# stochsys.py - stochastic systems module

# RMM, 16 Mar 2022

#

# This module contains functions that are intended to be used for analysis

# and design of stochastic control systems, mainly involving Kalman

# filtering and its variants.

#

"""The :mod:`~control.stochsys` module contains functions for analyzing and

designing stochastic (control) systems, including white noise processes and

Kalman filtering.

"""

__license__ = "BSD"

__maintainer__ = "Richard Murray"

__email__ = "murray@cds.caltech.edu"

import numpy as np

import scipy as sp

from math import sqrt

from .iosys import InputOutputSystem, NonlinearIOSystem

from .lti import LTI

from .namedio import isctime, isdtime

from .mateqn import care, dare, _check_shape

from .statesp import StateSpace, _ssmatrix

from .exception import ControlArgument, ControlNotImplemented

__all__ = ['lqe', 'dlqe', 'create_estimator_iosystem', 'white_noise',

'correlation']

# contributed by Sawyer B. Fuller <minster@uw.edu>

def lqe(*args, **kwargs):

"""lqe(A, G, C, QN, RN, [, NN])

Linear quadratic estimator design (Kalman filter) for continuous-time

systems. Given the system

.. math::

x &= Ax + Bu + Gw \\\\

y &= Cx + Du + v

with unbiased process noise w and measurement noise v with covariances

.. math:: E{ww'} = QN, E{vv'} = RN, E{wv'} = NN

The lqe() function computes the observer gain matrix L such that the

stationary (non-time-varying) Kalman filter

.. math:: x_e = A x_e + B u + L(y - C x_e - D u)

produces a state estimate x_e that minimizes the expected squared error

using the sensor measurements y. The noise cross-correlation `NN` is

set to zero when omitted.

The function can be called with either 3, 4, 5, or 6 arguments:

* ``L, P, E = lqe(sys, QN, RN)``

* ``L, P, E = lqe(sys, QN, RN, NN)``

* ``L, P, E = lqe(A, G, C, QN, RN)``

* ``L, P, E = lqe(A, G, C, QN, RN, NN)``

where `sys` is an `LTI` object, and `A`, `G`, `C`, `QN`, `RN`, and `NN`

are 2D arrays or matrices of appropriate dimension.

Parameters

----------

A, G, C : 2D array_like

Dynamics, process noise (disturbance), and output matrices

sys : LTI (StateSpace or TransferFunction)

Linear I/O system, with the process noise input taken as the system

input.

QN, RN : 2D array_like

Process and sensor noise covariance matrices

NN : 2D array, optional

Cross covariance matrix. Not currently implemented.

method : str, optional

Set the method used for computing the result. Current methods are

'slycot' and 'scipy'. If set to None (default), try 'slycot' first

and then 'scipy'.

Returns

-------

L : 2D array (or matrix)

Kalman estimator gain

P : 2D array (or matrix)

Solution to Riccati equation

.. math::

A P + P A^T - (P C^T + G N) R^{-1} (C P + N^T G^T) + G Q G^T = 0

E : 1D array

Eigenvalues of estimator poles eig(A - L C)

Notes

-----

1. If the first argument is an LTI object, then this object will be used

to define the dynamics, noise and output matrices. Furthermore, if

the LTI object corresponds to a discrete time system, the ``dlqe()``

function will be called.

2. The return type for 2D arrays depends on the default class set for

state space operations. See :func:`~control.use_numpy_matrix`.

Examples

--------

>>> L, P, E = lqe(A, G, C, QN, RN)

>>> L, P, E = lqe(A, G, C, Q, RN, NN)

See Also

--------

lqr, dlqe, dlqr

"""

# TODO: incorporate cross-covariance NN, something like this,

# which doesn't work for some reason

# if NN is None:

# NN = np.zeros(QN.size(0),RN.size(1))

# NG = G @ NN

#

# Process the arguments and figure out what inputs we received

#

# If we were passed a discrete time system as the first arg, use dlqe()

if isinstance(args[0], LTI) and isdtime(args[0], strict=True):

# Call dlqe

return dlqe(*args, **kwargs)

# Get the method to use (if specified as a keyword)

method = kwargs.pop('method', None)

if kwargs:

raise TypeError("unrecognized keyword(s): ", str(kwargs))

# Get the system description

if (len(args) < 3):

raise ControlArgument("not enough input arguments")

# If we were passed a state space system, use that to get system matrices

if isinstance(args[0], StateSpace):

A = np.array(args[0].A, ndmin=2, dtype=float)

G = np.array(args[0].B, ndmin=2, dtype=float)

C = np.array(args[0].C, ndmin=2, dtype=float)

index = 1

elif isinstance(args[0], LTI):

# Don't allow other types of LTI systems

raise ControlArgument("LTI system must be in state space form")

else:

# Arguments should be A and B matrices

A = np.array(args[0], ndmin=2, dtype=float)

G = np.array(args[1], ndmin=2, dtype=float)

C = np.array(args[2], ndmin=2, dtype=float)

index = 3

# Get the weighting matrices (converting to matrices, if needed)

QN = np.array(args[index], ndmin=2, dtype=float)

RN = np.array(args[index+1], ndmin=2, dtype=float)

# Get the cross-covariance matrix, if given

if (len(args) > index + 2):

NN = np.array(args[index+2], ndmin=2, dtype=float)

raise ControlNotImplemented("cross-covariance not implemented")

else:

# For future use (not currently used below)

NN = np.zeros((QN.shape[0], RN.shape[1]))

# Check dimensions of G (needed before calling care())

_check_shape("QN", QN, G.shape[1], G.shape[1])

# Compute the result (dimension and symmetry checking done in care())

P, E, LT = care(A.T, C.T, G @ QN @ G.T, RN, method=method,

B_s="C", Q_s="QN", R_s="RN", S_s="NN")

return _ssmatrix(LT.T), _ssmatrix(P), E

# contributed by Sawyer B. Fuller <minster@uw.edu>

def dlqe(*args, **kwargs):

"""dlqe(A, G, C, QN, RN, [, N])

Linear quadratic estimator design (Kalman filter) for discrete-time

systems. Given the system

.. math::

x[n+1] &= Ax[n] + Bu[n] + Gw[n] \\\\

y[n] &= Cx[n] + Du[n] + v[n]

with unbiased process noise w and measurement noise v with covariances

.. math:: E{ww'} = QN, E{vv'} = RN, E{wv'} = NN

The dlqe() function computes the observer gain matrix L such that the

stationary (non-time-varying) Kalman filter

.. math:: x_e[n+1] = A x_e[n] + B u[n] + L(y[n] - C x_e[n] - D u[n])

produces a state estimate x_e[n] that minimizes the expected squared error

using the sensor measurements y. The noise cross-correlation `NN` is

set to zero when omitted.

Parameters

----------

A, G : 2D array_like

Dynamics and noise input matrices

QN, RN : 2D array_like

Process and sensor noise covariance matrices

NN : 2D array, optional

Cross covariance matrix (not yet supported)

method : str, optional

Set the method used for computing the result. Current methods are

'slycot' and 'scipy'. If set to None (default), try 'slycot' first

and then 'scipy'.

Returns

-------

L : 2D array (or matrix)

Kalman estimator gain

P : 2D array (or matrix)

Solution to Riccati equation

.. math::

A P + P A^T - (P C^T + G N) R^{-1} (C P + N^T G^T) + G Q G^T = 0

E : 1D array

Eigenvalues of estimator poles eig(A - L C)

Notes

-----

The return type for 2D arrays depends on the default class set for

state space operations. See :func:`~control.use_numpy_matrix`.

Examples

--------

>>> L, P, E = dlqe(A, G, C, QN, RN)

>>> L, P, E = dlqe(A, G, C, QN, RN, NN)

See Also

--------

dlqr, lqe, lqr

"""

#

# Process the arguments and figure out what inputs we received

#

# Get the method to use (if specified as a keyword)

method = kwargs.pop('method', None)

if kwargs:

raise TypeError("unrecognized keyword(s): ", str(kwargs))

# Get the system description

if (len(args) < 3):

raise ControlArgument("not enough input arguments")

# If we were passed a continus time system as the first arg, raise error

if isinstance(args[0], LTI) and isctime(args[0], strict=True):

raise ControlArgument("dlqr() called with a continuous time system")

# If we were passed a state space system, use that to get system matrices

if isinstance(args[0], StateSpace):

A = np.array(args[0].A, ndmin=2, dtype=float)

G = np.array(args[0].B, ndmin=2, dtype=float)

C = np.array(args[0].C, ndmin=2, dtype=float)

index = 1

elif isinstance(args[0], LTI):

# Don't allow other types of LTI systems

raise ControlArgument("LTI system must be in state space form")

else:

# Arguments should be A and B matrices

A = np.array(args[0], ndmin=2, dtype=float)

G = np.array(args[1], ndmin=2, dtype=float)

C = np.array(args[2], ndmin=2, dtype=float)

index = 3

# Get the weighting matrices (converting to matrices, if needed)

QN = np.array(args[index], ndmin=2, dtype=float)

RN = np.array(args[index+1], ndmin=2, dtype=float)

# TODO: incorporate cross-covariance NN, something like this,

# which doesn't work for some reason

# if NN is None:

# NN = np.zeros(QN.size(0),RN.size(1))

# NG = G @ NN

if len(args) > index + 2:

NN = np.array(args[index+2], ndmin=2, dtype=float)

raise ControlNotImplemented("cross-covariance not yet implememented")

# Check dimensions of G (needed before calling care())

_check_shape("QN", QN, G.shape[1], G.shape[1])

# Compute the result (dimension and symmetry checking done in dare())

P, E, LT = dare(A.T, C.T, G @ QN @ G.T, RN, method=method,

B_s="C", Q_s="QN", R_s="RN", S_s="NN")

return _ssmatrix(LT.T), _ssmatrix(P), E

# Function to create an estimator

def create_estimator_iosystem(

sys, QN, RN, P0=None, G=None, C=None,

state_labels='xhat[{i}]', output_labels='xhat[{i}]',

covariance_labels='P[{i},{j}]', sensor_labels=None):

"""Create an I/O system implementing a linqear quadratic estimator

This function creates an input/output system that implements a

state estimator of the form

xhat[k + 1] = A x[k] + B u[k] - L (C xhat[k] - y[k])

P[k + 1] = A P A^T + F QN F^T - A P C^T Reps^{-1} C P A

L = A P C^T Reps^{-1}

where Reps = RN + C P C^T. It can be called in the form

estim = ct.create_estimator_iosystem(sys, QN, RN)

where ``sys`` is the process dynamics and QN and RN are the covariance

of the disturbance noise and sensor noise. The function returns the

estimator ``estim`` as I/O system with a parameter ``correct`` that can

be used to turn off the correction term in the estimation (for forward

predictions).

Parameters

----------

sys : InputOutputSystem

The I/O system that represents the process dynamics. If no estimator

is given, the output of this system should represent the full state.

QN, RN : ndarray

Process and sensor noise covariance matrices.

P0 : ndarray, optional

Initial covariance matrix. If not specified, defaults to the steady

state covariance.

G : ndarray, optional

Disturbance matrix describing how the disturbances enters the

dynamics. Defaults to sys.B.

C : ndarray, optional

If the system has all full states output, define the measured values

to be used by the estimator. Otherwise, use the system output as the

measured values.

{state, covariance, sensor, output}_labels : str or list of str, optional

Set the name of the signals to use for the internal state, covariance,

sensors, and outputs (state estimate). If a single string is

specified, it should be a format string using the variable ``i`` as an

index (or ``i`` and ``j`` for covariance). Otherwise, a list of

strings matching the size of the respective signal should be used.

Default is ``'xhat[{i}]'`` for state and output labels, ``'y[{i}]'``

for output labels and ``'P[{i},{j}]'`` for covariance labels.

Returns

-------

estim : InputOutputSystem

Input/output system representing the estimator. This system takes the

system input and output and generates the estimated state.

Notes

-----

This function can be used with the ``create_statefbk_iosystem()`` function

to create a closed loop, output-feedback, state space controller:

K, _, _ = ct.lqr(sys, Q, R)

est = ct.create_estimator_iosystem(sys, QN, RN, P0)

ctrl, clsys = ct.create_statefbk_iosystem(sys, K, estimator=est)

The estimator can also be run on its own to process a noisy signal:

resp = ct.input_output_response(est, T, [Y, U], [X0, P0])

If desired, the ``correct`` parameter can be set to ``False`` to allow

prediction with no additional sensor information:

resp = ct.input_output_response(

est, T, 0, [X0, P0], param={'correct': False)

"""

# Make sure that we were passed an I/O system as an input

if not isinstance(sys, InputOutputSystem):

raise ControlArgument("Input system must be I/O system")

# Extract the matrices that we need for easy reference

A, B = sys.A, sys.B

# Set the disturbance and output matrices

G = sys.B if G is None else G

if C is not None:

# Make sure that we have the full system output

if not np.array_equal(sys.C, np.eye(sys.nstates)):

raise ValueError("System output must be full state")

# Make sure that the output matches the size of RN

if C.shape[0] != RN.shape[0]:

raise ValueError("System output is the wrong size for C")

else:

# Use the system outputs as the sensor outputs

C = sys.C

if sensor_labels is None:

sensor_labels = sys.output_labels

# Initialize the covariance matrix

if P0 is None:

# Initalize P0 to the steady state value

_, P0, _ = lqe(A, G, C, QN, RN)

# Figure out the labels to use

if isinstance(state_labels, str):

# Generate the list of labels using the argument as a format string

state_labels = [state_labels.format(i=i) for i in range(sys.nstates)]

if isinstance(covariance_labels, str):

# Generate the list of labels using the argument as a format string

covariance_labels = [

covariance_labels.format(i=i, j=j) \

for i in range(sys.nstates) for j in range(sys.nstates)]

if isinstance(output_labels, str):

# Generate the list of labels using the argument as a format string

output_labels = [output_labels.format(i=i) for i in range(sys.nstates)]

sensor_labels = 'y[{i}]' if sensor_labels is None else sensor_labels

if isinstance(sensor_labels, str):

# Generate the list of labels using the argument as a format string

sensor_labels = [sensor_labels.format(i=i) for i in range(C.shape[0])]

if isctime(sys):

raise NotImplementedError("continuous time not yet implemented")

else:

# Create an I/O system for the state feedback gains

# Note: reshape vectors into column vectors for legacy np.matrix

def _estim_update(t, x, u, params):

# See if we are estimating or predicting

correct = params.get('correct', True)

# Get the state of the estimator

xhat = x[0:sys.nstates].reshape(-1, 1)

P = x[sys.nstates:].reshape(sys.nstates, sys.nstates)

# Extract the inputs to the estimator

y = u[0:C.shape[0]].reshape(-1, 1)

u = u[C.shape[0]:].reshape(-1, 1)

# Compute the optimal again

Reps_inv = np.linalg.inv(RN + C @ P @ C.T)

L = A @ P @ C.T @ Reps_inv

# Update the state estimate

dxhat = A @ xhat + B @ u # prediction

if correct:

dxhat -= L @ (C @ xhat - y) # correction

# Update the covariance

dP = A @ P @ A.T + G @ QN @ G.T

if correct:

dP -= A @ P @ C.T @ Reps_inv @ C @ P @ A.T

# Return the update

return np.hstack([dxhat.reshape(-1), dP.reshape(-1)])

def _estim_output(t, x, u, params):

return x[0:sys.nstates]

# Define the estimator system

return NonlinearIOSystem(

_estim_update, _estim_output, states=state_labels + covariance_labels,

inputs=sensor_labels + sys.input_labels, outputs=output_labels,

dt=sys.dt)

def white_noise(T, Q, dt=0):

"""Generate a white noise signal with specified intensity.

This function generates a (multi-variable) white noise signal of

specified intensity as either a sampled continous time signal or a

discrete time signal. A white noise signal along a 1D array

of linearly spaced set of times T can be computing using

V = ct.white_noise(T, Q, dt)

where Q is a positive definite matrix providing the noise intensity.

In continuous time, the white noise signal is scaled such that the

integral of the covariance over a sample period is Q, thus approximating

a white noise signal. In discrete time, the white noise signal has

covariance Q at each point in time (without any scaling based on the

sample time).

"""

# Convert input arguments to arrays

T = np.atleast_1d(T)

Q = np.atleast_2d(Q)

# Check the shape of the input arguments

if len(T.shape) != 1:

raise ValueError("Time vector T must be 1D")

if len(Q.shape) != 2 or Q.shape[0] != Q.shape[1]:

raise ValueError("Covariance matrix Q must be square")

# Figure out the time increment

if dt != 0:

# Discrete time system => white noise is not scaled

dt = 1

else:

dt = T[1] - T[0]

# Make sure data points are equally spaced

if not np.allclose(np.diff(T), T[1] - T[0]):

raise ValueError("Time values must be equally spaced.")

# Generate independent white noise sources for each input

W = np.array([

np.random.normal(0, 1/sqrt(dt), T.size) for i in range(Q.shape[0])])

# Return a linear combination of the noise sources

return sp.linalg.sqrtm(Q) @ W

def correlation(T, X, Y=None, squeeze=True):

"""Compute the correlation of time signals.

For a time series X(t) (and optionally Y(t)), the correlation() function

computes the correlation matrix E(X'(t+tau) X(t)) or the cross-correlation

matrix E(X'(t+tau) Y(t)]:

tau, Rtau = correlation(T, X[, Y])

The signal X (and Y, if present) represent a continuous time signal

sampled at times T. The return value provides the correlation Rtau

between X(t+tau) and X(t) at a set of time offets tau.

Parameters

----------

T : 1D array_like

Sample times for the signal(s).

X : 1D or 2D array_like

Values of the signal at each time in T. The signal can either be

scalar or vector values.

Y : 1D or 2D array_like, optional

If present, the signal with which to compute the correlation.

Defaults to X.

squeeze : bool, optional

If True, squeeze Rtau to remove extra dimensions (useful if the

signals are scalars).

Returns

-------

"""

T = np.atleast_1d(T)

X = np.atleast_2d(X)

Y = np.atleast_2d(Y) if Y is not None else X

# Check the shape of the input arguments

if len(T.shape) != 1:

raise ValueError("Time vector T must be 1D")

if len(X.shape) != 2 or len(Y.shape) != 2:

raise ValueError("Signals X and Y must be 2D arrays")

if T.shape[0] != X.shape[1] or T.shape[0] != Y.shape[1]:

raise ValueError("Signals X and Y must have same length as T")

# Figure out the time increment

dt = T[1] - T[0]

# Make sure data points are equally spaced

if not np.allclose(np.diff(T), T[1] - T[0]):

raise ValueError("Time values must be equally spaced.")

# Compute the correlation matrix

R = np.array(

[[sp.signal.correlate(X[i], Y[j])

for i in range(X.shape[0])] for j in range(Y.shape[0])]

) * dt / (T[-1] - T[0])

# From scipy.signal.correlation_lags (for use with older versions)

# tau = sp.signal.correlation_lags(len(X[0]), len(Y[0])) * dt

tau = np.arange(-len(Y[0]) + 1, len(X[0])) * dt

return tau, R.squeeze() if squeeze else R