OBSERVER LINEARIZATION OF NONLINEAR SYSTEMS BY GENERALIZED TANSFORMATIONS

In this paper, we study the problem of observer linearization for single output dynamical systems in the presence of an output‐dependent time‐scaling transformation and a simultaneous output diffeomorphism. The approach, based on an exterior calculus approach, provides a constructive approach to the problem of equivalence of a locally observable nonlinear system to a linear observer form by means of an output dependent time‐scale transformation, an output diffeomorphism and a state‐space diffeomorphism. A generalization of existing results is obtained which allows the treatment of a larger class of locally observable nonlinear systems.     

Reduced-order dynamic observer error linearisation: a natural extension of observer error linearisation

This paper introduces the concept of reduced-order dynamic observer error linearisation (RDOEL) for multi-output systems, which is the problem of transforming a nonlinear system into a nonlinear observer canonical form with the aid of auxiliary dynamics. The proposed RDOEL framework is not only a modified version of dynamic observer error linearisation (DOEL) but also a natural extension of observer error linearisation (OEL). We provide three necessary conditions for the RDOEL problem, and then derive a necessary and sufficient condition described in terms of Lie algebras of vector fields. Furthermore, from the result, we also give a geometric necessary and sufficient condition for the OEL problem, which has not yet been completely established in the case where a diffeomorphism on the output of general form is considered.     

On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form

This article deals with the transformation of a class of nonlinear systems into an extended nonlinear observable canonical form (ENOCF) by dynamic extension. The dynamic extension is obtained by adding auxiliary dynamics and virtual outputs to the original system, thus giving rise to a higher dimensional system in an extended state space. Sufficient geometrical conditions to guarantee the existence of a local diffeomorphism which allow the transformation of the extended system into the ENOCF are given. In particular, an algorithm that permits us to compute such a diffeomorphism is derived.     

Fast diffeomorphic matching to learn globally asymptotically stable nonlinear dynamical systems

We propose a new diffeomorphic matching algorithm and use it to learn nonlinear dynamical systems with the guarantee that the learned systems have global asymptotic stability. For a given set of demonstration trajectories, and a reference globally asymptotically stable time-invariant system, we compute a diffeomorphism that maps forward orbits of the reference system onto the demonstrations. The same diffeomorphism deforms the whole reference system into one that reproduces the demonstrations, and is still globally asymptotically stable.     

Linearization through state immersion of nonlinear systems admitting Lie symmetries

In this paper, it is shown that if a nonlinear system admits a Lie symmetry that can be transformed into its Poincaré-Dulac normal form by a state diffeomorphism, then, under some technical conditions, such a nonlinear system can be immersed into a linear one. This allows us to compute in closed-form the flow, all algebraic invariant curves (through semi-invariants) of the nonlinear system, and Lyapunov functions to study stability properties.     

高阶系统方法—Ⅲ.能观性与观测器设计

平行于第Ⅰ部分中提出的非线性系统的全驱性概念, 本文提出了非线性系统的全量测性概念.首先给出了非线性系统的一种能观规范型, 并证明了任何与该类非线性系统能观规范型等价的系统, 以及任何能观的线性系统, 都等价于一个高阶全量测系统.然后据此提出了一般动态系统的完全能观性定义, 同时指出线性系统的完全能观性等同于其通常意义下的能观性.最后提出了非线性全量测系统观测器设计的一种简洁方法.基于这种设计, 可以使观测误差系统为线性定常系统, 并且可以任意配置其特征多项式的系数矩阵.     

High-gain observer for a class of time-delay nonlinear systems

This article proposes a high-gain observer design for a class of nonlinear systems with multiple known time-varying delays intervening in the states and the inputs. In the free delay case, the class of systems under consideration coincides with a canonical form characterising a class of multi-output nonlinear systems, which are observable for any input. The underlying high-gain design has been mainly motivated by its inherent simplicity from both design and implementation points of view. Indeed, the observer gain is determined from an explicit resolution of a time-invariant Lyapunov algebraic equation up to the specification of a single design parameter. An academic observation problem is addressed to illustrate the effectiveness of the proposed observer.     

Reduction of the Rosenbrock matrix in analysis of invariant zeros of the linear MIMO-System

The problem is considered of reduction of the Rosenbrock matrix in analysis of invariant zeros of a linear multidimensional dynamical system with many inputs and many outputs. Approaches to the Rosenbrock matrix reduction are compared, which are carried out with the aid of the transformation of initial system equations to the Yokoyama canonical form and on the basis of zero divisors of numerical matrices. It is shown that the approach to the Rosenbrock matrix reduction on the basis of zero divisors is less arduous and offers the possibility of obtaining a well-posed problem in eigenvalues of the generalized linear bundle, while the reduction on the basis of subspaces of A.N. Krylov leads to the problem in eigenvalues of the nonlinear (polynomial) bundle of matrices, and its stipulation is generally rather poor. Various versions of the reduced matrices are presented, which are obtained in accordance with properties of the dynamical system. Properties of the Rosenbrock reduced matrices are analyzed.     

Approximate Feedback Linearisation Using Multilayer Neural Networks

In this paper we introduce the approximate feedback linearisation using multilayer feedforward neural networks. We propose to approximate a basis of the one-dimensional codistribution of an affine nonlinear system with the derivative of a multilayer neural network [6] and form a change of coordinates with n multilayer neural networks [5]. In this paper we will prove that the transformation can define a local diffeomorphism, with which a local stabilising feedback law can be designed for a kind of non-linearisable nonlinear systems.     

Synthesis of input/output matrices for a multi-input multi-output dynamical system by given zeros of transfer matrix

The problem of synthesizing a linear multi-input multi-output dynamical system with equal numbers of inputs and outputs that has given transfer zeros is solved by applying the matrix canonization technique. Algorithms for constructing input and output matrices of the dynamical system model ensuring given location of transfer (system) zeros are proposed.     

基于反推方法的一类自适应神经网络容错控制

针对一类可控标准型基础上添加非线性模型误差与故障项的MIMO非线性系统,结合反推技术,提出了神经网络自适应控制方案,对模型误差与故障项进行在线估计。文中鲁棒项用于补偿逼近模型误差,当检测出系统故障时,通过调整各步骤的虚拟控制量来补偿故障项,消除故障项对系统的影响。通过理论证明实现了提出的控制方法使得各残差信号一致有界,并最终收敛到一个小的邻域内。实例仿真表明该方案的可行性。     

Semi-global observer for multi-output nonlinear systems

We present an explicit form of nonlinear observers for a class of multi-input multi-output systems. Observer construction for multi-output nonlinear systems is not a trivial extension of single output case, especially when the global error convergence is of interest. In this paper, we consider a class of systems in which the subsystem for each output has a triangular dependence on the states of that subsystem itself, and the overall system has a block triangular form for each subsystem. Hence, the contribution is to extend the results existing in the literature in that interconnections between the subsystems are allowed. The construction is based on the saturation of some estimates, which is originated by Khalil and Esfandiari for the use of semi-global output feedback control.     

动态反馈解耦规范型时域结构特征分析及变换矩阵的构造

从可解耦线性多输入多输出(Multi-input multi-output, MIMO)系统的结构特性指数出发, 根据此类系统解耦后系统的可观测矩阵与基本向量矩阵的秩的关系, 提出了按照这种关系将解耦规范型划分为4大类的观点, 同时给出了一种针对各类积分型解耦系统构造相应的非奇异变换矩阵的构造方法. 分析了解耦规范型及其变换矩阵的时域结构形式, 通过一系列定理的证明,从一般意义上解释了解耦规范型的结构与变换矩阵的关系, 并通过一个数值实例验证了所提出方法的正确性及可行性.     

Stabilisation of state‐and‐input constrained nonlinear systems via diffeomorphisms: A Sontag's formula approach with an actual application

In this work, we provide a new and constructive outlook for the control of state‐and‐input constrained nonlinear systems. Previously, explicit solutions have been mainly focused on the finding of a barrier‐like Lyapunov function, whereas we propose the construction of a diffeomorphism to map all the trajectories of the constrained dynamics into an unconstrained one. Careful analysis has revealed that only some foundations of differential geometry and a technical assumption are necessary to construct the proposed methodology based on the well‐established theories of control Lyapunov functions and Sontag's universal formulae. Altogether, it allows us to obtain an explicit solution that even includes bounded constraints in the control action, giving the designer a way to decide (to some extent) the trade‐off between control saturations and robustness. Moreover, this approach does not rely on the own structure of the system dynamics, therefore covering a broad class of nonlinear systems. The main advantage of this approach is that the use of a diffeomorphism allows the splitting of the mathematical treatment of the constraint and the Lyapunov controller design. The result has been successfully applied to solve the dynamic positioning of an actual ship, where the nonlinear state constraints describe a strait. This approach enabled us to design a control Lyapunov function and thereby use Sontag's formula to solve the stabilisation problem. Realistic simulations have been executed in a real scenario on the simulator owned by an international shipbuilding company.     

基于高增益观测技术的高精度感应电机磁链观测器研究

针对感应电机磁链观测的一般方法具有观测精度不高、使用的电机参数多、易受影响等问题,利用非线性微分同胚变换对感应电机强耦合非线性系统进行转化,提出一种基于高增益观测技术的感应电机磁链观测器.该观测器不需要满足状态能观的条件且计算量小,通过调节增益能够达到任意精度误差.对非线性规范型的高增益观测器进行了稳定性分析,证明了收敛性.仿真和实验结果验证了该观测器的高观测精度和所提方法的可行性和有效性.     

Invariant codistributions and the feedforward form for discrete-time nonlinear systems

The goal of this paper is two-fold. First, given an arbitrary n-dimensional discrete-time nonlinear dynamical system, necessary and sufficient conditions for the existence of a one-dimensional invariant codistribution are obtained. Second, it is shown that the previous conditions can be used iteratively to obtain a nested sequence of n invariant codistributions with the properties that each codistribution contains the previous one and the last one coincides with the cotangent bundle of the state manifold. As a byproduct, necessary and sufficient conditions are obtained for a discrete-time nonlinear dynamical system to be equivalent to the so-called feedforward form.     

Stabilization of a class of switched nonlinear systems

The stabilization of a class of switched nonlinear systems is investigated in the paper. The systems concerned are of （generalized） switched Byrnes-Isidori canonical form, which has all switched models in （generalized） Byrnes- Isidori canonical form. First, a stability result of switched systems is obtained. Then it is used to solve the stabilization problem of the switched nonlinear control systems. In addition, necessary and sufficient conditions are obtained for a switched affine nonlinear system to be feedback equivalent to （generalized） switched Byrnes-Isidori canonical systems are presented. Finally, as an application the stability of switched lorenz systems is investigated.     

Stabilization of a class of switched nonlinear systems

The stabilization of a class of switched nonlinear systems is investigated in the paper. The systems con- cerned are of (generalized) switched Byrnes-Isidori canonical form, which has all switched models in (generalized) Byrnes- Isidori canonical form. First, a stability result of switched systems is obtained. Then it is used to solve the stabilization prob- lem of the switched nonlinear control systems. In addition, necessary and sufficient conditions are obtained for a switched affine nonlinear system to be feedback equivalent to (generalized) switched Byrnes-Isidori canonical systems are presented. Finally, as an application the stability of switched lorenz systems is investigated.     

Nonlinear observer design by dynamic observer error linearization

In this note, we address the problem of transforming a nonlinear system into nonlinear observer canonical form in the extended state-space with the aid of dynamic system extension and introduction of virtual outputs. As an intermediate step for the general problem, we consider a restricted structure of dynamic extension, which is obtained, roughly speaking, by adding the chains of integrators to the outputs of original system. We propose sufficient conditions which can be verified using the system dynamics expressed in their original coordinates. An illustrative example is included that demonstrates the advantage of the proposed method over the conventional method.