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GeezerVison is showing github.com — python-control-2/control/modelsimp.py at main · basicmachines/python-control-2 Expand file tree/ Copy pathmodelsimp.pyMore file actions 558 lines (476 loc) · 18 KB / Copy pathmodelsimp.pyFile metadata and controls558 lines (476 loc) · 18 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 #! TODO: add module docstring # modelsimp.py - tools for model simplification # # Author: Steve Brunton, Kevin Chen, Lauren Padilla # Date: 30 Nov 2010 # # This file contains routines for obtaining reduced order models # # Copyright (c) 2010 by California Institute of Technology # All rights reserved. # # Redistribution and use in source and binary forms, with or without # modification, are permitted provided that the following conditions # are met: # # 1. Redistributions of source code must retain the above copyright # notice, this list of conditions and the following disclaimer. # # 2. Redistributions in binary form must reproduce the above copyright # notice, this list of conditions and the following disclaimer in the # documentation and/or other materials provided with the distribution. # # 3. Neither the name of the California Institute of Technology nor # the names of its contributors may be used to endorse or promote # products derived from this software without specific prior # written permission. # # THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS # "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT # LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS # FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL CALTECH # OR THE CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, # SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT # LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF # USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND # ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, # OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT # OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF # SUCH DAMAGE. # # $Id$ # External packages and modules import numpy as np import warnings from .exception import ControlSlycot, ControlMIMONotImplemented, \ ControlDimension from .iosys import isdtime, isctime from .statesp import StateSpace from .statefbk import gram __all__ = ['hsvd', 'balred', 'modred', 'era', 'markov', 'minreal'] # Hankel Singular Value Decomposition # # The following returns the Hankel singular values, which are singular values # of the matrix formed by multiplying the controllability and observability # Gramians def hsvd(sys): """Calculate the Hankel singular values. Parameters ---------- sys : StateSpace A state space system Returns ------- H : array A list of Hankel singular values See Also -------- gram Notes ----- The Hankel singular values are the singular values of the Hankel operator. In practice, we compute the square root of the eigenvalues of the matrix formed by taking the product of the observability and controllability gramians. There are other (more efficient) methods based on solving the Lyapunov equation in a particular way (more details soon). Examples -------- >>> G = ct.tf2ss([1], [1, 2]) >>> H = ct.hsvd(G) >>> H[0] 0.25 """ # TODO: implement for discrete time systems if (isdtime(sys, strict=True)): raise NotImplementedError("Function not implemented in discrete time") Wc = gram(sys, 'c') Wo = gram(sys, 'o') WoWc = Wo @ Wc w, v = np.linalg.eig(WoWc) hsv = np.sqrt(w) hsv = np.array(hsv) hsv = np.sort(hsv) # Return the Hankel singular values, high to low return hsv[::-1] def modred(sys, ELIM, method='matchdc'): """ Model reduction of `sys` by eliminating the states in `ELIM` using a given method. Parameters ---------- sys: StateSpace Original system to reduce ELIM: array Vector of states to eliminate method: string Method of removing states in `ELIM`: either ``'truncate'`` or ``'matchdc'``. Returns ------- rsys: StateSpace A reduced order model Raises ------ ValueError Raised under the following conditions: * if `method` is not either ``'matchdc'`` or ``'truncate'`` * if eigenvalues of `sys.A` are not all in left half plane (`sys` must be stable) Examples -------- >>> G = ct.rss(4) >>> Gr = ct.modred(G, [0, 2], method='matchdc') >>> Gr.nstates 2 """ # Check for ss system object, need a utility for this? # TODO: Check for continous or discrete, only continuous supported for now # if isCont(): # dico = 'C' # elif isDisc(): # dico = 'D' # else: if (isctime(sys)): dico = 'C' else: raise NotImplementedError("Function not implemented in discrete time") # Check system is stable if np.any(np.linalg.eigvals(sys.A).real >= 0.0): raise ValueError("Oops, the system is unstable!") ELIM = np.sort(ELIM) # Create list of elements not to eliminate (NELIM) NELIM = [i for i in range(len(sys.A)) if i not in ELIM] # A1 is a matrix of all columns of sys.A not to eliminate A1 = sys.A[:, NELIM[0]].reshape(-1, 1) for i in NELIM[1:]: A1 = np.hstack((A1, sys.A[:, i].reshape(-1, 1))) A11 = A1[NELIM, :] A21 = A1[ELIM, :] # A2 is a matrix of all columns of sys.A to eliminate A2 = sys.A[:, ELIM[0]].reshape(-1, 1) for i in ELIM[1:]: A2 = np.hstack((A2, sys.A[:, i].reshape(-1, 1))) A12 = A2[NELIM, :] A22 = A2[ELIM, :] C1 = sys.C[:, NELIM] C2 = sys.C[:, ELIM] B1 = sys.B[NELIM, :] B2 = sys.B[ELIM, :] if method == 'matchdc': # if matchdc, residualize # Check if the matrix A22 is invertible if np.linalg.matrix_rank(A22) != len(ELIM): raise ValueError("Matrix A22 is singular to working precision.") # Now precompute A22\A21 and A22\B2 (A22I = inv(A22)) # We can solve two linear systems in one pass, since the # coefficients matrix A22 is the same. Thus, we perform the LU # decomposition (cubic runtime complexity) of A22 only once! # The remaining back substitutions are only quadratic in runtime. A22I_A21_B2 = np.linalg.solve(A22, np.concatenate((A21, B2), axis=1)) A22I_A21 = A22I_A21_B2[:, :A21.shape[1]] A22I_B2 = A22I_A21_B2[:, A21.shape[1]:] Ar = A11 - A12 @ A22I_A21 Br = B1 - A12 @ A22I_B2 Cr = C1 - C2 @ A22I_A21 Dr = sys.D - C2 @ A22I_B2 elif method == 'truncate': # if truncate, simply discard state x2 Ar = A11 Br = B1 Cr = C1 Dr = sys.D else: raise ValueError("Oops, method is not supported!") rsys = StateSpace(Ar, Br, Cr, Dr) return rsys def balred(sys, orders, method='truncate', alpha=None): """Balanced reduced order model of sys of a given order. States are eliminated based on Hankel singular value. If sys has unstable modes, they are removed, the balanced realization is done on the stable part, then reinserted in accordance with the reference below. Reference: Hsu,C.S., and Hou,D., 1991, Reducing unstable linear control systems via real Schur transformation. Electronics Letters, 27, 984-986. Parameters ---------- sys: StateSpace Original system to reduce orders: integer or array of integer Desired order of reduced order model (if a vector, returns a vector of systems) method: string Method of removing states, either ``'truncate'`` or ``'matchdc'``. alpha: float Redefines the stability boundary for eigenvalues of the system matrix A. By default for continuous-time systems, alpha <= 0 defines the stability boundary for the real part of A's eigenvalues and for discrete-time systems, 0 <= alpha <= 1 defines the stability boundary for the modulus of A's eigenvalues. See SLICOT routines AB09MD and AB09ND for more information. Returns ------- rsys: StateSpace A reduced order model or a list of reduced order models if orders is a list. Raises ------ ValueError If `method` is not ``'truncate'`` or ``'matchdc'`` ImportError if slycot routine ab09ad, ab09md, or ab09nd is not found ValueError if there are more unstable modes than any value in orders Examples -------- >>> G = ct.rss(4) >>> Gr = ct.balred(G, orders=2, method='matchdc') >>> Gr.nstates 2 """ if method != 'truncate' and method != 'matchdc': raise ValueError("supported methods are 'truncate' or 'matchdc'") elif method == 'truncate': try: from slycot import ab09md, ab09ad except ImportError: raise ControlSlycot( "can't find slycot subroutine ab09md or ab09ad") elif method == 'matchdc': try: from slycot import ab09nd except ImportError: raise ControlSlycot("can't find slycot subroutine ab09nd") # Check for ss system object, need a utility for this? # TODO: Check for continous or discrete, only continuous supported for now # if isCont(): # dico = 'C' # elif isDisc(): # dico = 'D' # else: dico = 'C' job = 'B' # balanced (B) or not (N) equil = 'N' # scale (S) or not (N) if alpha is None: if dico == 'C': alpha = 0. elif dico == 'D': alpha = 1. rsys = [] # empty list for reduced systems # check if orders is a list or a scalar try: order = iter(orders) except TypeError: # if orders is a scalar orders = [orders] for i in orders: n = np.size(sys.A, 0) m = np.size(sys.B, 1) p = np.size(sys.C, 0) if method == 'truncate': # check system stability if np.any(np.linalg.eigvals(sys.A).real >= 0.0): # unstable branch Nr, Ar, Br, Cr, Ns, hsv = ab09md( dico, job, equil, n, m, p, sys.A, sys.B, sys.C, alpha=alpha, nr=i, tol=0.0) else: # stable branch Nr, Ar, Br, Cr, hsv = ab09ad( dico, job, equil, n, m, p, sys.A, sys.B, sys.C, nr=i, tol=0.0) rsys.append(StateSpace(Ar, Br, Cr, sys.D)) elif method == 'matchdc': Nr, Ar, Br, Cr, Dr, Ns, hsv = ab09nd( dico, job, equil, n, m, p, sys.A, sys.B, sys.C, sys.D, alpha=alpha, nr=i, tol1=0.0, tol2=0.0) rsys.append(StateSpace(Ar, Br, Cr, Dr)) # if orders was a scalar, just return the single reduced model, not a list if len(orders) == 1: return rsys[0] # if orders was a list/vector, return a list/vector of systems else: return rsys def minreal(sys, tol=None, verbose=True): ''' Eliminates uncontrollable or unobservable states in state-space models or cancelling pole-zero pairs in transfer functions. The output sysr has minimal order and the same response characteristics as the original model sys. Parameters ---------- sys: StateSpace or TransferFunction Original system tol: real Tolerance verbose: bool Print results if True Returns ------- rsys: StateSpace or TransferFunction Cleaned model ''' sysr = sys.minreal(tol) if verbose: print("{nstates} states have been removed from the model".format( nstates=len(sys.poles()) - len(sysr.poles()))) return sysr def era(YY, m, n, nin, nout, r): """Calculate an ERA model of order `r` based on the impulse-response data `YY`. .. note:: This function is not implemented yet. Parameters ---------- YY: array `nout` x `nin` dimensional impulse-response data m: integer Number of rows in Hankel matrix n: integer Number of columns in Hankel matrix nin: integer Number of input variables nout: integer Number of output variables r: integer Order of model Returns ------- sys: StateSpace A reduced order model sys=ss(Ar,Br,Cr,Dr) Examples -------- >>> rsys = era(YY, m, n, nin, nout, r) # doctest: +SKIP """ raise NotImplementedError('This function is not implemented yet.') def markov(Y, U, m=None, transpose=False): """Calculate the first `m` Markov parameters [D CB CAB ...] from input `U`, output `Y`. This function computes the Markov parameters for a discrete time system .. math:: x[k+1] &= A x[k] + B u[k] \\\\ y[k] &= C x[k] + D u[k] given data for u and y. The algorithm assumes that that C A^k B = 0 for k > m-2 (see [1]_). Note that the problem is ill-posed if the length of the input data is less than the desired number of Markov parameters (a warning message is generated in this case). Parameters ---------- Y : array_like Output data. If the array is 1D, the system is assumed to be single input. If the array is 2D and transpose=False, the columns of `Y` are taken as time points, otherwise the rows of `Y` are taken as time points. U : array_like Input data, arranged in the same way as `Y`. m : int, optional Number of Markov parameters to output. Defaults to len(U). transpose : bool, optional Assume that input data is transposed relative to the standard :ref:`time-series-convention`. Default value is False. Returns ------- H : ndarray First m Markov parameters, [D CB CAB ...] References ---------- .. [1] J.-N. Juang, M. Phan, L. G. Horta, and R. W. Longman, Identification of observer/Kalman filter Markov parameters - Theory and experiments. Journal of Guidance Control and Dynamics, 16(2), 320-329, 2012. http://doi.org/10.2514/3.21006 Notes ----- Currently only works for SISO systems. This function does not currently comply with the Python Control Library :ref:`time-series-convention` for representation of time series data. Use `transpose=False` to make use of the standard convention (this will be updated in a future release). Examples -------- >>> T = np.linspace(0, 10, 100) >>> U = np.ones((1, 100)) >>> T, Y = ct.forced_response(ct.tf([1], [1, 0.5], True), T, U) >>> H = ct.markov(Y, U, 3, transpose=False) """ # Convert input parameters to 2D arrays (if they aren't already) Umat = np.array(U, ndmin=2) Ymat = np.array(Y, ndmin=2) # If data is in transposed format, switch it around if transpose: Umat, Ymat = np.transpose(Umat), np.transpose(Ymat) # Make sure the system is a SISO system if Umat.shape[0] != 1 or Ymat.shape[0] != 1: raise ControlMIMONotImplemented # Make sure the number of time points match if Umat.shape[1] != Ymat.shape[1]: raise ControlDimension( "Input and output data are of differnent lengths") n = Umat.shape[1] # If number of desired parameters was not given, set to size of input data if m is None: m = Umat.shape[1] # Make sure there is enough data to compute parameters if m > n: warnings.warn("Not enough data for requested number of parameters") # # Original algorithm (with mapping to standard order) # # RMM note, 24 Dec 2020: This algorithm sets the problem up correctly # until the final column of the UU matrix is created, at which point it # makes some modifications that I don't understand. This version of the # algorithm does not seem to return the actual Markov parameters for a # system. # # # Create the matrix of (shifted) inputs # UU = np.transpose(Umat) # for i in range(1, m-1): # # Shift previous column down and add a zero at the top # newCol = np.vstack((0, np.reshape(UU[0:n-1, i-1], (-1, 1)))) # UU = np.hstack((UU, newCol)) # # # Shift previous column down and add a zero at the top # Ulast = np.vstack((0, np.reshape(UU[0:n-1, m-2], (-1, 1)))) # # # Replace the elements of the last column new values (?) # # Each row gets the sum of the rows above it (?) # for i in range(n-1, 0, -1): # Ulast[i] = np.sum(Ulast[0:i-1]) # UU = np.hstack((UU, Ulast)) # # # Solve for the Markov parameters from Y = H @ UU # # H = [[D], [CB], [CAB], ..., [C A^{m-3} B], [???]] # H = np.linalg.lstsq(UU, np.transpose(Ymat))[0] # # # Markov parameters are in rows => transpose if needed # return H if transpose else np.transpose(H) # # New algorithm - Construct a matrix of control inputs to invert # # This algorithm sets up the following problem and solves it for # the Markov parameters # # [ y(0) ] [ u(0) 0 0 ] [ D ] # [ y(1) ] [ u(1) u(0) 0 ] [ C B ] # [ y(2) ] = [ u(2) u(1) u(0) ] [ C A B ] # [ : ] [ : : : : ] [ : ] # [ y(n-1) ] [ u(n-1) u(n-2) u(n-3) ... u(n-m) ] [ C A^{m-2} B ] # # Note: if the number of Markov parameters (m) is less than the size of # the input/output data (n), then this algorithm assumes C A^{j} B = 0 # for j > m-2. See equation (3) in # # J.-N. Juang, M. Phan, L. G. Horta, and R. W. Longman, Identification # of observer/Kalman filter Markov parameters - Theory and # experiments. Journal of Guidance Control and Dynamics, 16(2), # 320-329, 2012. http://doi.org/10.2514/3.21006 # # Create matrix of (shifted) inputs UU = Umat for i in range(1, m): # Shift previous column down and add a zero at the top new_row = np.hstack((0, UU[i-1, 0:-1])) UU = np.vstack((UU, new_row)) UU = np.transpose(UU) # Invert and solve for Markov parameters YY = np.transpose(Ymat) H, _, _, _ = np.linalg.lstsq(UU, YY, rcond=None) # Return the first m Markov parameters return H if transpose else np.transpose(H) |
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