Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 3863147, 9 pages

http://dx.doi.org/10.1155/2016/3863147

## Optimization of the Parameters of RISE Feedback Controller Using Genetic Algorithm

^{1}Faculty of Engineering, King Abdulaziz University, Rabigh 21911, Saudi Arabia^{2}Princess Fatima Alnijiris’s Research Chair for Advanced Manufacturing Technology (FARCAMT), King Saud University, Riyadh 11421, Saudi Arabia^{3}Industrial Engineering Department, King Saud University, Riyadh 11421, Saudi Arabia

Received 4 March 2016; Revised 22 May 2016; Accepted 7 June 2016

Academic Editor: Yan-Jun Liu

Copyright © 2016 Fayiz Abu Khadra 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

A control methodology based on a nonlinear control algorithm and optimization technique is presented in this paper. A controller called “the robust integral of the sign of the error” (in short, RISE) is applied to control chaotic systems. The optimum RISE controller parameters are obtained via genetic algorithm optimization techniques. RISE control methodology is implemented on two chaotic systems, namely, the Duffing-Holms and Van der Pol systems. Numerical simulations showed the good performance of the optimized RISE controller in tracking task and its ability to ensure robustness with respect to bounded external disturbances.

#### 1. Introduction

Chaos is the complex, unpredictable, and irregular behavior of systems. The response of a chaotic system is sensitive to a change in its initial conditions. Chaos can be found in many applications such as oscillators, biology, chemical reactions, robotics, lasers, and many other applications. For example, Kengne et al. [1] considered the dynamics and synchronization of improved Colpitts oscillators designed to operate in ultrahigh frequency range. Also, two-well Duffing oscillator with nonlinear damping term proportional to the power of velocity was considered in [2]. Novel swarm dynamics and their applications in automated multiagent systems biology were presented [3]. Also, application of chaos theory to the molecular biology of aging was presented [4]. For chemical reactions, Petrov et al. [5] applied map-based, proportional-feedback algorithm to stabilize the behavior in the chaotic regime of an oscillatory chemical system. Gaspard [6] showed that, for different chemical reactions, the reaction rate can be related to the characteristic quantities of chaos. In the field of robotics, Volos et al. [7] experimentally investigated the coverage performance of a chaotic autonomous mobile robot. A smart scheme for chaotic signal generation in a semiconductor ring laser with optical feedback was proposed in [8].

Many studies have been conducted to analyze and control chaotic systems. Chaotic systems are utilized as a benchmark for testing the performance of controller. Different control techniques have been tried to control uncertain nonlinear systems. Shi et al. [9] designed adaptive delay feedback controllers to control and suppress chaos in ultrasonic motor speed control system. In [10], a nonlinear feedback linearization control method combined with a modified adaptive control strategy was designed to synchronize the two unidirectional coupled neurons and stabilize the chaotic trajectory of the slave system to desired periodic orbit of the master system. Sundarapandian [11] proposed explicit state feedback control laws to regulate the output of the Tigan system so as to track constant reference signals. Furthermore, a new state feedback control law to regulate the output of the Sprott-G chaotic system was derived [12]. Also, Yu et al. [13] proposed a fuzzy adaptive control approach based on a modular design for uncertain chaotic Duffing oscillators. In [14], sliding mode adaptive controllers were proposed for synchronization of uncertain chaotic systems. The active backstepping technique was used for synchronization between two Josephson junction systems evolving from different initial conditions [15].

Fuzzy logic has been largely employed in the last decade for control and identification of nonlinear systems. A fuzzy adaptive controller combined with state observer was used for nonlinear discrete-time systems with input constraint [16]. Adaptive fuzzy control was employed to control unknown nonlinear dynamical systems [17]. Fuzzy adaptive inverse compensation method was proposed for a tracking control problem of uncertain nonlinear systems with generalized actuator dead zone [18]. A control method based on a neural network adaptive leader-following consensus control for second-order nonlinear multiagent systems was proposed in [19]. Fuzzy neural network-based adaptive control for a class of uncertain nonlinear stochastic systems was proposed [20].

A controller for uncertain nonlinear systems called the “robust integral of the sign of the error” (RISE) was proposed in [21]. This method utilizes a continuous control signal to compensate for bounded external disturbances and uncertainties of dynamic system. This robust controller is suitable for nonlinear systems whose dynamics have continuous derivative [22]. The RISE control method differs from the first-order sliding mode control by the use of the integral of the signum of the error. This idea enables an asymptotic tracking and eliminates chattering that is a problem in conventional sliding mode controllers. The RISE controller has been applied to different types of systems, such as autonomous flight control [23], control of an autonomous underwater vehicle [24], control of special classes of multiple input multiple output nonlinear systems [25, 26], and uncertain nonlinear system with unknown state delays [27], and compensates for structured and unstructured uncertainties. An integration of multilayer feedforward neural network with RISE feedback was proposed in [28]. A control method based on RISE feedback and NN feedforward for nonlinear systems with uncertainty was proposed in [29].

Many optimization techniques were introduced to optimize the controller parameters of continuous time nonlinear systems. For example, adaptive parameter control was presented for nonlinear parameter systems in many research studies [30–35]. Furthermore, nonlinear systems with completely unknown dynamic were optimized by using intelligent control-based adaptive design such as the fuzzy control system [17, 36–39] and the neural network control systems [19, 40–45]. On the other hand, the adaptive control of discrete-time nonlinear systems was also built in many other studies by utilizing the fuzzy logic systems [16, 46–48] and the neural networks [49–52]. Liu et al. [48] have presented an adaptive fuzzy controller for nonlinear discrete-time systems with unknown functions and bounded disturbances. Liu and Tong [38] have extended the previous work for a class of multi-input-multioutput (MIMO) problem. Moreover, an adaptive fuzzy controller design for a specific division of nonlinear MIMO systems in an interconnected form was explored by Liu and Tong [17]. Furthermore, Li et al. [53] presented a study for the menace of fuzzy control for nonlinear networked control systems with packet dropouts and uncertainties in parameters based on the interval type 2 fuzzy model based approach. Finally, model identification and adaptive control design are performed on Denavit-Hartenberg model of a humanoid robot. The study focused on the modeling of the 6-degree-of-freedom upper limb of the robot using recursive Newton-Euler formula for the coordinate frame of each joint. It also utilized the particle swarm optimization method to optimize the trajectory of each joint [54].

The aim of this study is to test the performance of the RISE controller in controlling chaotic systems. The RISE controller requires only the error (the difference between the reference set point and the output of the system). To obtain the first needed derivative of the error, a real time differentiator is used. Genetic algorithm (GA) is utilized to obtain the optimum controller parameters. The fitness function used is a combination of the integral of the absolute error and the integral of absolute of the control signal. The remaining structure of this paper is as follows. In the next section, the basics of RISE controller are explained. In Section 3, the characteristics of the GA are given. In Section 4, the simulation results from the application of the controller are presented and discussed. Finally, Section 5 concludes this paper.

#### 2. RISE Controller

Uncertain nonlinear systems can be described as [55] where is the state vector, is a given nonlinear function and is the control input, being the unmodeled dynamics of the system, and is the varying external disturbance with time. The superscript denotes the order of differentiation. is a control signal.

In general, the parameter uncertainty and the external disturbance are assumed to be bounded.

The objective of the control problem is to ensure that in spite of the external disturbances and modeling uncertainties the state will follow a desired reference signal in the state space. The output tracking error is given bywhere represent the reference trajectory which is assumed to be bounded continuous time derivatives. The main control objective is to ensure that the output tracking error converges asymptotically to zero; that is, as by designing a continuous robust control law.

To facilitate the control design, auxiliary error signals, denoted as , are defined in the following manner [21]:For a second-order system, the auxiliary error signal is given by [21]

The RISE controller to control system (1) for is as follows [23]:where , , , and are the controller parameters, all the parameters are constant and positive, and is the known sign function, which can be defined as

The first derivative of the error can be calculated in real time by a differentiator. A recommended real time differentiator for industrial application can be defined by its transfer function as follows: where is a time constant and , are the signal and its derivative, respectively. To attenuate high-frequency noises, the differentiator has a low pass filter (LPF) . An accurate estimation can be obtained by choosing a small in the noise-free case.

The saturation block imposes upper and lower limits on the control signal. Output the signal, but only up to some limited magnitude, and then cap the output to a value of . The saturation function is an odd function. The saturation function is given by

Figure 1 shows a flow chart detailing the implementation of the above described procedures of the closed loop control system.