Asymptotic linearization and stabilization for a class of nonlinear systems

This paper presents a nonlinear output feedback which asymptotically linearizes the class of nonlinear, continuous-time, affine in the control systems having relative degree equal to the state space dimension. Moreover, we show that any set of eigenvalues can be assigned for the asymptotic closed-loop linear system. The controller is based on a nonlinear observer, presented in a previous paper, and on the linearizing state feedback proposed by Isidori and computed in the estimated state. The main result obtained is equivalent to the separation theorem in the linear case.This work was supported by the Italian Ministry for University and Scientific and Technological Research.     

Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology

We associate to each infinite primitive Lie pseudogroup a Hopf algebra of ‘transverse symmetries,’ by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a ‘quantum group’ counterpart of the infinite-dimensional primitive Lie algebra of the pseudogroup. It is first constructed via its action on the étale groupoid associated to the pseudogroup, and then realized as a bicrossed product of a universal enveloping algebra by a Hopf algebra of regular functions on a formal group. The bicrossed product structure allows to express its Hopf cyclic cohomology in terms of a bicocyclic bicomplex analogous to the Chevalley-Eilenberg complex. As an application, we compute the relative Hopf cyclic cohomology modulo the linear isotropy for the Hopf algebra of the general pseudogroup, and find explicit cocycle representatives for the universal Chern classes in Hopf cyclic cohomology. As another application, we determine all Hopf cyclic cohomology groups for the Hopf algebra associated to the pseudogroup of local diffeomorphisms of the line.     

正则型控制系统的全局反馈镇定

在对非线性控制系统全局镇定的研究中 ,Byrness,Isidori讨论了光滑非线性系统的光滑反馈与全局正则型的等价条件 ;Kokotovic,Sussmann则在讨论了全局镇定的正实条件后 ,得到了一个判定系统为全局光滑可镇定的重要条件 .本文则考察一类正则型控制系统 ,通过变换系统和构造全局反馈镇定律的方法 ,得到全局光滑镇定     

The explicit synthesis of stabilizing (time optimal) feedback controls for the attitude control of a rotating satelite

The attitude stabilization problem for a spinning satellite controlled by two small jets may be modelled as a four-dimensional, nonlinear control system, linear in the controls. The recent feedback linearization theorem of Hunt and Su may be applied to transform this system, via state feedback and a local coordinate change, to a pair of uncoupled, two-dimensional, linear systems. Feedback controls for the problem of time optimal transfer to the origin for these linear systems are explicitly calculated and then transformed to give explicit feedback controls for time optimal stabilization in the original nonlinear problem. The theory is illustrated by sample calculations.     

基于输出反馈的一类大型互联非线性系统的鲁棒全局指数稳定

研究基于输出反馈的一类新的大型互联非线性不确定系统的鲁棒全局指数稳定问题,通过构造每个子系统收敛的状态观测器,并对观测器的状态作线性变换,得到鲁棒分散输出反馈控制器.当该反馈控制律作用于该系统时,闭环系统是全局指数稳定的.     

基于输出反馈的一类大型互联Holder非线性系统的全局实际镇定

本文研究基于输出反馈的一类大型互联Holder连续非线性系统的全局实际镇定问题.通过构造每个子系统的状态观测器,并对观测器的状态作线性变换,得到分散输出反馈控制器.当输出反馈控制律作用于该系统时,闭环系统是全局实际稳定的.     

On control and synchronization in chaotic and hyperchaotic systems via linear feedback control

This paper presents the control and synchronization of chaos by designing linear feedback controllers. The linear feedback control problem for nonlinear systems has been formulated under optimal control theory viewpoint. Asymptotic stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution of the Hamilton–Jacobi–Bellman equation thus guaranteeing both stability and optimality. The formulated theorem expresses explicitly the form of minimized functional and gives the sufficient conditions that allow using the linear feedback control for nonlinear system. The numerical simulations were provided in order to show the effectiveness of this method for the control of the chaotic Rössler system and synchronization of the hyperchaotic Rössler system.     

基于虚拟完整约束的欠驱动起重机控制方法

欠驱动系统的控制是非线性控制的一个重要领域,欠驱动系统指系统控制输入个数小于自由度个数的非线性系统.目前,欠驱动非线性系统动力学和控制研究的主要方法包括线性二次型最优控制方法和部分反馈线性化方法等,如何使系统持续的稳定在平衡位置一直是研究的难点.虚拟约束方法是指通过选择一个周期循环变化的变量作为自变量来设计系统的周期运动.该文以典型的欠驱动模型起重机为例,采用虚拟约束方法,使系统能够在平衡位置稳定或周期振荡运动.首先,通过建立虚拟约束,减少系统自由度变量;然后,通过部分反馈线性化理论推导出系统的状态方程;最后,通过线性二次调节器设计反馈控制器.仿真结果表明,重物在反馈控制下可以在竖直位置的附近达到稳定状态,反映了虚拟约束方法对欠驱动系统的有效性.     

Approximately linear tracking control of nonlinear systems

An elegant approach to control nonlinear state-space systems is the exact input-to-state linearization, where a nonlinear change of coordinates combined with a nonlinear feedback law yields a linear controllable system. In this contribution, we treat the single-input case, where input-to-state linearizability is equivalent to flatness. Sufficient and necessary existence conditions are well-known, but quite restrictive. We propose the design of a tracking controller for flat single-input systems, where the explicit knowledge of the flat output is not required. Our approach is based on a series expansion of the tracking error along a given reference trajectory. The controller gain can even be computed for non-flat systems with regular controllability matrix. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear feedback controllers and compensators: a state-dependent Riccati equation approach

State-dependent Riccati equation (SDRE) techniques are rapidly emerging as general design and synthesis methods of nonlinear feedback controllers and estimators for a broad class of nonlinear regulator problems. In essence, the SDRE approach involves mimicking standard linear quadratic regulator (LQR) formulation for linear systems. In particular, the technique consists of using direct parameterization to bring the nonlinear system to a linear structure having state-dependent coefficient matrices. Theoretical advances have been made regarding the nonlinear regulator problem and the asymptotic stability properties of the system with full state feedback. However, there have not been any attempts at the theory regarding the asymptotic convergence of the estimator and the compensated system. This paper addresses these two issues as well as discussing numerical methods for approximating the solution to the SDRE. The Taylor series numerical methods works only for a certain class of systems, namely with constant control coefficient matrices, and only in small regions. The interpolation numerical method can be applied globally to a much larger class of systems. Examples will be provided to illustrate the effectiveness and potential of the SDRE technique for the design of nonlinear compensator-based feedback controllers.     

Realistic feedback control of two interconnected turbogenerators

In this paper, the optimal control of a system with two identical interconnected turbogenerators, which are connected to an infinite bus, is considered. The alternators are controlled through a linear feedback of the state variables. The feedback parameters are obtained by solving a nonlinear, two-point boundary-value problem. The values obtained for these parameters depend on the strength and duration of the disturbance, since the model is nonlinear, contrary to the usual feedback control of a linear model. In contrast to the model used in Ref. 1, the model used here includes the transfer function of the governors, the turbines, and the voltage regulators.This work was supported in part by the National Science and Engineering Research Council of Canada, Grant No. A4146. The authors wish to express their appreciation to Mr. T. L. Gan for his help in computations.     

A Probability Approach to the Method of Vanishing Viscosity

We clarify conditions under which solutions to the Cauchy problem for a general (fully nondiagonal) system of linear and nonlinear parabolic equations admit probability representations. Such representations are also used for constructing and studying solutions to the Cauchy problem for nonlinear hyperbolic systems. Bibliography: 26 titles.     

New Approach to Linear Exact Model Matching for a Class of Nonlinear Systems

In this paper, a new approach to the linear exact model matching problem for a class of nonlinear systems, using static state feedback, is presented. This approach reduces the problem of determining the state feedback control law to that of solving a system of first-order partial differential equations. Based on these equations, two major issues are resolved: the necessary and sufficient conditions for the problem to have a solution and the general analytical expression for the feedback control law. Furthermore, the proposed approach is extended to solve the same problem via static output feedback.     

Exponential stabilization by delay feedback control for highly nonlinear hybrid stochastic functional differential equations with infinite delay

Given an unstable hybrid stochastic functional differential equation, how to design a delay feedback controller to make it stable? Some results have been obtained for hybrid systems with finite delay. However, the state of many stochastic differential equations are related to the whole history of the system, so it is necessary to discuss the feedback control of stochastic functional differential equations with infinite delay. On the other hand, in many practical stochastic models, the coefficients of these systems do not satisfy the linear growth condition, but are highly nonlinear. In this paper, the delay feedback controls are designed for a class of infinite delay stochastic systems with highly nonlinear and the influence of switching state.     

Tracking control of nonlinear chaotic systems with dynamics uncertainties

This paper deals with the tracking control of nonlinear chaotic systems with dynamics uncertainties. A robust control strategy is developed to control a class of nonlinear chaotic systems with uncertainties. The proposed strategy is an input-output control scheme which comprises an uncertainty estimator and a linearizing-like feedback. The control time is explicitly computed. Computer simulations of the Duffing system are provided to verify the validity of the proposed control scheme.     

Composite nonlinear feedback based discrete integral sliding mode controller for uncertain systems

In this paper, a discrete integral sliding mode (ISM) controller based on composite nonlinear feedback (CNF) method is proposed. The aim of the controller is to improve the transient performance of uncertain systems. The CNF based discrete ISM controller consists of a linear and a nonlinear term. The linear control law is used to decrease the damping ratio of the closed-loop system for yielding a quick transient response. The nonlinear feedback control law is used to increase the damping ratio with an aim to reduce the overshoot of the closed-loop system as it approaches the desired reference position. It is observed that the discrete CNF-ISM controller produces superior transient performance as compared to the discrete ISM controller. The closed-loop control system remains stable during the sliding condition. Simulation results demonstrate the effectiveness of the proposed controller.     

Synchronization of chaotic systems using feedback controller: An application to Diffie–Hellman key exchange protocol and ElGamal public key cryptosystem

In this paper, designing an appropriate linear and nonlinear feedback control, the two identical integer order chaotic systems are synchronized by analytically and numerically. It has been realizing that, synchronization using linear feedback control method is efficient than nonlinear feedback control method due to the less computational complexity and the synchronization error. ElGamal public key cryptosystem is described through the proposed Diffie–Hellman key exchange protocol based on the synchronized chaotic systems using linear feedback control and their security are analyzed. The numerical simulations are given to validate the correctness of the proposed synchronization of chaotic systems and the ElGamal cryptosystem.     

Realistic feedback control of turbogenerators

In this paper, the optimal control of a turboalternator connected to an infinite bus is considered. The alternator is controlled through a linear feedback of the state variables. The feedback parameters are obtained by solving a two-point nonlinear boundary-value problem. The values obtained for these parameters depend on the strength and duration of the disturbance, since the model is nonlinear, contrary to the usual feedback control of a linear model. In contrast to the model used in Ref. 1, the model used here include the transfer functions of the governor, the turbine, and the voltage regulator.This work was supported in part by the National Research Council of Canada, Grant No. A-4146.     

Stability performance for primary frequency regulation of hydro-turbine governing system with surge tank

This paper aims to study the stability for primary frequency regulation of hydro-turbine governing system with surge tank. Firstly, a novel nonlinear mathematical model of hydro-turbine governing system considering the nonlinear characteristic of penstock head loss is introduced. The nonlinear state equations under opening control mode and power control mode are derived. Then, the nonlinear dynamic performance of nonlinear hydro-turbine governing system is investigated based on the stable domain for primary frequency regulation. New feature of the nonlinear hydro-turbine governing system caused by the nonlinear characteristic of penstock head loss is described by comparing with a linear model, and the effect mechanism of nonlinear characteristic of penstock head loss is revealed. Finally, the concept of critical stable sectional area of surge tank for primary frequency regulation is proposed and the analytical solution is derived. The combined tuning and optimization method of governor parameters and sectional area of surge tank is proposed. The results indicate that for the primary frequency regulation under opening control mode and power control mode, the nonlinear hydro-turbine governing system is absolutely stable and conditionally stable, respectively. The stability of the nonlinear hydro-turbine governing system and linear hydro-turbine governing system is the same under opening control model and different under power control model. The nonlinear characteristic of penstock head loss mainly affects the initial stage of dynamic response process of power output, and then changes the stability of the nonlinear system. The critical stable sectional area of surge tank makes the system reach critical stable state. The governor parameters and critical stable sectional area of surge tank jointly determine the distributions of stability states.     

Implementation of dynamic programming for chaos control in discrete systems

In this paper the control of discrete chaotic systems by designing linear feedback controllers is presented. The linear feedback control problem for nonlinear systems has been formulated under the viewpoint of dynamic programming. For suppressing chaos with minimum control effort, the system is stabilized on its first order unstable fixed point (UFP). The presented method also could be employed to make any desired nth order fixed point of the system, stable. Two different methods for higher order UFPs stabilization are suggested. Afterwards, these methods are applied to two well-known chaotic discrete systems: the Logistic and the Henon Maps. For each of them, the first and second UFPs in their chaotic regions are stabilized and simulation results are provided for the demonstration of performance.