# scipy/scipy

3646 lines (3029 sloc) 120 KB
 """ 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. # Apr 2011: Jeffrey Armstrong # Added dlsim, dstep, dimpulse, cont2discrete # Aug 2013: Juan Luis Cano # Rewrote abcd_normalize. # Jan 2015: Irvin Probst irvin DOT probst AT ensta-bretagne DOT fr # Added pole placement # Mar 2015: Clancy Rowley # Rewrote lsim # May 2015: Felix Berkenkamp # Split lti class into subclasses # Merged discrete systems and added dlti import warnings # np.linalg.qr fails on some tests with LinAlgError: zgeqrf returns -7 # use scipy's qr until this is solved from scipy.linalg import qr as s_qr from scipy import integrate, interpolate, linalg from scipy.interpolate import interp1d from scipy._lib.six import xrange from .filter_design import tf2zpk, zpk2tf, normalize, freqs, freqz from .lti_conversion import (tf2ss, abcd_normalize, ss2tf, zpk2ss, ss2zpk, cont2discrete) import numpy import numpy as np from numpy import (real, atleast_1d, atleast_2d, squeeze, asarray, zeros, dot, transpose, ones, zeros_like, linspace, nan_to_num) import copy __all__ = ['lti', 'dlti', 'TransferFunction', 'ZerosPolesGain', 'StateSpace', 'lsim', 'lsim2', 'impulse', 'impulse2', 'step', 'step2', 'bode', 'freqresp', 'place_poles', 'dlsim', 'dstep', 'dimpulse', 'dfreqresp', 'dbode'] class LinearTimeInvariant(object): def __new__(cls, *system, **kwargs): """Create a new object, don't allow direct instances.""" if cls is LinearTimeInvariant: raise NotImplementedError('The LinearTimeInvariant class is not ' 'meant to be used directly, use lti ' 'or dlti instead.') return super(LinearTimeInvariant, cls).__new__(cls) def __init__(self): """ Initialize the lti baseclass. The heavy lifting is done by the subclasses. """ super(LinearTimeInvariant, self).__init__() self.inputs = None self.outputs = None self._dt = None @property def dt(self): """Return the sampling time of the system, None for lti systems.""" return self._dt @property def _dt_dict(self): if self.dt is None: return {} else: return {'dt': self.dt} @property def num(self): """Numerator of the TransferFunction system.""" warnings.warn('Cross-class properties have been deprecated in scipy ' '0.18.0 and will be removed in a future version of ' 'scipy. Please use sys.to_tf().numinstead.', DeprecationWarning) return self.to_tf().num @num.setter def num(self, num): warnings.warn('Cross-class setters have been deprecated in scipy ' '0.18.0 and will be removed in a future version of ' 'scipy. Please convert your system with sys.to_tf() ' 'before setting num.', DeprecationWarning) obj = self.to_tf() obj.num = num source_class = type(self) self._copy(source_class(obj)) @property def den(self): """Denominator of the TransferFunction system.""" warnings.warn('Cross-class properties have been deprecated in scipy ' '0.18.0 and will be removed in a future version of ' 'scipy. Please use sys.to_tf().deninstead.', DeprecationWarning) return self.to_tf().den @den.setter def den(self, den): warnings.warn('Cross-class setters have been deprecated in scipy ' '0.18.0 and will be removed in a future version of ' 'scipy. Please convert your system with sys.to_tf() ' 'before setting den.', DeprecationWarning) obj = self.to_tf() obj.den = den source_class = type(self) self._copy(source_class(obj)) @property def zeros(self): """Zeros of the system.""" return self.to_zpk().zeros @zeros.setter def zeros(self, zeros): warnings.warn('Cross-class setters have been deprecated in scipy ' '0.18.0 and will be removed in a future version of ' 'scipy. Please convert your system with sys.to_zpk() ' 'before setting zeros.', DeprecationWarning) obj = self.to_zpk() obj.zeros = zeros source_class = type(self) self._copy(source_class(obj)) @property def poles(self): """Poles of the system.""" return self.to_zpk().poles @poles.setter def poles(self, poles): warnings.warn('Cross-class setters have been deprecated in scipy ' '0.18.0 and will be removed in a future version of ' 'scipy. Please convert your system with sys.to_zpk() ' 'before setting poles.', DeprecationWarning) obj = self.to_zpk() obj.poles = poles source_class = type(self) self._copy(source_class(obj)) @property def gain(self): """Gain of the ZerosPolesGain system.""" warnings.warn('Cross-class properties have been deprecated in scipy ' '0.18.0 and will be removed in a future version of ' 'scipy. Please use sys.to_zpk().gaininstead.', DeprecationWarning) return self.to_zpk().gain @gain.setter def gain(self, gain): warnings.warn('Cross-class setters have been deprecated in scipy ' '0.18.0 and will be removed in a future version of ' 'scipy. Please convert your system with sys.to_zpk() ' 'before setting gain.', DeprecationWarning) obj = self.to_zpk() obj.gain = gain source_class = type(self) self._copy(source_class(obj)) @property def A(self): """State matrix of the StateSpace system.""" warnings.warn('Cross-class properties have been deprecated in scipy ' '0.18.0 and will be removed in a future version of ' 'scipy. Please use sys.to_ss().Ainstead.', DeprecationWarning) return self.to_ss().A @A.setter def A(self, A): warnings.warn('Cross-class setters have been deprecated in scipy ' '0.18.0 and will be removed in a future version of ' 'scipy. Please convert your system with sys.to_ss() ' 'before setting A.', DeprecationWarning) obj = self.to_ss() obj.A = A source_class = type(self) self._copy(source_class(obj)) @property def B(self): """Input matrix of the StateSpace system.""" warnings.warn('Cross-class properties have been deprecated in scipy ' '0.18.0 and will be removed in a future version of ' 'scipy. Please use sys.to_ss().Binstead.', DeprecationWarning) return self.to_ss().B @B.setter def B(self, B): warnings.warn('Cross-class setters have been deprecated in scipy ' '0.18.0 and will be removed in a future version of ' 'scipy. Please convert your system with sys.to_ss() ' 'before setting B.', DeprecationWarning) obj = self.to_ss() obj.B = B source_class = type(self) self._copy(source_class(obj)) @property def C(self): """Output matrix of the StateSpace system.""" warnings.warn('Cross-class properties have been deprecated in scipy ' '0.18.0 and will be removed in a future version of ' 'scipy. Please use sys.to_ss().Cinstead.', DeprecationWarning) return self.to_ss().C @C.setter def C(self, C): warnings.warn('Cross-class setters have been deprecated in scipy ' '0.18.0 and will be removed in a future version of ' 'scipy. Please convert your system with sys.to_ss() ' 'before setting C.', DeprecationWarning) obj = self.to_ss() obj.C = C source_class = type(self) self._copy(source_class(obj)) @property def D(self): """Feedthrough matrix of the StateSpace system.""" warnings.warn('Cross-class properties have been deprecated in scipy ' '0.18.0 and will be removed in a future version of ' 'scipy. Please use sys.to_ss().Dinstead.', DeprecationWarning) return self.to_ss().D @D.setter def D(self, D): warnings.warn('Cross-class setters have been deprecated in scipy ' '0.18.0 and will be removed in a future version of ' 'scipy. Please convert your system with sys.to_ss() ' 'before setting D.', DeprecationWarning) obj = self.to_ss() obj.D = D source_class = type(self) self._copy(source_class(obj)) def _as_ss(self): """Convert to StateSpace system, without copying. Returns ------- sys: StateSpace The StateSpace system. If the class is already an instance of StateSpace then this instance is returned. """ if isinstance(self, StateSpace): return self else: return self.to_ss() def _as_zpk(self): """Convert to ZerosPolesGain system, without copying. Returns ------- sys: ZerosPolesGain The ZerosPolesGain system. If the class is already an instance of ZerosPolesGain then this instance is returned. """ if isinstance(self, ZerosPolesGain): return self else: return self.to_zpk() def _as_tf(self): """Convert to TransferFunction system, without copying. Returns ------- sys: ZerosPolesGain The TransferFunction system. If the class is already an instance of TransferFunction then this instance is returned. """ if isinstance(self, TransferFunction): return self else: return self.to_tf() class lti(LinearTimeInvariant): """ Continuous-time linear time invariant system base class. Parameters ---------- *system : arguments The lti class can be instantiated with either 2, 3 or 4 arguments. The following gives the number of arguments and the corresponding continuous-time subclass that is created: * 2: TransferFunction: (numerator, denominator) * 3: ZerosPolesGain: (zeros, poles, gain) * 4: StateSpace: (A, B, C, D) Each argument can be an array or a sequence. See Also -------- ZerosPolesGain, StateSpace, TransferFunction, dlti Notes ----- lti instances do not exist directly. Instead, lti creates an instance of one of its subclasses: StateSpace, TransferFunction or ZerosPolesGain. If (numerator, denominator) is passed in for *system, coefficients for both the numerator and denominator should be specified in descending exponent order (e.g., s^2 + 3s + 5 would be represented as [1, 3, 5]). Changing the value of properties that are not directly part of the current system representation (such as the zeros of a StateSpace system) is very inefficient and may lead to numerical inaccuracies. It is better to convert to the specific system representation first. For example, call sys = sys.to_zpk() before accessing/changing the zeros, poles or gain. Examples -------- >>> from scipy import signal >>> signal.lti(1, 2, 3, 4) StateSpaceContinuous( array([[1]]), array([[2]]), array([[3]]), array([[4]]), dt: None ) >>> signal.lti([1, 2], [3, 4], 5) ZerosPolesGainContinuous( array([1, 2]), array([3, 4]), 5, dt: None ) >>> signal.lti([3, 4], [1, 2]) TransferFunctionContinuous( array([ 3., 4.]), array([ 1., 2.]), dt: None ) """ def __new__(cls, *system): """Create an instance of the appropriate subclass.""" if cls is lti: N = len(system) if N == 2: return TransferFunctionContinuous.__new__( TransferFunctionContinuous, *system) elif N == 3: return ZerosPolesGainContinuous.__new__( ZerosPolesGainContinuous, *system) elif N == 4: return StateSpaceContinuous.__new__(StateSpaceContinuous, *system) else: raise ValueError("system needs to be an instance of lti " "or have 2, 3 or 4 arguments.") # __new__ was called from a subclass, let it call its own functions return super(lti, cls).__new__(cls) def __init__(self, *system): """ Initialize the lti baseclass. The heavy lifting is done by the subclasses. """ super(lti, self).__init__(*system) def impulse(self, X0=None, T=None, N=None): """ Return the impulse response of a continuous-time system. See impulse for details. """ return impulse(self, X0=X0, T=T, N=N) def step(self, X0=None, T=None, N=None): """ Return the step response of a continuous-time system. See step for details. """ return step(self, X0=X0, T=T, N=N) def output(self, U, T, X0=None): """ Return the response of a continuous-time system to input U. See lsim for details. """ return lsim(self, U, T, X0=X0) def bode(self, w=None, n=100): """ Calculate Bode magnitude and phase data of a continuous-time system. Returns a 3-tuple containing arrays of frequencies [rad/s], magnitude [dB] and phase [deg]. See bode for details. Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> sys = signal.TransferFunction([1], [1, 1]) >>> w, mag, phase = sys.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 freqresp for details. """ return freqresp(self, w=w, n=n) def to_discrete(self, dt, method='zoh', alpha=None): """Return a discretized version of the current system. Parameters: See cont2discrete for details. Returns ------- sys: instance of dlti """ raise NotImplementedError('to_discrete is not implemented for this ' 'system class.') class dlti(LinearTimeInvariant): """ Discrete-time linear time invariant system base class. Parameters ---------- *system: arguments The dlti class can be instantiated with either 2, 3 or 4 arguments. The following gives the number of arguments and the corresponding discrete-time subclass that is created: * 2: TransferFunction: (numerator, denominator) * 3: ZerosPolesGain: (zeros, poles, gain) * 4: StateSpace: (A, B, C, D) Each argument can be an array or a sequence. dt: float, optional Sampling time [s] of the discrete-time systems. Defaults to True (unspecified sampling time). Must be specified as a keyword argument, for example, dt=0.1. See Also -------- ZerosPolesGain, StateSpace, TransferFunction, lti Notes ----- dlti instances do not exist directly. Instead, dlti creates an instance of one of its subclasses: StateSpace, TransferFunction or ZerosPolesGain. Changing the value of properties that are not directly part of the current system representation (such as the zeros of a StateSpace system) is very inefficient and may lead to numerical inaccuracies. It is better to convert to the specific system representation first. For example, call sys = sys.to_zpk() before accessing/changing the zeros, poles or gain. If (numerator, denominator) is passed in for *system, coefficients for both the numerator and denominator should be specified in descending exponent order (e.g., z^2 + 3z + 5 would be represented as [1, 3, 5]). .. versionadded:: 0.18.0 Examples -------- >>> from scipy import signal >>> signal.dlti(1, 2, 3, 4) StateSpaceDiscrete( array([[1]]), array([[2]]), array([[3]]), array([[4]]), dt: True ) >>> signal.dlti(1, 2, 3, 4, dt=0.1) StateSpaceDiscrete( array([[1]]), array([[2]]), array([[3]]), array([[4]]), dt: 0.1 ) >>> signal.dlti([1, 2], [3, 4], 5, dt=0.1) ZerosPolesGainDiscrete( array([1, 2]), array([3, 4]), 5, dt: 0.1 ) >>> signal.dlti([3, 4], [1, 2], dt=0.1) TransferFunctionDiscrete( array([ 3., 4.]), array([ 1., 2.]), dt: 0.1 ) """ def __new__(cls, *system, **kwargs): """Create an instance of the appropriate subclass.""" if cls is dlti: N = len(system) if N == 2: return TransferFunctionDiscrete.__new__( TransferFunctionDiscrete, *system, **kwargs) elif N == 3: return ZerosPolesGainDiscrete.__new__(ZerosPolesGainDiscrete, *system, **kwargs) elif N == 4: return StateSpaceDiscrete.__new__(StateSpaceDiscrete, *system, **kwargs) else: raise ValueError("system needs to be an instance of dlti " "or have 2, 3 or 4 arguments.") # __new__ was called from a subclass, let it call its own functions return super(dlti, cls).__new__(cls) def __init__(self, *system, **kwargs): """ Initialize the lti baseclass. The heavy lifting is done by the subclasses. """ dt = kwargs.pop('dt', True) super(dlti, self).__init__(*system, **kwargs) self.dt = dt @property def dt(self): """Return the sampling time of the system.""" return self._dt @dt.setter def dt(self, dt): self._dt = dt def impulse(self, x0=None, t=None, n=None): """ Return the impulse response of the discrete-time dlti system. See dimpulse for details. """ return dimpulse(self, x0=x0, t=t, n=n) def step(self, x0=None, t=None, n=None): """ Return the step response of the discrete-time dlti system. See dstep for details. """ return dstep(self, x0=x0, t=t, n=n) def output(self, u, t, x0=None): """ Return the response of the discrete-time system to input u. See dlsim for details. """ return dlsim(self, u, t, x0=x0) def bode(self, w=None, n=100): """ Calculate Bode magnitude and phase data of a discrete-time system. Returns a 3-tuple containing arrays of frequencies [rad/s], magnitude [dB] and phase [deg]. See dbode for details. Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt Transfer function: H(z) = 1 / (z^2 + 2z + 3) with sampling time 0.5s >>> sys = signal.TransferFunction([1], [1, 2, 3], dt=0.5) Equivalent: signal.dbode(sys) >>> w, mag, phase = sys.bode() >>> plt.figure() >>> plt.semilogx(w, mag) # Bode magnitude plot >>> plt.figure() >>> plt.semilogx(w, phase) # Bode phase plot >>> plt.show() """ return dbode(self, w=w, n=n) def freqresp(self, w=None, n=10000, whole=False): """ Calculate the frequency response of a discrete-time system. Returns a 2-tuple containing arrays of frequencies [rad/s] and complex magnitude. See dfreqresp for details. """ return dfreqresp(self, w=w, n=n, whole=whole) class TransferFunction(LinearTimeInvariant): r"""Linear Time Invariant system class in transfer function form. Represents the system as the continuous-time transfer function :math:H(s)=\sum_{i=0}^N b[N-i] s^i / \sum_{j=0}^M a[M-j] s^j or the discrete-time transfer function :math:H(s)=\sum_{i=0}^N b[N-i] z^i / \sum_{j=0}^M a[M-j] z^j, where :math:b are elements of the numerator num, :math:a are elements of the denominator den, and N == len(b) - 1, M == len(a) - 1. TransferFunction systems inherit additional functionality from the lti, respectively the dlti classes, depending on which system representation is used. Parameters ---------- *system: arguments The TransferFunction class can be instantiated with 1 or 2 arguments. The following gives the number of input arguments and their interpretation: * 1: lti or dlti system: (StateSpace, TransferFunction or ZerosPolesGain) * 2: array_like: (numerator, denominator) dt: float, optional Sampling time [s] of the discrete-time systems. Defaults to None (continuous-time). Must be specified as a keyword argument, for example, dt=0.1. See Also -------- ZerosPolesGain, StateSpace, lti, dlti tf2ss, tf2zpk, tf2sos Notes ----- Changing the value of properties that are not part of the TransferFunction system representation (such as the A, B, C, D state-space matrices) is very inefficient and may lead to numerical inaccuracies. It is better to convert to the specific system representation first. For example, call sys = sys.to_ss() before accessing/changing the A, B, C, D system matrices. If (numerator, denominator) is passed in for *system, coefficients for both the numerator and denominator should be specified in descending exponent order (e.g. s^2 + 3s + 5 or z^2 + 3z + 5 would be represented as [1, 3, 5]) Examples -------- Construct the transfer function: .. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1} >>> from scipy import signal >>> num = [1, 3, 3] >>> den = [1, 2, 1] >>> signal.TransferFunction(num, den) TransferFunctionContinuous( array([ 1., 3., 3.]), array([ 1., 2., 1.]), dt: None ) Contruct the transfer function with a sampling time of 0.5 seconds: .. math:: H(z) = \frac{z^2 + 3z + 3}{z^2 + 2z + 1} >>> signal.TransferFunction(num, den, dt=0.1) TransferFunctionDiscrete( array([ 1., 3., 3.]), array([ 1., 2., 1.]), dt: 0.1 ) """ def __new__(cls, *system, **kwargs): """Handle object conversion if input is an instance of lti.""" if len(system) == 1 and isinstance(system[0], LinearTimeInvariant): return system[0].to_tf() # Choose whether to inherit from lti or from dlti if cls is TransferFunction: if kwargs.get('dt') is None: return TransferFunctionContinuous.__new__( TransferFunctionContinuous, *system, **kwargs) else: return TransferFunctionDiscrete.__new__( TransferFunctionDiscrete, *system, **kwargs) # No special conversion needed return super(TransferFunction, cls).__new__(cls) def __init__(self, *system, **kwargs): """Initialize the state space LTI system.""" # Conversion of lti instances is handled in __new__ if isinstance(system[0], LinearTimeInvariant): return # Remove system arguments, not needed by parents anymore super(TransferFunction, self).__init__(**kwargs) self._num = None self._den = None self.num, self.den = normalize(*system) def __repr__(self): """Return representation of the system's transfer function""" return '{0}(\n{1},\n{2},\ndt: {3}\n)'.format( self.__class__.__name__, repr(self.num), repr(self.den), repr(self.dt), ) @property def num(self): """Numerator of the TransferFunction system.""" return self._num @num.setter def num(self, num): self._num = atleast_1d(num) # Update dimensions if len(self.num.shape) > 1: self.outputs, self.inputs = self.num.shape else: self.outputs = 1 self.inputs = 1 @property def den(self): """Denominator of the TransferFunction system.""" return self._den @den.setter def den(self, den): self._den = atleast_1d(den) def _copy(self, system): """ Copy the parameters of another TransferFunction object Parameters ---------- system : TransferFunction The StateSpace system that is to be copied """ self.num = system.num self.den = system.den def to_tf(self): """ Return a copy of the current TransferFunction system. Returns ------- sys : instance of TransferFunction The current system (copy) """ return copy.deepcopy(self) def to_zpk(self): """ Convert system representation to ZerosPolesGain. Returns ------- sys : instance of ZerosPolesGain Zeros, poles, gain representation of the current system """ return ZerosPolesGain(*tf2zpk(self.num, self.den), **self._dt_dict) def to_ss(self): """ Convert system representation to StateSpace. Returns ------- sys : instance of StateSpace State space model of the current system """ return StateSpace(*tf2ss(self.num, self.den), **self._dt_dict) @staticmethod def _z_to_zinv(num, den): """Change a transfer function from the variable z to z**-1. Parameters ---------- num, den: 1d array_like Sequences representing the coefficients of the numerator and denominator polynomials, in order of descending degree of 'z'. That is, 5z**2 + 3z + 2 is presented as [5, 3, 2]. Returns ------- num, den: 1d array_like Sequences representing the coefficients of the numerator and denominator polynomials, in order of ascending degree of 'z**-1'. That is, 5 + 3 z**-1 + 2 z**-2 is presented as [5, 3, 2]. """ diff = len(num) - len(den) if diff > 0: den = np.hstack((np.zeros(diff), den)) elif diff < 0: num = np.hstack((np.zeros(-diff), num)) return num, den @staticmethod def _zinv_to_z(num, den): """Change a transfer function from the variable z to z**-1. Parameters ---------- num, den: 1d array_like Sequences representing the coefficients of the numerator and denominator polynomials, in order of ascending degree of 'z**-1'. That is, 5 + 3 z**-1 + 2 z**-2 is presented as [5, 3, 2]. Returns ------- num, den: 1d array_like Sequences representing the coefficients of the numerator and denominator polynomials, in order of descending degree of 'z'. That is, 5z**2 + 3z + 2 is presented as [5, 3, 2]. """ diff = len(num) - len(den) if diff > 0: den = np.hstack((den, np.zeros(diff))) elif diff < 0: num = np.hstack((num, np.zeros(-diff))) return num, den class TransferFunctionContinuous(TransferFunction, lti): r""" Continuous-time Linear Time Invariant system in transfer function form. Represents the system as the transfer function :math:H(s)=\sum_{i=0}^N b[N-i] s^i / \sum_{j=0}^M a[M-j] s^j, where :math:b are elements of the numerator num, :math:a are elements of the denominator den, and N == len(b) - 1, M == len(a) - 1. Continuous-time TransferFunction systems inherit additional functionality from the lti class. Parameters ---------- *system: arguments The TransferFunction class can be instantiated with 1 or 2 arguments. The following gives the number of input arguments and their interpretation: * 1: lti system: (StateSpace, TransferFunction or ZerosPolesGain) * 2: array_like: (numerator, denominator) See Also -------- ZerosPolesGain, StateSpace, lti tf2ss, tf2zpk, tf2sos Notes ----- Changing the value of properties that are not part of the TransferFunction system representation (such as the A, B, C, D state-space matrices) is very inefficient and may lead to numerical inaccuracies. It is better to convert to the specific system representation first. For example, call sys = sys.to_ss() before accessing/changing the A, B, C, D system matrices. If (numerator, denominator) is passed in for *system, coefficients for both the numerator and denominator should be specified in descending exponent order (e.g. s^2 + 3s + 5 would be represented as [1, 3, 5]) Examples -------- Construct the transfer function: .. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1} >>> from scipy import signal >>> num = [1, 3, 3] >>> den = [1, 2, 1] >>> signal.TransferFunction(num, den) TransferFunctionContinuous( array([ 1., 3., 3.]), array([ 1., 2., 1.]), dt: None ) """ def to_discrete(self, dt, method='zoh', alpha=None): """ Returns the discretized TransferFunction system. Parameters: See cont2discrete for details. Returns ------- sys: instance of dlti and StateSpace """ return TransferFunction(*cont2discrete((self.num, self.den), dt, method=method, alpha=alpha)[:-1], dt=dt) class TransferFunctionDiscrete(TransferFunction, dlti): r""" Discrete-time Linear Time Invariant system in transfer function form. Represents the system as the transfer function :math:H(z)=\sum_{i=0}^N b[N-i] z^i / \sum_{j=0}^M a[M-j] z^j, where :math:b are elements of the numerator num, :math:a are elements of the denominator den, and N == len(b) - 1, M == len(a) - 1. Discrete-time TransferFunction systems inherit additional functionality from the dlti class. Parameters ---------- *system: arguments The TransferFunction class can be instantiated with 1 or 2 arguments. The following gives the number of input arguments and their interpretation: * 1: dlti system: (StateSpace, TransferFunction or ZerosPolesGain) * 2: array_like: (numerator, denominator) dt: float, optional Sampling time [s] of the discrete-time systems. Defaults to True (unspecified sampling time). Must be specified as a keyword argument, for example, dt=0.1. See Also -------- ZerosPolesGain, StateSpace, dlti tf2ss, tf2zpk, tf2sos Notes ----- Changing the value of properties that are not part of the TransferFunction system representation (such as the A, B, C, D state-space matrices) is very inefficient and may lead to numerical inaccuracies. If (numerator, denominator) is passed in for *system, coefficients for both the numerator and denominator should be specified in descending exponent order (e.g., z^2 + 3z + 5 would be represented as [1, 3, 5]). Examples -------- Construct the transfer function with a sampling time of 0.5 seconds: .. math:: H(z) = \frac{z^2 + 3z + 3}{z^2 + 2z + 1} >>> from scipy import signal >>> num = [1, 3, 3] >>> den = [1, 2, 1] >>> signal.TransferFunction(num, den, 0.5) TransferFunctionDiscrete( array([ 1., 3., 3.]), array([ 1., 2., 1.]), dt: 0.5 ) """ pass class ZerosPolesGain(LinearTimeInvariant): r""" Linear Time Invariant system class in zeros, poles, gain form. Represents the system as the continuous- or discrete-time transfer function :math:H(s)=k \prod_i (s - z[i]) / \prod_j (s - p[j]), where :math:k is the gain, :math:z are the zeros and :math:p are the poles. ZerosPolesGain systems inherit additional functionality from the lti, respectively the dlti classes, depending on which system representation is used. Parameters ---------- *system : arguments The ZerosPolesGain class can be instantiated with 1 or 3 arguments. The following gives the number of input arguments and their interpretation: * 1: lti or dlti system: (StateSpace, TransferFunction or ZerosPolesGain) * 3: array_like: (zeros, poles, gain) dt: float, optional Sampling time [s] of the discrete-time systems. Defaults to None (continuous-time). Must be specified as a keyword argument, for example, dt=0.1. See Also -------- TransferFunction, StateSpace, lti, dlti zpk2ss, zpk2tf, zpk2sos Notes ----- Changing the value of properties that are not part of the ZerosPolesGain system representation (such as the A, B, C, D state-space matrices) is very inefficient and may lead to numerical inaccuracies. It is better to convert to the specific system representation first. For example, call sys = sys.to_ss() before accessing/changing the A, B, C, D system matrices. Examples -------- >>> from scipy import signal Transfer function: H(s) = 5(s - 1)(s - 2) / (s - 3)(s - 4) >>> signal.ZerosPolesGain([1, 2], [3, 4], 5) ZerosPolesGainContinuous( array([1, 2]), array([3, 4]), 5, dt: None ) Transfer function: H(z) = 5(z - 1)(z - 2) / (z - 3)(z - 4) >>> signal.ZerosPolesGain([1, 2], [3, 4], 5, dt=0.1) ZerosPolesGainDiscrete( array([1, 2]), array([3, 4]), 5, dt: 0.1 ) """ def __new__(cls, *system, **kwargs): """Handle object conversion if input is an instance of lti""" if len(system) == 1 and isinstance(system[0], LinearTimeInvariant): return system[0].to_zpk() # Choose whether to inherit from lti or from dlti if cls is ZerosPolesGain: if kwargs.get('dt') is None: return ZerosPolesGainContinuous.__new__( ZerosPolesGainContinuous, *system, **kwargs) else: return ZerosPolesGainDiscrete.__new__( ZerosPolesGainDiscrete, *system, **kwargs ) # No special conversion needed return super(ZerosPolesGain, cls).__new__(cls) def __init__(self, *system, **kwargs): """Initialize the zeros, poles, gain system.""" # Conversion of lti instances is handled in __new__ if isinstance(system[0], LinearTimeInvariant): return super(ZerosPolesGain, self).__init__(**kwargs) self._zeros = None self._poles = None self._gain = None self.zeros, self.poles, self.gain = system def __repr__(self): """Return representation of the ZerosPolesGain system.""" return '{0}(\n{1},\n{2},\n{3},\ndt: {4}\n)'.format( self.__class__.__name__, repr(self.zeros), repr(self.poles), repr(self.gain), repr(self.dt), ) @property def zeros(self): """Zeros of the ZerosPolesGain system.""" return self._zeros @zeros.setter def zeros(self, zeros): self._zeros = atleast_1d(zeros) # Update dimensions if len(self.zeros.shape) > 1: self.outputs, self.inputs = self.zeros.shape else: self.outputs = 1 self.inputs = 1 @property def poles(self): """Poles of the ZerosPolesGain system.""" return self._poles @poles.setter def poles(self, poles): self._poles = atleast_1d(poles) @property def gain(self): """Gain of the ZerosPolesGain system.""" return self._gain @gain.setter def gain(self, gain): self._gain = gain def _copy(self, system): """ Copy the parameters of another ZerosPolesGain system. Parameters ---------- system : instance of ZerosPolesGain The zeros, poles gain system that is to be copied """ self.poles = system.poles self.zeros = system.zeros self.gain = system.gain def to_tf(self): """ Convert system representation to TransferFunction. Returns ------- sys : instance of TransferFunction Transfer function of the current system """ return TransferFunction(*zpk2tf(self.zeros, self.poles, self.gain), **self._dt_dict) def to_zpk(self): """ Return a copy of the current 'ZerosPolesGain' system. Returns ------- sys : instance of ZerosPolesGain The current system (copy) """ return copy.deepcopy(self) def to_ss(self): """ Convert system representation to StateSpace. Returns ------- sys : instance of StateSpace State space model of the current system """ return StateSpace(*zpk2ss(self.zeros, self.poles, self.gain), **self._dt_dict) class ZerosPolesGainContinuous(ZerosPolesGain, lti): r""" Continuous-time Linear Time Invariant system in zeros, poles, gain form. Represents the system as the continuous time transfer function :math:H(s)=k \prod_i (s - z[i]) / \prod_j (s - p[j]), where :math:k is the gain, :math:z are the zeros and :math:p are the poles. Continuous-time ZerosPolesGain systems inherit additional functionality from the lti class. Parameters ---------- *system : arguments The ZerosPolesGain class can be instantiated with 1 or 3 arguments. The following gives the number of input arguments and their interpretation: * 1: lti system: (StateSpace, TransferFunction or ZerosPolesGain) * 3: array_like: (zeros, poles, gain) See Also -------- TransferFunction, StateSpace, lti zpk2ss, zpk2tf, zpk2sos Notes ----- Changing the value of properties that are not part of the ZerosPolesGain system representation (such as the A, B, C, D state-space matrices) is very inefficient and may lead to numerical inaccuracies. It is better to convert to the specific system representation first. For example, call sys = sys.to_ss() before accessing/changing the A, B, C, D system matrices. Examples -------- >>> from scipy import signal Transfer function: H(s) = 5(s - 1)(s - 2) / (s - 3)(s - 4) >>> signal.ZerosPolesGain([1, 2], [3, 4], 5) ZerosPolesGainContinuous( array([1, 2]), array([3, 4]), 5, dt: None ) """ def to_discrete(self, dt, method='zoh', alpha=None): """ Returns the discretized ZerosPolesGain system. Parameters: See cont2discrete for details. Returns ------- sys: instance of dlti and ZerosPolesGain """ return ZerosPolesGain( *cont2discrete((self.zeros, self.poles, self.gain), dt, method=method, alpha=alpha)[:-1], dt=dt) class ZerosPolesGainDiscrete(ZerosPolesGain, dlti): r""" Discrete-time Linear Time Invariant system in zeros, poles, gain form. Represents the system as the discrete-time transfer function :math:H(s)=k \prod_i (s - z[i]) / \prod_j (s - p[j]), where :math:k is the gain, :math:z are the zeros and :math:p are the poles. Discrete-time ZerosPolesGain systems inherit additional functionality from the dlti class. Parameters ---------- *system : arguments The ZerosPolesGain class can be instantiated with 1 or 3 arguments. The following gives the number of input arguments and their interpretation: * 1: dlti system: (StateSpace, TransferFunction or ZerosPolesGain) * 3: array_like: (zeros, poles, gain) dt: float, optional Sampling time [s] of the discrete-time systems. Defaults to True (unspecified sampling time). Must be specified as a keyword argument, for example, dt=0.1. See Also -------- TransferFunction, StateSpace, dlti zpk2ss, zpk2tf, zpk2sos Notes ----- Changing the value of properties that are not part of the ZerosPolesGain system representation (such as the A, B, C, D state-space matrices) is very inefficient and may lead to numerical inaccuracies. It is better to convert to the specific system representation first. For example, call sys = sys.to_ss() before accessing/changing the A, B, C, D system matrices. Examples -------- >>> from scipy import signal Transfer function: H(s) = 5(s - 1)(s - 2) / (s - 3)(s - 4) >>> signal.ZerosPolesGain([1, 2], [3, 4], 5) ZerosPolesGainContinuous( array([1, 2]), array([3, 4]), 5, dt: None ) Transfer function: H(z) = 5(z - 1)(z - 2) / (z - 3)(z - 4) >>> signal.ZerosPolesGain([1, 2], [3, 4], 5, dt=0.1) ZerosPolesGainDiscrete( array([1, 2]), array([3, 4]), 5, dt: 0.1 ) """ pass def _atleast_2d_or_none(arg): if arg is not None: return atleast_2d(arg) class StateSpace(LinearTimeInvariant): r""" Linear Time Invariant system in state-space form. Represents the system as the continuous-time, first order differential equation :math:\dot{x} = A x + B u or the discrete-time difference equation :math:x[k+1] = A x[k] + B u[k]. StateSpace systems inherit additional functionality from the lti, respectively the dlti classes, depending on which system representation is used. Parameters ---------- *system: arguments The StateSpace class can be instantiated with 1 or 3 arguments. The following gives the number of input arguments and their interpretation: * 1: lti or dlti system: (StateSpace, TransferFunction or ZerosPolesGain) * 4: array_like: (A, B, C, D) dt: float, optional Sampling time [s] of the discrete-time systems. Defaults to None (continuous-time). Must be specified as a keyword argument, for example, dt=0.1. See Also -------- TransferFunction, ZerosPolesGain, lti, dlti ss2zpk, ss2tf, zpk2sos Notes ----- Changing the value of properties that are not part of the StateSpace system representation (such as zeros or poles) is very inefficient and may lead to numerical inaccuracies. It is better to convert to the specific system representation first. For example, call sys = sys.to_zpk() before accessing/changing the zeros, poles or gain. Examples -------- >>> from scipy import signal >>> a = np.array([[0, 1], [0, 0]]) >>> b = np.array([[0], [1]]) >>> c = np.array([[1, 0]]) >>> d = np.array([[0]]) >>> sys = signal.StateSpace(a, b, c, d) >>> print(sys) StateSpaceContinuous( array([[0, 1], [0, 0]]), array([[0], [1]]), array([[1, 0]]), array([[0]]), dt: None ) >>> sys.to_discrete(0.1) StateSpaceDiscrete( array([[ 1. , 0.1], [ 0. , 1. ]]), array([[ 0.005], [ 0.1 ]]), array([[1, 0]]), array([[0]]), dt: 0.1 ) >>> a = np.array([[1, 0.1], [0, 1]]) >>> b = np.array([[0.005], [0.1]]) >>> signal.StateSpace(a, b, c, d, dt=0.1) StateSpaceDiscrete( array([[ 1. , 0.1], [ 0. , 1. ]]), array([[ 0.005], [ 0.1 ]]), array([[1, 0]]), array([[0]]), dt: 0.1 ) """ def __new__(cls, *system, **kwargs): """Create new StateSpace object and settle inheritance.""" # Handle object conversion if input is an instance of lti if len(system) == 1 and isinstance(system[0], LinearTimeInvariant): return system[0].to_ss() # Choose whether to inherit from lti or from dlti if cls is StateSpace: if kwargs.get('dt') is None: return StateSpaceContinuous.__new__(StateSpaceContinuous, *system, **kwargs) else: return StateSpaceDiscrete.__new__(StateSpaceDiscrete, *system, **kwargs) # No special conversion needed return super(StateSpace, cls).__new__(cls) def __init__(self, *system, **kwargs): """Initialize the state space lti/dlti system.""" # Conversion of lti instances is handled in __new__ if isinstance(system[0], LinearTimeInvariant): return # Remove system arguments, not needed by parents anymore super(StateSpace, self).__init__(**kwargs) self._A = None self._B = None self._C = None self._D = None self.A, self.B, self.C, self.D = abcd_normalize(*system) def __repr__(self): """Return representation of the StateSpace system.""" return '{0}(\n{1},\n{2},\n{3},\n{4},\ndt: {5}\n)'.format( self.__class__.__name__, repr(self.A), repr(self.B), repr(self.C), repr(self.D), repr(self.dt), ) @property def A(self): """State matrix of the StateSpace system.""" return self._A @A.setter def A(self, A): self._A = _atleast_2d_or_none(A) @property def B(self): """Input matrix of the StateSpace system.""" return self._B @B.setter def B(self, B): self._B = _atleast_2d_or_none(B) self.inputs = self.B.shape[-1] @property def C(self): """Output matrix of the StateSpace system.""" return self._C @C.setter def C(self, C): self._C = _atleast_2d_or_none(C) self.outputs = self.C.shape[0] @property def D(self): """Feedthrough matrix of the StateSpace system.""" return self._D @D.setter def D(self, D): self._D = _atleast_2d_or_none(D) def _copy(self, system): """ Copy the parameters of another StateSpace system. Parameters ---------- system : instance of StateSpace The state-space system that is to be copied """ self.A = system.A self.B = system.B self.C = system.C self.D = system.D def to_tf(self, **kwargs): """ Convert system representation to TransferFunction. Parameters ---------- kwargs : dict, optional Additional keywords passed to ss2zpk Returns ------- sys : instance of TransferFunction Transfer function of the current system """ return TransferFunction(*ss2tf(self._A, self._B, self._C, self._D, **kwargs), **self._dt_dict) def to_zpk(self, **kwargs): """ Convert system representation to ZerosPolesGain. Parameters ---------- kwargs : dict, optional Additional keywords passed to ss2zpk Returns ------- sys : instance of ZerosPolesGain Zeros, poles, gain representation of the current system """ return ZerosPolesGain(*ss2zpk(self._A, self._B, self._C, self._D, **kwargs), **self._dt_dict) def to_ss(self): """ Return a copy of the current StateSpace system. Returns ------- sys : instance of StateSpace The current system (copy) """ return copy.deepcopy(self) class StateSpaceContinuous(StateSpace, lti): r""" Continuous-time Linear Time Invariant system in state-space form. Represents the system as the continuous-time, first order differential equation :math:\dot{x} = A x + B u. Continuous-time StateSpace systems inherit additional functionality from the lti class. Parameters ---------- *system: arguments The StateSpace class can be instantiated with 1 or 3 arguments. The following gives the number of input arguments and their interpretation: * 1: lti system: (StateSpace, TransferFunction or ZerosPolesGain) * 4: array_like: (A, B, C, D) See Also -------- TransferFunction, ZerosPolesGain, lti ss2zpk, ss2tf, zpk2sos Notes ----- Changing the value of properties that are not part of the StateSpace system representation (such as zeros or poles) is very inefficient and may lead to numerical inaccuracies. It is better to convert to the specific system representation first. For example, call sys = sys.to_zpk() before accessing/changing the zeros, poles or gain. Examples -------- >>> from scipy import signal >>> a = np.array([[0, 1], [0, 0]]) >>> b = np.array([[0], [1]]) >>> c = np.array([[1, 0]]) >>> d = np.array([[0]]) >>> sys = signal.StateSpace(a, b, c, d) >>> print(sys) StateSpaceContinuous( array([[0, 1], [0, 0]]), array([[0], [1]]), array([[1, 0]]), array([[0]]), dt: None ) """ def to_discrete(self, dt, method='zoh', alpha=None): """ Returns the discretized StateSpace system. Parameters: See cont2discrete for details. Returns ------- sys: instance of dlti and StateSpace """ return StateSpace(*cont2discrete((self.A, self.B, self.C, self.D), dt, method=method, alpha=alpha)[:-1], dt=dt) class StateSpaceDiscrete(StateSpace, dlti): r""" Discrete-time Linear Time Invariant system in state-space form. Represents the system as the discrete-time difference equation :math:x[k+1] = A x[k] + B u[k]. StateSpace systems inherit additional functionality from the dlti class. Parameters ---------- *system: arguments The StateSpace class can be instantiated with 1 or 3 arguments. The following gives the number of input arguments and their interpretation: * 1: dlti system: (StateSpace, TransferFunction or ZerosPolesGain) * 4: array_like: (A, B, C, D) dt: float, optional Sampling time [s] of the discrete-time systems. Defaults to True (unspecified sampling time). Must be specified as a keyword argument, for example, dt=0.1. See Also -------- TransferFunction, ZerosPolesGain, dlti ss2zpk, ss2tf, zpk2sos Notes ----- Changing the value of properties that are not part of the StateSpace system representation (such as zeros or poles) is very inefficient and may lead to numerical inaccuracies. It is better to convert to the specific system representation first. For example, call sys = sys.to_zpk() before accessing/changing the zeros, poles or gain. Examples -------- >>> from scipy import signal >>> a = np.array([[1, 0.1], [0, 1]]) >>> b = np.array([[0.005], [0.1]]) >>> c = np.array([[1, 0]]) >>> d = np.array([[0]]) >>> signal.StateSpace(a, b, c, d, dt=0.1) StateSpaceDiscrete( array([[ 1. , 0.1], [ 0. , 1. ]]), array([[ 0.005], [ 0.1 ]]), array([[1, 0]]), array([[0]]), dt: 0.1 ) """ pass 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: * 1: (instance of lti) * 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 (num, den) is passed in for system, coefficients for both the numerator and denominator should be specified in descending exponent order (e.g. s^2 + 3s + 5 would be represented as [1, 3, 5]). """ if isinstance(system, lti): sys = system._as_ss() elif isinstance(system, dlti): raise AttributeError('lsim2 can only be used with continuous-time ' 'systems.') else: sys = lti(*system)._as_ss() 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 exception; 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 _cast_to_array_dtype(in1, in2): """Cast array to dtype of other array, while avoiding ComplexWarning. Those can be raised when casting complex to real. """ if numpy.issubdtype(in2.dtype, numpy.float): # dtype to cast to is not complex, so use .real in1 = in1.real.astype(in2.dtype) else: in1 = in1.astype(in2.dtype) return in1 def lsim(system, U, T, X0=None, interp=True): """ 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: * 1: (instance of lti) * 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. If U = 0 or None, a zero input is used. T : array_like The time steps at which the input is defined and at which the output is desired. Must be nonnegative, increasing, and equally spaced. X0 : array_like, optional The initial conditions on the state vector (zero by default). interp : bool, optional Whether to use linear (True, the default) or zero-order-hold (False) interpolation for the input array. Returns ------- T : 1D ndarray Time values for the output. yout : 1D ndarray System response. xout : ndarray Time evolution of the state vector. Notes ----- If (num, den) is passed in for system, coefficients for both the numerator and denominator should be specified in descending exponent order (e.g. s^2 + 3s + 5 would be represented as [1, 3, 5]). Examples -------- Simulate a double integrator y'' = u, with a constant input u = 1 >>> from scipy import signal >>> system = signal.lti([[0., 1.], [0., 0.]], [[0.], [1.]], [[1., 0.]], 0.) >>> t = np.linspace(0, 5) >>> u = np.ones_like(t) >>> tout, y, x = signal.lsim(system, u, t) >>> import matplotlib.pyplot as plt >>> plt.plot(t, y) """ if isinstance(system, lti): sys = system._as_ss() elif isinstance(system, dlti): raise AttributeError('lsim can only be used with continuous-time ' 'systems.') else: sys = lti(*system)._as_ss() T = atleast_1d(T) if len(T.shape) != 1: raise ValueError("T must be a rank-1 array.") A, B, C, D = map(np.asarray, (sys.A, sys.B, sys.C, sys.D)) n_states = A.shape[0] n_inputs = B.shape[1] n_steps = T.size if X0 is None: X0 = zeros(n_states, sys.A.dtype) xout = zeros((n_steps, n_states), sys.A.dtype) if T[0] == 0: xout[0] = X0 elif T[0] > 0: # step forward to initial time, with zero input xout[0] = dot(X0, linalg.expm(transpose(A) * T[0])) else: raise ValueError("Initial time must be nonnegative") no_input = (U is None or (isinstance(U, (int, float)) and U == 0.) or not np.any(U)) if n_steps == 1: yout = squeeze(dot(xout, transpose(C))) if not no_input: yout += squeeze(dot(U, transpose(D))) return T, squeeze(yout), squeeze(xout) dt = T[1] - T[0] if not np.allclose((T[1:] - T[:-1]) / dt, 1.0): warnings.warn("Non-uniform timesteps are deprecated. Results may be " "slow and/or inaccurate.", DeprecationWarning) return lsim2(system, U, T, X0) if no_input: # Zero input: just use matrix exponential # take transpose because state is a row vector expAT_dt = linalg.expm(transpose(A) * dt) for i in xrange(1, n_steps): xout[i] = dot(xout[i-1], expAT_dt) yout = squeeze(dot(xout, transpose(C))) return T, squeeze(yout), squeeze(xout) # Nonzero input U = atleast_1d(U) if U.ndim == 1: U = U[:, np.newaxis] if U.shape[0] != n_steps: raise ValueError("U must have the same number of rows " "as elements in T.") if U.shape[1] != n_inputs: raise ValueError("System does not define that many inputs.") if not interp: # Zero-order hold # Algorithm: to integrate from time 0 to time dt, we solve # xdot = A x + B u, x(0) = x0 # udot = 0, u(0) = u0. # # Solution is # [ x(dt) ] [ A*dt B*dt ] [ x0 ] # [ u(dt) ] = exp [ 0 0 ] [ u0 ] M = np.vstack([np.hstack([A * dt, B * dt]), np.zeros((n_inputs, n_states + n_inputs))]) # transpose everything because the state and input are row vectors expMT = linalg.expm(transpose(M)) Ad = expMT[:n_states, :n_states] Bd = expMT[n_states:, :n_states] for i in xrange(1, n_steps): xout[i] = dot(xout[i-1], Ad) + dot(U[i-1], Bd) else: # Linear interpolation between steps # Algorithm: to integrate from time 0 to time dt, with linear # interpolation between inputs u(0) = u0 and u(dt) = u1, we solve # xdot = A x + B u, x(0) = x0 # udot = (u1 - u0) / dt, u(0) = u0. # # Solution is # [ x(dt) ] [ A*dt B*dt 0 ] [ x0 ] # [ u(dt) ] = exp [ 0 0 I ] [ u0 ] # [u1 - u0] [ 0 0 0 ] [u1 - u0] M = np.vstack([np.hstack([A * dt, B * dt, np.zeros((n_states, n_inputs))]), np.hstack([np.zeros((n_inputs, n_states + n_inputs)), np.identity(n_inputs)]), np.zeros((n_inputs, n_states + 2 * n_inputs))]) expMT = linalg.expm(transpose(M)) Ad = expMT[:n_states, :n_states] Bd1 = expMT[n_states+n_inputs:, :n_states] Bd0 = expMT[n_states:n_states + n_inputs, :n_states] - Bd1 for i in xrange(1, n_steps): xout[i] = (dot(xout[i-1], Ad) + dot(U[i-1], Bd0) + dot(U[i], Bd1)) yout = (squeeze(dot(xout, transpose(C))) + squeeze(dot(U, transpose(D)))) 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 : array_like 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. # TODO: 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 impulse(system, X0=None, T=None, N=None): """Impulse response of continuous-time system. Parameters ---------- system : an instance of the LTI class or a tuple of array_like describing the system. The following gives the number of elements in the tuple and the interpretation: * 1 (instance of lti) * 2 (num, den) * 3 (zeros, poles, gain) * 4 (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). Notes ----- If (num, den) is passed in for system, coefficients for both the numerator and denominator should be specified in descending exponent order (e.g. s^2 + 3s + 5 would be represented as [1, 3, 5]). """ if isinstance(system, lti): sys = system._as_ss() elif isinstance(system, dlti): raise AttributeError('impulse can only be used with continuous-time ' 'systems.') else: sys = lti(*system)._as_ss() if X0 is None: X = squeeze(sys.B) else: X = squeeze(sys.B + X0) if N is None: N = 100 if T is None: T = _default_response_times(sys.A, N) else: T = asarray(T) _, h, _ = lsim(sys, 0., T, X, interp=False) 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 of array_like describing the system. The following gives the number of elements in the tuple and the interpretation: * 1 (instance of lti) * 2 (num, den) * 3 (zeros, poles, gain) * 4 (A, B, C, D) X0 : 1-D array_like, optional The initial condition of the state vector. Default: 0 (the zero vector). 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. 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. If (num, den) is passed in for system, coefficients for both the numerator and denominator should be specified in descending exponent order (e.g. s^2 + 3s + 5 would be represented as [1, 3, 5]). .. 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._as_ss() elif isinstance(system, dlti): raise AttributeError('impulse2 can only be used with continuous-time ' 'systems.') else: sys = lti(*system)._as_ss() 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(). ic = B + X0 Tr, Yr, Xr = lsim2(sys, T=T, X0=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 of array_like describing the system. The following gives the number of elements in the tuple and the interpretation: * 1 (instance of lti) * 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, optional 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 Notes ----- If (num, den) is passed in for system, coefficients for both the numerator and denominator should be specified in descending exponent order (e.g. s^2 + 3s + 5 would be represented as [1, 3, 5]). """ if isinstance(system, lti): sys = system._as_ss() elif isinstance(system, dlti): raise AttributeError('step can only be used with continuous-time ' 'systems.') else: sys = lti(*system)._as_ss() if N is None: N = 100 if T is None: T = _default_response_times(sys.A, N) else: T = asarray(T) U = ones(T.shape, sys.A.dtype) vals = lsim(sys, U, T, X0=X0, interp=False) 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 of array_like describing the system. The following gives the number of elements in the tuple and the interpretation: * 1 (instance of lti) * 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, optional Number of time points to compute if T is not given. kwargs : various types 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 ----- If (num, den) is passed in for system, coefficients for both the numerator and denominator should be specified in descending exponent order (e.g. s^2 + 3s + 5 would be represented as [1, 3, 5]). .. versionadded:: 0.8.0 """ if isinstance(system, lti): sys = system._as_ss() elif isinstance(system, dlti): raise AttributeError('step2 can only be used with continuous-time ' 'systems.') else: sys = lti(*system)._as_ss() if N is None: N = 100 if T is None: T = _default_response_times(sys.A, N) else: T = asarray(T) 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. 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: * 1 (instance of lti) * 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 an interval chosen to include the influence of the poles and zeros of the system. Returns ------- w : 1D ndarray Frequency array [rad/s] mag : 1D ndarray Magnitude array [dB] phase : 1D ndarray Phase array [deg] Notes ----- If (num, den) is passed in for system, coefficients for both the numerator and denominator should be specified in descending exponent order (e.g. s^2 + 3s + 5 would be represented as [1, 3, 5]). .. versionadded:: 0.11.0 Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> sys = signal.TransferFunction([1], [1, 1]) >>> w, mag, phase = signal.bode(sys) >>> plt.figure() >>> plt.semilogx(w, mag) # Bode magnitude plot >>> plt.figure() >>> plt.semilogx(w, phase) # Bode phase plot >>> plt.show() """ w, y = freqresp(system, w=w, n=n) mag = 20.0 * numpy.log10(abs(y)) phase = numpy.unwrap(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: * 1 (instance of lti) * 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 an interval chosen to include the influence of the poles and zeros of the system. Returns ------- w : 1D ndarray Frequency array [rad/s] H : 1D ndarray Array of complex magnitude values Notes ----- If (num, den) is passed in for system, coefficients for both the numerator and denominator should be specified in descending exponent order (e.g. s^2 + 3s + 5 would be represented as [1, 3, 5]). Examples -------- Generating the Nyquist plot of a transfer function >>> from scipy import signal >>> import matplotlib.pyplot as plt Transfer function: H(s) = 5 / (s-1)^3 >>> s1 = signal.ZerosPolesGain([], [1, 1, 1], [5]) >>> 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._as_tf() elif isinstance(system, dlti): raise AttributeError('freqresp can only be used with continuous-time ' 'systems.') else: sys = lti(*system)._as_tf() if sys.inputs != 1 or sys.outputs != 1: raise ValueError("freqresp() requires a SISO (single input, single " "output) system.") if w is not None: worN = w else: worN = n # In the call to freqs(), sys.num.ravel() is used because there are # cases where sys.num is a 2-D array with a single row. w, h = freqs(sys.num.ravel(), sys.den, worN=worN) return w, h # This class will be used by place_poles to return its results # see http://code.activestate.com/recipes/52308/ class Bunch: def __init__(self, **kwds): self.__dict__.update(kwds) def _valid_inputs(A, B, poles, method, rtol, maxiter): """ Check the poles come in complex conjugage pairs Check shapes of A, B and poles are compatible. Check the method chosen is compatible with provided poles Return update method to use and ordered poles """ poles = np.asarray(poles) if poles.ndim > 1: raise ValueError("Poles must be a 1D array like.") # Will raise ValueError if poles do not come in complex conjugates pairs poles = _order_complex_poles(poles) if A.ndim > 2: raise ValueError("A must be a 2D array/matrix.") if B.ndim > 2: raise ValueError("B must be a 2D array/matrix") if A.shape[0] != A.shape[1]: raise ValueError("A must be square") if len(poles) > A.shape[0]: raise ValueError("maximum number of poles is %d but you asked for %d" % (A.shape[0], len(poles))) if len(poles) < A.shape[0]: raise ValueError("number of poles is %d but you should provide %d" % (len(poles), A.shape[0])) r = np.linalg.matrix_rank(B) for p in poles: if sum(p == poles) > r: raise ValueError("at least one of the requested pole is repeated " "more than rank(B) times") # Choose update method update_loop = _YT_loop if method not in ('KNV0','YT'): raise ValueError("The method keyword must be one of 'YT' or 'KNV0'") if method == "KNV0": update_loop = _KNV0_loop if not all(np.isreal(poles)): raise ValueError("Complex poles are not supported by KNV0") if maxiter < 1: raise ValueError("maxiter must be at least equal to 1") # We do not check rtol <= 0 as the user can use a negative rtol to # force maxiter iterations if rtol > 1: raise ValueError("rtol can not be greater than 1") return update_loop, poles def _order_complex_poles(poles): """ Check we have complex conjugates pairs and reorder P according to YT, ie real_poles, complex_i, conjugate complex_i, .... The lexicographic sort on the complex poles is added to help the user to compare sets of poles. """ ordered_poles = np.sort(poles[np.isreal(poles)]) im_poles = [] for p in np.sort(poles[np.imag(poles) < 0]): if np.conj(p) in poles: im_poles.extend((p, np.conj(p))) ordered_poles = np.hstack((ordered_poles, im_poles)) if poles.shape[0] != len(ordered_poles): raise ValueError("Complex poles must come with their conjugates") return ordered_poles def _KNV0(B, ker_pole, transfer_matrix, j, poles): """ Algorithm "KNV0" Kautsky et Al. Robust pole assignment in linear state feedback, Int journal of Control 1985, vol 41 p 1129->1155 http://la.epfl.ch/files/content/sites/la/files/ users/105941/public/KautskyNicholsDooren """ # Remove xj form the base transfer_matrix_not_j = np.delete(transfer_matrix, j, axis=1) # If we QR this matrix in full mode Q=Q0|Q1 # then Q1 will be a single column orthogonnal to # Q0, that's what we are looking for ! # After merge of gh-4249 great speed improvements could be achieved # using QR updates instead of full QR in the line below # To debug with numpy qr uncomment the line below # Q, R = np.linalg.qr(transfer_matrix_not_j, mode="complete") Q, R = s_qr(transfer_matrix_not_j, mode="full") mat_ker_pj = np.dot(ker_pole[j], ker_pole[j].T) yj = np.dot(mat_ker_pj, Q[:, -1]) # If Q[:, -1] is "almost" orthogonal to ker_pole[j] its # projection into ker_pole[j] will yield a vector # close to 0. As we are looking for a vector in ker_pole[j] # simply stick with transfer_matrix[:, j] (unless someone provides me with # a better choice ?) if not np.allclose(yj, 0): xj = yj/np.linalg.norm(yj) transfer_matrix[:, j] = xj # KNV does not support complex poles, using YT technique the two lines # below seem to work 9 out of 10 times but it is not reliable enough: # transfer_matrix[:, j]=real(xj) # transfer_matrix[:, j+1]=imag(xj) # Add this at the beginning of this function if you wish to test # complex support: # if ~np.isreal(P[j]) and (j>=B.shape[0]-1 or P[j]!=np.conj(P[j+1])): # return # Problems arise when imag(xj)=>0 I have no idea on how to fix this def _YT_real(ker_pole, Q, transfer_matrix, i, j): """ Applies algorithm from YT section 6.1 page 19 related to real pairs """ # step 1 page 19 u = Q[:, -2, np.newaxis] v = Q[:, -1, np.newaxis] # step 2 page 19 m = np.dot(np.dot(ker_pole[i].T, np.dot(u, v.T) - np.dot(v, u.T)), ker_pole[j]) # step 3 page 19 um, sm, vm = np.linalg.svd(m) # mu1, mu2 two first columns of U => 2 first lines of U.T mu1, mu2 = um.T[:2, :, np.newaxis] # VM is V.T with numpy we want the first two lines of V.T nu1, nu2 = vm[:2, :, np.newaxis] # what follows is a rough python translation of the formulas # in section 6.2 page 20 (step 4) transfer_matrix_j_mo_transfer_matrix_j = np.vstack(( transfer_matrix[:, i, np.newaxis], transfer_matrix[:, j, np.newaxis])) if not np.allclose(sm[0], sm[1]): ker_pole_imo_mu1 = np.dot(ker_pole[i], mu1) ker_pole_i_nu1 = np.dot(ker_pole[j], nu1) ker_pole_mu_nu = np.vstack((ker_pole_imo_mu1, ker_pole_i_nu1)) else: ker_pole_ij = np.vstack(( np.hstack((ker_pole[i], np.zeros(ker_pole[i].shape))), np.hstack((np.zeros(ker_pole[j].shape), ker_pole[j])) )) mu_nu_matrix = np.vstack( (np.hstack((mu1, mu2)), np.hstack((nu1, nu2))) ) ker_pole_mu_nu = np.dot(ker_pole_ij, mu_nu_matrix) transfer_matrix_ij = np.dot(np.dot(ker_pole_mu_nu, ker_pole_mu_nu.T), transfer_matrix_j_mo_transfer_matrix_j) if not np.allclose(transfer_matrix_ij, 0): transfer_matrix_ij = (np.sqrt(2)*transfer_matrix_ij / np.linalg.norm(transfer_matrix_ij)) transfer_matrix[:, i] = transfer_matrix_ij[ :transfer_matrix[:, i].shape[0], 0 ] transfer_matrix[:, j] = transfer_matrix_ij[ transfer_matrix[:, i].shape[0]:, 0 ] else: # As in knv0 if transfer_matrix_j_mo_transfer_matrix_j is orthogonal to # Vect{ker_pole_mu_nu} assign transfer_matrixi/transfer_matrix_j to # ker_pole_mu_nu and iterate. As we are looking for a vector in # Vect{Matker_pole_MU_NU} (see section 6.1 page 19) this might help # (that's a guess, not a claim !) transfer_matrix[:, i] = ker_pole_mu_nu[ :transfer_matrix[:, i].shape[0], 0 ] transfer_matrix[:, j] = ker_pole_mu_nu[ transfer_matrix[:, i].shape[0]:, 0 ] def _YT_complex(ker_pole, Q, transfer_matrix, i, j): """ Applies algorithm from YT section 6.2 page 20 related to complex pairs """ # step 1 page 20 ur = np.sqrt(2)*Q[:, -2, np.newaxis] ui = np.sqrt(2)*Q[:, -1, np.newaxis] u = ur + 1j*ui # step 2 page 20 ker_pole_ij = ker_pole[i] m = np.dot(np.dot(np.conj(ker_pole_ij.T), np.dot(u, np.conj(u).T) - np.dot(np.conj(u), u.T)), ker_pole_ij) # step 3 page 20 e_val, e_vec = np.linalg.eig(m) # sort eigenvalues according to their module e_val_idx = np.argsort(np.abs(e_val)) mu1 = e_vec[:, e_val_idx[-1], np.newaxis] mu2 = e_vec[:, e_val_idx[-2], np.newaxis] # what follows is a rough python translation of the formulas # in section 6.2 page 20 (step 4) # remember transfer_matrix_i has been split as # transfer_matrix[i]=real(transfer_matrix_i) and # transfer_matrix[j]=imag(transfer_matrix_i) transfer_matrix_j_mo_transfer_matrix_j = ( transfer_matrix[:, i, np.newaxis] + 1j*transfer_matrix[:, j, np.newaxis] ) if not np.allclose(np.abs(e_val[e_val_idx[-1]]), np.abs(e_val[e_val_idx[-2]])): ker_pole_mu = np.dot(ker_pole_ij, mu1) else: mu1_mu2_matrix = np.hstack((mu1, mu2)) ker_pole_mu = np.dot(ker_pole_ij, mu1_mu2_matrix) transfer_matrix_i_j = np.dot(np.dot(ker_pole_mu, np.conj(ker_pole_mu.T)), transfer_matrix_j_mo_transfer_matrix_j) if not np.allclose(transfer_matrix_i_j, 0): transfer_matrix_i_j = (transfer_matrix_i_j / np.linalg.norm(transfer_matrix_i_j)) transfer_matrix[:, i] = np.real(transfer_matrix_i_j[:, 0]) transfer_matrix[:, j] = np.imag(transfer_matrix_i_j[:, 0]) else: # same idea as in YT_real transfer_matrix[:, i] = np.real(ker_pole_mu[:, 0]) transfer_matrix[:, j] = np.imag(ker_pole_mu[:, 0]) def _YT_loop(ker_pole, transfer_matrix, poles, B, maxiter, rtol): """ Algorithm "YT" Tits, Yang. Globally Convergent Algorithms for Robust Pole Assignment by State Feedback http://drum.lib.umd.edu/handle/1903/5598 The poles P have to be sorted accordingly to section 6.2 page 20 """ # The IEEE edition of the YT paper gives useful information on the # optimal update order for the real poles in order to minimize the number # of times we have to loop over all poles, see page 1442 nb_real = poles[np.isreal(poles)].shape[0] # hnb => Half Nb Real hnb = nb_real // 2 # Stick to the indices in the paper and then remove one to get numpy array # index it is a bit easier to link the code to the paper this way even if it # is not very clean. The paper is unclear about what should be done when # there is only one real pole => use KNV0 on this real pole seem to work if nb_real > 0: #update the biggest real pole with the smallest one update_order = [[nb_real], [1]] else: update_order = [[],[]] r_comp = np.arange(nb_real+1, len(poles)+1, 2) # step 1.a r_p = np.arange(1, hnb+nb_real % 2) update_order[0].extend(2*r_p) update_order[1].extend(2*r_p+1) # step 1.b update_order[0].extend(r_comp) update_order[1].extend(r_comp+1) # step 1.c r_p = np.arange(1, hnb+1) update_order[0].extend(2*r_p-1) update_order[1].extend(2*r_p) # step 1.d if hnb == 0 and np.isreal(poles[0]): update_order[0].append(1) update_order[1].append(1) update_order[0].extend(r_comp) update_order[1].extend(r_comp+1) # step 2.a r_j = np.arange(2, hnb+nb_real % 2) for j in r_j: for i in range(1, hnb+1): update_order[0].append(i) update_order[1].append(i+j) # step 2.b if hnb == 0 and np.isreal(poles[0]): update_order[0].append(1) update_order[1].append(1) update_order[0].extend(r_comp) update_order[1].extend(r_comp+1) # step 2.c r_j = np.arange(2, hnb+nb_real % 2) for j in r_j: for i in range(hnb+1, nb_real+1): idx_1 = i+j if idx_1 > nb_real: idx_1 = i+j-nb_real update_order[0].append(i) update_order[1].append(idx_1) # step 2.d if hnb == 0 and np.isreal(poles[0]): update_order[0].append(1) update_order[1].append(1) update_order[0].extend(r_comp) update_order[1].extend(r_comp+1) # step 3.a for i in range(1, hnb+1): update_order[0].append(i) update_order[1].append(i+hnb) # step 3.b if hnb == 0 and np.isreal(poles[0]): update_order[0].append(1) update_order[1].append(1) update_order[0].extend(r_comp) update_order[1].extend(r_comp+1) update_order = np.array(update_order).T-1 stop = False nb_try = 0 while nb_try < maxiter and not stop: det_transfer_matrixb = np.abs(np.linalg.det(transfer_matrix)) for i, j in update_order: if i == j: assert i == 0, "i!=0 for KNV call in YT" assert np.isreal(poles[i]), "calling KNV on a complex pole" _KNV0(B, ker_pole, transfer_matrix, i, poles) else: transfer_matrix_not_i_j = np.delete(transfer_matrix, (i, j), axis=1) # after merge of gh-4249 great speed improvements could be # achieved using QR updates instead of full QR in the line below #to debug with numpy qr uncomment the line below #Q, _ = np.linalg.qr(transfer_matrix_not_i_j, mode="complete") Q, _ = s_qr(transfer_matrix_not_i_j, mode="full") if np.isreal(poles[i]): assert np.isreal(poles[j]), "mixing real and complex " + \ "in YT_real" + str(poles) _YT_real(ker_pole, Q, transfer_matrix, i, j) else: assert ~np.isreal(poles[i]), "mixing real and complex " + \ "in YT_real" + str(poles) _YT_complex(ker_pole, Q, transfer_matrix, i, j) det_transfer_matrix = np.max((np.sqrt(np.spacing(1)), np.abs(np.linalg.det(transfer_matrix)))) cur_rtol = np.abs( (det_transfer_matrix - det_transfer_matrixb) / det_transfer_matrix) if cur_rtol < rtol and det_transfer_matrix > np.sqrt(np.spacing(1)): # Convergence test from YT page 21 stop = True nb_try += 1 return stop, cur_rtol, nb_try def _KNV0_loop(ker_pole, transfer_matrix, poles, B, maxiter, rtol): """ Loop over all poles one by one and apply KNV method 0 algorithm """ # This method is useful only because we need to be able to call # _KNV0 from YT without looping over all poles, otherwise it would # have been fine to mix _KNV0_loop and _KNV0 in a single function stop = False nb_try = 0 while nb_try < maxiter and not stop: det_transfer_matrixb = np.abs(np.linalg.det(transfer_matrix)) for j in range(B.shape[0]): _KNV0(B, ker_pole, transfer_matrix, j, poles) det_transfer_matrix = np.max((np.sqrt(np.spacing(1)), np.abs(np.linalg.det(transfer_matrix)))) cur_rtol = np.abs((det_transfer_matrix - det_transfer_matrixb) / det_transfer_matrix) if cur_rtol < rtol and det_transfer_matrix > np.sqrt(np.spacing(1)): # Convergence test from YT page 21 stop = True nb_try += 1 return stop, cur_rtol, nb_try def place_poles(A, B, poles, method="YT", rtol=1e-3, maxiter=30): """ Compute K such that eigenvalues (A - dot(B, K))=poles. K is the gain matrix such as the plant described by the linear system AX+BU will have its closed-loop poles, i.e the eigenvalues A - B*K, as close as possible to those asked for in poles. SISO, MISO and MIMO systems are supported. Parameters ---------- A, B : ndarray State-space representation of linear system AX + BU. poles : array_like Desired real poles and/or complex conjugates poles. Complex poles are only supported with method="YT" (default). method: {'YT', 'KNV0'}, optional Which method to choose to find the gain matrix K. One of: - 'YT': Yang Tits - 'KNV0': Kautsky, Nichols, Van Dooren update method 0 See References and Notes for details on the algorithms. rtol: float, optional After each iteration the determinant of the eigenvectors of A - B*K is compared to its previous value, when the relative error between these two values becomes lower than rtol the algorithm stops. Default is 1e-3. maxiter: int, optional Maximum number of iterations to compute the gain matrix. Default is 30. Returns ------- full_state_feedback : Bunch object full_state_feedback is composed of: gain_matrix : 1-D ndarray The closed loop matrix K such as the eigenvalues of A-BK are as close as possible to the requested poles. computed_poles : 1-D ndarray The poles corresponding to A-BK sorted as first the real poles in increasing order, then the complex congugates in lexicographic order. requested_poles : 1-D ndarray The poles the algorithm was asked to place sorted as above, they may differ from what was achieved. X : 2-D ndarray The transfer matrix such as X * diag(poles) = (A - B*K)*X (see Notes) rtol : float The relative tolerance achieved on det(X) (see Notes). rtol will be NaN if it is possible to solve the system diag(poles) = (A - B*K), or 0 when the optimization algorithms can't do anything i.e when B.shape[1] == 1. nb_iter : int The number of iterations performed before converging. nb_iter will be NaN if it is possible to solve the system diag(poles) = (A - B*K), or 0 when the optimization algorithms can't do anything i.e when B.shape[1] == 1. Notes ----- The Tits and Yang (YT), [2]_ paper is an update of the original Kautsky et al. (KNV) paper [1]_. KNV relies on rank-1 updates to find the transfer matrix X such that X * diag(poles) = (A - B*K)*X, whereas YT uses rank-2 updates. This yields on average more robust solutions (see [2]_ pp 21-22), furthermore the YT algorithm supports complex poles whereas KNV does not in its original version. Only update method 0 proposed by KNV has been implemented here, hence the name 'KNV0'. KNV extended to complex poles is used in Matlab's place function, YT is distributed under a non-free licence by Slicot under the name robpole. It is unclear and undocumented how KNV0 has been extended to complex poles (Tits and Yang claim on page 14 of their paper that their method can not be used to extend KNV to complex poles), therefore only YT supports them in this implementation. As the solution to the problem of pole placement is not unique for MIMO systems, both methods start with a tentative transfer matrix which is altered in various way to increase its determinant. Both methods have been proven to converge to a stable solution, however depending on the way the initial transfer matrix is chosen they will converge to different solutions and therefore there is absolutely no guarantee that using 'KNV0' will yield results similar to Matlab's or any other implementation of these algorithms. Using the default method 'YT' should be fine in most cases; 'KNV0' is only provided because it is needed by 'YT' in some specific cases. Furthermore 'YT' gives on average more robust results than 'KNV0' when abs(det(X)) is used as a robustness indicator. [2]_ is available as a technical report on the following URL: http://drum.lib.umd.edu/handle/1903/5598 References ---------- .. [1] J. Kautsky, N.K. Nichols and P. van Dooren, "Robust pole assignment in linear state feedback", International Journal of Control, Vol. 41 pp. 1129-1155, 1985. .. [2] 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 simple example demonstrating real pole placement using both KNV and YT algorithms. This is example number 1 from section 4 of the reference KNV publication ([1]_): >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> A = np.array([[ 1.380, -0.2077, 6.715, -5.676 ], ... [-0.5814, -4.290, 0, 0.6750 ], ... [ 1.067, 4.273, -6.654, 5.893 ], ... [ 0.0480, 4.273, 1.343, -2.104 ]]) >>> B = np.array([[ 0, 5.679 ], ... [ 1.136, 1.136 ], ... [ 0, 0, ], ... [-3.146, 0 ]]) >>> P = np.array([-0.2, -0.5, -5.0566, -8.6659]) Now compute K with KNV method 0, with the default YT method and with the YT method while forcing 100 iterations of the algorithm and print some results after each call. >>> fsf1 = signal.place_poles(A, B, P, method='KNV0') >>> fsf1.gain_matrix array([[ 0.20071427, -0.96665799, 0.24066128, -0.10279785], [ 0.50587268, 0.57779091, 0.51795763, -0.41991442]]) >>> fsf2 = signal.place_poles(A, B, P) # uses YT method >>> fsf2.computed_poles array([-8.6659, -5.0566, -0.5 , -0.2 ]) >>> fsf3 = signal.place_poles(A, B, P, rtol=-1, maxiter=100) >>> fsf3.X array([[ 0.52072442+0.j, -0.08409372+0.j, -0.56847937+0.j, 0.74823657+0.j], [-0.04977751+0.j, -0.80872954+0.j, 0.13566234+0.j, -0.29322906+0.j], [-0.82266932+0.j, -0.19168026+0.j, -0.56348322+0.j, -0.43815060+0.j], [ 0.22267347+0.j, 0.54967577+0.j, -0.58387806+0.j, -0.40271926+0.j]]) The absolute value of the determinant of X is a good indicator to check the robustness of the results, both 'KNV0' and 'YT' aim at maximizing it. Below a comparison of the robustness of the results above: >>> abs(np.linalg.det(fsf1.X)) < abs(np.linalg.det(fsf2.X)) True >>> abs(np.linalg.det(fsf2.X)) < abs(np.linalg.det(fsf3.X)) True Now a simple example for complex poles: >>> A = np.array([[ 0, 7/3., 0, 0 ], ... [ 0, 0, 0, 7/9. ], ... [ 0, 0, 0, 0 ], ... [ 0, 0, 0, 0 ]]) >>> B = np.array([[ 0, 0 ], ... [ 0, 0 ], ... [ 1, 0 ], ... [ 0, 1 ]]) >>> P = np.array([-3, -1, -2-1j, -2+1j]) / 3. >>> fsf = signal.place_poles(A, B, P, method='YT') We can plot the desired and computed poles in the complex plane: >>> t = np.linspace(0, 2*np.pi, 401) >>> plt.plot(np.cos(t), np.sin(t), 'k--') # unit circle >>> plt.plot(fsf.requested_poles.real, fsf.requested_poles.imag, ... 'wo', label='Desired') >>> plt.plot(fsf.computed_poles.real, fsf.computed_poles.imag, 'bx', ... label='Placed') >>> plt.grid() >>> plt.axis('image') >>> plt.axis([-1.1, 1.1, -1.1, 1.1]) >>> plt.legend(bbox_to_anchor=(1.05, 1), loc=2, numpoints=1) """ # Move away all the inputs checking, it only adds noise to the code update_loop, poles = _valid_inputs(A, B, poles, method, rtol, maxiter) # The current value of the relative tolerance we achieved cur_rtol = 0 # The number of iterations needed before converging nb_iter = 0 # Step A: QR decomposition of B page 1132 KN # to debug with numpy qr uncomment the line below # u, z = np.linalg.qr(B, mode="complete") u, z = s_qr(B, mode="full") rankB = np.linalg.matrix_rank(B) u0 = u[:, :rankB] u1 = u[:, rankB:] z = z[:rankB, :] # If we can use the identity matrix as X the solution is obvious if B.shape[0] == rankB: # if B is square and full rank there is only one solution # such as (A+BK)=inv(X)*diag(P)*X with X=eye(A.shape[0]) # i.e K=inv(B)*(diag(P)-A) # if B has as many lines as its rank (but not square) there are many # solutions and we can choose one using least squares # => use lstsq in both cases. # In both cases the transfer matrix X will be eye(A.shape[0]) and I # can hardly think of a better one so there is nothing to optimize # # for complex poles we use the following trick # # |a -b| has for eigenvalues a+b and a-b # |b a| # # |a+bi 0| has the obvious eigenvalues a+bi and a-bi # |0 a-bi| # # e.g solving the first one in R gives the solution # for the second one in C diag_poles = np.zeros(A.shape) idx = 0 while idx < poles.shape[0]: p = poles[idx] diag_poles[idx, idx] = np.real(p) if ~np.isreal(p): diag_poles[idx, idx+1] = -np.imag(p) diag_poles[idx+1, idx+1] = np.real(p) diag_poles[idx+1, idx] = np.imag(p) idx += 1 # skip next one idx += 1 gain_matrix = np.linalg.lstsq(B, diag_poles-A)[0] transfer_matrix = np.eye(A.shape[0]) cur_rtol = np.nan nb_iter = np.nan else: # step A (p1144 KNV) and begining of step F: decompose # dot(U1.T, A-P[i]*I).T and build our set of transfer_matrix vectors # in the same loop ker_pole = [] # flag to skip the conjugate of a complex pole skip_conjugate = False # select orthonormal base ker_pole for each Pole and vectors for # transfer_matrix for j in range(B.shape[0]): if skip_conjugate: skip_conjugate = False continue pole_space_j = np.dot(u1.T, A-poles[j]*np.eye(B.shape[0])).T # after QR Q=Q0|Q1 # only Q0 is used to reconstruct the qr'ed (dot Q, R) matrix. # Q1 is orthogonnal to Q0 and will be multiplied by the zeros in # R when using mode "complete". In default mode Q1 and the zeros # in R are not computed # To debug with numpy qr uncomment the line below # Q, _ = np.linalg.qr(pole_space_j, mode="complete") Q, _ = s_qr(pole_space_j, mode="full") ker_pole_j = Q[:, pole_space_j.shape[1]:] # We want to select one vector in ker_pole_j to build the transfer # matrix, however qr returns sometimes vectors with zeros on the # same line for each pole and this yields very long convergence # times. # Or some other times a set of vectors, one with zero imaginary # part and one (or several) with imaginary parts. After trying # many ways to select the best possible one (eg ditch vectors # with zero imaginary part for complex poles) I ended up summing # all vectors in ker_pole_j, this solves 100% of the problems and # is a valid choice for transfer_matrix. # This way for complex poles we are sure to have a non zero # imaginary part that way, and the problem of lines full of zeros # in transfer_matrix is solved too as when a vector from # ker_pole_j has a zero the other one(s) when # ker_pole_j.shape[1]>1) for sure won't have a zero there. transfer_matrix_j = np.sum(ker_pole_j, axis=1)[:, np.newaxis] transfer_matrix_j = (transfer_matrix_j / np.linalg.norm(transfer_matrix_j)) if ~np.isreal(poles[j]): # complex pole transfer_matrix_j = np.hstack([np.real(transfer_matrix_j), np.imag(transfer_matrix_j)]) ker_pole.extend([ker_pole_j, ker_pole_j]) # Skip next pole as it is the conjugate skip_conjugate = True else: # real pole, nothing to do ker_pole.append(ker_pole_j) if j == 0: transfer_matrix = transfer_matrix_j else: transfer_matrix = np.hstack((transfer_matrix, transfer_matrix_j)) if rankB > 1: # otherwise there is nothing we can optimize stop, cur_rtol, nb_iter = update_loop(ker_pole, transfer_matrix, poles, B, maxiter, rtol) if not stop and rtol > 0: # if rtol<=0 the user has probably done that on purpose, # don't annoy him err_msg = ( "Convergence was not reached after maxiter iterations.\n" "You asked for a relative tolerance of %f we got %f" % (rtol, cur_rtol) ) warnings.warn(err_msg) # reconstruct transfer_matrix to match complex conjugate pairs, # ie transfer_matrix_j/transfer_matrix_j+1 are # Re(Complex_pole), Im(Complex_pole) now and will be Re-Im/Re+Im after transfer_matrix = transfer_matrix.astype(complex) idx = 0 while idx < poles.shape[0]-1: if ~np.isreal(poles[idx]): rel = transfer_matrix[:, idx].copy() img = transfer_matrix[:, idx+1] # rel will be an array referencing a column of transfer_matrix # if we don't copy() it will changer after the next line and # and the line after will not yield the correct value transfer_matrix[:, idx] = rel-1j*img transfer_matrix[:, idx+1] = rel+1j*img idx += 1 # skip next one idx += 1 try: m = np.linalg.solve(transfer_matrix.T, np.dot(np.diag(poles), transfer_matrix.T)).T gain_matrix = np.linalg.solve(z, np.dot(u0.T, m-A)) except np.linalg.LinAlgError: raise ValueError("The poles you've chosen can't be placed. " "Check the controllability matrix and try " "another set of poles") # Beware: Kautsky solves A+BK but the usual form is A-BK gain_matrix = -gain_matrix # K still contains complex with ~=0j imaginary parts, get rid of them gain_matrix = np.real(gain_matrix) full_state_feedback = Bunch() full_state_feedback.gain_matrix = gain_matrix full_state_feedback.computed_poles = _order_complex_poles( np.linalg.eig(A - np.dot(B, gain_matrix))[0] ) full_state_feedback.requested_poles = poles full_state_feedback.X = transfer_matrix full_state_feedback.rtol = cur_rtol full_state_feedback.nb_iter = nb_iter return full_state_feedback def dlsim(system, u, t=None, x0=None): """ Simulate output of a discrete-time linear system. Parameters ---------- system : tuple of array_like or instance of dlti A tuple describing the system. The following gives the number of elements in the tuple and the interpretation: * 1: (instance of dlti) * 3: (num, den, dt) * 4: (zeros, poles, gain, dt) * 5: (A, B, C, D, dt) 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, optional The time steps at which the input is defined. If t is given, it must be the same length as u, and the final value in t determines the number of steps returned in the output. x0 : array_like, optional The initial conditions on the state vector (zero by default). Returns ------- tout : ndarray Time values for the output, as a 1-D array. yout : ndarray System response, as a 1-D array. xout : ndarray, optional Time-evolution of the state-vector. Only generated if the input is a StateSpace system. See Also -------- lsim, dstep, dimpulse, cont2discrete Examples -------- A simple integrator transfer function with a discrete time step of 1.0 could be implemented as: >>> from scipy import signal >>> tf = ([1.0,], [1.0, -1.0], 1.0) >>> t_in = [0.0, 1.0, 2.0, 3.0] >>> u = np.asarray([0.0, 0.0, 1.0, 1.0]) >>> t_out, y = signal.dlsim(tf, u, t=t_in) >>> y.T array([[ 0., 0., 0., 1.]]) """ # Convert system to dlti-StateSpace if isinstance(system, lti): raise AttributeError('dlsim can only be used with discrete-time dlti ' 'systems.') elif not isinstance(system, dlti): system = dlti(*system[:-1], dt=system[-1]) # Condition needed to ensure output remains compatible is_ss_input = isinstance(system, StateSpace) system = system._as_ss() u = np.atleast_1d(u) if u.ndim == 1: u = np.atleast_2d(u).T if t is None: out_samples = len(u) stoptime = (out_samples - 1) * system.dt else: stoptime = t[-1] out_samples = int(np.floor(stoptime / system.dt)) + 1 # Pre-build output arrays xout = np.zeros((out_samples, system.A.shape[0])) yout = np.zeros((out_samples, system.C.shape[0])) tout = np.linspace(0.0, stoptime, num=out_samples) # Check initial condition if x0 is None: xout[0, :] = np.zeros((system.A.shape[1],)) else: xout[0, :] = np.asarray(x0) # Pre-interpolate inputs into the desired time steps if t is None: u_dt = u else: if len(u.shape) == 1: u = u[:, np.newaxis] u_dt_interp = interp1d(t, u.transpose(), copy=False, bounds_error=True) u_dt = u_dt_interp(tout).transpose() # Simulate the system for i in range(0, out_samples - 1): xout[i+1, :] = (np.dot(system.A, xout[i, :]) + np.dot(system.B, u_dt[i, :])) yout[i, :] = (np.dot(system.C, xout[i, :]) + np.dot(system.D, u_dt[i, :])) # Last point yout[out_samples-1, :] = (np.dot(system.C, xout[out_samples-1, :]) + np.dot(system.D, u_dt[out_samples-1, :])) if is_ss_input: return tout, yout, xout else: return tout, yout def dimpulse(system, x0=None, t=None, n=None): """ Impulse response of discrete-time system. Parameters ---------- system : tuple of array_like or instance of dlti A tuple describing the system. The following gives the number of elements in the tuple and the interpretation: * 1: (instance of dlti) * 3: (num, den, dt) * 4: (zeros, poles, gain, dt) * 5: (A, B, C, D, dt) 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 ------- tout : ndarray Time values for the output, as a 1-D array. yout : ndarray Impulse response of system. Each element of the tuple represents the output of the system based on an impulse in each input. See Also -------- impulse, dstep, dlsim, cont2discrete """ # Convert system to dlti-StateSpace if isinstance(system, dlti): system = system._as_ss() elif isinstance(system, lti): raise AttributeError('dimpulse can only be used with discrete-time ' 'dlti systems.') else: system = dlti(*system[:-1], dt=system[-1])._as_ss() # Default to 100 samples if unspecified if n is None: n = 100 # If time is not specified, use the number of samples # and system dt if t is None: t = np.linspace(0, n * system.dt, n, endpoint=False) else: t = np.asarray(t) # For each input, implement a step change yout = None for i in range(0, system.inputs): u = np.zeros((t.shape[0], system.inputs)) u[0, i] = 1.0 one_output = dlsim(system, u, t=t, x0=x0) if yout is None: yout = (one_output[1],) else: yout = yout + (one_output[1],) tout = one_output[0] return tout, yout def dstep(system, x0=None, t=None, n=None): """ Step response of discrete-time system. Parameters ---------- system : tuple of array_like A tuple describing the system. The following gives the number of elements in the tuple and the interpretation: * 1: (instance of dlti) * 3: (num, den, dt) * 4: (zeros, poles, gain, dt) * 5: (A, B, C, D, dt) 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 ------- tout : ndarray Output time points, as a 1-D array. yout : ndarray Step response of system. Each element of the tuple represents the output of the system based on a step response to each input. See Also -------- step, dimpulse, dlsim, cont2discrete """ # Convert system to dlti-StateSpace if isinstance(system, dlti): system = system._as_ss() elif isinstance(system, lti): raise AttributeError('dstep can only be used with discrete-time dlti ' 'systems.') else: system = dlti(*system[:-1], dt=system[-1])._as_ss() # Default to 100 samples if unspecified if n is None: n = 100 # If time is not specified, use the number of samples # and system dt if t is None: t = np.linspace(0, n * system.dt, n, endpoint=False) else: t = np.asarray(t) # For each input, implement a step change yout = None for i in range(0, system.inputs): u = np.zeros((t.shape[0], system.inputs)) u[:, i] = np.ones((t.shape[0],)) one_output = dlsim(system, u, t=t, x0=x0) if yout is None: yout = (one_output[1],) else: yout = yout + (one_output[1],) tout = one_output[0] return tout, yout def dfreqresp(system, w=None, n=10000, whole=False): """ Calculate the frequency response of a discrete-time system. Parameters ---------- system : an instance of the dlti class or a tuple describing the system. The following gives the number of elements in the tuple and the interpretation: * 1 (instance of dlti) * 2 (numerator, denominator, dt) * 3 (zeros, poles, gain, dt) * 4 (A, B, C, D, dt) w : array_like, optional Array of frequencies (in radians/sample). 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 an interval chosen to include the influence of the poles and zeros of the system. whole : bool, optional Normally, if 'w' is not given, frequencies are computed from 0 to the Nyquist frequency, pi radians/sample (upper-half of unit-circle). If whole is True, compute frequencies from 0 to 2*pi radians/sample. Returns ------- w : 1D ndarray Frequency array [radians/sample] H : 1D ndarray Array of complex magnitude values Notes ----- If (num, den) is passed in for system, coefficients for both the numerator and denominator should be specified in descending exponent order (e.g. z^2 + 3z + 5 would be represented as [1, 3, 5]). .. versionadded:: 0.18.0 Examples -------- Generating the Nyquist plot of a transfer function >>> from scipy import signal >>> import matplotlib.pyplot as plt Transfer function: H(z) = 1 / (z^2 + 2z + 3) >>> sys = signal.TransferFunction([1], [1, 2, 3], dt=0.05) >>> w, H = signal.dfreqresp(sys) >>> plt.figure() >>> plt.plot(H.real, H.imag, "b") >>> plt.plot(H.real, -H.imag, "r") >>> plt.show() """ if isinstance(system, dlti): system = system._as_tf() elif isinstance(system, lti): raise AttributeError('dfreqresp can only be used with discrete-time ' 'systems.') else: system = dlti(*system[:-1], dt=system[-1])._as_tf() if system.inputs != 1 or system.outputs != 1: raise ValueError("dfreqresp requires a SISO (single input, single " "output) system.") if w is not None: worN = w else: worN = n # Convert numerator and denominator from polynomials in the variable 'z' # to polynomials in the variable 'z^-1', as freqz expects. num, den = TransferFunction._z_to_zinv(system.num.ravel(), system.den) return freqz(num, den, worN=worN, whole=whole) def dbode(system, w=None, n=100): """ Calculate Bode magnitude and phase data of a discrete-time system