Exact finite-dimensional filters via systems realization for a class of discrete-time nonlinear systems

We obtain necessary and sufficient conditions for the existence of a finite-dimensional filter for the discrete-time nonlinear system (ε x_k+1 =φ(x_k), y_k = h(x_k)+η(x_k)ν_k, K=0, 1,…. This system is distinguished by the absence of noise in the dynamic and by the correlation between the state and the intensity of noise in the observations.The necessary and sufficient condition provides an explicit formula for the minimal filter and various system-theoretic properties of (ε) and of the minimal filter.     

On input-output linearization of discrete-time nonlinear systems

We consider a nonlinear discrete-time system of the form Σ: x(t+1)=f(x(t), u(t)), y(t) =h(x(t)), where x ε R^N, u ε R^m, y ε R^q and f and h are analytic. Necessary and sufficient conditions for local input-output linearizability are given. We show that these conditions are also sufficient for a formal solution to the global input-output linearization problem. Finally, we show that zeros at infinity of ε can be obtained by the structure algorithm for locally input-output linearizable systems.     

A new adaptive scheme for a class of discrete-time nonlinear systems

A new adaptive control scheme for discrete-time systems is proposed. The objective is the tracking of the trajectory. Global boundedness convergence and boundedness are obtained for a certain subclass of nonlinear systems.     

On H∞-control of discrete-time nonlinear systems

Following recent works on continuous-time nonlinear H_∞-control, where connections with game theory and passivity conditions have been set, the present paper studies the corresponding problem for discrete-time systems. The paper describes sufficient conditions for the existence and the construction of a feedback law which imposes a prescribed level of disturbance attenuation with internal stability. Both cases of state feedback and measurement feedback are considered.     

On the realization problem for stationary,homogeneous, discrete-time systems

For degree 2, stationary, homogeneous systems, we consider realizations in the form of state-affine difference equations, and in the form of feedback-free interconnections of stationary linear systems, adders, and multipliers. Realizability conditions are presented in terms of the symmetric transfer function (or kernel), and in terms of a new representation called the regular transfer function (or kernel). A procedure is given for constructing a minimal dimension state-affine realization, and the corresponding interconnection.     

Volterra filter modeling of a nonlinear discrete-time system based on a ranked differential evolution algorithm

This paper presents a ranked differential evolution （RDE） algorithm for solving the identification problem of non- linear discrete-time systems based on a Volterra filter model. In the improved method, a scale factor, generated by combining a sine function and randomness, effectively keeps a balance between the global seareh and the local search. Also, the mutation operation is modified after ranking all candidate solutions of the population to help avoid the occurrence of premature convergence. Finally, two examples including a highly nonlinear discrete-time rational system and a real heat exchanger are used to evaluate the per- formance of the RDE algorithm and five other approaches. Numerical experiments and comparisons demonstrate that the RDE algorithm performs better than the other approaches in most cases.     

一类非线性离散时间系统的模糊辨识

对一类非线性离散时间系统提出了模糊辨识方法,此方法用与未知参数向量成线性关系的模糊逻辑系统作为辨识模型,并通过自适应学习律对此模糊逻辑系统中的未知参数进行自适应调节,文中证明了此方法可使辨识误差收敛到原点的一个邻域内。仿真结果验证了此方法的有效性。     

On finite dimensional realization theory of discrete time nonlinear systems

We characterize finite dimensional realizability of discrete time nonlinear systems which have a Volterra series development with separable structure of the Volterra kernels.     

Nonsmooth stabilizability and feedback linearization of discrete-time nonlinear systems

We consider the problem of stabilizing a discrete-time nonlinear system using a feedback which is not necessarily smooth. A sufficient condition for global dynamical stabilizability of single-input triangular systems is given. We obtain conditions expressed in terms of distributions for the nonsmooth feedback triangularization and linearization of discrete-time systems. Relations between stabilization and linearization of discrete-time systems are given.     

非线性离散时间系统的自适应模糊补偿控制

针对一类非线性离散时间系统,提出一种自适应模糊逻辑补偿控制方案.控制律由跟踪控制律和逼近误差补偿控制律两部分组成,利用模糊逻辑系统对系统参数扰动和外界干扰进行自适应补偿,由模糊滑模控制律实现对模糊逻辑系统逼近误差的进一步补偿.所设计的控制器可保证闭环系统一致最终有界.将该控制器用于月球探测车动态转向系统中,仿真结果表明了该方法的有效性.     

非线性离散系统的近似最优跟踪控制

研究非线性离散系统的最优跟踪控制问题. 通过在由最优控制问题所导致的非线性两点边值问题中引入灵敏度参数, 并对它进行Maclaurin级数展开, 将原最优跟踪控制问题转化为一族非齐次线性两点边值问题. 得到的最优跟踪控制由解析的前馈反馈项和级数形式的补偿项组成. 解析的前馈反馈项可以由求解一个Riccati差分方程和一个矩阵差分方程得到. 级数补偿项可以由一个求解伴随向量的迭代算法近似求得. 以连续槽式反应器为例进行仿真验证了该方法的有效性．     

A new symbolic calculus for the response of nonlinear systems

We present a new operational calculus for computing the response of nonlinear systems to various deterministic excitations. The use of a new tool: noncommutative generating power series, allows us to derive, by simple algebraic manipulations, the Volterra functional series of the solution of a large class of nonlinear forced differential equations. The symbolic calculus introduced appears as a natural generalization to the nonlinear domain, of the well known Heaviside operational calculus. Moreover, this method has the advantage of allowing the use of a computer.     

Adaptive discrete-time sliding-mode control of nonlinear systems described by Wiener models

In this paper, we propose an adaptive control scheme that can be applied to nonlinear systems with unknown parameters. The considered class of nonlinear systems is described by the block-oriented models, specifically, the Wiener models. These models consist of dynamic linear blocks in series with static nonlinear blocks. The proposed adaptive control method is based on the inverse of the nonlinear function block and on the discrete-time sliding-mode controller. The parameters adaptation are performed using a new recursive parametric estimation algorithm. This algorithm is developed using the adjustable model method and the least squares technique. A recursive least squares (RLS) algorithm is used to estimate the inverse nonlinear function. A time-varying gain is proposed, in the discrete-time sliding mode controller, to reduce the chattering problem. The stability of the closed-loop nonlinear system, with the proposed adaptive control scheme, has been proved. An application to a pH neutralisation process has been carried out and the simulation results clearly show the effectiveness of the proposed adaptive control scheme.     

On the immersion of a discrete-time polynomial analytic system into a polynomial affine one

The concept of immersion, just recently introduced by Fliess and Kupka, is related to the capability of reproducing the input-output maps of a given system by means of another one. In this paper we give necessary and sufficient conditions under which a polynomial analytic system, i.e. a system whose dynamics are polynomial with respect to the input and nonlinear analytic with respect to the state, is immersed into a polynomial affine system, i.e. a system whose dynamics are polynomial in the input and linear in the state. An application to the problem of the reproducibility of a linear analytic state feedback acting on a given system by means of a preprocessor suitably initialized is also performed.     

Multiple model-based event-triggered adaptive control of a class of discrete-time nonlinear systems

In this study, the problem of event-triggered-based adaptive control (ETAC) for a class of discrete-time nonlinear systems with unknown parameters and nonlinear uncertainties is considered. Both neural network (NN) based and linear identifiers are used to approximate the unknown system dynamics. The feedback output signals are transmitted, and the parameters and the NN weights of the identifiers are tuned in an aperiodic manner at the event sample instants. A switching mechanism is provided to evaluate the approximate performance of each identifier and decide which estimated output is utilised for the event-triggered controller design, during any two events. The linear identifier with an auxiliary output and an improved adaptive law is introduced so that the nonlinear uncertainties are no longer assumed to be Lipschitz. The number of transmission times are significantly reduced by incorporating multiple model schemes into ETAC. The boundedness of both the parameters of identifiers and the system outputs is demonstrated though the Lyapunov approach. Simulation results demonstrate the effectiveness of the proposed method.     

一种新的非线性离散时间系统的模糊辨识方法

对一类非线性离散时间系统提出一种新的模糊的辨识方法。该方法在假设逼近误差界已知的情况下,基于死区函数对模糊逻辑系统中的未知参数设计自适应学习律;在逼近误差界未知的情况下,基于时变死区函数对模糊逻辑系统中的未知参数设计自适应学习律,并对时变死区进行自适应调节。证明了所设计的自适应学习律均可使辨识误差收敛到原点的一个小邻域内。仿真结果表明了该算法的有效性。     

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.     

On the positive invariance of polyhedral sets for discrete-time systems

In this note, necessary and sufficient conditions for a polyhedral set to be a positively invariant set of a linear discrete-time system are established.