Why is the jacobian function so slow for calculating jacobian matrix with many variables?

I have 40 variables, define them in a symbolic vector and then use this vector to bulid a scalar cost-function. Now I want to use the jacobian function to calculate the jacobian matrix of this cost function. I think I will get the jacobian matrix with symbolics variables. But the computation is too slow. I guess maybe i've done something wrong somewhere.

In my code I set N_p to 20 and what i want to get the jacobian matrix of cost_fun about u.

The following is my main code:

x0 = sym('x0',[8 1],'real');
u = sym('u',[2*N_p 1],'real');
kappa = sym('kappa','real');
x = sym(zeros(8*N_p,1));
u0 = sym('u0',[2 1],'real');
f = cost_fun(x0,u,m,I_z,lf,lr,Cf,Cr,N_p,t,Q,x_ref,R1,R2,F_deltau,kappa,u0);
g=jacobian(f,u);

The following is my cost-function:

function f = cost_fun(x0,u,m,I_z,lf,lr,Cf,Cr,N_p,t,Q,x_ref,R1,R2,F_deltau,kappa,u0)
% cost function

x(1:8,1) = x0 + t * bicycle_model_2(x0,u(1:2,1),m,I_z,lf,lr,Cf,Cr,kappa);
for i=1:1:(N_p-1)
    x((8*i+1):(8*i+8),1) = x((8*i-7):(8*i),1) + t * bicycle_model_2(x((8*i-7):(8*i),1),u((2*i+1):(2*i+2),1),m,I_z,lf,lr,Cf,Cr,kappa);
end
% u_0 = sym('u_0',[2*N_p 1],'real');
u_0(1:2,1)=u0;
u_0(3:2*N_p,1)=zeros(2*N_p-2,1);
f=(x-x_ref)'*Q*(x-x_ref)+u'*R1*u+(F_deltau*u-u_0)'*R2*(F_deltau*u-u_0);
end

function f=bicycle_model_2(x,u,m,I_z,lf,lr,Cf,Cr,kappa)

delta_f=u(1);F_xT=1000*u(2);
x_dot=x(1)+eps;y_dot=x(2);phi_dot=x(3);phi=x(4);X=x(5);Y=x(6);e_phi=x(7);e_y=x(8);
f1=(m*y_dot*phi_dot+F_xT)/m;
f2=(-m*x_dot*phi_dot+2*Cf*(delta_f-(y_dot+lf*phi_dot)/x_dot)+2*Cr*(-(y_dot-lr*phi_dot)/x_dot))/m;
f3=(lf*2*Cf*(delta_f-(y_dot+lf*phi_dot)/x_dot)-lr*(-(y_dot-lr*phi_dot)/x_dot))/I_z;
f4=phi_dot;
f5=x_dot*cos(phi)-y_dot*sin(phi);
f6=x_dot*sin(phi)+y_dot*cos(phi);
f7=phi_dot-kappa*x_dot;
f8=x_dot*sin(e_phi)+y_dot*cos(e_phi);
f=[f1 f2 f3 f4 f5 f6 f7 f8]';

In fact, I also want to change the result of jacobian matrix with symbolic variables into a function handle. But the matlabFunction costs much time, that can be unacceptable. Has someone some suggestions?

[Tan Shuwen is pig]

Have you solve this problem?

[Torsten]

Don't use symbolic computation if speed is an issue. When the Jacobian is needed, use a finite-difference approximation for the derivatives.

Answer 1

[Matt J]

Use matlabFunction to compute the jacobian from symbolic to numerical form, so that when it comes time to use it repeatedly, it will do so faster.