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_at_mercer.edu
• http//faculty.mercer.edu/olivier_pd

2
Introduction

• Introduction
• Laguerre expansions
• Robust stabilization
• Example
• Conclusions

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

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
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
Laguerre Expansions
7
Q (or Youla) Parameterization Theorem

• Consider the SISO feedback system in the
  figure. Let the possibly unstable rational plant
  have stable coprime factorization GN/D with
  stable auxiliary functions U and V such that
  UNVDI. All stabilizing controllers have the
  form CUDQ/V-NQ for some stable proper Q.
  (There is a MIMO version.)

8
Small Gain Theorem

• Suppose that M is stable and that Minf lt 1
  then (IM)-1 is also stable.

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Robust Stabilization Theorem

• Consider a (potentially unstable and
  distributed parameter) plant with
  Gd(NEN)/(DED) 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
  UNVD1. All controllers of the form
  CUDQ/V-NQ internally stabilize the unity
  gain negative feedback system with either G or Gd
  provided ED(V-NQ)EN(UDQ)inf lt 1.

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
Example
Find a controller that stabilizes the unstable
distributed parameter plant and provides zero
steady-state error due to step inputs.
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Example (Cont)
Zero steady state error due to a step input gt
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.
13
Example (cont)

• Does it track a step input?
• YES

14
Example (cont)

• How conservative?
• This theorem implies a stability margin of
  about Einf-max /Einf1/.33432.991
• Theorem in Francis, Doyle, Tannenbaum (can be
  viewed as a corollary of this one) implies a
  stability margin of about
• Nearly 50 improvement

15
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