Controltools¶

This file contains some functions that are quite helpful when designing feedback laws. This collection is not complete and does not aim to be so. For a more sophisticated collection have a look at the symbtools (https://github.com/TUD-RST/symbtools) or control package which are not used in this package to keep a small footprint.

pymoskito.controltools.char_coefficients(poles)[source]¶

Calculate the coefficients of a characteristic polynomial.

Parameters:	poles (list or `numpy.ndarray`) – pol configuration
Returns:	coefficients
Return type:	`numpy.ndarray`

pymoskito.controltools.place_siso(a_mat, b_mat, poles)[source]¶

Place poles for single input single output (SISO) systems:

pol placement for state feedback: $A$ and $b$

pol placement for observer: $A^T$ and $c$

Parameters:	a_mat (`numpy.ndarray`) – System matrix.:math:A b_mat (`numpy.ndarray`) – Input vector $b$ or Output matrix $c$ . poles (list or `numpy.ndarray`) – Desired poles.
Returns:	Feedback vector or $k$ or observer gain $l^T$ .
Return type:	`numpy.ndarray`

pymoskito.controltools.calc_prefilter(a_mat, b_mat, c_mat, k_mat=None)[source]¶

Calculate the prefilter matrix

$\boldsymbol{V} = - \left[\boldsymbol{C} \left(\boldsymbol{A} - \boldsymbol{B}\boldsymbol{K}\right)^{-1}\boldsymbol{B}\right]^{-1}$

Parameters:	a_mat (`numpy.ndarray`) – system matrix b_mat (`numpy.ndarray`) – input matrix c_mat (`numpy.ndarray`) – output matrix k_mat (`numpy.ndarray`) – control matrix
Returns:	Prefilter matrix
Return type:	`numpy.ndarray`

pymoskito.controltools.controllability_matrix(a_mat, b_mat)[source]¶

Calculate controllability matrix and check controllability of the system.

$\boldsymbol{Q_{c}} = \begin{pmatrix} \boldsymbol{B} & \boldsymbol{A}\boldsymbol{B} & \boldsymbol{A}^{2} \boldsymbol{B} & \cdots & \boldsymbol{A}^{n-1}\boldsymbol{B}\\ \end{pmatrix}$

Parameters:	a_mat (`numpy.ndarray`) – system matrix b_mat (`numpy.ndarray`) – manipulating matrix
Returns:	controllability matrix $\boldsymbol{Q_{c}}$
Return type:	`numpy.ndarray`

pymoskito.controltools.observability_matrix(a_mat, c_mat)[source]¶

Calculate observability matrix and check observability of the system.

$\boldsymbol{Q_{o}} = \begin{pmatrix} \boldsymbol{C}\\ \boldsymbol{C}\boldsymbol{A}\\ \boldsymbol{C}\boldsymbol{A}^{2}\\ \vdots \\ \boldsymbol{C}\boldsymbol{A}^{n-1}\\ \end{pmatrix}$

Parameters:	a_mat (`numpy.ndarray`) – system matrix c_mat (`numpy.ndarray`) – output matrix
Returns:	observability matrix $\boldsymbol{Q_{o}}$
Return type:	`numpy.ndarray`

pymoskito.controltools.lie_derivatives(h, f, x, order=1)[source]¶

Calculates the Lie-Derivative from a scalar field $h(x)$ along a vector field $f(x)$ .

Parameters:	h (sympy.matrix) – scalar field f (sympy.matrix) – vector field x (sympy.matrix) – symbolic representation of the states order (int) – order
Returns:	lie derivatives in ascending order
Return type:	list of sympy.matrix