Chaos suppression on a class of uncertain nonlinear chaotic systems using an optimal H∞ adaptive PID controller

This paper introduces an optimal H_∞ adaptive PID (OHAPID) control scheme for a class of nonlinear chaotic system in the presence system uncertainties and external disturbances. Based on Lyapunov stability theory, it is shown that the proposed control scheme can guarantee the stability robustness of closed-loop system with H_∞ tracking performance. In the core of proposed controller, to achieve an optimal performance of OHAPID, the Particle Swarm Optimization (PSO) algorithm is utilized. To show the feasibility of proposed OHAPID controller, it is applied on the chaotic gyro system. Simulation results demonstrate that it has highly effective in providing an optimal performance.     

基于极大极小方法的一类非线性系统的自适应控制

研究一类具有非线性不确定参数的非线性系统的自适应模型参考跟踪问题.假设系统的非线性项关于不确定参数是凸或凹的.去掉了在先前有关研究中要求参考模型矩阵有小于零的实特征值的条件.既考虑了状态反馈控制方式,也考虑了输出反馈控制方式.在采用输出反馈控制时,假设非线性项满足李普希兹条件,但李普希兹常数未知.基于一种极大极小方法,提出了一种自适应控制器的设计方法.控制器是连续的,能保证闭环系统的所有变量有界,并且渐近精确跟踪参考模型.举例说明了本结论的有用性.     

Factorization in Hurwitz series domain

Let A be a commutative ring with unit and HA the set of formal expressions of the type \(f=\sum_{i:0}^{\infty}a_{i}X^{i}\) where a _i ∈A. When \(g=\sum_{i:0}^{\infty}b_{i}X^{i}\) then \(f+g=\sum_{i:0}^{\infty}(a_{i}+b_{i})X^{i}\) and \(f*g=\sum_{n:0}^{\infty}c_{n}X^{n}\) with \(c_{n}=\sum_{i:0}^{n}C_{n}^{i}a_{i}b_{n-i}\), where \(C_{n}^{i}={n!\over i!(n-i)!}\). With these two operations HA is a commutative ring with identity. It was introduced and studied by Keigher in 1997. In this note we continue the investigation and we focus on factorization in HA and its sub-ring hA of Hurwitz polynomials. We recall from Benhissi (Contrib. Algebra. Geom. 48(1):251–256, 2007, Proposition 1.1) and Keigher (Commun. Algebra 25(6):1845–1859, 1997, Corollary 2.8) that HA is an integral domain if and only if A is an integral domain with zero characteristic. Let π ₀:HA?A be the natural ring homomorphism that assigns to each series its constant term. The key property is that a series f∈HA is a unit in HA if and only if π ₀(f) is a unit in A, Keigher (Commun. Algebra 25(6):1845–1859, 1997, Proposition 2.5).     

Design of robust sub-optimal nonlinear H∞ controllers with some applications

The problem of nonlinear sub-optimal H_∞ controller design with some applications is addressed in the paper. Nonlinear H_∞ control has robust performance in response to external disturbances and parameter uncertainty as well as capability in dealing with nonlinear systems. In order to obtain the nonlinear H_∞ control law, some partial differential inequalities so-called Hamilton–Jacobi–Isaacs (HJI) should be solved. There are some approximate solutions, which are generally based on the approximation of nonlinear parts of HJI inequalities. Using the Taylor series expansion, a sub-optimal solution for the HJI inequalities will be obtained. To assess the performance of the method, two applications are considered: the tracking problem of a two-degree-of-freedom rigid robot manipulator and speed control in a permanent magnet synchronous (PMS) motors. Simulation results show superior performance for higher order approximate controllers than that of lower order ones.     

A model of parameter adaptive law with time varying function for robot control

In this article, a model of adaptive control law for controlling robot manipulators using the Lyapunov based theory of guaranteed the stability of uncertain a system is derived. The novelty of obtained result is that the adaptive control algorithm is developed using a parameter estimation rule depending on manipulator kinematic, dynamic parameters and tracking error. This study is supported by a computer simulation and tracking performance has been improved.     

On the matrix equations AH + HA1 = A1H + HA = I

This paper considers the simultaneous solution of the matrix equations AH + HA¹ = A¹H + HA = I, where H is Hermitian. A full characterization of A in terms of a given H is obtained. Various results are also obtained for those H satisfying the equations with a given A.     

Novel design of adaptive neural network controller for a class of non-affine nonlinear systems

The tracking control problem is studied for a class of uncertain non-affine systems. Based on the principle of sliding mode control (SMC), using the neural networks (NNs) and the property of the basis function, a novel adaptive design scheme is proposed. A novel Lyapunov function, which depends on both system states and control input variable, is used for the development of the control law and the adaptive law. The approach overcomes the drawback in the literature. In addition, the lumped disturbances are taken in account. By theoretical analysis, it is proved that tracking errors asymptotically converge to zero. Finally, simulation results demonstrate the effectiveness of the proposed approach.     

基于模糊与神经网络技术的复杂非线性跟踪控制

针对一类具有不确定性、多重时延和状态未知的复杂非线性系统,把模糊T-S模型和RBF神经网络结合起来,提出了一种基于观测器的跟踪控制方案.首先,应用模糊T-S模型对非线性系统建模,设计观测器用来观测系统状态,并由线性矩阵不等式得到模糊模型的控制律;其次,构建了自适应RBF神经网络,应用自适应RBF神经网络作为补偿器来补偿建模误差和不确定非线性部分.证明了闭环系统满足期望的跟踪性能.示例仿真结果表明了该方案的有效性.     

Observer-based robust adaptive variable universe fuzzy control for chaotic system

A novel observer-base output feedback variable universe adaptive fuzzy controller is investigated in this paper. The contraction and expansion factor of variable universe fuzzy controller is on-line tuned and the accuracy of the system is improved. With the state-observer, a novel type of adaptive output feedback control is realized. A supervisory controller is used to force the states to be within the constraint sets. In order to attenuate the effect of both external disturbance and variable parameters on the tracking error and guarantee the states to be within the constraint sets, a robust controller is appended to the variable universe fuzzy controller. Thus, the robustness of system is improved. By Lyapunov method, the observer-controller system is shown to be stable. The overall adaptive control algorithm can guarantee the global stability of the resulting closed-loop system in the sense that all signals involved are uniformly bounded. In the paper, we apply the proposed control algorithms to control the Duffing chaotic system and Chua’s chaotic circuit. Simulation results confirm that the control algorithm is feasible for practical application.     

Generalizations of theorems of Lyapunov and Stein

The matrix equation $\text{f}{}_{\mathrm{H}}\text{(A)=∑C}{}_{\mathrm{ij}}\text{A}{}^{\mathrm{1i}}\text{HA}{}^{\mathrm{j}}\text{=W, H >0}$, W ?0, is studied. In the case A¹H+HA = W[H?A¹HA = W], the controllability matrix of (A¹,W) is used to determine the number of eigenvalues of A on the imaginary axis [on the unit circle]. As an application a result of Pták on the critical exponent of the spectral norm is obtained. Estimates for the eigenvalues of A satisfying f_H(A) = M are given.     

Inertia theorems for matrices,controllability, and linear vibrations

If H is a Hermitian matrix and $\text{W = AH + HA}{}^1$ is positive definite, then A has as many eigenvalues with positive (negative) real part as H has positive (negative) eigenvalues [5]. Theorems of this type are known as inertia theorems. In this note the rank of the controllability matrix of A and W is used to derive a new inertia theorem. As an application, a result in [8] and [4] on a damping problem of the equation $\text{M}\text{x}\text{?}\text{+ (D + G) xdot; + Kx = 0}$ is extended.     

Comment on “Based on interval type-2 adaptive fuzzy H∞ tracking controller for SISO time-delay nonlinear systems”

In this comment, we point out the inappropriateness of Theorem 1 in the article [Tsung-Chih Lin, Mehdi Roopaei. Based on interval type-2 adaptive fuzzy H^∞ tracking controller for SISO time-delay nonlinear systems. Commun Nonlinear Sci Numer Simulat 2010;15:4065–75]. For solving this problem, some formular mistakes are corrected and novel parameter adaptive laws of interval type-2 fuzzy neural network system are given.     

Decentralized Adaptive Robust Tracking and Model Following for Large-Scale Systems Including Delayed State Perturbations in the Interconnections

The problem of decentralized robust tracking and model following is considered for a class of uncertain large-scale systems including delayed state perturbations in the interconnections. In this paper, it is assumed that the upper bounds of the delayed state perturbations, uncertainties, and external disturbances are unknown. A modified adaptation law with σ-modification is introduced to estimate such unknown bounds, and on the basis of the updated values of these unknown bounds, a class of decentralized local memoryless state feedback controllers is constructed for robust tracking of dynamical signals. The proposed decentralized adaptive robust tracking controllers can guarantee that the tracking errors between each time-delay subsystem and the corresponding local reference model without time-delay decrease uniformly asymptotically to zero. Finally, a numerical example is given to demonstrate the validity of the results.     

ℋ-theories, fragments of HA and PA-normality

For a classical theory T, ℋ(T) denotes the intuitionistic theory of T-normal (i.e. locally T) Kripke structures. S. Buss has asked for a characterization of the theories in the range of ℋ and raised the particular question of whether HA is an ℋ-theory. We show that T ⁱ ∈ range(ℋ) iff T ⁱ = ℋ(T). As a corollary, no fragment of HA extending iΠ ₁ belongs to the range of ℋ. A. Visser has already proved that HA is not in the range of H by different methods. We provide more examples of theories not in the range of ℋ. We show PA-normality of once-branching Kripke models of HA + MP, where it is not known whether the same holds if MP is dropped. Received: 15 August 1999 / Published online: 3 October 2001     

不确定非线性系统的周期信号自适应跟踪

考虑不确定非线性系统的周期信号的自适应跟踪问题. 系统的不确定性不能参数化,周期信号由一非线性系统产生.提出了跟踪周期信号的自适应控制律. 此控制律保证了闭环系统所有的信号有界和跟踪误差趋于零. 已有的有关的周期信号跟踪控制律只能保证跟踪误差的平方在一周期上的积分趋于零.     

Development of a new method for online parameter identification in seismically excited smart building structures using virtual synchronization and adaptive control design

In this paper a new method for online parameter identification and damage detection in smart building structures that are subjected to arbitrary seismic excitation is proposed. It uses real-time measurements of a structure's motion to identify its unknown constant or piecewise constant parameters such as stiffness, damping and mass over the time. The method is based on elements of system synchronization and adaptive control theories. First, a computational system, called the virtual system, is defined. Next, by using properly designed controller and estimations for the unknown parameters, the state of the virtual system is forced to follow the measured motion of the real structure. The mentioned estimations are computed from a proposed update law which depends on the measured motion of the real structure and the virtual system’s state. A major theoretical novelty of this paper is a proposed convergence condition which is applicable in case of arbitrary external forces or ground acceleration. It is shown that upon the satisfaction of that condition, as the synchronization completes, the computed estimation function converges to the true value of the vector of unknown parameters. In addition, an important practical contribution presented in this study is the introduction of a technique called scale factors. It helps to use available initial guesses of the unknown parameters to improve the speed of online identification. Numerical examples show that the proposed method is promising and has a good performance in both online identification and online damage detection problems.     

Uniformly Hyperarchimedean Lattice-Ordered Groups

An abelian ?-group with strong unit (\(\user1{\mathcal{L}}_1 \)-object) G is hyperarchimedean (HA) iff G?≤?C(YG) (the ?-group of real continuous functions on the maximal ideal space, YG) with λ(g)?=?inf{?∣?g(x)?∣?≠?0}?>?0 for each 0?≠?g?∈?G. In case inf{λ(g):0?≠?g?∈?G}?>?0, we call G uniformly hyperarchimedean (UHA). This paper: examines the structure of the UHA groups in detail; shows that UHA solves the problem: when is an \(\user1{\mathcal{L}}_1 \)-product HA?; describes completely the \(\user1{\mathcal{L}}_1 \)???HSP-classes which are contained in HA. Final remarks detail the connection with MV-algebras.     

Adaptive PID-tracking control of muscle-like actuated compliant robotic systems with input constraints

In this paper, adaptive control strategies are analyzed to control a compliant robot arm in form of a double pendulum structure. Powered by antagonistic, muscle-like actuators, the controller shall enable the system to track prescribed trajectories for a desired movement of the arm, as quickly and precisely as possible. But, due to the natural muscle behavior, the control variables underly prescribed bounds: the limited control inputs are generated by muscle intensities. Therefore, feedback strategies are sought which have to be limited/saturated from the very beginning. For this, at first, simulation studies of adaptive λ-tracking controllers are performed to apply the best-rated ones to the compliant robotic system. After some model adjustments concerning the mechanics of a prototype (for experimental verification), an intensity control is determined, to divide the controller variables to the respective muscle pairs. Various numerical simulations of a system with DoF=2 show the effectiveness of the saturated controllers in set-point and tracking control. The efficiency of these controllers is verified and proven in experiments of a prototype.     

具有类反斜线回滞执行器的非线性系统的自适应控制

针对类反斜线回滞非线性系统,设计了自适应控制器,保持系统稳定并实现对参考信号的任意精度跟踪.在控制器设计中,通过构造足够光滑的非线性函数作为符号函数的逼近,从而解决了传统自适应控制器设计中由于符号函数的引入而导致控制器不连续的问题,避免了因此而产生的抖震现象.在系统模型中充分考虑未建模动态,使模型更具一般性.最后应用MATLAB软件进行仿真实验,结果进一步验证了该控制器的有效性.