1. Philip D Olivier Mercer University IASTED Controls and Applications 2004 1 ROBUST CONTROL OF UNSTABLE DISTRIBUTED PARAMETER SYSTEMS Philip D. Olivier Mercer University Macon, GA 31207 United States of America Olivier_pd@mercer.edu http://faculty.mercer.edu/olivier_pd

2. Philip D Olivier Mercer University IASTED Controls and Applications 2004 2 Introduction Laguerre expansions Robust stabilization Example Conclusions

3. Philip D Olivier Mercer University IASTED Controls and Applications 2004 3 Introduction G d (s) unstable, distributed C(s) = ? So that –System is stable –Satisfies other design objectives Input tracking, disturbance rejection, etc Most design procedures are for lumped systems

4. Philip D Olivier Mercer University IASTED Controls and Applications 2004 4 Introduction (continued) Laguerre series can be used directly to approximate stable systems Many recent papers on applying Laguerre series to controls (see e.g. [1-17]) This paper shows how Laguerre series can be used to design controllers for unstable distributed parameter systems Further, it addresses the issue of how good is “good enough”: i.e. robustness

5. Philip D Olivier Mercer University IASTED Controls and Applications 2004 5 Introduction (Continued) Conclusion: Laguerre series are convenient and natural for approximating unstable distributed parameter systems in terms of stable lumped parameter systems in a way that conveniently allows for –Application of well established design procedures (most of which apply to lumped parameter systems) –Robustness analysis –Easy extension to MIMO systems

6. Philip D Olivier Mercer University IASTED Controls and Applications 2004 6 Laguerre Expansions

7. Philip D Olivier Mercer University IASTED Controls and Applications 2004 7 Q (or Youla) Parameterization Theorem Consider the SISO feedback system in the figure. Let the possibly unstable rational plant have stable coprime factorization G=N/D with stable auxiliary functions U and V such that UN+VD=I. All stabilizing controllers have the form C=[U+DQ]/[V-NQ] for some stable proper Q. (There is a MIMO version.)

8. Philip D Olivier Mercer University IASTED Controls and Applications 2004 8 Small Gain Theorem Suppose that M is stable and that ||M|| inf < 1 then (I+M) -1 is also stable.

9. Philip D Olivier Mercer University IASTED Controls and Applications 2004 9 Robust Stabilization Theorem Consider a (potentially unstable and distributed parameter) plant with G d =(N+E N )/(D+E D ) where N and D are stable, rational, proper, coprime transfer functions and E N and E D are the stable errors. Let U and V be stable rational auxiliary functions such that UN+VD=1. All controllers of the form C=[U+DQ]/[V-NQ] internally stabilize the unity gain negative feedback system with either G or G d provided ||E D (V-NQ)+E N (U+DQ)|| inf < 1.

10. Philip D Olivier Mercer University IASTED Controls and Applications 2004 10 Proof Recognize that the numerator and denominator are algebraic expressions of stable factors/terms. Hence each is stable. Apply Small gain Theorem to denominator.

11. Philip D Olivier Mercer University IASTED Controls and Applications 2004 11 Example Find a controller that stabilizes the unstable distributed parameter plant and provides zero steady-state error due to step inputs.

12. Philip D Olivier Mercer University IASTED Controls and Applications 2004 12 Example (Cont) Zero steady state error due to a step input => T(0)=1, C(0) = inf, So choose simplest Q(s) Does the resulting C(s) stabilize both G(s) and G d (s)? Robust stabilization theorem says YES.

13. Philip D Olivier Mercer University IASTED Controls and Applications 2004 13 Example (cont) Does it track a step input? YES

14. Philip D Olivier Mercer University IASTED Controls and Applications 2004 14 Example (cont) How conservative? This theorem implies a “stability margin” of about ||E|| inf-max /||E|| inf =1/.3343=2.991 Theorem in Francis, Doyle, Tannenbaum (can be viewed as a corollary of this one) implies a “stability margin” of about Nearly 50% improvement

15. Philip D Olivier Mercer University IASTED Controls and Applications 2004 15 Conclusions Laguerre expansions provide stable approximations of stable functions with additive errors When combined with coprime factorizations and Youla parameterizations provides yields nice, less conservative, robust stabilization theorem If robust stabilization check fails, add more terms to Laguerre expansion to reduce errors. Other design constraints are easy to include