python-control · bnavigator · Jul 30, 2022 · Jun 18, 2022 · Jul 11, 2022 · Jul 11, 2022
diff --git a/control/__init__.py b/control/__init__.py
@@ -72,6 +72,7 @@
 from .config import *
 from .sisotool import *
 from .iosys import *
+from .passivity import *
 
 # Exceptions
 from .exception import *

diff --git a/control/passivity.py b/control/passivity.py
@@ -1,42 +1,72 @@
-'''
-Author: Mark Yeatman  
-Date: May 15, 2022
-'''
+"""
+Functions for passive control.
+
+Author: Mark Yeatman
+Date: July 17, 2022
+"""
 
 import numpy as np
-from control import statesp as ss
+from control import statesp
+from control.exception import ControlArgument, ControlDimension
 
 try:
     import cvxopt as cvx
-except ImportError as e:
+except ImportError:
     cvx = None
 
+eps = np.nextafter(0, 1)
+
+__all__ = ["get_output_fb_index", "get_input_ff_index",  "ispassive",
+           "solve_passivity_LMI"]
+
+
+def solve_passivity_LMI(sys, rho=None, nu=None):
+    """Compute passivity indices and/or solves feasiblity via a LMI.
 
-def ispassive(sys):
-    '''
-    Indicates if a linear time invariant (LTI) system is passive
+    Constructs a linear matrix inequality (LMI) such that if a solution exists
+    and the last element of the solution is positive, the system `sys` is
+    passive. Inputs of None for `rho` or `nu` indicate that the function should
+    solve for that index (they are mutually exclusive, they can't both be
+    None, otherwise you're trying to solve a nonconvex bilinear matrix
+    inequality.) The last element of `solution` is either the output or input
+    passivity index, for `rho` = None and  `nu` = None respectively.
 
-    Constructs a linear matrix inequality and a feasibility optimization
-    such that if a solution exists, the system is passive.
+    The sources for the algorithm are:
 
-    The source for the algorithm is: 
-    McCourt, Michael J., and Panos J. Antsaklis. 
-        "Demonstrating passivity and dissipativity using computational methods." ISIS 8 (2013).
+    McCourt, Michael J., and Panos J. Antsaklis
+        "Demonstrating passivity and dissipativity using computational
+        methods."
+
+    Nicholas Kottenstette and Panos J. Antsaklis
+        "Relationships Between Positive Real, Passive Dissipative, & Positive
+        Systems", equation 36.
 
     Parameters
     ----------
-    sys: A continuous LTI system
-        System to be checked.
+    sys : LTI
+        System to be checked
+    rho : float or None
+        Output feedback passivity index
+    nu : float or None
+        Input feedforward passivity index
 
     Returns
     -------
-    bool: 
-        The input system passive.
-    '''
+    solution : ndarray
+        The LMI solution
+    """
     if cvx is None:
         raise ModuleNotFoundError("cvxopt required for passivity module")
 
-    sys = ss._convert_to_statespace(sys)
+    if sys.ninputs != sys.noutputs:
+        raise ControlDimension(
+            "The number of system inputs must be the same as the number of "
+            "system outputs.")
+
+    if rho is None and nu is None:
+        raise ControlArgument("rho or nu must be given a numerical value.")
+
+    sys = statesp._convert_to_statespace(sys)
 
     A = sys.A
     B = sys.B
@@ -45,43 +75,218 @@ def ispassive(sys):
 
     # account for strictly proper systems
     [n, m] = D.shape
-    D = D + np.nextafter(0, 1)*np.eye(n, m)
+    D = D + eps * np.eye(n, m)
 
     [n, _] = A.shape
-    A = A - np.nextafter(0, 1)*np.eye(n)
-
-    def make_LMI_matrix(P):
-        V = np.vstack((
-            np.hstack((A.T @ P + P@A, P@B)),
-            np.hstack((B.T@P, np.zeros_like(D))))
-        )
-        return V
-
-    matrix_list = []
-    state_space_size = sys.nstates
-    for i in range(0, state_space_size):
-        for j in range(0, state_space_size):
-            if j <= i:
-                P = np.zeros_like(A)
-                P[i, j] = 1.0
-                P[j, i] = 1.0
-                matrix_list.append(make_LMI_matrix(P).flatten())
-
-    coefficents = np.vstack(matrix_list).T
-
-    constants = -np.vstack((
-        np.hstack((np.zeros_like(A),  - C.T)),
-        np.hstack((- C, -D - D.T)))
-    )
+    A = A - eps*np.eye(n)
+
+    def make_LMI_matrix(P, rho, nu, one):
+        q = sys.noutputs
+        Q = -rho*np.eye(q, q)
+        S = 1.0/2.0*(one+rho*nu)*np.eye(q)
+        R = -nu*np.eye(m)
+        if sys.isctime():
+            off_diag = P@B - (C.T@S + C.T@Q@D)
+            return np.vstack((
+                np.hstack((A.T @ P + P@A - C.T@Q@C,  off_diag)),
+                np.hstack((off_diag.T, -(D.T@Q@D + D.T@S + S.T@D + R)))
+            ))
+        else:
+            off_diag = A.T@P@B - (C.T@S + C.T@Q@D)
+            return np.vstack((
+                np.hstack((A.T @ P  @ A - P - C.T@Q@C,  off_diag)),
+                np.hstack((off_diag.T, B.T@P@B-(D.T@Q@D + D.T@S + S.T@D + R)))
+            ))
+
+    def make_P_basis_matrices(n, rho, nu):
+        """Make list of matrix constraints for passivity LMI.
+
+        Utility function to make basis matrices for a LMI from a
+        symmetric matrix P of size n by n representing a parametrized symbolic
+        matrix
+        """
+        matrix_list = []
+        for i in range(0, n):
+            for j in range(0, n):
+                if j <= i:
+                    P = np.zeros((n, n))
+                    P[i, j] = 1
+                    P[j, i] = 1
+                    matrix_list.append(make_LMI_matrix(P, 0, 0, 0).flatten())
+        zeros = eps*np.eye(n)
+        if rho is None:
+            matrix_list.append(make_LMI_matrix(zeros, 1, 0, 0).flatten())
+        elif nu is None:
+            matrix_list.append(make_LMI_matrix(zeros, 0, 1, 0).flatten())
+        return matrix_list
+
+
+    def P_pos_def_constraint(n):
+        """Make a list of matrix constraints for P >= 0.
+
+        Utility function to make basis matrices for a LMI that ensures
+        parametrized symbolic matrix of size n by n is positive definite
+        """
+        matrix_list = []
+        for i in range(0, n):
+            for j in range(0, n):
+                if j <= i:
+                    P = np.zeros((n, n))
+                    P[i, j] = -1
+                    P[j, i] = -1
+                    matrix_list.append(P.flatten())
+        if rho is None or nu is None:
+            matrix_list.append(np.zeros((n, n)).flatten())
+        return matrix_list
+
+    n = sys.nstates
+
+    # coefficents for passivity indices and feasibility matrix
+    sys_matrix_list = make_P_basis_matrices(n, rho, nu)
 
+    # get constants for numerical values of rho and nu
+    sys_constants = list()
+    if rho is not None and nu is not None:
+        sys_constants = -make_LMI_matrix(np.zeros_like(A), rho, nu, 1.0)
+    elif rho is not None:
+        sys_constants = -make_LMI_matrix(np.zeros_like(A), rho, eps, 1.0)
+    elif nu is not None:
+        sys_constants = -make_LMI_matrix(np.zeros_like(A), eps, nu, 1.0)
+
+    sys_coefficents = np.vstack(sys_matrix_list).T
+
+    # LMI to ensure P is positive definite
+    P_matrix_list = P_pos_def_constraint(n)
+    P_coefficents = np.vstack(P_matrix_list).T
+    P_constants = np.zeros((n, n))
+
+    # cost function
     number_of_opt_vars = int(
-        (state_space_size**2-state_space_size)/2 + state_space_size)
+        (n**2-n)/2 + n)
     c = cvx.matrix(0.0, (number_of_opt_vars, 1))
 
+    #we're maximizing a passivity index, include it in the cost function
+    if rho is None or nu is None:
+        c = cvx.matrix(np.append(np.array(c), -1.0))
+
+    Gs = [cvx.matrix(sys_coefficents)] + [cvx.matrix(P_coefficents)]
+    hs = [cvx.matrix(sys_constants)] + [cvx.matrix(P_constants)]
+
     # crunch feasibility solution
     cvx.solvers.options['show_progress'] = False
-    sol = cvx.solvers.sdp(c,
-                          Gs=[cvx.matrix(coefficents)],
-                          hs=[cvx.matrix(constants)])
+    sol = cvx.solvers.sdp(c, Gs=Gs, hs=hs)
+    return sol["x"]
+
+
+def get_output_fb_index(sys):
+    """Return the output feedback passivity (OFP) index for the system.
+
+    The OFP is the largest gain that can be placed in positive feedback
+    with a system such that the new interconnected system is passive.
+
+    Parameters
+    ----------
+    sys : LTI
+        System to be checked
+
+    Returns
+    -------
+    float
+        The OFP index
+    """
+    sol = solve_passivity_LMI(sys, nu=eps)
+    if sol is None:
+        raise RuntimeError("LMI passivity problem is infeasible")
+    else:
+        return sol[-1]
+
+
+def get_input_ff_index(sys):
+    """Return the input feedforward passivity (IFP) index for the system.
 
-    return (sol["x"] is not None)
+    The IFP is the largest gain that can be placed in negative parallel
+    interconnection with a system such that the new interconnected system is
+    passive.
+
+    Parameters
+    ----------
+    sys : LTI
+        System to be checked.
+
+    Returns
+    -------
+    float
+        The IFP index
+    """
+    sol = solve_passivity_LMI(sys, rho=eps)
+    if sol is None:
+        raise RuntimeError("LMI passivity problem is infeasible")
+    else:
+        return sol[-1]
+
+
+def get_relative_index(sys):
+    """Return the relative passivity index for the system.
+
+    (not implemented yet)
+    """
+    raise NotImplementedError("Relative passivity index not implemented")
+
+
+def get_combined_io_index(sys):
+    """Return the combined I/O passivity index for the system.
+
+    (not implemented yet)
+    """
+    raise NotImplementedError("Combined I/O passivity index not implemented")
+
+
+def get_directional_index(sys):
+    """Return the directional passivity index for the system.
+
+    (not implemented yet)
+    """
+    raise NotImplementedError("Directional passivity index not implemented")
+
+
+def ispassive(sys, ofp_index=0, ifp_index=0):
+    r"""Indicate if a linear time invariant (LTI) system is passive.
+
+    Checks if system is passive with the given output feedback (OFP) and input
+    feedforward (IFP) passivity indices.
+
+    Parameters
+    ----------
+    sys : LTI
+        System to be checked
+    ofp_index : float
+        Output feedback passivity index
+    ifp_index : float
+        Input feedforward passivity index
+
+    Returns
+    -------
+    bool
+        The system is passive.
+
+    Notes
+    -----
+    Querying if the system is passive in the sense of
+
+    .. math:: V(x) >= 0 \land \dot{V}(x) <= y^T u
+
+    is equivalent to the default case of `ofp_index` = 0 and `ifp_index` = 0.
+    Note that computing the `ofp_index` and `ifp_index` for a system, then
+    using both values simultaneously as inputs to this function is not
+    guaranteed to have an output of True (the system might not be passive with
+    both indices at the same time).
+
+    For more details, see [1].
+
+    References
+    ----------
+    .. [1] McCourt, Michael J., and Panos J. Antsaklis
+          "Demonstrating passivity and dissipativity using computational
+          methods."
+    """
+    return solve_passivity_LMI(sys, rho=ofp_index, nu=ifp_index) is not None