Appendix A MATLAB Codes for Parameter Identiﬁcation of …978-3-319-58319-8/1.pdf · 127 Ob29 = tanh(r*dq2); 128 129 130 %% Discrete filtering of the regression matricesOmega_a

Appendix AMATLAB Codes for Parameter Identificationof the Underactuated Mechanical Systemsin Chap. 3

A.1 Identification of the Furuta Pendulum

This section contains theMATLABcode for the identification of the Furuta pendulumin Sect. 3.2. The data measured from the identification experiment of the Furutapendulum can be downloaded from https://www.dropbox.com/s/ahs8c9z9swl1l54/furutadataexp.mat?dl=1. The identification procedure is performed in the followingorder: (i) The measured position signals are filtered; (ii) The velocity estimationalgorithm for the computation of the first and second joint velocities is applied;(iii) The filtered regression model is computed; and (iv) The LS algorithm is appliedand the estimated parameters θ are obtained.

1 % Motion Control of Underactuated Mechanical Systems %2 % Parameter identification of the Furuta Pendulum %3 % Created: 12/Oct /2016 %4 % Chapter 3, Section 3.2 ...

%5

6 %% Data from the identification experiment are loaded %7 clc8 clear all9 % The file furutadataexp.mat containing the data measured from %

10 % the identification experiment of the Furuta pendulum can be %11 % downloaded from:12 % www.dropbox.com/s/ahs8c9z9swl1l54/furutadataexp.mat?dl=1 %13

14 load furutadataexp.mat15

16 time =20.0; % Time of the identification experiment %17 T=0.001; % Sample time %18 l=time/T; % Number of samples to be considered %19 t=Posicion.time (1:l); % Time vector from experimental data %20

21 % Position signals from experimental data %22 q1=Posicion.signals (1,2) .values (1:l,1); % Pendulum position %23 q2=Posicion.signals (1,3) .values (1:l,1); % Wheel position %24

25 % Torque constant of the DC motor model MBR3410NI %26 KT=0.1757;27 KI=2.0; % Volatje/current relation of the servoamplifier %

© Springer International Publishing AG 2018J. Moreno-Valenzuela and C. Aguilar-Avelar, Motion Control of UnderactuatedMechanical Systems, Intelligent Systems, Control and Automation: Scienceand Engineering 88, DOI 10.1007/978-3-319-58319-8

189

http://dx.doi.org/10.1007/978-3-319-58319-8_3

http://dx.doi.org/10.1007/978-3-319-58319-8_3

https://www.dropbox.com/s/ahs8c9z9swl1l54/furutadataexp.mat?dl=1

https://www.dropbox.com/s/ahs8c9z9swl1l54/furutadataexp.mat?dl=1

190 Appendix A: MATLAB Codes for Parameter Identification …

28 % Applied torque of the arm u_1(t) from the experimental data %29 % The torque applied to the arm is tau(t)=KT*KI*v(t) %30 u1=Posicion.signals (1,1) .values (1:l,1)*KI*KT;31 u2=zeros(l,1); % Input signal of the pendulum %32

33

34 %% Filtering of the joint position measurements q1 and q2 to %35 % reduce the quantization error from the experimental data %36 % Filter design %37 N=30; % Filter order %38 Fc=0.07; % Normalized cutoff frequency %39 flag='scale ';40 win=nuttallwin(N+1);41 b=fir1(N,Fc ,'low',win ,flag); % Numerator coefficients %42 a=1; % Denominator coefficients of the filter %43 % Filtering of the joint position %44 qf1=filtfilt(b,a,q1); % Filtered pendulum position qf1(t) %45 qf2=filtfilt(b,a,q2); % Filtered wheel position qf2(t) %46

47 % Example: Position q_1(t) after and before the filtering %48 f1=figure (1);49 hold all; grid on;50 plot(t,q1,'Color ' ,[1 0 0],'LineWidth ',1.5);51 plot(t,qf1 ,'Color ' ,[0 0 1],'LineWidth ',1.5);52 xlabel('Time [s]','FontSize ',14,...53 'FontName ','Times New Roman ');54 ylabel('Position [rad]','FontSize ',14,...55 'FontName ','Times New Roman ');56 % Create legend57 leg = legend('$q_1$ ','$q_{f1}$','Location ',...58 'NorthEast ','Orientation ','horizontal ');59 set(leg ,'Interpreter ','latex ','FontSize ',18,...60 'FontName ','Times New Roman ');61 axis ([6.24 6.325 1.925 1.9425 ]);62

63

64 %% Procedure to compute joint velocities dot(q)_1 and dot(q)_2 %65 % using the central difference algorithm %66

67 % The first values of the velocity vectors are assigned to zero%68 dq1(1) = 0.0;69 dq2(1) = 0.0;70

71 % Central difference algorithm %72 for i=2:l-173 dq1(i) = (qf1(i+1)-qf1(i-1))/(2*T);74 dq2(i) = (qf2(i+1)-qf2(i-1))/(2*T);75 end

76 % The last values of the velocity vectors are copied from their%77 % respective penultimate values to complete vectors dimensions %78 dq1(l) = dq1(l-1);79 dq2(l) = dq2(l-1);80 dq1 = dq1 ';81 dq2 = dq2 ';82

83

84 %% Construction of the regression matrices Omega_a and Omega_b %85 r=100.0; % Constant to approximate the sign function with tanh %86

87 % Matrix Omega_a(q,dq) %88 Oa11 = dq1;89 Oa12 = (dq1.* sin(qf2)).* sin(qf2);90 Oa13 = dq2.*cos(qf2);91 Oa14 = zeros(size(qf1));92 Oa15 = zeros(size(qf1));

Appendix A: MATLAB Codes for Parameter Identification … 191

93 Oa16 = zeros(size(qf1));94 Oa17 = zeros(size(qf1));95 Oa18 = zeros(size(qf1));96 Oa19 = zeros(size(qf1));97

98 Oa21 = zeros(size(qf1));99 Oa22 = zeros(size(qf1));100 Oa23 = dq1.*cos(qf2);101 Oa24 = dq2;102 Oa25 = zeros(size(qf1));103 Oa26 = zeros(size(qf1));104 Oa27 = zeros(size(qf1));105 Oa28 = zeros(size(qf1));106 Oa29 = zeros(size(qf1));107

108 % Matrix Omega_b(q,dq) %109 Ob11 = zeros(size(qf1));110 Ob12 = zeros(size(qf1));111 Ob13 = zeros(size(qf1));112 Ob14 = zeros(size(qf1));113 Ob15 = zeros(size(qf1));114 Ob16 = dq1;115 Ob17 = zeros(size(qf1));116 Ob18 = tanh(r*dq1);117 Ob19 = zeros(size(qf1));118

119 Ob21 = zeros(size(qf1));120 Ob22 = -0.5*(dq1.*dq1).*sin(2*qf2);121 Ob23 = (dq2.*dq1).*sin(qf2);122 Ob24 = zeros(size(qf1));123 Ob25 = -sin(qf2);124 Ob26 = zeros(size(qf1));125 Ob27 = dq2;126 Ob28 = zeros(size(qf1));127 Ob29 = tanh(r*dq2);128

129

130 %% Discrete filtering of the regression matrices Omega_a and %131 % Omega_b in order to compute the filtered matrices Omega_{af} %132 % and Omega_{bf} %133

134 % Filter h_D(z) design for matrix Omega_{af}(q,dq) %135 Lambda =30;136 B1 = [Lambda -Lambda ];137 A1 = [1 -exp(-Lambda*T)];138

139 % Matrix Omega_{af}(q,dq) %140 Oaf11 = filter(B1 ,A1,Oa11);141 Oaf12 = filter(B1 ,A1,Oa12);142 Oaf13 = filter(B1 ,A1,Oa13);143 Oaf14 = filter(B1 ,A1,Oa14);144 Oaf15 = filter(B1 ,A1,Oa15);145 Oaf16 = filter(B1 ,A1,Oa16);146 Oaf17 = filter(B1 ,A1,Oa17);147 Oaf18 = filter(B1 ,A1,Oa18);148 Oaf19 = filter(B1 ,A1,Oa19);149

150 Oaf21 = filter(B1 ,A1,Oa21);151 Oaf22 = filter(B1 ,A1,Oa22);152 Oaf23 = filter(B1 ,A1,Oa23);153 Oaf24 = filter(B1 ,A1,Oa24);154 Oaf25 = filter(B1 ,A1,Oa25);155 Oaf26 = filter(B1 ,A1,Oa26);156 Oaf27 = filter(B1 ,A1,Oa27);157 Oaf28 = filter(B1 ,A1,Oa28);158 Oaf29 = filter(B1 ,A1,Oa29);159


160 % Filter f_D(z) design for matrix Omega_{bf}(q,dq) %161 B2 = [0 1-exp(-Lambda*T)];162 A2 = [1 -exp(-Lambda*T)];163

164 % Matrix Omega_{bf}(q,dq)165 Obf11 = filter(B2 ,A2,Ob11);166 Obf12 = filter(B2 ,A2,Ob12);167 Obf13 = filter(B2 ,A2,Ob13);168 Obf14 = filter(B2 ,A2,Ob14);169 Obf15 = filter(B2 ,A2,Ob15);170 Obf16 = filter(B2 ,A2,Ob16);171 Obf17 = filter(B2 ,A2,Ob17);172 Obf18 = filter(B2 ,A2,Ob18);173 Obf19 = filter(B2 ,A2,Ob19);174

175 Obf21 = filter(B2 ,A2,Ob21);176 Obf22 = filter(B2 ,A2,Ob22);177 Obf23 = filter(B2 ,A2,Ob23);178 Obf24 = filter(B2 ,A2,Ob24);179 Obf25 = filter(B2 ,A2,Ob25);180 Obf26 = filter(B2 ,A2,Ob26);181 Obf27 = filter(B2 ,A2,Ob27);182 Obf28 = filter(B2 ,A2,Ob28);183 Obf29 = filter(B2 ,A2,Ob29);

184

185 % Computation of the elements of filtered regression matrix %186 % Omega_f(kT)=Omega_{af}+ Omega_{bf}187 Of11=Oaf11+Obf11;188 Of12=Oaf12+Obf12;189 Of13=Oaf13+Obf13;190 Of14=Oaf14+Obf14;191 Of15=Oaf15+Obf15;192 Of16=Oaf16+Obf16;193 Of17=Oaf17+Obf17;194 Of18=Oaf18+Obf18;195 Of19=Oaf19+Obf19;196

197 Of21=Oaf21+Obf21;198 Of22=Oaf22+Obf22;199 Of23=Oaf23+Obf23;200 Of24=Oaf24+Obf24;201 Of25=Oaf25+Obf25;202 Of26=Oaf26+Obf26;203 Of27=Oaf27+Obf27;204 Of28=Oaf28+Obf28;205 Of29=Oaf29+Obf29;206

207 % Filtered input vector u_f(kT) obtained from filtering the %208 % input vector u(t) using the filter f_D(z) %209 uf1=filter(B2 ,A2 ,u1);210 uf2=filter(B2 ,A2 ,u2);211

212

213 %% Computation of the estimated parameters \hat{\theta} using %214 %the least squares algorithm. With this method only the final %215 %value of each parameter is obtained in a single operation %216

217 Off=[Of11 (1:l) Of12 (1:l) Of13 (1:l) Of14 (1:l) Of15 (1:l)...218 Of16 (1:l) Of17 (1:l) Of18 (1:l) Of19 (1:l);219 Of21 (1:l) Of22 (1:l) Of23 (1:l) Of24 (1:l) Of25 (1:l)...220 Of26 (1:l) Of27 (1:l) Of28 (1:l) Of29 (1:l)];221 TauF=[uf1(1:l); uf2(1:l)];222

223

224 disp('Values of the estimated parameters:');


225 format short G226 % Least squares algorithm using pseudoinverse %227 theta_mat =(Off '*Off)^(-1)*(Off '*TauF)228

229

230 %% Computation of the estimated parameters \hat{\theta} using %231 %the least squares algorithm. With this method the time %232 %evolution of the estimated parameters is computed and plotted %233

234

235 % Construction of the filtered regression matrix Omega_f %236 % and the filtered input vector u_f %237

238 Of=zeros (2,9); % Initialization of Omega_f %239 Uf=zeros (2,1); % Initialization of u_f %240

241 for i=1:l242 Of(:,:,i)=[Of11(i) Of12(i) Of13(i) Of14(i) Of15(i)...243 Of16(i) Of17(i) Of18(i) Of19(i);244 Of21(i) Of22(i) Of23(i) Of24(i) Of25(i) Of26(i)...245 Of27(i) Of28(i) Of29(i)];246 Uf(:,:,i)=[uf1(i);uf2(i)];247 end248

249 P=zeros (9,9); % Initialization of P %250 Z=zeros (9,1); % Initialization of Z %251

252 % Least squares algorithm %253 for i=1:l254 P=P+(Of(:,:,i)')*Of(:,:,i);255 Z=Z+(Of(:,:,i)')*Uf(:,:,i);256 thetai(:,i)=P^(-1)*Z;257 end258

259 % Time evolution of the estimated parameters to be plotted %260 theta1=thetai (1,:);261 theta2=thetai (2,:);262 theta3=thetai (3,:);263 theta4=thetai (4,:);264 theta5=thetai (5,:);265 theta6=thetai (6,:);266 theta7=thetai (7,:);267 theta8=thetai (8,:);268 theta9=thetai (9,:);269

270

271 % Display of the time evolution of the estimated parameters %272 % Figure dimensions and properties %273 x0=0;y0=0; width =800; height =600;274 f2 = figure('position ',[x0 ,y0 ,width ,height],'Color ' ,[1 1 1]);275 set(f2 ,'PaperPositionMode ','auto')

276

277 % Subplot of \hat{\theta}_1 %278 f = subplot (3,3,1);279 hold all; box on; grid on280 plot(t,theta1 ,'Color ' ,[0 0 0],'LineWidth ',1.5);281 set(f,'FontName ','Times New Roman ','FontSize ' ,12);282 xlabel({'Time [s]'},'FontSize ',12,...283 'FontName ','Times New Roman ');284 ylabel('${\hat{\theta}_1}$','Interpreter ','latex ',...285 'FontSize ',14,'FontName ','Times New Roman ','rotation ' ,0);286 axis ([0 time 0 0.1]);287

288 % Subplot of \hat{\theta}_2 %289 f = subplot (3,3,2);


290 hold all; box on; grid on291 plot(t,theta2 ,'Color ' ,[0 0 0],'LineWidth ',1.5);292 set(f,'FontName ','Times New Roman ','FontSize ' ,12);293 xlabel({'Time [s]'},'FontSize ',...294 12,'FontName ','Times New Roman ');295 ylabel('${\hat{\theta}_2}$','Interpreter ','latex ',...296 'FontSize ',14,'FontName ','Times New Roman ','rotation ' ,0);297 axis ([0 time -0.1 0.1]);298

299 % Subplot of \hat{\theta}_3 %300 f = subplot (3,3,3);301 hold all; box on; grid on302 plot(t,theta3 ,'Color ' ,[0 0 0],'LineWidth ',1.5);303 set(f,'FontName ','Times New Roman ','FontSize ' ,12);304 xlabel({'Time [s]'},'FontSize ',12,...305 'FontName ','Times New Roman ');306 ylabel('${\hat{\theta}_3}$','Interpreter ','latex ',...307 'FontSize ',14,'FontName ','Times New Roman ','rotation ' ,0);308 axis ([0 time -0.01 0.03]);309


321 % Subplot of \hat{\theta}_5 %322 f = subplot (3,3,5);323 hold all; box on; grid on324 plot(t,theta5 ,'Color ' ,[0 0 0],'LineWidth ',1.5);325 set(f,'FontName ','Times New Roman ','FontSize ' ,12);326 xlabel({'Time [s]'},'FontSize ',...327 12,'FontName ','Times New Roman ');328 ylabel('${\hat{\theta}_5}$','Interpreter ','latex ',...329 'FontSize ',14,'FontName ','Times New Roman ','rotation ' ,0);330 axis ([0 time -0.1 0.7]);331



354 % Subplot of \hat{\theta}_8 %355 f = subplot (3,3,8);356 hold all; box on; grid on


357 plot(t,theta8 ,'Color ' ,[0 0 0],'LineWidth ',1.5);358 set(f,'FontName ','Times New Roman ','FontSize ' ,12);359 xlabel({'Time [s]'},'FontSize ',12,...360 'FontName ','Times New Roman ');361 ylabel('${\hat{\theta}_8}$','Interpreter ','latex ',...362 'FontSize ',14,'FontName ','Times New Roman ','rotation ' ,0);363 axis ([0 time -0.01 0.4]);364


376 % End of the code %

A.2 Identification of the Inertia Wheel Pendulum

This section contains theMATLAB code for the identification of the IWP in Sect. 3.3.The data measured from the identification experiment can be downloaded fromhttps://www.dropbox.com/s/5hco66oaik4ovw2/iwpdataexp.mat?dl=1. The identifi-cation procedure is performed in the following order: (i) The measured positionsignals are filtered; (ii) The velocity estimation algorithm for the computation ofthe first and second joint velocities is applied; (iii) The filtered regression model iscomputed; and (iv) The LS algorithm is applied and the estimated parameters θ areobtained.

1 % Motion Control of Underactuated Mechanical Systems %2 % Parameter identification of the Inertia Wheel Pendulum %3 % Created: 12/Oct /2016 %4 % Chapter 3, Section 3.3 ...

%5

6 %% Data from the identification experiment are loaded %7 clc8 clear all9

10 % The file iwpdataexp.mat containing the data measured from the%11 % identification experiment of the IWP can be downloaded from: %12 % www.dropbox.com/s/5 hco66oaik4ovw2/iwpdataexp.mat?dl=1 %13 load iwpdataexp.mat14

15 time =10.0; % Time of the identification experiment %16 T=0.001; % Sample time %17 l=time/T; % Number of samples to be considered %18 t=DataIdenExp.time (1:l); % Time vector from experimental data %19

20 % Position signals from experimental data %21 q1=DataIdenExp.signals (1,2) .values (1:l); % Pendulum position %

http://dx.doi.org/10.1007/978-3-319-58319-8_3

https://www.dropbox.com/s/5hco66oaik4ovw2/iwpdataexp.mat?dl=1


22 q2=DataIdenExp.signals (1,3) .values (1:l); % Wheel position %23

24 %Torque/current constant of the 24VDC motor Leadshine DCM20202A%25 KT=0.0551;26 u1=zeros(l,1); % Input vector of the pendulum %27 % Applied torque of the wheel u_2(t) from the experimental data%28 % The torque applied to the wheel is computed as tau(t)=KT*v(t)%29 %since the servoamplifier voltage -current relation is 1A per 1V%30 u2=KT.*DataIdenExp.signals (1,1) .values (1:l);31

32

33 %% Filtering of the joint position measurements q1 and q2 to %34 % reduce the quantization error from the experimental data %35 % Filter design %36 N=30; % Filter order %37 Fc=0.07; %Normalized cutoff frequency %38 flag='scale ';39 win=nuttallwin(N+1);40 b=fir1(N,Fc ,'low',win ,flag); % Numerator coefficients %41 a=1; % Denominator coefficients of the filter %42 % Filtering of the joint position %43 qf1=filtfilt(b,a,q1); % Filtered pendulum position qf1(t) %44 qf2=filtfilt(b,a,q2); % Filtered wheel position qf2(t) %45

46

47 %% Procedure to compute joint velocities dot(q)_1 and dot(q)_2 %48 % using the central difference algorithm %49

50 % The first values of the velocity vectors are assigned to zero%51 dq1(1)=0.0;52 dq2(1)=0.0;53

54 % Central difference algorithm %55 for i=2:l-156 dq1(i)=(qf1(i+1)-qf1(i-1))/(2*T);57 dq2(i)=(qf2(i+1)-qf2(i-1))/(2*T);58 end59

60 % The last values of the velocity vectors are copied from their%61 % respective penultimate values to complete vectors dimensions %62 dq1(l)=dq1(l-1);63 dq1=dq1 ';64 dq2(l)=dq2(l-1);65 dq2=dq2 ';66

67

68 %% Construction of the regression matrices Omega_a and Omega_b %69 g=9.81; % Gravitational acceleration constant %70 r=100.0; % Constant to approximate the sign function with tanh %71

72 % Matrix Omega_a(q,dq) %73 Oa11=dq1;74 Oa12=dq2;75 Oa13=zeros(l,1);76 Oa14=zeros(l,1);77 Oa15=zeros(l,1);78 Oa16=zeros(l,1);79 Oa17=zeros(l,1);80

81 Oa21=zeros(l,1);82 Oa22=dq1+dq2;83 Oa23=zeros(l,1);84 Oa24=zeros(l,1);85 Oa25=zeros(l,1);86 Oa26=zeros(l,1);87 Oa27=zeros(l,1);88


89 % Matrix Omega_b(q,dq) %90 Ob11=zeros(l,1);91 Ob12=zeros(l,1);92 Ob13=-g*sin(q1);93 Ob14=dq1;94 Ob15=zeros(l,1);95 Ob16=zeros(l,1);96 Ob17=zeros(l,1);97

98 Ob21=zeros(l,1);99 Ob22=zeros(l,1);100 Ob23=zeros(l,1);101 Ob24=zeros(l,1);102 Ob25=dq2;103 Ob26=0.5+0.5*tanh(r*dq2);104 Ob27=-0.5 -0.5*tanh(-r*dq2);105

106

107 %% Discrete filtering of the regression matrices Omega_a and %108 % Omega_b in order to compute the filtered matrices Omega_{af} %109 % and Omega_{bf} %110

111 % Second order filter h_D(z) design for matrix Omega_{af}(q,dq)%112 Lambda =8;113 B1=[ Lambda*Lambda -Lambda*Lambda ];114 A1=[T*((1/T)+Lambda)*((1/T)+Lambda) -2*((1/T)+Lambda) (1/T)];115

116 % Matrix Omega_{af}(q,dq) %117 Oaf11=filter(B1 ,A1 ,Oa11);118 Oaf12=filter(B1 ,A1 ,Oa12);119 Oaf13=filter(B1 ,A1 ,Oa13);120 Oaf14=filter(B1 ,A1 ,Oa14);121 Oaf15=filter(B1 ,A1 ,Oa15);122 Oaf16=filter(B1 ,A1 ,Oa16);123 Oaf17=filter(B1 ,A1 ,Oa17);124

125 Oaf21=filter(B1 ,A1 ,Oa21);126 Oaf22=filter(B1 ,A1 ,Oa22);127 Oaf23=filter(B1 ,A1 ,Oa23);128 Oaf24=filter(B1 ,A1 ,Oa24);129 Oaf25=filter(B1 ,A1 ,Oa25);130 Oaf26=filter(B1 ,A1 ,Oa26);131 Oaf27=filter(B1 ,A1 ,Oa27);

132

133 % Second order filter f_D(z) design for matrix Omega_{bf}(q,dq)%134 B2=[ Lambda*Lambda ];135 A2 =[((1/T)+Lambda)*((1/T)+Lambda) ...136 -2*((1/T)+Lambda)*(1/T) (1/(T*T))];137

138 % Matrix Omega_{bf}(q,dq)139 Obf11=filter(B2 ,A2 ,Ob11);140 Obf12=filter(B2 ,A2 ,Ob12);141 Obf13=filter(B2 ,A2 ,Ob13);142 Obf14=filter(B2 ,A2 ,Ob14);143 Obf15=filter(B2 ,A2 ,Ob15);144 Obf16=filter(B2 ,A2 ,Ob16);145 Obf17=filter(B2 ,A2 ,Ob17);146

147 Obf21=filter(B2 ,A2 ,Ob21);148 Obf22=filter(B2 ,A2 ,Ob22);149 Obf23=filter(B2 ,A2 ,Ob23);150 Obf24=filter(B2 ,A2 ,Ob24);151 Obf25=filter(B2 ,A2 ,Ob25);152 Obf26=filter(B2 ,A2 ,Ob26);


153 Obf27=filter(B2 ,A2 ,Ob27);154

155 % Computation of the elements of filtered regression matrix %156 % Omega_f(kT)=Omega_{af}+ Omega_{bf}157 Of11=Oaf11+Obf11;158 Of12=Oaf12+Obf12;159 Of13=Oaf13+Obf13;160 Of14=Oaf14+Obf14;161 Of15=Oaf15+Obf15;162 Of16=Oaf16+Obf16;163 Of17=Oaf17+Obf17;164

165 Of21=Oaf21+Obf21;166 Of22=Oaf22+Obf22;167 Of23=Oaf23+Obf23;168 Of24=Oaf24+Obf24;169 Of25=Oaf25+Obf25;170 Of26=Oaf26+Obf26;171 Of27=Oaf27+Obf27;172

173 % Filtered input vector u_f(kT) obtained from filtering the %174 % input vector u(t) using the filter f_D(z) %175 uf1=filter(B2 ,A2 ,u1);176 uf2=filter(B2 ,A2 ,u2);177

178

179 %% Computation of the estimated parameters \hat{\theta} using %180 %the least squares algorithm. With this method only the final %181 %value of each parameter is obtained in a single operation %182

183 Off=[Of11 (1:l) Of12 (1:l) Of13 (1:l) Of14 (1:l) ...184 Of15 (1:l) Of16 (1:l) Of17 (1:l);185 Of21 (1:l) Of22 (1:l) Of23 (1:l) Of24 (1:l) ...186 Of25 (1:l) Of26 (1:l) Of27 (1:l)];187 TauF=[uf1(1:l); uf2(1:l)];188

189 disp('Values of the estimated parameters:');190 format short G191 % Least squares algorithm using pseudoinverse %192 theta_mat =(Off '*Off)^(-1)*(Off '*TauF)193

194

195 %% Computation of the estimated parameters \hat{\theta} using %196 %the least squares algorithm. With this method the time %197 %evolution of the estimated parameters is computed and plotted %198

199 % Construction of the filtered regression matrix Omega_f %200 % and the filtered input vector u_f %201

202 Of=zeros (2,7); % Initialization of Omega_f %203 Uf=zeros (2,1); % Initialization of u_f %204

205 for i=1:l206 Of(:,:,i)=[Of11(i) Of12(i) Of13(i) ...207 Of14(i) Of15(i) Of16(i) Of17(i);208 Of21(i) Of22(i) Of23(i) ...209 Of24(i) Of25(i) Of26(i) Of27(i)];210 Uf(:,:,i)=[uf1(i);uf2(i)];211 end212

213 P=zeros (7,7); % Initialization of P %214 Z=zeros (7,1); % Initialization of Z %215

216 % Least squares algorithm %217 for i=1:l218 P=P+(Of(:,:,i)')*Of(:,:,i);219 Z=Z+(Of(:,:,i)')*Uf(:,:,i);


220 thetai(:,i)=P^(-1)*Z;221 end222

223 % Time evolution of the estimated parameters to be plotted %224 theta1=thetai (1,:);225 theta2=thetai (2,:);226 theta3=thetai (3,:);227 theta4=thetai (4,:);228 theta5=thetai (5,:);229 theta6=thetai (6,:);230 theta7=thetai (7,:);

231

232 % Display of the time evolution of the estimated parameters %233 % Figure dimensions and properties %234 x0=0;y0=0; width =800; height =600;235 f1 = figure('position ',[x0 ,y0 ,width ,height],'Color ' ,[1 1 1]);236 set(f1 ,'PaperPositionMode ','auto')237





282 % Subplot of \hat{\theta}_5 %283 f = subplot (3,3,5);284 hold all; box on; grid on285 plot(t,theta5 ,'Color ' ,[0 0 0],'LineWidth ',1.5);


286 set(f,'FontName ','Times New Roman ','FontSize ' ,12);287 xlabel({'Time [s]'},'FontSize ',12,...288 'FontName ','Times New Roman ');289 ylabel('${\hat{\theta}_5}$','Interpreter ','latex ',...290 'FontSize ',14,'FontName ','Times New Roman ','rotation ' ,0);291 axis ([0 time -2e-03 2e-03]);292



315 % End of the code %

Appendix BConditions to Ensure that Matrix Ain Chaps. 5 and 6 is Hurwitz

This appendix is devoted to deriving the conditions for ensuring that matrix A intro-duced in both (5.22) and (6.47) for Chaps. 5 and 6, respectively, is a Hurwitz matrix.With this aim, the explicit expressions of the internal dynamics functions w1(ζ1,η)

and w2(t, ζ1,η), introduced in both (5.20) and (6.42), are given.The explicit expression of w1(ζ1,η) is given by

w1(ζ1,η) = [w11 w12 w13

]T, (B.1)

where

w11 = −�1θ4η1 + �2θ4η2 + θ4η3 − ζ1θ4

θ4 − θ3 cos(η2),

w12 = [�1η1 + �2η2 − ζ1]θ3 cos(η2) + θ4η3

θ4 − θ3 cos(η2),

w13 = −θ3

θ4sin(η2)η1η2 + θ2

2θ4sin(2η2)η

21 + θ5

θ4sin(η2) − θ7

θ4η2.

The explicit expression of w2(t, ζ1,η) is given by

w2(t, ζ1,η) =⎡

⎣00

θ3θ4cos(η2)qd1 + θ2

2θ4sin(2η2)q2

d1 − θ2θ4sin(2η2)qd1η1

⎤

⎦ , (B.2)

with η1 and η2 given byw11 andw12 in (B.1), respectively. The total internal dynamicsis given by

η = w1(ζ1,η) + w2(t, ζ1,η).


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202 Appendix B: Conditions to Ensure that Matrix A …

The explicit expression of matrix A, which is defined as

A = ∂w1(0,η)

∂η

∣∣∣η=0

, (B.3)

can be obtained from the definition of w1(ζ1,η) in (B.1) with ζ1 = 0 as follows:

A =⎡

⎣�1G10 �2G10 G10

−�1G20 −�2G20 −G10

θ−14 θ7�1G20 θ−1

4 [θ5 + θ7�2G20] θ−14 θ7G10

⎤

⎦ , (B.4)

where the Furuta pendulum parameters θi are positive constants, the parameters �1

and �2 are control gains,

G10 = −θ4

θ4 − θ3, and G20 = −θ3

θ4 − θ3.

The characteristic polynomial of matrix A is given by

P = λ3 + �2θ3 − �1θ4 − θ7

θ3 − θ4λ2 + θ5 − �1θ7

θ3 − θ4λ + �1θ5

θ3 − θ4. (B.5)

Considering that the characteristic polynomial of matrix A is a third-order system ofthe form

a3λ3 + a2λ

2 + a1λ + a0,

necessary and sufficient conditions for A to be Hurwitz are that the coefficients a0,a1, a2, and a3 be positive and a2a1 − a0a3 > 0 [208].

In order to apply the Routh–Hurwitz criterion, it is necessary to assume thatθ3 > θ4, which is satisfied for our experimental case. Hence, the conditions for A tobe Hurwitz are the following:

• From a0 > 0, we have that the control gain �1 must be positive.• From a1 > 0, we have that �1 < (θ5/θ7).• From a2 > 0, we have that

�2 > κ1 = �1θ4 + θ7

θ3(B.6)

must be satisfied.• From a2a1 − a0a3 > 0, we have

(�2θ3 − �1θ4 − θ7)(θ5 − �1θ7)

(θ3 − θ4)2− �1θ5

θ3 − θ4> 0,

Appendix B: Conditions to Ensure that Matrix A … 203

and considering that θ3 > θ4, it can be written that

(�2θ3 − �1θ4 − θ7)(θ5 − �1θ7) > �1θ5(θ3 − θ4)

�2θ3θ5 − �1θ4θ5 − θ7θ5 − �1�2θ3θ7 + �21θ4θ7 + �1θ

27 > �1θ3θ5 − �1θ4θ5,

and solving for �2, we obtain

�2 >�1θ3θ5 − �1θ4θ5 + �1θ4θ5 + θ7θ5 − �2

1θ4θ7 − �1θ27

θ3θ5 − �1θ3θ7,

and finally, after some simplifications, the inequality

�2 > κ2 = �1θ4 + θ7

θ3+ �1θ5(θ3 − θ4)

θ3(θ5 − �1θ7)(B.7)

is obtained.

If all the conditions mentioned above are satisfied, then κ2 in (B.7) is greater thanκ1 in (B.6). Therefore, necessary and sufficient conditions for ensuring that all theroots of the matrix A are in the left-half of the complex plane are as follows:

θ3 > θ4, (B.8)

0 < �1 <θ5

θ7, (B.9)

�2 >�1θ4 + θ7

θ3+ �1θ5(θ3 − θ4)

θ3(θ5 − �1θ7). (B.10)

Note that the conditions (B.8)–(B.10) correspond to (5.48)–(5.50).

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Appendix CConvergence Proof of the Swing-up Controllerfor the IWP in Chaps. 7 and 8

In this appendix, the swing-up scheme introduced in [13, 14], which was used inChaps. 7 and 8, is reviewed. In particular, the controller is rewritten in the notationfor the system (7.2) and no change of coordinates is necessary.

The swing-up controller is obtained from the assumption that the system is notaffected by friction, which in state-space form is as follows:

d

dtq1 = q1,

d

dtq1 = g0θ3θ2 sin(q1)

|M | − θ2

|M |τ ,

d

dtq2 = q2,

d

dtq2 = −g0θ3θ2 sin(q1)

|M | + θ1

|M |τ ,

where |M | = θ1θ2 − θ22 > 0.By invoking the ideas in [13], we define the output

y = q2 ,

which is the wheel velocity whose time derivative is then given by

˙y2 = q2 ,

= −θ2θ3g0 sin(q1)

|M | + θ1

|M |τ .

The control input τ is obtained by using the collocated partial feedback lineariza-tion controller


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206 Appendix C: Convergence Proof of the Swing-up Controller …

τ = θ2

θ1θ3g0 sin(q1) + |M |

θ1u, (C.1)

which produces the following closed-loop system:

θ1q1 − θ3g0 sin(q1) = −θ2u , (C.2)

q2 = u. (C.3)

Once the collocated partial feedback linearization technique is used, a procedurefor obtaining the swing-up controller is given such as is achieved in [14]. In otherwords, we show the design of the signal u for the system given by (C.2)–(C.3). Theswing-up controller is based on passivity properties.

First, the energy of the pendulum (C.2) is given by

E = 1

2θ1q1

2 + θ3g0(cos(q1) − 1) ,

whose time derivative is expressed as

E = θ1q1q1 − θ3g0 sin(q1)q1 ,

= θ1q1

[θ3g0 sin(q1)

θ1− θ2

θ1u

]− θ3g0 sin(q1)q1 ,

= −q1θ2u.

Therefore, the pendulum is passive with input −θ2u and output y1 = q1.On the other hand, the wheel energy is given by

Er = 1

2q2

2 ,

whose time derivative is

Er = q2q2 ,

= q2u.

In this case, the wheel is passive with input u and output y2 = q2.It will be useful to define

Eo = E − Eref ,

where Eref is constant and Eo ∈ IR is the energy error.

Appendix C: Convergence Proof of the Swing-up Controller … 207

Now let us consider the Lyapunov function

V (q1, q1, q2) = 1

2ke Eo

2 + 1

2kvq2

2,

with strictly positive constants ke and kv .The time derivative of V is computed as follows:

V = ke Eo Eo + kvq2q2 ,

= −ke Eoq1θ2u + kvq2u ,

= − [ke Eoq1θ2 − kvq2] u.

If the input u is introduced with form

u = ke Eoq1θ2 − kvq2, (C.4)

then

V = −u2 ≤ 0.

Multiplying equation (C.4) times a gain ku , which is an additional gain to modifythe answer of the controller, we have

u = ku [ke Eoq1θ2 − kvq2] . (C.5)

Hence,

V = −kuu2 ≤ 0.

So, V = 0 for S = S1⋃

S2, where

S1 ={[

qq

]∈ IR4 : q ∈ IR2 and q = 0 ∈ IR2

},

S2 ={[

qq

]∈ IR4 : 1

2θ1q1

2 + θ3g0(cos(q1) − 1) = Eref and q2 = 0 ∈ IR2

}.

By LaSalle’s invariance principle [41], the solutions [q(t)T q(t)T ]T converge to thebiggest invariant set in S. To ensure that a solution remains in S, the following shouldbe accomplished:

u(t) = 0, ∀ t ≥ 0, (C.6)

208 Appendix C: Convergence Proof of the Swing-up Controller …

which also implies

˙u(t) = 0, ∀ t ≥ 0. (C.7)

By expanding (C.7) and replacing the closed-loop equations (C.2)–(C.3), as well as(C.6), it follows that

θ2ke E0(t)θ3g0

θ1sin(q1(t)) = 0, ∀ t ≥ 0,

which is simplified to

[E(t) − Eref ] sin(q1(t)) = 0, ∀ t ≥ 0.

As pointed out in [14], this equation says that the closed-loop system trajectories willconverge to either E = Eref or sin(q1) = 0. In the first case, E = Eref , it follows, inaddition, that q2 = 0. In the second case, it follows that q1 = nπ and q1 = constant.

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Index

AAbsolute value, 13Acrobot, 1, 2Adaptation law, 98, 170Adaptive control, 6, 159, 167Additive Gaussian noise, 114Aircraft, 1

BBackpropagation, 109, 110Backstepping, 5Ball and beam system, 1, 3Barbalat’s lemma, 18, 169, 170Boundedness, 18Butterworth filter, 35

CCart-pole system, 1, 2Characteristic polynomial, 202Composite control, 51, 119Control problem, 53, 72, 96, 122, 145, 163Convey-crane system, 2

DDAQ, 33, 44DC motor, 30, 33, 39, 41, 44Decoupling control, 22Decoupling matrix, 22Degrees of freedom, 1Determinant, 13Discrete derivative algorithm, 28

Discretization, 32, 43Disturbance rejection, 110, 113Dynamics

closed-loop, 14error, 71, 96, 144, 151external, 5, 74, 101, 103, 104, 147, 179Internal, 201internal, 5, 55, 74, 101, 146, 152, 164,179

open-loop, 14output, 55, 98, 145, 164, 178zero, 5, 164

EEigenvalue, 13Energy function, 5, 56Energy-based control, 6Equilibrium, 14Euler-Lagrange, 28, 52Experimental platform, 7, 33, 44

FFeedback linearization, 5, 6, 19, 54, 72, 96,

145, 163Collocated partial, 205

Filter Design and Analysis Tool, 35, 45Filtered regression model, 30, 42Finite escape time, 5Flat system, 2Flexible joint robot, 182

model, 182Flexible link robots, 1


221

222 Index

Frictioncoulomb, 29, 41viscous, 29, 41

Furuta pendulum, 2, 8, 189dynamic model, 28, 52, 70, 94

HHelicopter, 1Homogeneity, 3Hurwitz matrix, 15, 77, 80, 105, 148, 180,

201

IIdentity matrix, 13Induced norm, 13Inertia wheel pendulum, 1, 2, 8

dynamic model, 121, 143, 160Integral of absolute error, 89Integral of squared error, 89

JJacobian, 98, 148

LLaSalle’s invariance principle, 207Layer

hidden, 23input, 23outout, 23

Least squares, 6, 27, 36, 46, 189, 195Lie derivative, 20Limit cycles, 5Linear quadratic regulator, 121, 131Linear regression model, 7, 28Linearity in the parameters, 7, 30Linearization, 3

full-state, 20input-output, 20input-state, 20

Lipschitzcondition, 15constant, 15, 76, 102function, 15, 76, 103

Low-pass filter, 31second-order, 42

Lyapunov, 5direct method, 16equation, 77, 105, 186function, 17, 56, 149, 207

MMATLAB, 28, 33, 37, 189, 195Mean absolute value, 85Mean Value Theorem, 15, 75, 76, 102, 103,

148Motion control, 7, 28, 34, 45, 142Multi-input–multi-output, 21

NNegative definite, 5Neural networks, 160

adaptive, 6, 93artifitial, 22control, 97, 166two-layer, 23

Normal form, 99Null matrix, 13

OOutput design, 178, 181Output function, 54, 162Output tracking controller, 61, 81

PParameter identification, 6, 27, 189, 195Parameter vector, 30, 42Passivity, 206Passivity-based control, 5PD controller, 45Pendubot, 1, 2Perceptron, 22

two-layer, 23PID controller, 80, 108, 171Positive definite, 5, 13, 17

function, 60, 104, 105, 169, 170matrix, 107, 149

RReal-Time Windows Target, 33, 44Regression matrix, 30, 42Relative degree, 20, 22, 72, 145, 146, 151Root mean square, 66, 86, 113, 136, 173Rotary inverted pendulum, 1Routh–Hurwitz criterion, 152, 202

SSatellite, 1, 3Simulink, 33Single-input–single-output, 19Sliding mode control, 5

Index 223

Spacecraft, 1, 3Stability, 15, 17

asymptotic, 16, 17exponential, 16, 17global, 16local, 16lyapunov, 16

Stabilization, 7State feedback control, 150State space

form, 53, 71, 95, 121, 144, 161realization, 31

State-space, 13form, 205

Superposition, 3Surface vessel, 1Swing-up, 7, 205

control, 132Sylvester’s criterion, 56Symetric matrix, 13System

autonomous, 14non-autonomous, 14time-invariant, 14time-varying, 14underactuated, 1unforced, 14

TTaylor series, 97Trajectory, 14

desired, 7, 45, 54, 63, 72, 82, 96, 111,114, 122, 134, 145, 155, 163, 171

planning, 7tracking, 6, 7

Translational oscillator with rotational actu-ator, 1, 2

UUltimate bound, 78Underactuated systems, 1, 2Underwater vehicle, 1, 3Uniformly bounded, 18Uniformly ultimately bounded, 18, 79, 108,

150Universal approximation property, 6, 24, 165Unstable, 16

VVector norm, 13Vertical takeoff and landing, 1, 3

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Appendix A MATLAB Codes for Parameter Identiﬁcation of …978-3-319-58319-8/1.pdf · 127 Ob29 = tanh(r*dq2); 128 129 130 %% Discrete filtering of the regression matricesOmega_a