An Advanced Extremum Seeking Scheme for the Target Trajectory in Electromagnetic Micromirror System

Now two problems result in bad control in the development of the electromagnetic micromirror system. One is that theoretical model in electromagnetic micromirror system is difficult to be determined; Another is that parameters in common control need to be tuned according to the experience. In this paper, cost function concept is proposed to determine the model order in slow-scan axis control of the electromagnetic micromirror. Then recursive least square scheme is built to off-line identify this model. Furthermore, an advanced extremum seeking scheme along with backtracking line search is exploited, which can automatically identify the best parameter value before each extremum search to improve the controllability based on this model for the target trajectory in slow-scan axis control. And the convergence of it is proved. Finally, the experiments and the simulations verify this method proposed valid.


I. INTRODUCTION
LECTROMAGNETIC micromirror as a MEMS (Micro-Electro-Mechanical System) device, which has a prominent advantage over other micromirrors in aspects of lower power consumption, larger deflection angle and so on [1]- [3]. The attempt has been made by some researchers now. Design and fabrication of electromagnetic micromirror were focused on in Ji's research [4], but the control was located in open-loop control. The sensitivity of four terminal piezo-resistive sensor on electromagnetic micromirror was emphasized in Chen's study [5], but control was still in open-loop control. Newton's method to determine the harmonic coefficients of electromagnetic micromirror was focused on in Steve's paper [6], but controlling the mirror presented a significant engineering challenge. PID control and a low pass filter (LPF) were proposed in Han's research [7], but choosing appropriate gains for PID controller is very difficult to manage and the rings from driving coils were difficult to be eliminated. In our previous work [8], it showed that the slow axis in electromagnetic micromirror system participated in several resonant motions under the signals of resonant frequencies, especially in the resonant movement on slow-scan axis shown in fig. 1, which is the main reason of the instability in the control of electromagnetic micromirror. Therefore, there is a requirement of reliability in slowscan axis control compared to the one of reliability in quick-scan axis control and a more advanced control method need to be exploited for the slow-scan axis control compared to common control at present, in which the parameters are tuned by experience. Extremum seeking control has an advantage to autonomously finding an optimal system behavior (e.g., set point or trajectory to be tracked) for the system in the control field, while at the same time maintaining stability and boundedness of signals [9], [10]. So it is first considered for the slow-scan axis control in this paper. Now two problems are faced in exploiting extremum seeking control for slow-scan axis control. One is that theoretical model is difficult to be determined due to the structure and the fabrication [5] of it; Another is that parameters in extremum control [9] on it need to be tuned according to the experience.  Target Trajectory in Electromagnetic  Micromirror System In this work, schematic in electromagnetic micromirror system is proposed and cost function concept is first used to determine model-order [11] in slow-scan axis control. Then from the mathematical point of view, recursive least square algorithm is applied in identifying this model. So the first problem can be solved mathematically. Furthermore, a new extremum seeking scheme with backtracking line search is exploited to solve the problem of the instability caused by resonant movement in slow-scan axis control and to improve the ability of extremum seeking control based on this model for the target trajectory in this slow-scan axis control. This method can automatically identify the best extremum step value in candidate step lengths at the beginning of each extremum search. The second problem can also be solved. Finally, the simulations off-line show that this algorithm is effective by using the experiment data.

A. Schematic in Electromagnetic Micromirror System
Schematic in electromagnetic micromirror system shown in a of Fig. 2  V t are the input voltage () ut and the output voltage () yt of quick axis in this electromagnetic micromirror system, respectively. The work in this paper revolves around this schematic (see Fig. 2.a) in slow-scan axis control. Input voltage () ut as tuning voltage from DA is linearly converted to drive electric current () It to drive the slow-scan axis of the electromagnetic micromirror. Four terminal piezo-resistive sensor is used to monitor the angle of electromagnetic micromirror. Meanwhile, it provides the output voltage () yt as feedback voltage for the slow-scan axis control after being scaled up linearly. Its simplified schematic of electromagnetic micromirror system is seen in b of Fig. 2 in order to mathematically identify its model.

B. Model-order Identification
Suppose this model is linear discrete time invariant single input single output system based on this structure (Fig. 2), which can be described as (2) transposed parameter vector 12 12 and data vector

C. Recursive Least Square Identification
Recursive least square algorithm as the method on the identification of dynamic process is widely used in the engineering application of system. [12,13] Based on this schematic (Fig. 2) and Sec. II-B, we considers the application of the recursive least square method to the problem of identifying the model in slow-scan axis control of electromagnetic micromirror. It can update the coefficients recursively that minimize the cost function corresponding to output voltage () yk of electromagnetic micromirror system at time step k in the process of identification. Recursive least square algorithm can be described as Where () Kk is a 21 m × gain matrix at time step k , () Pk is a 22 mm × covariance matrix at time step k and ˆ( ) k θ is a parameter vector at time step k . We set the transposed consistent estimation as This recursive least square algorithm can give strongly consistent estimation [12], that is ˆ( ) () .

A. Extremum Seeking Control Algorithm
Relative differential equation of slow-scan axis in electromagnetic micromirror system based on the model (see Sec. II) is depicted by and The corresponding discrete-time state space representation in slow-scan axis control is given as Define the cost function ( 1) i s Jk + in extremum seeking control of slow-scan axis in ith iteration at step time ( 1) ()

B. Automatically Choosing the Best Extremum Step Length
Before each extremum seeking algorithm in slow-scan axis control at time step k , backtracking line search technique [14] is applied in choosing the best extremum step length k s λ during the candidate step lengths, and its essential backtracking proceed is shown in Fig. 4. In the flow diagram (see Fig. 4), after starting,

IV. EXPERIMENT AND SIMULATION RESULTS
In this section, experiments are all based on dsPACE platform including DA (DS2012) and AD (DS2004). Its connection between dsPACE and electromagnetic micromirror system (Fig. 2) is shown in Fig. 5. Sampling frequencies in this section are all 5 kHz .

A. Second Order Model in Slow-scan axis control
The electromagnetic micromirror system model in slow scan-axis was modelled with the parameters ( ) (45) The corresponding performance in off-line calculations of recursive least square algorithm is shown in Fig. 6. In the recursive least square algorithm of slow-scan axis control, the parameters     52)) mathematically. Now two aspects are focused on to show the performance of the advanced extremum seeking scheme. Compared with the secondary filtering [8] (seen in Fig. 9 and Fig. 10), this advanced extremum seeking algorithm plays a better role in filtering (seen from Fig. 7 to Fig. 10). Based on this model, the proposed method (see Sec. III) of slow-scan axis succeed in keeping the original frequency 60 Hz for target trajectory with an amplitude 0.1 V under external disturbance, especially under electromagnetic interference similar to cell phone calls ( Operating Performance in Fig. 7 and Spectrogram Performance in Fig. 8). This is the first advantage. The second advantage is that the proposed method can automatically identify the best extremum step value in candidate step lengths at the beginning of each extremum search to improve the tracking for the target trajectory in slow-scan axis control. Furthermore, tracking performance such as deflection angle, tuning voltage, state variables and so on in off-line iteration and recursion of this advanced extremum seeking scheme in slow-scan axis control is shown in Fig. 11 and in Fig. 12, in the case that input sine voltage is with an amplitude of 2 V and is of the frequency 60 Hz . The resolution of four terminal piezo-resistive sensor for slow-scan axis for this electromagnetic system is 1 @ V 3 deg . Maximum absolute deflection angle error 0.3113  appears in 0.0002 second of this new extremum seeking scheme (see the second row in Fig. 11). And then it is limited in 0.0191  . Compared with the maximum absolute error of 0.67  in open-loop control (see Fig. 13), this advanced extremum seeking scheme in slowscan axis control succeeds in regulating the tuning voltage more quickly and more accurately. (see Fig. 12). Although the maximum absolute error is also 0.3113  in the extremum seeking with fixed step length (see Fig. 14 and Fig. 15) through adjusting the fixed step length by experience. After 0.0002 second, it is only limited in 0.18  . So the advanced extremum seeking scheme show stronger tracking the target angle than the extremum seeking with fixed step length (see Fig. 11). Maximum deflection angle in this new extremum seeking scheme of slow-scan axis control is 6 ±  .  The proposed scheme presents an advanced extremum seeking scheme with backtracking line search based on recursive least square algorithm in slow-scan axis control for electromagnetic micromirror. Furthermore, the results in experiment and simulation show that the performance of the proposed method not only keep the original frequency of target trajectory but also track the target angle through the optimal actuation voltage in best extremum step length.