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GeezerVison is showing github.com — python-control/control/sysnorm.py at main · python-control/python-control Expand file tree/ Copy pathsysnorm.pyMore file actions 329 lines (283 loc) · 11.4 KB / Copy pathsysnorm.pyFile metadata and controls329 lines (283 loc) · 11.4 KB 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 # sysnorm.py - functions for computing system norms # # Initial author: Henrik Sandberg # Creation date: 21 Dec 2023 """Functions for computing system norms.""" import warnings import numpy as np import numpy.linalg as la import control as ct __all__ = ['system_norm', 'norm'] #------------------------------------------------------------------------------ def _h2norm_slycot(sys, print_warning=True): """H2 norm of a linear system. For internal use. Requires Slycot. See Also -------- slycot.ab13bd """ # See: https://github.com/python-control/Slycot/issues/199 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 system_norm(system, p=2, tol=1e-6, print_warning=True, method=None): """Computes the input/output norm of system. Parameters ---------- system : LTI (`StateSpace` or `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) at the origin unless Slycot is used. Examples -------- >>> Gc = ct.tf([1], [1, 2, 1]) >>> round(ct.norm(Gc, 2), 3) 0.5 >>> round(ct.norm(Gc, 'inf', tol=1e-5, method='scipy'), 3) np.float64(1.0) """ 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: # Solve for controllability Gramian P = ct.lyap(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( "There appears to be poles close to the " "imaginary axis. Norm value may be uncertain.", UserWarning) return float('inf') else: # Argument in sqrt should be non-negative norm_value = np.sqrt(np.trace(C@P@C.T)) 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( "There appears to be poles close to the complex " "unit circle. Norm value may be uncertain.", UserWarning) return float('inf') else: # Argument in sqrt should be non-negative norm_value = np.sqrt(np.trace(C@P@C.T + D@D.T)) 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.") norm = system_norm |
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