424. SUBMARINE SLIDING MODE CONTROLLER OPTIMISATION USING GENETIC ALGORITHMS. E. W. McGookin, D. J. Murray-Smith and Y. Li. University of ...
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SUBMARINE SLIDING MODE CONTROLLER OPTIMISATION USING GENETIC ALGORITHMS
E. W. McGookin, D. J. Murray-Smith and Y. Li University of Glasgow, Scotland, UK.
ABSTRACT
2. GENETIC ALGORITHMS
A nonlinear control application of Genetic Algorithm (GA) optimisation techniques is stuhed in this paper. The theory of the application is presented from its basis in Darwinian evolution to the development of the form of optimisation algorithm used in the application. This involves Sliding Mode (SM) controllers for the diving and heading manoeuvres of a linear submarine model. The GA optimised results are compared with handtuned results in terms of performance and acquisition time. Post-optimisation analysis of the GA results has shown that in order to obtain good performance the GA solution involves controllers which operate in a specific region. The selection of this region effectively changes the structure of the controllers.
Genetic algorithms (GAS) provide a basis for a search method which is thought to be one of the most powerful [1,2,4,5,6] and their use has increased dramatically in the last few years due to favourable publicity. This method is based on the natural selection process which was outlined by the Darwinian principle of survival of the Jittest. This stated that species evolve through their fittest genetic variation until the species reaches its evolutionary optimum.
1. INTRODUCTION
The process of obtaining an optimum nonlinear controller by the hand-tuning of parameters can be Micult and tedous due to interactions between the controller parameters. This is particularly the case when the designer is unfamiliar with the controller structure and the design method. In recent years many optimisation techniques have emerged that can tune these parameters automatically and thus avoid the tedium of hand-tuning. The most widely used of these are called Genetic Algorithms [ 1,2]. These are methods which investigate the problem search space by simulating the evolution of a population of parameters as it goes from generation to generation. The approach follows the Darwinian theory of survival of the jttest which was applied to the evolution of species. The evolution of the parameters is achieved by the reproduction of encoded chromosomes made up of integers. This paper analyses the optimisation of Sliding Mode (SM) controllers for a submarine [ 3 ] . Its structure is as follows. Section 2 outlines the theory of genetic algorithms. Section 3 describes the submarine control systems and the process of evaluation. Section 4 shows the simulation results obtained by hand-tuning the controllers and by optimising them using a GA routine. Finally, section 5 concludes this paper by discussing the findings drawn from these results.
Simulating this process, GA methods also use nomenclature from natural genetics to define its component parts and operations [1,2,4,5,6]. The GA searches the problem solution space by using strings to represent the parameters to be optimised. These strings are called chromosomes and their individual components are called genes (each having an integer value range from 0 to 9 [4,6] in the current work). A number of these chromosomes can be initially generated at random. This is called the population and the number of chromosomes is the population size. This initial population is the first in the generation and may be evaluated by the following steps (see Figure 1) [ 1,2,4,5,6]. Firstly, the indwidual chromosomes are decoded from their integer representation into the form used in the optimisation problem (usually real numbers). The decoded parameters are then used in the problem in question (usually by simulation) to obtain evaluations whch show how close the chromosomes' responses are to the desired response of the problem in question. In this GA application the evaluation is achieved by using a cost function that must be minimised in order to obtain optimum results. The cost function is defined by the difference between the desired and actual responses obtained from the evaluations and the GA looks for an optimum with a low cost (i.e. low cost means that the actual response is close to the desired). In a GA the parameter values are varied by the operations of Reproduction, Crossover and Mutation [ 1,2,4,5,6] which change the chromosomes and allow the GA to search in different areas of the search space. After the chromosomes are altered to form the new population, they are incorporated in the optimisation process which is repeated a set number of times. T h s
UKACC International Conference on CONTROL '96,2-5 September 1996, Conference Publication No. 427 0 IEE 1996
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POPULATION & DECODING
’ t
EVALUATE COST
4
SORTINTOORDER
Figure 2: Subsystem Configuration
CROSSOVER, MUTATION &,DECODING
submatrix (vector foam) of the subsystem) and x d is the desired state vlxtor [3]. This controller form provides hard switching due to the q sgn(o) whch is a characteristic of this type of SMC [3]. It is called hard switching because it does not provide smooth transition between the switching states. The magnitude of this switching action is determined by q , the switching gain and the switchng action is governed by the sliding surface (3 [3,8,9] where:
I SIMULATE APPLICATION
I
+ EVALUATE COST
REPEAT U N n L GENERATION SEE IS REACHED
Figure 1: GA Algorithm number is called the generation size and when it is reached, the GA has reached the optimum. 3. SUBMARINE APPLICATION
The application in this study is the control system of a linearised military submarine model. This model may be represented by the following state space equation [3 J (see Appendix for model definition)
o = h’ ( x - x ~ )= hTAx
(3)
The use of the signum function causes the controller to oscillate rapidly (chuttering) [31, which is undesirable. To remove this, the signum function is replaced by a saturation function [3,7,8,9]i.e. u=-kx+(hTb)-’ h T X d - q s a l ( ~ ~ l (4) ’J where
X=Ax+Bu
(1)
The control system consists of two sliding mode (SM) controllers which are optimised by the GA. Each of these controls a separate motion of the submarine which in this case are the diving and heading motions [3]. In order to apply these controllers, the Multi-Input Multi-Output (MIMO) submarine model has been divided into two Single Input Single Output (SISO) subsystems which describe the dynamics of each individual motion being controlled [3,7,8,9]. The signals from these controllers are generated from step responses and then applied to the submarine model as elements of the input vector U in equation (1) (see Figure 2). The resulting controllers have been developed from the following form [3,8,9]
(5)
Tlus provides a gradual transition between the switching states for so$ switching. The transition region is called the boundary layer and 4 is the boundary layer thickness. However, this function is discontinuous and it has been noted that the continuous hyperbolic tan function gives almost the same response (see Figure 3) [3,7,8]. Therefore, the controllers used in thls study are of the following form [3,7,8,9] U = -kx
u=--l
