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GeezerVison is showing github.com — python-control/control/statefbk.py at main · ControlCorp/python-control Expand file tree/ Copy pathstatefbk.pyMore file actions 1140 lines (945 loc) · 40.2 KB / Copy pathstatefbk.pyFile metadata and controls1140 lines (945 loc) · 40.2 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 # statefbk.py - tools for state feedback control # # Author: Richard M. Murray, Roberto Bucher # Date: 31 May 2010 # # This file contains routines for designing state space controllers # # 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 scipy as sp import warnings from . import statesp from .mateqn import care, dare, _check_shape from .statesp import StateSpace, _ssmatrix, _convert_to_statespace from .lti import LTI from .namedio import isdtime, isctime, _process_indices, _process_labels from .iosys import InputOutputSystem, NonlinearIOSystem, LinearIOSystem, \ interconnect, ss from .exception import ControlSlycot, ControlArgument, ControlDimension, \ ControlNotImplemented from .config import _process_legacy_keyword # Make sure we have access to the right slycot routines try: from slycot import sb03md57 # wrap without the deprecation warning def sb03md(n, C, A, U, dico, job='X',fact='N',trana='N',ldwork=None): ret = sb03md57(A, U, C, dico, job, fact, trana, ldwork) return ret[2:] except ImportError: try: from slycot import sb03md except ImportError: sb03md = None try: from slycot import sb03od except ImportError: sb03od = None __all__ = ['ctrb', 'obsv', 'gram', 'place', 'place_varga', 'lqr', 'dlqr', 'acker', 'create_statefbk_iosystem'] # Pole placement def place(A, B, p): """Place closed loop eigenvalues K = place(A, B, p) Parameters ---------- A : 2D array_like Dynamics matrix B : 2D array_like Input matrix p : 1D array_like Desired eigenvalue locations Returns ------- K : 2D array (or matrix) Gain such that A - B K has eigenvalues given in p Notes ----- Algorithm This is a wrapper function for :func:`scipy.signal.place_poles`, which implements the Tits and Yang algorithm [1]_. It will handle SISO, MISO, and MIMO systems. If you want more control over the algorithm, use :func:`scipy.signal.place_poles` directly. Limitations The algorithm will not place poles at the same location more than rank(B) times. The return type for 2D arrays depends on the default class set for state space operations. See :func:`~control.use_numpy_matrix`. References ---------- .. [1] A.L. Tits and Y. Yang, "Globally convergent algorithms for robust pole assignment by state feedback, IEEE Transactions on Automatic Control, Vol. 41, pp. 1432-1452, 1996. Examples -------- >>> A = [[-1, -1], [0, 1]] >>> B = [[0], [1]] >>> K = ct.place(A, B, [-2, -5]) See Also -------- place_varga, acker """ from scipy.signal import place_poles # Convert the system inputs to NumPy arrays A_mat = np.array(A) B_mat = np.array(B) if (A_mat.shape[0] != A_mat.shape[1]): raise ControlDimension("A must be a square matrix") if (A_mat.shape[0] != B_mat.shape[0]): err_str = "The number of rows of A must equal the number of rows in B" raise ControlDimension(err_str) # Convert desired poles to numpy array placed_eigs = np.atleast_1d(np.squeeze(np.asarray(p))) result = place_poles(A_mat, B_mat, placed_eigs, method='YT') K = result.gain_matrix return _ssmatrix(K) def place_varga(A, B, p, dtime=False, alpha=None): """Place closed loop eigenvalues K = place_varga(A, B, p, dtime=False, alpha=None) Required Parameters ---------- A : 2D array_like Dynamics matrix B : 2D array_like Input matrix p : 1D array_like Desired eigenvalue locations Optional Parameters --------------- dtime : bool False for continuous time pole placement or True for discrete time. The default is dtime=False. alpha : double scalar If `dtime` is false then place_varga will leave the eigenvalues with real part less than alpha untouched. If `dtime` is true then place_varga will leave eigenvalues with modulus less than alpha untouched. By default (alpha=None), place_varga computes alpha such that all poles will be placed. Returns ------- K : 2D array (or matrix) Gain such that A - B K has eigenvalues given in p. Algorithm --------- This function is a wrapper for the slycot function sb01bd, which implements the pole placement algorithm of Varga [1]. In contrast to the algorithm used by place(), the Varga algorithm can place multiple poles at the same location. The placement, however, may not be as robust. [1] Varga A. "A Schur method for pole assignment." IEEE Trans. Automatic Control, Vol. AC-26, pp. 517-519, 1981. Notes ----- The return type for 2D arrays depends on the default class set for state space operations. See :func:`~control.use_numpy_matrix`. Examples -------- >>> A = [[-1, -1], [0, 1]] >>> B = [[0], [1]] >>> K = place_varga(A, B, [-2, -5]) See Also: -------- place, acker """ # Make sure that SLICOT is installed try: from slycot import sb01bd except ImportError: raise ControlSlycot("can't find slycot module 'sb01bd'") # Convert the system inputs to NumPy arrays A_mat = np.array(A) B_mat = np.array(B) if (A_mat.shape[0] != A_mat.shape[1] or A_mat.shape[0] != B_mat.shape[0]): raise ControlDimension("matrix dimensions are incorrect") # Compute the system eigenvalues and convert poles to numpy array system_eigs = np.linalg.eig(A_mat)[0] placed_eigs = np.atleast_1d(np.squeeze(np.asarray(p))) # Need a character parameter for SB01BD if dtime: DICO = 'D' else: DICO = 'C' if alpha is None: # SB01BD ignores eigenvalues with real part less than alpha # (if DICO='C') or with modulus less than alpha # (if DICO = 'D'). if dtime: # For discrete time, slycot only cares about modulus, so just make # alpha the smallest it can be. alpha = 0.0 else: # Choosing alpha=min_eig is insufficient and can lead to an # error or not having all the eigenvalues placed that we wanted. # Evidently, what python thinks are the eigs is not precisely # the same as what slicot thinks are the eigs. So we need some # numerical breathing room. The following is pretty heuristic, # but does the trick alpha = -2*abs(min(system_eigs.real)) elif dtime and alpha < 0.0: raise ValueError("Discrete time systems require alpha > 0") # Call SLICOT routine to place the eigenvalues A_z, w, nfp, nap, nup, F, Z = \ sb01bd(B_mat.shape[0], B_mat.shape[1], len(placed_eigs), alpha, A_mat, B_mat, placed_eigs, DICO) # Return the gain matrix, with MATLAB gain convention return _ssmatrix(-F) # Contributed by Roberto Bucher <[email protected]> def acker(A, B, poles): """Pole placement using Ackermann method Call: K = acker(A, B, poles) Parameters ---------- A, B : 2D array_like State and input matrix of the system poles : 1D array_like Desired eigenvalue locations Returns ------- K : 2D array (or matrix) Gains such that A - B K has given eigenvalues Notes ----- The return type for 2D arrays depends on the default class set for state space operations. See :func:`~control.use_numpy_matrix`. """ # Convert the inputs to matrices a = _ssmatrix(A) b = _ssmatrix(B) # Make sure the system is controllable ct = ctrb(A, B) if np.linalg.matrix_rank(ct) != a.shape[0]: raise ValueError("System not reachable; pole placement invalid") # Compute the desired characteristic polynomial p = np.real(np.poly(poles)) # Place the poles using Ackermann's method # TODO: compute pmat using Horner's method (O(n) instead of O(n^2)) n = np.size(p) pmat = p[n-1] * np.linalg.matrix_power(a, 0) for i in np.arange(1, n): pmat = pmat + p[n-i-1] * np.linalg.matrix_power(a, i) K = np.linalg.solve(ct, pmat) K = K[-1][:] # Extract the last row return _ssmatrix(K) def lqr(*args, **kwargs): """lqr(A, B, Q, R[, N]) Linear quadratic regulator design The lqr() function computes the optimal state feedback controller u = -K x that minimizes the quadratic cost .. math:: J = \\int_0^\\infty (x' Q x + u' R u + 2 x' N u) dt The function can be called with either 3, 4, or 5 arguments: * ``K, S, E = lqr(sys, Q, R)`` * ``K, S, E = lqr(sys, Q, R, N)`` * ``K, S, E = lqr(A, B, Q, R)`` * ``K, S, E = lqr(A, B, Q, R, N)`` where `sys` is an `LTI` object, and `A`, `B`, `Q`, `R`, and `N` are 2D arrays or matrices of appropriate dimension. Parameters ---------- A, B : 2D array_like Dynamics and input matrices sys : LTI StateSpace system Linear system Q, R : 2D array State and input weight matrices N : 2D array, optional Cross weight matrix integral_action : ndarray, optional If this keyword is specified, the controller includes integral action in addition to state feedback. The value of the `integral_action` keyword should be an ndarray that will be multiplied by the current state to generate the error for the internal integrator states of the control law. The number of outputs that are to be integrated must match the number of additional rows and columns in the `Q` matrix. 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 ------- K : 2D array (or matrix) State feedback gains S : 2D array (or matrix) Solution to Riccati equation E : 1D array Eigenvalues of the closed loop system See Also -------- lqe, dlqr, dlqe Notes ----- 1. If the first argument is an LTI object, then this object will be used to define the dynamics and input matrices. Furthermore, if the LTI object corresponds to a discrete time system, the ``dlqr()`` function will be called. 2. The return type for 2D arrays depends on the default class set for state space operations. See :func:`~control.use_numpy_matrix`. Examples -------- >>> K, S, E = lqr(sys, Q, R, [N]) # doctest: +SKIP >>> K, S, E = lqr(A, B, Q, R, [N]) # doctest: +SKIP """ # # Process the arguments and figure out what inputs we received # # If we were passed a discrete time system as the first arg, use dlqr() if isinstance(args[0], LTI) and isdtime(args[0], strict=True): # Call dlqr return dlqr(*args, **kwargs) # Get the system description if (len(args) < 3): raise ControlArgument("not enough input arguments") # If we were passed a state space system, use that to get system matrices if isinstance(args[0], StateSpace): A = np.array(args[0].A, ndmin=2, dtype=float) B = np.array(args[0].B, ndmin=2, dtype=float) index = 1 elif isinstance(args[0], LTI): # Don't allow other types of LTI systems raise ControlArgument("LTI system must be in state space form") else: # Arguments should be A and B matrices A = np.array(args[0], ndmin=2, dtype=float) B = np.array(args[1], ndmin=2, dtype=float) index = 2 # Get the weighting matrices (converting to matrices, if needed) Q = np.array(args[index], ndmin=2, dtype=float) R = np.array(args[index+1], ndmin=2, dtype=float) if (len(args) > index + 2): N = np.array(args[index+2], ndmin=2, dtype=float) else: N = None # # Process keywords # # Get the method to use (if specified as a keyword) method = kwargs.pop('method', None) # See if we should augment the controller with integral feedback integral_action = kwargs.pop('integral_action', None) if integral_action is not None: # Figure out the size of the system nstates = A.shape[0] ninputs = B.shape[1] # Make sure that the integral action argument is the right type if not isinstance(integral_action, np.ndarray): raise ControlArgument("Integral action must pass an array") elif integral_action.shape[1] != nstates: raise ControlArgument( "Integral gain size must match system state size") # Process the states to be integrated nintegrators = integral_action.shape[0] C = integral_action # Augment the system with integrators A = np.block([ [A, np.zeros((nstates, nintegrators))], [C, np.zeros((nintegrators, nintegrators))] ]) B = np.vstack([B, np.zeros((nintegrators, ninputs))]) if kwargs: raise TypeError("unrecognized keywords: ", str(kwargs)) # Compute the result (dimension and symmetry checking done in care()) X, L, G = care(A, B, Q, R, N, None, method=method, S_s="N") return G, X, L def dlqr(*args, **kwargs): """dlqr(A, B, Q, R[, N]) Discrete-time linear quadratic regulator design The dlqr() function computes the optimal state feedback controller u[n] = - K x[n] that minimizes the quadratic cost .. math:: J = \\sum_0^\\infty (x[n]' Q x[n] + u[n]' R u[n] + 2 x[n]' N u[n]) The function can be called with either 3, 4, or 5 arguments: * ``dlqr(dsys, Q, R)`` * ``dlqr(dsys, Q, R, N)`` * ``dlqr(A, B, Q, R)`` * ``dlqr(A, B, Q, R, N)`` where `dsys` is a discrete-time :class:`StateSpace` system, and `A`, `B`, `Q`, `R`, and `N` are 2d arrays of appropriate dimension (`dsys.dt` must not be 0.) Parameters ---------- A, B : 2D array Dynamics and input matrices dsys : LTI :class:`StateSpace` Discrete-time linear system Q, R : 2D array State and input weight matrices N : 2D array, optional Cross weight matrix integral_action : ndarray, optional If this keyword is specified, the controller includes integral action in addition to state feedback. The value of the `integral_action` keyword should be an ndarray that will be multiplied by the current state to generate the error for the internal integrator states of the control law. The number of outputs that are to be integrated must match the number of additional rows and columns in the `Q` matrix. 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 ------- K : 2D array (or matrix) State feedback gains S : 2D array (or matrix) Solution to Riccati equation E : 1D array Eigenvalues of the closed loop system See Also -------- lqr, lqe, dlqe Notes ----- The return type for 2D arrays depends on the default class set for state space operations. See :func:`~control.use_numpy_matrix`. Examples -------- >>> K, S, E = dlqr(dsys, Q, R, [N]) # doctest: +SKIP >>> K, S, E = dlqr(A, B, Q, R, [N]) # doctest: +SKIP """ # # Process the arguments and figure out what inputs we received # # Get the system description if (len(args) < 3): raise ControlArgument("not enough input arguments") # If we were passed a continus time system as the first arg, raise error if isinstance(args[0], LTI) and isctime(args[0], strict=True): raise ControlArgument("dsys must be discrete time (dt != 0)") # If we were passed a state space system, use that to get system matrices if isinstance(args[0], StateSpace): A = np.array(args[0].A, ndmin=2, dtype=float) B = np.array(args[0].B, ndmin=2, dtype=float) index = 1 elif isinstance(args[0], LTI): # Don't allow other types of LTI systems raise ControlArgument("LTI system must be in state space form") else: # Arguments should be A and B matrices A = np.array(args[0], ndmin=2, dtype=float) B = np.array(args[1], ndmin=2, dtype=float) index = 2 # Get the weighting matrices (converting to matrices, if needed) Q = np.array(args[index], ndmin=2, dtype=float) R = np.array(args[index+1], ndmin=2, dtype=float) if (len(args) > index + 2): N = np.array(args[index+2], ndmin=2, dtype=float) else: N = np.zeros((Q.shape[0], R.shape[1])) # # Process keywords # # Get the method to use (if specified as a keyword) method = kwargs.pop('method', None) # See if we should augment the controller with integral feedback integral_action = kwargs.pop('integral_action', None) if integral_action is not None: # Figure out the size of the system nstates = A.shape[0] ninputs = B.shape[1] if not isinstance(integral_action, np.ndarray): raise ControlArgument("Integral action must pass an array") elif integral_action.shape[1] != nstates: raise ControlArgument( "Integral gain size must match system state size") else: nintegrators = integral_action.shape[0] C = integral_action # Augment the system with integrators A = np.block([ [A, np.zeros((nstates, nintegrators))], [C, np.eye(nintegrators)] ]) B = np.vstack([B, np.zeros((nintegrators, ninputs))]) if kwargs: raise TypeError("unrecognized keywords: ", str(kwargs)) # Compute the result (dimension and symmetry checking done in dare()) S, E, K = dare(A, B, Q, R, N, method=method, S_s="N") return _ssmatrix(K), _ssmatrix(S), E # Function to create an I/O sytems representing a state feedback controller def create_statefbk_iosystem( sys, gain, integral_action=None, estimator=None, controller_type=None, xd_labels=None, ud_labels=None, gainsched_indices=None, gainsched_method='linear', control_indices=None, state_indices=None, name=None, inputs=None, outputs=None, states=None, **kwargs): """Create an I/O system using a (full) state feedback controller This function creates an input/output system that implements a state feedback controller of the form u = ud - K_p (x - xd) - K_i integral(C x - C x_d) It can be called in the form ctrl, clsys = ct.create_statefbk_iosystem(sys, K) where `sys` is the process dynamics and `K` is the state (+ integral) feedback gain (eg, from LQR). The function returns the controller `ctrl` and the closed loop systems `clsys`, both as I/O systems. A gain scheduled controller can also be created, by passing a list of gains and a corresponding list of values of a set of scheduling variables. In this case, the controller has the form u = ud - K_p(mu) (x - xd) - K_i(mu) integral(C x - C x_d) where mu represents the scheduling variable. Parameters ---------- sys : InputOutputSystem The I/O system that represents the process dynamics. If no estimator is given, the output of this system should represent the full state. gain : ndarray or tuple If an array is given, it represents the state feedback gain (K). This matrix defines the gains to be applied to the system. If `integral_action` is None, then the dimensions of this array should be (sys.ninputs, sys.nstates). If `integral action` is set to a matrix or a function, then additional columns represent the gains of the integral states of the controller. If a tuple is given, then it specifies a gain schedule. The tuple should be of the form `(gains, points)` where gains is a list of gains :math:`K_j` and points is a list of values :math:`\\mu_j` at which the gains are computed. The `gainsched_indices` parameter should be used to specify the scheduling variables. xd_labels, ud_labels : str or list of str, optional Set the name of the signals to use for the desired state and inputs. If a single string is specified, it should be a format string using the variable `i` as an index. Otherwise, a list of strings matching the size of xd and ud, respectively, should be used. Default is "xd[{i}]" for xd_labels and "ud[{i}]" for ud_labels. These settings can also be overriden using the `inputs` keyword. integral_action : ndarray, optional If this keyword is specified, the controller can include integral action in addition to state feedback. The value of the `integral_action` keyword should be an ndarray that will be multiplied by the current and desired state to generate the error for the internal integrator states of the control law. estimator : InputOutputSystem, optional If an estimator is provided, use the states of the estimator as the system inputs for the controller. gainsched_indices : int, slice, or list of int or str, optional If a gain scheduled controller is specified, specify the indices of the controller input to use for scheduling the gain. The input to the controller is the desired state xd, the desired input ud, and the system state x (or state estimate xhat, if an estimator is given). If value is an integer `q`, the first `q` values of the [xd, ud, x] vector are used. Otherwise, the value should be a slice or a list of indices. The list of indices can be specified as either integer offsets or as signal names. The default is to use the desired state xd. gainsched_method : str, optional The method to use for gain scheduling. Possible values are 'linear' (default), 'nearest', and 'cubic'. More information is available in :func:`scipy.interpolate.griddata`. For points outside of the convex hull of the scheduling points, the gain at the nearest point is used. controller_type : 'linear' or 'nonlinear', optional Set the type of controller to create. The default for a linear gain is a linear controller implementing the LQR regulator. If the type is 'nonlinear', a :class:NonlinearIOSystem is created instead, with the gain `K` as a parameter (allowing modifications of the gain at runtime). If the gain parameter is a tuple, then a nonlinear, gain-scheduled controller is created. Returns ------- ctrl : InputOutputSystem Input/output system representing the controller. This system takes as inputs the desired state `xd`, the desired input `ud`, and either the system state `x` or the estimated state `xhat`. It outputs the controller action `u` according to the formula :math:`u = u_d - K(x - x_d)`. If the keyword `integral_action` is specified, then an additional set of integrators is included in the control system (with the gain matrix `K` having the integral gains appended after the state gains). If a gain scheduled controller is specified, the gain (proportional and integral) are evaluated using the scheduling variables specified by `gainsched_indices`. clsys : InputOutputSystem Input/output system representing the closed loop system. This systems takes as inputs the desired trajectory `(xd, ud)` and outputs the system state `x` and the applied input `u` (vertically stacked). Other Parameters ---------------- control_indices : int, slice, or list of int or str, optional Specify the indices of the system inputs that should be determined by the state feedback controller. If value is an integer `m`, the first `m` system inputs are used. Otherwise, the value should be a slice or a list of indices. The list of indices can be specified as either integer offsets or as system input signal names. If not specified, defaults to the system inputs. state_indices : int, slice, or list of int or str, optional Specify the indices of the system (or estimator) outputs that should be used by the state feedback controller. If value is an integer `n`, the first `n` system states are used. Otherwise, the value should be a slice or a list of indices. The list of indices can be specified as either integer offsets or as estimator/system output signal names. If not specified, defaults to the system states. inputs, outputs : str, or list of str, optional List of strings that name the individual signals of the transformed system. If not given, the inputs and outputs are the same as the original system. name : string, optional System name. If unspecified, a generic name <sys[id]> is generated with a unique integer id. """ # Make sure that we were passed an I/O system as an input if not isinstance(sys, InputOutputSystem): raise ControlArgument("Input system must be I/O system") # Process (legacy) keywords controller_type = _process_legacy_keyword( kwargs, 'type', 'controller_type', controller_type) if kwargs: raise TypeError("unrecognized keywords: ", str(kwargs)) # Figure out what inputs to the system to use control_indices = _process_indices( control_indices, 'control', sys.input_labels, sys.ninputs) sys_ninputs = len(control_indices) # Decide what system is going to pass the states to the controller if estimator is None: estimator = sys # Figure out what outputs (states) from the system/estimator to use state_indices = _process_indices( state_indices, 'state', estimator.state_labels, sys.nstates) sys_nstates = len(state_indices) # Make sure the system/estimator states are proper dimension if estimator.noutputs < sys_nstates: # If no estimator, make sure that the system has all states as outputs raise ControlArgument( ("system" if estimator == sys else "estimator") + " output must include the full state") elif estimator == sys: # Issue a warning if we can't verify state output if (isinstance(sys, NonlinearIOSystem) and sys.outfcn is not None) or \ (isinstance(sys, StateSpace) and not (np.all(sys.C[np.ix_(state_indices, state_indices)] == np.eye(sys_nstates)) and np.all(sys.D[state_indices, :] == 0))): warnings.warn("cannot verify system output is system state") # See whether we should implement integral action nintegrators = 0 if integral_action is not None: if not isinstance(integral_action, np.ndarray): raise ControlArgument("Integral action must pass an array") elif integral_action.shape[1] != sys_nstates: raise ControlArgument( "Integral gain size must match system state size") else: nintegrators = integral_action.shape[0] C = integral_action else: # Create a C matrix with no outputs, just in case update gets called C = np.zeros((0, sys_nstates)) # Check to make sure that state feedback has the right shape if isinstance(gain, np.ndarray): K = gain if K.shape != (sys_ninputs, estimator.noutputs + nintegrators): raise ControlArgument( f'control gain must be an array of size {sys_ninputs}' f' x {sys_nstates}' + (f'+{nintegrators}' if nintegrators > 0 else '')) gainsched = False elif isinstance(gain, tuple): # Check for gain scheduled controller if len(gain) != 2: raise ControlArgument("gain must be a 2-tuple for gain scheduling") gains, points = gain[0:2] # Stack gains and points if past as a list gains = np.stack(gains) points = np.stack(points) gainsched=True else: raise ControlArgument("gain must be an array or a tuple") # Decide on the type of system to create if gainsched and controller_type == 'linear': raise ControlArgument( "controller_type 'linear' not allowed for" " gain scheduled controller") elif controller_type is None: controller_type = 'nonlinear' if gainsched else 'linear' elif controller_type not in {'linear', 'nonlinear'}: raise ControlArgument(f"unknown controller_type '{controller_type}'") # Figure out the labels to use xd_labels = _process_labels( xd_labels, 'xd', ['xd[{i}]'.format(i=i) for i in range(sys_nstates)]) ud_labels = _process_labels( ud_labels, 'ud', ['ud[{i}]'.format(i=i) for i in range(sys_ninputs)]) # Create the signal and system names if inputs is None: inputs = xd_labels + ud_labels + estimator.output_labels if outputs is None: outputs = [sys.input_labels[i] for i in control_indices] if states is None: states = nintegrators # Process gainscheduling variables, if present if gainsched: # Create a copy of the scheduling variable indices (default = xd) gainsched_indices = _process_indices( gainsched_indices, 'gainsched', inputs, sys_nstates) # If points is a 1D list, convert to 2D if points.ndim == 1: points = points.reshape(-1, 1) # Make sure the scheduling variable indices are the right length if len(gainsched_indices) != points.shape[1]: raise ControlArgument( "length of gainsched_indices must match dimension of" " scheduling variables") # Create interpolating function if points.shape[1] < 2: _interp = sp.interpolate.interp1d( points[:, 0], gains, axis=0, kind=gainsched_method) _nearest = sp.interpolate.interp1d( points[:, 0], gains, axis=0, kind='nearest') elif gainsched_method == 'nearest': _interp = sp.interpolate.NearestNDInterpolator(points, gains) def _nearest(mu): raise SystemError(f"could not find nearest gain at mu = {mu}") elif gainsched_method == 'linear': _interp = sp.interpolate.LinearNDInterpolator(points, gains) _nearest = sp.interpolate.NearestNDInterpolator(points, gains) elif gainsched_method == 'cubic': _interp = sp.interpolate.CloughTocher2DInterpolator(points, gains) _nearest = sp.interpolate.NearestNDInterpolator(points, gains) else: raise ControlArgument( f"unknown gain scheduling method '{gainsched_method}'") def _compute_gain(mu): K = _interp(mu) if np.isnan(K).any(): K = _nearest(mu) return K # Define the controller system if controller_type == 'nonlinear': # Create an I/O system for the state feedback gains def _control_update(t, states, inputs, params): # Split input into desired state, nominal input, and current state xd_vec = inputs[0:sys_nstates] x_vec = inputs[-sys_nstates:] # Compute the integral error in the xy coordinates return C @ (x_vec - xd_vec) def _control_output(t, states, inputs, params): if gainsched: mu = inputs[gainsched_indices] K_ = _compute_gain(mu) else: K_ = params.get('K') # Split input into desired state, nominal input, and current state xd_vec = inputs[0:sys_nstates] ud_vec = inputs[sys_nstates:sys_nstates + sys_ninputs] x_vec = inputs[-sys_nstates:] # Compute the control law u = ud_vec - K_[:, 0:sys_nstates] @ (x_vec - xd_vec) if nintegrators > 0: u -= K_[:, sys_nstates:] @ states return u params = {} if gainsched else {'K': K} ctrl = NonlinearIOSystem( _control_update, _control_output, name=name, inputs=inputs, outputs=outputs, states=states, params=params) elif controller_type == 'linear' or controller_type is None: # Create the matrices implementing the controller if isctime(sys): # Continuous time: integrator A_lqr = np.zeros((C.shape[0], C.shape[0])) else: # Discrete time: summer A_lqr = np.eye(C.shape[0]) B_lqr = np.hstack([-C, np.zeros((C.shape[0], sys_ninputs)), C]) C_lqr = -K[:, sys_nstates:] D_lqr = np.hstack([ K[:, 0:sys_nstates], np.eye(sys_ninputs), -K[:, 0:sys_nstates] ]) ctrl = ss( A_lqr, B_lqr, C_lqr, D_lqr, dt=sys.dt, name=name, inputs=inputs, outputs=outputs, states=states) else: raise ControlArgument(f"unknown controller_type '{controller_type}'") # Define the closed loop system inplist=inputs[:-sys.nstates] input_labels=inputs[:-sys.nstates] outlist=sys.output_labels + [sys.input_labels[i] for i in control_indices] output_labels=sys.output_labels + \ [sys.input_labels[i] for i in control_indices] closed = interconnect( [sys, ctrl] if estimator == sys else [sys, ctrl, estimator], name=sys.name + "_" + ctrl.name, add_unused=True, inplist=inplist, inputs=input_labels, outlist=outlist, outputs=output_labels ) return ctrl, closed def ctrb(A, B): """Controllabilty matrix Parameters ---------- A, B : array_like or string Dynamics and input matrix of the system Returns ------- C : 2D array (or matrix) Controllability matrix Notes ----- The return type for 2D arrays depends on the default class set for state space operations. See :func:`~control.use_numpy_matrix`. Examples -------- >>> G = ct.tf2ss([1], [1, 2, 3]) >>> C = ct.ctrb(G.A, G.B) >>> np.linalg.matrix_rank(C) 2 """ # Convert input parameters to matrices (if they aren't already) amat = _ssmatrix(A) bmat = _ssmatrix(B) n = np.shape(amat)[0] # Construct the controllability matrix ctrb = np.hstack( [bmat] + [np.linalg.matrix_power(amat, i) @ bmat for i in range(1, n)]) return _ssmatrix(ctrb) def obsv(A, C): |
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