MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
2.010 Modeling Dynamics and Control III
Fall 2000

Notes for PS #7

• Here's the one-page correction to the 2.004 final SOLUTIONS for the piezoelectric actuator problem (Problem 4 on the final, on which problem 7.2 in this week's homework is based.)
  • fin2.pdf , fin2.ps
  (Use this to help formulate the system model (part a) for problem 7.2...)
  
  
• Tutorial #7 Notes are now available via this link.
  
  
• HERE IS THE LINK TO THE FIGURES FOR PROBLEM 7.2:
  Figures for Piezoelectric Acuator, Problem 7.2
  
  
• The hand-out on specifications in the s-plane has been modified.
  Here is the REVISED version:
  splane.pdf or splane.ps
  
  
• For problem 7.2, you may find it useful to reference solutions to PROBLEM 4 on last term's 2.004 Final Exam. [This was the problem involving PZT piezoelectric stuff (as an aside, 'PZT' stands for the chemical composition: 'lead-zirconia-titanium') ...]
  
  Here are links to both the questions and solutions from the final:
  • final.pdf , final.ps
  • finalsol.pdf , finalsol.ps
  
  
• With Lecture #15 (Nov.2), we have begun studying ROOT LOCUS techniques.
  
  The MATLAB graphical tool Prof. Hogan demonstrated in class is called rltool. (Just type 'rltool' at a MATLAB prompt to invoke.)
  MATLAB also has other tools (like 'rlocus') which you can tuck away into your arsenal of plotting functions (along with step, bode, pzmap, et al).
  
  A root locus is like a roadmap (for closed-loop poles) in the s-plane (based on poles and zeros of the open-loop system).
  As you increase the GAIN of a system, the poles travel along little roadways (lines and curves) -- and you can create this map of roadways which they will follow in a systematic, procedural manner. [In practice, some of these roadmaps end up looking like schematic attack plans from a sadistic football coach, with poles seeming to curve gratuitously, and sometimes apparently trying to 'sack' some zero... or sometimes zipping off into the distance.]
  
  Essentially, root locus provides a GRAPHICAL way of understanding how cranking the gain of a system up or down will change the location of poles (and correspondingly alter the transient response characteristics [and perhaps stability] of this system).
  
  Here are two sets of reference material:
  • http://me.mit.edu/lectures/rlocus - MIT Mechanical Engineering Dept. on-line lecture on ROOT LOCUS
    (I literally 'stumbled' onto this while searching google.com for root locus references. It provides a fairly succinct overview description and includes both examples and an overall 'quiz' to gauge comprehension afterward...)
  • ...And some suggested readings from Nise and Ogata.
    
    

    Suggested Reading

    Book	Chapter	Sections	Topics
    Nise	Chapter 8	8.1 (pp. 380-384)	Root Locus: Introduction
    Nise	Chapter 8	8.4 (pp. 390-394)	Root Locus: Sketching (5 rules, as per lecture)
    Nise	Appendix C	pp. 774-775	Root Locus: MATLAB Examples
    Ogata	Chapter 6	6-2 (pp. 319-330)	Root Locus: Introduction
    Ogata	Chapter 6	6-3 (pp. 330-338)	Root Locus: Sketching (an 8-rule summary)
    Ogata	Chapter 6	6.4 (pp. 338-346)	Root Locus: MATLAB Commands (pp. 338-346)

    
    You can skim all of Chapter 8 in Nise, if you'd like a thorough overview of root locus.
    
    Again, these are the suggested references for the class:
    

    Textbook References:	Palm, MATLAB for Engineering Applications, McGraw-Hill, 1999.
    	Ogata, Modern Control Engineering, 3rd Ed., Prentice-Hall, 1997.
    	Nise, Control Systems Engineering, 2nd Ed., Addison-Wesley, 1995.
    	Dorf and Bishop, Modern Control Systems, Addison-Wesley, 7th Ed., 1995.

    
    These are available (on reserve) at Barker library.
    
    
  Feel free, as always, to email me any time with questions or comments-
  
  - Katie (gonzo@mit.edu)

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