Identification and Adaptive Fuzzy Control for Navigation Systems of Autonomus Vehicles

Indentificación y control difuso adaptativo para sistemas de navegación de vehículos autónomos

Juan Contreras¹
Manuel Ariza²
Luís Velásquez³
Alcides Ramos⁴
Vladimir Díaz⁵
Julián Arbeláez⁶

Abstract

This article presents a simple method to generate the automatic course control of surface naval vehicles by using fuzzy logic to adjust the parameters of the PID controller. The fuzzy systems constructed to automatically adapt the parameters of the PID controller are Mamdani type and use functions of triangular relevance for the antecedent and the consequent. An application to a real model is presented, exposing the identification process to obtain the mathematical model from experimental data, the temporal analysis from the mathematical model, obtain the parameters from the PI controller, and the results on course changes between 125° and 305°. Thereafter, and bearing in mind the range of Kp and Ti parameters in which the system is stable, a structure of a self-tunable fuzzy PI controller was presented and implemented with notable improvement in the response system.

Key words: identification, root locus, PID control, fuzzy PI control, self-tuning

Resumen

Este artículo presenta un método sencillo para la generación del control automático del rumbo de vehículos navales de superficie empleando la lógica difusa para ajustar los parámetros del controlador PID. El sistema difuso construido para adaptar automáticamente los parámetros del controlador PID es del tipo Mamdani y emplea funciones de pertenencia triangulares para el antecedente y el consecuente. Se presenta una aplicación a un modelo real, exponiendo el proceso de identificación para la obtención del modelo matemático a partir de datos experimentales, el análisis temporal del modelo matemático, la obtención de los parámetros del controlador PI, y los resultados ante cambios de rumbo entre 125° y 305°. Posteriormente, y teniendo en cuenta el rango de los parámetros Kp y Ti en que el sistema es estable, se presentó e implementó una estructura de un controlador PI difuso autosintonizable con una mejora notable en la respuesta del sistema.

Palabras claves: identificación, lugar de las raíces, control PID, control PI difuso, autosintonización

Date Received: December 13th, 2012 - Fecha de recepción: 13 de Diciembre de 2012
Date Accepted: February 19th, 2013 - Fecha de aceptación: 19 de Febrero de 2013

________________________

¹Faculty of Naval Engineering - Escuela Naval Almirante Padilla, Cartagena de Indias, Colombia. e-mail: epcontrerasj@ieee.org
²Faculty of Naval Engineering - Escuela Naval Almirante Padilla, Cartagena de Indias, Colombia. e-mail: amatgwill@hotmail.com
³Faculty of Naval Engineering - Escuela Naval Almirante Padilla, Cartagena de Indias, Colombia. e-mail: jdesem@gmail.com
⁴Faculty of Naval Engineering - Escuela Naval Almirante Padilla, Cartagena de Indias, Colombia. e-mail: alcidesramos@yahoo.es
⁵COTECMAR. Research, Development, and Innovation Department. Cartagena, Colombia. e-mail: vdiaz@cotecmar.com
⁶Faculty of Naval Engineering - Escuela Naval Almirante Padilla, Cartagena de Indias, Colombia. e-mail: yeha8040@hotmail.com

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Introduction

An autopilot is one of the most important devices on board a ship; it is connected to its governor system and makes course corrections based on information from the compass, GPS, and digital compass, among other sensors. An autopilot with adaptable parameters and optimization techniques incorporated can contribute to maintaining or following a desired course in such an efficient manner that it can reduce the route by up to 5%, thus, also reducing fuel consumption (Witkowska and Smierzchalski, 2008; Krishna et al., 2011).

The Proportional-Integral-Derivative (PID) control technique has been widely used in autopilots because of its simplicity and ease of implementation. However, the performance of PID controllers has not shown good results in the changing environmental conditions that can be encountered by the ship in its navigation route (Nguyen et al., 2003; Loo and Mastorakis, 2007).

To design the course controller, it is necessary to have a mathematical model that describes, as closely as possible, the ship’s dynamics. The motion of the ship at sea is described by a set of six complicated differential equations related to its six degrees of freedom (Tzeng and Chen, 1999; Fossen, 2002). However, it is a common practice to consider only the horizontal plane, which is why the degrees of freedom are reduced to three and, thereby, the mathematical model is simplified (Krishna et al., 2011). One of the most used models for the preliminary design of the course controller is the Nomoto model (Velazco et al., 2008; Chang et al., 2010; Krishna et al., 2011) due to its simplicity.

Fuzzy logic has turned out to be a very effective technique to overcome inconveniences presented by PID control, permitting the generation of fuzzy controllers with great adaptation capacity. In some cases, it is used as automatic adjustment mechanism (selftuning) of PID controller parameters (Velagic et al., 2003; Nguyen et al., 2003; Loo et al., 2007; Contreras, 2011; Contreras et al., 2011).

This article is organized as follows. Section 2 presents a brief description of the scale model (surface vehicle) used during the identification and control tests; Section 3 describes the identification process used to obtain the mathematical model of the surface vehicle; Section 4 analyzes the temporal behavior of the mathematical model obtained and proceeds to calculate the parameters of the PID controller; Section 5 gives the structure of the self-tunable fuzzy PI controller and the results obtained, followed by the conclusions.

Description of the scale model

The surface vehicle used belongs to the Almirante Padilla Naval School, located in Cartagena, Colombia, and has a wooden hull, with a length of 0.78 m and a 0.22-m breadth. The dimensions of the model are presented in Table I and Fig. 1.

The surface vehicle contains, among others, the following elements on board: electronic compass, GPS; accelerometers, propulsion motors, servomotors and rudder system, sonar, video camera, and RF communication module.

Indentification of the Dynamics the Surface Model

A simple model describing the dynamic behavior of a ship can be expressed through Nomoto’s third-order model (Tzeng and Chen, 1999; Du and Guo, 2004):

where is the course angle and is the rudder angle. If we assume initial conditions null, the Nomoto equation can be represented in the Laplace domain through the following transference function:

The prior equation could be obtained through identification techniques using experimental data and selecting one of the structures of models included in the IDET identification tool by MATLAB (arx, armax, oe, etc.) that best approaches the dynamics represented by the experimental data.

To identify the course a maneuver similar to the zig-zag curve was used, varying the rudder angle between 20° and -20° (Fig. 3), which causes course changes of the physical model to port and starboard. The platform maintains approximately constant velocity during the whole maneuver. Several tests were conducted in the Bay of Cartagena de Indias, in El Laguito sector, seeking to obtain a mathematical model with enough precision to proceed to design the course control system.

We sought to obtain a low-order model, but due to the delay presented by the model’s response it was not possible to reach a second-order model that permitted an analogy of the coeffcients with the structure of the model by Nomoto. The best model was obtained with the AutoRegressive Moving Average eXogen (ARMAX) structure, whose discrete time representation is shown by the following:

The mathematical model obtained is of fourth order, with a pole close to the origin (s1=0.0067), but on the positive side of the complex S plane, which indicates a tendency to instability. It presents a pole on the real negative axis (s2 = -18.3478) and two complex poles conjugated with real negative part (s3 = -33.3394 + j0.7993; s4 = -33.3394 - j0.7993), which give an indication that the system can have damped oscillations in its response (as long as it is taken out of the instability zone).

Design of the Pid Controller

After obtaining the mathematical model, we proceeded to design a PID controller to regulate the course of the surface vehicle. The first step was to obtain the system’s Root locus (vehicle) to know its temporal behavior (Fig. 5).

The Root locus shows that the system has a pole that at low system gain is located near the origin but on the positive side of the S plane, which generates unstable behavior. "is pole can be displaced to the left side of the S plane if the system’s closed loop gain is greater than 0.0153. If the closed loop gain increases above 90, the system will again fall in instability. For the closed loop gain of 90, the system will have a critically stable behavior, which permits applying the Ziegler – Nichols method to design the PID controller.

Fig. 6 shows the application of a gain variable to the system to determine the critical gain, Kcr, which would bring the system to the critically stable state.

The value of Kcr is 12.7. The critical period is obtained from Fig. 7. The critical period, Pcr, is 0.93 seconds. With the value of the critical gain, Kcr, value of 12.7, we proceeded to seek the controller parameters in Table 2.

That is, to obtain the PI controller, the parameters will be:

It should be noted that the SIMULINK PID equation di%ers from the conventional equation, thus:

Conventional equation of the PID:

SIMULINK PID equation:

That is, Kp = P; I = Kp/Ti; D = Kp*Td. Hence, for the SIMULINK simulation use P = 6.35 and I = 6.35/0.775 = 8.19

The system response with PID controller upon a 20-degree reference signal is shown in Fig. 8.

An error is reached in stationary state equal to zero; however, the over impulse is high. The results obtained in the tests on the prototype resulted with a higher over impulse due to delay in controller data transmission from the PC.

The proportional gain was adjusted to diminish over impulse and a better response was obtained with P = 3 (Fig. 9).

The results of the vehicle’s real response to reference signal changes (desired course) from 300° to 125°, regulated with the PI controller, are shown in Fig. 10.

The real system response, shown in Fig. 10, presents oscillations, especially when the course must be maintained. Hence, proceeding to include an adaptive fuzzy system that adjusts the gain from the PI controller according to the error size so that at high error values the controller action was higher than at low values, where the course must be maintained.

Fig. 11 shows the structure of the self-tunable fuzzy PI controller:

The range of the output universe of the proportional fuzzy adaptor was defined between 0.1 and 6 for the Kp generated not to bring the system to the instability zone. The range of the output universe of the integral fuzzy adaptor was defined between 1 and 100 for the Ki generated not to cause high oscillations.

The range of the error input variable was estimated between -180° and 80°. For both fuzzy adaptation systems, five fuzzy triangular sets were considered for the error input variable, symmetrically distributed so that the minimum values (-180°) and maximum values (180°) coincided with the modal value of the sets of two of the extremes. Likewise, three triangular sets were considered for the output variables. For each case, five fuzzy rules were considered (Table 3).

Fig. 12 shows the surface of the adaptive fuzzy system for the controller’s proportional gain.

It can be noted from Fig. 12 that Kp takes values according to the error value, that is, depending on how off course the vehicle is.

Results

Results of the vehicle’s real response upon reference signal changes (desired course) from 305° to 125°, regulated with self-tunable fuzzy PI controller, are shown in Fig. 13. Note a response without over impulses or significant oscillations, although with a slightly higher setting time than with the PI controller, given that Kp and Ti values diminish as the desired response (course) is reached.

Conclusions

This work presented the identification process of a surface vehicle scale model; analysis of the temporal response from the information delivered by the root locus of the mathematical model obtained; and, with the temporal analysis proceeded to obtain the parameters of a PI controller to regulate the vehicle’s change of course, employing Ziegler Nichols second method. The data obtained from the application of the PI controller in a real operation of the vehicle permitted seeing that the vehicle followed the change of course ordered, but presented high oscillations in the permanent response (maintaining the course).

A simple structure was presented of a self-tunable fuzzy PI controller in which it is only necessary to indicate the range in which the Kp and Ti parameters can vary. Thus, if the error diminishes, then the gain (Kp and Ti) also diminishes, considerably reducing the oscillations in stable or permanent state.

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