Figure 8.11

Figure 8.11:

Gauss-Newton iterations for the direct multiple-shooting method

Code for Figure 8.11

dms_gn.py


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180 # Direct multiple-shooting Gauss-Newton SQP solver using CasADi/qpoases.
#
# Mirrors functions/dms_gn.m. ocp must contain CasADi symbolic expressions:
#     'x'   : casadi.SX/MX state vector
#     'u'   : casadi.SX/MX control vector
#     'ode' : casadi.SX/MX state derivative
#     'lsq' : casadi.SX/MX least-squares residuals
#
# data dict keys: 'T', 'x0', 'xN', 'x_min', 'x_max', 'x_guess',
#                 'u_min', 'u_max', 'u_guess'
# opts dict keys: 'N', 'verbose' (optional)
import numpy as np
import casadi


class DmsGn:
    def __init__(self, ocp, data, opts):
        self.ocp = ocp
        self.data = data
        self.opts = opts
        self.verbose = opts.get('verbose', True)

        # Problem dimensions
        self.N = opts['N']
        self.nx = ocp['x'].numel()
        self.nu = ocp['u'].numel()

        # Time grid
        self.t = np.linspace(0, data['T'], self.N + 1)

        # Interval length
        self.dt = data['T'] / self.N

        # Containers populated by rk4/transcribe/init_gauss_newton
        self.fun = {}
        self.nlp = {}
        self.sol = {}

        # Discrete-time dynamics (RK4) and the least-squares residual function
        self._rk4()

        # Transcribe to NLP
        self._transcribe()

        # Initialise decision variable and trajectories
        self.sol['w'] = self.nlp['w0']
        x_traj, u_traj = self.fun['traj'](self.sol['w'])
        self.sol['x'] = np.array(x_traj)
        self.sol['u'] = np.array(u_traj)

        # Build Gauss-Newton QP solver
        self._init_gauss_newton()

    def _rk4(self):
        x = self.ocp['x']
        u = self.ocp['u']
        ode = self.ocp['ode']
        f = casadi.Function('f', [x, u], [ode], ['x', 'p'], ['ode'])

        k1 = f(x, u)
        k2 = f(x + 0.5 * self.dt * k1, u)
        k3 = f(x + 0.5 * self.dt * k2, u)
        k4 = f(x + self.dt * k3, u)
        xk = x + self.dt / 6.0 * (k1 + 2 * k2 + 2 * k3 + k4)

        self.fun['F'] = casadi.Function('RK4', [x, u], [xk],
                                        ['x0', 'p'], ['xf'])

        lsq = self.ocp['lsq']
        self.fun['Lsq'] = casadi.Function('Lsq', [x, u], [lsq],
                                          ['x', 'p'], ['lsq'])

    def _transcribe(self):
        w = []
        g = []
        M = []
        lbw = []
        ubw = []
        w0 = []

        x_plot = []
        u_plot = []

        xk = casadi.MX.sym('x0', self.nx)
        w.append(xk)
        lbw.append(self.data['x0'])
        ubw.append(self.data['x0'])
        w0.append(self.data['x_guess'])
        x_plot.append(xk)

        for k in range(self.N):
            uk = casadi.MX.sym('u{}'.format(k), self.nu)
            w.append(uk)
            lbw.append(self.data['u_min'])
            ubw.append(self.data['u_max'])
            w0.append(self.data['u_guess'])
            u_plot.append(uk)

            Fk = self.fun['F'](x0=xk, p=uk)
            x_next = Fk['xf']

            M.append(self.fun['Lsq'](xk, uk))

            xk = casadi.MX.sym('x{}'.format(k + 1), self.nx)
            w.append(xk)
            if k == self.N - 1:
                lbw.append(self.data['xN'])
                ubw.append(self.data['xN'])
            else:
                lbw.append(self.data['x_min'])
                ubw.append(self.data['x_max'])
            w0.append(self.data['x_guess'])
            x_plot.append(xk)

            g.append(xk - x_next)

        self.nlp['w'] = casadi.vertcat(*w)
        self.nlp['g'] = casadi.vertcat(*g)
        self.nlp['M'] = casadi.vertcat(*M)
        self.nlp['lbw'] = casadi.vertcat(*lbw)
        self.nlp['ubw'] = casadi.vertcat(*ubw)
        self.nlp['w0'] = casadi.vertcat(*w0)

        self.fun['traj'] = casadi.Function(
            'traj', [self.nlp['w']],
            [casadi.horzcat(*x_plot), casadi.horzcat(*u_plot)],
            ['w'], ['x', 'u'])

    def _init_gauss_newton(self):
        self.nlp['J'] = casadi.jacobian(self.nlp['g'], self.nlp['w'])
        self.nlp['JM'] = casadi.jacobian(self.nlp['M'], self.nlp['w'])

        self.nlp['H'] = self.nlp['JM'].T @ self.nlp['JM']
        self.nlp['c'] = self.nlp['JM'].T @ self.nlp['M']

        self.fun['g'] = casadi.Function('g', [self.nlp['w']], [self.nlp['g']],
                                        ['w'], ['g'])
        self.fun['J'] = casadi.Function('J', [self.nlp['w']], [self.nlp['J']],
                                        ['w'], ['J'])
        self.fun['H'] = casadi.Function('H', [self.nlp['w']],
                                        [self.nlp['H'], self.nlp['c']],
                                        ['w'], ['H', 'c'])

        qp = dict(h=self.nlp['H'].sparsity(), a=self.nlp['J'].sparsity())
        qp_options = dict()
        if not self.verbose:
            qp_options['printLevel'] = 'none'
        self.fun['qp_solver'] = casadi.conic('qp_solver', 'qpoases', qp,
                                             qp_options)

        self.n_iter = 0
        self.sol['norm_dw'] = np.inf

    def sqpstep(self):
        self.n_iter += 1

        w = self.sol['w']
        self.sol['g'] = self.fun['g'](w)
        self.sol['J'] = self.fun['J'](w)
        H_, c_ = self.fun['H'](w)
        self.sol['H'] = H_
        self.sol['c'] = c_

        qp_solution = self.fun['qp_solver'](
            a=self.sol['J'], h=self.sol['H'], g=self.sol['c'],
            lbx=self.nlp['lbw'] - w, ubx=self.nlp['ubw'] - w,
            lba=-self.sol['g'], uba=-self.sol['g'], x0=0)
        dw = np.array(qp_solution['x']).flatten()

        self.sol['norm_dw'] = float(np.linalg.norm(dw))
        self.sol['w'] = w + dw

        x_traj, u_traj = self.fun['traj'](self.sol['w'])
        self.sol['x'] = np.array(x_traj)
        self.sol['u'] = np.array(u_traj)

        if self.verbose or self.n_iter % 10 == 1:
            print('-' * 70)
            print('{:>15s} {:>15s}'.format('SQP iteration', 'norm(dw)'))
        print('{:15d} {:15g}'.format(self.n_iter, self.sol['norm_dw']))

main.py


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57 # [makes] vdp_gauss_newton.mat
# [depends] functions/dms_gn.py
# Solves the minimization problem with T=10
#  minimize       1/2*integral{t=0 until t=T}(x1^2 + x2^2 + u^2) dt
#  subject to     dot(x1) = (1-x2^2)*x1 - x2 + u,   x1(0)=0, x1(T)=0
#                 dot(x2) = x1,                     x2(0)=1, x2(T)=0
#                 x1(t) >= -0.25, 0<=t<=T
# Joel Andersson, UW Madison 2017
import os
import sys
sys.path.insert(0, os.path.join(os.path.dirname(os.path.abspath(__file__)), 'functions'))

import numpy as np
import casadi
from scipy.io import savemat
from dms_gn import DmsGn

# States and control
x1 = casadi.SX.sym('x1')
x2 = casadi.SX.sym('x2')
u = casadi.SX.sym('u')

# Model equations
x1_dot = (1 - x2**2) * x1 - x2 + u
x2_dot = x1

# Least squares objective terms
lsq = casadi.vertcat(x1, x2, u)

# Problem structure
ocp = dict(x=casadi.vertcat(x1, x2), u=u,
           ode=casadi.vertcat(x1_dot, x2_dot), lsq=lsq)

# Problem data
data = dict(T=10.0,
            x0=np.array([0.0, 1.0]),
            xN=np.array([0.0, 0.0]),
            x_min=np.array([-0.25, -np.inf]),
            x_max=np.array([np.inf, np.inf]),
            x_guess=np.array([0.0, 0.0]),
            u_min=-1.0, u_max=1.0, u_guess=0.0)

# Solver options
opts = dict(N=20, verbose=False)

# Create an OCP solver instance
s = DmsGn(ocp, data, opts)

# Track u iterates across SQP iterations
u_all = s.sol['u'][0, :].reshape(1, -1)

while s.sol['norm_dw'] > 1e-8:
    s.sqpstep()
    u_all = np.vstack((u_all, s.sol['u'][0, :].reshape(1, -1)))

tgrid = s.t
savemat('vdp_gauss_newton.mat', {'tgrid': tgrid, 'u_all': u_all})