Model-Reference Adaptive Control (MRAC) SISO, direct Lyapunov rule

Code

  1# mrac_siso_lyapunov.py
  2# Johannes Kaisinger, 3 July 2023
  3#
  4# Demonstrate a MRAC example for a SISO plant using Lyapunov rule.
  5# Based on [1] Ex 5.7, Fig 5.12 & 5.13.
  6# Notation as in [2].
  7#
  8# [1] K. J. Aström & B. Wittenmark "Adaptive Control" Second Edition, 2008.
  9#
 10# [2] Nhan T. Nguyen "Model-Reference Adaptive Control", 2018.
 11
 12import numpy as np
 13import scipy.signal as signal
 14import matplotlib.pyplot as plt
 15import os
 16
 17import control as ct
 18
 19# Plant model as linear state-space system
 20A = -1
 21B = 0.5
 22C = 1
 23D = 0
 24
 25io_plant = ct.ss(A, B, C, D,
 26                 inputs=('u'), outputs=('x'), states=('x'), name='plant')
 27
 28# Reference model as linear state-space system
 29Am = -2
 30Bm = 2
 31Cm = 1
 32Dm = 0
 33
 34io_ref_model = ct.ss(Am, Bm, Cm, Dm,
 35                     inputs=('r'), outputs=('xm'), states=('xm'), name='ref_model')
 36
 37# Adaptive control law, u = kx*x + kr*r
 38kr_star = (Bm)/B
 39print(f"Optimal value for {kr_star = }")
 40kx_star = (Am-A)/B
 41print(f"Optimal value for {kx_star = }")
 42
 43def adaptive_controller_state(_t, xc, uc, params):
 44    """Internal state of adaptive controller, f(t,x,u;p)"""
 45    
 46    # Parameters
 47    gam = params["gam"]
 48    signB = params["signB"]
 49    
 50    # Controller inputs
 51    r = uc[0]
 52    xm = uc[1]
 53    x = uc[2]
 54    
 55    # Controller states
 56    # x1 = xc[0] # kr
 57    # x2 = xc[1] # kx
 58    
 59    # Algebraic relationships
 60    e = xm - x
 61
 62    # Controller dynamics
 63    d_x1 = gam*r*e*signB
 64    d_x2 = gam*x*e*signB
 65    
 66    return [d_x1, d_x2]
 67
 68def adaptive_controller_output(_t, xc, uc, params):
 69    """Algebraic output from adaptive controller, g(t,x,u;p)"""
 70
 71    # Controller inputs
 72    r = uc[0]
 73    #xm = uc[1]
 74    x = uc[2]
 75    
 76    # Controller state
 77    kr = xc[0]
 78    kx = xc[1]
 79
 80    # Control law
 81    u = kx*x + kr*r
 82
 83    return [u]
 84
 85params={"gam":1, "Am":Am, "Bm":Bm, "signB":np.sign(B)}
 86
 87io_controller = ct.nlsys(
 88    adaptive_controller_state,
 89    adaptive_controller_output,
 90    inputs=('r', 'xm', 'x'),
 91    outputs=('u'),
 92    states=2,
 93    params=params,
 94    name='control',
 95    dt=0
 96)
 97
 98# Overall closed loop system
 99io_closed = ct.interconnect(
100    [io_plant, io_ref_model, io_controller],
101    connections=[
102        ['plant.u', 'control.u'],
103        ['control.xm', 'ref_model.xm'],
104        ['control.x', 'plant.x']
105    ],
106    inplist=['control.r', 'ref_model.r'],
107    outlist=['plant.x', 'control.u'],
108    dt=0
109)
110
111# Set simulation duration and time steps
112Tend = 100
113dt = 0.1
114
115# Define simulation time 
116t_vec = np.arange(0, Tend, dt)
117
118# Define control reference input
119r_vec = np.zeros((2, len(t_vec)))
120rect = signal.square(2 * np.pi * 0.05 * t_vec)
121r_vec[0, :] = rect
122r_vec[1, :] = r_vec[0, :]
123
124plt.figure(figsize=(16,8))
125plt.plot(t_vec, r_vec[0,:])
126plt.title(r'reference input $r$')
127plt.show()
128
129# Set initial conditions, io_closed
130X0 = np.zeros((4, 1))
131X0[0] = 0 # state of plant, (x)
132X0[1] = 0 # state of ref_model, (xm)
133X0[2] = 0 # state of controller, (kr)
134X0[3] = 0 # state of controller, (kx)
135
136# Simulate the system with different gammas
137tout1, yout1, xout1 = ct.input_output_response(io_closed, t_vec, r_vec, X0,
138                                               return_x=True, params={"gam":0.2})
139tout2, yout2, xout2 = ct.input_output_response(io_closed, t_vec, r_vec, X0,
140                                               return_x=True, params={"gam":1.0})
141tout3, yout3, xout3 = ct.input_output_response(io_closed, t_vec, r_vec, X0,
142                                               return_x=True, params={"gam":5.0})
143
144plt.figure(figsize=(16,8))
145plt.subplot(2,1,1)
146plt.plot(tout1, yout1[0,:], label=r'$x_{\gamma = 0.2}$')
147plt.plot(tout2, yout2[0,:], label=r'$x_{\gamma = 1.0}$')
148plt.plot(tout2, yout3[0,:], label=r'$x_{\gamma = 5.0}$')
149plt.plot(tout1, xout1[1,:] ,label=r'$x_{m}$', color='black', linestyle='--')
150plt.legend(fontsize=14)
151plt.title(r'system response $x, (x_m)$')
152plt.subplot(2,1,2)
153plt.plot(tout1, yout1[1,:], label=r'$u_{\gamma = 0.2}$')
154plt.plot(tout2, yout2[1,:], label=r'$u_{\gamma = 1.0}$')
155plt.plot(tout3, yout3[1,:], label=r'$u_{\gamma = 5.0}$')
156plt.legend(loc=4, fontsize=14)
157plt.title(r'control $u$')
158
159plt.figure(figsize=(16,8))
160plt.subplot(2,1,1)
161plt.plot(tout1, xout1[2,:], label=r'$k_{r, \gamma = 0.2}$')
162plt.plot(tout2, xout2[2,:], label=r'$k_{r, \gamma = 1.0}$')
163plt.plot(tout3, xout3[2,:], label=r'$k_{r, \gamma = 5.0}$')
164plt.hlines(kr_star, 0, Tend, label=r'$k_r^{\ast}$', color='black', linestyle='--')
165plt.legend(loc=4, fontsize=14)
166plt.title(r'control gain $k_r$ (feedforward)')
167plt.subplot(2,1,2)
168plt.plot(tout1, xout1[3,:], label=r'$k_{x, \gamma = 0.2}$')
169plt.plot(tout2, xout2[3,:], label=r'$k_{x, \gamma = 1.0}$')
170plt.plot(tout3, xout3[3,:], label=r'$k_{x, \gamma = 5.0}$')
171plt.hlines(kx_star, 0, Tend, label=r'$k_x^{\ast}$', color='black', linestyle='--')
172plt.legend(loc=4, fontsize=14)
173plt.title(r'control gain $k_x$ (feedback)')
174if 'PYCONTROL_TEST_EXAMPLES' not in os.environ:
175    plt.show()

Notes

1. The environment variable PYCONTROL_TEST_EXAMPLES is used for testing to turn off plotting of the outputs.