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System norms 2#971
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a1f27db New function for LTI system norm computation
e8cca34 * Updated documentation of function norm
f60c83a Added:
99e49d9 Added in sysnorm.py:
b39710f * In sysnorm: Warnings now of type UserWarning
bad229a * sysnorm: Changed default tolerance in norm to 1e-6.
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3 changes: 3 additions & 0 deletions.gitignore
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1 change: 1 addition & 0 deletionscontrol/__init__.py
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294 changes: 294 additions & 0 deletionscontrol/sysnorm.py
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,294 @@ | ||
| # -*- coding: utf-8 -*- | ||
| """sysnorm.py | ||
| Functions for computing system norms. | ||
| Routine in this module: | ||
| norm | ||
| Created on Thu Dec 21 08:06:12 2023 | ||
| Author: Henrik Sandberg | ||
| """ | ||
| import numpy as np | ||
| import scipy as sp | ||
| import numpy.linalg as la | ||
| import warnings | ||
| import control as ct | ||
| __all__ = ['norm'] | ||
| #------------------------------------------------------------------------------ | ||
| def _h2norm_slycot(sys, print_warning=True): | ||
| """H2 norm of a linear system. For internal use. Requires Slycot. | ||
| See also | ||
| -------- | ||
| ``slycot.ab13bd`` : the Slycot routine that does the calculation | ||
| https://github.com/python-control/Slycot/issues/199 : Post on issue with ``ab13bf`` | ||
| """ | ||
| try: | ||
| from slycot import ab13bd | ||
| except ImportError: | ||
| ct.ControlSlycot("Can't find slycot module ``ab13bd``!") | ||
| try: | ||
| from slycot.exceptions import SlycotArithmeticError | ||
| except ImportError: | ||
| raise ct.ControlSlycot("Can't find slycot class ``SlycotArithmeticError``!") | ||
| A, B, C, D = ct.ssdata(ct.ss(sys)) | ||
| n = A.shape[0] | ||
| m = B.shape[1] | ||
| p = C.shape[0] | ||
| dico = 'C' if sys.isctime() else 'D' # Continuous or discrete time | ||
| jobn = 'H' # H2 (and not L2 norm) | ||
| if n == 0: | ||
| # ab13bd does not accept empty A, B, C | ||
| if dico == 'C': | ||
| if any(D.flat != 0): | ||
| if print_warning: | ||
| warnings.warn("System has a direct feedthrough term!", UserWarning) | ||
| return float("inf") | ||
| else: | ||
| return 0.0 | ||
| elif dico == 'D': | ||
| return np.sqrt(D@D.T) | ||
| try: | ||
| norm = ab13bd(dico, jobn, n, m, p, A, B, C, D) | ||
| except SlycotArithmeticError as e: | ||
| if e.info == 3: | ||
| if print_warning: | ||
| warnings.warn("System has pole(s) on the stability boundary!", UserWarning) | ||
| return float("inf") | ||
| elif e.info == 5: | ||
| if print_warning: | ||
| warnings.warn("System has a direct feedthrough term!", UserWarning) | ||
| return float("inf") | ||
| elif e.info == 6: | ||
| if print_warning: | ||
| warnings.warn("System is unstable!", UserWarning) | ||
| return float("inf") | ||
| else: | ||
| raise e | ||
| return norm | ||
| #------------------------------------------------------------------------------ | ||
| def norm(system, p=2, tol=1e-6, print_warning=True, method=None): | ||
| """Computes norm of system. | ||
| Parameters | ||
| ---------- | ||
| system : LTI (:class:`StateSpace` or :class:`TransferFunction`) | ||
| System in continuous or discrete time for which the norm should be computed. | ||
| p : int or str | ||
| Type of norm to be computed. ``p=2`` gives the H2 norm, and ``p='inf'`` gives the L-infinity norm. | ||
| tol : float | ||
| Relative tolerance for accuracy of L-infinity norm computation. Ignored | ||
| unless p='inf'. | ||
| print_warning : bool | ||
| Print warning message in case norm value may be uncertain. | ||
| 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 | ||
| ------- | ||
| norm_value : float | ||
| Norm value of system. | ||
| Notes | ||
| ----- | ||
| Does not yet compute the L-infinity norm for discrete time systems with pole(s) in z=0 unless Slycot is used. | ||
| Examples | ||
| -------- | ||
| >>> Gc = ct.tf([1], [1, 2, 1]) | ||
| >>> ct.norm(Gc, 2) | ||
| 0.5000000000000001 | ||
| >>> ct.norm(Gc, 'inf', tol=1e-11, method='scipy') | ||
| 1.000000000007276 | ||
| """ | ||
| if not isinstance(system, (ct.StateSpace, ct.TransferFunction)): | ||
| raise TypeError('Parameter ``system``: must be a ``StateSpace`` or ``TransferFunction``') | ||
| G = ct.ss(system) | ||
| A = G.A | ||
| B = G.B | ||
| C = G.C | ||
| D = G.D | ||
| # Decide what method to use | ||
| method = ct.mateqn._slycot_or_scipy(method) | ||
| # ------------------- | ||
| # H2 norm computation | ||
| # ------------------- | ||
| if p == 2: | ||
| # -------------------- | ||
| # Continuous time case | ||
| # -------------------- | ||
| if G.isctime(): | ||
| # Check for cases with infinite norm | ||
| poles_real_part = G.poles().real | ||
| if any(np.isclose(poles_real_part, 0.0)): # Poles on imaginary axis | ||
| if print_warning: | ||
| warnings.warn("Poles close to, or on, the imaginary axis. Norm value may be uncertain.", UserWarning) | ||
| return float('inf') | ||
| elif any(poles_real_part > 0.0): # System unstable | ||
| if print_warning: | ||
| warnings.warn("System is unstable!", UserWarning) | ||
| return float('inf') | ||
| elif any(D.flat != 0): # System has direct feedthrough | ||
| if print_warning: | ||
| warnings.warn("System has a direct feedthrough term!", UserWarning) | ||
| return float('inf') | ||
| else: | ||
| # Use slycot, if available, to compute (finite) norm | ||
| if method == 'slycot': | ||
| return _h2norm_slycot(G, print_warning) | ||
| # Else use scipy | ||
| else: | ||
| P = ct.lyap(A, B@B.T, method=method) # Solve for controllability Gramian | ||
| # System is stable to reach this point, and P should be positive semi-definite. | ||
| # Test next is a precaution in case the Lyapunov equation is ill conditioned. | ||
| if any(la.eigvals(P).real < 0.0): | ||
| if print_warning: | ||
| warnings.warn("There appears to be poles close to the imaginary axis. Norm value may be uncertain.", UserWarning) | ||
| return float('inf') | ||
| else: | ||
| norm_value = np.sqrt(np.trace(C@P@C.T)) # Argument in sqrt should be non-negative | ||
| if np.isnan(norm_value): | ||
| raise ct.ControlArgument("Norm computation resulted in NaN.") | ||
| else: | ||
| return norm_value | ||
| # ------------------ | ||
| # Discrete time case | ||
| # ------------------ | ||
| elif G.isdtime(): | ||
| # Check for cases with infinite norm | ||
| poles_abs = abs(G.poles()) | ||
| if any(np.isclose(poles_abs, 1.0)): # Poles on imaginary axis | ||
| if print_warning: | ||
| warnings.warn("Poles close to, or on, the complex unit circle. Norm value may be uncertain.", UserWarning) | ||
| return float('inf') | ||
| elif any(poles_abs > 1.0): # System unstable | ||
| if print_warning: | ||
| warnings.warn("System is unstable!", UserWarning) | ||
| return float('inf') | ||
| else: | ||
| # Use slycot, if available, to compute (finite) norm | ||
| if method == 'slycot': | ||
| return _h2norm_slycot(G, print_warning) | ||
| # Else use scipy | ||
| else: | ||
| P = ct.dlyap(A, B@B.T, method=method) | ||
| # System is stable to reach this point, and P should be positive semi-definite. | ||
| # Test next is a precaution in case the Lyapunov equation is ill conditioned. | ||
| if any(la.eigvals(P).real < 0.0): | ||
| if print_warning: | ||
| warnings.warn("Warning: There appears to be poles close to the complex unit circle. Norm value may be uncertain.", UserWarning) | ||
| return float('inf') | ||
| else: | ||
| norm_value = np.sqrt(np.trace(C@P@C.T + D@D.T)) # Argument in sqrt should be non-negative | ||
| if np.isnan(norm_value): | ||
| raise ct.ControlArgument("Norm computation resulted in NaN.") | ||
| else: | ||
| return norm_value | ||
| # --------------------------- | ||
| # L-infinity norm computation | ||
| # --------------------------- | ||
| elif p == "inf": | ||
| # Check for cases with infinite norm | ||
| poles = G.poles() | ||
| if G.isdtime(): # Discrete time | ||
| if any(np.isclose(abs(poles), 1.0)): # Poles on unit circle | ||
| if print_warning: | ||
| warnings.warn("Poles close to, or on, the complex unit circle. Norm value may be uncertain.", UserWarning) | ||
| return float('inf') | ||
| else: # Continuous time | ||
| if any(np.isclose(poles.real, 0.0)): # Poles on imaginary axis | ||
| if print_warning: | ||
| warnings.warn("Poles close to, or on, the imaginary axis. Norm value may be uncertain.", UserWarning) | ||
| return float('inf') | ||
| # Use slycot, if available, to compute (finite) norm | ||
| if method == 'slycot': | ||
| return ct.linfnorm(G, tol)[0] | ||
| # Else use scipy | ||
| else: | ||
| # ------------------ | ||
| # Discrete time case | ||
| # ------------------ | ||
| # Use inverse bilinear transformation of discrete time system to s-plane if no poles on |z|=1 or z=0. | ||
| # Allows us to use test for continuous time systems next. | ||
| if G.isdtime(): | ||
| Ad = A | ||
| Bd = B | ||
| Cd = C | ||
| Dd = D | ||
| if any(np.isclose(la.eigvals(Ad), 0.0)): | ||
| raise ct.ControlArgument("L-infinity norm computation for discrete time system with pole(s) in z=0 currently not supported unless Slycot installed.") | ||
| # Inverse bilinear transformation | ||
| In = np.eye(len(Ad)) | ||
| Adinv = la.inv(Ad+In) | ||
| A = 2*(Ad-In)@Adinv | ||
| B = 2*Adinv@Bd | ||
| C = 2*Cd@Adinv | ||
| D = Dd - Cd@Adinv@Bd | ||
| # -------------------- | ||
| # Continuous time case | ||
| # -------------------- | ||
| def _Hamilton_matrix(gamma): | ||
| """Constructs Hamiltonian matrix. For internal use.""" | ||
| R = Ip*gamma**2 - D.T@D | ||
| invR = la.inv(R) | ||
| return np.block([[A+B@invR@D.T@C, B@invR@B.T], [-C.T@(Ip+D@invR@D.T)@C, -(A+B@invR@D.T@C).T]]) | ||
| gaml = la.norm(D,ord=2) # Lower bound | ||
| gamu = max(1.0, 2.0*gaml) # Candidate upper bound | ||
| Ip = np.eye(len(D)) | ||
| while any(np.isclose(la.eigvals(_Hamilton_matrix(gamu)).real, 0.0)): # Find actual upper bound | ||
| gamu *= 2.0 | ||
| while (gamu-gaml)/gamu > tol: | ||
| gam = (gamu+gaml)/2.0 | ||
| if any(np.isclose(la.eigvals(_Hamilton_matrix(gam)).real, 0.0)): | ||
| gaml = gam | ||
| else: | ||
| gamu = gam | ||
| return gam | ||
| # ---------------------- | ||
| # Other norm computation | ||
| # ---------------------- | ||
| else: | ||
| raise ct.ControlArgument(f"Norm computation for p={p} currently not supported.") | ||
74 changes: 74 additions & 0 deletionscontrol/tests/sysnorm_test.py
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,74 @@ | ||
| # -*- coding: utf-8 -*- | ||
| """ | ||
| Tests for sysnorm module. | ||
| Created on Mon Jan 8 11:31:46 2024 | ||
| Author: Henrik Sandberg | ||
| """ | ||
| import control as ct | ||
| import numpy as np | ||
| import pytest | ||
| def test_norm_1st_order_stable_system(): | ||
| """First-order stable continuous-time system""" | ||
| s = ct.tf('s') | ||
| G1 = 1/(s+1) | ||
| assert np.allclose(ct.norm(G1, p='inf'), 1.0) # Comparison to norm computed in MATLAB | ||
| assert np.allclose(ct.norm(G1, p=2), 0.707106781186547) # Comparison to norm computed in MATLAB | ||
| Gd1 = ct.sample_system(G1, 0.1) | ||
| assert np.allclose(ct.norm(Gd1, p='inf'), 1.0) # Comparison to norm computed in MATLAB | ||
| assert np.allclose(ct.norm(Gd1, p=2), 0.223513699524858) # Comparison to norm computed in MATLAB | ||
| def test_norm_1st_order_unstable_system(): | ||
| """First-order unstable continuous-time system""" | ||
| s = ct.tf('s') | ||
| G2 = 1/(1-s) | ||
| assert np.allclose(ct.norm(G2, p='inf'), 1.0) # Comparison to norm computed in MATLAB | ||
| with pytest.warns(UserWarning, match="System is unstable!"): | ||
| assert ct.norm(G2, p=2) == float('inf') # Comparison to norm computed in MATLAB | ||
| Gd2 = ct.sample_system(G2, 0.1) | ||
| assert np.allclose(ct.norm(Gd2, p='inf'), 1.0) # Comparison to norm computed in MATLAB | ||
| with pytest.warns(UserWarning, match="System is unstable!"): | ||
| assert ct.norm(Gd2, p=2) == float('inf') # Comparison to norm computed in MATLAB | ||
| def test_norm_2nd_order_system_imag_poles(): | ||
| """Second-order continuous-time system with poles on imaginary axis""" | ||
| s = ct.tf('s') | ||
| G3 = 1/(s**2+1) | ||
| with pytest.warns(UserWarning, match="Poles close to, or on, the imaginary axis."): | ||
| assert ct.norm(G3, p='inf') == float('inf') # Comparison to norm computed in MATLAB | ||
| with pytest.warns(UserWarning, match="Poles close to, or on, the imaginary axis."): | ||
| assert ct.norm(G3, p=2) == float('inf') # Comparison to norm computed in MATLAB | ||
| Gd3 = ct.sample_system(G3, 0.1) | ||
| with pytest.warns(UserWarning, match="Poles close to, or on, the complex unit circle."): | ||
| assert ct.norm(Gd3, p='inf') == float('inf') # Comparison to norm computed in MATLAB | ||
| with pytest.warns(UserWarning, match="Poles close to, or on, the complex unit circle."): | ||
| assert ct.norm(Gd3, p=2) == float('inf') # Comparison to norm computed in MATLAB | ||
| def test_norm_3rd_order_mimo_system(): | ||
| """Third-order stable MIMO continuous-time system""" | ||
| A = np.array([[-1.017041847539126, -0.224182952826418, 0.042538079149249], | ||
| [-0.310374015319095, -0.516461581407780, -0.119195790221750], | ||
| [-1.452723568727942, 1.7995860837102088, -1.491935830615152]]) | ||
| B = np.array([[0.312858596637428, -0.164879019209038], | ||
| [-0.864879917324456, 0.627707287528727], | ||
| [-0.030051296196269, 1.093265669039484]]) | ||
| C = np.array([[1.109273297614398, 0.077359091130425, -1.113500741486764], | ||
| [-0.863652821988714, -1.214117043615409, -0.006849328103348]]) | ||
| D = np.zeros((2,2)) | ||
| G4 = ct.ss(A,B,C,D) # Random system generated in MATLAB | ||
| assert np.allclose(ct.norm(G4, p='inf'), 4.276759162964244) # Comparison to norm computed in MATLAB | ||
| assert np.allclose(ct.norm(G4, p=2), 2.237461821810309) # Comparison to norm computed in MATLAB | ||
| Gd4 = ct.sample_system(G4, 0.1) | ||
| assert np.allclose(ct.norm(Gd4, p='inf'), 4.276759162964228) # Comparison to norm computed in MATLAB | ||
| assert np.allclose(ct.norm(Gd4, p=2), 0.707434962289554) # Comparison to norm computed in MATLAB |
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