This blog describes a novel method of nonlinear feedback control of a MEMS mirror. The idea presented here is covered in my patent that expires the end of 2022, link. The technique employs a method of input linearization.
Introduction
The equation of motion of a 1D Micro-electro-mechanical (MEMS) mirror is derived from first principles. The result will show MEMS rotation is a nonlinear function of the applied voltage. I will then show the derivation of a novel method to linearize the applied torque to control the position. This method increases the operating control range of the MEMS device.
First Principles of a MEMS Device
The MEMS device we are looking at is depicted in Figure 1.
Figure - 1 MEMS Structure
The moving plate rotates around the x axis (2). The moving plate has hinges (1) which allow the plate to rotate. One side of the moving plate is an optical mirror and the opposite side (shown) has two metal electrodes (4) which allow application of a voltage v1 on the left plate and v2 on the right plate. The silicon MEMS has a restoring torque when the moving plate is not at zero angle.
The separation between the rotating plate and the ground plane beneath it is d at zero deflection. Assume each electrode has a dimension H long by W wide (with area $A = H\cdot W$) and is centered L from the axis of rotation.
When energized each electrode will create a torque on the silicon mirror and there will be a restoring torque from the silicon hinges given by the stiffness of the hinges, kss. The net torque accelerates the inertia of the MEMS given by the dynamic equation below.
$\begin{equation} J_m \cdot \ddot{\theta} = T_1(V_1) - T_2(V_2) - kss \cdot \theta - kv \cdot \dot{\theta}Â \equiv T_e(V_1,V_2,\theta)Â - kss \cdot \theta - kv \cdot \dot{\theta} \end{equation}$Â Â Â [1]
We start out by looking at the simplest case of the force across the plates of a capacitor. Let the spacing of the plates be 'd' when the voltage across the plates is zero. Now if we apply a voltage V across the plates the added charge Q generates a force that attracts the plates. The equation for charge is Q = V C and capacitance is C = e A/ d where d is the plate separation and A is the area.
The force acting on the negatively charged plate is F-= Q E+, where E+ is the magnitude of the electric intensity on the positively charged plate. Likewise F+ = Q E- is the force on the negatively charged plate that has electric intensity magnitude E-.
Both E+ and E- vectors are of equal magnitude and the same direction so we can write E+ = E- = E/2, where E is the electric intensity between the two plates. The electric field between the plates is given by E = V/d.
We substitue the electric intensity into the equation for electostatic force and get Fe= Q V / 2d. We use the equation for charge Q and obtain Fe = V^2 C/2d. Finally we arrive at the force as
$\begin{equation} F_e = \epsilon A \cdot V^2 / 2d^2 \end{equation}$Â Â Â [2]
This is the force acting on one electrode. If the distance from the center of rotation to the application of the force vector is L, then the torque generated by the electostatic force is given by [3]
$\begin{equation} T_e = \epsilon A \cdot L \cdot V^2 / 2d^2 \end{equation}$Â Â Â [3]
When the MEMS is rotated the separation will be $d - L\cdot sin(\theta)$ on one electrode and $d + L\cdot sin(\theta)$ on the other electrode. The total electrostatic torque is given by [4]
$\begin{equation} {T_e}(V_1,V_2,\theta) = (\epsilon A \cdot L)/2 \cdot ( {V_1}^2 / (d-L \cdot sin(\theta))^2 -Â {V_2}^2 / (d+L \cdot sin(\theta))^2 ) \end{equation}$Â Â Â [4]
Using a small angle approximation the net electrostatic torque becomes this:
$\begin{equation} {T_e}(V_1,V_2,\theta) = (\epsilon A \cdot L)/2 \cdot ( {V_1}^2 / (d-L \cdot \theta)^2 -Â {V_2}^2 / (d+L \cdot \theta)^2 ) \end{equation}$Â Â Â [5]
Equation [5] is obviously non-linear, so now we will proceed to use input linearization on system [1] and [5].
Model Linearization
In the dynamic equation [5] we can replace $\begin{equation} T_e(V_1,V_2,\theta)Â \end{equation}$ with $\zeta$ to realize a new dynamic equation given in [6] below.
$\begin{equation} J_m \cdot \ddot{\theta} = \zeta - kss \cdot \theta - kv \cdot \dot{\theta} \end{equation}$   [6]
System [6] is now linear with input $\zeta$. Measurements or estimates of $\theta$ and $\dot{\theta}$ can be used with a controller, such as a PID, to compute a desired value for the input $\zeta$. A block diagram of the system is shown below.
Figure - 2 Linearized Input Control Loop
In the MEMS control the electrode voltage is comprised of a fixed positive bias voltage, $V_b$ plus a signed control voltage. The two electrode voltages are $V_1 = V_b + v_c$ and $V_2 = V_b - v_c$.
Each sample time of the control loop we will update the estimator values $\theta$ and $\dot{\theta}$ and then compute a $\zeta$ that satisfies the PID control loop. The new values of $\zeta$ and $\theta$ are then used in [7] to compute a new value of $v_c$.
$\begin{equation} 0 = (\epsilon A \cdot L)/2 \cdot ( {(V_b + v_c)^2} / (d-L \cdot \theta)^2 -Â {(V_b - v_c)^2}/ (d+L \cdot \theta)^2 ) - \zeta \end{equation}$Â Â Â [7]
Finally, from the value of $v_c$ the electrode voltages $V_1$ and $V_2$ are computed and applied to the MEMS.
Parting Note
The above architecture was implemented about 2002 on an Analog Devices SHARC Blackfin processor. In that instance it was used to control a 2D MEMS mirror and the two axes, x and y, were treated as a decoupled system. In that implementation the MEMS structure was that shown in Figure 3. The concept of input linearization is a useful technique.
Figure 3 - 2D MEMS
