On the realization of nonlinear discrete-time systems

It is shown, on the basis of a compact expression of the kernels of the discrete Volterra series associated to a linear analytic discrete-time system, that the kernels satisfy a suitable property which enables their inductive characterization. We also give a first result on the realization of a given family of functions by a linear analytic discrete-time system with linear invertible drift term.     

Model structure simplification of Nonlinear Systems via immersion

This paper concerns the simplification of model structures of nonlinear systems while preserving their input-output maps. The basic technique is an immersion of a system, which is a mapping of the initial state from the original system to another system of an identical input-output map. The necessary and sufficient conditions are presented for the immersibility of a given system into a state-space representation of such particular structures as rational functions or polynomials with respect to the state. The conditions are so mild that many types of nonlinear systems can be represented by rational or polynomial model structures. Moreover, it is also shown that a polynomial state-space representation can always be constructed to be at most quadratic with respect to the state.     

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.     

Exact Search Directions for Optimization of Linear and Nonlinear Models Based on Generalized Orthonormal Functions

A novel technique for selecting the poles of orthonormal basis functions (OBF) in Volterra models of any order is presented. It is well-known that the usual large number of parameters required to describe the Volterra kernels can be significantly reduced by representing each kernel using an appropriate basis of orthonormal functions. Such a representation results in the so-called OBF Volterra model, which has a Wiener structure consisting of a linear dynamic generated by the orthonormal basis followed by a nonlinear static mapping given by the Volterra polynomial series. Aiming at optimizing the poles that fully parameterize the orthonormal bases, the exact gradients of the outputs of the orthonormal filters with respect to their poles are computed analytically by using a back-propagation-through-time technique. The expressions relative to the Kautz basis and to generalized orthonormal bases of functions (GOBF) are addressed; the ones related to the Laguerre basis follow straightforwardly as a particular case. The main innovation here is that the dynamic nature of the OBF filters is fully considered in the gradient computations. These gradients provide exact search directions for optimizing the poles of a given orthonormal basis. Such search directions can, in turn, be used as part of an optimization procedure to locate the minimum of a cost-function that takes into account the error of estimation of the system output. The Levenberg-Marquardt algorithm is adopted here as the optimization procedure. Unlike previous related work, the proposed approach relies solely on input-output data measured from the system to be modeled, i.e., no information about the Volterra kernels is required. Examples are presented to illustrate the application of this approach to the modeling of dynamic systems, including a real magnetic levitation system with nonlinear oscillatory behavior.      

Commutativity of Immersion and Linearization

A given nonlinear system can be represented via an immersion as rational or polynomial functions, thus leading to a simplified model structure. An immersion is a mapping of the initial state from the original state space to another state space, while exactly preserving the input-output map. In this note we show that the linearization of the system after immersion has an identical input-output map to the linearization of the original system before immersion. In other words, immersion and linearization commute. This is potentially useful for applications such as linear control design and sensitivity analysis after nonlinear identification, and has important implications for system approximation by linearization.     

Comment on ‘ Tolerance analysis using fuzzy sets ’ and ‘ A procedure for multiple aspect decision making ’

The problem of tracking a reference vector variable from a given class is considered for discrete time linear multiple input-output plants. The plant and the reference are both described by an input-output relation and the objective is to track so that a quadratic criterion is minimized. This tracking problem is solved by reformulating it as a regulator problem for an augmented system. The optimal control law is shown to contain both feedback and feedforward terms and it is obtained by applying polynomial matrix techniques. The design procedure consists in spectral factorization and the solution of linear equations in polynomial matrices. The case of unstable references is included and a natural solvability condition is derived in the form of divisibility of polynomial matrices.     

On the Input-Output Representation of Piecewise Affine State Space Models

This paper addresses the conversion of discrete-time piecewise affine (PWA) state space models into input-output form. Necessary and sufficient conditions for the existence of equivalent input-output representations of a given PWA state space model are derived. Connections to the observability properties of PWA models are investigated. Under a technical assumption, it is shown that every finite-time observable PWA model admits an equivalent input-output representation. When an equivalent input-output model exists, a constructive procedure is presented to derive its equations. Several examples illustrate the proposed results.      

Realization and generalized frequency response for non-linear input-output maps

A realization theory is developed and the ‘generalized frequency response’ is presented for an input-output map introduced in a previous work. Relations with the realization of finite Volterra series are outlined. A simple example is given to show that the new representations have improved convergence properties.     

Modeling of nonlinear nonstationary dynamic systems with a novelclass of artificial neural networks

This paper introduces a novel neural-network architecture that can be used to model time varying Volterra systems from input-output data. The Volterra systems constitute a very broad class of stable nonlinear dynamic systems that can be extended to cover nonstationary (time-varying) cases. This novel architecture is composed of parallel subnets of three-layer perceptrons with polynomial activation functions, with the output of each subnet modulated by an appropriate time function that gives the summative output its time-varying characteristics. The paper shows the equivalence between this network architecture and the class of time-varying Volterra systems, and demonstrates the range of applicability of this approach with computer-simulated examples and real data. Although certain types of nonstationarities may not be amenable to this approach, it is hoped that this methodology will provide the practical tools for modeling some broad classes of nonlinear, nonstationary systems from input-output data, thus advancing the state of the art in a problem area that is widely viewed as a daunting challenge.     

Nonlinear predictive controller based on S-PARAFAC Volterra models applied to a communicating two-tank system

This paper proposes a new predictive controller approach for nonlinear process based on a reduced complexity homogeneous, quadratic discrete-time Volterra model called quadratic S-PARAFAC Volterra model. The proposed model is yielded by using the symmetry property of the Volterra kernels and their tensor decomposition using the PARAFAC technique that provides a parametric reduction compared to the conventional Volterra model. This property allows synthesising a new nonlinear-model-based predictive control (NMBPC). We develop the general form of a new predictor, and therefore, we propose an optimisation algorithm formulated as a quadratic programming under linear and nonlinear constraints. The performances of the proposed quadratic S-PARAFAC Volterra model and the developed NMBPC algorithm are illustrated on a numerical simulation and validated on a benchmark as a continuous stirred-tank reactor system. Moreover, the efficiency of the proposed quadratic S-PARAFAC Volterra model and the NMBPC approach are validated on an experimental communicating two-tank system.     

Measurement of the Kernels of a Non-linear System of Finite Order†

A method of measuring the Volterra kernels of a finite-order non-linear system is presented. The kernels are obtained individually as a multi-dimensional impulse response. The multi-dimensional kernel transforms also can be obtained by the method described. As an extension, a technique of obtaining the Volterra kernels from a multi-dimensional step response is presented. This technique is useful for non-linear systems which can be considered to be of a given finite order for only a limited range of input amplitudes.     

Abstract bilinear systems: The forward shift approach

A bilinear realization theory for a Volterra series input-output map is given. The approach involves a special transform representation for a Volterra series and certain shift operators on a Fock space. The approach yields in a very simple manner a theory of span reachability, observability and minimality for bilinear systems.     

The existence and uniqueness of Volterra series for nonlinear systems

Given an input-output map described by a nonlinear control systemdot{x}=f(x,u)and nonlinear outputy=h(x), we present a simple straightforward means for obtaining a series representation of the outputy(t)in terms of the inputu(t). When the control enters linearly,dot{x} =f(x)+ ug(x), the method yields the existence of a Volterra series representation. The proof is constructive and explicitly exhibits the kernels. It depends on standard mathematical tools such as the Fundamental Theorem of Calculus and the Cauchy estimates for the Taylor series coefficients of analytic functions. In addition, the uniqueness of Volterra series representations is discussed.     

Volterra series and geometric control theory

It is shown here that controlled differential equations which are analytic in the state and linear in the control have solutions which can be expanded in a Volterra series provided there is no finite escape time. The Volterra kernels are computed in terms of the power series expansion of the functions defining the differential equation. We also give necessary and sufficient conditions for a Volterra series to be realizable by a linear-analytic system. These conditions are particularly easy to test if the Volterra series is finite; a complete theory is worked out for this case. In the final section some applications are considered to singular control, multilinear realization theory, etc.     

基于输入输出线性化方法的一类多输入多输出非线性系统的观测器设计

研究一类多输入多输出仿射非线性系统的状态观测器设计问题. 基于输入输出线性化方法提出了一类多输入多输出仿射非线性系统的状态观测器设计的新方法, 并给出保证状态估计误差渐近趋于零的充分条件. 算例表明了所得结果的有效性.     

Separating the Drift from Affine Control Systems via Feedbacks

This paper investigates the use of state feedbacks to transform an affine control system into a special form with a driftless property. First, we consider the regular dynamic state feedback for an affine control system. Both local and global sufficient and necessary conditions are given on the affine control system such that it admits the driftless property. An algorithm is provided to compute explicitly the desired coordinate transformation. Then, the nonregular cases are studied. Some necessary conditions, as well as sufficient conditions, for nonregular feedback are presented. We illustrate with some examples.     

基于Volterra级数模型的非线性系统自适应控制稳定性研究

研究一类单输入单输出非线性系统的自适应控制问题, 这类系统能用有限阶离散Volterra级数模型表示. 采用递推最小二乘算法进行参数估计, 并通过解高次方程得到控制律. 结合反馈型Volterra级数系统的局部L稳定理论, 证明了算法的局部收敛性质. 通过对一个化工连续搅拌反应器 (CSTR)的过程控制进一步验证了该算法的有效性.     

Robust adaptive control of nonaffine nonlinear plants with small input signal changes

Assuming small input signal magnitudes, ARMA models can approximate the NARMA model of nonaffine plants. Recently, NARMA-L1 and NARMA-L2 approximate models were introduced to relax such input magnitude restrictions. However, some applications require larger input signals than allowed by ARMA, NARMA-L1 and NARMA-L2 models. Under certain assumptions, we recently developed an affine approximate model that eliminates the small input magnitude restriction and replaces it with a requirement of small input changes. Such a model complements existing models. Using this model, we present an adaptive controller for discrete nonaffine plants with unknown system equations, accessible input-output signals, but inaccessible states. Our approximate model is realized by a neural network that learns the unknown input-output map online. A deadzone is used to make the weight update algorithm robust against modeling errors. A control law is developed for asymptotic tracking of slowly varying reference trajectories.     

A Dissipation Inequality for the Minimum Phase Property

The minimum phase property is an important notion in systems and control theory. In this paper, a characterization of the minimum phase property of nonlinear control systems in terms of a dissipation inequality is derived. It is shown that this dissipation inequality is equivalent to the classical definition of the minimum phase property in the sense of Byrnes and Isidori, if the control system is affine in the input and the so-called input-output normal form exists.