ó ˽÷Xc@`sCdZddlmZmZmZddlZddlZddlmZmZm Z m Z m Z m Z m Z mZmZmZddlmZddlmZmZmZdd d d d d gZd„Zd„Zd„Zd„Zd„Zd„Zddddd„Zdd„Z d„Z!dd„Z"ddd„Z#dS(sr ltisys -- a collection of functions to convert linear time invariant systems from one representation to another. i(tdivisiontprint_functiontabsolute_importN( tr_teyet atleast_2dtpolytdottasarraytproducttzerostarraytouter(tlinalgi(ttf2zpktzpk2tft normalizettf2sstabcd_normalizetss2tftzpk2sstss2zpkt cont2discretec C`sWt||ƒ\}}t|jƒ}|dkrHt|g|jƒ}n|jd}t|ƒ}||kr‚d}t|ƒ‚n|dksš|dkrÎtgtƒtgtƒtgtƒtgtƒfStdt |jd||f|jƒ|f}|jddkr0t |dd…dfƒ}ntdggtƒ}|dkrŸ|j |jƒ}t dƒt d|jdfƒt |jddfƒ|fSt|dgƒ }t|t |d|dƒf}t |ddƒ} |dd…dd…ft |dd…df|dƒ} |j | jd| jdfƒ}|| | |fS( s½Transfer function to state-space representation. Parameters ---------- num, den : array_like Sequences representing the coefficients of the numerator and denominator polynomials, in order of descending degree. The denominator needs to be at least as long as the numerator. Returns ------- A, B, C, D : ndarray State space representation of the system, in controller canonical form. Examples -------- Convert the transfer function: .. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1} >>> num = [1, 3, 3] >>> den = [1, 2, 1] to the state-space representation: .. math:: \dot{\textbf{x}}(t) = \begin{bmatrix} -2 & -1 \\ 1 & 0 \end{bmatrix} \textbf{x}(t) + \begin{bmatrix} 1 \\ 0 \end{bmatrix} \textbf{u}(t) \\ \textbf{y}(t) = \begin{bmatrix} 1 & 2 \end{bmatrix} \textbf{x}(t) + \begin{bmatrix} 1 \end{bmatrix} \textbf{u}(t) >>> from scipy.signal import tf2ss >>> A, B, C, D = tf2ss(num, den) >>> A array([[-2., -1.], [ 1., 0.]]) >>> B array([[ 1.], [ 0.]]) >>> C array([[ 1., 2.]]) >>> D array([[ 1.]]) is7Improper transfer function. `num` is longer than `den`.is-1iÿÿÿÿNi(ii(RtlentshapeRtdtypet ValueErrorR tfloatRR RtreshapeRR ( tnumtdentnntMtKtmsgtDtfrowtAtBtC((s:/tmp/pip-build-7oUkmx/scipy/scipy/signal/lti_conversion.pyRs48    $0 !=#cC`s|dkrtdƒS|SdS(Ni(ii(tNoneR (targ((s:/tmp/pip-build-7oUkmx/scipy/scipy/signal/lti_conversion.pyt_none_to_empty_2dus  cC`s|dk rt|ƒSdS(N(R(R(R)((s:/tmp/pip-build-7oUkmx/scipy/scipy/signal/lti_conversion.pyt_atleast_2d_or_none|s cC`s|dk r|jSdSdS(Ni(N(NN(R(R(R ((s:/tmp/pip-build-7oUkmx/scipy/scipy/signal/lti_conversion.pyt_shape_or_nones cG`s%x|D]}|dk r|SqWdS(N(R((targsR)((s:/tmp/pip-build-7oUkmx/scipy/scipy/signal/lti_conversion.pyt_choice_not_noneˆs  cC`s?|jdkrt|ƒS|j|kr7tdƒ‚n|SdS(Nis*The input arrays have incompatible shapes.(ii(RR R(R R((s:/tmp/pip-build-7oUkmx/scipy/scipy/signal/lti_conversion.pyt_restoreŽs  cC`s]tt||||fƒ\}}}}t|ƒ\}}t|ƒ\}}t|ƒ\}} t|ƒ\} } t||| ƒ} t|| ƒ} t|| ƒ}| dksÃ| dksÃ|dkrÒtdƒ‚ntt||||fƒ\}}}}t|| | fƒ}t|| | fƒ}t||| fƒ}t||| fƒ}||||fS(s¿Check state-space matrices and ensure they are two-dimensional. If enough information on the system is provided, that is, enough properly-shaped arrays are passed to the function, the missing ones are built from this information, ensuring the correct number of rows and columns. Otherwise a ValueError is raised. Parameters ---------- A, B, C, D : array_like, optional State-space matrices. All of them are None (missing) by default. See `ss2tf` for format. Returns ------- A, B, C, D : array Properly shaped state-space matrices. Raises ------ ValueError If not enough information on the system was provided. s%Not enough information on the system.N(tmapR+R,R.R(RR*R/(R%R&R'R#tMAtNAtMBtNBtMCtNCtMDtNDtptqtr((s:/tmp/pip-build-7oUkmx/scipy/scipy/signal/lti_conversion.pyR—s '$'c C`st||||ƒ\}}}}|j\}}||krKtdƒ‚n|dd…||d…f}|dd…||d…f}yt|ƒ}Wntk r´d}nXt|jddƒdkrCt|jddƒdkrCtj|ƒ}t|jddƒdkr9t|jddƒdkr9g}n||fS|jd} |dd…df|dd…df|ddd…f|} tj|| df| jƒ}x]t |ƒD]O} t || dd…fƒ} t|t || ƒƒ|| d||| >> A = [[-2, -1], [1, 0]] >>> B = [[1], [0]] # 2-dimensional column vector >>> C = [[1, 2]] # 2-dimensional row vector >>> D = 1 to the transfer function: .. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1} >>> from scipy.signal import ss2tf >>> ss2tf(A, B, C, D) (array([[1, 3, 3]]), array([ 1., 2., 1.])) s)System does not have the input specified.Nitaxisi( RRRRR tnumpytravelR RtrangeRR( R%R&R'R#tinputtnouttninRRt num_statest type_testtktCk((s:/tmp/pip-build-7oUkmx/scipy/scipy/signal/lti_conversion.pyRÆs,:!     66   B1cC`stt|||ƒŒS(s:Zero-pole-gain representation to state-space representation Parameters ---------- z, p : sequence Zeros and poles. k : float System gain. Returns ------- A, B, C, D : ndarray State space representation of the system, in controller canonical form. (RR(tzR9RE((s:/tmp/pip-build-7oUkmx/scipy/scipy/signal/lti_conversion.pyRscC`stt||||d|ƒŒS(sªState-space representation to zero-pole-gain representation. A, B, C, D defines a linear state-space system with `p` inputs, `q` outputs, and `n` state variables. Parameters ---------- A : array_like State (or system) matrix of shape ``(n, n)`` B : array_like Input matrix of shape ``(n, p)`` C : array_like Output matrix of shape ``(q, n)`` D : array_like Feedthrough (or feedforward) matrix of shape ``(q, p)`` input : int, optional For multiple-input systems, the index of the input to use. Returns ------- z, p : sequence Zeros and poles. k : float System gain. R@(RR(R%R&R'R#R@((s:/tmp/pip-build-7oUkmx/scipy/scipy/signal/lti_conversion.pyR3stzohc C`sît|ƒdkr|jƒSt|ƒdkr„tt|d|dƒ|d|d|ƒ}t|d|d|d|dƒ|fSt|ƒdkrótt|d|d|dƒ|d|d|ƒ}t|d|d|d|dƒ|fSt|ƒdkr|\}}}}n tdƒ‚|d krw|dkrMtd ƒ‚qw|dkse|dkrwtd ƒ‚qwn|d kr4t j |j dƒ|||} t j | t j |j dƒd |||ƒ} t j | ||ƒ} t j | jƒ|jƒƒ} | jƒ} ||t j|| ƒ} n§|d ksL|dkret||dd ddƒS|dks}|dkr–t||dd ddƒS|dkr»t||dd dd ƒS|dkrËt j||fƒ}t jt j|j d|j dfƒt j|j d|j dfƒfƒ}t j||fƒ}t j||ƒ}|d|j d…dd…f}|dd…d|j d…f} |dd…|j dd…f} |} |} ntd|ƒ‚| | | | |fS(sþ Transform a continuous to a discrete state-space system. Parameters ---------- system : a tuple describing the system or an instance of `lti` 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) dt : float The discretization time step. method : {"gbt", "bilinear", "euler", "backward_diff", "zoh"}, optional Which method to use: * gbt: generalized bilinear transformation * bilinear: Tustin's approximation ("gbt" with alpha=0.5) * euler: Euler (or forward differencing) method ("gbt" with alpha=0) * backward_diff: Backwards differencing ("gbt" with alpha=1.0) * zoh: zero-order hold (default) alpha : float within [0, 1], optional The generalized bilinear transformation weighting parameter, which should only be specified with method="gbt", and is ignored otherwise Returns ------- sysd : tuple containing the discrete system Based on the input type, the output will be of the form * (num, den, dt) for transfer function input * (zeros, poles, gain, dt) for zeros-poles-gain input * (A, B, C, D, dt) for state-space system input Notes ----- By default, the routine uses a Zero-Order Hold (zoh) method to perform the transformation. Alternatively, a generalized bilinear transformation may be used, which includes the common Tustin's bilinear approximation, an Euler's method technique, or a backwards differencing technique. The Zero-Order Hold (zoh) method is based on [1]_, the generalized bilinear approximation is based on [2]_ and [3]_. References ---------- .. [1] http://en.wikipedia.org/wiki/Discretization#Discretization_of_linear_state_space_models .. [2] http://techteach.no/publications/discretetime_signals_systems/discrete.pdf .. [3] G. Zhang, X. Chen, and T. Chen, Digital redesign via the generalized bilinear transformation, Int. J. Control, vol. 82, no. 4, pp. 741-754, 2009. (http://www.ece.ualberta.ca/~gfzhang/research/ZCC07_preprint.pdf) iiitmethodtalphaiisKFirst argument must either be a tuple of 2 (tf), 3 (zpk), or 4 (ss) arrays.tgbtsUAlpha parameter must be specified for the generalized bilinear transform (gbt) methodsDAlpha parameter must be within the interval [0,1] for the gbt methodgð?tbilinearttustingà?teulert forward_diffgt backward_diffRHNs"Unknown transformation method '%s'(Rt to_discreteRRRRRRR(tnpRRR tsolvet transposeRthstackR tvstacktexpm(tsystemtdtRIRJtsysdtatbtctdtimatadtbdtcdtddtem_uppertem_lowertemtms((s:/tmp/pip-build-7oUkmx/scipy/scipy/signal/lti_conversion.pyRQsX= # *$*    "2   &)### ($t__doc__t __future__RRRR=RRRRRRRRR R R R tscipyR t filter_designRRRt__all__RR*R+R,R.R/R(RRRRR(((s:/tmp/pip-build-7oUkmx/scipy/scipy/signal/lti_conversion.pyts&  F  a     / Y