Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 167852, 9 pages

http://dx.doi.org/10.1155/2015/167852

## Second Order Sliding Mode Control of the Coupled Tanks System

^{1}Faculty of Engineering, King Abdulaziz University, Rabigh 21911, Saudi Arabia^{2}FARCAMT, Advanced Manufacturing Institute, King Saud University, Riyadh 11421, Saudi Arabia

Received 7 March 2015; Revised 4 April 2015; Accepted 5 April 2015

Academic Editor: Qingling Zhang

Copyright © 2015 Fayiz Abu Khadra and Jaber Abu Qudeiri. 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

Four classes of second order sliding mode controllers (2-SMC) have been successfully applied to regulate the liquid level in the second tank of a coupled tanks system. The robustness of these classes of 2-SMC is investigated and their performances are compared with a first order controller to show the merits of these controllers. The effectiveness of these controllers is verified through computer simulations. Comparison between the controllers is based on the time domain performance measures such as rise time, settling time, and the integral absolute error. Results showed that controllers are able to regulate the liquid level with small differences in their performance.

#### 1. Introduction

The control of liquid level in multiple connected tanks by controlling the liquid flow is a typical nonlinear control problem in the field of process control. It is present in many industrial processes. Many researchers around the world have attempted the design and implementation of controllers for the liquid level of a coupled tanks system. Some of the controllers used to control coupled tanks systems include Proportional-Integral-Derivative (PID) type controllers [1], a parallel structure of fuzzy PID control systems [2], a nonlinear constrained predictive algorithms based on feedback linearization control [3], and fractional PID controller [4].

Sliding mode control is an efficient method for robust control of uncertain systems [5–7]. The basic idea of the first order sliding mode control (1-SMC) is to let the system converge towards a selected surface and then to stay there in spite of uncertainties and disturbances. The 1-SMC method can be designed by performing two steps. The first step is to select an appropriate sliding surface to constrain the state trajectory on it. The second step includes designing of a discontinuous control law to force the system state to reach the designed surface preferably in finite time. 1-SMC requires sliding variable relative degree (the relative degree is defined as the order of the derivative of the controlled variable, in which the control input appears explicitly) to be equal to one with respect to the control input which limits the choice of the sliding variable. The 1-SMC is also used to regulate the liquid level. An input-dependent sliding surface has been used in [8] to regulate the liquid level in a coupled tanks system. A sliding mode controller, which has a state varying sliding surface parameter, has been designed in [9]. A neuro-fuzzy-sliding mode controller using nonlinear sliding surface has been proposed in [10].

In addition to the restriction regarding the relative degree, 1-SMC also has the drawback of chattering due to high switching frequency of the control. The drawbacks of 1-SMC can be successfully eliminated by the use of higher order sliding mode controllers (HOSMC). HOSMC force the sliding variable and its successive derivatives to zero. There is no restriction on the relative degrees. As the high frequency control switching is pushed in the higher derivative of the sliding variable, chattering is significantly reduced. Another feature of HOSMC is that the detailed mathematical model of the plant is not required. The most widely used HOSMC are second order sliding mode controllers (2-SMC). Examples of 2-SMC are widely used twisting controllers and its modified variant the super twisting controllers, the quasi-continuous controllers, the suboptimal control algorithm, and the control algorithm with prescribed convergence law.

Khan and Spurgeon [11] applied a second order sliding mode control idea to control a coupled tank system.

The super twisting and adaptive super twisting control algorithms are developed for the two-spacecraft formation flying system in [12]. A 2-SMC is proposed for second order uncertain plants using equivalent control approach to improve the performance of control systems in [13]. A discrete integral sliding mode controller based on composite nonlinear feedback method to improve the transient performance of uncertain systems is proposed in [14]. A second order sliding mode controller using nonlinear sliding surface to guarantee stability as well as to enhance the transient performance of uncertain linear systems with parametric uncertainty has been proposed in [15]. An adaptive second order sliding mode controller with a nonlinear sliding surface that consists of a gain matrix having a variable damping ratio has been presented in [16]. A higher order sliding mode control algorithm is described for a class of uncertain multi-input multioutput nonlinear systems and the developed algorithm was applied on a hovercraft vessel control [17].

Despite the many existing publications related to the 2-SMC, there is a lack of articles that compare performance of different types of 2-SMC in water level of the second tank in the coupled tanks system from one side and between them and the first order controller from the other side. Moreover, to the best of authors’ knowledge, there is no published work concerning the 2-SMC especially, which contains the detailed analysis of the time domain control measures and the tracking performance of the well-known controller.

In this paper, robustness of four classes of 2-SMC, namely, twisting (TA), the super twisting (STC), prescribed convergence law controller (PCL), and the quasi-continuous controller (QCC), to regulate the water level of the second tank in the coupled tanks system is introduced and their performances are compared. Moreover, the performances of the four controllers mentioned above are compared with a first order controller to show the merits of these controllers.

The remaining structure of this paper is as follows. In the next section, the dynamic model of the coupled tank system will be explained. Section 3 briefly provides the basics of the 1-SMC controller and the 2-SMC. In Section 4, 2-SMC will be described briefly. In Section 5, the simulation results from the application of the controller will be presented and discussed. Finally, Section 6 includes the concluding remarks based on the results obtained.

#### 2. Mathematical Modeling of the Coupled Tanks System

Figure 1 shows a schematic diagram of the two-coupled tanks system. The tanks system consists of two connected tanks. A pump supplies the water into the first tank. The second tank is filled from the first tank via a connecting pipe. An outlet is located at the bottom of the second tank to change the output flow . The mathematical model of the coupled tanks system is nonlinear and can be derived as follows.