Complexity

Volume 2018, Article ID 8953035, 12 pages

https://doi.org/10.1155/2018/8953035

## Control of Complex Nonlinear Dynamic Rational Systems

^{1}School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China^{2}Department of Engineering Design and Mathematics, University of the West of England, Frenchay Campus, Coldharbour Lane, Bristol BS16 1QY, UK^{3}Department of Automation, Shanghai Jiao Tong University, 800 Dongchuan Rd., Minhang, Shanghai 200240, China

Correspondence should be addressed to Li Liu; nc.ude.btsu@iluil

Received 11 April 2018; Accepted 6 May 2018; Published 14 June 2018

Academic Editor: Zhile Yang

Copyright © 2018 Quanmin Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Nonlinear rational systems/models, also known as total nonlinear dynamic systems/models, in an expression of a ratio of two polynomials, have roots in describing general engineering plants and chemical reaction processes. The major challenge issue in the control of such a system is the control input embedded in its denominator polynomials. With extensive searching, it could not find any systematic approach in designing this class of control systems directly from its model structure. This study expands the U-model-based approach to establish a platform for the first layer of feedback control and the second layer of adaptive control of the nonlinear rational systems, which, in principle, separates control system design (without involving a plant model) and controller output determination (with solving inversion of the plant U-model). This procedure makes it possible to achieve closed-loop control of nonlinear systems with linear performance (transient response and steady-state accuracy). For the conditions using the approach, this study presents the associated stability and convergence analyses. Simulation studies are performed to show off the characteristics of the developed procedure in numerical tests and to give the general guidelines for applications.

#### 1. Introduction

This section justifies the reasons for designing controllers for rational models by introducing model expression and representations, achieved results in model identification, and a critical review of controller-designing approaches.

##### 1.1. Nonlinear Dynamic Rational Systems

*Definition 1 [1]. *Assign a triplet , where is an irreducible real affine variety and are mapping functions. A system , with input and output , is defined as polynomial/rational, while the functions and both on are mappings from input space to state space and from state space to output space polynomial/rational, respectively. That is, for polynomial systems, for all where is the algebra of all polynomials on the variety , and for rational systems, for all where is the algebra of all rational functions on the variety .

For a single-input and single-output (SISO) nonlinear dynamic rational system, it can be generally modelled with a ratio of two polynomials [1, 2]. where , , and denote measured output, input, and model error/noise/uncertainties, respectively, at time instant . and are real valued and smooth numerator and denominator polynomials, respectively. , , and denote the delayed outputs, inputs, and model noises, respectively. and for regression terms and , respectively, are the coefficients and and for numbers of total regression terms of the polynomials. The major properties of the rational model (1) are summarised below:

It is also defined as a total nonlinear model [2] as it covers many different linear and nonlinear models as its subsets (such as NARMAX (nonlinear autoregressive moving average with exogenous input) models [3] and intelligent models for neurofuzzy systems [4]). Rational systems have been observed in general engineering, chemical processes, physics, biological reactions, and econometrics; for example, rational models are a class of mechanistic models in describing catalytic reactions in chemical kinetics [5, 6]; metabolic, signal, and genetic networks in systems biology [1]; and movement of satellites in Earth orbit [1]. There have also been reports of rational modelling applications [7–9].

This is more concise in structure than a polynomial; the example below uses a Taylor series expansion to approximate a simple rational model below.

The other characteristic of the rational models is the power to quickly change the model output while input has small variations. Consider a simple system output below

Clearly the model output will be dramatically increased, as the input approaches −1. This comes from the function of the denominator.

Introducing a denominator polynomial makes the model concise in describing complexity and adds more functions in describing nonlinearities. On the other side, in contrast to polynomial systems, this makes identification and control system design noticeably different and more difficult with the inherent nonlinear parameters and control inputs [2]. Therefore, comprehensive studies of this class of systems in theoretical and application aspects are required. This study takes the pioneer step towards the control of rational systems.

##### 1.2. Model Identification

Model identification has been relatively mutual to some extent. So far, the identification aspect has gone through data-driven model structure detection, parameter estimation, and model validation from noise-contaminated input and output data. The major work on rational model identification is summarised in the following categories: linear least squares (LLS) algorithms for parameter estimation—extended LLS estimator [10], recursive LLS estimator [11], orthogonal LLS structure detector and estimator [12], fast orthogonal algorithm [13], and implicit least squares algorithm [14], and nonlinear least squares algorithms—prediction error estimator [15] and globally consistent nonlinear least squares estimator [16]. Other algorithms include the following categories: back propagation (BP) algorithm [17] and enhanced linear Kalman filter (EnLKF) [18].

There are two model validation methods: higher order correlation tests [19] and omnidirectional cross-correlation tests [20].

A summary of the representative publications till 2015 can be found in a survey of rational model identification [2].

##### 1.3. Controller Design

As surveyed above, rational models have been increasingly used to represent nonlinear dynamic plants. Consequently, the control system design should have been considered on the agenda in the follow-up studies. However, up to now, there is no reference found for designing such controllers directly referred to the model analytical knowledge. The paramount difficulty is that part of the controller output is embedded in the denominator polynomial . For example, . With extensive investigations through major academic publication searching engines, it can be concluded that this study is the first trial with analytical approaches to design a controller for rational systems.

Regarding controller design approaches possibly referred to the rational systems, these could be the reduction of rational model structure complexity, which are neural network models, linear approximation models, linearization, and iterative learning control and U-model enhanced control. A brief critical review of the approaches is presented.

Reference [21] on neural controllers is probably the first publication relating to control of rational models. However, the design approach has merely used rational models as extreme nonlinear examples; it has not designed controllers by taking the model structure into consideration (even if known in advance), except for taking the models as the representatives of complex nonlinear dynamic systems.

Piecewise linearization [22, 23] around operating points has been widely studied to simplify controller-designed procedures when plants are subject to mild nonlinear dynamics. It should be mentioned that a group of piecewise linear models can be admitted as a linear model, with varying order and parameters in different operating intervals. The promising property is using linear control design strategies directly. However, it could induce inaccuracy and dynamic uncertainty because of ignoring some inherent nonlinearities from their original nonlinear representations. Further, this method may also increase computational burden/complexity while overborrowing piecewise linear intervals to match severe nonlinearities.

Pointwise linearization has been claimed by neural network-based control and/or adaptive control, which uses linear models to approximate predominant dynamics around an operating point or every input-output dynamic gain at each time instance and then employs a neural network to determine the error induced by the linearization [24, 25]. Once again, it uses linear control system design to construct nonlinear control systems. However, this involves online neural network learning or online model iden parameter estimation, and therefore, the constructed nonlinear control system is operated under adaptive principles (the controller parameters are updated with the neural network output), even for deterministic nonlinear plants. The other related issue is the selection of neural network topology, which has no systematic procedure available to find the best fitted neural network representative.

Feedback linearization is a well-developed subject [26]. A general SISO nonlinear system is described as where is the state vector and and are the input and output, respectively. , , and are real valued and smooth mapping functions. With this model structure, a series of analogies with some fundamental features of linear control systems have been established, which provides a very useful concept in the design of nonlinear control systems using linear design methodologies. Obviously, the model has in an explicit position. The studied nonlinear rational model has no such explicit expression for input to be designed, and this immediately reveals that the methodologies rooted in the approach, although useful references, are not directly applicable in designing control of nonlinear rational systems. The other input-output linearization techniques [27] have had similar requirements for an explicit expression and special skills for state variable transformation.

Iterative learning/data-driven control/model-free control is another possible control system design methodology in avoiding model structure complexity. The approaches do not require a clear plant model structure but still need plants with some mild conditions in control [28, 29]. Again, if a rational model is available, it is wasteful without using the model information in the control system design. It is believed, particularly for man-made engineering systems/products (built up by rules/models), that any repetitive process and motion has a model existing in operation even though the model is yet to be identified.

U-Model-based control has claimed to radically relieve the dependence of plant model-oriented design foundation. The use of the plant model is effectively reduced as a reference for converting to U-model and accordingly to work out the control output [30]. U-Model-based control assumes the feasibility of using linear system design procedures to design the control of nonlinear dynamic plants with assigned response performances. The U-model control platform is illustrated in Figure 1.