This annual report summarizes the research activities that were performed from 26 Jun. 1993 to 28 Feb. 1994. We continued to investigate the Robust Stability of Systems where transfer functions or characteristic polynomials are affine multilinear functions of parameters. An approach that differs from 'Stability by Linear Process' and that reduces the computational burden of checking the robust stability of the system with multilinear uncertainty was found for low order, 2-order, and 3-order cases. We proved a crucial theorem, the so-called Face Theorem. Previously, we have proven Kharitonov's Vertex Theorem and the Edge Theorem by Bartlett. The detail of this proof is contained in the Appendix. This Theorem provides a tool to describe the boundary of the image of the affine multilinear function. For SPR design, we have developed some new results. The third objective for this period is to design a controller for IHM by the H-infinity optimization technique. The details are presented in the Appendix

Xu, Zhong Ling

English

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NASA-CR-195136
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1993 ANNUAL STATUS REPORT FOR
NASA Grant NAG 5-2210
Exploiting Structure: Introduction and Motivation
Zhong Ling Xu
Department of Mathematics
University of Texas at Brownsville
80 Fort Brown
Tel: (210) 544-8901, Fax: 544-8988
E-mail: BINET%"ZHONG2UTB" SMTP%"ZHONG@UTB. EDU"
Submitted to
NASA Goddard Space Flight Center
Greenbelt, MD 20771
NASA Technical Officer: Mr. Gary E. Mosier
Report Period: March i, 1993 - February 28, 1994
February 1994
(NASA-CR-19513o) EXPLOITING
STRUCTURE: INTRODUCTION AND
MOTIVATION Annual Stdtus Report, I
Mar. 1993 - 28 Feb. 1994 (Texas
Univ.) i0 p
N94-24847
Unclas
G3/64 0205069
https://ntrs.nasa.gov/search.jsp?R=19940020374 2020-06-16T16:32:59+00:00Z
Contents
1. Summary
2. Progress of Research
2.1 Parametric Robust Stability
2.2 Strictly Positive Real for Robust Design
2.3 ACS
3. Appendix
1. Summary
This annual report briefly summarizes research activities performed during the period of June
26, 1993 through February 28, 1994. We continue to develop the Robust Stability of Systems
where transfer function or characteristic polynomial are affine multilinear functions of
parameters of interest in two directions, Algorithmic and Theoretical. In the algorithmic
direction, a approach different form "Stability by Linear Process" that reduces the computational
burden of checking the robust stability of the system with multilinear uncertainty was found for
low order, 2-order and 3-order, cases. In the analysis, we proved a crucial theorem so-called
Face Theorem as we have had the Kharitonov's Vertex Theorem, the Edge Theorem by Bartlett
else. The detail of proof please see it in the Appendix. This Theorem provides a tool to describe
the boundary of the image of the affine multilinear function. Thus, we will finally complete our
paper. For SPR design, we develop some new results based on our work in semiannual report
and the paper is in progress. The third work during this period is to design a controller for IHM
by the H= optimization technique. The details is shown in the Appendix.
0
2.1 Parametric Robust Stability
Let p(s, Q) = Eni:o ai(q)s% qeQ C R _
and a_(q) are affine multilinear functions. As we know that
p(s,Q) e H if and only if 0 /_ p(jco, Q)
Where H is Hurwitz space.
The objective of our research is to develop this criterion by algorithm and theoretical analysis.
In the algorithmic side, we gave a computational feasible algorithm so called "stability
by linear process" to reduce the heavy computational burden. The speed of the algorithm can
be determined by examining the best/worst case scenario. The average case requires (n+ 1)/2
iterations, and one iteration includes 5 steps calculations. The linear property in multilinear
provides not enough informations to reduce a lot of computational burden and to prove our
conjectures in our semiannual report.
According to the line of research of this field, we have had Kharitonov's Vertex Theorem
(1978), the Edge Theorem by Bartlett, Hollot and Huang (1986) and they reduced the heavy
computational burden for linear case. Logically, this motivated us to consider a Face Theorem
Analytically. In our study, we found that the image of F(Q) = p(j_0, Q) is the union of the
image of all possible 2-dimensional exposed edge of Q. There is a image of F(Q) for n = 4
shown in Appendix. Thus a crucial Face Theorem turns out.
Face Theorem for Q c R".
If F is a holomorphic function in C whose real part and imaginary part are affine
multilinear functions of Q c R n, and f_, i = .1, 2, ..., m are the two-dimensional exposed faces
of Q with m = C2,2 ha, then U m=_ F(f_) = Im F(Q).
For this, we proved this theorem in the case Q c R 3 first, and then obtained the Face
Theorem for Q c R". The details is shown in Appendix. This theorem will lead us to classify
the boundary of F(Q) because we can get the information about F(Q) from the image of 2-dim
exposed edges. That is to say that the study of the boundary of F(Q) becomes the study of the
boundary of F(f_). The boundary of F(Q) could be classified as follows:
Such fact provides the possibility to reduce Computational burden.
1. The boundary of F(Q) with l-dim vertex as the ends is a line segment. The proof was
done.
2. The boundary of F(Q) with 2-dim vertex as the ends is a parabola. The proof was done.
3. The boundary of F(Q) with 3-dim vertex as the ends is a cubic algebraic curve which is
a set of points on the boundary of F(Q) with 2-dim vertex. The proof was done.
4. The boundary of F(Q) with k-dim vertex as the end is k-order algebraic curve. We have
something else to be done.
We are doing now.
1
2.2 Strictly Positive RealFunctions(SPR)for RobustDesign
The basicproblemin SPRisto determineif for a givenHurwitz Polynomialset f_, there
is a Hurwitz Polynomiald(s) suchthat
(n(s), d(s)) e SPR V n(s) _ fl
and under which condition on f_ can be found to guarantee the existence of such a polynomial
d(s).
In our 1993 semiannual report we completed the case for which fl is a line segment, ie.,
= {n(s)cR "÷l : n(s) = _,nl(s)+(1-_,)n2(s) for 0 < _, _< l, nt(s), n2(s) _ H }.
We proved that there exist d(s) e H such that {X n_ (s) + (l-X) n2(s), d(s)} e SPR
if and only if Xnt(s) + (1-h)n2(s) e H.
In the proof, we apply its equivalent condition SPR {n_(s)} N SPR {n2(s)} ;_ ,_ .
In order to generalize the set _, we need to describe SPR{n(s)} based on the coefficients
of n(s), in geometric terms.
Description of SPR{n(s)}.
Let n(s) = I;"i_ 0 ais i in H, and n(s) = ah(s2) + Sag(S2)
where a_(s 2) = ao + a2s2 + a4 $4 -4- ..., and ag(S2) = a_ + a3 s2 4- ass 4 4- ..,
Then SPR {n(s)} = B ¢3 I.
where B={(bo, bl, • • ", b°) e R ".1 : Eni,E ah(-oJ2)_oibi+a_(-_o2)_oibi+t > 0,
vo_eR, E is even integers}
and I = {(bo, bl, -" ", b,) _ R "÷_ : bi_> 0 for i=0,1,2,. • -,n}
is the first octant of the coefficient space.
It is evidence that SPR{n(s)} is a convex cone in R "÷_
The Generalizations
1. For the case in which fl is a Kharitonov Box, we have that there exist d(s) _ H such that
{Es_=t h., n_(s), d(s)} e SPR, and E8_=_ X_ = 1, for the vertex polynomial n,(s) c H
if and only if I28i=1Xi n,(s)_ H.
For this, it is sufficient to showthat
rq8_=_SPR(n_(s)) # 4,
Indeed, this is equivalent to the existence of a solution to the system of inequalities
_ni_ E alh(-W2)c0ibi -t-- alg(-W2)wibi+i > 0
I_"i,E a2h(-co2)_oibi+ a2g(-Oj2)coibi+i > 0
]_ni, E aSh(-W2)wibi + aSg(-_o2)wibi+, > 0, and
b, > 0 for i = 0, 1, • - -, n.
This solution does exist.
2. Consider the case in which fl is a polytope. If v represents the set of vertices of fl, we
have that there exist d(s) e H such that
{1;_,v X,n,(s), d(s)}ESPR, where IS_,vX, = 1, for the vertex polynomial n_(s)eH
if and only if
13i,v Xi ni(s) e H.
For this, we proved that ('1,,,, SPR (n_(s) # 4,
This is equivalent to the existence of a solution to the system of inequalities
_ni, E a|h(-W2)wibi + alg(-_o2)6oibi+l > 0
_ni, E a2h(-C02)wibi -t- a2g(-W2)wibi+l > 0
:C"i,E aVh(-_o2)_0"oi+ aVg(-Co2)coibi+|
b,
and this system has a solution.
> 0, and
> 0, fori =0, 1, , n
3. For more generalizations of _, we have something else to be done.
The significance of this work is for proving the convergence of some algorithms in the
identification Of systems and designing a transfer function which is SPR invariant in the
parameter identification problem.
2.3 ACS
2.3.1 The Robust Stability and Robust Performance of the Inertial Hold Mode (IHM)
In accordance with the requirement from Mr. Mosier, the main objective of this research
is to design a controller for the Inertial Hold Mode (IHM). The controlIer will stabilize the plant
with uncertainties from the natural frequencies of the flexible body, and the very weak structural
damping value that results in hormonic vibration of the spacecraft. Also, it must perform as a
good regulator to provide an adequate rejection to the torque disturbance acting on the
spacecraft.
In the IHM, we are facing the control of a systems with flexible body having parametrically
defined uncertainty in the natural frequencies coi and the external torque disturbances. It is well-
known that the H= design methodology is tailor-made for handling ignored high frequency
dynamics and thus is ideally suited for the control of flexible structures. However, this does not
automatically guarantee robustness to the low frequency parametric uncertainty in IHM. The
problem is how to develop techniques for modeling the parametric uncertainty so that:
The H= design technique can be used.
The level of robust performance is maximized.
Our main work is to modify IHM into a typical Mixed-Sensitivity Problem in H= and to
design a controller K(s) such that the nominal system, G(s), is stable and such that
{[[ W_ (1 + GK)-' II= + II GK(1 + GK) _ II=}< 1.
In this work, our main contributions are as follows:
(1) Modeling the parametric uncertainty with a single non-parametric block and retaining
the simplicity and intuition of single-loop control. That is, we give a non-parametric maximum
additive perturbation A_(s) to cover the entire flexible body as an additive perturbation, and then
turn out a multiplicative perturbation Am(S) for convenience making the nominal system, G(s),
and the adequate weighting function, W3(s), for maximizing the robust stability.
(2) Defining the weighting function Wt(s) for maximizing the robust performance and
getting an adequate rejection to torque disturbance.
(3) In order to employ the Ho_ optimization algorithm directly, we found a suitable
transformation of the regulator to shift the imaginary axis.
(4) In order to obtain good robust stability and robust performance, we adjust W_ and W3
alternatively to simulate the controller.
5
(5) To reducethe order of the controller, we eliminate thosezeros and poles of the
controller that are far away from the imaginary axis or the origin. This is for maintaining
robustness(stabilityandperformance)of thesystem,ie., maintaininga goodperformanceof the
open-loop transfer function of the system with reducedcontroller in the low and middle
frequencybandwidths.
Finally, let us makea comparisonbetweenthesimulationresultsof the H= designand thePID
design.
HooDesign PID Design
(a) GainMargins < -24.04dband> 24.04db < -21dband> 14db
(b) PhaseMargins > 55.03° > 45°
(c) Robustnesswithparametric
uncertainty+/- 25% Good Notgood
(d) Attenuation 150Seconds > 200Seconds(e) Controller'sOrder > 4 2
2.3.2 The ControlIer Reduction
The controller in IHM is designed and satisfied to the specific requirements but its order is
too high.
In general, the order of the controller designed by the H= technique is guaranteed to be not
higher than that of the generalized plant. However, in many engineering problems, the order of
generalized plant could be very high. This is due to the fact that the generalized plant consists
of the original plant as well as all the weighting matrices which are chosen to meet certain
design specification. Hence, the order of the H® controller obtained by the standard State-Space
approach is usually too high to be implemented in practice, and therefore a systematical
methodology of designing a reduced-order Ho. robust controller is desired.
According to our investigation of the literature, several existing reduction methods have large
error. The Balanced Stochastic Truncation (BST) method is good in high frequency bandwidths
but is bad in the low and middle frequency bandwidths. We develop such reduction method for
which the open-loop frequency characteristic of the system with reduced controller approaches
to the original as close as possible and the stability and performance are retained perfectly. In
terms of a formula,
IIKG-K G I1 ,= IIcK-I )G II-
is as small as possible, where K, is a reduced controller, and the generalized plant G(s) becomes
a frequency weighted function in practice. The Frequency Weighted Balanced Truncation method
by Ends could be used. The details in the paper for controller reduction is in progress and will
be submitted to CDC of IEEE by March 1, 1994.
Appendix
1. Image of F(Q) for Q c R4
2. The Face Theorem
3. Ho. Optimal Controller Design and Reduction for IHM


Exploiting structure: Introduction and motivation

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