Figure 2.5:

Closed-loop economic MPC versus tracking MPC starting at x=(-8,8) with optimal steady state (8,4). Both controllers asymptotically stabilize the steady state. Dashed contours show cost functions for each controller.

Code for Figure 2.5

Text of the GNU GPL.

main.py

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# Compare economic vs. tracking mpc.
# [makes] econvstracking.mat

import numpy as np
import scipy.io as sio
import matplotlib.pyplot as plt
import mpctools as mpc
from scipy.linalg import solve_discrete_lyapunov
import os

Nx = 2
Ny = Nx
Nu = 1
hor = 50

N = {}
N['x'] = Nx
N['y'] = Ny
N['u'] = Nu
N['t'] = hor

# Linear model
A = np.array([[0.5, 1], [0, 0.75]])
B = np.array([[0], [1]])
f = mpc.getCasadiFunc(lambda x, u: A @ x + B @ u, [Nx, Nu], ['x', 'u'], 'f')

# Objective function
q = np.array([[-2], [2]])
r = -10 * np.eye(Nu)
lecon = mpc.getCasadiFunc(lambda x, u: q.T @ x + r @ u, [Nx, Nu], ['x', 'u'], 'lecon')

Q = np.array([[10, 0], [0, 10]])
R = np.array([[1]])
ltrack = mpc.getCasadiFunc(lambda x, u, x_sp, u_sp: (x - x_sp).T @ Q @ (x - x_sp) + (u - u_sp).T @ R @ (u - u_sp),
    [Nx, Nu, Nx, Nu], ['x', 'u', 'x_sp', 'u_sp'], 'ltrack')

# Bounds
lb, ub = {}, {}
lb['u'] = -np.ones(Nu)
ub['u'] = np.ones(Nu)
lb['x'] = np.ones(Nx) * -10
ub['x'] = np.ones(Nx) * 10


# Target finder
target = mpc.sstarg(f=f,
                    h=None,
                    l=lecon,
                    N=N,
                    lb=lb,
                    ub=ub,
                    verbosity=0,
                    inferargs=True)
target.solve()
xs = np.squeeze(target.var['x', 0])
us = np.squeeze(target.var['u', 0])
ls = target.obj

# Set-up controllers
init = {}
init['xs'] = xs
init['us'] = us

# Define equality constraint for terminal state xf
# NOTE: keep some wiggle room, may encounter solver problems otherwise
wiggle = 1e-5
lb['xf'] = xs * (1 - wiggle)
ub['xf'] = xs * (1 + wiggle)

econ = mpc.nmpc(f=f,
                l=lecon,
                N=N,
                lb=lb, ub=ub,
                verbosity=0,
                inferargs=True)

track = mpc.nmpc(f=f,
                l=ltrack,
                N=N,
                par={'xsp': xs, 'usp': us},
                lb=lb, ub=ub,
                verbosity=0,
                inferargs=True)

# Simulate both
Nsim = 25
x0 = np.array([-8, 8])
controllers = [econ, track]
names = ['econ', 'track']
data = {}
for i, controller in enumerate(controllers):
    print(f'Simulating {names[i]} MPC\n')
    x = np.zeros((Nx, Nsim + 1))
    x[:, 0] = x0 # Initial condition
    u = np.zeros((Nu, Nsim))
    for t in range(Nsim):
        controller.fixvar('x', 0, x[:, t])
        controller.solve()
        u[:, t] = np.squeeze(controller.var['u', 0])
        x[:, t + 1] = np.squeeze(controller.var['x', 1])
        controller.saveguess()
    data[names[i]] = {}
    data[names[i]]['x'] = x
    data[names[i]]['u'] = u

# get data for objective function contours
x1 = np.linspace(-10, 15, 251)
x2 = np.linspace(0, 10, 101)

xx1, xx2 = np.meshgrid(x1, x2)
dx1 = xx1 - xs[0]
dx2 = xx2 - xs[1]
Vecon = q[0] * dx1 + q[1] * dx2
Vtrack = Q[0, 0] * (dx1 ** 2) + 2 * Q[1, 0] * dx1 * dx2 + Q[1, 1] * (dx2 ** 2)
data['contours'] = {}
data['contours']['x1'] = x1
data['contours']['x2'] = x2
data['contours']['Vecon'] = Vecon
data['contours']['Vtrack'] = Vtrack

if not os.getenv('OMIT_PLOTS') == 'true':
    plt.figure()
    plt.contour(xx1, xx2, Vecon, levels=10)
    plt.contour(xx1, xx2, Vtrack, levels=10)
    colors = ['blue', 'red']
    for i, name in enumerate(names):
        plt.plot(data[name]['x'][0, :], data[name]['x'][1, :], '-o', color=colors[i], mfc='none')

    plt.xlabel(r'$x_1$')
    plt.ylabel(r'$x_2$', rotation=0)
    plt.show()

# Find rotated cost function
mu = -np.linalg.solve((np.eye(Nx) - A).T, q)
alpha = 0.01
Mu = solve_discrete_lyapunov(A.T, alpha * np.eye(Nx))

lam = lambda x: mu.T @ (x - xs) + (x - xs).T @ Mu @ (x - xs)

lrot = lambda x, u: q.T @ (x - xs) + r @ (u - us) + lam(x) - lam(A@x + B@u) - alpha * (x - xs).T @ (x - xs)

x1_vals = np.linspace(lb['x'][0], ub['x'][0], 21)
x2_vals = np.linspace(lb['x'][1], ub['x'][1], 21)
u_vals = np.linspace(lb['u'], ub['u'], 21)
x1, x2, u = np.meshgrid(x1_vals, x2_vals, u_vals, indexing='ij')
l = np.empty_like(x1)
for i in range(21):
    for j in range(21):
        for k in range(21):
            x = np.array([x1[i, j, k], x2[i, j, k]])
            l[i, j, k] = lrot(x, np.array([u[i, j, k]])).item()
print(f"Minimum of dissipation inequality: {l.min():.6g}")

# Simulate multiple economic MPC trajectories
econ.saveguess(default=True)
Nx0 = 16
colors = plt.cm.jet(np.linspace(0, 1, Nx0))
theta = np.linspace(0, 2 * np.pi, Nx0 + 1)[1:]
x0 = np.vstack((8 * np.cos(theta), 8 * np.sin(theta)))
Nsim = 21
x = np.full((Nx, Nsim, Nx0), np.nan)
u = np.full((Nu, Nsim, Nx0), np.nan)
V = np.full((Nsim, Nx0), np.nan)
if not os.getenv('OMIT_PLOTS') == 'true':
    fig, axs = plt.subplots(2, 2, figsize=(10, 6))
t = np.arange(Nsim)
print('Simulating more economic MPC', end='', flush=True)
for j in range(Nx0):
    x[:, 0, j] = x0[:, j]
    for i in range(Nsim):
        econ.fixvar('x', 0, x[:, i, j])
        econ.solve()
        if econ.stats['status'] != 'Solve_Succeeded':
            print(f'Controller failed at time {i} with controller status {econ.stats["status"]}')
            break
        u[:, i, j] = np.squeeze(econ.var['u', 0])
        V[i, j] = econ.obj + lam(x[:, i,j]).item() - N['t'] * ls
        if i < Nsim-1:
            x[:, i + 1, j] = np.squeeze(econ.var['x', 1])
    print('.', end='', flush=True)

    if not os.getenv('OMIT_PLOTS') == 'true':
        axs[0, 0].plot(t, x[0, :, j], color=colors[j])
        axs[0, 0].set_ylabel(r'$x_1$', rotation=0)

        axs[0, 1].plot(t, u[0, :, j], color=colors[j])
        axs[0, 1].set_ylabel(r'$u$', rotation=0)

        axs[1, 0].plot(t, x[1, :, j], color=colors[j])
        axs[1, 0].set_ylabel(r'$x_2$', rotation=0)
        axs[1, 0].set_xlabel('Time')

        axs[1, 1].plot(t, V[:Nsim, j], color=colors[j])
        axs[1, 1].set_ylabel(r'$V$', rotation=0)
        axs[1, 1].set_xlabel('Time')

if not os.getenv('OMIT_PLOTS') == 'true':
    plt.tight_layout()
    plt.show()

data['phase'] = {}
data['phase']['x'] = x
data['phase']['u'] = u
data['phase']['V'] = V