B &]\6@sdZddlmZmZmZddlZddlZddlmZmZm Z m Z m Z m Z m Z mZmZmZddlmZddlmZmZmZdd d d d d gZddZddZddZddZddZddZddd Zd dd Zdd Z d!dd Z!d"dd Z"dS)#zr ltisys -- a collection of functions to convert linear time invariant systems from one representation to another. )divisionprint_functionabsolute_importN) r_eye atleast_2dpolydotasarrayproductzerosarrayouter)linalg)tf2zpkzpk2tf normalizetf2ssabcd_normalizess2tfzpk2ssss2zpk cont2discretec Cst||\}}t|j}|dkr.t|g|j}|jd}t|}||krTd}t||dksd|dkrtgttgttgttgtfStdt |jd||f|j|f}|jddkrt |dddf}ntdggt}|dkr | |j}t dt d|jdft |jddf|fSt|ddg }t|t |d|df}t |dd} |ddddft |dddf|dd} | | jd| jdf}|| | |fS) aTransfer 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.]]) rz7Improper transfer function. `num` is longer than `den`.rz-1N)rr)rlenshaper dtype ValueErrorr floatrr rZreshaperr) numdenZnnMKmsgDZfrowABCr*:lib/python3.7/site-packages/scipy/signal/lti_conversion.pyrs48   $  2cCs|dkrtdS|SdS)N)rr)r )argr*r*r+_none_to_empty_2dusr-cCs|dk rt|SdS)N)r)r,r*r*r+_atleast_2d_or_none|sr.cCs|dk r|jSdSdS)N)NN)r)r#r*r*r+_shape_or_nonesr/cGsx|D]}|dk r|SqWdS)Nr*)argsr,r*r*r+_choice_not_nones r1cCs,|jdkrt|S|j|kr$td|SdS)N)rrz*The input arrays have incompatible shapes.)rr r)r#rr*r*r+_restores   r2cCstt||||f\}}}}t|\}}t|\}}t|\}} t|\} } t||| } t|| } t|| }| dks| dks|dkrtdtt||||f\}}}}t|| | f}t|| | f}t||| f}t||| f}||||fS)aCheck 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. Nz%Not enough information on the system.)mapr.r/r1rr-r2)r'r(r)r&ZMAZNAZMBZNBZMCZNCZMDZNDpqrr*r*r+rs        c Cst||||\}}}}|j\}}||kr0td|dd||df}|dd||df}y t|}Wntk rd}YnXt|jdddkrt|jdddkrt|}t|jdddkrt|jdddkrg}||fS|jd} |dddf|dddf|dddf|} t|| df| j}xLt |D]@} t || ddf} t|t || || d||| <qBW||fS)aState-space to transfer function. 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 ------- num : 2-D ndarray Numerator(s) of the resulting transfer function(s). `num` has one row for each of the system's outputs. Each row is a sequence representation of the numerator polynomial. den : 1-D ndarray Denominator of the resulting transfer function(s). `den` is a sequence representation of the denominator polynomial. Examples -------- Convert 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) >>> 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.])) z)System does not have the input specified.Nrr)Zaxis) rrrrr numpyZravelr rrangerr ) r'r(r)r&inputZnoutZninr"r!Z num_statesZ type_testkZCkr*r*r+rs,:   $ $ 4,cCstt|||S)a: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)zr4r:r*r*r+rscCstt|||||dS)aState-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. )r9)rr)r'r(r)r&r9r*r*r+r3szohcCst|dkr|St|dkrbtt|d|d|||d}t|d|d|d|d|fSt|dkrtt|d|d|d|||d}t|d|d|d|d|fSt|dkr|\}}}}ntd|dkr|d krtd n|dks|dkrtd |dkrt |j d|||} t | t |j dd |||} t | ||} t | | } | } ||t|| } n4|d ks|dkrt||dddS|dks|dkrt||dddS|dkrt||dd dS|dkrt||f}tt|j d|j dft|j d|j dff}t||f}t ||}|d |j dd d f}|d d d|j df} |d d |j dd f} |} |} n td|| | | | |fS)a 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] https://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. (https://www.mypolyuweb.hk/~magzhang/Research/ZCC09_IJC.pdf) rrr)methodalphazKFirst argument must either be a tuple of 2 (tf), 3 (zpk), or 4 (ss) arrays.ZgbtNzUAlpha parameter must be specified for the generalized bilinear transform (gbt) methodzDAlpha parameter must be within the interval [0,1] for the gbt methodg?ZbilinearZtusting?ZeulerZ forward_diffgZ backward_diffr<z"Unknown transformation method '%s')rZ to_discreterrrrrrnprrrZsolveZ transposer Zhstackr ZvstackZexpm)systemZdtr=r>ZsysdabcdZimaZadZbdZcdZddZem_upperZem_lowerZemZmsr*r*r+rQsX=  $  $    (   )NNNN)r)r)r<N)#__doc__Z __future__rrrr7rArrrrr r r r r rZscipyrZ filter_designrrr__all__rr-r.r/r1r2rrrrrr*r*r*r+s&0  a / Y