In this paper we analyze asymptotic stability, instability and stabilization for the relative equilibria, i.e. equilibria modulo a group action, of natural mechanical systems. The practical applications of these results are to rotating mechanical systems where the group is the rotation group. We use a modification of the Energy-Casimir and Energy-Momentum methods for Hamiltonian systems to analyze systems with dissipation. Our work couples the modern theory of block diagonalization to the classical work of Chetaev

Bloch, A. M.

Krishnaprasad, P. S.

Marsden, J. E.

Ratiu, T. S.

English

A. M. Bloch

P. S. Krishnaprasad

J. E. Marsden

T. S. Ratiu

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Asymptotic Stability, Instability and Stabilization of Relative Equilibria

Caltech Authors - Main

1 
ASYMPTOTIC STABILITY, INSTABILITY 
STABILIZATION RELATIVE 
A. M. Bloch! 
Department of Mathematics 
The Ohio State University 
Columbus, OH 43210 
J. E. Marsden3 
Dept. of Mathematics 
University of California 
Berkeley, CA 94720 
ABSTRACT 
In this paper we analyze asymptotic stability, in-
stability and stabilization for the relative equilibria, 
i.e. equilibria modulo a group action, of natural me-
chanical systems. The practical applications of these 
results are to rotating mechanical systems where the 
group is the rotation group. We use a modifica-
tion of the Energy-Casimir and Energy-Momentum 
methods for Hamiltonian systems to analyze systems 
with dissipation. Our work couples the modern the-
ory of block diagonalization to the classical work of 
Chetaev. 
1. INTRODUCTION 
A central problem in control theory is that of 
stabilizing nonlinear systems. Very often the sys-
tems one is interested in are mechanical systems, 
which, in the absence of dissipation, are Hamilton-
ian in nature. Further, the equilibria we wish to 
stabilize are relative equilibria-equilibria modulo a 
group action, usually the rotation group. Recently 
two distinct but related methods have been devel-
oped to analyze the stability of the relative equilib-
ria of Hamiltonian systems. The first, the "Energy-
Casimir" method, originally goes back to Arnold [2] 
and was developed and formalized in Holm, Mars-
den, Ratiu and Weinstein [8] and Krishnaprasad 
1 Supported in part by NSF Grant DMS-90-02136 and a 
Seed Grant from Ohio State University. 
2 Supported in part by AFOSR-87-0073 and AFOSR-90-
0105 and NSF Engineering Resarch Program grant AFOSR-
87-0073. 
3 Supported in part by DOE Contract DE-FG03-88ER-
25064. 
4 Supported in part by NSF Grant DMS-8922699. 
P. S. Krishnaprasad2 
Dept. of Electrical Engineering and 
Systems Research Center 
University of Maryland 
College Park, MD 20742 
T. S. Ratiu4 
Dept. of Mathematics 
University of California 
Santa Cruz, CA 95064 
1120 
and Marsden [10] and related papers. While the anal-
ysis in this method takes place in the body frame, the 
analysis in the "Energy-Momentum" method takes 
place in the material frame - see Marsden, Simo, 
Lewis and Posbergh [11] and Simo, Posbergh and 
Marsden [14]. Both these techniques use a combi-
nation of energy and other conserved quantitites to 
prove nonlinear (Lyapunov) stability. 
In the papers Bloch and Marsden [4] and Bloch, 
Krishnaprasad, Marsden and Sanchez de Alvarez [5] 
we showed that for some systems the Hamiltonian 
structure could be preserved under feedback enabling 
these techniques to be used to analyze feedback sta-
bilization. 
In this paper we consider asymptotic stability 
and instability results associated with the Energy-
Casimir and Energy-Momentum methods. We con-
sider first the problem of stabilizing a spacecraft 
(rigid body) with momentum wheels. We show 
that when the Energy-Casimir method gives nonlin-
ear stability, the addition of velocity feedback yields 
asymptotic stability, as demonstrated by a Routh-
Hurwitz analysis. (See also [15].) This indicates how 
one can use Hamiltonian methods in asymptotic sta-
bilization - first one achieves nonlinear stability in 
the Hamiltonian context then one adds a dissipative 
velocity feedback. 
We then turn to the question of instability. Our 
main point of interest here is to determine if a nonlin-
ear system becomes unstable if the Lyapunov func-
tion in the Energy-Casimir or Energy-Momentum 
methods becomes indefinite. We do the analysis here 
in the context of the Energy-Momentum method as 
it has been shov/n - see Marsden et. al. [11], Simo 
et. al. [14] - that this method yields a normal form 
for the linearized equations of motion that may be 
analyzed in quite general fashion. 
This normal form corresponds to a block diag-
onalization of the second variation of the energy-
momentum function. For the case of 8 1 symmetry 
it turns out that the normal form is precisely that of 
the general linear mechanical system with gyroscopic 
forces analyzed by Chetaev [6]. We discuss Chetaev's 
result that a gyroscopically stabilized system (i.e. a 
system that is unstable in the absence of gyroscopic 
forces) becomes unstable in the presence of damp-
ing. In the two degree of freedom case we show how 
this can be demonstrated by a Routh analysis. In the 
general case this can be done by Chetaev's Lyapunov-
type analysis. We also discuss extensions to infinite 
dimensions and the application of a theorem due in 
different contexts to Chetaev [6], Oh [12] and others 
- relating the oddness of the number of eigenvalues 
of the Hessian of the Lyapunov function to spectral 
instability. 
2. THE ENERGy-CASIMIR METHOD 
AND ASYMPTOTIC STABILIZATION 
OF A DUAL SPIN SATTELITE. 
In this section we describe briefly the Energy-
Casimir method for stabilization. We then describe 
how it was used (see Bloch et. al [5]) to show how a 
dual spin satellite can be stabilized about its interme-
diate axis of inertia by quadratic feedback in a single 
rotor. We demonstrate by the Routh scheme that 
the addition of velocity feedback to the rotor then 
gives asymptotic stability. (An alternate approach is 
discussed in [9].) 
The Energy-Casimir method is as follows: Write 
the equations of motion (in the body frame) as u = 
F(u) on a given space P. Find a conserved func-
tion H for the system. H is usually taken to be 
the Hamiltonian and the equations are in Hamilton-
ian form F = {F, H} where { , } is the Lie-Poisson 
bracket on P, a Poisson manifold. Then find a family 
of constants of motion C for the system. Often these 
are taken to be Casimirs - functions that commute 
with every other function under the bracket. Now 
choose C such that H + C has a critical point at the 
equilibrium U e of interest. Finally show that the sec-
ond variation of H + C is definite at U e • This proves 
Lyapunov stability (in finite dimensions - in infinite 
dimensions some a priori estimates are needed). 
Consider now the equations for a rigid body with 
a single rotor. Let the rigid body have moments of 
inertia II > 12 > 13 and suppose the symmetric rotor 
is aligned with the third principal axis and has mo-
ments of inertia J1 = J2 and J3 . Let Wi, i = 1,2,3, 
denote the carrier body angular velocities and let a 
denote that of the rotor (relative to a frame fixed on 
1121 
the carrier body). Let 
diag( AI, A2, A3) 
= diag(J1 + h, J2 + 12, J3 + 13) (2.1) 
be the locked inertia tensor. The natural momenta 
for the system are: 
mi = (Ji + h )Wi = Ai 
m3 = I 3w3+£3 
£3 = J3(W3 + a). 
i = 1,2 
(2.2) 
Then one can show (see Bloch et. al. [5]) that the 
equations of motion are 
where U is the rotor torque. Choosing u = ka3m1m2 
where a3 = (12 - 11)' the equations reduce to the 
system 
(2.4) 
which are Hamiltonian on 80(3)* with resepct to the 
standard Lie-Poisson structure with Hamiltonian 
H = ~ (mr + m~ + ((1 - k)m3 - p)2) (2.5) 
2 Al A2 (1-k)I3 
where p is a constant integral of the motion. 
Now using the Energy-Casimir function H + C 
where C = <p(mr + m~ + m~), <p a smooth function, 
we can prove 
Theorem 2.1. For k > 1 - :f; (and p = 0) the sys-
tem (2.4) is stabilized about the relative equlibrium 
(0, M, 0). 
For details of the proof see Bloch et. al. [5]. 
This yields Lyapunov stability for the given equi-
librium. We would like to obtain asymptotic stability. 
Let us add therefore velocity feedback a in the rotor. 
Since a = j; - W3 this yields the equations 
for some k, r> O. 
To analyze stability for the system we linearize 
(2.4) about the given equilibrium (0, M, 0). This 
yields a system with characteristic polynomial 
). { ).' + ).2rv +). [~: ka. - M 2a.!'] 
+M2a. [-!'rv + ~]} (2.7) 
where I' = (fa -1:J, V = (Ja + t;). Now for r = 
o we see the system has two zero eigenvalues and 
eigenvalues in the left and right half-planes for k < 
1- f;, while it has two zero eigenvalues and a pair on 
the imaginary axis for k > 1 - f;, as we expect from 
the Energy-Casimir analysis. For r > 0 we apply the 
Routh test. Writing (2.7) as 
(2.8) 
the Routh criterion for having all eigenvalues of the 
system in the left half-plane (see e.g. Gantmacher 
[7]) is that there should be no changes of sign in the 
sequence 
(2.9) 
(2.10) 
This requires Ie - -11 > 0 and p1 - ltv > 0, yielding 
all " a 
Ie > J3 
J3 +13 
(13 - 12) + (J3 - J2) > O. 
Thus we have 
(2.11) 
(2.12) 
1122 
Theorem 2.2. The system (2.6) is asymptotically 
stable under the conditions (2.11), (2.12). 
Combining (2.11) and (2.12) yields precisely the 
condition k > 1- f; of Theorem 2.1. 
This example is suggestive of the general technique 
for "assessing asymptotic stability via the Energy-
Casimir method. We will discuss this in detail in 
a forthcoming publication. 
3. THE ENERGy-MOMENTUM METHOD 
In order to analyze stability we turn now to the 
Energy-Momentum method where, as mentioned ear-
lier, the analysis takes place in the spatial frame of 
reference. 
In the Energy-Momentum method (Marsden et. 
al. [11]) one considers a symplectic manifold (P, n) 
and a Lie group G acting symplectically on P with 
equivariant momentum mapping J : P ---* g*. If H : 
P ---* JR is a G-invariant Hamiltonian, Ze E P is called 
a relative equilibrium if there is a e E 9 such that 
for all t E JR, z(t) = exp(te)ze, where z(t) is the 
dynamical orbit of XH, the Hamiltonian vector field 
of H with z(O) = Ze. Now one can show that Ze is 
a relative equilibrium if and only if there is a e E 9 
such that Ze is a critical point of He(z) = H(z) -
(J(z) -I'e, e) where I'e = J(ze). 
The key to the Energy-Momentum method is that 
one" can find a subspace S C ker dJ(ze) such that 
definiteness of {)2 He(ze) restricted to S yields sta-
bility, and, moreover, this second variation block-
diagonalizes on S. For this analysis one considers 
systems where P = T*Q, the cotangent bundle of Q, 
the configuration space of a given mechanical system 
with Hamiltonian H = I«q,p) + V(q), where f{ is 
a quadratic form in the momentum variables p, and 
V(q) is the potential energy. 
In this paper we will not describe the details of 
the block diagonalization, (we refer to Marsden et. 
al. [11]), but state merely that one can reduce {)2 He 
to a block diagonal matrix of the form 
A o o 
o A o (3.1) 
o o 
where A is a positive definite co-adjoint orbit block 
(2 x 2 in the case of G = 80(3)), A corresponds to the 
second variation of the augmented potential energy 
and M- 1 to the inertia matrix. 
N ow to get the linearized dynamics we need the 
corresponding symplectic form for the linearized dy-
namics, which is given by 
L c 
° 
s 1 (3.2) 
° 
-1 
° 
where S is skew-symmetric. We remark that in (3.1) 
and (3.2) the upper block corresponds to the "rota-
tional" dynamics (L is in fact the co-adjoint orbit 
symplectic form for G) while the two lower blocks 
correspond to the "internal" dynamics. In (3.2) C 
represents coupling between the internal and rota-
tional dynamics, while S gives the Coriolis or gyro-
scopic forces. 
The corresponding linearized Hamiltonian vec-
tor field is then given by XH = (0- 1 )T'\1 H = 
(O-l)T {)2 He, which a computation (that we omit 
here) reveals to be 
o 
= o o M- 1 
-A -SM-1 
(3.3) 
where S = S + aT L- 1C. 
To add damping to the "internal" variables (but 
not the rotational variables) we add a term - RM- 1 
to the (3,3) block. This R is the Rayleigh dissipation 
matrix. 
We restrict ourselves here to consideration of the 
case G = S1, an abelian group, in which case the 
(1, 1) block A vanishes. This corresponds, for exam-
ple, to the analysis of planar rotating systems, such 
as in Oh et. al. [13]. The general case will be dis-
cussed in a forthcoming paper. 
Taking M = I, we obtain the linear system 
x=v 
v = -Ax - Rv + Sv (3.4) 
1123 
where R = RT ;::: ° is the Rayleigh dissipation ma-
trix and S = -sT gives the gyroscopic forces in the 
system. Note that for 
where n is the Rayleigh dissipation function. 
Systems of this type were analyzed by Chetaev 
and Thompson (Lord Kelvin) and we shall call this 
the Chetaev-Thompson normal form. 
Two questions of interest to us that were analyzed 
by Chetaev are: a) if the system (3.4) is stable for 
R = S = 0, does it retain stability for S =j:. 0, R> 0; 
and b) if the system is only gyroscopically stable, i.e. 
it is unstable for R = S = 0 and neutrally stable for 
S =j:. 0, does it become unstable for R> O? 
The answer to both these questions is in the affir-
mative, and we shall concern ourselves here with the 
latter question. 
This question is of interst to us because it indeed 
shows that by examining the A-block of {)2 He one can 
deduce instability for the linearized system without 
finding the spectrum of the system. 
It is instructive to examine first the two degrees-
of-freedom system 
x - gy + ,x + ax = 0 
ii + gx + 8y + f3y = 0 (3.7) 
with, ;::: 0, 8 ;::: 0. Here g represents the intensity of 
gyroscopic forces, , and {) the damping, and a and 13 
the stiffness. (See also Baillicul and Levi [3].) 
The characteristic polynomial for the system is 
peA) = A4 + A3 (, + 8) + A2(g2 + a + 13 + ,8) 
+ A( ,13 + 8a) + a{3. (3.8) 
For, = 8 = 0, it is simple to calculate the eigen-
values and one deduces that 
(i) for a, 13 > ° the system is spectrally stable 
(ii) for a > 0, 13 < 0 the system is unstable (a 
special case of Oh's lemma - see later) 
(iii) for a < 0, 13 < ° the system is spectrally stable 
for g2 + (a + (3) ;::: 21~1, unstable otherwise. 
To analyze the dissipative case we employ Routh's 
scheme as in section 2. We can show in fact 
Proposition 3.1. For a, 13 < 0 and one of" 8 > ° 
the null solution is unstable for system (3.7). 
Proof. Wrhe the characteristic polynomial as 
as before. 
The number of right half plane eigenvalues then 
equals the number of sign changes in the sequence 
1 PIP2 - P3 
,PI, , 
PI 
P3PIP2 - P~ - P4pr 
PIP2 - P3 
(3.9) 
From the assumptions of the theorem PI = ('Y + 
8) > 0, P2 = (g2+a+.B+'Y6) > 0 P3 = ('Y.B+a8) < 0 
and P4 = a.B > O. This yields the sign sequence 
{+, +, +, -, +}, giving the result. 0 
Consider now the general case. We have the fol-
lowing result, which is due to Chetaev. Our proof is 
a slight modification of his which extends to infinite 
dimensions (see below). 
Proposition 3.2. Suppose A has one or more eigen-
values in the left-half-plane. Then the system (3.4) 
is unstable. 
Proof. We use a Lyapunov instability argument. Let 
W = H +.BBx· v 
1 f!..B 2 
f!..B' A 2 
and f3 be determined. Then 
= 
~(R+ S)B 
+ 
. Then, for f3 sufficiently 
..... "" ... .,.t-nr'" aeli1111.te, but W has at least one nega-
tive Hence Lyapunov's instability 
theorem we have nonlinear instability. 0 
goes through for (3.4) defined 
Hibert space for A-I compact and 
bounded. 
hO'we'ver that the above result proves non-
the linear system (3.4), not spec-
is easy to construct an of 
such a system without dissipation (i.e. when R = 0) 
which is nonlinearly unstable but has eigenvalues on 
the imaginary axis. In such a case we have no infor-
mation on the stability of the nonlinear system which 
has (3.4) as its linearization. Spectral instability will 
be discussed in a forthcoming paper. 
However, if A has odd index we can deduce spectral 
instability by the following argument (see Chetaev [6] 
and also Oh [12]). 
The characteristic polynomial of the system (3.4). 
IS 
.6.(J\) = det(J\2 I + J\(S - R) + J\). 
For J\ = 0 we have .6.(0) = det(A) < O. Now as J\ -? 
00, .6.(A) -? J\2 I -? +00. Hence there exists a positive 
(real) A* such that .6.(J\*) = 0, i.e. there exists a right 
half plane eigenvalue and we have spectral instability. 
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11 


A stability criterion for Hamiltonian systems with symmetry,

Constrained Relative Motion in Rotational Mechanics, to appear in

Foundations of Mechanics,

Nonlinear stability of elasticity and geo metrically exact rods by the energy-momentum method Physics Reports,

Nonlinear stability of elasticity and geometrically exact rods by the energy-momentum method Physics Reports,

Stabilization of rigid body by internal and external torques, submitted for also Systems Research Center

Sur la geometrie differentielle des groups de Lie de dimension infinie et ses appli cations it l'hydrodynamique des fluids parfait,

Sur la geometrie differentielle des groups de Lie de dimension infinie et ses applications it l'hydrodynamique des fluids parfait,

The Dynamics of Coupled Pla nar Rigid Bodies, Part 2: Bifurcations, Periodic Solutions and Chaos, Dynamics and Differential Equations,

The Dynamics of Coupled Planar Rigid Bodies, Part 2: Bifurcations, Periodic Solutions and Chaos, Dynamics and Differential Equations,

http://authors.library.caltech.edu/20390/1/BlKrMaRa1991.pdf

Asymptotic Stability, Instability and Stabilization of Relative Equilibria

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