`doc_disturbance_rejection_kang`

Syntax


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doc_disturbance_rejection_kang(A, B, D, Sw)

Description


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rho = doc_disturbance_rejection_kang(A, B, D, Sw)

This function is used to compute the new measure representing degree of controllability for disturbance rejection.

For the following LTI system

\dot{x} = A x (t) + B u (t) + D w (t)

$\mathbf{w}\left(t\right)$ is the disturbance vector. $\mathbf{S}_w$ .

A new measure representing degree of controllability for disturbance rejection in presented in [1].

The controllability Grammian of the system can be calculated by solving the following differential equation:

\dot{W} (t) = A W (t) + W (t) A^{'} + B B^{'}

Similarly, the disturbance-sensitivity Grammian satisfies the following differential equation:

\dot{Σ} (t) = A Σ (t) + Σ (t) A^{'} + D S_{w} D^{'}

Then,

ρ = tr {W {(T)}^{- 1} \cdot Σ (T)}

$T$ , consider steady-state solutions of Eqs. (5) and (6), satisfying Eqs. (16) and (17) in [1] for asymptotically stable systems:

\begin{aligned} A \bar{W} + \bar{W} A^{'} + B B^{'} & = 0 \\ A \bar{Σ} + \bar{Σ} A^{'} + D S_{ω} D^{'} & = 0 \end{aligned}

Then

ρ = tr {{\bar{W}}^{- 1} \cdot \bar{Σ}}

Details of the proof and other details can be found in the original literature.

Examples


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

>>> A = np.array(
        [
            [0, 0, 0, 1, 0, 0],
            [0, 0, 0, 0, 1, 0],
            [0, 0, 0, 0, 0, 1],
            [-2, 1, 0, -2, 1, 0],
            [1, -2, 1, 1, -2, 1],
            [0, 1, -1, 0, 1, -1],
        ]
    )
>>> B = np.array([[0, 0, 0, 0, 1, 0]]).T
>>> D = np.array([[0, 0, 0, 0, 1, 0]]).T
>>> Sw = np.array([[1]])

>>> doc_disturbance_rejection_kang(A, B, D, Sw)

5.999999999999998

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-r matrix.

D —— Disturbance matrix, specified as an n-by-l matrix.

Sw —— Covariance matrix of disturbance vectors, specified as an l-by-l square matrix.

Properties of Arguments

Name of the parameters	Is optional?	Source, dialog or input port?
`A`	No	Input port
`B`	No	Input port
`D`	No	Input port
`Sw`	No	Input port

References

[1] O. Kang, Y. Park, Y. S. Park, M. Suh, "New measure representing degree of controllability for disturbance rejection," Journal of Guidance, Control, and Dynamics, vol. 32, no. 5, pp. 1658-1661, 2009. DOI: 10.2514/1.43864.