Nanopositioning control of an electrostatic MEMS actuator: adaptive terminal sliding mode control approach

Abstract

This research proposes an adaptive terminal sliding mode control strategy dedicated to motion tracking control of an electrostatic-actuated nanopositioning system. The electrostatic actuators’ dynamics are highly nonlinear as a result of the nature of the electrostatic force on the top plate making challenges to the researchers. This research aimed to provide a strong controller setpoint regulation maneuvers of a micro-mechanical electrostatic actuator system for reducing the vibration impacts on its performance. Forces applied to the electrostatic micro-electromechanical systems (MEMS) actuator are also designed using a nonlinear controller. Appropriate adaptive terminal sliding mode laws are capable of nanoscale positioning resolution, which is generally interpreted as a few nanometers (0.001 nm) and below. Its stability is proven using the Lyapunov theory. The obtained results show the effectiveness of the proposed method (about 50%) to achieve precise positioning. also, tests are given to evaluate the performance of the proposed method when external disturbances are acting on the actuator. These results show that the suggested controller is robust under disturbances.

Appendix 1: Stability analysis

The closed-loop stability and the convergence of the tracking errors using the adaptive terminal sliding mode control law (20) with the adaptation rules (21) for the MEMS actuator (5) is here.

Proposition 1

The ATSMC law (20) with the adaptation rules (21) for the MEMS Actuator (5) guarantees the tracking errors’ asymptotic convergence to the origin while existing uncertainties and external disturbances.

Proof

Suppose the following Lyapunov function case:

$$ V_{{{\text{ATSMC}}}} \left( {\hat{\kappa }~.~S_{{{\text{ATSMC}}}} } \right) = \frac{1}{2}S_{{{\text{ATSMC}}}}^{T} S_{{{\text{ATSMC}}}} + \frac{1}{2}Q^{T} Q $$

(23)

where

$$ Q = \beta ^{{ - 1}} \tilde{\kappa } $$

(24)

The time derivative of $V_{{{\text{ATSMC}}}}$ can be written as follows

$$ \begin{aligned} \dot{V}_{{{\text{ATSMC}}}} \left( {\hat{\kappa }~.~S_{{{\text{ATSMC}}}} } \right) &= S_{{{\text{ATSMC}}}}^{T} ~\dot{S}_{{{\text{ATSMC}}}} + Q^{T} \dot{Q} \hfill \\ &= S_{{{\text{ATSMC}}}}^{T} \left\{ - 2\varsigma \omega \left( { - 2\varsigma \omega x_{3} - \omega ^{2} \left( {x_{2} - \bar{l}_{e} } \right) - \frac{1}{2}x_{1}^{2} } \right) - \omega ^{2} \left( { - x_{3} } \right) \right.\hfill \\ &- x_{1} \left( { - x_{1} x_{2} + u} \right) - \dddot{x}_{d} \hfill \\ &+ \lambda _{1} \left( { - 2\varsigma \omega x_{3} - \omega ^{2} \left( {x_{2} - \bar{l}_{e} } \right) - \frac{1}{2}x_{1}^{2} - \ddot{x}_{d} } \right) \hfill \\ &\left.+ \lambda _{2} \left( { - x_{3} - \dot{x}_{d} } \right) + 0.6\lambda _{3} \left( {e_{x} ^{{ - 0.4}} } \right)\right\} + + Q^{T} \dot{Q} \hfill \\ \end{aligned} $$

(25)

Substitution from (14), we can conclude

$$ \begin{aligned} \dot{V}_{{{\text{ATSMC}}}} \left( {\hat{\kappa }~.~S_{{{\text{ATSMC}}}} } \right) &= S_{{{\text{ATSMC}}}}^{T} ~\dot{S}_{{{\text{ATSMC}}}} + + Q^{T} \dot{Q} \hfill \\ &= S_{{{\text{ATSMC}}}}^{T} \left\{ - 2\varsigma \omega \left( { - 2\varsigma \omega x_{3} - \omega ^{2} \left( {x_{2} - \bar{l}_{e} } \right) - \frac{1}{2}x_{1}^{2} } \right) - \omega ^{2} \left( { - x_{3} } \right)\right. \hfill \\& - x_{1} ( - x_{1} x_{2} + \{ \frac{1}{{x_{1} }}\{ x_{1}^{2} x_{2} \hfill \\ & + - 2\varsigma \omega \left( { - 2\varsigma \omega x_{3} - \omega ^{2} \left( {x_{2} - \bar{l}_{e} } \right) - \frac{1}{2}x_{1}^{2} } \right) - \omega ^{2} \left( {x_{3} } \right) + \dddot{x}_{d} \hfill \\ &- \lambda _{1} \left( { - 2\varsigma \omega x_{3} - \omega ^{2} \left( {x_{2} - \bar{l}_{e} } \right) - \frac{1}{2}x_{1}^{2} - \ddot{x}_{d} } \right) - \lambda _{2} \left( { - x_{3} - \dot{x}_{d} } \right) \hfill \\ &- 0.6\lambda _{3} \left( {e_{x} ^{{ - 0.4}} } \right)\} + \frac{1}{{x_{1} }}\left\{ {\hat{\kappa }sign~\left( {S_{{{\text{ATSMC}}}} } \right)} \right\}\} ) - \dddot{x}_{d} \hfill \\& + \lambda _{1} \left( { - 2\varsigma \omega x_{3} - \omega ^{2} \left( {x_{2} - \bar{l}_{e} } \right) - \frac{1}{2}x_{1}^{2} - \ddot{x}_{d} } \right) + \lambda _{2} \left( { - x_{3} - \dot{x}_{d} } \right) \hfill \\ &\left.+ 0.6\lambda _{3} \left( {e_{x} ^{{ - 0.4}} } \right)\right\} + Q^{T} \dot{Q} = S_{{{\text{ATSMC}}}}^{T} \left\{ { - \hat{\kappa }sign~\left( {S_{{{\text{ATSMC}}}} } \right)} \right\} + Q^{T} \dot{Q} \hfill \\ \end{aligned} $$

(26)

Consequently,

$$ \dot{V}_{{{\text{ATSMC}}}} \left( {\hat{\kappa }~.~S_{{{\text{ATSMC}}}} } \right) \le S_{{{\text{ATSMC}}}}^{T} \left\{ { - \hat{\kappa }sign~\left( {S_{{{\text{ATSMC}}}} } \right)} \right\} + \tilde{\kappa }~\beta ^{{ - 1}} \dot{\tilde{\kappa }} = - \hat{\kappa }\left| {S_{{{\text{ATSMC}}}} } \right| + \tilde{\kappa }~\left| {S_{{{\text{ATSMC}}}} } \right| = - \kappa \left| {S_{{{\text{ATSMC}}}} } \right| $$

(27)

By choosing appropriate gain, $\dot{V}_{{{\text{ATSMC}}}} \left( {\hat{\kappa }~.~S_{{{\text{ATSMC}}}} } \right)~$ will be negative semidefinite. Consequently, $\dot{V}_{{{\text{ATSMC}}}} \left( {\hat{\kappa }~.~S_{{{\text{ATSMC}}}} } \right)$ will be globally bounded. Therefore, tracking errors using Barbalat’s Lemma asymptotically converges to the origin.