LIDS-P-1137
August, 1981
ASYMPTOTIC U1NBOUNDED ROOT LOCI - FORMULAE AND COMPUTATION
S.S. Sastry*
C.A. Desoer*
L.I.D.S., M.I.T.
Cambridge, 1A 02139
E.R.L., Univ. of Calif.
Berkeley, CA 94720
(i.e., there are no generalized eigenvectors associIt is clear that if
ated with the eigenvalue X=0).
has simple null structure, the number
A(mod S )
2 S
l
(counting multiplicities) of its non-zero eigenvalues
is equal to its rank, or the dimension of its range.
We present a new geometric way of calculating
the asymptotic behavior of unbounded root loci of a
strictly proper, linear, time invariant control
o
system as loop gainrr . The asymptotic behavior
of unbounded root loci has been studied extensively
by Kouvaritakis, Shaked, Edmunds, Owens and others
is
[1,2,3,4]. Our approach, we feel, leads to more
explicit formulae and our methods have applications
in other asymptotic calculations as well for example, we have applied them to the hierarchical
multiple time-scales aggregation of Markov chains
The details of our approach
with weak coupling [5].
are presented in [6] - here we only state the main
reesults.
are
6]here we onlystatethemain
results. ented in
of a Linear
1.
Restrictions
Map
1. Restrictions of a Linear Ma p
1.t* -and two subGiven a linear map A from in to n
spaces S1 , S2 of complementary dimension the linear
map A(mod S2) I from S1 to ccS 2 is defined by
IS
'l
i
2.
System Description, Assumptions and Main Formulae
The system under study is the system of Figurel,
where G(s) is the mxm transfer function matrix of a
linear, time-invariant, strictly proper control
-system assumed to have Taylor expansion about s
G
A .>
is said to be an eigenvalue of A(mod S2)
(A-AIYx e S2
.
if
3
A(mod S2)
i
n/s 2
a:
1
This termi-
where p
p
(parameterized by k) traced
,
(k)l/n + O(k0
)
0
# 0 and 0(k ) is a term of order 1.
the coefficient of its asymptotic value.
Theorem (Generalized eigenvalue problem for n th
order unbounded root locus).
1/n
nC
n = (-X)
e /c is the coefficient of the asymptotic
value of an nth order unbounded
a non-zero solution of
'~2
root locus iff X is
is said to have simple null structure if
2 lG-
3x e
+
n
n
as follows:
2
I >__ n
S i4
On
I 1 C.n
I· *
(k) = p
s
1itbfhn
2
1n there is a natural isomorphism I between 1 S1 and
= P*
_I
G
2
3
+
on an appropriately defined Riemann surface, by the
closed loop eigenvalues are referred to as multivariable root loci. An unbounded multivariable root
locuss
(k) is said to be an nth order unbounded
n
root locus (n=1,2,3 ....) if asymptotically
(Here i stands for the inclusion map and P for the
canonical
projection).
canonicral projection).
an/S
2
2
The one-parameter curves
P
_ C"niS
A(mod S2
non-zero x e S1 such that
+
n
1
A(mod S2 =s
P.A.i
1
= s
G(s)
i
S1 . Xn
said to have simple structure as-
is said to have simple structure asA(mod S2)
I 1
has
sociates with an eigenvalue A if A-XI(mod S
IS
simple null structure.
A(mod S2)x f
AXI
G
G
Gn-l
Sl such that
0 and A(mod S2 )I
-A(mod S2)x = 0
Gn-1
G1
e1
Gn2
n-2
0
e
e2
~~~~~~~~~~~~1
= 0
G
G1
by NSF under grant
Research
ENG-78-09032-AO1 (ERL, UCB) and by NASA under grant
NGL-22-009-124 (LIDS, MIT).
----
~~~wit e
·------
with el m
I
~-0,
~
n e,
...,
2e
ensupported
0
°
O. e2,-
en
e
¢
(2.1)
2
Comments:
(i) There are n separate nth order root
loci corresponding to the distinct nth roots of 1
associated with each solution of (2.1).
Assumption 1 (non degeneracy)
G(s) has normal (or generic)
(ii)
The matrix of (2.1) is a triangular, block
Toeplitz matrix and so must admit
of simplication:
2.1
Assumption 2 (Simple Null Structure)
Formulaue for the asymptotic values of the unbounded root loci first-order.
Assume
G
1
These are clearly the negatives of the nonzero eigenvalues of G1.
G2
G
Second Order
G3=
G3-G2G G2
These
mod
(G
mod R(G 1 ) mod R(G2)
are given by
l/2
Si,
rank m.
= (-Xi,2k)
0
+ O(k)
to have simple null structure.
Comment:
where Ai,G2 imo
is a non-zero eigenvalue
of
i~e,2
G2(mod R(Gl))IC
G
m
N(
G 1) m
1
--z -G
,...
N (G2
Under Assumption 2, Ill
below are isomorphisms.
-*figure
12, 13 in the
2' i.e.
=
]RI
2
>
m
G>6.
1)_G;R"
-RmIR(Gl)
G2
12
Third Order
~
R (G
(Rm I
R (G1 )) R(a )
2
'3
3
Theorem
These are given by
Under Assumptions 1 and 2 the only unbounded
S
= (-X
k)1/3 + 0 (k)
i,3
i,3
root loci of the system of Figure 1 are the
2nd ... , n0 th order unbounded root loci.
where ki 3 is a non-zero eigenvalue of
G3 -
+
(Here G+ stands for
G
)1,
= :G3
N(ie)
2failure
(any) right pseudo-inverse of
(ii)
i.e.
3.
(N2)e N(GA )
2-
_
~'?t~~~
1
mm
G3
2
2>
formulae for non-integral order root loci under
of Assumption 2 are discussed in [6].
Pivots for the Asymptotic' Root Loci
~Under
a further assumption (AssuLmption 3)
JRmIR(CG
G3
.
S.1k
)
~]Rm[R(G1)/R&(2)
Higher order root loci formulae may also be
written. The basic idea is to solve the triangular
system using right pseudo-inverses. The Fredholm
alternative is kept track off through the computation by restricting in the domain successively to
N(G1) D N(G)
...
and modding out in the range
R(G 1 ),
2.2
R(aG2
,..
Simple Null Structure and Integer Order
Unbounded Root Loci
the
asymptotic series for the integral order unbounded
root loci is given by
Rmm
7
l'
st,
Remarks: (i) nO in the above theorem is the degree
of the Smith McMillan zero of G(l/s) at s=O
(see [6] and [7] for details).
G2G1G (mod R(G1)) (mod R(G2))I
2
1
~
RmjR(G)R"a
Sin
i
l/n
+ c. + o(l)
n=l,...,n
(3.1)
The leading term in the expansion of the 0(1) term
in
is
in (3.1)
31
is referred
referred to
to as
as the
the pivot
pivot of
of the
the
Assume that G1, G2, G3..... Gn
have simple
structure associated with all their eigenvalues.
for All
In general, the branches at s_- of the algebraic
function defined by det(I+kG(s)) = 0 have asymptotic
n
expansion with leading telrm of the form X(k)m/
showing possible non-integral order unbounded root
loci.
Using tcchniques of
van Dooren, et al [7]
and some simple assumptions it may be shown that
there are only integral order root loci.
Theorem (Expression for the pivots)
Theorem (Expression for the pivots)
Under assumptions 1,2,3 the nth order asymptotic
unbounded root loci for the system of Figure 1 have
the fom (3.1) with c. given by
i
3
G
c in
G n+l
G
-kXI
n
i
G n -X i i
G1
e.1
G
0
e2
·
*
n-1
e
.1l
with el
3.1
0, e2.
e
.
1
Proceedings of the Conf. on Dec. and Control,
San Diego, Dec. 1981.
[6]
Sastry, S.S., and Desoer, C.A., "Unbounded
Root Loci-Formulae and Computation", UCB
ERL Memo No. M81/6, Dec. 1980.
[7]
van Dooren, P.M., Dewilde, P. and Vanderwalle,
J., "On the Determination of the Smith Mac
Millan form of a Rational Matrix from its
Laurent Expansion", IEEE trans. on Circuits
and Systems, Vol. CAS-26 (1979), pp. 180-189.
=0
J
.( .
Formulae for the Pivots of the Unbounded Root
Loci
First Order
1 .
C i is X times an eigenvalue of
12~
X.
G(s
I(Gi-A.I)
Second Order
c
is
(G3 -
1
2-
Fig. 1:
times an eigenvalue of
))
(G2-XI)G- (G
2
mod R(G 2 -A.I)
nod
i
R(G 1 )IN (2- i)
2i
where G2 -AiI:
G2 -AiI
mod R(G
and so on.
1)j
N (G
In [6] we have discussed how the computation
of the formulae given above can be mechanized in
a numerically robust fashion using orthogonal
rojections and the singular value decomposition.
References:
1.
Kouvaritakis, B. and Skaked, U. "Asymptotic
Behavior of Root Loci of Linear Multivariable
Systems", Intl. J. of Control, Vol. 23 (1976),
pp. 297-340.
2.
Kouvaritakis,
B., "Gain Margins and Root Locus
Asymptotic Behavior in Multivariable Design,
Part I: Prcperties of Markov Parameters and the
Use of High Gain Feedback", Intl. J. of Control,
Vol. 27 (1978), pp. 705-729.
3.
Kouvaritaskis, B. and Edmunds, J.M., "multivariable Loot Loci:
A Unified Approach to
Finite and Infinite
Zeros", Intl.
J.
of Control
Vol. 29 (1979), pp. 393-428.
4.
Owens, D.H., "Multivariable Root Loci-an
Emerging Design Tool", University of Sheffield,
preprint 1930.
5.
Coderch, M., Willsky, A., Sastry, S. and
Castanon, D., "Hierarchical Aggregation of
Linear Systems with Multiple Time Scales,
System Configuration