Control of MEMS
EE 689 LABORATORY, ASSIGNMENT and HW #1 (30+15 extra points)
When: Next week, R (11/11/2014). Students must do HW (as assigned), solve problems, and, be prepared
Assignment Cover Sheet
Analysis and Control of MEMS Using MATLAB
Written By ( up to 2 students in the group): ____________________
Date Due: Next Class Note: Soft Copy files for all 2 Labs and Matlab files are required
Checklist _____ Text, model developed and used, MATLAB scripts and SIMULINK diagrams, results _____ Analytical results and numerical data. Developed MATLAB and SIMULINK files ______ Report (EME Technical Memo or IEEE format with the assignemnts as below) ______ “Soft Copy” of 2 Lab Reports (MS Word and Matlab files MUST be provided on a CD)
Grades will be based on: Organization and clarity of analytical results, correctness of deviations and model, organization and clarity of MATLAB / SIMULINK files, numerical results, evaluation and interpretation of results, accuracy of results, experiments (if any), conclusions, references and citations
Laboratory Assignment (Report) Contents - IEEE or EE Technical Memo format (typed Report is required) Abstract: Introduction
1. Problems 1.a and 1.b. (Start on a New page). ………
4. Conclusions (Start on a New page) References (textbook, books, the IEEE Explore http://wally.rit.edu/electronic/xplore/xplore.html, etc.)
Problem: Consider a MEMS electrostatic/electromagnetic actuator (microspeaker, hard-drive actuator or others, as we studied). Design and analyze closed-loop systems solving nonlinear optimization problem (per Sections 8.11 and 8.12 of a textbook, which is below). Simulate a closed-loop system with nonlinear control laws. Examine and evaluate the closed-loop system performance.
Problems to Solve:
Use the results of the Control chapter (Sections 8.11 & 8.12).
For three MEMS models per
Examples 8.20, e.g. ,
Example 8.21, e.g., the torque/force model ,, please do:
1. Derive nonlinear control laws by using:
(a) the arctan(x) function with limits 1. E.g. use (x)= 2arctan(x)/;
(b) the Guderman function gdx with limits 1. E.g. use (x)= 2gdx/
(c) The generalized logistic, sigmoid or other functions, such as a=1 and b=5 – 15 Extra-points
Results MUST be reported for two cases, e.g., two analyses, Simulink diagrams, closed-loop systems, etc.
2. Using the Lyapunov stability theory examine the stability of the closed-loop systems (plot the Lyapunov and dV/dt functions as per Section 8.11 and others);
3. Develop the SIMULINK models and perform coherent simulations (ALL parameters, if any, used must be reported as the “comments” (text) in your Simulink files);
4. Analyze analytic and numeric results and report conclusions. Emphasize practicality of your results to analysis, control and optimization of MEMS systems.
Note: The Gudermannian function is with the inverse function as gd–1x=arsinh(tanx)=artanh(sinx)
The derivative of the Gudermannian and its inverse are d(gdx)/dx=sechx and d(gd–1x)/dx=secx
Recall: The following MATLAB statement is used to perform calculations and plotting plot(ry(:,1),ry(:,2), ry(:,1),ry(:,3),'LineWidth',2); title('Dynamics of y(t), Evolution of r(t)','FontSize',14); xlabel('Time [seconds]','FontSize',14);
plot(ry(:,1),ry(:,2), ry(:,1),ry(:,4),'LineWidth',2); title('Dynamics of y(t), Evolution of r(t)','FontSize',14); xlabel('Time [seconds]','FontSize',14);
% Plotting statement plot(k,y,'o',k,y,'--',k,r,':'); title('Angular Displacement, y(t)=0.01\theta_r, r(t)=1 [rad]','FontSize',14); xlabel('Discrete Time k, Continuous Time t=kT_s [seconds]','FontSize',14); ylabel('Output y(k) and Reference r(k)','FontSize',14); axis([0 20,0 1.2]) % Axis limits
