flts - time response (discrete time, sampled system)
State-space form:
sl is a syslin list containing the matrices of the following linear system
sl=syslin('d',A,B,C,D) (see syslin):
x[t+1] = A x[t] + B u[t] y[t] = C x[t] + D u[t]
or, more generally, if D is a polynomial matrix (p = degree(D(z))) :
Transfer form:
y=flts(u,sl[,past]). Here sl is a linear system in transfer matrix representation i.e
sl=syslin('d',transfer_matrix) (see syslin).
is the matrix of past values of u and y.
nd is the maximum of degrees of lcm's of each row of the denominator matrix of sl.
u=[u0 u1 ... un] (input) y=[y0 y1 ... yn] (output)
p is the difference between maximum degree of numerator and maximum degree of denominator
sl=syslin('d',1,1,1);u=1:10;
y=flts(u,sl);
plot2d2("onn",(1:size(u,'c'))',y')
[y1,x1]=flts(u(1:5),sl);y2=flts(u(6:10),sl,x1);
y-[y1,y2]
//With polynomial D:
z=poly(0,'z');
D=1+z+z^2; p =degree(D);
sl=syslin('d',1,1,1,D);
y=flts(u,sl);[y1,x1]=flts(u(1:5),sl);
y2=flts(u(5-p+1:10),sl,x1); // (update)
y-[y1,y2]
//Delay (transfer form): flts(u,1/z)
// Usual responses
z=poly(0,'z');
h=(1-2*z)/(z^2+0.3*z+1)
u=zeros(1,20);u(1)=1;
imprep=flts(u,tf2ss(h)); //Impulse response
plot2d2("onn",(1:size(u,'c'))',imprep')
u=ones(1,20);
stprep=flts(u,tf2ss(h)); //Step response
plot2d2("onn",(1:size(u,'c'))',stprep')
//
// Other examples
A=[1 2 3;0 2 4;0 0 1];B=[1 0;0 0;0 1];C=eye(3,3);Sys=syslin('d',A,B,C);
H=ss2tf(Sys); u=[1;-1]*(1:10);
//
yh=flts(u,H); ys=flts(u,Sys);
norm(yh-ys,1)
//hot restart
[ys1,x]=flts(u(:,1:4),Sys);ys2=flts(u(:,5:10),Sys,x);
norm([ys1,ys2]-ys,1)
//
yh1=flts(u(:,1:4),H);yh2=flts(u(:,5:10),H,[u(:,2:4);yh(:,2:4)]);
norm([yh1,yh2]-yh,1)
//with D<>0
D=[-3 8;4 -0.5;2.2 0.9];
Sys=syslin('d',A,B,C,D);
H=ss2tf(Sys); u=[1;-1]*(1:10);
rh=flts(u,H); rs=flts(u,Sys);
norm(rh-rs,1)
//hot restart
[ys1,x]=flts(u(:,1:4),Sys);ys2=flts(u(:,5:10),Sys,x);
norm([ys1,ys2]-rs,1)
//With H:
yh1=flts(u(:,1:4),H);yh2=flts(u(:,5:10),H,[u(:,2:4); yh1(:,2:4)]);
norm([yh1,yh2]-rh)