-
Notifications
You must be signed in to change notification settings - Fork 2
/
nonlinear_KF.m
193 lines (162 loc) · 4.95 KB
/
nonlinear_KF.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
clc;clear;close all;
%% Nonlinear KF Initialization
% Variable Init
Sig_w = 2.5; %Process Noise
Sig_v = 1; %Sensor Noise
%Initial State
x = 2+randn(1); xhat = 2; u = 0;
SigX = 1;
%Variable Store
iteration = 50;
xstore = zeros(1,iteration+1);
xstore(1,1) = x; % initial value
xhatstore = zeros(1,iteration);
SigXstore = zeros(1^2,iteration);
% Input, Output store for comparison:
ustore = zeros(1,iteration+1);
ustore(1,1) = u;
zstore = zeros(1,iteration);
%nonlinear System x = (4+x)^(1/3)+w+2*u, x^3+3*v
%% Extended KF
for k=1:iteration
%true value simulation
u_prev = u;
u = sin(k/pi)/3;
w = chol(Sig_w)'*randn(1);
v = chol(Sig_v)*randn(1);
z = x^3+3*v;
x = (4+x)^(1/3)+w+2*u;
%KF Prediction
Aty = (1/3)*(4+x)^(-2/3);
Bty = 1;
xhat = (4+xhat)^(1/3)+2*u_prev;
SigX = Aty*SigX*Aty'+Bty*Sig_w*Bty';
Cty = 3*xhat^2;
Dty = 3;
zhat = xhat^3;
%KF Correction
L = SigX*Cty'/(Cty*SigX*Cty'+Dty*Sig_v*Dty');
xhat = xhat+L*(z-zhat);
xhat = max(-4,xhat);
SigX = SigX-L*Cty*SigX;
% Store Data for Plotting
xstore(1,k+1) = x(:);
xhatstore(1,k) = xhat;
SigXstore(1,k) = SigX(:);
% Later Use for Comparison
ustore(1,k+1)=u;
zstore(1,k) = z;
end
%% Central Difference Kalman Filter
Nxa = 3;
h = sqrt(3);
alpham(1) = (h*h-Nxa)/(h*h);
alpham(2) = 1/(2*h*h);
alphamk = alpham(2);
alphac = alpham;
alpha = [alpham(1) alphamk(:,ones([1 2*Nxa]))]';
xhat2 = 2;
SigX2 = 1;
xhatstore2 = zeros(iteration,length(xhat2));
SigXstore2 = zeros(iteration,length(xhat2)^2);
for k=1:iteration
% Prediction - Obtain Sigma Points
xhat_aug = [xhat2;0;0];
Sig_aug = blkdiag(SigX2,Sig_w,Sig_v);
sqr_Sig_aug=chol(Sig_aug,'lower');
X = xhat_aug(:,ones([1 2*Nxa+1]))+h*[zeros(Nxa,1), sqr_Sig_aug,-sqr_Sig_aug];
% Prediction - xhat, zhat
X_x = (4+X(1,:)).^(1/3)+X(2,:)+2*ustore(1,k);
xhat2 = X_x*alpha;
square_X = (X_x(:,2:end) - xhat2(:,ones(1,2*Nxa)))*sqrt(alphac(2));
square_X1 =X_x(:,1)-xhat2;
SigX2 = square_X*square_X'+alphac(1)*square_X1*square_X1;
Z = X_x.^3+3*X(3,:);
zhat2 = Z*alpha;
% Correction
square_Z = (Z(:,2:end)-zhat2*ones(1,2*Nxa))*sqrt(alphac(2));
square_Z1 = Z(:,1)-zhat2;
SigZ = square_Z*square_Z'+alphac(1)*square_Z1*square_Z1';
SigXZ = square_X*square_Z'+alphac(1)*square_X1*square_Z1';
Lsp = SigXZ/SigZ;
xhat2 = xhat2+Lsp*(zstore(1,k)-zhat2);
SigX2 = SigX2-Lsp*SigZ*Lsp';
xhatstore2(k,:) = xhat2;
SigXstore2(k,:) = SigX2(:);
end
%% Unscented Kalman Filter
nxa = 3;
alp = 0.2;
kepa = 3-nxa;
beta = 2;
lambda = alp^2*(nxa+kepa)-nxa;
alpham1 = lambda/(nxa+lambda);
alphamk = 1/(2*(nxa+lambda));
alphac1 = lambda/(nxa+lambda)+(1-alp^2+beta);
alphack = 1/(2*(nxa+lambda));
alpha_mean = [alpham1 alphamk(:,ones(1,2*Nxa))]';
alpha_cov = [alphac1 alphack(:,ones(1,2*nxa))]';
xhat3 = 2;
SigX3 = 1;
xhatstore3 = zeros(iteration,length(xhat3));
SigXstore3 = zeros(iteration,length(xhat3)^2);
for k=1:iteration
% Prediction - Obtain Sigma Points
xhat_aug = [xhat3;0;0];
Sig_aug = blkdiag(SigX3,Sig_w,Sig_v);
sqr_Sig_aug=chol(Sig_aug,'lower');
X = xhat_aug(:,ones([1 2*nxa+1]))+sqrt(nxa+lambda)*[zeros(nxa,1), sqr_Sig_aug,-sqr_Sig_aug];
% Prediction - xhat, zhat
X_x = (4+X(1,:)).^(1/3)+X(2,:)+2*ustore(1,k);
xhat3 = X_x*alpha_mean;
square_X = (X_x(:,2:end) - xhat3(:,ones(1,2*nxa)))*sqrt(alphack);
square_X1 =X_x(:,1)-xhat3;
SigX3 = square_X*square_X'+alphac1*square_X1*square_X1;
Z = X_x.^3+3*X(3,:);
zhat3 = Z*alpha_mean;
% Correction
square_Z = (Z(:,2:end)-zhat3*ones(1,2*nxa))*sqrt(alphack);
square_Z1 = Z(:,1)-zhat3;
SigZ = square_Z*square_Z'+alphac1*square_Z1*square_Z1';
SigXZ = square_X*square_Z'+alphac1*square_X1*square_Z1';
L_unscent = SigXZ/SigZ;
xhat3 = xhat3+L_unscent*(zstore(1,k)-zhat3);
SigX3 = SigX3-L_unscent*SigZ*L_unscent';
xhatstore3(k,:) = xhat3;
SigXstore3(k,:) = SigX3(:);
end
%% Plot
figure(1)
subplot(3,1,1)
hold on
plot(0:iteration-1,xstore(1:iteration)','k-')
plot(0:iteration-1,xhatstore','r-')
plot(0:iteration-1,xhatstore'+3*sqrt(SigXstore'), 'b--', ...
0:iteration-1,xhatstore'-3*sqrt(SigXstore'), 'b--')
grid on
legend('true x','estimate x (Extended KF)','error bounds')
title("Extended Kalman Filter")
ylim([-2,13])
hold off
subplot(3,1,2)
hold on
plot(0:iteration-1,xstore(1:iteration)','k-')
plot(0:iteration-1,xhatstore2','r-')
plot(0:iteration-1,xhatstore2'+3*sqrt(SigXstore2'), 'b--', ...
0:iteration-1,xhatstore2'-3*sqrt(SigXstore2'), 'b--')
grid on
legend('true x','estimate x (CDKF)','error bounds')
ylim([-2,13])
title("Central Difference Kalman Filter")
hold off
subplot(3,1,3)
hold on
plot(0:iteration-1,xstore(1:iteration)','k-')
plot(0:iteration-1,xhatstore3','r-')
plot(0:iteration-1,xhatstore3'+3*sqrt(SigXstore3'), 'b--', ...
0:iteration-1,xhatstore3'-3*sqrt(SigXstore3'), 'b--')
grid on
legend('true x','estimate x (UKF)','error bounds')
ylim([-2,13])
title("Unscented Kalman Filter")
hold off