Supplementary Document for:
Health Aware Planning Under
Uncertainty for
Collaborating Heterogeneous Teams of
Mobile Agents
N. Kemal Ure∗, Girish Chowdhary,†Jonathan P. How,‡John Vian§
October 4, 2013
Proof of Theorem 2
Proof. Let ζ = [e, vec(K), Kr ]T and define the following Lyapunov candidate,
1
1
1
T −1
(1)
V (ζ) = eT P e + tr(K̃ T Γ−1
x K̃) + tr(K̃r Γr K̃r ).
2
2
2
It is possible to bound the Lyapunov candidate above and below with the
following positive definite functions.
1
−1
2
min(λmin (P ), λmin (Γ−1
x ), Γr )kζk ≤ V (ζ)
2
1
−1
2
≤ max(λmax (P ), λmax (Γ−1
x ), Γr )kζk
2
Let [t1 , t2 , ..., tp ], ti+1 > ti be the sequence of times where each data was
recorded. Taking the derivative of the Lyapunov candidate along trajectories
∗
PhD. Candidate, Aerospace Controls Laboratory (ACL), Department of Aeronautics and Astronautics, Massachusetts Institute of Technology (MIT), Cambridge, MA,
ure@mit.edu
†
Assistant Professor, Oklahoma State University, girish.chowdhary@okstate.edu
‡
Richard C. Maclaurin Professor of Aeronautics and Astronautics, ACL (Director),
MIT, jhow@mit.edu
§
Technical Fellow at Boeing Research and Technology, Seattle, john.vian@boeing.com
1
of the system for each interval [ti , ti+1 ] and performing simplifications, we
get:
1
T
V̇ (ζ) = − eT Qe + (K̃ T xeT + K̃r reT )P (B − B̂)
2
p
p
X
T X
T
T
−tr(K̃
xj ˆKj ) − tr(K̃r
rj ˆTKr )
j=1
j=1
Define, ˜ = ˆ − , B̃ = B̂ − B,
1
T
V̇ (ζ) = − eT Qe − (K̃ T xeT + K̃r reT )P B̃
2
p
p
X
T X
T
T
T T
−tr(K̃
xj xj K̃ ) − tr(K̃r
rj rjT K̃r )
j=1
−tr(K̃ T
j=1
p
X
xj ˜TKj ) − tr(K̃r
j=1
T
p
X
rj ˜TKr ).
j=1
P
Note that the matrix Ω = pj=1 xj xTj is positive definite. Bounding the
inequality above using norms and the triangle inequality,
1
1
V̇ (ζ) ≤ − λmin (Q)kek2 − λmin (Ω)kK̃k2
2
2
p
X
1
− (
rj2 )kK̃r k2 + kK̃kkxrm kkekkP kkB̃k
2
(2)
j=1
+kK̃kkek2 kP kkB̃k + kK̃r kkrkkekkP kkB̃k
p
p
X
X
+kK̃kkP kk
xj ˜K k + kK̃r kkP kk
rj ˜Kr k
j=1
j=1
Note that as long as the matrix Ω is full ranked (which can be guaranteed if the reference signal r is exciting over a finite interval [0, T ] [1]) and
sgn(B) = sgn(B̂), the first three terms on the right hand side of the inequality above are negative-definite. Conservative bounds on the rest of the right
hand side terms can be found as follows: The matrix Arm of the reference
model is assumed to be Hurwitz and the reference signal r is always bounded,
therefore there exist scalars cr , cxrm > 0 such that kxrm k < crm , krk < cr .
Note that kB̃k is assumed to be bounded, therefore there exists a scalar
cB such that kP kkB̃k < cB . Finally, note that the error terms ˆ are functions of recorded data xj , rj , which are bounded by assumption and do
2
not evolve
P there exist scalars cx , cr > 0 such that
P with time. Therefore
kP kk pj=1 xj ˜K k < cx , kP kk pj=1 rj ˜Kr k < cr . Hence,
1
1
V̇ (ζ) ≤ − λmin (Q)kek2 − λmin (Ω)kK̃k2
2
2
p
X
1
− (
rj2 )kK̃r k2 + crm cB kK̃kkek + cB kK̃kkek2
2
(3)
j=1
+cr cB kK̃r kkek + cx kK̃k + cr kK̃r k.
P
Therefore, for sufficiently large λmin (Q), λmin (Ω), and pj=1 rj2 , V̇ (ζ) ≤ 0
outside of a compact set. To see that the set is indeed compact, note that the
terms on the right hand side of Eq. 3 yield three quadratic inequalities in kek,
kK̃k and kK̃r k. A conservative estimate of the positively invariant set within
which the solutions are bounded can be found by solving these quadratic
inequalities for each variable while assuming that other two variables are
non-zero. First we check the case where kK̃k > 0, kK̃r k > 0. In this case,
kek ≥
−be +
p
b2e − 4ae ce
2ae
(4)
1
ae = − λmin (Q) + cb kK̃k
2
be = crm cB kK̃k + cr cB kK̃r k
p
1
1 X 2
2
ce = − λmin (Ω)kK̃k − (
rj )kK̃r k2
2
2
j=1
+cx kK̃k + cr kK̃r k
then V̇ (ζ) ≤ 0.
Secondly, we consider the case kek ≥ 0, kK̃r k ≥ 0. In this case, V̇ (ζ) ≤ 0
if
q
−bk + b2k − 4ak ck
(5)
kK̃k ≥
2ak
1
ak = − λmin (Ω),
2
bk = crm cB kek + cB kek2 + cx
p
1 X 2
1
ck = − λmin (Q)kek2 − (
rj )kK̃r k2
2
2
j=1
+cr cB kK̃r kkek + cr kK̃r k
3
(6)
Then we check the case where kek ≥ 0, kK̃k ≥ 0.
kK̃r k ≥
−bkr +
q
b2kr − 4akr ckr
2akr
(7)
p
akr
1 X 2
= − (
rj ),
2
j=1
bkr
ckr
= cr cB kek + cr
1
1
= − λmin (Q)kek2 − λmin (Ω)kK̃k2
2
2
crm cB kK̃kkek + cB kK̃kkek2 + cx kK̃k.
(8)
Inequalities 4–7 characterize the compact set outside of which V̇ (ζ) ≤ 0.
Therefore, all solutions will eventually end up within this set, which in turn
proves that the system [e, K̃, K̃r ] is uniformly ultimately bounded.
References
[1] G. Chowdhary, T. Yucelen, M. Mühlegg, and E. N. Johnson, “Concurrent learning adaptive control of linear systems with exponentially convergent bounds,” International Journal of Adaptive Control and Signal
Processing, 2012.
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