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% Radial Basis Function Neural Network for Tanker Ship Heading Regulation
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%
% By: Kevin Passino
% Version: 1/12/00
%
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clear % Clear all variables in memory
pause off
% Initialize ship parameters
% (can test two conditions, "ballast" or "full"):
ell=350; % Length of the ship (in meters)
u=5; % Nominal speed (in meters/sec)
%u=3; % A lower speed where the ship is more difficult to control
abar=1; % Parameters for nonlinearity
bbar=1;
% The parameters for the tanker under "ballast" conditions
% (a heavy ship) are:
K_0=5.88;
tau_10=-16.91;
tau_20=0.45;
tau_30=1.43;
% The parameters for the tanker under "full" conditions (a ship
% that weighs less than one under "ballast" conditions) are:
%K_0=0.83;
%tau_10=-2.88;
%tau_20=0.38;
%tau_30=1.07;
% Some other plant parameters are:
K=K_0*(u/ell);
tau_1=tau_10*(ell/u);
tau_2=tau_20*(ell/u);
tau_3=tau_30*(ell/u);
% Parameters for the radial basis function neural network
% Define parameters of the approximator
nG=11; % The number of partitions on each edge of the grid
nR=nG^2; % The number of receptive field units in the RBF
n=2; % The number of inputs
tempe=(-pi/2):(pi)/(nG-1):pi/2; % Defines a uniformly spaced vector roughly on the input domain
% that is used to form the uniform grid on the (e,c) space
tempc=(-0.01):(0.02)/(nG-1):0.01;
k=0; % Counter for centers below
% Place the centers on a grid
for i=1:length(tempe)
for j=1:length(tempc)
k=k+1;
center(1,k)=tempe(i);
center(2,k)=tempc(j);
end
end
% Plot the center points of the grid
% Convert to degrees:
centerd=center*(180/pi);
figure(1)
clf
plot(centerd(1,:),centerd(2,:),'ko')
grid on
xlabel('Error e (deg.)')
ylabel('Change in error, c (deg./sec.)')
title('Grid of receptive field unit centers (each "o" is a center)')
axis([-110 110 -.8 .8])
hold on
istar=61; % Fix a special point where you will plot a RBF - and designate its center here
% Plot an dark o over the center point of the middle RBF, and some of its neighbors
neighbors=plot(centerd(1,istar),centerd(2,istar),'ko',centerd(1,istar+1),centerd(2,istar+1),'ko',centerd(1,istar+11),centerd(2,istar+11),'ko',centerd(1,istar+12),centerd(2,istar+12),'ko')
set(neighbors,'LineWidth',2);
hold off
% Next, plot a radial basis function to show what it looks like - a Gaussian
% Define spreads of Gaussian functions
sigmae=0.7*((pi/nG)); % Use same value for all on e domain
sigmac=0.7*((0.02)/nG);
% First, compute vectors with points over the whole range of
% the neural controller inputs
e_input=(-pi/2):(pi)/50:(pi/2);
c_input=(-0.01):(0.02)/50:(0.01);
% Next, compute the neural controller output for all these inputs
for jj=1:length(e_input)
for ii=1:length(c_input)
% Pick the special RBFs
rbfistar1(ii,jj)=2*exp(-(((e_input(jj)-center(1,istar))^2)/sigmae^2)-(((c_input(ii)-center(2,istar))^2)/sigmac^2));
rbfistar2(ii,jj)=exp(-(((e_input(jj)-center(1,istar+11))^2)/sigmae^2)-(((c_input(ii)-center(2,istar+11))^2)/sigmac^2));
rbfistar3(ii,jj)=2*exp(-(((e_input(jj)-center(1,istar+1))^2)/sigmae^2)-(((c_input(ii)-center(2,istar+1))^2)/sigmac^2));
rbfistar4(ii,jj)=exp(-(((e_input(jj)-center(1,istar+12))^2)/sigmae^2)-(((c_input(ii)-center(2,istar+12))^2)/sigmac^2));
end
end
% Convert from radians to degrees:
e_inputd=e_input*(180/pi);
c_inputd=c_input*(180/pi);
% Plot a receptive field unit (one that is not scaled)
figure(2)
clf
surf(e_inputd,c_inputd,rbfistar4);
view(145,30);
colormap(white);
xlabel('Heading error (e), deg.');
ylabel('Change in heading error (c), deg.');
zlabel('R_7_3(e,c)');
title('Receptive field unit R_7_3(e,c)');
rotate3d
% Next plot several receptive field units scalied and added together (RBF output)
figure(3)
clf
surf(e_inputd,c_inputd,rbfistar1+rbfistar2+rbfistar3+rbfistar4);
view(145,30);
colormap(white);
xlabel('Heading error (e), deg.');
ylabel('Change in heading error (c), deg.');
zlabel('Radial basis function neural network output');
title('Radial basis function neural network output, 2R_6_1(e,c)+R_6_2(e,c)+2R_7_2(e,c)+R_7_3(e,c)');
rotate3d
zoom
% Next, pick the strengths for the RBF
temp=(-((nG-1)/2)):1:((nG-1)/2);
for i=1:length(temp) % Across the e dimension
for j=1:length(temp) % Across the c dimension
thetamat(i,j)=-((1/10)*(200*(pi/180))*temp(i)+(1/10)*(200*(pi/180))*temp(j));
% Saturate it between max and min possible inputs to the plant
thetamat(i,j)=max([-80*(pi/180), min([80*(pi/180), thetamat(i,j)])]);
% Note that there are only nR "stregths" to adjust - here we choose them
% according to this mathematical formula to get an appropriately shaped surface
end
end
% And, put them in a vector
k=0; % Counter for centers below
for i=1:length(temp)
for j=1:length(temp)
k=k+1;
theta(k,1)=thetamat(i,j);
end
end
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% Next, provide a plot of the RBF neural controller surface:
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for jj=1:length(e_input)
for ii=1:length(c_input)
for i=1:nR
phit(i,1)=exp(-(((e_input(jj)-center(1,i))^2)/sigmae^2)-(((c_input(ii)-center(2,i))^2)/sigmac^2));
end
delta_output(ii,jj)=theta'*phit(:,1); % Performs summing and scaling of receptive field units
end
end
% Plot the controller map
delta_output=delta_output*(180/pi);
figure(4)
clf
surf(e_inputd,c_inputd,delta_output);
view(145,30);
colormap(white);
xlabel('Heading error (e), deg.');
ylabel('Change in heading error (c), deg.');
zlabel('Controller output (\delta), deg.');
title('Radial basis function neural network controller mapping between inputs and output');
rotate3d
zoom
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% Simulate the RBF regulating the ship heading
% Next, we initialize the simulation:
t=0; % Reset time to zero
index=1; % This is time's index (not time, its index).
tstop=4000; % Stopping time for the simulation (in seconds)
step=1; % Integration step size
T=10; % The controller is implemented in discrete time and
% this is the sampling time for the controller.
% Note that the integration step size and the sampling
% time are not the same. In this way we seek to simulate
% the continuous time system via the Runge-Kutta method and
% the discrete time controller as if it were
% implemented by a digital computer. Hence, we sample
% the plant output every T seconds and at that time
% output a new value of the controller output.
counter=10; % This counter will be used to count the number of integration
% steps that have been taken in the current sampling interval.
% Set it to 10 to begin so that it will compute a controller
% output at the first step.
% For our example, when 10 integration steps have been
% taken we will then we will sample the ship heading
% and the reference heading and compute a new output
% for the controller.
eold=0; % Initialize the past value of the error (for use
% in computing the change of the error, c). Notice
% that this is somewhat of an arbitrary choice since
% there is no last time step. The same problem is
% encountered in implementation.
x=[0;0;0]; % First, set the state to be a vector
x(1)=0; % Set the initial heading to be zero
x(2)=0; % Set the initial heading rate to be zero.
% We would also like to set x(3) initially but this
% must be done after we have computed the output
% of the controller. In this case, by
% choosing the reference trajectory to be
% zero at the beginning and the other initial conditions
% as they are, and the controller as designed,
% we will know that the output of the controller
% will start out at zero so we could have set
% x(3)=0 here. To keep things more general, however,
% we set the intial condition immediately after
% we compute the first controller output in the
% loop below.
% Next, we start the simulation of the system. This is the main
% loop for the simulation of the control system.
while t <= tstop
% First, we define the reference input psi_r (desired heading).
if t<100, psi_r(index)=0; end % Request heading of 0 deg
if t>=100, psi_r(index)=45*(pi/180); end % Request heading of 45 deg
if t>2000, psi_r(index)=0; end % Then request heading of 0 deg
%if t>4000, psi_r(index)=45*(pi/180); end % Then request heading of 45 deg
%if t>6000, psi_r(index)=0; end % Then request heading of 0 deg
%if t>8000, psi_r(index)=45*(pi/180); end % Then request heading of 45 deg
%if t>10000, psi_r(index)=0; end % Then request heading of 0 deg
%if t>12000, psi_r(index)=45*(pi/180); end % Then request heading of 45 deg
% Next, suppose that there is sensor noise for the heading sensor with that is
% additive, with a uniform distribution on [- 0.01,+0.01] deg.
%s(index)=0.01*(pi/180)*(2*rand-1);
s(index)=0; % This allows us to remove the noise.
psi(index)=x(1)+s(index); % Heading of the ship (possibly with sensor noise).
if counter == 10, % When the counter reaches 10 then execute the
% controller
counter=0; % First, reset the counter
% Radial basis function neural network controller calculations:
e(index)=psi_r(index)-psi(index); % Computes error (first layer of perceptron)
c(index)=(e(index)-eold)/T; % Sets the value of c
eold=e(index); % Save the past value of e for use in the above
% computation the next time around the loop
% Next, compute the RBF output
for i=1:nR
phi(i,1)=exp(-(((e(index)-center(1,i))^2)/sigmae^2)-(((c(index)-center(2,i))^2)/sigmac^2));
end
delta(index)=theta'*phi(:,1); % Performs summing and scaling of receptive field units
%
% A conventinal proportional controller:
%delta(index)=-e(index);
else % This goes with the "if" statement to check if the counter=10
% so the next lines up to the next "end" statement are executed
% whenever counter is not equal to 10
% Now, even though we do not compute the neural controller at each
% time instant, we do want to save the data at its inputs and output at
% each time instant for the sake of plotting it. Hence, we need to
% compute these here (note that we simply hold the values constant):
e(index)=e(index-1);
c(index)=c(index-1);
delta(index)=delta(index-1);
end % This is the end statement for the "if counter=10" statement
% Next, comes the plant:
% Now, for the first step, we set the initial condition for the
% third state x(3).
if t==0, x(3)=-(K*tau_3/(tau_1*tau_2))*delta(index); end
% Next, the Runge-Kutta equations are used to find the next state.
% Clearly, it would be better to use a Matlab "function" for
% F (but here we do not, so we can have only one program).
time(index)=t;
% First, we define a wind disturbance against the body of the ship
% that has the effect of pressing water against the rudder
%w(index)=0.5*(pi/180)*sin(2*pi*0.001*t); % This is an additive sine disturbance to
% the rudder input. It is of amplitude of
% 0.5 deg. and its period is 1000sec.
%delta(index)=delta(index)+w(index);
% Next, implement the nonlinearity where the rudder angle is saturated
% at +-80 degrees
if delta(index) >= 80*(pi/180), delta(index)=80*(pi/180); end
if delta(index) <= -80*(pi/180), delta(index)=-80*(pi/180); end
% Next, we use the formulas to implement the Runge-Kutta method
% (note that here only an approximation to the method is implemented where
% we do not compute the function at multiple points in the integration step size).
F=[ x(2) ;
x(3)+ (K*tau_3/(tau_1*tau_2))*delta(index) ;
-((1/tau_1)+(1/tau_2))*(x(3)+ (K*tau_3/(tau_1*tau_2))*delta(index))-...
(1/(tau_1*tau_2))*(abar*x(2)^3 + bbar*x(2)) + (K/(tau_1*tau_2))*delta(index) ];
k1=step*F;
xnew=x+k1/2;
F=[ xnew(2) ;
xnew(3)+ (K*tau_3/(tau_1*tau_2))*delta(index) ;
-((1/tau_1)+(1/tau_2))*(xnew(3)+ (K*tau_3/(tau_1*tau_2))*delta(index))-...
(1/(tau_1*tau_2))*(abar*xnew(2)^3 + bbar*xnew(2)) + (K/(tau_1*tau_2))*delta(index) ];
k2=step*F;
xnew=x+k2/2;
F=[ xnew(2) ;
xnew(3)+ (K*tau_3/(tau_1*tau_2))*delta(index) ;
-((1/tau_1)+(1/tau_2))*(xnew(3)+ (K*tau_3/(tau_1*tau_2))*delta(index))-...
(1/(tau_1*tau_2))*(abar*xnew(2)^3 + bbar*xnew(2)) + (K/(tau_1*tau_2))*delta(index) ];
k3=step*F;
xnew=x+k3;
F=[ xnew(2) ;
xnew(3)+ (K*tau_3/(tau_1*tau_2))*delta(index) ;
-((1/tau_1)+(1/tau_2))*(xnew(3)+ (K*tau_3/(tau_1*tau_2))*delta(index))-...
(1/(tau_1*tau_2))*(abar*xnew(2)^3 + bbar*xnew(2)) + (K/(tau_1*tau_2))*delta(index) ];
k4=step*F;
x=x+(1/6)*(k1+2*k2+2*k3+k4); % Calculated next state
t=t+step; % Increments time
index=index+1; % Increments the indexing term so that
% index=1 corresponds to time t=0.
counter=counter+1; % Indicates that we computed one more integration step
end % This end statement goes with the first "while" statement
% in the program so when this is complete the simulation is done.
%
% Next, we provide plots of the input and output of the ship
% along with the reference heading that we want to track.
%
% First, we convert from rad. to degrees
psi_r=psi_r*(180/pi);
psi=psi*(180/pi);
delta=delta*(180/pi);
e=e*(180/pi);
c=c*(180/pi);
% Next, we provide plots of data from the simulation
figure(5)
clf
subplot(211)
plot(time,psi,'k-',time,psi_r,'k--')
grid on
xlabel('Time (sec)')
title('Ship heading (solid) and desired ship heading (dashed), deg.')
subplot(212)
plot(time,delta,'k-')
grid on
xlabel('Time (sec)')
title('Rudder angle (\delta), deg.')
zoom
figure(6)
clf
subplot(211)
plot(time,e,'k-')
grid on
xlabel('Time (sec)')
title('Ship heading error between ship heading and desired heading, deg.')
subplot(212)
plot(time,c,'k-')
grid on
xlabel('Time (sec)')
title('Change in ship heading error, deg./sec')
zoom
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% End of program %
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