Linearize nonlinear state space equation in MATLAB at steady state setpoint

July 30, 2018 0 comments Article Uncategorized nspo

This post shows one way to linearize a nonlinear state equation $\dot{x} = f(x,u)$ at a steady state setpoint $(x_0, u_0)$ in MATLAB. It is assumed that a function ode.m exists in which the state equation is implemented:

function dx = ode(t, x, u)

 % example ODE
dx1 = tan(x(4)) * x(2) + 2*u(2) - 1;
dx2 = x(1) - sin(x(2));
dx3 = 13 * x(4) + u(1) + 1;
dx4 = 2*x(1);

dx = [dx1; dx2; dx3; dx4];
end

function dx = ode(t, x, u)

% example ODE

dx1 = tan(x(4)) * x(2) + 2*u(2) - 1;

dx2 = x(1) - sin(x(2));

dx3 = 13 * x(4) + u(1) + 1;

dx4 = 2*x(1);

dx = [dx1; dx2; dx3; dx4];

end

If the desired setpoint is not known, a steady state setpoint could be calculated with a desired objective function J in CasADi like this:

close all
clear variables
clc

import casadi.*;
ocp = casadi.Opti();

nx = 4;
nu = 2;

X = ocp.variable(nx);
U = ocp.variable(nu);


 % dynamics
ocp.subject_to([0;0;0;0] == ode(0, X, U));

 % add cost function to minimize input
J = U(1,1)^2 + U(2,1)^2;
ocp.minimize( J );

 % specify solver
ocp.solver( 'ipopt' );

 % solve OCP 
sol = ocp.solve();

x_ss = sol.value(X);
u_ss = sol.value(U);

close all

clear variables

clc

import casadi.*;

ocp = casadi.Opti();

nx = 4;

nu = 2;

X = ocp.variable(nx);

U = ocp.variable(nu);

% dynamics

ocp.subject_to([0;0;0;0] == ode(0, X, U));

% add cost function to minimize input

J = U(1,1)^2 + U(2,1)^2;

ocp.minimize( J );

% specify solver

ocp.solver( 'ipopt' );

% solve OCP

sol = ocp.solve();

x_ss = sol.value(X);

u_ss = sol.value(U);

The example yields the following setpoint: $x_0 = [0, 0, 0, -0.0769]^\mathrm{T}, u_0 = [0, 0.5]^\mathrm{T}$ .

Now we can linearize the model at the setpoint using the MATLAB Symbolic Math Toolbox:

 % setpoint
t = 0;
x=[0; 0; 0; -0.0769]; % state
u=[0;0.5]; % control input

 % for eval later
x1 = x(1); x2 = x(2); x3 = x(3); x4 = x(4);
u1 = u(1); u2 = u(2);

 % symbolic vars
x_sym = sym('x', [4 1]);
u_sym = sym('u', [2 1]);
ode_sym = ode(t,x_sym,u_sym);

 % calculate system and input matrices
Jac_x = jacobian(ode_sym, x_sym);
A = eval(subs(Jac_x));

Jac_u = jacobian(ode_sym, u_sym);
B = eval(subs(Jac_u));

% setpoint

t = 0;

x=[0; 0; 0; -0.0769]; % state

u=[0;0.5]; % control input

% for eval later

x1 = x(1); x2 = x(2); x3 = x(3); x4 = x(4);

u1 = u(1); u2 = u(2);

% symbolic vars

x_sym = sym('x', [4 1]);

u_sym = sym('u', [2 1]);

ode_sym = ode(t,x_sym,u_sym);

% calculate system and input matrices

Jac_x = jacobian(ode_sym, x_sym);

A = eval(subs(Jac_x));

Jac_u = jacobian(ode_sym, u_sym);

B = eval(subs(Jac_u));

This linearization yields the system matrix $A = \begin{bmatrix} 0& -0.0771& 0& 0 \\ 1& -1& 0& 0 \\ 0& 0& 0& 13 \\ 2& 0& 0& 0 \end{bmatrix}$
and the input matrix $B = \begin{bmatrix} 0& 2\\ 0& 0\\ 1& 0\\ 0& 0 \end{bmatrix}$ which could be used in a linear model approximation.

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Linearize nonlinear state space equation in MATLAB at steady state setpoint

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