Tartalomjegyzék

ccs_nonline_Polcz_Ex1b_feedbacklin

Teljes Matlab script kiegészítő függvényekkel.

File: ccs_nonline_Polcz_Ex1b_feedbacklin.m
Directory: 4_gyujtemegy/11_CCS/_2_nonlin-pannon/2018
Author: Peter Polcz (ppolcz@gmail.com)

Created on 2018. August 13.

Model description

syms t u v real
syms x1 x2 x3 real
syms z1 z2 z3 real

x = [x1 ; x2 ; x3];
z = [z1 ; z2 ; z3];

d = @(g) jacobian(g,x);
Lie = @(f,g) d(g)*f;
br = @(f,g) d(g)*f - d(f)*g;

f = [
    x1^2 / (x1 + 1)
    (x1 + x3) / (x1 + 1) - x1
    x2*x3 / (x1 + 1)
    ];

g = [
    0
    0
    x2 / (x1 + 1)
    ];

h = x1 + x2;

phi3 = x2;

Phi = [
    h
    Lie(f,h)
    phi3
    ];

% Quick check: should be zero
% Lie(g,phi3)

\begin{align}\Phi(x) = \left(\begin{array}{c} x_{1}+x_{2}\\ \frac{{x_{1}}^2}{x_{1}+1}-x_{1}+\frac{x_{1}+x_{3}}{x_{1}+1}\\ x_{2} \end{array}\right)\end{align}

Quick check

\begin{align}L_g \phi_3(x) = 0\end{align}

$\phi_3(x)$ is given by Isidori, but anyone can check that this is indeed a solution for the PDE. This solution can also be computed by Mathematica.

Phi_inv = struct2cell(solve(z-Phi, x1, x2, x3));
Phi_inv = [ Phi_inv{:} ]';

\begin{align}\Phi^{-1}(z) = \left(\begin{array}{c} z_{1}-z_{3}\\ z_{3}\\ z_{2}+z_{1}\,z_{2}-z_{2}\,z_{3} \end{array}\right)\end{align}

Coordinate transformation

b = simplify(Lie(f, Lie(f,h)));
a = simplify(Lie(g, Lie(f,h)));
q3 = simplify(Lie(f, Phi(3)));

\begin{align}b(x) = \frac{x_{3}\,\left(-{x_{1}}^2+x_{2}\,x_{1}+x_{2}\right)}{{\left(x_{1}+1\right)}^3},~ a(x) = \frac{x_{2}}{{\left(x_{1}+1\right)}^2},~ q_3(x) = \frac{x_{3}-{x_{1}}^2}{x_{1}+1}\end{align}

b = simplify(subs(b, x, Phi_inv));
a = simplify(subs(a, x, Phi_inv));
q3 = simplify(subs(q3, x, Phi_inv));

\begin{align}b(z) = \frac{z_{2}\,\left(-{z_{1}}^2+3\,z_{1}\,z_{3}-2\,{z_{3}}^2+z_{3}\right)}{{\left(z_{1}-z_{3}+1\right)}^2},~ a(z) = \frac{z_{3}}{{\left(z_{1}-z_{3}+1\right)}^2},~ q_3(z) = \frac{z_{2}-{\left(z_{1}-z_{3}\right)}^2+z_{1}\,z_{2}-z_{2}\,z_{3}}{z_{1}-z_{3}+1}\end{align}

dz = simplify([ z2 ; collect(b+a*u,u) ; q3 ]);

System dynamics in the new coordinates

\begin{align}\dot z = \left(\begin{array}{c} z_{2}\\ \frac{u\,z_{3}+z_{2}\,z_{3}-{z_{1}}^2\,z_{2}-2\,z_{2}\,{z_{3}}^2+3\,z_{1}\,z_{2}\,z_{3}}{{\left(z_{1}-z_{3}+1\right)}^2}\\ \frac{z_{2}-{\left(z_{1}-z_{3}\right)}^2+z_{1}\,z_{2}-z_{2}\,z_{3}}{z_{1}-z_{3}+1} \end{array}\right)\end{align}

Linearizing feedback design

u_fdb = a \ (-b + v);

\begin{align}u = \frac{\left(v-\frac{z_{2}\,\left(-{z_{1}}^2+3\,z_{1}\,z_{3}-2\,{z_{3}}^2+z_{3}\right)}{{\left(z_{1}-z_{3}+1\right)}^2}\right)\,{\left(z_{1}-z_{3}+1\right)}^2}{z_{3}}\end{align}

Closed loop

dz_fdb = simplify(subs(dz, u, u_fdb));

\begin{align}\dot z = \left(\begin{array}{c} z_{2}\\ v\\ \frac{z_{2}-{\left(z_{1}-z_{3}\right)}^2+z_{1}\,z_{2}-z_{2}\,z_{3}}{z_{1}-z_{3}+1} \end{array}\right)\end{align}

Zero dynamics

Zero_dynamics = simplify(subs(dz_fdb, [z1 z2 v], [0 0 0]));
ezplot(Zero_dynamics(3)), grid on

return

The zero dynamics is stable in the origin.

pcz_align(['\dot z_3 = ' latex(Zero_dynamics(3))])

Linearized dynamics

Overall dynamics without the zero dynamics.

A = [
    0 1
    0 0
    ];

B = [
    0
    1
    ];

C = [ 1 0 ];

sys = ss(A,B,C,0);

Pole placement controller

Download model here: Polcz_Ex1a_state_feedback.slx

% State feedback gain designed by LQR
K = lqr(A,B,0.1*eye(2),1);

EXTRACT-IO

pcz_print_model('Polcz_Ex1a_state_feedback', 'simulate', true)