Mathematical Problems in Engineering

Volume 2016, Article ID 2926914, 11 pages

http://dx.doi.org/10.1155/2016/2926914

## Smooth Adaptive Internal Model Control Based on Model for Nonlinear Systems with Dynamic Uncertainties

^{1}Engineering Research Institute, University of Science and Technology Beijing, Beijing 100083, China^{2}School of Automation & Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China

Received 22 June 2016; Accepted 27 September 2016

Academic Editor: Tarek Ahmed-Ali

Copyright © 2016 Li Zhao 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

An improved smooth adaptive internal model control based on model control method is presented to simplify modeling structure and parameter identification for a class of uncertain dynamic systems with unknown model parameters and bounded external disturbances. Differing from traditional adaptive methods, the proposed controller can simplify the identification of time-varying parameters in presence of bounded external disturbances. Combining the small gain theorem and the virtual equivalent system theory, learning rate of smooth adaptive internal model controller has been analyzed and the closed-loop virtual equivalent system based on discrete model has been constructed as well. The convergence of this virtual equivalent system is proved, which further shows the convergence of the complex closed-loop discrete model system. Finally, simulation and experimental results on a typical nonlinear dynamic system verified the feasibility of the proposed algorithm. The proposed method is shown to have lighter identification burden and higher control accuracy than the traditional adaptive controller.

#### 1. Introduction

During the past few decades, the rapid development of modern industry urgently needs the improvement of the production efficiency and quality. And the modern industrial control process is a class of nonlinear dynamic systems, which typically have characteristics of time-varying complexity and diversity [1]. Therefore, a nonlinear model with wide applicability and simple structure is essential to solve control system design problems caused by various uncertainties.

In order to improve the dynamic performance and control precision for industrial systems, it is important to establish an accurate dynamic model which can well represent its dynamic characteristics [2]. An accurate dynamics model is a precondition of dynamic performance analysis and precise control. The system dynamic performance can be calculated by mechanism modeling [3]. Meanwhile, the accurate parameters of the process system can be obtained via system identification methods. Therefore, modeling technology has become an important research content in the field of industrial system control.

However, the precise mechanism of modeling based on the energy conservation law and classical physics cannot resolve complex industrial process completely [4]. Traditional mechanism modeling of numerical calculation has difficulties to deal with a variety of information in complex industrialized environmental systems and to fully reflect the industrial process systems [5]. For systems with unknown parameters and bounded disturbances, the modern system modeling developed mathematical modeling and model validation.

A suit of considerable theoretical methods have been developed for nonlinear system modeling, such as the nonlinear FIR model, finite Volterra model [6], Hammerstein model [7], and Wiener model [8]. In 2002, professor Zhu first proposed the theory of model [9], which can be seen as a special deformation structure of nonlinear autoregressive moving average with exogenous inputs (NARMAX) model, with characteristics of simple structure and wide range [10]. In 2005 Shafiq and Haseebuddin proposed -model-based internal model control for nonlinear dynamic plants, and the system achieved good robustness for both linear systems and nonlinear systems [11]. The model control system was expanded from single input single output (SISO) nonlinear system to multi-input multioutput (MIMO) nonlinear system based on MIMO model control strategy proposed by Azhar in 2008. But the MIMO model system needs a lot of data identification calculations, which costs huge time and mechanical energy [12]. Up to now model control algorithms have obtained numerous achievements; however, the convergence study of the closed-loop system is still an open problem for the model theoretical research [13].

How to combine the modeling theory and the intelligent control theory for industrial process control is a practical problem, and it is a theoretical problem to analyze the convergence of industrial control system [14]. A smooth adaptive internal model control method based on model is proposed. For time-varying parameters of nonlinear dynamic system, the proposed control method based on model simplifies the structure of the nonlinear system. By establishing parameter error of the discrete model system and using the small gain theorem, the learning rate’s stability is analyzed. Meanwhile, based on the closed-loop virtual equivalent system theory [15], the convergence proof of the model closed-loop system is completed. Besides, the simulation and experimental results show the effectiveness of the proposed algorithm.

The rest of the paper is organized as follows. In Section 2, the model structure is presented. In Section 3, a smooth adaptive internal model control system is proposed for model system, including the controller design and control input solutions. The details of convergence proof for the closed-loop model system are described in Section 4. Finally, the effectiveness of the proposed control algorithm is verified by the simulation and experimental results in Section 5 and a brief conclusion is given in Section 6.

#### 2. Mathematical Description of Model

The common nonlinear dynamic system is described as a form of NARMAX model as follows: where and denote the discrete time system output and input at time , respectively. is the error signal caused by the system volume measured noise, model error, and uncertain dynamic factors. stands for the maximum delay of the whole system. denotes the unknown nonlinear smoothing function with any known delay.

*Assumption 1. *System inputs and outputs are measureable, and the output noise is bounded, but the function is unknown.

*Assumption 2. *The highest order of the system input is known.

The control goal is to keep the system output signal tracking the reference signal under initial condition via control input design.

By extending the nonlinear function , the control input is expressed as the power series expansion at time . Meanwhile, the other signals are integrated into time-varying parameters of the system input, and expression (1) is reconstructed as follows:where denotes the degree of the control input . The time-varying parameter is the nonlinear function including the past time output , the past time input , and error signal which can be described as

In order to consult the design of linear control system, a new variable is introduced in the nonlinear model as follows:

It is worth to notice that (5) is denoted as model or the time-varying polynomial function model [9].

In order to identify the dynamic parameters in the model, the weighted recursive least square (RLS) method is used. Since the “data saturation” may occur when the data is growing, the updating function of the parameter estimates will be much weaker, so a forgetting parameter is introduced [16]:where denotes the system iterative weighted coefficient. denotes error, including the output error and identification error of the system. According to formula (5), denotes identification vector of the system parameters . denotes the system input vector defined by at time . denotes covariance matrix.

The energy performance index can be derived:

It should be noticed that when the forgetting parameter ) equals 1, formula (6) will be simplified to a weighted recursive least square method.

The online identification results of model are obtained: model is a special structure form of NARMAX model, which not only has a simple structure but also can be widely applied to many nonlinear systems. Furthermore, the design of the model control system can refer to the design of linear control system for its special structure.

#### 3. Design of Smooth Adaptive Internal Model Controller Based on Model

The nonlinear controller designed for the model takes advantage of internal model controller with simple calculation and good tracking performance. Because the model controller contains a pseudo input signal, the solution block is used to solve the real control input from a nonlinear polynomial function. The smooth adaptive internal model controller and the solution block of nonlinear equation constitute the model general controller. The design of the proposed control system is shown in Figure 1 [17].