Tutorial Design via State Space

clear
clc
sympref('FloatingPointOutput',true)
syms s k1 k2 k3 k4 ke zeros;

%% Input:

% Problem 3
numg = [20]; % Define numerator of G(s)
deng = poly([2 4 8]); % Define denominator of G(s)
Os = 15; % Overshoot (%)
Ts = 0.75;  % Time settling

% Problem 4
%numg = [20 40]; % Define numerator of G(s)
%deng = poly([0 5 7]); % Define denominator of G(s)
%Os = 10; % Overshoot (%)
%Ts = 2;  % Time settling

% Problem 5
%numg = [1 6]; % Define numerator of G(s)
%deng = poly([3 8 10]); % Define denominator of G(s)
%Os = 10; % Overshoot (%)
%Ts = 1;  % Time settling
%zeros = -6

zeta = (-log(Os/100))/(sqrt(pi^2+log(Os/100)^2));
Wn = 4/(zeta*Ts);

zeros =  real(solve(s^2 + 2*zeta*Wn*s + Wn^2))*10;  % Third pole / Open loop zero
%speed = 10 % observer speed

%% Process

G = tf(numg,deng)   % Generate transfer function G(s)
[Ac Bc Cc Dc] = tf2ss(numg,deng);
P = [0 0 1; 0 1 0; 1 0 0];
A = inv(P)*Ac*P
B = inv(P)*Bc
C = Cc*P
D = Dc

Cm = det(ctrb(A,B))      % Controlbility
Om = det(obsv(A,C))      % Observability

%% controller

%state-feedback control
pole1 = s^2 + 2*zeta*Wn*s + Wn^2
pole2 = s-zeros(1)
Anew = A-(B*[k1 k2 k3])

%Integral control
%pole1 = s^2 + 2*zeta*Wn*s + Wn^2
%pole2 = s-zeros(1)
%a11 = A-(B*[k1 k2 k3]);
%a12 = B*ke;
%a21 = -C;
%Anew = [a11 a12; a21 0];

% Observer
%pole1 = s+zeta*Wn*speed + (Wn*sqrt(1-zeta^2))*speed*1i;
%pole2 = s+zeta*Wn*speed - (Wn*sqrt(1-zeta^2))*speed*1i;
%pole3 = s+zeta*Wn*speed*10;
%Anew = A-[L1; L2; L3]*C;

Ds = expand(pole1*pole2)
Ts = det(s*eye(3)-Anew)

% Solve
Dsm = coeffs(Ds,s);
Tsm = coeffs(Ts,s);
k = vpa(struct2cell((solve(Dsm==Tsm))))