o ֭j>@sdZddlZddlmZmZmZmZmZmZm Z m Z m Z ddl m Z ddlmZmZmZgdZdd Zd d Zd d ZddZddZddZdddZd ddZddZd ddZd!ddZdS)"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 cont2discretec Cst||\}}t|j}|dkrt|g|j}|jd}t|}||kr*d}t||dks2|dkrDtgttgttgttgtfSt tj |jd||f|jd|f}|jddkrjt |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.]]) r z7Improper transfer function. `num` is longer than `den`.r)dtypeN)r r )rlenshaperr ValueErrorr floatnphstackrrreshaperrr ) numdennnMKmsgDfrowABCr+Y/var/www/html/CropPilot/venv/lib/python3.10/site-packages/scipy/signal/_lti_conversion.pyrs88  ( 2 rcCs|durtdS|S)Nrr)rargr+r+r,_none_to_empty_2dssr0cCs|durt|SdSN)rr.r+r+r,_atleast_2d_or_nonezsr2cCs|dur|jSdS)N)NN)r)r#r+r+r,_shape_or_nonesr3cGs|D] }|dur |SqdSr1r+)argsr/r+r+r,_choice_not_nones r5cCs(|jdkr t|S|j|krtd|S)Nr-z*The input arrays have incompatible shapes.)rrr)r#rr+r+r,_restores  r6cCstt||||f\}}}}t|\}}t|\}}t|\}} t|\} } t||| } t|| } t|| }| dusA| dusA|durEtdtt||||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.)mapr2r3r5rr0r6)r(r)r*r&MANAMBNBMCNCMDNDpqrr+r+r,rs         rc Csjt||||\}}}}|j\}}||krtd|dd||df}|dd||df}zt|}Wn tyAd}Ynw|jdkra|jdkrat|}|jdkr]|jdkr]g}||fS|jd} |dddf|dddf|dddf|d} t|| df| j}t |D]} t || ddf} t|t || || d||| <q||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_testkCkr+r+r,rs.:     8 (rcCstt|||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@rMr+r+r,rsrcCstt|||||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. )rH)r r)r(r)r*r&rHr+r+r,r1srzohcCst|dkr |St|dkr1tt|d|d|||d}t|d|d|d|d|fSt|dkr[tt|d|d|d|||d}t|d|d|d|d|fSt|dkrh|\}}}}ntd|dkr|d urxtd |dks|dkrtd |dkrt |j d|||} t | t |j dd |||} t | ||} t | | } | } ||t|| } n)|d ks|dkrt||dddS|dks|dkrt||dddS|dkrt||dd dS|dkr[t||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|dkr|j d}|j d}t t||g|t |}t||d|f}t|g|gg}t |}|d |d|f}|d ||||f}|d |||d f}|} ||||} |} |||} n.|dkrt|dstdt ||} | ||} |} |||} 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_diffrC backward_diffrPfohimpulsezs", a /Y