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GeezerVison is showing github.com — python-control/control/canonical.py at 0.9.1 · python-control/python-control Expand file tree/ Copy pathcanonical.pyMore file actions 450 lines (353 loc) · 13.7 KB / Copy pathcanonical.pyFile metadata and controls450 lines (353 loc) · 13.7 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 # canonical.py - functions for converting systems to canonical forms # RMM, 10 Nov 2012 from .exception import ControlNotImplemented, ControlSlycot from .lti import issiso from .statesp import StateSpace, _convert_to_statespace from .statefbk import ctrb, obsv import numpy as np from numpy import zeros, zeros_like, shape, poly, iscomplex, vstack, hstack, \ transpose, empty, finfo, float64 from numpy.linalg import solve, matrix_rank, eig from scipy.linalg import schur __all__ = ['canonical_form', 'reachable_form', 'observable_form', 'modal_form', 'similarity_transform', 'bdschur'] def canonical_form(xsys, form='reachable'): """Convert a system into canonical form Parameters ---------- xsys : StateSpace object System to be transformed, with state 'x' form : str Canonical form for transformation. Chosen from: * 'reachable' - reachable canonical form * 'observable' - observable canonical form * 'modal' - modal canonical form Returns ------- zsys : StateSpace object System in desired canonical form, with state 'z' T : (M, M) real ndarray Coordinate transformation matrix, z = T * x """ # Call the appropriate tranformation function if form == 'reachable': return reachable_form(xsys) elif form == 'observable': return observable_form(xsys) elif form == 'modal': return modal_form(xsys) else: raise ControlNotImplemented( "Canonical form '%s' not yet implemented" % form) # Reachable canonical form def reachable_form(xsys): """Convert a system into reachable canonical form Parameters ---------- xsys : StateSpace object System to be transformed, with state `x` Returns ------- zsys : StateSpace object System in reachable canonical form, with state `z` T : (M, M) real ndarray Coordinate transformation: z = T * x """ # Check to make sure we have a SISO system if not issiso(xsys): raise ControlNotImplemented( "Canonical forms for MIMO systems not yet supported") # Create a new system, starting with a copy of the old one zsys = StateSpace(xsys) # Generate the system matrices for the desired canonical form zsys.B = zeros_like(xsys.B) zsys.B[0, 0] = 1.0 zsys.A = zeros_like(xsys.A) Apoly = poly(xsys.A) # characteristic polynomial for i in range(0, xsys.nstates): zsys.A[0, i] = -Apoly[i+1] / Apoly[0] if (i+1 < xsys.nstates): zsys.A[i+1, i] = 1.0 # Compute the reachability matrices for each set of states Wrx = ctrb(xsys.A, xsys.B) Wrz = ctrb(zsys.A, zsys.B) if matrix_rank(Wrx) != xsys.nstates: raise ValueError("System not controllable to working precision.") # Transformation from one form to another Tzx = solve(Wrx.T, Wrz.T).T # matrix right division, Tzx = Wrz * inv(Wrx) # Check to make sure inversion was OK. Note that since we are inverting # Wrx and we already checked its rank, this exception should never occur if matrix_rank(Tzx) != xsys.nstates: # pragma: no cover raise ValueError("Transformation matrix singular to working precision.") # Finally, compute the output matrix zsys.C = solve(Tzx.T, xsys.C.T).T # matrix right division, zsys.C = xsys.C * inv(Tzx) return zsys, Tzx def observable_form(xsys): """Convert a system into observable canonical form Parameters ---------- xsys : StateSpace object System to be transformed, with state `x` Returns ------- zsys : StateSpace object System in observable canonical form, with state `z` T : (M, M) real ndarray Coordinate transformation: z = T * x """ # Check to make sure we have a SISO system if not issiso(xsys): raise ControlNotImplemented( "Canonical forms for MIMO systems not yet supported") # Create a new system, starting with a copy of the old one zsys = StateSpace(xsys) # Generate the system matrices for the desired canonical form zsys.C = zeros_like(xsys.C) zsys.C[0, 0] = 1 zsys.A = zeros_like(xsys.A) Apoly = poly(xsys.A) # characteristic polynomial for i in range(0, xsys.nstates): zsys.A[i, 0] = -Apoly[i+1] / Apoly[0] if (i+1 < xsys.nstates): zsys.A[i, i+1] = 1 # Compute the observability matrices for each set of states Wrx = obsv(xsys.A, xsys.C) Wrz = obsv(zsys.A, zsys.C) # Transformation from one form to another Tzx = solve(Wrz, Wrx) # matrix left division, Tzx = inv(Wrz) * Wrx if matrix_rank(Tzx) != xsys.nstates: raise ValueError("Transformation matrix singular to working precision.") # Finally, compute the output matrix zsys.B = Tzx @ xsys.B return zsys, Tzx def similarity_transform(xsys, T, timescale=1, inverse=False): """Perform a similarity transformation, with option time rescaling. Transform a linear state space system to a new state space representation z = T x, or x = T z, where T is an invertible matrix. Parameters ---------- xsys : StateSpace object System to transform T : (M, M) array_like The matrix `T` defines the new set of coordinates z = T x. timescale : float, optional If present, also rescale the time unit to tau = timescale * t inverse: boolean, optional If True (default), transform so z = T x. If False, transform so x = T z. Returns ------- zsys : StateSpace object System in transformed coordinates, with state 'z' """ # Create a new system, starting with a copy of the old one zsys = StateSpace(xsys) T = np.atleast_2d(T) # Define a function to compute the right inverse (solve x M = y) def rsolve(M, y): return transpose(solve(transpose(M), transpose(y))) # Update the system matrices if not inverse: zsys.A = rsolve(T, T @ zsys.A) / timescale zsys.B = T @ zsys.B / timescale zsys.C = rsolve(T, zsys.C) else: zsys.A = solve(T, zsys.A) @ T / timescale zsys.B = solve(T, zsys.B) / timescale zsys.C = zsys.C @ T return zsys _IM_ZERO_TOL = np.finfo(np.float64).eps ** 0.5 _PMAX_SEARCH_TOL = 1.001 def _bdschur_defective(blksizes, eigvals): """Check for defective modal decomposition Parameters ---------- blksizes: (N,) int ndarray size of Schur blocks eigvals: (M,) real or complex ndarray Eigenvalues Returns ------- True iff Schur blocks are defective. blksizes, eigvals are the 3rd and 4th results returned by mb03rd. """ if any(blksizes > 2): return True if all(blksizes == 1): return False # check eigenvalues associated with blocks of size 2 init_idxs = np.cumsum(np.hstack([0, blksizes[:-1]])) blk_idx2 = blksizes == 2 im = eigvals[init_idxs[blk_idx2]].imag re = eigvals[init_idxs[blk_idx2]].real if any(abs(im) < _IM_ZERO_TOL * abs(re)): return True return False def _bdschur_condmax_search(aschur, tschur, condmax): """Block-diagonal Schur decomposition search up to condmax Iterates mb03rd with different pmax values until: - result is non-defective; - or condition number of similarity transform is unchanging despite large pmax; - or condition number of similarity transform is close to condmax. Parameters ---------- aschur: (N, N) real ndarray Real Schur-form matrix tschur: (N, N) real ndarray Orthogonal transformation giving aschur from some initial matrix a condmax: float Maximum condition number of final transformation. Must be >= 1. Returns ------- amodal: (N, N) real ndarray block diagonal Schur form tmodal: (N, N) real ndarray similarity transformation give amodal from aschur blksizes: (M,) int ndarray Array of Schur block sizes eigvals: (N,) real or complex ndarray Eigenvalues of amodal (and a, etc.) Notes ----- Outputs as for slycot.mb03rd aschur, tschur are as returned by scipy.linalg.schur. """ try: from slycot import mb03rd except ImportError: raise ControlSlycot("can't find slycot module 'mb03rd'") # see notes on RuntimeError below pmaxlower = None # get lower bound; try condmax ** 0.5 first pmaxlower = condmax ** 0.5 amodal, tmodal, blksizes, eigvals = mb03rd(aschur.shape[0], aschur, tschur, pmax=pmaxlower) if np.linalg.cond(tmodal) <= condmax: reslower = amodal, tmodal, blksizes, eigvals else: pmaxlower = 1.0 amodal, tmodal, blksizes, eigvals = mb03rd(aschur.shape[0], aschur, tschur, pmax=pmaxlower) cond = np.linalg.cond(tmodal) if cond > condmax: msg = 'minimum cond={} > condmax={}; try increasing condmax'.format(cond, condmax) raise RuntimeError(msg) pmax = pmaxlower # phase 1: search for upper bound on pmax for i in range(50): amodal, tmodal, blksizes, eigvals = mb03rd(aschur.shape[0], aschur, tschur, pmax=pmax) cond = np.linalg.cond(tmodal) if cond < condmax: pmaxlower = pmax reslower = amodal, tmodal, blksizes, eigvals else: # upper bound found; go to phase 2 pmaxupper = pmax break if _bdschur_defective(blksizes, eigvals): pmax *= 2 else: return amodal, tmodal, blksizes, eigvals else: # no upper bound found; return current result return reslower # phase 2: bisection search for i in range(50): pmax = (pmaxlower * pmaxupper) ** 0.5 amodal, tmodal, blksizes, eigvals = mb03rd(aschur.shape[0], aschur, tschur, pmax=pmax) cond = np.linalg.cond(tmodal) if cond < condmax: if not _bdschur_defective(blksizes, eigvals): return amodal, tmodal, blksizes, eigvals pmaxlower = pmax reslower = amodal, tmodal, blksizes, eigvals else: pmaxupper = pmax if pmaxupper / pmaxlower < _PMAX_SEARCH_TOL: # hit search limit return reslower else: raise ValueError('bisection failed to converge; pmaxlower={}, pmaxupper={}'.format(pmaxlower, pmaxupper)) def bdschur(a, condmax=None, sort=None): """Block-diagonal Schur decomposition Parameters ---------- a : (M, M) array_like Real matrix to decompose condmax : None or float, optional If None (default), use 1/sqrt(eps), which is approximately 1e8 sort : {None, 'continuous', 'discrete'} Block sorting; see below. Returns ------- amodal : (M, M) real ndarray Block-diagonal Schur decomposition of `a` tmodal : (M, M) real ndarray Similarity transform relating `a` and `amodal` blksizes : (N,) int ndarray Array of Schur block sizes Notes ----- If `sort` is None, the blocks are not sorted. If `sort` is 'continuous', the blocks are sorted according to associated eigenvalues. The ordering is first by real part of eigenvalue, in descending order, then by absolute value of imaginary part of eigenvalue, also in decreasing order. If `sort` is 'discrete', the blocks are sorted as for 'continuous', but applied to log of eigenvalues (i.e., continuous-equivalent eigenvalues). """ if condmax is None: condmax = np.finfo(np.float64).eps ** -0.5 if not (np.isscalar(condmax) and condmax >= 1.0): raise ValueError('condmax="{}" must be a scalar >= 1.0'.format(condmax)) a = np.atleast_2d(a) if a.shape[0] == 0 or a.shape[1] == 0: return a.copy(), np.eye(a.shape[1], a.shape[0]), np.array([]) aschur, tschur = schur(a) amodal, tmodal, blksizes, eigvals = _bdschur_condmax_search(aschur, tschur, condmax) if sort in ('continuous', 'discrete'): idxs = np.cumsum(np.hstack([0, blksizes[:-1]])) ev_per_blk = [complex(eigvals[i].real, abs(eigvals[i].imag)) for i in idxs] if sort == 'discrete': ev_per_blk = np.log(ev_per_blk) # put most unstable first sortidx = np.argsort(ev_per_blk)[::-1] # block indices blkidxs = [np.arange(i0, i0+ilen) for i0, ilen in zip(idxs, blksizes)] # reordered permidx = np.hstack([blkidxs[i] for i in sortidx]) rperm = np.eye(amodal.shape[0])[permidx] tmodal = tmodal @ rperm amodal = rperm @ amodal @ rperm.T blksizes = blksizes[sortidx] elif sort is None: pass else: raise ValueError('unknown sort value "{}"'.format(sort)) return amodal, tmodal, blksizes def modal_form(xsys, condmax=None, sort=False): """Convert a system into modal canonical form Parameters ---------- xsys : StateSpace object System to be transformed, with state `x` condmax : None or float, optional An upper bound on individual transformations. If None, use `bdschur` default. sort : bool, optional If False (default), Schur blocks will not be sorted. See `bdschur` for sort order. Returns ------- zsys : StateSpace object System in modal canonical form, with state `z` T : (M, M) ndarray Coordinate transformation: z = T * x """ if sort: discrete = xsys.dt is not None and xsys.dt > 0 bd_sort = 'discrete' if discrete else 'continuous' else: bd_sort = None xsys = _convert_to_statespace(xsys) amodal, tmodal, _ = bdschur(xsys.A, condmax=condmax, sort=bd_sort) return similarity_transform(xsys, tmodal, inverse=True), tmodal |
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