Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 3835616, 9 pages

https://doi.org/10.1155/2017/3835616

## Control of Chaos Using the Controller Identification Technique

Department of Mathematics and Statistics, Federal University of Pelotas, Campus Universitário, No. 354, 96010-900 Pelotas, RS, Brazil

Correspondence should be addressed to Alexandre Molter

Received 10 July 2017; Revised 18 October 2017; Accepted 22 October 2017; Published 14 November 2017

Academic Editor: Jonathan N. Blakely

Copyright © 2017 Alexandre Molter and Fabricio B. Cabral. 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

Modeling and simulation of chaotic system with dynamic control have been extensively presented in the past decades. Several control techniques have been proposed for the control of chaos. One technique that has not been sufficiently explored for the control of nonlinear systems is the controller identification technique. This technique is based on the evaluation of controllers even if they are not online. This technique does not use a priori knowledge of the plant parameters. In this work, we propose a class of controllers candidates to follow desired trajectories. Simulation results are presented for the control of chaotic systems.

#### 1. Introduction

Phenomena which exhibit chaotic behavior appear in several areas, attracting researchers which try to describe this behavior through mathematical equations and to control its dynamic.

The search for efficient control techniques of nonlinear systems continues to be a challenge to researchers. Among a variety of controllers used in dynamical systems, proportional integrative derivative (PID) is the most common and practical to be adopted by control engineers. Since the PID controllers are commonly used in industrial control systems, they are usually adjusted by empirical methods. On the other hand, there are more sophisticated control techniques involving complex theoretical developments, which impose very restrictive hypothesis on the systems to be controlled, as nonlinear control techniques.

The theoretical success of robust control led many researchers to say that control should focus on the plant parameter estimation and its bounds. However, other researchers in the control area, such as Safonov and Tsao [1], become unsatisfied with the robust control paradigm, which required the control of a family of plants instead of the control of one, which would lead to conservative results. They stated that it was time to reformulate the control problem. A first formulation was the unfalsified control [1, 2]. This control has two major characteristics: it advances from plant parameter estimation to controller parameter estimation [3] and it considers model falsification in the sense of Popper instead of model validation [4, 5]. This was a first formulation of the controller identification problem, which in general considers a list of candidate controllers and a criterion to judge the performance of these controllers without needing to put them online [6].

In fact, there are few control techniques which require minimal information on the systems to be controlled. Among these techniques we can cite the ones based on the unfalsified control paradigm and the controller identification technique [2, 6]. The unfalsified control paradigm allows us to formulate the control problem based on experimental data [7]. The advantage of this technique, when compared to others, is that it does not require a priori knowledge of the state or physical properties of the plant. This fact illustrates a potential to be explored through the use of the controller identification technique, which can also be used for the control of nonlinear dynamical systems.

The main goal of this work is to use the controller identification technique to control the trajectories of dynamical systems with chaotic behavior. The novelty of this approach is that the plant is treated as a model free plant which accounts for any model mismatches and that we try to identify a low order controller for a complex system.

The proposed technique will be applied to control a Rössler [8], a Lotka–Volterra three-species system [9–11], and a Rössler hyperchaos model [12]. In the literature we can find several works dealing with control of chaos. We highlight those related to the control of the Rössler system [13, 14] and the predator-prey systems [15–17]. In general, one can observe in these applications that the controllers used are of proportional kind. In the same way, in this work we use the controller identification technique applied to the proportional case. More precisely, given a class of candidate controllers, we identify the ones that present the best performance. The proportional control parameter is periodically modified. This update is made in order to put a better controller online. There are several ways to update the control parameters, the one chosen in this work is the step; that is, after a certain time, the control parameters are updated.

Numerical simulation is presented to illustrate the effectiveness of the proposed technique. These simulations regard the systems previously cited with the requirement of following a desired trajectory. The controllability of each system is calculated, which allows us to show that the proposed control technique preserves the locally complete state controllability of the nonlinear controlled systems.

This paper is organized as follows. In Section 2, the technique of controller identification is presented. In Section 3, a brief explanation of local controllability is presented and the proposed technique is applied to control the chaotic Rössler, predator-prey, and Rössler hyperchaos systems with simulations. In Section 4, some concluding remarks are given.

#### 2. The Controller Identification Technique

A general overview of the controller identification technique can be found in [6]. According to this technique, the only plant information used is the plant experimental data. For the cases presented in this work, we need only the reference functions and the data (from the dynamical system) to obtain the control . Given a desired behavior, a class of candidate controllers is proposed. Then, a controller is selected through the use of a performance index and the fictitious reference concept. In this work, we apply the controller identification to a class of proportional controllers and the respective mathematical development follows below.

The control law for a proportional controller is given by and, consequently, the fictitious reference is given bywhere is a constant.

The performance index is given bywhere is a transfer function of desired behavior and is its inverse Laplace transform

Theorem 1. *Among the controllers of class (1), the one that minimizes the performance index (3) is given by where with “” denoting the convolution operation.*

*Proof. *From (1), the fictitious reference is given by (2), where is a constant.

Using (8) and (9) we have thatwhereThe minimization of the performance index means finding that satisfies the equation:which leads to our estimator for the proportional constant

One can note that is the multiplicative constant of control functions . In order to obtain the simulation results, will be periodically updated. The objective is to minimize the difference between the measured and the desired . We used a continuous time formulation for the dynamical systems.

From (1)–(4) and Theorem presented above, we obtained the constant that determines the proportionality constant of the optimal control function , given by (1). This development can be extended to any odd power of the control function . Thus, if is an odd number, we replace (1) and (2):Isolating produces

Then (8) and (10) are rewritten asrespectively. The other equations are not modified.

The controller identification technique can be translated to the following procedure.

*Step 1. *Define the space-state from the dynamical system.

*Step 2. *Obtain the experimental data of the system ; that is, define and choose the desired trajectories and initial .

*Step 3. *Integrate system (10)–(12).

*Step 4. *Find .

*Step 5. *Update the input of the system with results obtained in the previous step. Go to Step 3.

*Step 6. *After reaching the number of iterations previously stipulated to update , update it and go to Step 2.

*Step 7. *The simulation ends when the number of iterations has been reached.

Figure 1 shows a block diagram of the proposed control.