ROBUST CONTROL OF UNSTABLE DISTRIBUTED PARAMETER SYSTEMS

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 IASTED Controls and Applications 2004

2. Introduction • Introduction • Laguerre expansions • Robust stabilization • Example • Conclusions IASTED Controls and Applications 2004

3. Introduction • Gd(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 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 IASTED Controls and Applications 2004

5. Introduction (Continued) • Conclusion: Laguerre series are convenient and natural for approximating unstabledistributed 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 IASTED Controls and Applications 2004

6. Laguerre Expansions 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.) 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. IASTED Controls and Applications 2004

9. Robust Stabilization Theorem Consider a (potentially unstable and distributed parameter) plant with Gd=(N+EN)/(D+ED) where N and D are stable, rational, proper, coprime transfer functions and EN and ED 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 Gd provided ||ED(V-NQ)+EN(U+DQ)||inf < 1. 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. 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. 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 Gd(s)? Robust stabilization theorem says YES. IASTED Controls and Applications 2004

13. Example (cont) • Does it track a step input? • YES 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 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 IASTED Controls and Applications 2004