_Mh>dZddlZddlmZmZmZmZmZmZm Z m Z m Z ddl m Z ddlmZmZmZgdZdZd Zd Zd Zd Zd ZddZddZdZddZddZdS)zr ltisys -- a collection of functions to convert linear time invariant systems from one representation to another. N) r_eye atleast_2dpolydotasarrayzerosarrayouter)linalg)tf2zpkzpk2tf normalize)tf2ssabcd_normalizess2tfzpk2ssss2zpk cont2discretect||\}}t|j}|dkrt|g|j}|jd}t|}||krd}t ||dks|dkrRt gtt gtt gtt gtfStj tj |jd||z f|j|f}|jddkrt|dddf}nt dggt}|dkra| |j}tdtd|jdft|jddf|fSt |ddg }t|t|dz |dz f}t|dz d} |ddddft|dddf|ddz } | | 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.]]) r z7Improper transfer function. `num` is longer than `den`.r)dtypeN)r r )rlenshaperr ValueErrorr floatnphstackr rreshaperrr ) numdennnMKmsgDfrowABCs \/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/scipy/signal/_lti_conversion.pyrrsEpc""HC SYB QwwseSY'' ! A CA1uuGooAvvab%  %E"2"2E"e4D4Db%  " " )RXsy|QU339EEEsK L LC y}q s111a4y ! ! A3%  Avv IIci f ua_55qwqz1o&&+ + 3qrr7)   D 4QUAE"" "#A AE1 A AAAqrrE U3qqq!t9c!""g...A 171:qwqz*++A aA:c(|tdS|S)Nrr)r args r-_none_to_empty_2dr3ss {V}} r.c(|t|SdSN)rr1s r-_atleast_2d_or_noner6zs #r.c||jSdS)N)NN)r)r%s r-_shape_or_noner8s}w{r.c|D]}||cS dSr5)argsr2s r-_choice_not_noner<s. ?JJJ r.cn|jdkrt|S|j|krtd|S)Nr0z*The input arrays have incompatible shapes.)rr r)r%rs r-_restorer>s<w&U|| 7e  IJJ Jr.c2tt||||f\}}}}t|\}}t|\}}t|\}} t|\} } t||| } t|| } t|| }| | |t dtt ||||f\}}}}t || | f}t || | f}t ||| f}t ||| f}||||fS)aCheck state-space matrices and ensure they are 2-D. 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.)mapr6r8r<rr3r>)r*r+r,r(MANAMBNBMCNCMDNDpqrs r-rrs-2(1aA,77JAq!Q A  FB A  FB A  FB A  FBR$$AR  AR  AyAI@AAA&Aq! 55JAq!QQFAQFAQFAQFA aA:r.ct||||\}}}}|j\}}||krtd|dd||dzf}|dd||dzf} t|}n#t$rd}YnwxYw|jdkr;|jdkr0t j|}|jdkr |jdkrg}||fS|jd} |dddf|dddfz|dddfz|zdz} t j|| dzf| j}t|D]M} t|| ddf} t|t|| z || dz |zz|| <N||fS)aState-space to transfer function. A, B, C, D defines a linear state-space system with `p` inputs, `q` outputs, and `n` state variables. Parameters ---------- A : array_like State (or system) matrix of shape ``(n, n)`` B : array_like Input matrix of shape ``(n, p)`` C : array_like Output matrix of shape ``(q, n)`` D : array_like Feedthrough (or feedforward) matrix of shape ``(q, p)`` input : int, optional For multiple-input systems, the index of the input to use. Returns ------- num : 2-D ndarray Numerator(s) of the resulting transfer function(s). `num` has one row for each of the system's outputs. Each row is a sequence representation of the numerator polynomial. den : 1-D ndarray Denominator of the resulting transfer function(s). `den` is a sequence representation of the denominator polynomial. Examples -------- Convert the state-space representation: .. math:: \dot{\textbf{x}}(t) = \begin{bmatrix} -2 & -1 \\ 1 & 0 \end{bmatrix} \textbf{x}(t) + \begin{bmatrix} 1 \\ 0 \end{bmatrix} \textbf{u}(t) \\ \textbf{y}(t) = \begin{bmatrix} 1 & 2 \end{bmatrix} \textbf{x}(t) + \begin{bmatrix} 1 \end{bmatrix} \textbf{u}(t) >>> A = [[-2, -1], [1, 0]] >>> B = [[1], [0]] # 2-D column vector >>> C = [[1, 2]] # 2-D row vector >>> D = 1 to the transfer function: .. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1} >>> from scipy.signal import ss2tf >>> ss2tf(A, B, C, D) (array([[1., 3., 3.]]), array([ 1., 2., 1.])) z)System does not have the input specified.Nr r) rrrrsizerravelemptyrrangerr) r*r+r,r(inputnoutninr#r" num_states type_testkCks r-rrst 1a++JAq!QID# ||DEEE !!!U519_ A !!!U519_ A1gg  ! !&A++hqkk FaKKafkkCCxJ!!!Q$!AAAqD'!AadG+a/#5I (D*q.)9? ; ;C 4[[99 !QQQ$ a#a**n%%1S(88A 8OsA** A98A9c2tt|||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)zrIrWs r-rrs" &Aq// ""r.c 8tt|||||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. )rR)rr)r*r+r,r(rRs r-rr1s"6 5Aq!5111 22r.zohc t|dkr|St|dkr[tt|d|d|||}t |d|d|d|d|fzSt|dkrbtt |d|d|d|||}t |d|d|d|d|fzSt|dkr|\}}}}ntd|dkr,|td |dks|dkrtd |dkrtj |j d||z|zz } tj | tj |j dd |z |z|zz} tj | ||z} tj | | } | } ||tj|| zz} n|d ks|dkrt||ddS|dks|dkrt||ddS|dkrt||dd S|dkrtj||f}tjtj|j d|j dftj|j d|j dff}tj||f}tj||z}|d |j dd d f}|d d d|j df} |d d |j dd f} |} |} nP|dkr|j d}|j d}tjtj||g|ztj |}t!||d|zzf}tj|g|gg}tj|}|d |d|f}|d ||||zf}|d |||zd f}|} ||z ||zz} |} |||zz} ng|dkrNtj|dstdtj||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() r rr)methodalphazKFirst 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_diffrM backward_diffr\fohimpulsezrrrrrr:r.r-rs 11111111111111111111115555555555   ^^^B  ,,,,^VVVVr###(3333