis flexible robotic arm. The joint angle trajectory tracking is accomplished by proportional and derivative and feedforward controllers. Based on the pole placement technique, a linear stabilizer is designed for elastic mode stabilization in the plane perpendicular to each joint axis. The stabilizer is switched on when the trajectory reaches the vicinity of the terminal state. The effect of switching time of the stabilizer and varying payload on arm vibration are investigated. With the proposed control system, accurate joint angle tracking and elastic mode stabilization can be accomplished. © 1993 John Wiley & Sons, Inc.     

Optimal random perturbations for stochastic approximation using asimultaneous perturbation gradient approximation

The simultaneous perturbation stochastic approximation (SPSA) algorithm has attracted considerable attention for challenging optimization problems where it is difficult or impossible to obtain a direct gradient of the objective (say, loss) function. The approach is based on a highly efficient simultaneous perturbation approximation to the gradient based on loss function measurements. SPSA is based on picking a simultaneous perturbation (random) vector in a Monte Carlo fashion as part of generating the approximation to the gradient. This paper derives the optimal distribution for the Monte Carlo process. The objective is to minimize the mean square error of the estimate. The authors also consider maximization of the likelihood that the estimate be confined within a bounded symmetric region of the true parameter. The optimal distribution for the components of the simultaneous perturbation vector is found to be a symmetric Bernoulli in both cases. The authors end the paper with a numerical study related to the area of experiment design     

Magnetic sensor-based simultaneous state and parameter estimation using a nonlinear observer

This paper presents an observer design technique for a newly developed non-intrusive position estimation system based on magnetic sensors. Typically, the magnetic field of an object as a function of position needs to be represented by a highly nonlinear measurement equation. Previous results on observer design for nonlinear systems have mostly assumed that the measurement equation is linear, even if the process dynamics are nonlinear. Hence, a new nonlinear observer design method for a Wiener system composed of a linear process model together with a nonlinear measurement equation is developed in this paper. First, the design of a two degree-of-freedom nonlinear observer is proposed that relies on a Lure system representation of the observer error dynamics. To improve the performance in the presence of parametric uncertainty in the measurement model, the nonlinear observer is augmented to estimate both the state and unknown parameters simultaneously. A rigorous nonlinear observability analysis is also presented to show that a dual sensor configuration is a sufficient and necessary condition for simultaneous state and parameter estimation. Finally, the developed observer design technique is applied to non-intrusive position estimation of the piston inside a pneumatic cylinder. Experimental results show that both position and unknown parameters can be reliably estimated in this application.     

Infinite dimensional observer for a flexible robot arm with a tip load

This note addresses observer design for a flexible robot arm with a tip load. The robot arm is modeled as an Euler–Bernoulli beam. The beam is clamped to a motor at one end and attached to a force actuator at the other. Based on boundary measurements, an exponentially stable observer is proposed. The existence, uniqueness, and stability of solutions of the observer are proven using semigroup theory. The results are illustrated by simulation. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A robust recurrent simultaneous perturbation stochastic approximation training algorithm for recurrent neural networks

Training of recurrent neural networks (RNNs) introduces considerable computational complexities due to the need for gradient evaluations. How to get fast convergence speed and low computational complexity remains a challenging and open topic. Besides, the transient response of learning process of RNNs is a critical issue, especially for online applications. Conventional RNN training algorithms such as the backpropagation through time and real-time recurrent learning have not adequately satisfied these requirements because they often suffer from slow convergence speed. If a large learning rate is chosen to improve performance, the training process may become unstable in terms of weight divergence. In this paper, a novel training algorithm of RNN, named robust recurrent simultaneous perturbation stochastic approximation (RRSPSA), is developed with a specially designed recurrent hybrid adaptive parameter and adaptive learning rates. RRSPSA is a powerful novel twin-engine simultaneous perturbation stochastic approximation (SPSA) type of RNN training algorithm. It utilizes three specially designed adaptive parameters to maximize training speed for a recurrent training signal while exhibiting certain weight convergence properties with only two objective function measurements as the original SPSA algorithm. The RRSPSA is proved with guaranteed weight convergence and system stability in the sense of Lyapunov function. Computer simulations were carried out to demonstrate applicability of the theoretical results.     

Vibration control for a nonlinear three-dimensional flexible manipulator trajectory tracking

This paper proposes an innovative approach to the trajectory tracking in three-dimensional space and vibration control problems in the presence of a nonlinear three-dimensional flexible manipulator based on the partial differential equation model. Unlike two-dimensional plane, we select spherical coordinates to describe the position of the end point in three-dimensional space. This novel approach makes it possible to realise the trajectory tracking by controlling the two angles in spherical coordinates, meanwhile, a vibration control scheme is proposed to restrain vibrations. In addition, the existence and uniqueness of solutions are demonstrated. Finally, the performance of the desired trajectory tracking, the proposed vibration control scheme and their convergence properties are demonstrated by numerical simulations.     

Convergence of stochastic approximation coupled with perturbation analysis in a class of manufacturing flow control models

This paper deals with a class of piecewise determinstic control systems for which the optimal control can be approximated through the use of an optimization-by-simulation approach. The feedback control law is restricted to belong to an a priori fixed class of feedback control laws depending on a (small) finite set of parameters. Under some general conditions developed in this paper, infinitesimal perturbation analysis (IPA) can be used to estimate the gradient of the objective function with respect to these parameters for finite horizon simulation and the consistency of the IPA estimators, as the simulation length goes to infinity, is assured. Also, the parameters can be optimized through a stochastic approximation (SA) algorithm combined with IPA. We prove that in this context, under appropriate conditions, such an approach converges towards the optimum.     

Minimal-order observer and output-feedback stabilization control design of stochastic nonlinear systems

1IntroductionTheproblemofestimatingtheparametersofmultiplesinusoidsinnoisehasre-ceivedconsiderableattentioninthepastthirtyyears,andalotofalgorithmshavebeenestablishedtosolvetheproblem.Amongallofthealgorithms,themaximumlikelihood(ML)estimatorisaprominentone[1],andseveralalgorithms,suchasANP[2]andIMP[3,4]arerelatedtoML.ThedrawbackoftheMLestimatorisitshighcomputationalcomplexity,sothealternatingprojection(AP)algorithm[5]wasdevelopedtomakeitsrealtimerealizationpossible.Butthefundamentaldefic…     

Performance analysis of model-free PID tuning of MIMO systems based on simultaneous perturbation stochastic approximation

This paper addresses the performance comparison of simultaneous perturbation stochastic approximation (SPSA) based methods for PID tuning of MIMO systems. Four typical SPSA based methods, which are one-measurement SPSA (1SPSA), two-measurement SPSA (2SPSA), Global SPSA (GSPSA) and Adaptive SPSA (ASPSA) are examined. Their performances are evaluated by extensive simulation for several controller design examples, in terms of the stability of the closed-loop system, tracking performance and computation time. In addition, the performance of the SPSA based methods are compared to the other stochastic optimization based approaches. It turns out that the GSPSA based algorithm is the most practical in terms of the stability and the tracking performance.     

基于非线性能量阱的深海柔性张力腿的振动抑制

本文研究了基于非线性能量阱的深海柔性张力腿的振动抑制问题.考虑端部参数激励和非线性能量阱(NES)作用下的张力腿力学模型,采用哈密顿变分原理推导出非线性振动控制的运动微分方程,利用伽辽金法进行离散化.通过参数分析和数值仿真计算,得到柔性张力腿的横向位移模态振动响应,同时还对比分析了NES与调谐质量阻尼器的减振性能.结果表明,相同情况下NES具有更为显著的减振效果,并且可以通过调整NES吸振器的参数,达到最优振动控制效果.     

Experimental control of a single-link flexible robot arm using energy shaping

Flexible-link robotic manipulators are mechanical devices whose control can be rather challenging, among other reasons because of their intrinsic under-actuated nature. This paper presents the application of an energy-based control design methodology (the so-called IDA-PBC, interconnection and damping assignment passivity-based control) to a single-link flexible robotic arm. It is shown that the method is well suited to handle this kind of under-actuated device not only from a theoretical viewpoint but also in practice. A Lyapunov analysis of the closed-loop system stability is given and the design performance is illustrated by means of a set of simulations and laboratory control experiments, comparing the results with those obtained using conventional control schemes for mechanical manipulators.     

Multivariate stochastic approximation using a simultaneousperturbation gradient approximation

The problem of finding a root of the multivariate gradient equation that arises in function minimization is considered. When only noisy measurements of the function are available, a stochastic approximation (SA) algorithm for the general Kiefer-Wolfowitz type is appropriate for estimating the root. The paper presents an SA algorithm that is based on a simultaneous perturbation gradient approximation instead of the standard finite-difference approximation of Keifer-Wolfowitz type procedures. Theory and numerical experience indicate that the algorithm can be significantly more efficient than the standard algorithms in large-dimensional problems