Re-write MPC cost function to QP formulation

Question

MC 2017-4-29

1
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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/338012-re-write-mpc-cost-function-to-qp-formulation

回答： Nicolas Schmit 2017-9-5

Hello!

I've been trying to make a simple script to re-write the cost function in the usual MPC formulation to the QP formulation used to solve the optmization problem. My code is as follows: (Note its using symbolics toolbox)

xd = xda+gradxda*(x7-thetaka);
yd = yda+gradyda*(x7-thetaka);
x = [x1 x2 x3 x4 x5 x6 x7].';
u = [u1 u2 u3 u4 u5].';
z = [x;u];
e_c = sinx*(x1-xd)-cosx*(x2-yd);
e_l = -cosx*(x1-xd)-sinx*(x2-yd);
Q = [q_c 0; 0 q_l];
J = [e_c;e_l].'*Q*[e_c;e_l] - q_t*x7;
H = hessian(J,z);
c = gradient(J,z);
c = subs(c, z, [0;0;0;0;0;0;0;0;0;0;0;0]);
f = 0.5*z.'*H*z+c.'*z;
isequal(simplify(J), simplify(f))

Should not J and f be equal? The isequal command will yield a logical 0 for me. However I tested it on an example cost function, found here: https://se.mathworks.com/help/optim/ug/quadprog.html#bssh6y6-8 and using that cost function with my above code will yield a logical 1, as I expect. Have I misunderstood the connection between J and f or have I made a coding mistake?

Best regards MC