% ELEC-E8116 Model-based Control Systems
% Exercise 3
%%

clearvars;
close all;
clc;

%% 1.b: state-space
k1 = 1;
k2 = 4;
b1 = 0.2;
b2 = 0.1;
m1 = 1;
m2 = 2;

A = [0 0 1 0; 0 0 0 1; -k1/m1 k1/m1 -b1/m1 b1/m1; k1/m2 -(k1+k2)/m2 b1/m2 -(b1+b2)/m2];

B = [0 0;0 0; 1/m1 0; 0 1/m2];

C = [1 0 0 0; 0 1 0 0];

D  = [0 0 ; 0 0];

Gss = ss(A, B, C, D);

%% 1.c: singular values 
figure(1);clf;sigma(A, B, C, D);

%% 1.d.: H-inf norm

[Hinf,fpeak] = hinfnorm(Gss, 0.001)
% 20lgx = 21.2
x = 10^(21.2/20)

%% 2.
g11 = tf([10 10],[1 0.2 100]);
g12 = tf(1,[1 1]);
g21 = tf([1 2],[1 0.1 10]);
g22 = tf([5 5],[1 5 6]);

% transfer function
G = [g11, g12; g21, g22]

% state-space representation
Gss = ss(G)

% minimal realization
Gssm = minreal(G) % or Gss

% obtain matrices 
[Am, Bm, Cm, Dm] = ssdata(Gssm)

figure(2);clf;
sigma(Am, Bm, Cm, Dm)

%% 3. 

% a.
G1a1 = tf([2 6 4],[1 7 12 0]);
G1a2 = tf([1 2],[1 4 3]);
G1 = [G1a1, G1a2]

pole(G1)
tzero(G1)

% G must be in state-space form
G1ss = ss(G1)
G1m = minreal(G1ss)
pole(G1m)

% b. 
G2a11 = tf(1, [1 1]);
G2a12 = tf([1 3], [1 -1 -2]);
G2a21 = tf(10, [1 -2]);
G2a22 = tf(5, [1 3]);

G2 = [G2a11, G2a12; G2a21, G2a22]

pole(G2)
tzero(G2)

G2ss = ss(G2)
G2m = minreal(G2ss)
pole(G2m)
tzero(G2m)

%% 
Gsys = [0 100; 0 0];
% Gsys'
% eigenvalues
eig(Gsys)

% singular values 
svd(Gsys)
sqrt(eig(Gsys'*Gsys))