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Mark-YeatmanMark-Yeatman
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Passivity indices and support for discrete time systems. (#750)
This adds support for input and output passivity indices. This implementation ignores frequency bands, relative passivity index, combined I/O passivity, and directional passivity index. This also adds support for computing indices and "is passive or not" for discrete time systems. Authored-by: Mark <myeatman@oxefit.com>
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‎control/__init__.py

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from .config import *
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from .sisotool import *
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from .iosys import *
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from .passivity import *
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# Exceptions
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from .exception import *

‎control/passivity.py

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'''
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Author: Mark Yeatman
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Date: May 15, 2022
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'''
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"""
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Functions for passive control.
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Author: Mark Yeatman
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Date: July 17, 2022
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"""
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import numpy as np
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from control import statesp as ss
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from control import statesp
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from control.exception import ControlArgument, ControlDimension
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try:
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import cvxopt as cvx
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except ImportError as e:
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except ImportError:
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cvx = None
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eps = np.nextafter(0, 1)
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__all__ = ["get_output_fb_index", "get_input_ff_index", "ispassive",
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"solve_passivity_LMI"]
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def solve_passivity_LMI(sys, rho=None, nu=None):
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"""Compute passivity indices and/or solves feasiblity via a LMI.
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def ispassive(sys):
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'''
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Indicates if a linear time invariant (LTI) system is passive
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Constructs a linear matrix inequality (LMI) such that if a solution exists
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and the last element of the solution is positive, the system `sys` is
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passive. Inputs of None for `rho` or `nu` indicate that the function should
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solve for that index (they are mutually exclusive, they can't both be
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None, otherwise you're trying to solve a nonconvex bilinear matrix
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inequality.) The last element of `solution` is either the output or input
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passivity index, for `rho` = None and `nu` = None respectively.
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Constructs a linear matrix inequality and a feasibility optimization
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such that if a solution exists, the system is passive.
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The sources for the algorithm are:
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The source for the algorithm is:
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McCourt, Michael J., and Panos J. Antsaklis.
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"Demonstrating passivity and dissipativity using computational methods." ISIS 8 (2013).
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McCourt, Michael J., and Panos J. Antsaklis
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"Demonstrating passivity and dissipativity using computational
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methods."
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Nicholas Kottenstette and Panos J. Antsaklis
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"Relationships Between Positive Real, Passive Dissipative, & Positive
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Systems", equation 36.
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Parameters
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----------
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sys: A continuous LTI system
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System to be checked.
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sys : LTI
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System to be checked
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rho : float or None
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Output feedback passivity index
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nu : float or None
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Input feedforward passivity index
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Returns
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-------
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bool:
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The input system passive.
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'''
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solution : ndarray
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The LMI solution
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"""
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if cvx is None:
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raise ModuleNotFoundError("cvxopt required for passivity module")
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sys = ss._convert_to_statespace(sys)
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if sys.ninputs != sys.noutputs:
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raise ControlDimension(
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"The number of system inputs must be the same as the number of "
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"system outputs.")
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if rho is None and nu is None:
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raise ControlArgument("rho or nu must be given a numerical value.")
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sys = statesp._convert_to_statespace(sys)
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A = sys.A
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B = sys.B
@@ -45,43 +75,218 @@ def ispassive(sys):
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# account for strictly proper systems
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[n, m] = D.shape
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D = D + np.nextafter(0, 1)*np.eye(n, m)
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D = D + eps * np.eye(n, m)
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[n, _] = A.shape
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A = A - np.nextafter(0, 1)*np.eye(n)
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def make_LMI_matrix(P):
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V = np.vstack((
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np.hstack((A.T @ P + P@A, P@B)),
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np.hstack((B.T@P, np.zeros_like(D))))
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)
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return V
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matrix_list = []
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state_space_size = sys.nstates
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for i in range(0, state_space_size):
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for j in range(0, state_space_size):
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if j <= i:
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P = np.zeros_like(A)
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P[i, j] = 1.0
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P[j, i] = 1.0
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matrix_list.append(make_LMI_matrix(P).flatten())
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coefficents = np.vstack(matrix_list).T
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constants = -np.vstack((
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np.hstack((np.zeros_like(A), - C.T)),
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np.hstack((- C, -D - D.T)))
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)
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A = A - eps*np.eye(n)
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def make_LMI_matrix(P, rho, nu, one):
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q = sys.noutputs
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Q = -rho*np.eye(q, q)
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S = 1.0/2.0*(one+rho*nu)*np.eye(q)
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R = -nu*np.eye(m)
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if sys.isctime():
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off_diag = P@B - (C.T@S + C.T@Q@D)
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return np.vstack((
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np.hstack((A.T @ P + P@A - C.T@Q@C, off_diag)),
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np.hstack((off_diag.T, -(D.T@Q@D + D.T@S + S.T@D + R)))
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))
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else:
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off_diag = A.T@P@B - (C.T@S + C.T@Q@D)
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return np.vstack((
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np.hstack((A.T @ P @ A - P - C.T@Q@C, off_diag)),
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np.hstack((off_diag.T, B.T@P@B-(D.T@Q@D + D.T@S + S.T@D + R)))
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))
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def make_P_basis_matrices(n, rho, nu):
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"""Make list of matrix constraints for passivity LMI.
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Utility function to make basis matrices for a LMI from a
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symmetric matrix P of size n by n representing a parametrized symbolic
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matrix
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"""
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matrix_list = []
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for i in range(0, n):
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for j in range(0, n):
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if j <= i:
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P = np.zeros((n, n))
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P[i, j] = 1
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P[j, i] = 1
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matrix_list.append(make_LMI_matrix(P, 0, 0, 0).flatten())
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zeros = eps*np.eye(n)
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if rho is None:
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matrix_list.append(make_LMI_matrix(zeros, 1, 0, 0).flatten())
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elif nu is None:
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matrix_list.append(make_LMI_matrix(zeros, 0, 1, 0).flatten())
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return matrix_list
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def P_pos_def_constraint(n):
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"""Make a list of matrix constraints for P >= 0.
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Utility function to make basis matrices for a LMI that ensures
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parametrized symbolic matrix of size n by n is positive definite
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"""
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matrix_list = []
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for i in range(0, n):
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for j in range(0, n):
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if j <= i:
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P = np.zeros((n, n))
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P[i, j] = -1
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P[j, i] = -1
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matrix_list.append(P.flatten())
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if rho is None or nu is None:
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matrix_list.append(np.zeros((n, n)).flatten())
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return matrix_list
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n = sys.nstates
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# coefficents for passivity indices and feasibility matrix
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sys_matrix_list = make_P_basis_matrices(n, rho, nu)
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# get constants for numerical values of rho and nu
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sys_constants = list()
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if rho is not None and nu is not None:
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sys_constants = -make_LMI_matrix(np.zeros_like(A), rho, nu, 1.0)
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elif rho is not None:
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sys_constants = -make_LMI_matrix(np.zeros_like(A), rho, eps, 1.0)
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elif nu is not None:
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sys_constants = -make_LMI_matrix(np.zeros_like(A), eps, nu, 1.0)
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sys_coefficents = np.vstack(sys_matrix_list).T
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# LMI to ensure P is positive definite
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P_matrix_list = P_pos_def_constraint(n)
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P_coefficents = np.vstack(P_matrix_list).T
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P_constants = np.zeros((n, n))
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# cost function
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number_of_opt_vars = int(
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(state_space_size**2-state_space_size)/2 + state_space_size)
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(n**2-n)/2 + n)
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c = cvx.matrix(0.0, (number_of_opt_vars, 1))
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#we're maximizing a passivity index, include it in the cost function
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if rho is None or nu is None:
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c = cvx.matrix(np.append(np.array(c), -1.0))
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Gs = [cvx.matrix(sys_coefficents)] + [cvx.matrix(P_coefficents)]
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hs = [cvx.matrix(sys_constants)] + [cvx.matrix(P_constants)]
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# crunch feasibility solution
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cvx.solvers.options['show_progress'] = False
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sol = cvx.solvers.sdp(c,
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Gs=[cvx.matrix(coefficents)],
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hs=[cvx.matrix(constants)])
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sol = cvx.solvers.sdp(c, Gs=Gs, hs=hs)
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return sol["x"]
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def get_output_fb_index(sys):
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"""Return the output feedback passivity (OFP) index for the system.
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The OFP is the largest gain that can be placed in positive feedback
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with a system such that the new interconnected system is passive.
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Parameters
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----------
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sys : LTI
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System to be checked
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Returns
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-------
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float
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The OFP index
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"""
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sol = solve_passivity_LMI(sys, nu=eps)
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if sol is None:
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raise RuntimeError("LMI passivity problem is infeasible")
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else:
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return sol[-1]
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def get_input_ff_index(sys):
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"""Return the input feedforward passivity (IFP) index for the system.
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return (sol["x"] is not None)
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The IFP is the largest gain that can be placed in negative parallel
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interconnection with a system such that the new interconnected system is
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passive.
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Parameters
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----------
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sys : LTI
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System to be checked.
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Returns
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-------
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float
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The IFP index
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"""
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sol = solve_passivity_LMI(sys, rho=eps)
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if sol is None:
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raise RuntimeError("LMI passivity problem is infeasible")
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else:
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return sol[-1]
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def get_relative_index(sys):
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"""Return the relative passivity index for the system.
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(not implemented yet)
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"""
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raise NotImplementedError("Relative passivity index not implemented")
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def get_combined_io_index(sys):
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"""Return the combined I/O passivity index for the system.
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(not implemented yet)
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"""
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raise NotImplementedError("Combined I/O passivity index not implemented")
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def get_directional_index(sys):
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"""Return the directional passivity index for the system.
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(not implemented yet)
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"""
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raise NotImplementedError("Directional passivity index not implemented")
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def ispassive(sys, ofp_index=0, ifp_index=0):
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r"""Indicate if a linear time invariant (LTI) system is passive.
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Checks if system is passive with the given output feedback (OFP) and input
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feedforward (IFP) passivity indices.
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Parameters
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----------
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sys : LTI
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System to be checked
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ofp_index : float
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Output feedback passivity index
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ifp_index : float
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Input feedforward passivity index
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Returns
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-------
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bool
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The system is passive.
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Notes
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-----
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Querying if the system is passive in the sense of
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.. math:: V(x) >= 0 \land \dot{V}(x) <= y^T u
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is equivalent to the default case of `ofp_index` = 0 and `ifp_index` = 0.
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Note that computing the `ofp_index` and `ifp_index` for a system, then
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using both values simultaneously as inputs to this function is not
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guaranteed to have an output of True (the system might not be passive with
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both indices at the same time).
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For more details, see [1].
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References
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----------
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.. [1] McCourt, Michael J., and Panos J. Antsaklis
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"Demonstrating passivity and dissipativity using computational
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methods."
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"""
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return solve_passivity_LMI(sys, rho=ofp_index, nu=ifp_index) is not None

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