mpcgainOrigin.m

function[F,Phi,Phi_Phi,Phi_F,Phi_R] = mpcgainOrigin(Ap,Bp,Cp,Nc,Np,Rs,Q)

%%augmented matrix%%
[m1,n1]=size(Cp);
[n1,n_in]=size(Bp);
A_e=eye(n1+m1,n1+m1);
A_e(1:n1,1:n1)=Ap;
A_e(n1+1:n1+m1,1:n1)=Cp*Ap;
B_e=zeros(n1+m1,n_in);
B_e(1:n1,:)=Bp;
B_e(n1+1:n1+m1,:)=Cp*Bp;
C_e=zeros(m1,n1+m1);
C_e(:,n1+1:n1+m1)=eye(m1,m1);
n=n1+m1;
h(:,:)=C_e;
F(:,:)=C_e*A_e;
for kk=2:Np
h(kk,:)=h(kk-1,:)*A_e;
F(kk,:)= F(kk-1,:)*A_e;
end
v=h*B_e;
Phi=zeros(Np,Nc); %declare the dimension of Phi
Phi(:,1)=v; % first column of Phi
for i=2:Nc
Phi(:,i)=[zeros(i-1,1);v(1:Np-i+1,1)]; %Toeplitz matrix
end
BarRs=ones(Np,1);
Phi_Phi= Phi'*Phi;
Phi_F= Phi'*F;
Phi_R=Phi'*BarRs;
end