Figure 7.6:

Regions R_x,\ x \in X for a second-order example.

Code for Figure 7.6

Text of the GNU GPL.

solvempqp.py

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import os
import numpy as np
from scipy import linalg
import matplotlib.pyplot as plt
from datetime import datetime
import time

from removeredundantcon import removeredundantcon
from findinteriorpoint import findinteriorpoint
from halfspace2vertex import halfspace2vertex

# Module-level log file (persistent state across calls).
_logfile = [None]

def log_(msg=None, *args):
    """Write to log. Pass a file object to open it; call with no args to close."""
    if hasattr(msg, 'write'):
        _logfile[0] = msg
    elif msg is None:
        if _logfile[0] is not None:
            _logfile[0].close()
            _logfile[0] = None
    elif _logfile[0] is not None:
        _logfile[0].write(msg % args if args else msg)
        _logfile[0].flush()


def viridiscolors(n):
    """Returns n colors from the viridis colormap as an (n, 3) RGB array."""
    cmap = plt.cm.get_cmap('viridis', n)
    return np.array([cmap(i)[:3] for i in range(n)])


def bin2hex(bits):
    """Converts a boolean vector to a hexadecimal string."""
    hexchars = '0123456789ABCDEF'
    bits = np.asarray(bits, dtype=float).flatten()
    Npad = (-len(bits)) % 4
    bits = np.concatenate([np.zeros(Npad), bits])
    Nhex = len(bits) // 4
    digits = np.array([8, 4, 2, 1]) @ bits.reshape(4, Nhex)
    return ''.join(hexchars[int(d)] for d in digits)


def getcrid(active):
    """Returns the string ID of the given set of active constraints."""
    return 'CR_' + bin2hex(active)


def scalerows(A, b, zerotol=1e-12):
    """Scales rows of [A, b] to unit norm; zeros trivial rows."""
    b = np.asarray(b, dtype=float).flatten()
    Z = np.hstack([A, b[:, None]])
    scale = np.sqrt(np.sum(Z**2, axis=1))
    trivrows = (scale == 0)
    scale[trivrows] = 1.0
    A = A / scale[:, None]
    b = b / scale
    b[trivrows] = 1.0
    A[np.abs(A) < zerotol] = 0.0
    return A, b, scale


def checkcrfullrank(dmpqp, activecon):
    """Checks whether the active-constraint submatrix of H is positive definite."""
    i = activecon
    Hii = dmpqp['H'][np.ix_(np.where(i)[0], np.where(i)[0])]
    if Hii.size == 0:
        return True, Hii
    try:
        Hiihalf = linalg.cholesky(Hii, lower=False)  # upper triangular
        fullrank = bool(np.all(np.diag(Hiihalf) > 1e-6))
    except linalg.LinAlgError:
        Hiihalf = np.zeros_like(Hii)
        fullrank = False
    return fullrank, Hiihalf


def getcriticalregion(pmpqp, dmpqp, crset):
    """Returns the critical region and optimal control law for a given active set."""
    activecon = crset['active']
    if 'Hiihalf' in crset:
        Hiihalf = crset['Hiihalf']
        okay = True
    else:
        okay, Hiihalf = checkcrfullrank(dmpqp, activecon)

    cr = {}
    if not okay:
        return cr, False

    i = activecon
    j = ~activecon
    i_idx = np.where(i)[0]
    j_idx = np.where(j)[0]

    # Solve for dual variable as affine function of parameters:
    # Y * [p; 1] = lambda*(p), using Hiihalf (upper Cholesky of H[i,i]).
    RHS = np.column_stack([dmpqp['D'][i_idx, :], dmpqp['d'][i_idx]])
    Y = -linalg.cho_solve((Hiihalf, False), RHS)  # (Ni, Np+1)

    # Complementary slackness for inactive constraints.
    W = dmpqp['H'][np.ix_(j_idx, i_idx)] @ Y + \
        np.column_stack([dmpqp['D'][j_idx, :], dmpqp['d'][j_idx]])  # (Nj, Np+1)

    P = np.vstack([W, Y])        # (Nj+Ni, Np+1)
    A = -P[:, :-1]               # critical region: A*p <= b
    b = P[:, -1]

    # Screen out near-zero rows.
    Anorms = np.max(np.abs(A), axis=1)
    badrows = (Anorms < 1e-10)
    if np.any(b[badrows] < 0):
        return cr, False
    A[badrows, :] = 0.0
    b[badrows] = 1.0

    # Append parameter bounding region.
    Ne = len(pmpqp['e'])
    A = np.vstack([A, pmpqp['E']])
    b = np.concatenate([b, pmpqp['e']])
    A, b, _ = scalerows(A, b)

    x, okay, _, _ = findinteriorpoint(A, b)
    if not okay:
        return cr, False

    cr['A'] = A
    cr['b'] = b
    cr['cind'] = np.concatenate([j_idx, i_idx, -np.arange(1, Ne + 1)]).astype(int)
    cr['ctype'] = np.array(['w'] * len(j_idx) + ['y'] * len(i_idx) + ['e'] * Ne)
    cr['x'] = np.asarray(x).flatten()

    # Optimal control law: z*(p) = Z*p + z0.
    G_i = pmpqp['G'][i_idx, :]
    inner = G_i.T @ Y + np.column_stack([pmpqp['F'], pmpqp['c']])  # (Nu, Np+1)
    law = -linalg.cho_solve((pmpqp['Qhalf'], False), inner)         # (Nu, Np+1)
    cr['Z'] = law[:, :-1]
    cr['z0'] = law[:, -1]

    return cr, True


def solveprimalqp(x, pmpqp):
    """Solves the primal QP at a given parameter value x."""
    from scipy.optimize import minimize
    Q = pmpqp['Q']
    f = (pmpqp['F'] @ np.asarray(x).flatten()).flatten()
    A_ineq = pmpqp['G']
    b_ineq = (pmpqp['S'] @ np.asarray(x).flatten() + pmpqp['W']).flatten()
    n = Q.shape[0]

    result = minimize(
        fun=lambda z: 0.5 * z @ Q @ z + f @ z,
        jac=lambda z: Q @ z + f,
        x0=np.zeros(n),
        method='SLSQP',
        constraints={'type': 'ineq',
                     'fun': lambda z: b_ineq - A_ineq @ z,
                     'jac': lambda z: -A_ineq},
        options={'ftol': 1e-10, 'maxiter': 500, 'disp': False},
    )
    z = result.x if result.success else np.full(n, np.nan)
    feas = result.success
    slack = A_ineq @ z - b_ineq
    active = np.abs(slack) < 1e-5
    return z, feas, slack, active


def addcr(opensets, nopen, newcr):
    """Adds a new candidate critical region to the open set."""
    nopen += 1
    opensets[f'N{nopen}'] = newcr
    log_(f"        Adding {newcr['id']}\n")
    return opensets, nopen


def findadjacentcr(cr, pmpqp):
    """Finds an adjacent CR by stepping across the given facet and solving the QP."""
    ineq = np.ones(cr['A'].shape[0], dtype=bool)
    ineq[cr['facet']] = False
    Aeq = cr['A'][[cr['facet']], :]
    beq = cr['b'][[cr['facet']]]
    x0, okay, _, _ = findinteriorpoint(cr['A'][ineq, :], cr['b'][ineq], Aeq, beq)
    if okay:
        normal = cr['A'][cr['facet'], :]
        x0p = np.asarray(x0).flatten() + 1e-4 * normal / np.linalg.norm(normal)
        _, feas, _, newactive = solveprimalqp(x0p, pmpqp)
    else:
        feas = False
        newactive = np.array([], dtype=bool)
    if not okay or not feas:
        newactive = np.array([], dtype=bool)
    return newactive, feas


def solvempqp(prob, **kwargs):
    """
    Solves the multiparametric QP:

        min  0.5*z'*Q*z + p'*C'*z
         z
        s.t. A*z <= b + S*p

    for all p satisfying E*p <= e.

    prob must contain Q, C, A, b, S. E and e are optional.
    """
    defaults = {
        'display': True,
        'logfile': '',
        'vertices': np.nan,
        'plot': False,
        'maxiter': np.inf,
        'ascell': True,
        'maxtime': np.inf,
        'printevery': 100,
    }
    options = {**defaults, **kwargs}

    if options['logfile']:
        log_(open(options['logfile'], 'w'))
    log_(f"Log started {datetime.now()}\n")

    # Convert to Bemporad (2015) nomenclature.
    pmpqp = {}
    pmpqp['G'] = np.asarray(prob['A'])
    pmpqp['W'] = np.asarray(prob['b']).flatten()
    pmpqp['S'] = np.asarray(prob['S'])
    pmpqp['Q'] = np.asarray(prob['Q'])
    pmpqp['Q'] = 0.5 * (pmpqp['Q'] + pmpqp['Q'].T)
    pmpqp['F'] = np.asarray(prob['C'])
    pmpqp['c'] = np.zeros(pmpqp['F'].shape[0])
    pmpqp['Qhalf'] = linalg.cholesky(pmpqp['Q'], lower=False)  # upper triangular

    pmpqp['E'] = np.asarray(prob.get('E', np.zeros((0, pmpqp['S'].shape[1]))))
    if pmpqp['E'].size == 0:
        pmpqp['e'] = np.zeros(0)
    else:
        if 'e' not in prob:
            raise ValueError('prob["e"] must be provided if prob["E"] is given!')
        pmpqp['e'] = np.asarray(prob['e']).flatten()

    if np.isnan(options['vertices']):
        options['vertices'] = (pmpqp['S'].shape[1] == 2)

    # Dual problem matrices.
    Qinv_G = linalg.cho_solve((pmpqp['Qhalf'], False), pmpqp['G'].T)  # Q^{-1} G'
    dmpqp = {}
    dmpqp['H'] = pmpqp['G'] @ Qinv_G
    dmpqp['D'] = pmpqp['G'] @ linalg.cho_solve((pmpqp['Qhalf'], False), pmpqp['F']) + pmpqp['S']
    dmpqp['d'] = pmpqp['G'] @ linalg.cho_solve((pmpqp['Qhalf'], False), pmpqp['c']) + pmpqp['W']

    # qhull options (passed as dict to ConvexHull via removeredundantcon).
    qhullstr = 'Qt QbB Pp'
    if pmpqp['S'].shape[1] >= 5:
        qhullstr += ' Qx'
    qhullopts = {'qhull_options': qhullstr}

    # Initialise: no constraints active at the origin.
    Ncon = dmpqp['D'].shape[0]
    init_active = np.zeros(Ncon, dtype=bool)
    opensets = {'N1': {'active': init_active, 'id': getcrid(init_active)}}
    nopen = 1
    nfound = 0
    regions = {}
    probedregions = {opensets['N1']['id']: []}

    niter = 0
    starttime = time.time()
    maxtime = options['maxtime'] + starttime

    while nopen > 0 and niter < options['maxiter'] and time.time() < maxtime:
        niter += 1
        if options['display'] and niter % options['printevery'] == 0:
            print(f"Iteration {niter} ({nfound} found, {nopen} left to search).")
        log_(f"Iteration {niter}\n")

        activestr = f'N{nopen}'
        nopen -= 1
        currentset = opensets.pop(activestr)
        crid = currentset['id']

        log_(f"    Getting {crid}\n")
        cr, okay = getcriticalregion(pmpqp, dmpqp, currentset)

        if okay:
            log_("    Found representation\n")
            nonredundant, cr['A'], cr['b'], _, _ = removeredundantcon(
                cr['A'], cr['b'], cr['x'], None, qhullopts)
            cr['cind'] = cr['cind'][nonredundant]
            cr['ctype'] = cr['ctype'][nonredundant]

            if options['vertices']:
                cr['V'], _ = halfspace2vertex(cr['A'], cr['b'], cr['x'])

            regions[crid] = cr
            nfound += 1

            log_("    Searching for adjacent regions\n")
            for k in range(len(nonredundant)):
                n = cr['cind'][k]
                newactive = currentset['active'].copy()

                tryqpsearch = True
                if cr['ctype'][k] == 'w':
                    newactive[n] = True
                elif cr['ctype'][k] == 'y':
                    newactive[n] = False
                    tryqpsearch = False
                else:
                    continue

                keepgoing = True
                while keepgoing:
                    newcrid = getcrid(newactive)
                    if newcrid not in probedregions:
                        probedregions[newcrid] = []
                        newfullrank, newHiihalf = checkcrfullrank(dmpqp, newactive)
                        if newfullrank:
                            newcr = {'active': newactive, 'id': newcrid,
                                     'Hiihalf': newHiihalf}
                            opensets, nopen = addcr(opensets, nopen, newcr)
                            keepgoing = False
                        else:
                            log_(f"        {newcrid} is not full-rank\n")
                            if tryqpsearch:
                                tryqpsearch = False
                                log_("        Trying QP neighbor search\n")
                                cr['facet'] = k
                                newactive, keepgoing = findadjacentcr(cr, pmpqp)
                                if not keepgoing:
                                    log_("        No neighbor found\n")
                            else:
                                keepgoing = False
                    else:
                        keepgoing = False
        else:
            log_(f"    {crid} is infeasible\n")

        log_(f"    {nopen} regions open\n")

    log_(f"Finished with {nopen} open regions\n")
    log_()

    if options['display']:
        elapsed = time.time() - starttime
        print(f"Found {nfound} regions ({nopen} left to search) after "
              f"{niter} iterations (in {elapsed:.1f} s).")

    # Optional plot.
    if options['plot'] and pmpqp['S'].shape[1] == 2 and not os.getenv('OMIT_PLOTS') == 'true':
        print("Making plot (may take some time).")
        plt.figure()
        fields = list(regions.keys())
        colors = viridiscolors(len(fields))
        cperm = np.argsort(np.sin(np.arange(1, len(fields) + 1)))
        colors = colors[cperm]
        for i, f in enumerate(fields):
            plt.fill(regions[f]['V'][:, 0], regions[f]['V'][:, 1],
                     color=colors[i])
        plt.xlabel('p_1')
        plt.ylabel('p_2', rotation=0)

    # Return as list (mimicking MATLAB cell array).
    if options['ascell']:
        result = []
        for f, r in regions.items():
            region = r.copy()
            region['id'] = f
            result.append(region)
        return result

    return regions

main.py

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# Example explicit MPC for a 2D system From "Optimal control of constrained
# piecewise affine discrete-time systems" (Mayne and Rakovic, 2002).
# [depends] functions/solvempqp.py
# [makes] explicitmpc2d.mat
import os
import sys
import numpy as np
from scipy.linalg import solve_discrete_are
from scipy.io import savemat

sys.path.append('functions/')

from calcOinf import calcOinf
from ss2mpqp import ss2mpqp
from solvempqp import solvempqp, viridiscolors

# Choose system.
A = np.array([[1., 0.1], [0., 1.]])
B = np.array([[0.], [0.0787]])
Q = np.array([[1., 0.], [0., 0.]])
R = np.array([[0.1]])
ulb = np.array([-1.])
uub = np.array([1.])
N = 10

# Find LQR terminal cost and feedback gain.
P_are = solve_discrete_are(A, B, Q, R)
K_lqr = np.linalg.solve(R + B.T @ P_are @ B, B.T @ P_are @ A)
K = -K_lqr  # u = K*x (negative gain convention)

# Find terminal set Oinf for x^+ = (A + B*K)*x subject to |K*x| <= 1.
A_u = np.vstack([-K, K])
b_u = np.array([-float(ulb[0]), float(uub[0])])
A_lqr, b_lqr, _ = calcOinf(A + B @ K, A_u, b_u)
b_lqr = np.asarray(b_lqr).flatten()

# Build mpQP and solve.
prob = {'A': A, 'B': B, 'Q': Q, 'R': R, 'P': P_are, 'N': N,
        'ulb': ulb, 'uub': uub, 'Af': A_lqr, 'bf': b_lqr}
mpqp, _ = ss2mpqp(prob)
regions = solvempqp(mpqp, plot=True)