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# robust.py - tools for robust control
#
# Original authors: Steve Brunton, Kevin Chen, Lauren Padilla
# Creation date: 24 Dec 2010
"""
This file contains routines for obtaining reduced order models.
"""
import warnings
# External packages and modules
import numpy as np
from .exception import ControlSlycot
from .statesp import StateSpace
def h2syn(P, nmeas, ncon):
"""H_2 control synthesis for plant P.
Parameters
----------
P : StateSpace
Partitioned LTI plant (state-space system).
nmeas : int
Number of measurements (input to controller).
ncon : int
Number of control inputs (output from controller).
Returns
-------
K : StateSpace
Controller to stabilize `P`.
Raises
------
ImportError
if slycot routine sb10hd is not loaded
See Also
--------
StateSpace
Examples
--------
>>> # Unstable first order SISI system
>>> G = ct.tf([1], [1, -1], inputs=['u'], outputs=['y'])
>>> all(G.poles() < 0) # Is G stable?
False
>>> # Create partitioned system with trivial unity systems
>>> P11 = ct.tf([0], [1], inputs=['w'], outputs=['z'])
>>> P12 = ct.tf([1], [1], inputs=['u'], outputs=['z'])
>>> P21 = ct.tf([1], [1], inputs=['w'], outputs=['y'])
>>> P22 = G
>>> P = ct.interconnect([P11, P12, P21, P22],
... inplist=['w', 'u'], outlist=['z', 'y'])
>>> # Synthesize H2 optimal stabilizing controller
>>> K = ct.h2syn(P, nmeas=1, ncon=1)
>>> T = ct.feedback(G, K, sign=1)
>>> all(T.poles() < 0) # Is T stable?
True
"""
# Check for ss system object, need a utility for this?
# TODO: Check for continous or discrete, only continuous supported right now
try:
from slycot import sb10hd
except ImportError:
raise ControlSlycot("can't find slycot subroutine sb10hd")
n = np.size(P.A, 0)
m = np.size(P.B, 1)
np_ = np.size(P.C, 0)
out = sb10hd(n, m, np_, ncon, nmeas, P.A, P.B, P.C, P.D)
Ak = out[0]
Bk = out[1]
Ck = out[2]
Dk = out[3]
K = StateSpace(Ak, Bk, Ck, Dk)
return K
def hinfsyn(P, nmeas, ncon):
# TODO: document significance of rcond
"""H_{inf} control synthesis for plant P.
Parameters
----------
P : StateSpace
Partitioned LTI plant (state-space system).
nmeas : int
Number of measurements (input to controller).
ncon : int
Number of control inputs (output from controller).
Returns
-------
K : StateSpace
Controller to stabilize `P`.
CL : StateSpace
Closed loop system.
gam : float
Infinity norm of closed loop system.
rcond : list
4-vector, reciprocal condition estimates of:
1: control transformation matrix
2: measurement transformation matrix
3: X-Riccati equation
4: Y-Riccati equation
Raises
------
ImportError
if slycot routine sb10ad is not loaded
See Also
--------
StateSpace
Examples
--------
>>> # Unstable first order SISI system
>>> G = ct.tf([1], [1,-1], inputs=['u'], outputs=['y'])
>>> all(G.poles() < 0)
False
>>> # Create partitioned system with trivial unity systems
>>> P11 = ct.tf([0], [1], inputs=['w'], outputs=['z'])
>>> P12 = ct.tf([1], [1], inputs=['u'], outputs=['z'])
>>> P21 = ct.tf([1], [1], inputs=['w'], outputs=['y'])
>>> P22 = G
>>> P = ct.interconnect([P11, P12, P21, P22], inplist=['w', 'u'], outlist=['z', 'y'])
>>> # Synthesize Hinf optimal stabilizing controller
>>> K, CL, gam, rcond = ct.hinfsyn(P, nmeas=1, ncon=1)
>>> T = ct.feedback(G, K, sign=1)
>>> all(T.poles() < 0)
True
"""
# Check for ss system object, need a utility for this?
# TODO: Check for continous or discrete, only continuous supported right now
try:
from slycot import sb10ad
except ImportError:
raise ControlSlycot("can't find slycot subroutine sb10ad")
n = np.size(P.A, 0)
m = np.size(P.B, 1)
np_ = np.size(P.C, 0)
gamma = 1.e100
out = sb10ad(n, m, np_, ncon, nmeas, gamma, P.A, P.B, P.C, P.D)
gam = out[0]
Ak = out[1]
Bk = out[2]
Ck = out[3]
Dk = out[4]
Ac = out[5]
Bc = out[6]
Cc = out[7]
Dc = out[8]
rcond = out[9]
K = StateSpace(Ak, Bk, Ck, Dk)
CL = StateSpace(Ac, Bc, Cc, Dc)
return K, CL, gam, rcond
def _size_as_needed(w, wname, n):
"""Convert LTI object to appropriately sized StateSpace object.
Intended for use in .robust only
Parameters
----------
w: None, 1x1 LTI object, or mxn LTI object
wname: name of w, for error message
n: number of inputs to w
Returns
-------
w_: processed weighting function, a StateSpace object:
- if w is None, empty StateSpace object
- if w is scalar, w_ will be w * eye(n)
- otherwise, w as StateSpace object
Raises
------
ValueError
- if w is not None or scalar, and doesn't have n inputs
See Also
--------
augw
"""
from . import append, ss
if w is not None:
if not isinstance(w, StateSpace):
w = ss(w)
if 1 == w.ninputs and 1 == w.noutputs:
w = append(*(w,) * n)
else:
if w.ninputs != n:
msg = ("{}: weighting function has {} inputs, expected {}".
format(wname, w.ninputs, n))
raise ValueError(msg)
else:
w = ss([], [], [], [])
return w
def augw(g, w1=None, w2=None, w3=None):
"""Augment plant for mixed sensitivity problem.
If a weighting is None, no augmentation is done for it. At least
one weighting must not be None.
If a weighting w is scalar, it will be replaced by I*w, where I is
ny-by-ny for w1 and w3, and nu-by-nu for w2.
Parameters
----------
g : LTI object, ny-by-nu
Plant.
w1 : None, scalar, or k1-by-ny LTI object
Weighting on S.
w2 : None, scalar, or k2-by-nu LTI object
Weighting on KS.
w3 : None, scalar, or k3-by-ny LTI object
Weighting on T.
Returns
-------
p : StateSpace
Plant augmented with weightings, suitable for submission to hinfsyn or
h2syn.
Raises
------
ValueError
If all weightings are None
See Also
--------
h2syn, hinfsyn, mixsyn
"""
from . import append, connect, ss
if w1 is None and w2 is None and w3 is None:
raise ValueError("At least one weighting must not be None")
ny = g.noutputs
nu = g.ninputs
w1, w2, w3 = [_size_as_needed(w, wname, n)
for w, wname, n in zip((w1, w2, w3),
('w1', 'w2', 'w3'),
(ny, nu, ny))]
if not isinstance(g, StateSpace):
g = ss(g)
# w u
# z1 [ w1 | -w1*g ]
# z2 [ 0 | w2 ]
# z3 [ 0 | w3*g ]
# [------+--------- ]
# v [ I | -g ]
# error summer: inputs are -y and r=w
Ie = ss([], [], [], np.eye(ny))
# control: needed to "distribute" control input
Iu = ss([], [], [], np.eye(nu))
sysall = append(w1, w2, w3, Ie, g, Iu)
niw1 = w1.ninputs
niw2 = w2.ninputs
niw3 = w3.ninputs
now1 = w1.noutputs
now2 = w2.noutputs
now3 = w3.noutputs
q = np.zeros((niw1 + niw2 + niw3 + ny + nu, 2))
q[:, 0] = np.arange(1, q.shape[0] + 1)
# Ie -> w1
q[:niw1, 1] = np.arange(1 + now1 + now2 + now3,
1 + now1 + now2 + now3 + niw1)
# Iu -> w2
q[niw1:niw1 + niw2, 1] = np.arange(1 + now1 + now2 + now3 + 2 * ny,
1 + now1 + now2 + now3 + 2 * ny + niw2)
# y -> w3
q[niw1 + niw2:niw1 + niw2 + niw3, 1] = np.arange(1 + now1 + now2 + now3 + ny,
1 + now1 + now2 + now3 + ny + niw3)
# -y -> Iy; note the leading -
q[niw1 + niw2 + niw3:niw1 + niw2 + niw3 + ny, 1] = -np.arange(1 + now1 + now2 + now3 + ny,
1 + now1 + now2 + now3 + 2 * ny)
# Iu -> G
q[niw1 + niw2 + niw3 + ny:niw1 + niw2 + niw3 + ny + nu, 1] = np.arange(
1 + now1 + now2 + now3 + 2 * ny,
1 + now1 + now2 + now3 + 2 * ny + nu)
# input indices: to Ie and Iu
ii = np.hstack((np.arange(1 + now1 + now2 + now3,
1 + now1 + now2 + now3 + ny),
np.arange(1 + now1 + now2 + now3 + ny + nu,
1 + now1 + now2 + now3 + ny + nu + nu)))
# output indices
oi = np.arange(1, 1 + now1 + now2 + now3 + ny)
# Filter out known warning due to use of connect
with warnings.catch_warnings():
warnings.filterwarnings(
'ignore', message="`connect`", category=DeprecationWarning)
p = connect(sysall, q, ii, oi)
return p
def mixsyn(g, w1=None, w2=None, w3=None):
"""Mixed-sensitivity H-infinity synthesis.
mixsyn(g,w1,w2,w3) -> k,cl,info
Parameters
----------
g : LTI
The plant for which controller must be synthesized.
w1 : None, or scalar or k1-by-ny LTI
Weighting on S = (1+G*K)**-1.
w2 : None, or scalar or k2-by-nu LTI
Weighting on K*S.
w3 : None, or scalar or k3-by-ny LTI
Weighting on T = G*K*(1+G*K)**-1.
Returns
-------
k : StateSpace
Synthesized controller.
cl : StateSpace
Closed system mapping evaluation inputs to evaluation outputs.
Let p be the augmented plant, with::
[z] = [p11 p12] [w]
[y] [p21 g] [u]
then cl is the system from w->z with `u = -k*y`.
info : tuple
Two-tuple `(gamma, rcond)` containing additional information:
* gamma (scalar): H-infinity norm of cl.
* rcond (array): Estimates of reciprocal condition numbers computed
during synthesis. See hinfsyn for details.
See Also
--------
hinfsyn, augw
Notes
-----
If a weighting `w` is scalar, it will be replaced by `I*w`, where `I` is
ny-by-ny for `w1` and `w3`, and nu-by-nu for `w2`.
"""
nmeas = g.noutputs
ncon = g.ninputs
p = augw(g, w1, w2, w3)
k, cl, gamma, rcond = hinfsyn(p, nmeas, ncon)
info = gamma, rcond
return k, cl, info