doc_gramian

Syntax

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doc_gramian(A, B)

Compute the Gramian-matrix-based DOC.

Description

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rho1, rho2, rho3 = doc_gramian(A, B)

Computes the Gramian-matrix-based DOC of an LTI system, which refers to equation (2.9) in the paper [1]. The mathematical principle is available in Section 2.2 of [2].

For the above linear time-variant system, we recall the relation of the controllability problem to the fixed-time minimum-energy transfer control problem.

The optimization problem is solved for each initial condition $x_0$$x_0$ if and only if the system is completely controllable at $[t_0,t_1]$$\left[t_0,t_1\right]$. Then the minimum control energy is given by

where ${\mathbf{W}}_c$$\mathbf{W}_c$ a kind of the controllability matrix associated with $\mathbf{A}\left(t\right)$$\mathbf{A}\left(t\right)$ and $\mathbf{B}\left(t\right)$$\mathbf{B}\left(t\right)$.

Then the minimum eigenvalue, trace, and determinant of the ${\mathbf{W}}_c^{-1}(t_1,t_0)$$\mathbf{W}_c^{-1}\left(t_1,t_0\right)$ are just three proper physically meaningful measures of the quality of controllability.

• ${\lambda }_{min}\left({\mathbf{W}}_c\right)$$\lambda_{\min}\left(\mathbf{W}_c\right)$. Related to the maximum value of the minimum control energy over the unit ball $\{{\mathbf{x}}_0:\Vert {\mathbf{x}}_0\Vert =1\}$$\left\{\mathbf{x}_0:\left\|\mathbf{x}_0\right\|=1\right\}$.
• $\frac{n}{\text{tr}{\mathbf{W}}_c^{-1}}$$\frac{n}{\text{tr}\mathbf{W}_c^{-1}}$. Related to the average value of the minimum control energy over the unit hypersphere $\{{\mathbf{x}}_0:\Vert {\mathbf{x}}_0\Vert =1\}$$\left\{\mathbf{x}_0:\left\|\mathbf{x}_0\right\|=1\right\}$.
• $\text{det}{\mathbf{W}}_c$$\text{det}\mathbf{W}_c$. A characteristic of the attainable set of initial state vectors with prescribed maximum costs.

For the special case of an LTI system, the controllability matrix is

The three above measures become

Significantly, this function is only for LTI systems.

Examples

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>>> from OpenHA.assessment.attribute import doc_gramian
>>> import numpy as np

>>> A = np.array(
        [
            [0, 1, 0, 0, 0],
            [0, -0.089, 0.7132, 0, 0.8903],
            [0, 0, 0, 1, 0],
            [0, -0.2671, 31.5391, 0, 2.6708],
            [0, 0, 0, 0, -1],
        ]
    )
>>> B = np.array([0, 0, 0, 0, -1]).reshape((-1, 1))
>>> doc_gramian(A, B)

(0.204458903592688, 0.707999248988378, 29.896138372000475)

Input Arguments

A —— System transition matrix of the state-space model of an LTI system, specified as an n-by-n square matrix.

B —— Input coefficient matrix of the state-space model of an LIT system, specified as an n-by-p matrix.

Properties of Arguments

Output Arguments

A tuple (rho1, rho2, rho3), whose physical meanings are introduced above.

References

[1] G.-X. Du, Q. Quan, "Degree of Controllability and its Application in Aircraft Flight Control," Journal of Systems Science and Mathematical Sciences, vol. 34, no. 12, pp. 1578-1594, 2014.

[2] P.C. Müller, H.I. Weber, "Analysis and optimization of certain qualities of controllability and observability for linear dynamical systems," vol. 8, no. 3, pp. 237-246, 1972. DOI: 10.1016/0005-1098(72)90044-1.