Approximate observability of infinite dimensional bilinear systems using a Volterra series expansion

In this paper, the approximate observability of a class of infinite dimensional bilinear systems is investigated. The observability conditions are discussed based on a Volterra series representation of this class of systems. A testable observability criterion is derived for the case where the infinitesimal generator is self-adjoint or Riesz-spectral operator. The theoretical results are illustrated by examples.     

Nonlinear MPC based on a Volterra series model for greenhouse temperature control using natural ventilation

Suitable environmental conditions are a fundamental issue in greenhouse crop growth and can be achieved by advanced climate control strategies. In different climatic zones, natural ventilation is used to regulate both the greenhouse temperature and humidity. In mild climates, the greatest problem faced by far in greenhouse climate control is cooling, which, for dynamical reasons, leads to natural ventilation as a standard tool. This work addresses the design of a nonlinear model predictive control (NMPC) strategy for greenhouse temperature control using natural ventilation. The NMPC strategy is based on a second-order Volterra series model identified from experimental input/output data of a greenhouse. These models, representing the simple and logical extension of convolution models, can be used to approximate the nonlinear dynamic effect of the ventilation and other environmental conditions on the greenhouse temperature. The developed NMPC is applied to a greenhouse and the control performance of the proposed strategy will be illustrated by means of experimental results.     

Optimal expansions of discrete-time Volterra models using Laguerre functions

This work is concerned with the optimization of Laguerre bases for the orthonormal series expansion of discrete-time Volterra models. The aim is to minimize the number of Laguerre functions associated with a given series truncation error, thus reducing the complexity of the resulting finite-dimensional representation. Fu and Dumont (IEEE Trans. Automatic Control 38(6) (1993) 934) indirectly approached this problem in the context of linear systems by minimizing an upper bound for the error resulting from the truncated Laguerre expansion of impulse response models, which are equivalent to first-order Volterra models. A generalization of the work mentioned above focusing on Volterra models of any order is presented in this paper. The main result is the derivation of analytic strict global solutions for the optimal expansion of the Volterra kernels either using an independent Laguerre basis for each kernel or using a common basis for all the kernels.     

Nonlinear MPC for the airflow in a PEM fuel cell using a Volterra series model

The particular industrial interest in fuel cells is based on the possibility to generate clean energy both for stationary and automotive applications. Performance and safety issues of fuel cells are closely related to the control strategy used for the fuel and oxidant supply. In PEM (polymer electrolyte membrane or proton exchange membrane) fuel cells the oxygen excess ratio expresses the proportion between oxygen reacting in the cells and oxygen entering the stack and represents a decisive variable for the mentioned issues. This work is focused on the design of a nonlinear model predictive control (NMPC) strategy manipulating the air flow rate in order to maintain the oxygen excess ratio in a desired value, both for safety and performance reasons. The designed NMPC, based on a second order Volterra series model, was implemented on a commercial fuel cell. The proposed NMPC strategy is validated in experiments and compared to a linear model predictive controller and to the original built-in controller.     

Choice of free parameters in expansions of discrete-time Volterra models using Kautz functions

This work tackles the problem of modeling nonlinear systems using Volterra models based on Kautz functions. The drawback of requiring a large number of parameters in the representation of these models can be circumvented by describing every kernel using an orthonormal basis of functions, such as the Kautz basis. The resulting model, so-called Wiener/Volterra model, can be truncated into a few terms if the Kautz functions are properly designed. The underlying problem is how to select the free-design complex poles that fully parameterize these functions. A solution to this problem has been provided in the literature for linear systems, which can be represented by first-order Volterra models. A generalization of such strategy focusing on Volterra models of any order is presented in this paper. This problem is solved by minimizing an upper bound for the error resulting from the truncation of the kernel expansion into a finite number of functions. The aim is to minimize the number of two-parameter Kautz functions associated with a given series truncation error, thus reducing the complexity of the resulting finite-dimensional representation. The main result is the derivation of an analytical solution for a sub-optimal expansion of the Volterra kernels using a set of Kautz functions.     

Control relevant model reduction of Volterra series models

This paper presents a two-step method for control-relevant model reduction of Volterra series models. First, using the nonlinear IMC design as a basis, an explicit expression relating the closed-loop performance to the open-loop modeling error is obtained. Secondly, an optimization problem that seeks to minimize the closed-loop error subject to the restriction of a reduced-order model is posed. By showing that model reduction of kernels with different degrees can be decoupled in the problem formulation, the optimization problem is simplified into a mathematically more convenient form which can be solved with significantly less computational effort. The effectiveness of the proposed method is illustrated on a polymerization reactor example where a second-order Volterra model with 85 parameters is reduced to a Hammerstein model with 3 parameters. Despite the lower ‘open-loop’ predictive ability of the control-relevant model, the closed-loop performance of the reduced-order control system closely mimics that of the full order model.     

基于沃尔泰拉级数的模拟电路组合故障诊断法

针对模拟电路的固有复杂性及其传统故障检测方法延时大和正确识别率低的问题,借鉴基于隐马尔科夫模型改进最小二乘支持向量机以及Volterra级数原理,将二者组合进行故障诊断。该方法首先采用Volterra级数频域核对电路故障特征进行提取,再利用经隐马尔科夫模型改进的最小二乘支持向量机进行模态分类,最终完成故障诊断。仿真结果表明,与目前使用的BP神经网络诊断方法和LSSVM诊断方法相比,该方法不仅提高了系统故障辨识能力,还提高了系统故障诊断的速度。     

Nonlinear influence in the frequency domain: Alternating series

The nonlinear influence on system output spectrum is studied for a class of nonlinear systems which have Volterra series expansion. It is shown that under certain conditions the system output spectrum can be expressed in an alternating series with respect to some model parameters which define system nonlinearities. The magnitude of the system output spectrum can therefore be suppressed by exploiting the properties of alternating series. Sufficient (and necessary) conditions in which the output spectrum can be cast into an alternating series are studied. These results reveal a novel frequency-domain insight into the nonlinear influence on a system, and provide a new method for the analysis and design of nonlinear systems in the frequency domain. Examples are given to illustrate the results.     

Nonlinear thermal system identification using fractional Volterra series

Linear fractional differentiation models have already proven their efficacy in modeling thermal diffusive phenomena for small temperature variations involving constant thermal parameters such as thermal diffusivity and thermal conductivity. However, for large temperature variations, encountered in plasma torch or in machining in severe conditions, the thermal parameters are no longer constant, but vary along with the temperature. In such a context, thermal diffusive phenomena can no longer be modeled by linear fractional models. In this paper, a new class of nonlinear fractional models based on the Volterra series is proposed for modeling such nonlinear diffusive phenomena. More specifically, Volterra series are extended to fractional derivatives, and fractional orthogonal generating functions are used as Volterra kernels. The linear coefficients are estimated along with nonlinear fractional parameters of the Volterra kernels by nonlinear programming techniques. The fractional Volterra series are first used to identify thermal diffusion in an iron sample with data generated using the finite element method and temperature variations up to 700 K. For that purpose, the thermal properties of the iron sample have been characterized. Then, the fractional Volterra series are used to identify the thermal diffusion with experimental data obtained by injecting a heat flux generated by a 200 W laser beam in the iron sample with temperature variations of 150 K. It is shown that the identified model is always more accurate than the finite element model because it allows, in a single experiment, to take into account system uncertainties.     

Output frequency response function of nonlinear Volterra systems

An expression for the output frequency response function (OFRF), which defines the explicit analytical relationship between the output spectrum and the system parameters, is derived for nonlinear systems which can be described by a polynomial form differential equation model. An effective algorithm is developed to determine the OFRF directly from system simulation or experimental data. Simulation studies demonstrate the significance of the OFRF concept, and verify the effectiveness of the algorithm which evaluates the OFRF numerically. These new results provide an important basis for the analytical study and design of a wide class of nonlinear systems in the frequency domain.     

A note on the optimal expansion of Volterra models using Laguerre functions

This work tackles the problem of expanding Volterra models using Laguerre functions. A strict global optimal solution is derived when each multidimensional kernel of the model is decomposed into a set of independent orthonormal bases, each of which parameterized by an individual Laguerre pole intended for representing the dominant dynamic of the kernel along a particular dimension. It is proved that the solution derived minimizes the upper bound of the squared norm of the error resulting from the practical truncation of the Laguerre series expansion into a finite number of functions. This is an extension of the results in Campello, Favier and Amaral [(2004). Optimal expansions of discrete-time Volterra models using Laguerre functions. Automatica, 40, 815-822.], where an optimal solution was obtained for the usual yet particular case in which a single Laguerre pole is used for expanding a given kernel along all its dimensions. It is also proved that the particular and extended solutions are equivalent to each other when the Volterra kernels are symmetric.     

Robust nonlinear MPC based on Volterra series and polynomial chaos expansions

A robust nonlinear model predictive controller (NMPC) based on a Volterra series is proposed. Polynomial chaos expansions (PCE) are used to represent the uncertainty in the Volterra series coefficients and this uncertainty is then propagated onto the output predictions. The key advantage of the PCE is that it provides an analytical expression to compute the L²-norm of the output prediction error resulting in computational savings, compared to previously proposed techniques, which are essential for real time implementation. Terminal and input constraints based on Structured Singular Value based-norms are used to ensure convergence to a set-point and compliance with constraints in manipulated variables. The algorithm is applied to a multivariable pH neutralization system. A comparative study shows superior closed loop performance and computational efficiency of the proposed technique as compared to previously proposed algorithms.     

一类非线性系统的在线建模新方法

为减少非线性系统的Volterra级数模型在线建模的计算量，根据多模型合成的思想。提出一种基于预设模型在线合成被测系统当前Volterra级数模型的新方法。建立了模型合成的公式和方法．仿真实验表明。该方法具有较高的建模精度。能有效减少Volterra模型在线建模的计算量。并且易于工程实现．     

Series expansions for analytic systems linear in control

This paper presents a series expansion for the evolution of a class of nonlinear systems characterized by constant input vector fields. We present a series expansion that can be computed via explicit recursive expressions, and we derive sufficient conditions for uniform convergence over the finite- and infinite-time horizon. Furthermore, we present a simplified series and convergence analysis for the setting of second-order polynomial vector fields. The treatment only relies on elementary notions on analytic functions, number theory, and operator norms.     

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.     

Magnitude bounds of generalized frequency response functions for nonlinear Volterra systems described by NARX model

In order to reveal the relationship between system time domain model parameters and system frequency response functions, new magnitude bounds of frequency response functions for nonlinear Volterra systems described by NARX model are established. The magnitude bound of the nth-order generalized frequency response function (GFRF) can be expressed as a simple n-degree polynomial function of the magnitude of the first order GFRF, whose coefficients are functions of the model parameters and frequency variables. Thus the system output spectrum can also be bounded by a polynomial function of the magnitude of the first order GFRF. These results demonstrate explicitly the analytical relationship between model parameters and system frequency response functions, and provide a significant insight into the magnitude based analysis and synthesis of nonlinear systems in the frequency domain.     

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.     

基于Volterra泛函级数的非线性系统的鲁棒辨识

针对弱非线性系统的鲁棒建模问题, 基于Volterra泛函级数, 结合集员辨识理论, 提出了广义频率响应函数的鲁棒辨识方法, 形成了一套较完整的弱非线性系统的鲁棒建模方法, 仿真结果表明该方法是行之有效的.