Skip to content
This repository
Fetching contributors…

Octocat-spinner-32-eaf2f5

Cannot retrieve contributors at this time

file 1004 lines (841 sloc) 31.057 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 1001 1002 1003
"""
ltisys -- a collection of classes and functions for modeling linear
time invariant systems.
"""
from __future__ import division, print_function, absolute_import

#
# Author: Travis Oliphant 2001
#
# Feb 2010: Warren Weckesser
# Rewrote lsim2 and added impulse2.
#

from .filter_design import tf2zpk, zpk2tf, normalize, freqs
import numpy
from numpy import product, zeros, array, dot, transpose, ones, \
    nan_to_num, zeros_like, linspace
import scipy.interpolate as interpolate
import scipy.integrate as integrate
import scipy.linalg as linalg
from scipy.lib.six.moves import xrange
from numpy import r_, eye, real, atleast_1d, atleast_2d, poly, \
     squeeze, diag, asarray

__all__ = ['tf2ss', 'ss2tf', 'abcd_normalize', 'zpk2ss', 'ss2zpk', 'lti',
           'lsim', 'lsim2', 'impulse', 'impulse2', 'step', 'step2', 'bode',
           'freqresp']


def tf2ss(num, den):
    """Transfer function to state-space representation.

Parameters
----------
num, den : array_like
Sequences representing the numerator and denominator polynomials.
The denominator needs to be at least as long as the numerator.

Returns
-------
A, B, C, D : ndarray
State space representation of the system.

"""
    # Controller canonical state-space representation.
    # if M+1 = len(num) and K+1 = len(den) then we must have M <= K
    # states are found by asserting that X(s) = U(s) / D(s)
    # then Y(s) = N(s) * X(s)
    #
    # A, B, C, and D follow quite naturally.
    #
    num, den = normalize(num, den) # Strips zeros, checks arrays
    nn = len(num.shape)
    if nn == 1:
        num = asarray([num], num.dtype)
    M = num.shape[1]
    K = len(den)
    if M > K:
        msg = "Improper transfer function. `num` is longer than `den`."
        raise ValueError(msg)
    if M == 0 or K == 0: # Null system
        return array([], float), array([], float), array([], float), \
               array([], float)

    # pad numerator to have same number of columns has denominator
    num = r_['-1', zeros((num.shape[0], K - M), num.dtype), num]

    if num.shape[-1] > 0:
        D = num[:, 0]
    else:
        D = array([], float)

    if K == 1:
        return array([], float), array([], float), array([], float), D

    frow = -array([den[1:]])
    A = r_[frow, eye(K - 2, K - 1)]
    B = eye(K - 1, 1)
    C = num[:, 1:] - num[:, 0] * den[1:]
    return A, B, C, D


def _none_to_empty(arg):
    if arg is None:
        return []
    else:
        return arg


def abcd_normalize(A=None, B=None, C=None, D=None):
    """Check state-space matrices and ensure they are rank-2.

"""
    A, B, C, D = map(_none_to_empty, (A, B, C, D))
    A, B, C, D = map(atleast_2d, (A, B, C, D))

    if ((len(A.shape) > 2) or (len(B.shape) > 2) or \
        (len(C.shape) > 2) or (len(D.shape) > 2)):
        raise ValueError("A, B, C, D arrays can be no larger than rank-2.")

    MA, NA = A.shape
    MB, NB = B.shape
    MC, NC = C.shape
    MD, ND = D.shape

    if (MC == 0) and (NC == 0) and (MD != 0) and (NA != 0):
        MC, NC = MD, NA
        C = zeros((MC, NC))
    if (MB == 0) and (NB == 0) and (MA != 0) and (ND != 0):
        MB, NB = MA, ND
        B = zeros(MB, NB)
    if (MD == 0) and (ND == 0) and (MC != 0) and (NB != 0):
        MD, ND = MC, NB
        D = zeros(MD, ND)
    if (MA == 0) and (NA == 0) and (MB != 0) and (NC != 0):
        MA, NA = MB, NC
        A = zeros(MA, NA)

    if MA != NA:
        raise ValueError("A must be square.")
    if MA != MB:
        raise ValueError("A and B must have the same number of rows.")
    if NA != NC:
        raise ValueError("A and C must have the same number of columns.")
    if MD != MC:
        raise ValueError("C and D must have the same number of rows.")
    if ND != NB:
        raise ValueError("B and D must have the same number of columns.")

    return A, B, C, D


def ss2tf(A, B, C, D, input=0):
    """State-space to transfer function.

Parameters
----------
A, B, C, D : ndarray
State-space representation of linear system.
input : int, optional
For multiple-input systems, the input to use.

Returns
-------
num, den : 1D ndarray
Numerator and denominator polynomials (as sequences)
respectively.

"""
    # transfer function is C (sI - A)**(-1) B + D
    A, B, C, D = map(asarray, (A, B, C, D))
    # Check consistency and
    # make them all rank-2 arrays
    A, B, C, D = abcd_normalize(A, B, C, D)

    nout, nin = D.shape
    if input >= nin:
        raise ValueError("System does not have the input specified.")

    # make MOSI from possibly MOMI system.
    if B.shape[-1] != 0:
        B = B[:, input]
    B.shape = (B.shape[0], 1)
    if D.shape[-1] != 0:
        D = D[:, input]

    try:
        den = poly(A)
    except ValueError:
        den = 1

    if (product(B.shape, axis=0) == 0) and (product(C.shape, axis=0) == 0):
        num = numpy.ravel(D)
        if (product(D.shape, axis=0) == 0) and (product(A.shape, axis=0) == 0):
            den = []
        return num, den

    num_states = A.shape[0]
    type_test = A[:, 0] + B[:, 0] + C[0, :] + D
    num = numpy.zeros((nout, num_states + 1), type_test.dtype)
    for k in range(nout):
        Ck = atleast_2d(C[k, :])
        num[k] = poly(A - dot(B, Ck)) + (D[k] - 1) * den

    return num, den


def zpk2ss(z, p, k):
    """Zero-pole-gain representation to state-space representation

Parameters
----------
z, p : sequence
Zeros and poles.
k : float
System gain.

Returns
-------
A, B, C, D : ndarray
State-space matrices.

"""
    return tf2ss(*zpk2tf(z, p, k))


def ss2zpk(A, B, C, D, input=0):
    """State-space representation to zero-pole-gain representation.

Parameters
----------
A, B, C, D : ndarray
State-space representation of linear system.
input : int, optional
For multiple-input systems, the input to use.

Returns
-------
z, p : sequence
Zeros and poles.
k : float
System gain.

"""
    return tf2zpk(*ss2tf(A, B, C, D, input=input))


class lti(object):
    """Linear Time Invariant class which simplifies representation.

Parameters
----------
args : arguments
The `lti` class can be instantiated with either 2, 3 or 4 arguments.
The following gives the number of elements in the tuple and the
interpretation:

* 2: (numerator, denominator)
* 3: (zeros, poles, gain)
* 4: (A, B, C, D)

Each argument can be an array or sequence.

Notes
-----
`lti` instances have all types of representations available; for example
after creating an instance s with ``(zeros, poles, gain)`` the state-space
representation (numerator, denominator) can be accessed as ``s.num`` and
``s.den``.

"""
    def __init__(self, *args, **kwords):
        """
Initialize the LTI system using either:

- (numerator, denominator)
- (zeros, poles, gain)
- (A, B, C, D) : state-space.

"""
        N = len(args)
        if N == 2: # Numerator denominator transfer function input
            self.__dict__['num'], self.__dict__['den'] = normalize(*args)
            self.__dict__['zeros'], self.__dict__['poles'], \
            self.__dict__['gain'] = tf2zpk(*args)
            self.__dict__['A'], self.__dict__['B'], \
                                self.__dict__['C'], \
                                self.__dict__['D'] = tf2ss(*args)
            self.inputs = 1
            if len(self.num.shape) > 1:
                self.outputs = self.num.shape[0]
            else:
                self.outputs = 1
        elif N == 3: # Zero-pole-gain form
            self.__dict__['zeros'], self.__dict__['poles'], \
                                    self.__dict__['gain'] = args
            self.__dict__['num'], self.__dict__['den'] = zpk2tf(*args)
            self.__dict__['A'], self.__dict__['B'], \
                                self.__dict__['C'], \
                                self.__dict__['D'] = zpk2ss(*args)
            # make sure we have numpy arrays
            self.zeros = numpy.asarray(self.zeros)
            self.poles = numpy.asarray(self.poles)
            self.inputs = 1
            if len(self.zeros.shape) > 1:
                self.outputs = self.zeros.shape[0]
            else:
                self.outputs = 1
        elif N == 4: # State-space form
            self.__dict__['A'], self.__dict__['B'], \
                                self.__dict__['C'], \
                                self.__dict__['D'] = abcd_normalize(*args)
            self.__dict__['zeros'], self.__dict__['poles'], \
                                    self.__dict__['gain'] = ss2zpk(*args)
            self.__dict__['num'], self.__dict__['den'] = ss2tf(*args)
            self.inputs = self.B.shape[-1]
            self.outputs = self.C.shape[0]
        else:
            raise ValueError("Needs 2, 3, or 4 arguments.")

    def __setattr__(self, attr, val):
        if attr in ['num', 'den']:
            self.__dict__[attr] = val
            self.__dict__['zeros'], self.__dict__['poles'], \
                                    self.__dict__['gain'] = \
                                    tf2zpk(self.num, self.den)
            self.__dict__['A'], self.__dict__['B'], \
                                self.__dict__['C'], \
                                self.__dict__['D'] = \
                                tf2ss(self.num, self.den)
        elif attr in ['zeros', 'poles', 'gain']:
            self.__dict__[attr] = val
            self.__dict__['num'], self.__dict__['den'] = \
                                  zpk2tf(self.zeros,
                                         self.poles, self.gain)
            self.__dict__['A'], self.__dict__['B'], \
                                self.__dict__['C'], \
                                self.__dict__['D'] = \
                                zpk2ss(self.zeros,
                                       self.poles, self.gain)
        elif attr in ['A', 'B', 'C', 'D']:
            self.__dict__[attr] = val
            self.__dict__['zeros'], self.__dict__['poles'], \
                                    self.__dict__['gain'] = \
                                    ss2zpk(self.A, self.B,
                                           self.C, self.D)
            self.__dict__['num'], self.__dict__['den'] = \
                                  ss2tf(self.A, self.B,
                                        self.C, self.D)
        else:
            self.__dict__[attr] = val

    def impulse(self, X0=None, T=None, N=None):
        return impulse(self, X0=X0, T=T, N=N)

    def step(self, X0=None, T=None, N=None):
        return step(self, X0=X0, T=T, N=N)

    def output(self, U, T, X0=None):
        return lsim(self, U, T, X0=X0)

    def bode(self, w=None, n=100):
        """Calculate bode magnitude and phase data.

Returns a 3-tuple containing arrays of frequencies [rad/s], magnitude
[dB] and phase [deg]. See scipy.signal.bode for details.

.. versionadded:: 0.11.0

Examples
--------
>>> from scipy import signal
>>> import matplotlib.pyplot as plt

>>> s1 = signal.lti([1], [1, 1])
>>> w, mag, phase = s1.bode()

>>> plt.figure()
>>> plt.semilogx(w, mag) # bode magnitude plot
>>> plt.figure()
>>> plt.semilogx(w, phase) # bode phase plot
>>> plt.show()
"""
        return bode(self, w=w, n=n)

    def freqresp(self, w=None, n=10000):
        """Calculate the frequency response of a continuous-time system.

Returns a 2-tuple containing arrays of frequencies [rad/s] and
complex magnitude.
See scipy.signal.freqresp for details.

"""
        return freqresp(self, w=w, n=n)


def lsim2(system, U=None, T=None, X0=None, **kwargs):
    """
Simulate output of a continuous-time linear system, by using
the ODE solver `scipy.integrate.odeint`.

Parameters
----------
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and
the interpretation:

* 2: (num, den)
* 3: (zeros, poles, gain)
* 4: (A, B, C, D)

U : array_like (1D or 2D), optional
An input array describing the input at each time T. Linear
interpolation is used between given times. If there are
multiple inputs, then each column of the rank-2 array
represents an input. If U is not given, the input is assumed
to be zero.
T : array_like (1D or 2D), optional
The time steps at which the input is defined and at which the
output is desired. The default is 101 evenly spaced points on
the interval [0,10.0].
X0 : array_like (1D), optional
The initial condition of the state vector. If `X0` is not
given, the initial conditions are assumed to be 0.
kwargs : dict
Additional keyword arguments are passed on to the function
odeint. See the notes below for more details.

Returns
-------
T : 1D ndarray
The time values for the output.
yout : ndarray
The response of the system.
xout : ndarray
The time-evolution of the state-vector.

Notes
-----
This function uses `scipy.integrate.odeint` to solve the
system's differential equations. Additional keyword arguments
given to `lsim2` are passed on to `odeint`. See the documentation
for `scipy.integrate.odeint` for the full list of arguments.

"""
    if isinstance(system, lti):
        sys = system
    else:
        sys = lti(*system)

    if X0 is None:
        X0 = zeros(sys.B.shape[0], sys.A.dtype)

    if T is None:
        # XXX T should really be a required argument, but U was
        # changed from a required positional argument to a keyword,
        # and T is after U in the argument list. So we either: change
        # the API and move T in front of U; check here for T being
        # None and raise an excpetion; or assign a default value to T
        # here. This code implements the latter.
        T = linspace(0, 10.0, 101)

    T = atleast_1d(T)
    if len(T.shape) != 1:
        raise ValueError("T must be a rank-1 array.")

    if U is not None:
        U = atleast_1d(U)
        if len(U.shape) == 1:
            U = U.reshape(-1, 1)
        sU = U.shape
        if sU[0] != len(T):
            raise ValueError("U must have the same number of rows "
                             "as elements in T.")

        if sU[1] != sys.inputs:
            raise ValueError("The number of inputs in U (%d) is not "
                             "compatible with the number of system "
                             "inputs (%d)" % (sU[1], sys.inputs))
        # Create a callable that uses linear interpolation to
        # calculate the input at any time.
        ufunc = interpolate.interp1d(T, U, kind='linear',
                                     axis=0, bounds_error=False)

        def fprime(x, t, sys, ufunc):
            """The vector field of the linear system."""
            return dot(sys.A, x) + squeeze(dot(sys.B, nan_to_num(ufunc([t]))))
        xout = integrate.odeint(fprime, X0, T, args=(sys, ufunc), **kwargs)
        yout = dot(sys.C, transpose(xout)) + dot(sys.D, transpose(U))
    else:
        def fprime(x, t, sys):
            """The vector field of the linear system."""
            return dot(sys.A, x)
        xout = integrate.odeint(fprime, X0, T, args=(sys,), **kwargs)
        yout = dot(sys.C, transpose(xout))

    return T, squeeze(transpose(yout)), xout


def lsim(system, U, T, X0=None, interp=1):
    """
Simulate output of a continuous-time linear system.

Parameters
----------
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and
the interpretation:

* 2: (num, den)
* 3: (zeros, poles, gain)
* 4: (A, B, C, D)

U : array_like
An input array describing the input at each time `T`
(interpolation is assumed between given times). If there are
multiple inputs, then each column of the rank-2 array
represents an input.
T : array_like
The time steps at which the input is defined and at which the
output is desired.
X0 :
The initial conditions on the state vector (zero by default).
interp : {1, 0}
Whether to use linear (1) or zero-order hold (0) interpolation.

Returns
-------
T : 1D ndarray
Time values for the output.
yout : 1D ndarray
System response.
xout : ndarray
Time-evolution of the state-vector.

"""
    # system is an lti system or a sequence
    # with 2 (num, den)
    # 3 (zeros, poles, gain)
    # 4 (A, B, C, D)
    # describing the system
    # U is an input vector at times T
    # if system describes multiple inputs
    # then U can be a rank-2 array with the number of columns
    # being the number of inputs
    if isinstance(system, lti):
        sys = system
    else:
        sys = lti(*system)
    U = atleast_1d(U)
    T = atleast_1d(T)
    if len(U.shape) == 1:
        U = U.reshape((U.shape[0], 1))
    sU = U.shape
    if len(T.shape) != 1:
        raise ValueError("T must be a rank-1 array.")
    if sU[0] != len(T):
        raise ValueError("U must have the same number of rows "
                         "as elements in T.")
    if sU[1] != sys.inputs:
        raise ValueError("System does not define that many inputs.")

    if X0 is None:
        X0 = zeros(sys.B.shape[0], sys.A.dtype)

    xout = zeros((len(T), sys.B.shape[0]), sys.A.dtype)
    xout[0] = X0
    A = sys.A
    AT, BT = transpose(sys.A), transpose(sys.B)
    dt = T[1] - T[0]
    lam, v = linalg.eig(A)
    vt = transpose(v)
    vti = linalg.inv(vt)
    GT = dot(dot(vti, diag(numpy.exp(dt * lam))), vt).astype(xout.dtype)
    ATm1 = linalg.inv(AT)
    ATm2 = dot(ATm1, ATm1)
    I = eye(A.shape[0], dtype=A.dtype)
    GTmI = GT - I
    F1T = dot(dot(BT, GTmI), ATm1)
    if interp:
        F2T = dot(BT, dot(GTmI, ATm2) / dt - ATm1)

    for k in xrange(1, len(T)):
        dt1 = T[k] - T[k - 1]
        if dt1 != dt:
            dt = dt1
            GT = dot(dot(vti, diag(numpy.exp(dt * lam))),
                     vt).astype(xout.dtype)
            GTmI = GT - I
            F1T = dot(dot(BT, GTmI), ATm1)
            if interp:
                F2T = dot(BT, dot(GTmI, ATm2) / dt - ATm1)

        xout[k] = dot(xout[k - 1], GT) + dot(U[k - 1], F1T)
        if interp:
            xout[k] = xout[k] + dot((U[k] - U[k - 1]), F2T)

    yout = (squeeze(dot(U, transpose(sys.D))) +
            squeeze(dot(xout, transpose(sys.C))))
    return T, squeeze(yout), squeeze(xout)


def _default_response_times(A, n):
    """Compute a reasonable set of time samples for the response time.

This function is used by `impulse`, `impulse2`, `step` and `step2`
to compute the response time when the `T` argument to the function
is None.

Parameters
----------
A : ndarray
The system matrix, which is square.
n : int
The number of time samples to generate.

Returns
-------
t : ndarray
The 1-D array of length `n` of time samples at which the response
is to be computed.
"""
    # Create a reasonable time interval. This could use some more work.
    # For example, what is expected when the system is unstable?
    vals = linalg.eigvals(A)
    r = min(abs(real(vals)))
    if r == 0.0:
        r = 1.0
    tc = 1.0 / r
    t = linspace(0.0, 7 * tc, n)
    return t


def _default_response_frequencies(A, n):
    """Compute a reasonable set of frequency points for bode plot.

This function is used by `bode` to compute the frequency points (in rad/s)
when the `w` argument to the function is None.

Parameters
----------
A : ndarray
The system matrix, which is square.
n : int
The number of frequency samples to generate.

Returns
-------
w : ndarray
The 1-D array of length `n` of frequency samples (in rad/s) at which
the response is to be computed.
"""
    vals = linalg.eigvals(A)
    # Remove poles at 0 because they don't help us determine an interesting
    # frequency range. (And if we pass a 0 to log10() below we will crash.)
    poles = [pole for pole in vals if pole != 0]
    # If there are no non-zero poles, just hardcode something.
    if len(poles) == 0:
        minpole = 1
        maxpole = 1
    else:
        minpole = min(abs(real(poles)))
        maxpole = max(abs(real(poles)))
    # A reasonable frequency range is two orders of magnitude before the
    # minimum pole (slowest) and two orders of magnitude after the maximum pole
    # (fastest).
    w = numpy.logspace(numpy.log10(minpole) - 2, numpy.log10(maxpole) + 2, n)
    return w


def impulse(system, X0=None, T=None, N=None):
    """Impulse response of continuous-time system.

Parameters
----------
system : LTI class or tuple
If specified as a tuple, the system is described as
``(num, den)``, ``(zero, pole, gain)``, or ``(A, B, C, D)``.
X0 : array_like, optional
Initial state-vector. Defaults to zero.
T : array_like, optional
Time points. Computed if not given.
N : int, optional
The number of time points to compute (if `T` is not given).

Returns
-------
T : ndarray
A 1-D array of time points.
yout : ndarray
A 1-D array containing the impulse response of the system (except for
singularities at zero).

"""
    if isinstance(system, lti):
        sys = system
    else:
        sys = lti(*system)
    if X0 is None:
        B = sys.B
    else:
        B = sys.B + X0
    if N is None:
        N = 100
    if T is None:
        T = _default_response_times(sys.A, N)
    h = zeros(T.shape, sys.A.dtype)
    s, v = linalg.eig(sys.A)
    vi = linalg.inv(v)
    C = sys.C
    for k in range(len(h)):
        es = diag(numpy.exp(s * T[k]))
        eA = (dot(dot(v, es), vi)).astype(h.dtype)
        h[k] = squeeze(dot(dot(C, eA), B))
    return T, h


def impulse2(system, X0=None, T=None, N=None, **kwargs):
    """
Impulse response of a single-input, continuous-time linear system.


Parameters
----------
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and
the interpretation:

* 2 (num, den)
* 3 (zeros, poles, gain)
* 4 (A, B, C, D)

T : 1-D array_like, optional
The time steps at which the input is defined and at which the
output is desired. If `T` is not given, the function will
generate a set of time samples automatically.
X0 : 1-D array_like, optional
The initial condition of the state vector. Default: 0 (the
zero vector).
N : int, optional
Number of time points to compute. Default: 100.
kwargs : various types
Additional keyword arguments are passed on to the function
`scipy.signal.lsim2`, which in turn passes them on to
`scipy.integrate.odeint`; see the latter's documentation for
information about these arguments.

Returns
-------
T : ndarray
The time values for the output.
yout : ndarray
The output response of the system.

See Also
--------
impulse, lsim2, integrate.odeint

Notes
-----
The solution is generated by calling `scipy.signal.lsim2`, which uses
the differential equation solver `scipy.integrate.odeint`.

.. versionadded:: 0.8.0

Examples
--------
Second order system with a repeated root: x''(t) + 2*x(t) + x(t) = u(t)

>>> from scipy import signal
>>> system = ([1.0], [1.0, 2.0, 1.0])
>>> t, y = signal.impulse2(system)
>>> import matplotlib.pyplot as plt
>>> plt.plot(t, y)

"""
    if isinstance(system, lti):
        sys = system
    else:
        sys = lti(*system)
    B = sys.B
    if B.shape[-1] != 1:
        raise ValueError("impulse2() requires a single-input system.")
    B = B.squeeze()
    if X0 is None:
        X0 = zeros_like(B)
    if N is None:
        N = 100
    if T is None:
        T = _default_response_times(sys.A, N)
    # Move the impulse in the input to the initial conditions, and then
    # solve using lsim2().
    U = zeros_like(T)
    ic = B + X0
    Tr, Yr, Xr = lsim2(sys, U, T, ic, **kwargs)
    return Tr, Yr


def step(system, X0=None, T=None, N=None):
    """Step response of continuous-time system.

Parameters
----------
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and
the interpretation:

* 2 (num, den)
* 3 (zeros, poles, gain)
* 4 (A, B, C, D)

X0 : array_like, optional
Initial state-vector (default is zero).
T : array_like, optional
Time points (computed if not given).
N : int
Number of time points to compute if `T` is not given.

Returns
-------
T : 1D ndarray
Output time points.
yout : 1D ndarray
Step response of system.

See also
--------
scipy.signal.step2

"""
    if isinstance(system, lti):
        sys = system
    else:
        sys = lti(*system)
    if N is None:
        N = 100
    if T is None:
        T = _default_response_times(sys.A, N)
    U = ones(T.shape, sys.A.dtype)
    vals = lsim(sys, U, T, X0=X0)
    return vals[0], vals[1]


def step2(system, X0=None, T=None, N=None, **kwargs):
    """Step response of continuous-time system.

This function is functionally the same as `scipy.signal.step`, but
it uses the function `scipy.signal.lsim2` to compute the step
response.

Parameters
----------
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and
the interpretation:

* 2 (num, den)
* 3 (zeros, poles, gain)
* 4 (A, B, C, D)

X0 : array_like, optional
Initial state-vector (default is zero).
T : array_like, optional
Time points (computed if not given).
N : int
Number of time points to compute if `T` is not given.
kwargs :
Additional keyword arguments are passed on the function
`scipy.signal.lsim2`, which in turn passes them on to
`scipy.integrate.odeint`. See the documentation for
`scipy.integrate.odeint` for information about these arguments.

Returns
-------
T : 1D ndarray
Output time points.
yout : 1D ndarray
Step response of system.

See also
--------
scipy.signal.step

Notes
-----
.. versionadded:: 0.8.0
"""
    if isinstance(system, lti):
        sys = system
    else:
        sys = lti(*system)
    if N is None:
        N = 100
    if T is None:
        T = _default_response_times(sys.A, N)
    U = ones(T.shape, sys.A.dtype)
    vals = lsim2(sys, U, T, X0=X0, **kwargs)
    return vals[0], vals[1]


def bode(system, w=None, n=100):
    """Calculate bode magnitude and phase data of a continuous-time system.

.. versionadded:: 0.11.0

Parameters
----------
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and
the interpretation:

* 2 (num, den)
* 3 (zeros, poles, gain)
* 4 (A, B, C, D)

w : array_like, optional
Array of frequencies (in rad/s). Magnitude and phase data is calculated
for every value in this array. If not given a reasonable set will be
calculated.
n : int, optional
Number of frequency points to compute if `w` is not given. The `n`
frequencies are logarithmically spaced in the range from two orders of
magnitude before the minimum (slowest) pole to two orders of magnitude
after the maximum (fastest) pole.

Returns
-------
w : 1D ndarray
Frequency array [rad/s]
mag : 1D ndarray
Magnitude array [dB]
phase : 1D ndarray
Phase array [deg]

Examples
--------
>>> from scipy import signal
>>> import matplotlib.pyplot as plt

>>> s1 = signal.lti([1], [1, 1])
>>> w, mag, phase = signal.bode(s1)

>>> plt.figure()
>>> plt.semilogx(w, mag) # bode magnitude plot
>>> plt.figure()
>>> plt.semilogx(w, phase) # bode phase plot
>>> plt.show()
"""
    if isinstance(system, lti):
        sys = system
    else:
        sys = lti(*system)

    if w is None:
        w = _default_response_frequencies(sys.A, n)
    else:
        w = numpy.asarray(w)

    jw = w * 1j
    y = numpy.polyval(sys.num, jw) / numpy.polyval(sys.den, jw)
    mag = 20.0 * numpy.log10(abs(y))
    phase = numpy.arctan2(y.imag, y.real) * 180.0 / numpy.pi
    return w, mag, phase


def freqresp(system, w=None, n=10000):
    """Calculate the frequency response of a continuous-time system.

Parameters
----------
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and
the interpretation:

* 2 (num, den)
* 3 (zeros, poles, gain)
* 4 (A, B, C, D)

w : array_like, optional
Array of frequencies (in rad/s). Magnitude and phase data is
calculated for every value in this array. If not given a reasonable
set will be calculated.
n : int, optional
Number of frequency points to compute if `w` is not given. The `n`
frequencies are logarithmically spaced in the range from two orders of
magnitude before the minimum (slowest) pole to two orders of magnitude
after the maximum (fastest) pole.

Returns
-------
w : 1D ndarray
Frequency array [rad/s]
H : 1D ndarray
Array of complex magnitude values

Example
-------
# Generating the Nyquist plot of a transfer function

>>> from scipy import signal
>>> import matplotlib.pyplot as plt

>>> s1 = signal.lti([], [1, 1, 1], [5])
# transfer function: H(s) = 5 / (s-1)^3
>>> w, H = signal.freqresp(s1)

>>> plt.figure()
>>> plt.plot(H.real, H.imag, "b")
>>> plt.plot(H.real, -H.imag, "r")
>>> plt.show()
"""
    if isinstance(system, lti):
        sys = system
    else:
        sys = lti(*system)

    if w is not None:
        worN = w
    else:
        worN = n

    return freqs(sys.num, sys.den, worN=worN)
Something went wrong with that request. Please try again.