A New Finite Time Convergence Condition for Super‐Twisting Observer Based on Lyapunov Analysis

A new convergence condition is proposed for the super‐twisting sliding mode observer in this paper, where Lyapunov stability analysis is used as the main method to get the new convergence condition. The super‐twisting sliding mode observer is designed to obtain unknown system states of the second order nonlinear system with bounded uncertainties and disturbances. By involving a quadratic Lyapunov function, the Lyapunov approach is applied to the stability analysis of the super‐twisting observer, from which a new convergence condition is obtained to guarantee the finite time convergence of the observer. Simulation results of a pendulum and a rigid manipulator are included to demonstrate the effectiveness of the new convergence condition.     

OBE算法对误差界低估的鲁棒性

研究了具有未知但有界(UBB)误差系统辨识的最优定界椭球(OBE)算法对误差界 低估的鲁棒性.证明了在一定的条件下,即使误差界低估,任何OBE算法都能保持其收敛性. 这一结论可用于具有UBB误差的实际系统参数估计中,以期获得不太保守的结果.     

Approximation and Monotonicity of the Maximal Invariant Ellipsoid for Discrete-Time Systems by Bounded Controls

An analytic approximation of the maximal invariant ellipsoid for a discrete-time linear system with bounded controls is derived. The approximation is expressed explicitly in terms of the coefficient matrices of the system and the positive definite matrix that represents the shape of the invariant ellipsoid. It is shown that this approximation is very close to the exact maximal invariant ellipsoid obtained by solving either an LMI-based optimization problem or a nonlinear algebraic equation. Furthermore, the necessary and sufficient condition for such an approximation to be equal to the exact maximal invariant ellipsoid is established. On the other hand, the monotonicity of the maximal invariant ellipsoid resulting from the “minimal energy control with guaranteed convergence rate” problem is established that shows a trade-off between increasing the size of the invariant ellipsoid and increasing the convergence rate of the closed-loop system under a bounded control. Two illustrative examples demonstrate of the effectiveness of the results.      

On the stability and convergence rate analysis for the nonlinear uncertain systems based upon active disturbance rejection control

Emerging as an effective control method, active disturbance rejection control technique (ADRC) can well deal with disturbances and uncertainties. Then focusing on the general nonlinear uncertain systems, this article illustrates the relation among stability, uncertainties, and parameters of linear ADRC controller when perturbation occurs in control input. On the one hand, if the reference and uncertainties satisfy particular conditions, the estimation and output errors are proved to be ultimately and globally bounded by Lyapunov function and Gronwall‐Bellman inequality, together with rigorous mathematical deducing and simulation verification. On the other hand, a globally and asymptotically stable result is obtained applying Lyapunov method and Cauchy inequality, and the convergence rate can be estimated when uncertainties exist. Besides, numerical simulations are carried out to fully display the correctness and dependability of results and proofs.     

Adaptive fuzzy sliding mode control for electro-hydraulic servo mechanism

An innovative approach to adaptive fuzzy sliding mode control for a class of SISO continuous nonlinear systems with unknown dynamics and bounded disturbances is introduced in this paper. The main idea of the presented method consists in the introduction of the fuzzy self-tuning mechanism for adaptation of the sliding mode control parameters – extended feedback and switching gains. Such modification reduces the well-known chattering problem in classical sliding mode control. In comparison with the other algorithms eliminating this problem the proposed method results in faster convergence and more transparent and interpretable design of self-tuning mechanism. Moreover, the proposed method guaranteing the asymptotic reference signal tracking with bounded system signals can be easily implemented to high order systems. The performance of the presented control design is demonstrated on control of a nonlinear electro-hydraulic servo mechanism.     

Parameter estimation and compensation in systems with nonlinearly parameterized perturbations

We consider a class of systems influenced by perturbations that are nonlinearly parameterized by unknown constant parameters, and develop a method for estimating the unknown parameters. The method applies to systems where the states are available for measurement, and perturbations with the property that an exponentially stable estimate of the unknown parameters can be obtained if the whole perturbation is known. The main contribution is to introduce a conceptually simple, modular design that gives freedom to the designer in accomplishing the main task, which is to construct an update law to asymptotically invert a nonlinear equation. Compensation for the perturbations in the system equations is considered for a class of systems with uniformly globally bounded solutions, for which the origin is uniformly globally asymptotically stable when no perturbations are present. We also consider the case when the parameters can only be estimated when the controlled state is bounded away from the origin, and show that we may still be able to achieve convergence of the controlled state. We illustrate the method through examples, and apply it to the problem of downhole pressure estimation during oil well drilling.     

Adaptive Dual Control of Discrete‐Time Lqg Problems with Unknown‐But‐Bounded Parameter

The dual control for a class of discrete‐time linear‐quadratic Gaussian (LQG) problems is studied in this paper, where there exist unknown‐but‐bounded parameter uncertainties both in the state equation and the observation equation. It is assumed that the unknown parameter belongs to a known bounded interval. To achieve the adaptive dual control, a subdivision for the continuous bounded interval is used, thus, the computational tractability is ensured. Based on such subdivision, the adaptive dual controller developed can learn a more accurate interval that contains the true value of the unknown parameter with a learning error given in advance. The performance of the controller is illustrated with two examples.     

Fault detection and isolation for a wind turbine benchmark using a mixed Bayesian/Set-membership approach

This paper addresses the problem of fault detection and isolation of wind turbines using a mixed Bayesian/Set-membership approach. Modeling errors are assumed to be unknown but bounded, following the set-membership approach. On the other hand, measurement noise is also assumed to be bounded, but following a statistical distribution inside the bounds. To avoid false alarms, the fault detection problem is formulated in a set-membership context. Regarding fault isolation, a new fault isolation scheme that is inspired on the Bayesian fault isolation framework is developed. Faults are isolated by matching the fault detection test results, enhanced by a complementary consistency index that measures the certainty of not being in a fault situation, with the structural information about the faults stored in the theoretical fault signature matrix. The main difference with respect to the classical Bayesian approach is that only models of fault-free behavior are used. Finally, the proposed FDI method is assessed against the wind turbine FDI benchmark proposed in the literature, where a set of realistic fault scenarios in wind turbines are proposed.     

基于有限时间输出反馈的线性扩张状态观测器

为快速、准确地观测系统中的未知扰动及状态, 提出一种有限时间线性扩张状态观测器(Finite-time linear extended state observer, FT-LESO), 它具有期望的收敛性能且结构简单、易于设计. 假设系统的状态无法量测, 观测器设计问题转化为扰动下的输出反馈控制问题. 针对该问题, 提出一种扰动下的有限时间线性输出反馈控制方法, 得到控制器参数与闭环系统状态向量2-范数间的解析关系. 在此基础上, 提出有限时间线性扩张状态观测器, 得到观测器参数与观测误差收敛速度及稳态观测误差间的解析关系, 给出一充分条件保证观测误差有限时间有界、且能以不低于指数收敛的速度收敛到给定范围内, 为观测器参数设计提供理论依据. 通过数值仿真验证提出的观测器, 仿真结果与理论分析相符, 提出的观测器是有效的.     

一种快速收敛的混合遗传算法

利用遗传算法早熟的特点 ,构造出一种快速收敛的混合算法来求解优化问题 ,并分析了它的收敛性。它是使用遗传算法来生成搜索方向 ,从而保证了算法的收敛性。该算法利用遗传算法的全局搜索能力 ,并采用 Nelder- Mead单纯形法来加强算法的局部搜索能力 ,加快了算法的收敛速率。模拟实验表明 ,该方法具有高效性和鲁棒性     

Iterative Learning Control Using Adjoint Systems and Stable Inversion

In this paper, we investigate iterative learning control (ILC) for non‐minimum phase systems from a novel viewpoint. For non‐minimum phase systems, the magnitude of a desiredinput obtained by ILC using forward‐time updating and Silverman's inversion are too large because of the influence of the unstable zeros. On the other hand, stable inversion constructs a bounded desired input by using non‐causal inverse for non‐minimum phase systems. In this paper, we first clarify that ILC using an adjoint system achieves the desired input defined by stable inversion. Hence, ILC using an adjoint system is an effective method for the control of non‐minimum phase systems with uncertainty. However, a useful convergence condition of ILC using an adjoint system was not achieved. Next, we develop a simple convergence condition in the frequency domain.     

Adaptive sliding mode control – convergence and gain boundedness revisited

This paper reviews the main adaptive sliding mode controller (ASMC) designs for nonlinear systems with finite uncertainties of unknown bounds. Different statements of convergence referring to uniformly ultimate boundedness (UUB), asymptotic convergence (AC) and finite-time convergence (FTC) for ASMC shown in recent papers are analysed. Weaknesses and incomplete proofs apropos FTC are pointed out. Thereafter, a new approach is proposed to successfully demonstrate FTC of the so-called sliding variable. We identify a compensating phase and a reaching phase during the ASMC process. A new explicit form for estimating the upper-bound reaching time is provided for any bounded perturbation. An amended form of the real ASMC is recalled showing improved accuracy and chattering reduction. Finally, numerical and experimental applications are performed to convey the discussed results.     

Identification of nonlinear systems in the presence of unknown butbounded disturbances

This paper deals with recursive parameter estimation of nonlinear systems in the presence of unknown but bounded disturbances. The proposed technique is performed by the use of a modified extended least squares algorithm so that the estimated parameters are consistent with the measurements and the noise constraints while local asymptotic convergence, under very general conditions, is assured. One of the main features is to show from the stability analysis, connection between some arbitrary factors and basin of attraction. High performances, from the design and implementation point of view will be shown through a numerical example under severe conditions     

Adaptive fuzzy observer backstepping control for a class of uncertain nonlinear systems with unknown time-delay

In this paper, a new adaptive fuzzy backstepping control approach is developed for a class of nonlinear systems with unknown time-delay and unmeasured states. Using fuzzy logic systems to approximate the unknown nonlinear functions, a fuzzy state observer is designed for estimating the unmeasured states. On the basis of the state observer and applying the backstepping technique, an adaptive fuzzy observer control approach is developed. The main features of the proposed adaptive fuzzy control approach not only guarantees that all the signals of the closed-loop system are semiglobally uniformly ultimately bounded, but also contain less adaptation parameters to be updated on-line. Finally, simulation results are provided to show the effectiveness of the proposed approach.     

Numerical approximation of an elliptic-parabolic equation arising in environment

We prove the convergence of a finite volume scheme for the Richards equation β(p)_t-div(λ(β(p)) (νp-ρg) =0, together with a Dirichlet boundary condition and an initial condition, in a bounded domain. We consider the hydraulic charge u=-z as the main unknown function so that no upwinding is necessary. The convergence proof is based on the strong convergence in L² of the water saturation β(p), which one obtains by estimating differences of space and time translates and applying Kolmogorov’s theorem. Received: 30 April 1999 / Revised version: 17 June 1999     

Switching adaptive learning control for nonlinearly parameterized systems with disturbance of unknown periods

In this paper, a logic‐based switching adaptive learning control mechanism is proposed for a class of nonlinearly parameterized systems with disturbance of unknown periods. The switching algorithms include two parts: one is to stabilize the nonlinearly parameterized uncertainties, the other is to learn the periodic bounded disturbance. An adaptive control method with fully saturated periodic adaptation law is presented to take advantage of the periodic and bounded property of the disturbance. It is shown that under the proposed control designs, the asymptotic convergence is ensured irrespective of initial conditions with all the signals in the closed‐loop system bounded. An illustrative example is given to show the validity of the switching adaptive learning control. Copyright © 2014 John Wiley & Sons, Ltd.     

A result on exponential tracking error convergence and persistentexcitation

Conditions are investigated for exponential convergence of the tracking error in feedforward adaptive systems having insufficient excitation. Particular attention is paid to the adaptive gradient algorithm with periodic excitation in the overparameterized case. A main result is that for a bounded periodic regressor, the tracking error converges exponentially without regard to parameter convergence or to the degree of overparameterization. These results weaken the persistent excitation (PE) conditions and parameter convergence conditions generally considered necessary to ensure exponential tracking convergence in this class of systems     

具有量化输入和边界扰动的柔性臂边界振动滑模控制

本文针对量化输入和有界扰动下柔性臂系统的振动抑制和边界滑模控制器设计问题开展研究. 柔性臂的动态特性由偏微分方程表示的分布参数模型描述. 对于具有未知有界干扰的柔性臂系统, 其主要控制目标是减小干扰的影响, 使柔性臂到达期望角度并同时抑制系统的振动. 首先, 利用边界输出信号构造滑模函数和滑模面. 其次, 结合所构造的滑模面, 设计一种边界滑模控制器, 并利用算子半群理论证明了闭环系统的适定性. 所提出的边界滑模控制策略保证了系统状态能够在有限时间内到达滑模面, 并且系统状态在滑模面上是指数收敛的. 最后, 通过物理实验验证了所提出控制策略的有效性.     

Chebyshev collocation method for solving singular integral equation with cosecant kernel

A collocation method based on Chebyshev polynomials is proposed for solving cosecant-type singular integral equations (SIE). For solving SIE, difficulties lie in its singular term. In order to remove singular term, we introduce Gauss–Legendre integration and integral properties of the cosecant kernel. An advantage of this method is to approximate the best uniform approximation by the best square approximation to obtain the unknown coefficients in the method. On the other hand, the convergence is fast and the accuracy is high, which is verified by the final numerical experiments compared with the existing references.     

A convex optimization-based nonlinear filtering algorithm with applications to real-time sensing for patterned wafers

The paper is concerned with nonlinear filtering under unknown dynamics and high-complexity observations. We propose a convex optimization-based filtering algorithm, and show that the algorithm yields a bounded error if the disturbances are small and bounded, and if the observations are redundant. An experimental result is presented to demonstrate that the algorithm is capable of accurate real-time estimation of patterned wafer parameters in a plasma etching process with optical observation.