14 AutonomousrwaterVenicleS (AUVs) are robotic submarines that can be used for a

Question

14 AutonomousrwaterVenicleS (AUVs) are robotic submarines that can be used for a variety of studies of the underwater environment. The vertical and horizontal dynamics of the vehicle must be controlled to remotely operate the AUV. The INFANTE (Figure P7.14) is a research AUV operated by the Instituto Superior Tecnico of Lisbon, Portugal. The variables of interest in horizontal motion are the sway speed and the yaw angle. A linearized model of the horizontal plane motion of the vehicle is given by -0.14-0.69 0.01xil i21=1-0.19-0.048 0.01 1x21+1-0.23|u X3 ig 0.0 1.0 0.0] La 0.0 y2 r3 where xi is the sway speed, 2 is the yaw angle, x3 is the yaw rate, and u is the rudder deflection. Obtain the discrete state-space model for the system with a sampling period of 50 ms.

Explanation / Answer

I have used MATLAB software for convering the state space model from continuous domain to discrete domain.

The code is provided below in bold letters.

clc;
clear all;
close all;

A = [-0.14 -0.69 0; -0.19 -0.048 0; 0 1 0]; % define the matrix A
B = [0.056;-0.23;0]; % define the matrix B
C = [1 0 0; 0 1 0]; % define the matrix C
D = [0]; % define the matrix D
G = ss(A,B,C,D); % define continuous state space model, G
Gd = c2d(G,50e-3,'tustin') % convert from continuous state space model, G to discrete state space model, Gd

RESULT:

>> Gd = c2d(G,50e-3,'tustin')

a =
x1 x2 x3
x1 0.9932 -0.03434 0
x2 -0.009456 0.9978 0
x3 -0.0002364 0.04994 1

b =
u1
x1 0.05976
x2 -0.23
x3 -0.00575

c =
x1 x2 x3
y1 0.04983 -0.0008585 0
y2 -0.0002364 0.04994 0

d =
u1
y1 0.001494
y2 -0.00575

Here the a, b, c and d are the matrices which are discrete equivalents of their continuous counterparts A, B, C and D.

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