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pgp_ltvmpc_quad.m
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function [gppred, varargout] = ...
pgp_ltvmpc_quad(pathtype, mpciterations, systype)
%%
% stochastic mpc control using lqr based on local linearization technique
%%
defaultLearnPathType = pathtype;
% defining dataset for model learning
filename4learn1 = strcat('ds_nmpc_', defaultLearnPathType, '_trans');
filename4learn2 = strcat('ds_nmpc_', defaultLearnPathType, '_rotate');
% defining another pathtype to follow
path2follow1 = strcat('ds_nmpc_', pathtype, '_trans');
path2follow2 = strcat('ds_nmpc_', pathtype, '_rotate');
switch systype
case 'TRAN';
filename4learn = filename4learn1;
path2follow = path2follow1;
loadType = 'original'; % original self0109
H = 10;
Ts = 0.01;
switch pathtype
case 'Lorenz';
umax = [0 2 2];
umin = [-45 -2 -2];
case 'Elliptical';
umax = [100 0.2 0.2];
umin = [0 -0.2 -0.2];
otherwise;
umax = [Inf Inf Inf];
umin = [-Inf -Inf -Inf];
end
case 'ROTATE';
filename4learn = filename4learn2;
path2follow = path2follow2;
loadType = 'self0109';
H = 10;
Ts = 1;
umax = [0.9 0.9 0.9];
umin = [0.1 0.1 0.1];
otherwise;
end
tempRandSequences = randperm(mpciterations);
opt.dsSampleRange = sort(tempRandSequences(1:100));
opt.dsTestRange = 2:mpciterations;
opt.state = 1:9;
opt.in = 7:9;
opt.out = 10:15;
quad_ds = loadSampleData(filename4learn,loadType,opt);
% pathtargets = quad_ds.proData;
% numtem = numel(quad_ds);
dynmodel.inputs = quad_ds.proData(opt.dsSampleRange,opt.state);
dynmodel.targets = quad_ds.proData(opt.dsSampleRange,opt.out);
dynmodel.train = @train;
dynmodel. iNum =3;
dynmodel. sNum =6;
dynmodel. oNum =6;
tic
[dynmodel nlml] = dynmodel.train(dynmodel, [], 1000);
% 32 -> ascii code of 'blank space'
process_mes = strcat('Successfully Training', 32, systype, ' GP Model\n');
fprintf(process_mes);
toc % counting dynamic model learning time
pathobj = loadSampleData(path2follow,loadType,opt);
pathtargets = pathobj.proData;
controller = 'mycontroller'; % mycontroller jMPC
m = pathtargets(1, 1:dynmodel.sNum)';
s= diag([0.1*ones(1, dynmodel.sNum)].^2);
N =mpciterations-10;
u = zeros(dynmodel.iNum, 1);
deltaU = zeros(dynmodel.iNum*H,1);
tic
fprintf('LTVMPC Starts...\n');
for i=1:N
ref.x = pathtargets(i:i+H-1,opt.out);
ref.u = pathtargets(i:i+H-1,opt.in);
switch controller
case 'jMPC';
init0 =[m; s(:)];
[m0, s0, predobj]= pgppred(dynmodel, m, s, u(:,i), 1);
[ltvm.A, ltvm.B]= linearProbmodel(dynmodel, predobj.dMdm, ...
predobj.dMds, predobj.dSdm, predobj.dSds);
ltvm.C = diag(ones(length(m)+length(s(:)), 1)); ltvm.D=0;
plantobj.A = ltvm.A;
plantobj.B = ltvm.B;
plantobj.C = ltvm.C;
plantobj.D = ltvm.D;
plantobj.Ts = 0.1;
% con.u = [-0.5,0.5, Inf; -1,1, Inf; -1,1, Inf];
con.u = [];
con.y = [];
setp = [ref.x, repmat(0*s(:)', H, 1)];
plantobj.con = con;
plantobj.init0 = init0;
controlobj.H = H;
controlobj.ywt = ones(1, length(m)+length(s(:)));
controlobj.uwt = [21000 27000 21000];
controlobj.setpoints = setp;
[optu, yp] = useJmpc(plantobj, controlobj);
case 'mycontroller';
[m0, s0, predobj]= pgppred(dynmodel, m, s, u(:,i), 1);
[ltvm.A, ltvm.B]= linearProbmodel(dynmodel, predobj.dMdm, ...
predobj.dMds, predobj.dSdm, predobj.dSds,m0,s0);
[dA, dB] = c2d(ltvm.A, ltvm.B, Ts);
Q = diag(ones(dynmodel.sNum+dynmodel.sNum^2,1));
R = diag([27000 21000 21000]);
[tildeA, tildeB, tildeQ, tildeR] =...
extendMatrix(H, dA, dB, Q, R);
Bk = tildeB'*tildeQ*tildeB+ tildeR;
% [HBk,exitflag] = chol(Bk);
% if(exitflag)
% [Hes, jitter] = jit_chol(Bk);
% else
% Hes = HBk;
% end
issym=@(x) all(all(tril(x)==triu(x).'));
if(issym(Bk))
Bk = Bk;
else
Bk = (Bk+Bk')/2;
end
Hes = Bk;
setp = [ref.x, repmat(0*s(:)', H, 1)];
x0 =[m; s(:)]-setp(1,:)';
f = 2*(x0)'*tildeA'*tildeQ*tildeB;
lbc = [umin(1).*ones(H,1)-ref.u(:,1),...
umin(2).*ones(H,1)-ref.u(:,2),...
umin(3).*ones(H,1)-ref.u(:,3)];
ubc = [umax(1).*ones(H,1)-ref.u(:,1),...
umax(2).*ones(H,1)-ref.u(:,2),...
umax(3).*ones(H,1)-ref.u(:,3)];
lbc = reshape(lbc', H*dynmodel.iNum,1);
ubc = reshape(ubc', H*dynmodel.iNum,1);
optuu= quadprog(Hes,f,[],[],[],[],lbc,ubc,...
deltaU-reshape(ref.u', H*dynmodel.iNum,1));
optu= reshape(optuu', dynmodel.iNum,H)';
optu = optu(1,:);
yp=0;
end
% linearised model around the operation point (x0,u0).
u = [u, optu'+ref.u(1,:)'];
[m, s]= pgppred(dynmodel, m, s, u(:,i+1), 1);
Mmat(:,i) = m;
Smat(:,:,i) =s;
lMmat(:,i) = yp';
fval(i) = sum((m([1 3 5])-ref.x(1,[1 3 5])').^2);
fprintf('LTVMPC:Iter %6.0d, StagError %12.8d\n', i, fval(i));
end
toc
res = Mmat';
lsyspred=lMmat';
reff = pathtargets(:,opt.out);
switch loadType
case 'self0109';
range1 = quad_ds.r1;
range2 = quad_ds.r2;
res = normalmatrix('reverse',res,range1(opt.out,:),range2);
reff = normalmatrix('reverse',reff,range1(opt.out,:),range2);
u = normalmatrix('reverse',u',range1(opt.in,:),range2);
u=u';
case 'Gauss01';
otherwise;
end
gppred = res(:,[1 3 5]);
refpred = reff(:,[1 3 5]);
lsyspred = 0;
if nargout >1
varargout{1} = refpred;
varargout{2} = lsyspred;
varargout{3} = u;
varargout{4} = fval;
varargout{5} = Smat;
end
end
function [A, B] = ...
oldlinearProbmodel(dynmodel, dMdm, dMds, dSdm, dSds)
D = dynmodel.iNum;
E = dynmodel.sNum;
dmdM0 = [eye(E); zeros(D, E)];
dsdS0 = [eye(E*E); zeros(D*D+2*E*D, E*E)];
dMdM0 = dMdm*dmdM0;
dMdS0 = dMds*dsdS0;
dSdM0 = dSdm*dmdM0;
dSdS0 = dSds*dsdS0;
A = [dMdM0, dMdS0; dSdM0, dSdS0];
dmdu = [zeros(E, D); eye(D)];
dsdu = zeros((E+D)*(E+D), 3);
dMdu = dMdm*dmdu + dMds*dsdu;
dSdu = dSdm*dmdu + dSds*dsdu;
B = [dMdu; dSdu];
end
function [A, B] = ...
linearProbmodel(dynmodel, dMdm, dMds, dSdm, dSds, m0, s0)
D = dynmodel.iNum;
E = dynmodel.sNum;
dS0dsqrtS0 = 2*repmat(reshape(s0^(0.5), 1, numel(s0))', 1,numel(s0));
dS0dsqrtS0 = 1;
dS0S0 = 2*s0^(0.5);
dmdM0 = [eye(E); zeros(D, E)];
dsdS0 = [eye(E*E); zeros(D*D+2*E*D, E*E)];
dMdM0 = dMdm*dmdM0;
dMdS0 = dMds*dsdS0; dMdS0 = dMdS0*dS0dsqrtS0;
dSdM0 = dSdm*dmdM0; dSdM0 = (1./dS0dsqrtS0)'*dSdM0;
dSdS0 = dSds*dsdS0;
A = [dMdM0, dMdS0; dSdM0, dSdS0];
dmdu = [zeros(E, D); eye(D)];
dsdu = zeros((E+D)*(E+D), D);
dMdu = dMdm*dmdu + dMds*dsdu;
dSdu = dSdm*dmdu + dSds*dsdu; dSdu = (1./dS0dsqrtS0)'*dSdu;
B = [dMdu; dSdu];
end
function [A, B] = ...
linearModel(dynmodel, dMdm, dMds)
D = dynmodel.iNum;
E = dynmodel.sNum;
dmdM0 = [eye(E); zeros(D, E)];
dMdM0 = dMdm*dmdM0;
A = dMdM0;
dmdu = [zeros(E, D); eye(D)];
% dsdu = zeros((E+D)*(E+D), D);
% dMdu = dMdm*dmdu + dMds*dsdu;
dMdu = dMdm*dmdu;
B = dMdu;
end
function [M, S, varargout] =...
pgppred(dynmodel, m, s, u, H)
for i=1:H
[tileM, tileS] = jointDistribution(m, s, u(:,i));
[m s, V, dMdm, dSdm, dVdm, dMds, dSds, dVds] = gp0d(dynmodel, tileM, tileS);
M(:,i) = m;
S(:,:,i) = s;
predobj{i}.V = V;
predobj{i}.dMdm = dMdm;
predobj{i}.dSdm = dSdm;
predobj{i}.dVdm = dVdm;
predobj{i}.dMds = dMds;
predobj{i}.dSds = dSds;
predobj{i}.dVds = dVds;
end
if nargout > 2
varargout = predobj;
end
end
function [tileM, tileS] = jointDistribution(m, s, u)
dimS = size(s, 1);
dimU = size(u,1);
tileM = [m; u];
tileS = zeros(dimS+dimU,dimS+dimU);
tileS(1:dimS, 1:dimS) = s;
end
function [optu, opty] = useJmpc(plantobj, controlobj)
% jSS is based on using "jMPC" toolbox
plant = jSS(plantobj.A, plantobj.B, plantobj.C, plantobj.D);
plant = c2d(plant, plantobj.Ts);
model = plant;
plant.x0 = plantobj.init0;
model.x0 = plantobj.init0;
% setp = zeros(1,42);
% %=====================
% % when using specified QP solvers.
% % 'Matlab' --> all QP solvers
% % 'Mex' --> 'wright' and 'mehrotra'
% % ------------------------------------------------------
% opts.QPSolver = 'mehrotra';
% MPC = jMPC(model, H, H, uwt,ywt,con,[],opts);
% %=====================
MPC = jMPC(model, controlobj.H, controlobj.H,...
controlobj.uwt, controlobj.ywt, plantobj.con);
simopts = jSIM(MPC,plant, controlobj.H, controlobj.setpoints);
simresult = sim(MPC,simopts,'Mex');
optu = simresult.plotvec.u(2,:);
opty = simresult.plotvec.yp(2,:);
end
function [tildeA, tildeB, tildeQ, tildeR] = extendMatrix(H, A, B, Q, R)
tildeA = cell(H,1);
for i =1:H
tildeA{i} = A^H;
end
tildeA = cell2mat(tildeA);
tildeB = cell(H,H);
for j = 1:H
for i= 1:H
if i>=j
tildeB{i,j}= A^(i-j)*B;
else
tildeB{i,j}= 0.*A*B;
end
end
end
tildeB = cell2mat(tildeB);
tildeB = tril(tildeB);
tildeQ = diag(repmat(diag(Q),H,1));
tildeR = diag(repmat(diag(R),H,1));
end
function y = subsys_trans(x, u, noise)
phi = pi/4; theta = pi/4;
m = 0.74; g = 9.81;
y = zeros(6, 1);
y(1) = x(2);
y(2) = u(2)*u(1)/m + noise(1);
y(3) = x(4);
y(4) = u(3)*u(1)/m+ noise(2);
y(5) = x(6);
y(6) = g-(cos(phi)*cos(theta))*u(1)/m+ noise(3);
end
function y = subsys_rotate(x, u, noise)
quad_params;
y = zeros(6, 1);
y(1) = x(2);
y(2) = x(4)*x(6)*a1+x(4)*a2*OmegaR+b1*u(1)+ noise(1);
y(3) = x(4);
y(4) = x(2)*x(6)*a3-x(2)*a4*OmegaR+b2*u(2)+ noise(2);
y(5) = x(6);
y(6) = x(4)*x(2)*a5+b3*u(3)+ noise(3);
end