Fixed-time disturbance observer based on fractional-order state observer and super-twisting sliding mode control for a class of second-order of slotless self-bearing motor

Abstract

This paper proposes a new fixed-time disturbance observer (FTDOB) based on the fractional-order state observer (FOSOB) and a super-twisting sliding-mode control (STSMC) for a slotless self-bearing motor (SSBM). First, the slotless self-bearing motor with fully embedded disturbances and uncertainties is analyzed, the imperfection of the perturbation sources. Second, a fractional-order state observer was designed to estimate the velocities and accelerations of movement on $x - ,$ $y - ,$ and $\omega -$ axes, respectively. Third, a new concept of fixed-time disturbance observer was proposed for estimating the perturbations on three axes of the bearing motor system. Fourth, the super-twisting sliding-mode control was designed to control the positions and the rotational speed of the bearing motor system. Final, the stability of the proposed algorithms was theoretically verified via the Lyapunov condition. To visually present the effectiveness and originality of the proposed theories, the MATLAB simulation was used to show that the proposed control system consisted small overshoots, small settling times, and stable tracking error values in narrow gaps around zero.

Appendices

Appendix

The fixed-time stability

First, consider the system

$$ \dot{x}(t) = H(t,x) $$

(86)

Definition 1

[34] Fixed-time stability. System (80) is called fixed-time stable if the settling time is limited by a constant value as follows:

$$ \left\{ \begin{gathered} T(x_{0} ) < T_{\begin{gathered} \max \hfill \\ \hfill \\ \end{gathered} } \hfill \\ T_{\max } > 0 \hfill \\ \end{gathered} \right.,\quad \forall x(0) $$

(87)

Lemma 1 [34]: Consider the function as follows:

$$ \dot{s} = - \zeta_{1} \left| s \right|^{{\frac{n1}{{m1}}}} sign(s) - \zeta_{2} \left| s \right|^{{\frac{n2}{{m2}}}} sign(s) $$

(88)

where $n_{1} ,$ $n_{2} ,$ $m_{1} ,$ $m_{2}$ are positively defined. $n_{1} > m_{1} ,$ $m_{2} < m_{2} ,$ $\zeta_{1} > 0$ and $\zeta_{2} > 0$ The settling time can be calculated as follows:

$$ T < T_{\max } = \frac{1}{{\varsigma_{1} }}\frac{{m_{1} }}{{n_{1} - m_{1} }} + \frac{1}{{\varsigma_{2} }}\frac{{m_{2} }}{{m_{2} - n_{2} }} $$

(89)

The fractional calculus

Definition 2

[27]: The Euler’s Gamma function.

The Gamma function is rewritten as follows:

$$ \Gamma \left( \gamma \right) = \int\limits_{0}^{\infty } {t^{\gamma - 1} e^{ - t} } dt $$

(90)

where the $\gamma$ is the order of the operation and the operation time is $t.$

Definition 3

Definition 3 [27]: Fractional function derivatives and its integrals.

$$ D_{t}^{n} = \left\{ {\begin{array}{*{20}c} {\frac{{d^{n} }}{{dt^{n} }},n > 0} \\ {1,n = 0} \\ {\int\limits_{a}^{t} {(d\tau )^{ - n} ,n < 0} } \\ \end{array} } \right. $$

(91)

where t is boundary of operation and n is order of the operation.

Definition 4

[27]: The Caputo fractional derivative.

$$ D_{t}^{n} h(t) = \frac{1}{\Gamma (l - n)}\int\limits_{a}^{t} {\frac{{f^{n} (\tau )}}{{(t - \tau )^{n - l + 1} }}} d\tau $$

(92)

where $l - 1 < n < l.$ $n$ is real number, $l \in N$.

Definition 5

[27]: Stability of the fractional-order system.

First, consider

$$ D_{t}^{n} \chi = h\left( \chi \right) $$

(93)

where $\chi = [\chi_{1} , \ldots \chi_{l} ]^{T}$ is the system state vector, $h(\chi ) = [h_{1} (\chi ), \ldots h_{l} (\chi )]^{T}$ is the functions of the system. Where 0 < n < 1 is the fractional order. System (93) stable if the eigenvalues of Jacobian $J = \partial h(X)/\partial x$ are all located in the domain of

$$ \left| {\arg (eig(J))} \right| > n\frac{\pi }{2} $$

(94)

Some properties of fractional calculus are as follows:

Property 1

If n = 0 the operation is become

$$ D_{t}^{0} h\left( X \right) = h\left( X \right) $$

(95)

Property 2

Caputo operation can be applied linearization as

$$ D_{t}^{n} \left( {g\left( X \right) + h\left( X \right)} \right) = D_{t}^{n} g\left( X \right) + D_{t}^{n} h\left( X \right) $$

(96)

Property 3

The product of fractional derivative is calculated as follows:

$$ D_{t}^{n + m} h(X) = D_{t}^{n} D_{t}^{m} h(X) $$

(97)