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# mrac_siso_mit.py
# Johannes Kaisinger, 3 July 2023
#
# Demonstrate a MRAC example for a SISO plant using MIT rule.
# Based on [1] Ex 5.2, Fig 5.5 & 5.6.
# Notation as in [2].
#
# [1] K. J. Aström & B. Wittenmark "Adaptive Control" Second Edition, 2008.
#
# [2] Nhan T. Nguyen "Model-Reference Adaptive Control", 2018.
import numpy as np
import scipy.signal as signal
import matplotlib.pyplot as plt
import os
import control as ct
# Plant model as linear state-space system
A = -1.
B = 0.5
C = 1
D = 0
io_plant = ct.ss(A, B, C, D,
inputs=('u'), outputs=('x'), states=('x'), name='plant')
# Reference model as linear state-space system
Am = -2
Bm = 2
Cm = 1
Dm = 0
io_ref_model = ct.ss(Am, Bm, Cm, Dm,
inputs=('r'), outputs=('xm'), states=('xm'), name='ref_model')
# Adaptive control law, u = kx*x + kr*r
kr_star = (Bm)/B
print(f"Optimal value for {kr_star = }")
kx_star = (Am-A)/B
print(f"Optimal value for {kx_star = }")
def adaptive_controller_state(t, xc, uc, params):
"""Internal state of adaptive controller, f(t,x,u;p)"""
# Parameters
gam = params["gam"]
Am = params["Am"]
Bm = params["Bm"]
signB = params["signB"]
# Controller inputs
r = uc[0]
xm = uc[1]
x = uc[2]
# Controller states
x1 = xc[0] #
# x2 = xc[1] # kr
x3 = xc[2] #
# x4 = xc[3] # kx
# Algebraic relationships
e = xm - x
# Controller dynamics
d_x1 = Am*x1 + Am*r
d_x2 = - gam*x1*e*signB
d_x3 = Am*x3 + Am*x
d_x4 = - gam*x3*e*signB
return [d_x1, d_x2, d_x3, d_x4]
def adaptive_controller_output(t, xc, uc, params):
"""Algebraic output from adaptive controller, g(t,x,u;p)"""
# Controller inputs
r = uc[0]
# xm = uc[1]
x = uc[2]
# Controller state
kr = xc[1]
kx = xc[3]
# Control law
u = kx*x + kr*r
return [u]
params={"gam":1, "Am":Am, "Bm":Bm, "signB":np.sign(B)}
io_controller = ct.nlsys(
adaptive_controller_state,
adaptive_controller_output,
inputs=('r', 'xm', 'x'),
outputs=('u'),
states=4,
params=params,
name='control',
dt=0
)
# Overall closed loop system
io_closed = ct.interconnect(
[io_plant, io_ref_model, io_controller],
connections=[
['plant.u', 'control.u'],
['control.xm', 'ref_model.xm'],
['control.x', 'plant.x']
],
inplist=['control.r', 'ref_model.r'],
outlist=['plant.x', 'control.u'],
dt=0
)
# Set simulation duration and time steps
Tend = 100
dt = 0.1
# Define simulation time
t_vec = np.arange(0, Tend, dt)
# Define control reference input
r_vec = np.zeros((2, len(t_vec)))
square = signal.square(2 * np.pi * 0.05 * t_vec)
r_vec[0, :] = square
r_vec[1, :] = r_vec[0, :]
plt.figure(figsize=(16,8))
plt.plot(t_vec, r_vec[0,:])
plt.title(r'reference input $r$')
plt.show()
# Set initial conditions, io_closed
X0 = np.zeros((6, 1))
X0[0] = 0 # state of plant, (x)
X0[1] = 0 # state of ref_model, (xm)
X0[2] = 0 # state of controller,
X0[3] = 0 # state of controller, (kr)
X0[4] = 0 # state of controller,
X0[5] = 0 # state of controller, (kx)
# Simulate the system with different gammas
tout1, yout1, xout1 = ct.input_output_response(io_closed, t_vec, r_vec, X0,
return_x=True, params={"gam":0.2})
tout2, yout2, xout2 = ct.input_output_response(io_closed, t_vec, r_vec, X0,
return_x=True, params={"gam":1.0})
tout3, yout3, xout3 = ct.input_output_response(io_closed, t_vec, r_vec, X0,
return_x=True, params={"gam":5.0})
plt.figure(figsize=(16,8))
plt.subplot(2,1,1)
plt.plot(tout1, yout1[0,:], label=r'$x_{\gamma = 0.2}$')
plt.plot(tout2, yout2[0,:], label=r'$x_{\gamma = 1.0}$')
plt.plot(tout2, yout3[0,:], label=r'$x_{\gamma = 5.0}$')
plt.plot(tout1, xout1[1,:] ,label=r'$x_{m}$', color='black', linestyle='--')
plt.legend(fontsize=14)
plt.title(r'system response $x, (x_m)$')
plt.subplot(2,1,2)
plt.plot(tout1, yout1[1,:], label=r'$u_{\gamma = 0.2}$')
plt.plot(tout2, yout2[1,:], label=r'$u_{\gamma = 1.0}$')
plt.plot(tout3, yout3[1,:], label=r'$u_{\gamma = 5.0}$')
plt.legend(loc=4, fontsize=14)
plt.title(r'control $u$')
plt.figure(figsize=(16,8))
plt.subplot(2,1,1)
plt.plot(tout1, xout1[3,:], label=r'$k_{r, \gamma = 0.2}$')
plt.plot(tout2, xout2[3,:], label=r'$k_{r, \gamma = 1.0}$')
plt.plot(tout3, xout3[3,:], label=r'$k_{r, \gamma = 5.0}$')
plt.hlines(kr_star, 0, Tend, label=r'$k_r^{\ast}$', color='black', linestyle='--')
plt.legend(loc=4, fontsize=14)
plt.title(r'control gain $k_r$ (feedforward)')
plt.subplot(2,1,2)
plt.plot(tout1, xout1[5,:], label=r'$k_{x, \gamma = 0.2}$')
plt.plot(tout2, xout2[5,:], label=r'$k_{x, \gamma = 1.0}$')
plt.plot(tout3, xout3[5,:], label=r'$k_{x, \gamma = 5.0}$')
plt.hlines(kx_star, 0, Tend, label=r'$k_x^{\ast}$', color='black', linestyle='--')
plt.legend(loc=4, fontsize=14)
plt.title(r'control gain $k_x$ (feedback)')
if 'PYCONTROL_TEST_EXAMPLES' not in os.environ:
plt.show()