Nonlinear inputs (manipulated variables) in GEKKO

Question

I have a nonlinear system that has quadratic inputs (manipulated variables, [MV]). To do an MPC simulation of this system using GEKKO, I have defined the following function to handle the quadratic input:

def NTVin(T, u):
    n = T.shape[0]
    Tntv = zeros(n)
    u2 = zeros_like(u, dtype = float64)
    for i in range(len(u2)):
        u2[i] = copy(u[i].VALUE)
    for i in range(n):
        Tntv[i] = dot(dot(u2, T[i,:,:]), u2)

    return Tntv

I then define my system equations as follows:

from pylab import*
dt = .02 # sec = 20ms
m = GEKKO(remote = True)
ntg = int(tf/dt) + 1
m.time = linspace(0., tf, ntg)
m.options.CV_TYPE = 2 # 2 = squared error
x = m.Array(m.SV, Nnew, value = 0.)
y = m.Array(m.CV, Mm, value = 0.)
unb = m.Array(m.MV, B.shape[1], lb = 5.e5, ub = 2.25e6)
untv = m.Array(m.MV, Bntv.shape[1], value = 0.)
tau_sp = 3.e-2
for i in range(Bntv.shape[1]):
    untv[i].value = ic[i]
    untv[i].STATUS = 1
for i in range(B.shape[1]):
    unb[i].value = 0.
    unb[i].STATUS = 1
    unb[i].DCOST = 0.
    unb[i].COST = 0.
for i in range(N):
    x[i].value = ans0[i]
for i in range(Nnew):
    m.Equation(x[i].dt() == dot(A[i,:], x) + Bint[i] + dot(B[i,:], unb) - dot(Mnew[i,:], NTVin(Bntv, untv)))
for i in range(Mm):
    m.Equation(y[i] == dot(Phim[i,:], x))
    y[i].value = dot(Phim[i,:], ans0)
    y[i].STATUS = 1
    y[i].TR_INIT = 1
    y[i].SP = omegaR0[i]
    y[i].TAU = tau_sp
m.options.IMODE = 6
m.options.NODES = 2
m.solve(disp = True, GUI = False)

The simulation runs, but GEKKO never changes the MV untv (whatever is set for untv[i].value , remains at that value for the simulation), whereas the linear MV unb is optimized during the simulation and does vary in time. I suspect that my function NTVin is to blame (specifically in the line: u2[i] = copy(u[i].VALUE ). My question is then, can GEKKO handle nonlinear MVs, and if so, is there a way to handle this quadratic input when declaring the system equations?

Answer 1

Yes, Gekko and the solvers (APOPT, BPOPT, IPOPT) can solve nonlinear expressions. When you copy the u2[i] value, it is creating a numeric value instead of a Gekko variable. One idea for your quadratic objective term is to try the pre-built qobj function as described in the Model Building Functions of the documentation. You could also try the State Space object and set m.options.CV_TYPE=2 for a quadratic objective. There are options for continuous or discrete state space.