**Design of Optimal PID Controller Using NSGA-II**

Algorithm and Level Diagram

Algorithm and Level Diagram

**Amal MOHARAM, Mostafa A. El-HOSSEINI, Hesham A. ALI **

Computers Engineering and Control systems Dept., Faculty of Engineering,

Mansura University, Egypt

eng.amalmoharam@yahoo.com, melhosseini@gmail.com, h_arafat_ali@mans.edu.eg

**Abstract: **This paper introduces a design for multi-objective PID controller using non-dominated sorting genetic algorithm (NSGA-II). When selecting the objectives to be optimized, it is taken into account to cover some important characteristics of the system like performance, robustness and control signals’ smoothness. The decision making is done using Level diagram tool. Three tanks liquid level system control is discussed as a case study. The results show that this tool improves the process of decision making (DM). Also, comparisons with Ziegler and Nichols (Z-N) and different optimization methods are presented.

**Keywords: **Decision making (DM), Evolutionary algorithm (EA), Level Diagram (LD), Multi-objective optimization (MO), PID controller, Three Tanks Liquid Level System.

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**CITE THIS PAPER AS**:

Amal MOHARAM, Mostafa A. El-HOSSEINI, Hesham A. ALI, **Design of Optimal PID Controller Using NSGA-II Algorithm and Level Diagram**, *Studies in Informatics and Control*, ISSN 1220-1766, vol. 24 (3), pp. 301-308, 2015. https://doi.org/10.24846/v24i3y201507

**Introduction**

In multi-objective optimization (MO), there is more than one objective to be optimized. Usually these objectives contradicts each other (i.e. optimize of one objective cannot be achieved without degradation of other objective). Hence there is no longer a single solution (as in a mono-objective optimization) but a group of trade–off solutions called Pareto points. The need for multiple Pareto points make evolutionary algorithms (EAs) more suitable for MO since the EAs work in parallel and can get more solutions in a single run[1,2].

EAs should be modified to be suitable for MO. This is due to the fact that it is required to have numerous solutions and thus need more diversity. Non-dominated sorting genetic algorithm (NSGA-II) [3] is considered one of efficient Multi-objective Optimization Evolutionary Algorithms (MOEAs). It is introduced in 2002 to overcome some shortcomings of NSGA [4]. Since then, it has proved its efficiency in many MO branches [5-8].

Decision making (DM) (i.e. selecting preferred solution points) is an important step in MO. Indeed, this is not an easy task since there are multiple trade-off solutions. Also, DM gets more complicated with the increase of number of objectives. To be able to carry out DM effectively, graphical presentation, can be helpful tool in our analysis. Scatter diagrams and Parallel coordinates [9, 10] are the most common graphical techniques used in MO analysis. However, these techniques lose clarity with increasing number of objectives. To

overcome this difficulty, a new graphical technique called Level diagram (LD) [11] is introduced in 2008. Its idea is based on plotting each objective and decision variable in a separate sub-plot .These sub-plots are related to each other. This separation yields a good visibility for each objective and decision variable hence more capability on doing DM.

PID controller is the most widely used controller in industry because of its simplicity and robustness [12]. PID controller is still the perfect choice for many plants. However, finding the optimal parameters of PID controller is quite difficult especially in non-linear control system as in the liquid level control system. So, several methods have been proposed for tuning PID controller. One of these methods is Ziegler and Nichols method [13]. It is the oldest method and simplest one. Recently, many EAs such as genetic algorithm (GA) have been employed to tune PID controller in various plants [14-16]. For tuning PID controller, there are many different measures which can be used to compare the quality of controlled responses. These measures or objectives include time response specifications of the control system (i.e. overshoot, settling time…), integral performance indices and frequency domain objectives (i.e. sensitivity, complementary sensitivity…) [12, 17-19]. When designing a control system, these objectives should be selected carefully to represent demands of decision maker. In this paper, due to its efficiency, NSGA-II is used for tuning MO-PID controller in three tanks liquid level system.

For tuning PID controller, three objectives are selected to be minimized. The aim of selecting these objectives is trying to reach the best performance of the control system while keeping anti-disturbance ability and avoiding stress of the control actuator. The analysis of results is performed using a graphic LD tool.

The rest of the paper is organized as follows. Section 2 reviews related work. Section 3 describes a case study for tuning MO-PID controller in liquid level system. Section 4 analyzes the results using LD and comparisons held between NSGA-II and different optimization algorithms. Finally, section 5 concludes this paper.

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