iiGdZddlZddlmZmZmZmZmZmZm Z m Z m Z ddl m Z ddlmZmZddlmcmZddlmZmZmZgdZd Zdd Zdd Zd Zdd ZddZy)zr ltisys -- a collection of functions to convert linear time invariant systems from one representation to another. N) r_eye atleast_2dpolydotasarrayzerosarrayouter)linalg)array_namespacexp_size)tf2zpkzpk2tf normalize)tf2ssabcd_normalizess2tfzpk2ssss2zpk cont2discretecDt||\}}t|j}|dk(rt|g|j}|jd}t|}||kDr d}t ||dk(s|dk(r>t gtt gtt gtt gtfStjtj|jd||z f|j|f}|jddkDrt|dddf}nt dggt}|dk(rZ|j|j}tdtd|jdft|jddf|fSt |ddg }t|t|dz |dz f}t|dz d} |ddddft|dddf|ddz } |j| 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`.r)dtypeN)rr)rlenshaperr ValueErrorr floatnphstackr rreshaperrr ) numdennnMKmsgDfrowABCs q/Volumes/fast/ai/experiments/voice-extract-mac/.venv/lib/python3.12/site-packages/scipy/signal/_lti_conversion.pyrrspc"HC SYYB QwseSYY' ! A CA1uGoAvab% %E"2E"e4Db% " " ))RXXsyy|QU3399EsK LC yy}q s1a4y ! A3% Av IIcii f ua_5qwwqz1o&+ + 3qr7)  D 4QUAE" "#A AE1 A AqrE U3q!t9c!"g..A 1771:qwwqz*+A aA:c D|| | td| | td| | tdt||||fd||||fD\}}}}|jdxs$|jdxs|jdxsd}|jdxs|jdxsd}|jdxs|jdxsd}t|dk(rj ||fn|}t|dk(rj ||fn|}t|dk(rj ||fn|}t|dk(rj ||fn|}|j||fk7rtd|jd|d |d |j||fk7rtd |jd|d |d |j||fk7rtd |jd|d |d |j||fk7rtd |jd|d |d ||||fS)a Check state-space matrices compatibility and ensure they are 2d arrays. Converts input matrices into two-dimensional arrays as needed. Then the dimensions n, q, p are determined by investigating the non-zero entries of the array shapes. If a parameter is ``None``, or has shape (0, 0), it is set to a zero-array of compatible shape. Finally, it is verified that all parameter shapes are compatible to each other. If that fails, a ``ValueError`` is raised. Note that the dimensions n, q, p are allowed to be zero. Parameters ---------- A: array_like, optional Two-dimensional array of shape (n, n). B: array_like, optional Two-dimensional array of shape (n, p). C: array_like, optional Two-dimensional array of shape (q, n). D: array_like, optional Two-dimensional array of shape (q, p). Returns ------- A, B, C, D : array State-space matrices as two-dimensional arrays. Notes ----- The :ref:`tutorial_signal_state_space_representation` section of the :ref:`user_guide` presents the corresponding definitions of continuous-time and disrcete time state space systems. Raises ------ ValueError If the dimensions n, q, or p could not be determined or if the shapes are incompatible with each other. See Also -------- StateSpace: Linear Time Invariant system in state-space form. dlti: Discrete-time linear time invariant system base class. tf2ss: Transfer function to state-space representation. ss2tf: State-space to transfer function. ss2zpk: State-space representation to zero-pole-gain representation. cont2discrete: Transform a continuous to a discrete state-space system. Examples -------- The following example demonstrates that the passed lists are converted into two-dimensional arrays: >>> from scipy.signal import abcd_normalize >>> AA, BB, CC, DD = abcd_normalize(A=[[1, 2], [3, 4]], B=[[-1], [5]], ... C=[[4, 5]], D=2.5) >>> AA.shape, BB.shape, CC.shape, DD.shape ((2, 2), (2, 1), (1, 2), (1, 1)) In the following, the missing parameter C is assumed to be an array of zeros with shape (1, 2): >>> from scipy.signal import abcd_normalize >>> AA, BB, CC, DD = abcd_normalize(A=[[1, 2], [3, 4]], B=[[-1], [5]], D=2.5) >>> AA.shape, BB.shape, CC.shape, DD.shape ((2, 2), (2, 1), (1, 2), (1, 1)) >>> CC array([[0., 0.]]) z9Dimension n is undefined for parameters A = B = C = None!z5Dimension p is undefined for parameters B = D = None!z5Dimension q is undefined for parameters C = D = None!c3K|]=}|&tjj|dnjd?yw)Nr)ndim)rr)xpx atleast_ndrr ).0M_xps r/ z!abcd_normalize..sA;-9r=?N#..Ba8((6"#-9sAArrzParameter A has shape z but should be (z, z)!zParameter B has shape zParameter C has shape zParameter D has shape )rr rrr )r,r-r.r*npqr8s @r/rrusJH yQY19TUUyQYPQQyQYPQQ Aq! $B;./Aq\;JAq!Q  3aggaj3AGGAJ3!A  %aggaj%AA  %aggaj%AA#AJ!O!QA#AJ!O!QA#AJ!O!QA#AJ!O!QAww1a&1!'':J1#RPQsRTUVVww1a&1!'':J1#RPQsRTUVVww1a&1!'':J1#RPQsRTUVVww1a&1!'':J1#RPQsRTUVV aA:r0ct||||\}}}}|j\}}||k\r td|dd||dzf}|dd||dzf} t|}|jdk(rH|jdk(r9t j |}|jdk(r|jdk(rg}||fS|jd} |dddf|dddfz|dddfz|zdz} t j|| dzf| j}t|D];} t|| ddf} t|t|| z || dz |zz|| <=||fS#t$rd}Y wxYw)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-D column vector >>> C = [[1, 2]] # 2-D 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) rrrrsizer!ravelemptyrrangerr) r,r-r.r*inputnoutninr%r$ num_states type_testkCks r/rrst 1a+JAq!QID# |DEE !U519_ A !U519_ A1g ! !&&A+hhqk FFaKaffkCCxJ!Q$!AqD'!AadG+a/#5I ((D*q.)9?? ;C 4[ !Q$ a#a*n%1S(88A 8O! s E E%$E%c(tt|||S)a:Zero-pole-gain representation to state-space representation Parameters ---------- z, p : sequence Zeros and poles. k : float System gain. Returns ------- A, B, C, D : ndarray State space representation of the system, in controller canonical form. )rr)zr;rHs r/rr2s" &Aq/ ""r0c .tt|||||S)aState-space representation to zero-pole-gain representation. A, B, C, D defines a linear state-space system with `p` inputs, `q` outputs, and `n` state variables. Parameters ---------- A : array_like State (or system) matrix of shape ``(n, n)`` B : array_like Input matrix of shape ``(n, p)`` C : array_like Output matrix of shape ``(q, n)`` D : array_like Feedthrough (or feedforward) matrix of shape ``(q, p)`` input : int, optional For multiple-input systems, the index of the input to use. Returns ------- z, p : sequence Zeros and poles. k : float System gain. )rC)rr)r,r-r.r*rCs r/rrFs6 5Aq!51 22r0cF t|dr)t|jr|j|||St|dk(r=t t |d|d|||}t |d|d|d|d|fzSt|dk(rAt t|d|d|d|||}t|d|d|d|d|fzSt|dk(r|\}}}}n td |d k(r"| td |dks|dkDr td |d k(rtj|jd||z|zz } tj| tj|jdd|z |z|zz} tj| ||z} tj| j|j} | j} ||tj || zz} n|dk(s|dk(rt ||d dS|dk(s|dk(rt ||d dS|dk(rt ||d dS|dk(r tj"||f}tj"tj$|jd|jdftj$|jd|jdff}tj&||f}tj(||z}|d |jdd d f}|d d d|jdf} |d d |jdd f} |} |} nD|dk(r|jd}|jd}tj*tj,||g|ztj|}t%||d|zzf}tj,|g|gg}tj(|}|d |d|f}|d ||||zf}|d |||zd f}|} ||z ||zz} |} |||zz} n`|dk(rLtj.|ds tdtj(||z} | |z|z} |} ||z|z} ntd|d| | | | |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 : str, 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) * foh: first-order hold (*versionadded: 1.3.0*) * impulse: equivalent impulse response (*versionadded: 1.3.0*) 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]_, the First-Order Hold (foh) method is based on [4]_. 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) .. [4] G. F. Franklin, J. D. Powell, and M. L. Workman, Digital control of dynamic systems, 3rd ed. Menlo Park, Calif: Addison-Wesley, pp. 204-206, 1998. Examples -------- We can transform a continuous state-space system to a discrete one: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.signal import cont2discrete, lti, dlti, dstep Define a continuous state-space system. >>> A = np.array([[0, 1],[-10., -3]]) >>> B = np.array([[0],[10.]]) >>> C = np.array([[1., 0]]) >>> D = np.array([[0.]]) >>> l_system = lti(A, B, C, D) >>> t, x = l_system.step(T=np.linspace(0, 5, 100)) >>> fig, ax = plt.subplots() >>> ax.plot(t, x, label='Continuous', linewidth=3) Transform it to a discrete state-space system using several methods. >>> dt = 0.1 >>> for method in ['zoh', 'bilinear', 'euler', 'backward_diff', 'foh', 'impulse']: ... d_system = cont2discrete((A, B, C, D), dt, method=method) ... s, x_d = dstep(d_system) ... ax.step(s, np.squeeze(x_d), label=method, where='post') >>> ax.axis([t[0], t[-1], x[0], 1.4]) >>> ax.legend(loc='best') >>> fig.tight_layout() >>> plt.show() to_discrete)dtmethodalpharrr)rPrQzKFirst argument must either be a tuple of 2 (tf), 3 (zpk), or 4 (ss) arrays.gbtNzUAlpha parameter must be specified for the generalized bilinear transform (gbt) methodzDAlpha parameter must be within the interval [0,1] for the gbt methodg?bilineartusting?euler forward_diffr> backward_diffzohfohimpulsezrs\ 111:((55 ^BaHVr#(3