ROBUST CONTROL OF
UNSTABLE DISTRIBUTED
PARAMETER SYSTEMS
Philip D. Olivier
Mercer University
Macon, GA 31207
United States of America
[email protected]
http://faculty.mercer.edu/olivier_pd
Philip D Olivier
Mercer University
IASTED Controls and Applications
2004
1
Introduction
•
•
•
•
•
Introduction
Laguerre expansions
Robust stabilization
Example
Conclusions
Philip D Olivier
Mercer University
IASTED Controls and Applications
2004
2
Introduction
L(s)
R(s) + E(s)
-
C(s)
+
+
U(s)
Gd(s)
Y(s)
• 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
Philip D Olivier
Mercer University
IASTED Controls and Applications
2004
3
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
Philip D Olivier
Mercer University
IASTED Controls and Applications
2004
4
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
Philip D Olivier
Mercer University
IASTED Controls and Applications
2004
5
Laguerre Expansions
( s  p)
L {li (t )}  2 p
 Li (s)
i 1
( s  p)
i

 (t )   fi li (t )
i 0
fi  

0
1
 (t )li (t )dt 
2
Li L j 
Philip D Olivier
Mercer University

j
 j
( j ) Li ( j )d 
Li  j  Li  j 1
2p
IASTED Controls and Applications
2004
6
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.)
Philip D Olivier
Mercer University
IASTED Controls and Applications
2004
7
Small Gain Theorem
Suppose that M is stable and that ||M||inf < 1
then (I+M)-1 is also stable.
Philip D Olivier
Mercer University
IASTED Controls and Applications
2004
8
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.
Philip D Olivier
Mercer University
IASTED Controls and Applications
2004
9
Proof
N (U  DQ)  EN (U  DQ)
GC
T

1  GC 1  ED (V  NQ)  EN (U  DQ)
•Recognize that the numerator and
denominator are algebraic expressions of
stable factors/terms. Hence each is stable.
•Apply Small gain Theorem to
denominator.
Philip D Olivier
Mercer University
IASTED Controls and Applications
2004
10
Example
Find a controller that stabilizes the unstable distributed
parameter plant and provides zero steady-state error due to
step inputs.
s  e s
N ( s) N ( s)
Gd (s)  2


s  s  1 D( s ) D( s )
s  e s
N ( s) 
 N ( s )  EN ( s )
2
( s  1)
s2  s 1
D( s ) 
 D(s)  ED ( s)
2
( s  1)
N ( s)  .484L0  (.037) L1  .26L2  .087 L3
U ( s)  2.453L0  (.575) L1
D(s)  1  (1.061) L0  .345L1
V (s)  1  .887 L0  (.566) L1  .245L2  .141L3
UN  VD  1
Philip D Olivier
Mercer University
IASTED Controls and Applications
2004
11
Example (Cont)
U  QD
C ( s) 
V  QN
Q( s )  ?
Zero steady state error due to a step input =>
V (0)
T(0)=1, C(0) = inf,
Q( s)  Q(0) 
 5.908
N (0)
So choose simplest Q(s)
Does the resulting C(s) stabilize both G(s) and Gd(s)?
Robust stabilization theorem says YES.
Philip D Olivier
Mercer University
IASTED Controls and Applications
2004
12
• Does it track a step
input?
• YES
Example (cont)
1.6
1.4
step input and output
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
time,sec
Philip D Olivier
Mercer University
IASTED Controls and Applications
2004
13
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

 max
.1149

 2.0588
.0558
• Nearly 50% improvement
Philip D Olivier
Mercer University
IASTED Controls and Applications
2004
14
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
Philip D Olivier
Mercer University
IASTED Controls and Applications
2004
15