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GeezerVison is showing github.com — python-control/control/freqplot.py at 0.9.1 · python-control/python-control Expand file tree/ Copy pathfreqplot.pyMore file actions 1444 lines (1239 loc) · 56.2 KB / Copy pathfreqplot.pyFile metadata and controls1444 lines (1239 loc) · 56.2 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 # freqplot.py - frequency domain plots for control systems # # Author: Richard M. Murray # Date: 24 May 09 # # This file contains some standard control system plots: Bode plots, # Nyquist plots and pole-zero diagrams. The code for Nichols charts # is in nichols.py. # # 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. # # $Id$ import math import matplotlib as mpl import matplotlib.pyplot as plt import numpy as np import warnings from .ctrlutil import unwrap from .bdalg import feedback from .margins import stability_margins from .exception import ControlMIMONotImplemented from .statesp import StateSpace from .xferfcn import TransferFunction from . import config __all__ = ['bode_plot', 'nyquist_plot', 'gangof4_plot', 'singular_values_plot', 'bode', 'nyquist', 'gangof4'] # Default values for module parameter variables _freqplot_defaults = { 'freqplot.feature_periphery_decades': 1, 'freqplot.number_of_samples': 1000, 'freqplot.dB': False, # Plot gain in dB 'freqplot.deg': True, # Plot phase in degrees 'freqplot.Hz': False, # Plot frequency in Hertz 'freqplot.grid': True, # Turn on grid for gain and phase 'freqplot.wrap_phase': False, # Wrap the phase plot at a given value # deprecations 'deprecated.bode.dB': 'freqplot.dB', 'deprecated.bode.deg': 'freqplot.deg', 'deprecated.bode.Hz': 'freqplot.Hz', 'deprecated.bode.grid': 'freqplot.grid', 'deprecated.bode.wrap_phase': 'freqplot.wrap_phase', } # # Main plotting functions # # This section of the code contains the functions for generating # frequency domain plots # # # Bode plot # def bode_plot(syslist, omega=None, plot=True, omega_limits=None, omega_num=None, margins=None, method='best', *args, **kwargs): """Bode plot for a system Plots a Bode plot for the system over a (optional) frequency range. Parameters ---------- syslist : linsys List of linear input/output systems (single system is OK) omega : array_like List of frequencies in rad/sec to be used for frequency response dB : bool If True, plot result in dB. Default is false. Hz : bool If True, plot frequency in Hz (omega must be provided in rad/sec). Default value (False) set by config.defaults['freqplot.Hz'] deg : bool If True, plot phase in degrees (else radians). Default value (True) config.defaults['freqplot.deg'] plot : bool If True (default), plot magnitude and phase omega_limits : array_like of two values Limits of the to generate frequency vector. If Hz=True the limits are in Hz otherwise in rad/s. omega_num : int Number of samples to plot. Defaults to config.defaults['freqplot.number_of_samples']. margins : bool If True, plot gain and phase margin. method : method to use in computing margins (see :func:`stability_margins`) *args : :func:`matplotlib.pyplot.plot` positional properties, optional Additional arguments for `matplotlib` plots (color, linestyle, etc) **kwargs : :func:`matplotlib.pyplot.plot` keyword properties, optional Additional keywords (passed to `matplotlib`) Returns ------- mag : ndarray (or list of ndarray if len(syslist) > 1)) magnitude phase : ndarray (or list of ndarray if len(syslist) > 1)) phase in radians omega : ndarray (or list of ndarray if len(syslist) > 1)) frequency in rad/sec Other Parameters ---------------- grid : bool If True, plot grid lines on gain and phase plots. Default is set by `config.defaults['freqplot.grid']`. initial_phase : float Set the reference phase to use for the lowest frequency. If set, the initial phase of the Bode plot will be set to the value closest to the value specified. Units are in either degrees or radians, depending on the `deg` parameter. Default is -180 if wrap_phase is False, 0 if wrap_phase is True. wrap_phase : bool or float If wrap_phase is `False`, then the phase will be unwrapped so that it is continuously increasing or decreasing. If wrap_phase is `True` the phase will be restricted to the range [-180, 180) (or [:math:`-\\pi`, :math:`\\pi`) radians). If `wrap_phase` is specified as a float, the phase will be offset by 360 degrees if it falls below the specified value. Default to `False`, set by config.defaults['freqplot.wrap_phase']. The default values for Bode plot configuration parameters can be reset using the `config.defaults` dictionary, with module name 'bode'. Notes ----- 1. Alternatively, you may use the lower-level methods :meth:`LTI.frequency_response` or ``sys(s)`` or ``sys(z)`` or to generate the frequency response for a single system. 2. If a discrete time model is given, the frequency response is plotted along the upper branch of the unit circle, using the mapping ``z = exp(1j * omega * dt)`` where `omega` ranges from 0 to `pi/dt` and `dt` is the discrete timebase. If timebase not specified (``dt=True``), `dt` is set to 1. Examples -------- >>> sys = ss("1. -2; 3. -4", "5.; 7", "6. 8", "9.") >>> mag, phase, omega = bode(sys) """ # Make a copy of the kwargs dictionary since we will modify it kwargs = dict(kwargs) # Check to see if legacy 'Plot' keyword was used if 'Plot' in kwargs: import warnings warnings.warn("'Plot' keyword is deprecated in bode_plot; use 'plot'", FutureWarning) # Map 'Plot' keyword to 'plot' keyword plot = kwargs.pop('Plot') # Get values for params (and pop from list to allow keyword use in plot) dB = config._get_param( 'freqplot', 'dB', kwargs, _freqplot_defaults, pop=True) deg = config._get_param( 'freqplot', 'deg', kwargs, _freqplot_defaults, pop=True) Hz = config._get_param( 'freqplot', 'Hz', kwargs, _freqplot_defaults, pop=True) grid = config._get_param( 'freqplot', 'grid', kwargs, _freqplot_defaults, pop=True) plot = config._get_param('freqplot', 'plot', plot, True) margins = config._get_param( 'freqplot', 'margins', margins, False) wrap_phase = config._get_param( 'freqplot', 'wrap_phase', kwargs, _freqplot_defaults, pop=True) initial_phase = config._get_param( 'freqplot', 'initial_phase', kwargs, None, pop=True) omega_num = config._get_param('freqplot', 'number_of_samples', omega_num) # If argument was a singleton, turn it into a tuple if not hasattr(syslist, '__iter__'): syslist = (syslist,) omega, omega_range_given = _determine_omega_vector( syslist, omega, omega_limits, omega_num) if plot: # Set up the axes with labels so that multiple calls to # bode_plot will superimpose the data. This was implicit # before matplotlib 2.1, but changed after that (See # https://github.com/matplotlib/matplotlib/issues/9024). # The code below should work on all cases. # Get the current figure if 'sisotool' in kwargs: fig = kwargs['fig'] ax_mag = fig.axes[0] ax_phase = fig.axes[2] sisotool = kwargs['sisotool'] del kwargs['fig'] del kwargs['sisotool'] else: fig = plt.gcf() ax_mag = None ax_phase = None sisotool = False # Get the current axes if they already exist for ax in fig.axes: if ax.get_label() == 'control-bode-magnitude': ax_mag = ax elif ax.get_label() == 'control-bode-phase': ax_phase = ax # If no axes present, create them from scratch if ax_mag is None or ax_phase is None: plt.clf() ax_mag = plt.subplot(211, label='control-bode-magnitude') ax_phase = plt.subplot( 212, label='control-bode-phase', sharex=ax_mag) mags, phases, omegas, nyquistfrqs = [], [], [], [] for sys in syslist: if not sys.issiso(): # TODO: Add MIMO bode plots. raise ControlMIMONotImplemented( "Bode is currently only implemented for SISO systems.") else: omega_sys = np.asarray(omega) if sys.isdtime(strict=True): nyquistfrq = math.pi / sys.dt if not omega_range_given: # limit up to and including nyquist frequency omega_sys = np.hstack(( omega_sys[omega_sys < nyquistfrq], nyquistfrq)) else: nyquistfrq = None mag, phase, omega_sys = sys.frequency_response(omega_sys) mag = np.atleast_1d(mag) phase = np.atleast_1d(phase) # # Post-process the phase to handle initial value and wrapping # if initial_phase is None: # Start phase in the range 0 to -360 w/ initial phase = -180 # If wrap_phase is true, use 0 instead (phase \in (-pi, pi]) initial_phase = -math.pi if wrap_phase is not True else 0 elif isinstance(initial_phase, (int, float)): # Allow the user to override the default calculation if deg: initial_phase = initial_phase/180. * math.pi else: raise ValueError("initial_phase must be a number.") # Shift the phase if needed if abs(phase[0] - initial_phase) > math.pi: phase -= 2*math.pi * \ round((phase[0] - initial_phase) / (2*math.pi)) # Phase wrapping if wrap_phase is False: phase = unwrap(phase) # unwrap the phase elif wrap_phase is True: pass # default calculation OK elif isinstance(wrap_phase, (int, float)): phase = unwrap(phase) # unwrap the phase first if deg: wrap_phase *= math.pi/180. # Shift the phase if it is below the wrap_phase phase += 2*math.pi * np.maximum( 0, np.ceil((wrap_phase - phase)/(2*math.pi))) else: raise ValueError("wrap_phase must be bool or float.") mags.append(mag) phases.append(phase) omegas.append(omega_sys) nyquistfrqs.append(nyquistfrq) # Get the dimensions of the current axis, which we will divide up # TODO: Not current implemented; just use subplot for now if plot: nyquistfrq_plot = None if Hz: omega_plot = omega_sys / (2. * math.pi) if nyquistfrq: nyquistfrq_plot = nyquistfrq / (2. * math.pi) else: omega_plot = omega_sys if nyquistfrq: nyquistfrq_plot = nyquistfrq phase_plot = phase * 180. / math.pi if deg else phase mag_plot = mag if nyquistfrq_plot: # append data for vertical nyquist freq indicator line. # if this extra nyquist lime is is plotted in a single plot # command then line order is preserved when # creating a legend eg. legend(('sys1', 'sys2')) omega_nyq_line = np.array( (np.nan, nyquistfrq_plot, nyquistfrq_plot)) omega_plot = np.hstack((omega_plot, omega_nyq_line)) mag_nyq_line = np.array(( np.nan, 0.7*min(mag_plot), 1.3*max(mag_plot))) mag_plot = np.hstack((mag_plot, mag_nyq_line)) phase_range = max(phase_plot) - min(phase_plot) phase_nyq_line = np.array( (np.nan, min(phase_plot) - 0.2 * phase_range, max(phase_plot) + 0.2 * phase_range)) phase_plot = np.hstack((phase_plot, phase_nyq_line)) # # Magnitude plot # if dB: ax_mag.semilogx(omega_plot, 20 * np.log10(mag_plot), *args, **kwargs) else: ax_mag.loglog(omega_plot, mag_plot, *args, **kwargs) # Add a grid to the plot + labeling ax_mag.grid(grid and not margins, which='both') ax_mag.set_ylabel("Magnitude (dB)" if dB else "Magnitude") # # Phase plot # # Plot the data ax_phase.semilogx(omega_plot, phase_plot, *args, **kwargs) # Show the phase and gain margins in the plot if margins: # Compute stability margins for the system margin = stability_margins(sys, method=method) gm, pm, Wcg, Wcp = (margin[i] for i in (0, 1, 3, 4)) # Figure out sign of the phase at the first gain crossing # (needed if phase_wrap is True) phase_at_cp = phases[0][(np.abs(omegas[0] - Wcp)).argmin()] if phase_at_cp >= 0.: phase_limit = 180. else: phase_limit = -180. if Hz: Wcg, Wcp = Wcg/(2*math.pi), Wcp/(2*math.pi) # Draw lines at gain and phase limits ax_mag.axhline(y=0 if dB else 1, color='k', linestyle=':', zorder=-20) ax_phase.axhline(y=phase_limit if deg else math.radians(phase_limit), color='k', linestyle=':', zorder=-20) mag_ylim = ax_mag.get_ylim() phase_ylim = ax_phase.get_ylim() # Annotate the phase margin (if it exists) if pm != float('inf') and Wcp != float('nan'): if dB: ax_mag.semilogx( [Wcp, Wcp], [0., -1e5], color='k', linestyle=':', zorder=-20) else: ax_mag.loglog( [Wcp, Wcp], [1., 1e-8], color='k', linestyle=':', zorder=-20) if deg: ax_phase.semilogx( [Wcp, Wcp], [1e5, phase_limit + pm], color='k', linestyle=':', zorder=-20) ax_phase.semilogx( [Wcp, Wcp], [phase_limit + pm, phase_limit], color='k', zorder=-20) else: ax_phase.semilogx( [Wcp, Wcp], [1e5, math.radians(phase_limit) + math.radians(pm)], color='k', linestyle=':', zorder=-20) ax_phase.semilogx( [Wcp, Wcp], [math.radians(phase_limit) + math.radians(pm), math.radians(phase_limit)], color='k', zorder=-20) # Annotate the gain margin (if it exists) if gm != float('inf') and Wcg != float('nan'): if dB: ax_mag.semilogx( [Wcg, Wcg], [-20.*np.log10(gm), -1e5], color='k', linestyle=':', zorder=-20) ax_mag.semilogx( [Wcg, Wcg], [0, -20*np.log10(gm)], color='k', zorder=-20) else: ax_mag.loglog( [Wcg, Wcg], [1./gm, 1e-8], color='k', linestyle=':', zorder=-20) ax_mag.loglog( [Wcg, Wcg], [1., 1./gm], color='k', zorder=-20) if deg: ax_phase.semilogx( [Wcg, Wcg], [0, phase_limit], color='k', linestyle=':', zorder=-20) else: ax_phase.semilogx( [Wcg, Wcg], [0, math.radians(phase_limit)], color='k', linestyle=':', zorder=-20) ax_mag.set_ylim(mag_ylim) ax_phase.set_ylim(phase_ylim) if sisotool: ax_mag.text( 0.04, 0.06, 'G.M.: %.2f %s\nFreq: %.2f %s' % (20*np.log10(gm) if dB else gm, 'dB ' if dB else '', Wcg, 'Hz' if Hz else 'rad/s'), horizontalalignment='left', verticalalignment='bottom', transform=ax_mag.transAxes, fontsize=8 if int(mpl.__version__[0]) == 1 else 6) ax_phase.text( 0.04, 0.06, 'P.M.: %.2f %s\nFreq: %.2f %s' % (pm if deg else math.radians(pm), 'deg' if deg else 'rad', Wcp, 'Hz' if Hz else 'rad/s'), horizontalalignment='left', verticalalignment='bottom', transform=ax_phase.transAxes, fontsize=8 if int(mpl.__version__[0]) == 1 else 6) else: plt.suptitle( "Gm = %.2f %s(at %.2f %s), " "Pm = %.2f %s (at %.2f %s)" % (20*np.log10(gm) if dB else gm, 'dB ' if dB else '', Wcg, 'Hz' if Hz else 'rad/s', pm if deg else math.radians(pm), 'deg' if deg else 'rad', Wcp, 'Hz' if Hz else 'rad/s')) # Add a grid to the plot + labeling ax_phase.set_ylabel("Phase (deg)" if deg else "Phase (rad)") def gen_zero_centered_series(val_min, val_max, period): v1 = np.ceil(val_min / period - 0.2) v2 = np.floor(val_max / period + 0.2) return np.arange(v1, v2 + 1) * period if deg: ylim = ax_phase.get_ylim() ax_phase.set_yticks(gen_zero_centered_series( ylim[0], ylim[1], 45.)) ax_phase.set_yticks(gen_zero_centered_series( ylim[0], ylim[1], 15.), minor=True) else: ylim = ax_phase.get_ylim() ax_phase.set_yticks(gen_zero_centered_series( ylim[0], ylim[1], math.pi / 4.)) ax_phase.set_yticks(gen_zero_centered_series( ylim[0], ylim[1], math.pi / 12.), minor=True) ax_phase.grid(grid and not margins, which='both') # ax_mag.grid(which='minor', alpha=0.3) # ax_mag.grid(which='major', alpha=0.9) # ax_phase.grid(which='minor', alpha=0.3) # ax_phase.grid(which='major', alpha=0.9) # Label the frequency axis ax_phase.set_xlabel("Frequency (Hz)" if Hz else "Frequency (rad/sec)") if len(syslist) == 1: return mags[0], phases[0], omegas[0] else: return mags, phases, omegas # # Nyquist plot # # Default values for module parameter variables _nyquist_defaults = { 'nyquist.mirror_style': '--', 'nyquist.arrows': 2, 'nyquist.arrow_size': 8, 'nyquist.indent_radius': 1e-1, 'nyquist.indent_direction': 'right', } def nyquist_plot(syslist, omega=None, plot=True, omega_limits=None, omega_num=None, label_freq=0, color=None, return_contour=False, warn_nyquist=True, *args, **kwargs): """Nyquist plot for a system Plots a Nyquist plot for the system over a (optional) frequency range. The curve is computed by evaluating the Nyqist segment along the positive imaginary axis, with a mirror image generated to reflect the negative imaginary axis. Poles on or near the imaginary axis are avoided using a small indentation. The portion of the Nyquist contour at infinity is not explicitly computed (since it maps to a constant value for any system with a proper transfer function). Parameters ---------- syslist : list of LTI List of linear input/output systems (single system is OK). Nyquist curves for each system are plotted on the same graph. plot : boolean If True, plot magnitude omega : array_like Set of frequencies to be evaluated, in rad/sec. omega_limits : array_like of two values Limits to the range of frequencies. Ignored if omega is provided, and auto-generated if omitted. omega_num : int Number of frequency samples to plot. Defaults to config.defaults['freqplot.number_of_samples']. color : string Used to specify the color of the line and arrowhead. mirror_style : string or False Linestyle for mirror image of the Nyquist curve. If `False` then omit completely. Default linestyle ('--') is determined by config.defaults['nyquist.mirror_style']. return_contour : bool If 'True', return the contour used to evaluate the Nyquist plot. label_freq : int Label every nth frequency on the plot. If not specified, no labels are generated. arrows : int or 1D/2D array of floats Specify the number of arrows to plot on the Nyquist curve. If an integer is passed. that number of equally spaced arrows will be plotted on each of the primary segment and the mirror image. If a 1D array is passed, it should consist of a sorted list of floats between 0 and 1, indicating the location along the curve to plot an arrow. If a 2D array is passed, the first row will be used to specify arrow locations for the primary curve and the second row will be used for the mirror image. arrow_size : float Arrowhead width and length (in display coordinates). Default value is 8 and can be set using config.defaults['nyquist.arrow_size']. arrow_style : matplotlib.patches.ArrowStyle Define style used for Nyquist curve arrows (overrides `arrow_size`). indent_radius : float Amount to indent the Nyquist contour around poles that are at or near the imaginary axis. indent_direction : str For poles on the imaginary axis, set the direction of indentation to be 'right' (default), 'left', or 'none'. warn_nyquist : bool, optional If set to 'False', turn off warnings about frequencies above Nyquist. *args : :func:`matplotlib.pyplot.plot` positional properties, optional Additional arguments for `matplotlib` plots (color, linestyle, etc) **kwargs : :func:`matplotlib.pyplot.plot` keyword properties, optional Additional keywords (passed to `matplotlib`) Returns ------- count : int (or list of int if len(syslist) > 1) Number of encirclements of the point -1 by the Nyquist curve. If multiple systems are given, an array of counts is returned. contour : ndarray (or list of ndarray if len(syslist) > 1)), optional The contour used to create the primary Nyquist curve segment. To obtain the Nyquist curve values, evaluate system(s) along contour. Notes ----- 1. If a discrete time model is given, the frequency response is computed along the upper branch of the unit circle, using the mapping ``z = exp(1j * omega * dt)`` where `omega` ranges from 0 to `pi/dt` and `dt` is the discrete timebase. If timebase not specified (``dt=True``), `dt` is set to 1. 2. If a continuous-time system contains poles on or near the imaginary axis, a small indentation will be used to avoid the pole. The radius of the indentation is given by `indent_radius` and it is taken to the right of stable poles and the left of unstable poles. If a pole is exactly on the imaginary axis, the `indent_direction` parameter can be used to set the direction of indentation. Setting `indent_direction` to `none` will turn off indentation. If `return_contour` is True, the exact contour used for evaluation is returned. Examples -------- >>> sys = ss([[1, -2], [3, -4]], [[5], [7]], [[6, 8]], [[9]]) >>> count = nyquist_plot(sys) """ # Check to see if legacy 'Plot' keyword was used if 'Plot' in kwargs: warnings.warn("'Plot' keyword is deprecated in nyquist_plot; " "use 'plot'", FutureWarning) # Map 'Plot' keyword to 'plot' keyword plot = kwargs.pop('Plot') # Check to see if legacy 'labelFreq' keyword was used if 'labelFreq' in kwargs: warnings.warn("'labelFreq' keyword is deprecated in nyquist_plot; " "use 'label_freq'", FutureWarning) # Map 'labelFreq' keyword to 'label_freq' keyword label_freq = kwargs.pop('labelFreq') # Check to see if legacy 'arrow_width' or 'arrow_length' were used if 'arrow_width' in kwargs or 'arrow_length' in kwargs: warnings.warn( "'arrow_width' and 'arrow_length' keywords are deprecated in " "nyquist_plot; use `arrow_size` instead", FutureWarning) kwargs['arrow_size'] = \ (kwargs.get('arrow_width', 0) + kwargs.get('arrow_length', 0)) / 2 kwargs.pop('arrow_width', False) kwargs.pop('arrow_length', False) # Get values for params (and pop from list to allow keyword use in plot) omega_num = config._get_param('freqplot', 'number_of_samples', omega_num) mirror_style = config._get_param( 'nyquist', 'mirror_style', kwargs, _nyquist_defaults, pop=True) arrows = config._get_param( 'nyquist', 'arrows', kwargs, _nyquist_defaults, pop=True) arrow_size = config._get_param( 'nyquist', 'arrow_size', kwargs, _nyquist_defaults, pop=True) arrow_style = config._get_param('nyquist', 'arrow_style', kwargs, None) indent_radius = config._get_param( 'nyquist', 'indent_radius', kwargs, _nyquist_defaults, pop=True) indent_direction = config._get_param( 'nyquist', 'indent_direction', kwargs, _nyquist_defaults, pop=True) # If argument was a singleton, turn it into a list if not hasattr(syslist, '__iter__'): syslist = (syslist,) omega, omega_range_given = _determine_omega_vector( syslist, omega, omega_limits, omega_num) if not omega_range_given: # Start contour at zero frequency omega[0] = 0. # Go through each system and keep track of the results counts, contours = [], [] for sys in syslist: if not sys.issiso(): # TODO: Add MIMO nyquist plots. raise ControlMIMONotImplemented( "Nyquist plot currently only supports SISO systems.") # Figure out the frequency range omega_sys = np.asarray(omega) # Determine the contour used to evaluate the Nyquist curve if sys.isdtime(strict=True): # Transform frequencies in for discrete-time systems nyquistfrq = math.pi / sys.dt if not omega_range_given: # limit up to and including nyquist frequency omega_sys = np.hstack(( omega_sys[omega_sys < nyquistfrq], nyquistfrq)) # Issue a warning if we are sampling above Nyquist if np.any(omega_sys * sys.dt > np.pi) and warn_nyquist: warnings.warn("evaluation above Nyquist frequency") # do indentations in s-plane where it is more convenient splane_contour = 1j * omega_sys # Bend the contour around any poles on/near the imaginary axis # TODO: smarter indent radius that depends on dcgain of system # and timebase of discrete system. if isinstance(sys, (StateSpace, TransferFunction)) \ and indent_direction != 'none': if sys.isctime(): splane_poles = sys.pole() else: # map z-plane poles to s-plane, ignoring any at the origin # because we don't need to indent for them zplane_poles = sys.pole() zplane_poles = zplane_poles[~np.isclose(abs(zplane_poles), 0.)] splane_poles = np.log(zplane_poles)/sys.dt if splane_contour[1].imag > indent_radius \ and np.any(np.isclose(abs(splane_poles), 0)) \ and not omega_range_given: # add some points for quarter circle around poles at origin splane_contour = np.concatenate( (1j * np.linspace(0., indent_radius, 50), splane_contour[1:])) for i, s in enumerate(splane_contour): # Find the nearest pole p = splane_poles[(np.abs(splane_poles - s)).argmin()] # See if we need to indent around it if abs(s - p) < indent_radius: if p.real < 0 or (np.isclose(p.real, 0) \ and indent_direction == 'right'): # Indent to the right splane_contour[i] += \ np.sqrt(indent_radius ** 2 - (s-p).imag ** 2) elif p.real > 0 or (np.isclose(p.real, 0) \ and indent_direction == 'left'): # Indent to the left splane_contour[i] -= \ np.sqrt(indent_radius ** 2 - (s-p).imag ** 2) else: ValueError("unknown value for indent_direction") # change contour to z-plane if necessary if sys.isctime(): contour = splane_contour else: contour = np.exp(splane_contour * sys.dt) # Compute the primary curve resp = sys(contour) # Compute CW encirclements of -1 by integrating the (unwrapped) angle phase = -unwrap(np.angle(resp + 1)) count = int(np.round(np.sum(np.diff(phase)) / np.pi, 0)) counts.append(count) contours.append(contour) if plot: # Parse the arrows keyword if isinstance(arrows, int): N = arrows # Space arrows out, starting midway along each "region" arrow_pos = np.linspace(0.5/N, 1 + 0.5/N, N, endpoint=False) elif isinstance(arrows, (list, np.ndarray)): arrow_pos = np.sort(np.atleast_1d(arrows)) elif not arrows: arrow_pos = [] else: raise ValueError("unknown or unsupported arrow location") # Set the arrow style if arrow_style is None: arrow_style = mpl.patches.ArrowStyle( 'simple', head_width=arrow_size, head_length=arrow_size) # Save the components of the response x, y = resp.real, resp.imag # Plot the primary curve p = plt.plot(x, y, '-', color=color, *args, **kwargs) c = p[0].get_color() ax = plt.gca() _add_arrows_to_line2D( ax, p[0], arrow_pos, arrowstyle=arrow_style, dir=1) # Plot the mirror image if mirror_style is not False: p = plt.plot(x, -y, mirror_style, color=c, *args, **kwargs) _add_arrows_to_line2D( ax, p[0], arrow_pos, arrowstyle=arrow_style, dir=-1) # Mark the -1 point plt.plot([-1], [0], 'r+') # Label the frequencies of the points if label_freq: ind = slice(None, None, label_freq) for xpt, ypt, omegapt in zip(x[ind], y[ind], omega_sys[ind]): # Convert to Hz f = omegapt / (2 * np.pi) # Factor out multiples of 1000 and limit the # result to the range [-8, 8]. pow1000 = max(min(get_pow1000(f), 8), -8) # Get the SI prefix. prefix = gen_prefix(pow1000) # Apply the text. (Use a space before the text to # prevent overlap with the data.) # # np.round() is used because 0.99... appears # instead of 1.0, and this would otherwise be # truncated to 0. plt.text(xpt, ypt, ' ' + str(int(np.round(f / 1000 ** pow1000, 0))) + ' ' + prefix + 'Hz') if plot: ax = plt.gca() ax.set_xlabel("Real axis") ax.set_ylabel("Imaginary axis") ax.grid(color="lightgray") # "Squeeze" the results if len(syslist) == 1: counts, contours = counts[0], contours[0] # Return counts and (optionally) the contour we used return (counts, contours) if return_contour else counts # Internal function to add arrows to a curve def _add_arrows_to_line2D( axes, line, arrow_locs=[0.2, 0.4, 0.6, 0.8], arrowstyle='-|>', arrowsize=1, dir=1, transform=None): """ Add arrows to a matplotlib.lines.Line2D at selected locations. Parameters: ----------- axes: Axes object as returned by axes command (or gca) line: Line2D object as returned by plot command arrow_locs: list of locations where to insert arrows, % of total length arrowstyle: style of the arrow arrowsize: size of the arrow transform: a matplotlib transform instance, default to data coordinates Returns: -------- arrows: list of arrows Based on https://stackoverflow.com/questions/26911898/ """ if not isinstance(line, mpl.lines.Line2D): raise ValueError("expected a matplotlib.lines.Line2D object") x, y = line.get_xdata(), line.get_ydata() arrow_kw = { "arrowstyle": arrowstyle, } color = line.get_color() use_multicolor_lines = isinstance(color, np.ndarray) if use_multicolor_lines: raise NotImplementedError("multicolor lines not supported") else: arrow_kw['color'] = color linewidth = line.get_linewidth() if isinstance(linewidth, np.ndarray): raise NotImplementedError("multiwidth lines not supported") else: arrow_kw['linewidth'] = linewidth if transform is None: transform = axes.transData # Compute the arc length along the curve s = np.cumsum(np.sqrt(np.diff(x) ** 2 + np.diff(y) ** 2)) arrows = [] for loc in arrow_locs: n = np.searchsorted(s, s[-1] * loc) # Figure out what direction to paint the arrow if dir == 1: arrow_tail = (x[n], y[n]) arrow_head = (np.mean(x[n:n + 2]), np.mean(y[n:n + 2])) elif dir == -1: # Orient the arrow in the other direction on the segment arrow_tail = (x[n + 1], y[n + 1]) arrow_head = (np.mean(x[n:n + 2]), np.mean(y[n:n + 2])) else: raise ValueError("unknown value for keyword 'dir'") p = mpl.patches.FancyArrowPatch( arrow_tail, arrow_head, transform=transform, lw=0, **arrow_kw) axes.add_patch(p) arrows.append(p) return arrows # # Gang of Four plot # # TODO: think about how (and whether) to handle lists of systems def gangof4_plot(P, C, omega=None, **kwargs): """Plot the "Gang of 4" transfer functions for a system Generates a 2x2 plot showing the "Gang of 4" sensitivity functions [T, PS; CS, S] Parameters ---------- P, C : LTI Linear input/output systems (process and control) omega : array Range of frequencies (list or bounds) in rad/sec **kwargs : :func:`matplotlib.pyplot.plot` keyword properties, optional Additional keywords (passed to `matplotlib`) Returns ------- None """ if not P.issiso() or not C.issiso(): # TODO: Add MIMO go4 plots. raise ControlMIMONotImplemented( "Gang of four is currently only implemented for SISO systems.") # Get the default parameter values dB = config._get_param( 'freqplot', 'dB', kwargs, _freqplot_defaults, pop=True) Hz = config._get_param( 'freqplot', 'Hz', kwargs, _freqplot_defaults, pop=True) grid = config._get_param( 'freqplot', 'grid', kwargs, _freqplot_defaults, pop=True) # Compute the senstivity functions L = P * C S = feedback(1, L) T = L * S # Select a default range if none is provided # TODO: This needs to be made more intelligent if omega is None: omega = _default_frequency_range((P, C, S)) # Set up the axes with labels so that multiple calls to # gangof4_plot will superimpose the data. See details in bode_plot. plot_axes = {'t': None, 's': None, 'ps': None, 'cs': None} for ax in plt.gcf().axes: label = ax.get_label() if label.startswith('control-gangof4-'): key = label[len('control-gangof4-'):] if key not in plot_axes: raise RuntimeError( "unknown gangof4 axis type '{}'".format(label)) plot_axes[key] = ax # if any of the axes are missing, start from scratch if any((ax is None for ax in plot_axes.values())): plt.clf() plot_axes = {'s': plt.subplot(221, label='control-gangof4-s'), 'ps': plt.subplot(222, label='control-gangof4-ps'), 'cs': plt.subplot(223, label='control-gangof4-cs'), 't': plt.subplot(224, label='control-gangof4-t')} # # Plot the four sensitivity functions # omega_plot = omega / (2. * math.pi) if Hz else omega # TODO: Need to add in the mag = 1 lines mag_tmp, phase_tmp, omega = S.frequency_response(omega) mag = np.squeeze(mag_tmp) if dB: plot_axes['s'].semilogx(omega_plot, 20 * np.log10(mag), **kwargs) else: |
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