%%%  mass- spring-damper system
%%%  evaluate the free evolution and the step response
clear all
close all
clc

%%% define the parameters
b=1 ; % N/m*s
m=1;  % Kg
k=4; %N/m

% with these parameters zita=1/4 and omega_n=2

% define A, B, C and D
A=[0 1; -k/m -b/m];

B=[0;1/m];

C=[1 0];

D=0;

% define the sys by state space representation
sys=ss(A,B,C,D);

% evaluate static gain
k0=dcgain(sys);
% evaluate zita and omega_n
[wn,zeta,p] = damp(sys);
omega_n=wn(1);
zita=zeta(1);


% define the intial condition
x1_0=1;
x2_0=0;
x0=[x1_0; x2_0];

% open the simulink file with the implemented scheme for such system
mass_spring_damper_sys_scheme

figure(1)
hold on
initial(sys,x0)
grid on

figure (2)
hold on
step(sys)
grid on

% parameters for the qualitative step response for zita<<1
if zita<=sqrt(2)/2
    s_=exp(-pi*zita/sqrt(1-zita^2));
    t_max=pi/(omega_n*sqrt(1-zita^2));
    n_oscil=1/2/zita;
    ta5=3/zita/omega_n;
    T_oscil=2*pi/(omega_n*sqrt(1-zita^2));
    u0=1;
    y_ss=k0*u0;
    y_max=y_ss*(1+s_);
end