A Precise Identification and Control Method for the 6D Micro-Vibration Exciting System

Abstract

Purpose

A precise identification and control method for the 6D micro-vibration exciting system is proposed in this paper. The controlled object is a Stewart configuration platform, which can provide six degree-of-freedom (6DOF) micro-vibration excitation excited by six actuating legs.

Methods

In view of the amplitude nonlinearity in this kind of exciting system, a nonlinearly iterative identification method is introduced to establish an inverse model to decouple the system in different levels of magnitude. For the decoupled system, a multi-input multiple-output (MIMO) adaptive disturbance cancellation (ADC) control method is proposed for the time domain control of the exciting system. Furthermore, the feedforward control loop is added to MIMO ADC control method to improve the rapidity and stability of the control system.

Results

Experiments show that the nonlinearly iterative identification method can greatly improve the decoupling accuracy of the system, and then improve the accuracy of sinusoidal vibration control.

Conclusion

The precise identification and control method can achieve sinusoidal vibration control of high precision in the control frequency band up to 300 Hz.

Appendix

A.1 Transient Response of LMS ADC Control Method

Taking a single-DOF linear system with a transfer function of $H = Ae^{j\varphi }$ at frequency $\omega_{{0}}$ as an example, the transient response of the LMS ADC control method with feedforward is derived, which reflects the characteristics of the method in the convergence process.

To reflect the superiority of the method proposed in this paper over the traditional control method without feedforward, the transient response of the traditional LMS ADC method without feedforward is derived first.

In the traditional LMS ADC control method, the weight coefficient is generally defined as ${\varvec{w}}(n) = [w_{1} ,w_{2} ]^{T}$, the reference signal as ${\varvec{x}}(n) = [\cos (\omega_{0} n),\sin (\omega_{0} n)]^{T}$, the error signal as $e(n) = d(n) - y(n)$, the expected signal as $d(n) = D\cos (\omega_{0} n + \gamma )$, and the control signal as $u(n) = {\varvec{w}}^{T} (n){\varvec{x}}(n)$.

According to the literature [43, 44], the weight coefficient updating formula is as follows:

$$ {\varvec{w}}(n + 1) = {\varvec{w}}(n) + \mu {\varvec{x}}(n)e(n) $$

(52)

The output response signal is as follows:

$$ y(n) = Ae^{j\varphi } u(n) $$

(53)

By substituting control signal into the Eq. (53), the Eq. (54) is obtained as follows:

$$ y(n) = {\varvec{w}}^{T} (n)Ae^{j\varphi } {\varvec{x}}(n) $$

(54)

Since $Ae^{j\varphi } {\varvec{x}}(n)$ is constant, the response is only related to the weight coefficient. Therefore, it is more intuitive to analyze the transient response of the output signal with the convergence process of the weight coefficient. By substituting Eq. (54) into Eq. (52), the Eq. (55) is obtained as follows:

$$ {\varvec{w}}(n + 1) = {\varvec{w}}(n) + \mu \left[ {d(n) - {\varvec{w}}^{T} (n)Ae^{j\varphi } {\varvec{x}}(n)} \right]{\varvec{x}}(n) $$

(55)

To simply represent the process of transient response, the system is normalized. Make $A = D = 1$, and $\gamma = 0$. Then the invariant term in Eq. (55) can be expressed as:

$$ Ae^{j\varphi } {\varvec{x}}(n) = {\varvec{z}}(n) = [\cos (\omega_{0} n + \varphi ),\sin (\omega_{0} n + \varphi )]^{T} $$

(56)

By substituting Eq. (56) into Eq. (55) and taking mathematical expectation on both sides of Eq. (55), the Eq. (57) is obtained as follows:

$$ E[{\varvec{w}}(n + 1)] = E[{\varvec{w}}(n)] + \mu E\left[ {d(n){\varvec{x}}(n)} \right] - \mu E[{\varvec{w}}^{T} (n){\varvec{z}}(n){\varvec{x}}(n)] $$

(57)

Since it can be assumed that ${\varvec{w}}(n)$ is independent of ${\varvec{x}}(n)$, Eq. (57) can be simplified as follows:

$$ E[{\varvec{w}}(n + 1)] = E[{\varvec{w}}(n)] + \mu E\left[ {d(n){\varvec{x}}(n)} \right] - \mu E[{\varvec{x}}(n){\varvec{z}}^{T} (n)]E[{\varvec{w}}(n)] $$

(58)

The steady-state value can be obtained by calculating the limit on both sides of Eq. (58) as follows:

$$ {\varvec{w}}_{\infty } = E[{\varvec{x}}(n){\varvec{z}}^{T} (n)]^{ - 1} E\left[ {d(n){\varvec{x}}(n)} \right] $$

(59)

It is assumed that the initial value is ${\varvec{w}}(0) = {\varvec{w}}_{0}$, and the Z-transform is used to solve Eq. (59). The solution of Eq. (59) can be obtained as follows:

$$ E[{\varvec{w}}(n)] = {\varvec{w}}_{\infty } + \left( {{\mathbf{I}}_{2 \times 2} - \mu E[{\varvec{x}}(n){\varvec{z}}^{T} (n)]} \right)^{n} \left( {{\varvec{w}}_{0} - {\varvec{w}}_{\infty } } \right) $$

(60)

It is known that $E[{\varvec{x}}(n){\varvec{z}}^{T} (n)]$ in Eq. (60) is the correlation matrix of vector ${\varvec{x}}(n)$ and vector ${\varvec{z}}(n)$, which can be easily obtained by the theory of matrix and linear algebra as follows:

$$ E[{\varvec{x}}(n){\varvec{z}}^{T} (n)] = \left( {\begin{array}{*{20}c} {\cos \varphi } & {\sin \varphi } \\ { - \sin \varphi } & {\cos \varphi } \\ \end{array} } \right) $$

(61)

For a convergent LMS ADC control method, the weight coefficients eventually converge to the optimal value. Obviously, the relationship between the steady-state value and the optimal value of the weight coefficients is as follows:

$$ {\varvec{w}}_{\infty } \approx [\cos (\varphi ),\sin (\varphi )]^{T} $$

(62)

Therefore, by substituting Eq. (61) and Eq. (62) into the Eq. (60), the general term expression of transient response can be obtained as follows:

$$ E[{\varvec{w}}(n)] = \left( {\begin{array}{*{20}c} {\cos (\varphi )} \\ {\sin (\varphi )} \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} {1 - \mu \cos \varphi } & {1 - \mu \sin \varphi } \\ {1 + \mu \sin \varphi } & {1 - \mu \cos \varphi } \\ \end{array} } \right)^{n} \left( {{\varvec{w}}_{0} - \left( {\begin{array}{*{20}c} {\cos (\varphi )} \\ {\sin (\varphi )} \\ \end{array} } \right)} \right) $$

(63)

For the control system without feedforward, the initial value of weight coefficient is ${\varvec{w}}_{0} { = 0}$.

For the LMS ADC control method with feedforward, ${\varvec{w}}^{T} (n){\varvec{x}}(n) + {\varvec{d}}(n)$ is used as control signal. Since the control signal is always sinusoidal in the iterative process, it can be transformed into ${\varvec{w}}_{e}^{T} (n){\varvec{x}}(n)$, where ${\varvec{w}}_{e}^{T} (n)$ is the equivalent weight coefficient by trigonometric function theory. Compared with the weight coefficient of the system without feedforward, the initial value of the equivalent weight coefficient is different, but the steady-state solution is the same.

Therefore, the general term expression in Eq. (63) can be applicable to the LMS ADC control method with and without feedforward, and the difference only lies in the initial values of the weight coefficients.