examples/python/lqr_control.py

#
#     This file is part of CasADi.
#
#     CasADi -- A symbolic framework for dynamic optimization.
#     Copyright (C) 2010-2014 Joel Andersson, Joris Gillis, Moritz Diehl,
#                             K.U. Leuven. All rights reserved.
#     Copyright (C) 2011-2014 Greg Horn
#
#     CasADi is free software; you can redistribute it and/or
#     modify it under the terms of the GNU Lesser General Public
#     License as published by the Free Software Foundation; either
#     version 3 of the License, or (at your option) any later version.
#
#     CasADi is distributed in the hope that it will be useful,
#     but WITHOUT ANY WARRANTY; without even the implied warranty of
#     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
#     Lesser General Public License for more details.
#
#     You should have received a copy of the GNU Lesser General Public
#     License along with CasADi; if not, write to the Free Software
#     Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
#
#
from pylab import *
from numpy import *
from casadi import *
from casadi.tools import *
import scipy.linalg
import numpy as np

"""
This example how an Linear Quadratic Regulator (LQR) can be designed
and simulated for a linear-time-invariant (LTI) system
"""

# Problem statement
# -----------------------------------

# System dimensions
ns = 3  # number of states
nu = 2  # number of controls
ny = 2  # number of outputs
N  = 21 # number of control intervals
te = 10 # end time [s]

# The system equations:   x' = A.x + B.u
A = DM([[0.4,0.1,-2],[0,-0.3,4],[1,0,0]])
B = DM([[1,1],[0,1],[1,0]])

# Inspect the open-loop system
[D,V] = linalg.eig(A)
print('Open-loop eigenvalues: ', D)

tn = linspace(0,te,N)

# Check if the system is controllable
# -----------------------------------
R = []

p = B
for i in range(ns):
  R.append(p)
  p = mtimes([A,p])

R = horzcat(*R)

# R must be of full rank
_, s, _ = linalg.svd(R)
eps = 1e-9
rank =  len([x for x in s if abs(x) > eps])
assert(rank==ns)

# Simulation of the open-loop system
# -----------------------------------

y  = SX.sym('y',ns)
u  = SX.sym('u',nu)

x0 = DM([1,0,0])
# no control
u_ = repmat(DM([[ -1, 1 ],[1,-1]]),1,int((N-1)/2))

print(u_)

p = SX.sym('p')

# tn = np.linspace(0,te,N)
# cdae = SX.fun('cdae', controldaeIn(x=y,u=u),[mtimes(A,y)+mtimes(B,u)])

# opts = {}
# opts['integrator'] = 'cvodes'
# opts['integrator_options'] = {'fsens_err_con': True,'reltol':1e-12}
# opts['nf'] = 20

# sim = ControlSimulator('sim', cdae, tn, opts)
# sim.setInput(x0,'x0')
# sim.setInput(u_,'u')
# sim.evaluate()

# tf = sim.minorT()

# figure(1)
# plot(tf,sim.getOutput().T)
# legend(('s1', 's2','s3'))
# title('reference simulation, open-loop, zero controls')
# out = sim.getOutput()

# # Simulation of the open-loop system
# #   sensitivity for initial conditions
# # -----------------------------------

# x0_pert = DM([0,0,1])*1e-4
# sim.setInput(x0+x0_pert,'x0')
# sim.setInput(u_,'u')
# sim.evaluate()

# tf = list(sim.minorT())

# figure(2)
# title('Deviation from reference simulation, with perturbed initial condition')
# plot(tf,sim.getOutput().T-out.T,linewidth=3)

# # Not supported in current revision, cf. #929
# # jacsim = sim.jacobian_old(CONTROLSIMULATOR_X0,0)
# # jacsim.setInput(x0,'x0')
# # jacsim.setInput(u_,'u')

# # jacsim.evaluate()

# # dev_est = []
# # for i in range(len(tf)):
# #   dev_est.append(mtimes(jacsim.getOutput()[i*ns:(i+1)*ns,:],x0_pert))

# # dev_est = horzcat(dev_est).T
# # plot(tf,dev_est,'+k')
# # legend(('s1 dev', 's2 dev','s3 dev','s1 dev (est.)', 's2 dev (est.)','s3 dev (est.)'),loc='upper left')


# # M = jacsim.getOutput()[-ns:,:]
# # # In the case of zero input, we could also use the matrix exponential to obtain sensitivity
# # Mref = scipy.linalg.expm(A*te)

# # e = fabs(M - Mref)
# # e = max(e)/max(fabs(M))
# # print e
# # assert(e<1e-6)


# # Simulation of the open-loop system
# #   sensitivity  for controls
# # -----------------------------------
# # What if we perturb the input?

# u_perturb = DM(u_)
# u_perturb[0,N/5] = 1e-4
# sim.setInput(x0,'x0')
# sim.setInput(u_+u_perturb,'u')
# sim.evaluate()

# figure(3)
# title('Deviation from reference simulation, with perturbed controls')
# plot(tf,sim.getOutput().T-out.T,linewidth=3)


# # Not supported in current revision, cf. #929
# # jacsim = sim.jacobian_old(CONTROLSIMULATOR_U,0)
# # jacsim.setInput(x0,'x0')
# # jacsim.setInput(u_,'u')

# # jacsim.evaluate()

# # dev_est = []
# # for i in range(len(tf)):
# #   dev_est.append(mtimes(jacsim.getOutput()[i*ns:(i+1)*ns,:],vec(u_perturb)))

# # dev_est = horzcat(dev_est).T
# # plot(tf,dev_est,'+k')
# # legend(('s1 dev', 's2 dev','s3 dev','s1 dev (est.)', 's2 dev (est.)','s3 dev (est.)'),loc='upper left')


# # Find feedforward controls
# # -----------------------------------
# #
# #  Our goal is to reach a particular end-state xref_e
# #  We can find the necessary controls explicitly in continuous time
# #  with the controllability Gramian
# #
# #  http://www-control.eng.cam.ac.uk/jmm/3f2/handout4.pdf
# #  http://www.ece.rutgers.edu/~gajic/psfiles/chap5.pdf

# x0 = vertcat([1,0,0])
# xref_e = vertcat([1,0,0])

# states = struct_symSX([
#            entry('eAt',shape=(ns,ns)),
#            entry('Wt',shape=(ns,ns))
#          ])

# eAt = states['eAt']
# Wt  = states['Wt']

# # We will find the control that allow to reach xref_e at t1
# t1 = te

# # Initial conditions
# e = densify(DM.eye(ns))
# states_ = states(0)
# states_['eAt'] = e
# states_['Wt'] = 0

# rhs = struct_SX(states)
# rhs['eAt'] = mtimes(A,eAt)
# rhs['Wt']  = mtimes([eAt,B,B.T,eAt.T])

# dae = SX.fun('dae', daeIn(x=states),daeOut(ode=rhs))

# integrator = Integrator('integrator', 'cvodes', dae, {'tf':t1, 'reltol':1e-12})
# integrator.setInput(states_,'x0')
# integrator.evaluate()

# out = states(integrator.getOutput())

# Wt_  = out['Wt']
# eAt_ = out['eAt']

# # Check that eAt is indeed the matrix exponential
# e = max(fabs(eAt_-scipy.linalg.expm(numpy.array(A*te))))/max(fabs(eAt_))
# assert(e<1e-7)

# # Simulate with feedforward controls
# # -----------------------------------

# states = struct_symSX([
#           entry('y',shape=ns),      # The regular states of the LTI system
#           entry('eAt',shape=(ns,ns))  # The matrix exponential exp(A*(t1-t))
#          ])

# eAt = states['eAt']
# y   = states['y']

# # Initial conditions
# states_ = states(0)
# states_['y'] = x0
# states_['eAt'] = eAt_

# u = mtimes([B.T,eAt.T,inv(Wt_),xref_e-mtimes(eAt_,x0)])

# rhs = struct_SX(states)
# rhs['y']   = mtimes(A,y)+mtimes(B,u)
# rhs['eAt'] = -mtimes(A,eAt)

# cdae = SX.fun('cdae', controldaeIn(x=states),[rhs])

# # Output function
# out = SX.fun('out', controldaeIn(x=states),[states,u])

# opts = {}
# opts['integrator'] = 'cvodes'
# opts['integrator_options'] = {'fsens_err_con': True,'reltol':1e-12}
# opts['nf'] = 20

# sim = ControlSimulator('sim', cdae, out, tn, opts)
# sim.setInput(states_,'x0')
# sim.evaluate()

# e = sim.getOutput()[states.i['y'],-1] - xref_e
# assert(max(fabs(e))/max(fabs(xref_e))<1e-6)

# tf = sim.minorT()
# # This will be our reference trajectory for tracking

# figure(4)
# subplot(211)
# title('Feedforward control, states')
# plot(tf,sim.getOutput(0)[list(states.i['y']),:].T)
# for i,c in enumerate(['b','g','r']):
#   plot(t1,xref_e[i],c+'o')
# subplot(212)
# title('Control action')
# plot(tf,sim.getOutput(1).T)

# # Design an infinite horizon LQR
# # -----------------------------------

# # Weights for the infinite horizon LQR control
# Q = DM.eye(ns)
# R = DM.eye(nu)

# # Continuous Riccati equation
# P = SX.sym('P',ns,ns)

# ric = (Q + mtimes(A.T,P) + mtimes(P,A) - mtimes([P,B,inv(R),B.T,P]))

# dae = SX.fun('dae', daeIn(x=vec(P)),daeOut(ode=vec(ric)))

# # We solve the ricatti equation by simulating backwards in time until steady state is reached.
# opts = {'reltol':1e-16, 'stop_at_end':False, 'tf':1}
# integrator = Integrator('integrator', 'cvodes', dae, opts)

# # Start from P = identity matrix
# u = densify(DM.eye(ns))
# xe = vec(u)

# # Keep integrating until steady state is reached
# for i in range(1,40):
#   x0 = xe
#   xe = integrator({'x0':x0})['xf']
#   e = max(fabs(xe-x0))
#   print 'it. %02d - deviation from steady state: %.2e' % (i, e)
#   if e < 1e-11:
#     break

# # Obtain the solution of the ricatti equation
# P_ = xe.reshape((ns,ns))
# print 'P=', P_

# [D,V] = linalg.eig(P_)
# assert (min(real(D))>0)
# print '(positive definite)'


# # Check that it does indeed satisfy the ricatti equation
# dae.setInput(integrator.getOutput(),'x')
# dae.evaluate()
# print max(fabs(dae.getOutput()))
# assert(max(fabs(dae.getOutput()))<1e-8)

# # From P, obtain a feedback matrix K
# K = mtimes([inv(R),B.T,P_])

# print 'feedback matrix= ', K

# # Inspect closed-loop eigenvalues
# [D,V] = linalg.eig(A-mtimes(B,K))
# print 'Open-loop eigenvalues: ', D

# # Check what happens if we integrate the Riccati equation forward in time
# dae = SX.fun('dae', daeIn(x = vec(P)),daeOut(ode=vec(-ric)))

# integrator = Integrator('integrator', 'cvodes', dae, {'reltol':1e-16, 'stop_at_end':False, 'tf':0.4})
# x0_pert = vec(P_)
# x0_pert[0] += 1e-9 # Put a tiny perturbation
# xe = x0_pert

# for i in range(1,10):
#   x0 = xe
#   xe = integrator({'x0':x0})['xf']
#   e = max(fabs(xe-x0))
#   print 'Forward riccati simulation %d; error: %.2e' % (i, e)

# # We notice divergence. Why?
# stabric = SX.fun('stabric', [P],[jacobian(-ric,P)])
# stabric.setInput(P_)
# stabric.evaluate()

# S = stabric.getOutput()

# [D,V] = linalg.eig(S)

# print 'Forward riccati eigenvalues = ', D


# # Simulation of the closed-loop system:
# #  continuous control action, various continuous references
# # ---------------------------------------------------------
# #

# x0 = DM([1,0,0])

# y  = SX.sym('y',ns)

# C = DM([[1,0,0],[0,1,0]])
# D = DM([[0,0],[0,0]])

# temp = inv(blockcat([[A,B],[C,D]]))

# F = temp[:ns,-ny:]
# Nm = temp[ns:,-ny:]

# t = SX.sym('t')

# figure(6)

# for k,yref in enumerate([ vertcat([-1,sqrt(t)]) , vertcat([-1,-0.5]), vertcat([-1,sin(t)])]):
#   u = -mtimes(K,y) + mtimes(mtimes(K,F)+Nm,yref)
#   rhs = mtimes(A,y)+mtimes(B,u)
#   cdae = SX.fun('cdae', controldaeIn(t=t, x=y),[rhs])

#   # Output function
#   out = SX.fun('out', controldaeIn(t=t, x=y),[y,mtimes(C,y),u,yref])

#   opts = {}
#   opts['integrator'] = 'cvodes'
#   opts['integrator_options'] = {'fsens_err_con': True,'reltol':1e-12}
#   opts['nf'] = 200
#   sim = ControlSimulator('sim', cdae, out, tn, opts)
#   sim.setInput(x0,'x0')
#   #sim.setInput(yref_,'u')
#   sim.evaluate()

#   tf = sim.minorT()

#   subplot(3,3,1+k*3)
#   plot(tf,sim.getOutput(0).T)
#   subplot(3,3,2+k*3)
#   title('ref ' + str(yref))
#   for i,c in enumerate(['b','g']):
#     plot(tf,sim.getOutput(1)[i,:].T,c,linewidth=2)
#     plot(tf,sim.getOutput(3)[i,:].T,c+'-')
#   subplot(3,3,3+k*3)
#   plot(tf,sim.getOutput(2).T)


# # Simulation of the closed-loop system:
# #  continuous control action, continuous feedforward reference
# # -----------------------------------------------------------
# #  To obtain a continous tracking reference,
# #  we augment statespace to construct it on the fly

# x0 = vertcat([1,0,0])

# # Now simulate with open-loop controls
# states = struct_symSX([
#            entry('y',shape=ns), # The regular states of the LTI system
#            entry('yref',shape=ns), # States that constitute a tracking reference for the LTI system
#            entry('eAt',shape=(ns,ns)) # The matrix exponential exp(A*(t1-t))
#          ])

# y     = states['y']
# eAt   = states['eAt']

# # Initial conditions
# states_ = states(0)
# states_['y']    = 2*x0
# states_['yref'] = x0
# states_['eAt']  = eAt_


# param = struct_symSX([entry('K',shape=(nu,ns))])

# param_ = param(0)

# uref = mtimes([B.T,eAt.T,inv(Wt_),xref_e-mtimes(eAt_,x0)])
# u    = uref - mtimes(param['K'],y-states['yref'])

# rhs = struct_SX(states)
# rhs['y']      =  mtimes(A,y)+mtimes(B,u)
# rhs['yref']   =  mtimes(A,states['yref'])+mtimes(B,uref)
# rhs['eAt']    = -mtimes(A,eAt)

# cdae = SX.fun('cdae', controldaeIn(x=states, p=param),[rhs])

# # Output function
# out = SX.fun('out', controldaeIn(x=states, p=param),[states,u,uref,states['yref']])

# opts = {}
# opts['integrator'] = 'cvodes'
# opts['integrator_options'] = {'fsens_err_con': True,'reltol':1e-8}
# opts['nf'] = 20
# sim = ControlSimulator('sim', cdae, out, tn, opts)

# # Not supported in current revision, cf. #929
# # jacsim = sim.jacobian_old(CONTROLSIMULATOR_X0,0)

# figure(7)

# for k,(caption,K_) in enumerate([('K: zero',DM.zeros((nu,ns))),('K: LQR',K)]):
#   param_['K'] = K_

#   sim.setInput(states_,'x0')
#   sim.setInput(param_,'p')
#   sim.evaluate()
#   sim.getOutput()

#   tf = sim.minorT()

#   subplot(2,2,2*k+1)
#   title('states (%s)' % caption)
#   for i,c in enumerate(['b','g','r']):
#     plot(tf,sim.getOutput()[states.i['yref',i],:].T,c+'--')
#     plot(tf,sim.getOutput()[states.i['y',i],:].T,c,linewidth=2)
#   subplot(2,2,2*k+2)
#   for i,c in enumerate(['b','g']):
#     plot(tf,sim.getOutput(1)[i,:].T,c,linewidth=2)
#     plot(tf,sim.getOutput(2)[i,:].T,c+'--')
#   title('controls (%s)' % caption)

#   # Not supported in current revision, cf. #929
#   # # Calculate monodromy matrix
#   # jacsim.setInput(states_,'x0')
#   # jacsim.setInput(param_,'p')
#   # jacsim.evaluate()
#   # M = jacsim.getOutput()[-states.size:,:][list(states.i['y']),list(states.i['y'])]

#   # # Inspect the eigenvalues of M
#   # [D,V] = linalg.eig(M)
#   # print 'Spectral radius of monodromy (%s): ' % caption

#   # print max(abs(D))

# print 'Spectral radius of exp((A-BK)*te): '

# [D,V] = linalg.eig(scipy.linalg.expm(numpy.array((A-mtimes(B,K))*te)))
# print max(abs(D))

# # Simulation of the controller:
# # discrete reference, continuous control action
# # -----------------------------------------------------------

# # Get discrete reference from previous simulation
# mi = sim.getMajorIndex()
# controls_ = sim.getOutput(2)[:,mi[:-1]]
# yref_     = sim.getOutput(3)[:,mi[:-1]]

# u_ = vertcat([controls_,yref_])

# x0 = DM([1,0,0])

# controls = struct_symSX([
#              entry('uref',shape=nu),
#              entry('yref',shape=ns)
#            ])

# yref  = SX.sym('yref',ns)
# y     = SX.sym('y',ns)
# dy    = SX.sym('dy',ns)
# u     = controls['uref']-mtimes(param['K'],y-controls['yref'])
# rhs   = mtimes(A,y)+mtimes(B,u)

# cdae = SX.fun('cdae', controldaeIn(x=y, u=controls, p=param),[rhs])

# # Output function
# out = SX.fun('out', controldaeIn(x=y, u=controls, p=param),[y,u,controls['uref'],controls['yref']])

# opts = {}
# opts['integrator'] = 'cvodes'
# opts['integrator_options'] = {'fsens_err_con': True,'reltol':1e-8}
# opts['nf'] = 20
# sim = ControlSimulator('sim', cdae, out, tn, opts)
# sim.setInput(2*x0,'x0')
# sim.setInput(param_,'p')
# sim.setInput(u_,'u')
# sim.evaluate()

# tf = sim.minorT()

# figure(8)
# subplot(2,1,1)
# title('states (%s)' % caption)
# for i,c in enumerate(['b','g','r']):
#   plot(tf,sim.getOutput(3)[i,:].T,c+'--')
#   plot(tf,sim.getOutput()[i,:].T,c,linewidth=2)
# subplot(2,1,2)
# for i,c in enumerate(['b','g']):
#   plot(tf,sim.getOutput(1)[i,:].T,c,linewidth=2)
#   plot(tf,sim.getOutput(2)[i,:].T,c+'--')
# title('controls (%s)' % caption)

# # Not supported in current revision, cf. #929
# # jacsim = sim.jacobian_old(CONTROLSIMULATOR_X0,0)

# # # Calculate monodromy matrix
# # jacsim.setInput(x0,'x0')
# # jacsim.setInput(param_,'p')
# # jacsim.setInput(u_,'u')
# # jacsim.evaluate()
# # M = jacsim.getOutput()[-ns:,:]

# # # Inspect the eigenvalues of M
# # [D,V] = linalg.eig(M)
# # print 'Spectral radius of monodromy, discrete reference, continous control'

# # print max(abs(D))

# # Simulation of the controller:
# # discrete reference, discrete control action
# # -----------------------------------------------------------

# y0     = SX.sym('y0',ns)

# u     = controls['uref']-mtimes(param['K'],y0-controls['yref'])
# rhs   = mtimes(A,y)+mtimes(B,u)

# cdae = SX.fun('cdae', controldaeIn(x=y, x_major=y0, u=controls, p=param),[rhs])

# # Output function
# out = SX.fun('out', controldaeIn(x=y, x_major=y0, u=controls, p=param),[y,u,controls['uref'],controls['yref']])

# opts = {}
# opts['integrator'] = 'cvodes'
# opts['integrator_options'] = {'fsens_err_con': True,'reltol':1e-8}
# opts['nf'] = 20

# sim = ControlSimulator('sim', cdae, out, tn, opts)
# sim.setInput(2*x0,'x0')
# sim.setInput(param_,'p')
# sim.setInput(u_,'u')
# sim.evaluate()

# tf = sim.minorT()

# figure(9)
# subplot(2,1,1)
# title('states (%s)' % caption)
# for i,c in enumerate(['b','g','r']):
#   plot(tf,sim.getOutput(3)[i,:].T,c+'--')
#   plot(tf,sim.getOutput()[i,:].T,c,linewidth=2)
# subplot(2,1,2)
# for i,c in enumerate(['b','g']):
#   plot(tf,sim.getOutput(1)[i,:].T,c,linewidth=2)
#   plot(tf,sim.getOutput(2)[i,:].T,c+'--')
# title('controls (%s)' % caption)

# # Not supported in current revision, cf. #929
# # jacsim = sim.jacobian_old(CONTROLSIMULATOR_X0,0)

# # # Calculate monodromy matrix
# # jacsim.setInput(x0,'x0')
# # jacsim.setInput(param_,'p')
# # jacsim.setInput(u_,'u')
# # jacsim.evaluate()
# # M = jacsim.getOutput()[-ns:,:]

# # # Inspect the eigenvalues of M
# # [D,V] = linalg.eig(M)
# # print 'Spectral radius of monodromy, discrete reference, discrete control'

# # print max(abs(D))

# show()