# scipy/scipy

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 """ Continuous to discrete transformations for state-space and transfer function. """ # Author: Jeffrey Armstrong # March 29, 2011 import numpy as np from scipy import linalg from ltisys import tf2ss, ss2tf, zpk2ss, ss2zpk __all__ = ['cont2discrete'] def cont2discrete(sys, dt, method="zoh", alpha=None): """Transform a continuous to a discrete state-space system. Parameters ----------- sys : a tuple describing the system. The following gives the number of elements in the tuple and the interpretation: * 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"} 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] 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) """ if len(sys) == 2: sysd = cont2discrete(tf2ss(sys[0], sys[1]), dt, method=method, alpha=alpha) return ss2tf(sysd[0], sysd[1], sysd[2], sysd[3]) + (dt,) elif len(sys) == 3: sysd = cont2discrete(zpk2ss(sys[0], sys[1], sys[2]), dt, method=method, alpha=alpha) return ss2zpk(sysd[0], sysd[1], sysd[2], sysd[3]) + (dt,) elif len(sys) == 4: a, b, c, d = sys else: raise ValueError("First argument must either be a tuple of 2 (tf), " "3 (zpk), or 4 (ss) arrays.") if method == 'gbt': if alpha is None: raise ValueError("Alpha parameter must be specified for the " "generalized bilinear transform (gbt) method") elif alpha < 0 or alpha > 1: raise ValueError("Alpha parameter must be within the interval " "[0,1] for the gbt method") if method == 'gbt': # This parameter is used repeatedly - compute once here ima = np.eye(a.shape[0]) - alpha*dt*a ad = linalg.solve(ima, np.eye(a.shape[0]) + (1.0-alpha)*dt*a) bd = linalg.solve(ima, dt*b) # Similarly solve for the output equation matrices cd = linalg.solve(ima.transpose(), c.transpose()) cd = cd.transpose() dd = d + alpha*np.dot(c, bd) elif method == 'bilinear' or method == 'tustin': return cont2discrete(sys, dt, method="gbt", alpha=0.5) elif method == 'euler' or method == 'forward_diff': return cont2discrete(sys, dt, method="gbt", alpha=0.0) elif method == 'backward_diff': return cont2discrete(sys, dt, method="gbt", alpha=1.0) elif method == 'zoh': # Build an exponential matrix em_upper = np.hstack((a, b)) # Need to stack zeros under the a and b matrices em_lower = np.hstack((np.zeros((b.shape[1], a.shape[0])), np.zeros((b.shape[1], b.shape[1])) )) em = np.vstack((em_upper, em_lower)) ms = linalg.expm(dt * em) # Dispose of the lower rows ms = ms[:a.shape[0], :] ad = ms[:, 0:a.shape[1]] bd = ms[:, a.shape[1]:] cd = c dd = d else: raise ValueError("Unknown transformation method '%s'" % method) return ad, bd, cd, dd, dt
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