Flow Balancing Nonlinear Systems
Erik I. Verriest
W. Steven Gray
School of ECE
Georgia Institute of Technology
Atlanta, GA 30332-0250
ECE Department
Old Dominion University
Norfolk, VA 23529-0246
Keywords: balanced realizations, linear time-varying systems, nonlinear systems.
Abstract
This paper introduces a novel approach towards balancing
of nonlinear systems. It differs from other approaches in
two key ideas: The first is the observation that linear timeinvariant systems have a single equilibrium point (the origin),
and that the balanced realization for the class of a linear timeinvariant systems essentially relates to this equilibrium. This
sparked the idea to consider balancing in general as a property associated with invariant sets of a given nominal flow.
The second idea is to define a notion of global balancing
as the one that commutes with linearization. The existence
and construction of balanced and uncorrelated realizations
are discussed. Uncorrelatedness is a relaxed notion of balancedness, introduced here. Several examples, including the
Van der Pol oscillator are shown.
1
Introduction
In this paper we define a balanced realization for a nonlinear system. For linear time invariant systems, balanced
realizations, pioneered by B.C. Moore [18] have found
wide applicability in problems of identification, as investigated by Deistler, Hanzon and Ober, Maciejowski e.a.,
and Moonen and Ramos, [3, 4, 17, 24] parametrization,
[14, 15, 21, 22, 30] model reduction, [18] and robust design
[7, 31]. In view of these successes, various extensions
were derived: for linear time-varying systems, [30], linear
periodic systems, [26, 32] and singular systems. [9]. The
literature is already quite substantial. It is only natural to
extend these ideas further to the nonlinear realm. Indeed,
recently, various approaches were made towards the notion
of balancing for nonlinear systems. One of the first contributions was the Ph. D. Dissertation of J. Scherpen [25], where
the problem is approached using energy functions for input
and output. These ideas were later strengthened in joint
work with S. Gray. [12]. See also [10, 11]. Newman and
Krishnaprasad offer a stochastic approach [19, 20]. Some
alternative uses of balancing in a nonlinear context appeared
in [8, 28]. Finally, the idea of balancing along a trajectory
appeared in [16] and in previous work of these authors [13].
In this paper a different viewpoint is taken. Based on the
fact that linear time-invariant systems have a single equilibrium point (the origin) it is argued, that the balanced realization for the class of stable linear time-invariant systems essentially relates to this equilibrium (the stable attractor). This
sparked the idea to consider balancing in general as a property associated with invariant sets of a given nominal flow.
As any extension should, this method of balancing reduces
to the known balancing in the sense of Moore in the stable
LTI case. We also believe that it may be a computationally
attractive method for nonlinear balancing.
2 Flow Balancing
Let an affine nonlinear system be given in a particular coordinate form in IRn . Systems with states evolving on more
general manifolds can be treated as usual with coordinate
patches.
ẋ = f (x) + g(x)u
y = h(x)
(1)
(2)
It is further assumed that the nominal behavior of the system
is obtained for u = 0. Hence the given u may already be
a perturbation input. The ideas can actually be extended to
more general nonlinear systems, but we limit the discussion
to the above form as it already requires all necessary ingredients, and does not clutter up the principles with superfluous
notation. The basic idea of obtaining a balanced realization
spanning the whole state space (by assumption here IRn ) is
to consider the nominal flow, and require that for the nominal trajectory passing through the point (P, t) in space-time
space the globally balanced coordinate system ξ is such that
the linearized equations of the original system about (P, t)
in the perturbation x̃- coordinates has a locally balanced rep˜ which are exactly the linearized
resentation in coordinates ξ,
form of the globally balanced system, in coordinates ξ. Thus
it means that balancing and linearization commute as in the
following diagram
(f, g, h)
global balancing ↓
(fˆ, ĝ, ĥ)
linearization
−→
linearization
−→
(AP , bP , cP )
↓
local balancing
(ÂP , b̂P , ĉP )
(3)
At this point it needs to be pointed out that in general, linearization near a nominal trajectory will yield a time-varying
linear system. In order to balance such a system, it is necessary to use the extension of balanced realizations to timevariant systems as described in [30]. Finally, for the linearized equations to remain valid it is required that the actually perturbed state remains in the neighborhood of the
nominal state. This prompts the consideration of small timeintervals for the computation of the gramians of the perturbed
system. Essentially, the small-time gramians need to be used.
For time-invariant linear systems this restriction is not necessary, and algebraically it is simpler to compute the infinite
time gramians, as they only require the solution to a Lyapunov equation, at least for a stable system. Keeping the
interval length as a parameter, it is clear that this type of balancing will reduce to the usual (arbitrary interval length) balancing when restricted to linear systems. Moreover, the use
of small-time gramians may be favorable as the computation may be performed without the explicit need of the state
transition matrix. In order to focus on these ideas, the next
section prepares a simple first order example.
through that point. The solution for xn (t) = x is easily found
to be:
x
.
x(τ ) = p
1 − 2(t − τ )x2
Step 2: Linearization
One linearizes about the nominal trajectory: for x̃(τ ) =
x(τ ) − xn (τ ) and ỹ(τ ) = y(τ ) − yn (τ ),
˙ ) =
x̃(τ
ỹ(τ )
=
−3x2
x
p
u(τ )
2 x̃(τ ) +
1 − 2(t − τ )x
1 − 2(t − τ )x2
x̃(τ )
Thus, we obtain a linear time varying (in τ ) system with
−3x2
1 − 2(t − τ )x2
x
b(τ ) = p
1 − 2(t − τ )x2
c(τ ) = 1.
A(τ ) =
Notice that t is fixed in the above! The fundamental matrix
is computed as
3 First Order Example
Consider the nonlinear first order system
ẋ
y
−x3 + xu
=
=
Φx,t (σ, µ) = exp
šZ
σ
A(θ) dθ
µ
›
=
[1 − 2(t − µ)x2 ]3/2
.
[1 − 2(t − σ)x2 ]3/2
x.
The basic idea is to consider the family of all linearized
systems in the neighborhood of the nominal solution for
u = 0, passing through x at time t. Each member in
the family is obtained from a diffeomorphism giving an
alternate coordinate description of the given dynamics. Note
that in general these linearized systems will not be similar.
Among this class of linear systems a realization will exist
that is balanced in the linear (but time varying) sense. Let
these locally balanced coordinates ξ˜x be the perturbations
of the global system state ξ, obtained from x by the global
diffeomorphism x̂. Then the global coordinate system ξ will
be called the globally balanced coordinate system. If the
original system dynamics are time invariant, the globally
balancing transformation x̂ is fixed in time as well.
Step 3: Computation of the Gramians
The system obtained in step 2 may be unstable. (The example does not have to be restricted to a stable system!). Recalling that the reachability gramian R(t0 , t) has the property
that x00 R−1 (t0 , t)x0 is the amount of control energy required
to go from event (0, t0 ) to event (x, t). Hence, in order to
remain faithful to the original nonlinear system, only small
excursions away from the nominal trajectory should be considered. This justifies that one should only consider finite
time gramians, since in a small time the state is not expected
to wander far away from the nominal trajectory, and the linearized system remains a good approximation. We shall settle on the δ-gramians [30]. The reachability gramian is:
Rδ (x, t)
=
Step 1: Nominal trajectory
One finds
ẋn
=
yn
=
t
Φ(t, τ )b(τ )b0 (τ )Φ0 (t, τ ) dτ
t−δ
=
3.1 Conceptual Ideas
Z
1 2
x δ(3 − 6δx2 + 4δ 2 x4 ).
3
Likewise the observability gramian is
−x3n
xn
Oδ (x, t)
=
=
Compute the state on the trajectory going through event
(x, t). An event is a point in the state space together with
a particular time at which the state of the system passes
Z
t+δ
Φ0 (τ, t)c0 (τ )c(τ )Φ(τ, t) dτ
t
δ(1 + δx2 )
(1 + 2δx2 )2
Step 4: Balancing in the moving tangent space
We balance here with respect to the gramians computed
above: Oδ and Rδ . Not only is this possible for unstable systems, but it remains faithful to the original nonlinear systems
since only small perturbations are considered. The balancing
transformation is (from Tδ Rδ Tδ0 = Tδ−T Oδ Tδ−1 ),
Tδ4 (x) =
2
x (1 +
3(1 + δx2 )
.
− 6δx2 + 4δ 2 x4 )
2δx2 )2 (3
For sufficiently small δ , the (local) balancing transformation
and the canonical gramian are respectively
1
Tδ (x) = ± p
andΛδ (x) = |x|δ.
|x|
Note that there are two distinct balancing transformations.
Step 5: Global balancing
Note that we have so far only balanced the perturbation dynamics in the tangent space. What one wants to do next
is to perform a global coordinate transformation (topologically permitting) so that the balanced realization in the tangent space, is exactly the linearized system of the globally
transformed system. This makes balancing and linearization
commute. Thus, let the global transform be
ˆ
ξ = ξ(x).
Solving, one obtains the equation (now dropping the overbar)
p
∂ x̂(ξ)
= ± |x̂(ξ)|.
∂ξ
These equations can be integrated, however their solution
with x(0) = 0 is not unique. The equation with the “+” sign
has infinitely many solutions: Let c0 ≤ 0 ≤ c be arbitrary,
then

(ξ−c)2

if ξ > c ≥ 0

4
0
if c0 < ξ < c
x(ξ) =

 (ξ−c0 )2
− 4
if ξ < c0 ≤ 0
Likewise, with the “−” sign, all solutions are

(ξ−c)2

if ξ > c > 0
 − 4
0
if c0 < ξ < c
x(ξ) =

 (ξ−c0 )2
if ξ < c0 ≤ 0
4
for arbitrary c0 ≤ 0 ≤ c. The choice c = c0 = 0 makes both
global coordinate transformations one to one on all of IR,
although not differentiable at 0.
With the “+” transformation, the√global balancing transformation is:
√ for x > 0, ξ(x) = 2 x > 0, and for x < 0,
ξ(x) = −2 −x < 0. Combining:
p
ξ(x) = 2sgn(x) |x|
with inverse transformation
and its inverse
x = x̂(ξ).
x(ξ) =
We get
ξ˙ = −ξˆ0 (x̂(ξ))(x̂(ξ))3 + ξˆ0 (x̂(ξ))x̂(ξ)u
y = x̂(ξ).
ˆ
Now, if this system is linearized about ξ = ξ(x),
then
˙
ξ˜ = Ab (ξ)ξ˜ + Bb (ξ)u
˜
ỹ = Cb (ξ)ξ.
One identifies
Ab (ξ) =
Bb (ξ) =
C( ξ) =
i
d h ˆ0
ξ (x̂(ξ))(x̂(ξ))3
dξ
ξ
Œ
Œ
0
ˆ
ξ (x̂(ξ))x̂(ξ) Œξ
dx̂(ξ) ŒŒ
Œ
dξ ξ
−
The triple (Ab , Bb , Cb ) must be the balanced realization obtained in step 2. Identifying at the event (x, t), thus setting
τ = t, one gets
q
x
Bb (ξ) = Tδ x̂(ξ) = ± p
= ±sgn(x̂(ξ)) |x̂(ξ|)
|x|
Ab (ξ)
=
Tδ A(x̂(ξ))Tδ−1
Cb (ξ)
=
Tδ−1
Ṫδ Tδ−1
+
q
= ± |x̂(ξ)|.
ξ2
sgn(ξ).
4
This yields the globally balanced realization of the nonlinear
system:
1
ξ5
ξ˙ = − + ξu
32 2
ξ2
y = sgn(ξ) .
4
The use of the “-” local transformation leads to the global
coordinate change
p
ξ(x) = −2sgn(x) |x|
with inverse transformation
x(ξ) = −
ξ2
sgn(ξ).
4
giving the nonlinear balanced realization.
ξ˙ =
y
=
ξ5
1
+ ξu
32 2
ξ2
−sgn(ξ) .
4
−
We emphasize that in both cases, the nonlinear balanced
realization is defined on all of IR.
In the next section we shall verify that these realizations are
indeed balanced.
3.2
Verification of the Balanced Realization
One easily verifies that this has a linearized system equal to
the balanced realization for the δ-gramians.
Perform all the steps shown for the x-system in the previous
section for the ξ-system: (we discuss only one realization,
the other proceeding analogously.
−
y
ξ2
.
4
=
4
ξ5
1
+ ξu
32 2
ξ˙ =
are related by global homeomorphisms: ξ = −η. However
2
x = ξ4 sgn(ξ) is not a diffeomorphism.
Conceptually there is no problem in performing all operations shown. Taking δ not sufficiently small, may invalidate
the approximation by the linearized system.
Infinitesimal Balancing
(4)
(5)
In this section we develop the sliding interval balanced (SIB)
realization [30] for linear time-varying systems for the case
of small interval lengths. Let the linear system be given by
Step 1: Nominal trajectory through (t, ξ)
ξ˙n
y
= −
=
ẋ(t) = A(t)x(t) + B(t)u(t)
y(t) = C(t)x(t).
ξn5
32
ξn2
.
4
It is easily verified that the solution with the required (t, ξ)
condition is, taking τ as the independent dynamical variable,
ξ
ξn (τ ) =
4
(1 + 18 (τ − t)ξ )1/4
.
Step 2: Linearize: the tangent space moves with ξn (τ ). Letting
˜ ),
ξ(τ ) = ξn (τ ) + ξ(τ
one finds
˜˙ ) =
ξ(τ
ỹ(τ ) =
5 4
˜ ) + ξn (τ ) u
ξ (τ )ξ(τ
32 n
2
ξn (τ ) ˜
ξ(τ ),
2
−
which is a time varying system, parametrized by (t, ξ).
The gramians R (t − δ, t) and O (t, t + δ) are defined as
Rt
R (t − δ, t) = t−δ Φ(t, τ )B(τ )B 0 (τ )Φ0 (t, τ ) dτ (8)
R t+δ
O (t, t + δ) = t Φ0 (τ, t)C 0 (τ )C(τ )Φ(τ, t) dτ. (9)
The matrix Φ : IR × IR → IRn×n is the transition matrix
∂
Φ(t, τ ) = A(t)Φ(t, τ ) and Φ(t, t) = I. If it is
satisfying ∂t
desired to reach a final state xf at time t in δ units of time,
Rt
at least an energy, measured by t−δ ku(τ )k2 dτ , equal to
Eu = x0f R −1 (t−δ, t)xf is required. The finite time gramian
R (t−δ, t) characterizes therefore the reachability in the various directions in the tangent state space. The higher the required energy associated with one direction, the less reachable states in this direction are. Similarly, the observability gramian O (t, t + δ) characterizes the energy available in
the output of the system during its undriven evolution from
the state x0 at time t to time t + δ, as the quadratic form
Ey = x00 O (t, t + δ)x0 . With A(t), B(t) and C(t) analytic,
the Taylor series
Step 3: Small Time Gramians:
For the time varying system, in the neighborhood of (t, ξ),
2
Φ(t, τ )B(τ )
=
2
ξ 2 (t)
ξ
Oδ = n
= ,
4
4
ξ 2 (t)
ξ
Rδ = n
= ,
4
4
As the gramians are equal, and trivially diagonal, the realization in the (moving) tangent space is already balanced.
Consequently, the balancing transformation Tδ = 1, i.e., the
system was already in its balanced form.
(6)
(7)
0
0
Φ (τ, t)C (τ ) =
∞
X
ƒ
(t − τ )i ‚
(A − D)i B t
i!
i=0
∞
X
ƒ
(τ − t)i ‚ 0
(A + D)i C 0 t
i!
i=0
hold, where D is the time differential operator acting on
functions to the right of it. This may be expressed in matrix form as
Φ(t, τ )B(τ ) = R∞ (t)[T (t − τ ) ⊗ Im ]
(10)
Φ0 (τ, t)C 0 (t) = [T (t − τ ) ⊗ Ip ]O∞ (t)
(11)
and
3.3 Remarks
The three realizations
š
(
(
ẋ
y
= −x3 + xu
=
x
ξ˙
y
=
=
η̇
y
= − η32 + η2 u
2
= −sgn(η) η4
5
ξ
+ 2ξ u
− 32
2
sgn(ξ) ξ4
5
where we defined the instantaneous reachability and observability matrices
‚
ƒ
R∞ (t) = B : (A − D)B : (A − D)2 B : · · · t
(12)
‚ 0
ƒ
0
0
0
2 0
0
O∞ (t) = [C : (A + D)C : (A + D) C : · · · t(13)
and
”
•
(δ)2
: ··· ,
T (δ) = 1 : (δ) :
2!
(14)
and ⊗ is the Kronecker product. The significance of these
matrices is explained in [29]. If the input
the time varying
Pto
∞
system is impulsive of the form u(t) = i=0 gi δ (i−1) (t−τ ),
then the state jumps instantaneously, at time τ by the amount
C∞ (τ )g, where g = [g10 , g20 , · · · ]0 . Likewise, if the system is
at time τ in the state x(τ ), then the successive derivatives



Y(τ ) = 

y
y0
y (2)
..
.





of the output of the undriven system are given by Y(τ ) =
O∞ (τ )x(τ ). For the analytic system, this series completely
determines the output y(t). Substituting these expressions in
the gramians gives
R (t − δ, t)
=
=
Z
∞ X
∞
X
ƒ
(t − τ )i ‚
(A − D)i B t
i!
t−δ i=0 j=0
t
ƒ0
(t − τ )j ‚
·
(A − D)j B t dτ
j!
∞
∞ X
X
ƒ ‚
ƒ0
‚
(A − D)i B t (A − D)j B t
4.1
State transformation
Definition : The time varying system (A(t), B(t), C(t))
is called sliding interval balanced (SIB) if its infinitesimal
gramians satisfy R (t − δ, t) = O (t, t + δ) = Λδ (t) where
Λδ (·) is a diagonal matrix with nonnegative valued functions
on its diagonal.
If the diagonal elements λk (t) are all distinct at t, then
there exists a neighborhood of t for which a canonical
gramian may be defined as the gramian for which the values
on the diagonal are ordered for all τ in that interval, i.e.,
λ1 (τ ) > λ2 (τ ) > · · · , λn (τ ).
Theorem 1: The canonical gramian is an invariant (function)
for the system. Proof: Indeed, a time variant state transformation, T (t), changes the gramians to
R (t) =
O (t) =
Hence the product R (t)O (t) transforms by similarity, and
therefore has invariant eigenvalues λ2k (t), k = 1, . . . n.
Define the Hankel matrix H(t) by
H(t) = O∞ (t)R∞ (t)
i=0 j=0
δ i+j+1
i!j!(i + j + 1)
0
R∞ (t)∆m (δ)R∞
(t).
·
=
(15)
where ∆m (δ) is an infinite matix with blocks of size m × m.
The ij-th block is
i+j−1
[∆m (δ)]ij =
δ
Im .
(i − 1)!(j − 1)!(i + j − 1)
(16)
T (t)R (t)T 0 (t)
T −T (t)O (t)T −1 (t).
(18)
For a single input single output system, we have
Theorem 2: The Hankel matrix H(t) and the canonical
SIB(δ)-gramian Λδ (t) satisfy
Z δ
T 0 (τ )HT (τ ) dτ ≤ Tr Λδ
0
where equality holds for δ = δ0 if and only if the balanced
realization for δ0 has the additional symmetry
0
O∞ (t) = R∞
(t).
Likewise, the observability gramian is obtained as
Proof: See [27, Theorem 2, p. 139].
O (t, t + δ) =
0
O∞
(t)∆p (δ)O∞ (t)
(17)
It is easily verified that the gramians satisfy the following
functional equations
∂
R (t − δ, t) = A(t)R (t − δ, t) + R (t − δ, t)A0 (t)
∂t
+B(t)B 0 (t) − Φ(t, t − δ)B(t − δ)B 0 (t − δ)Φ0 (t, t − δ)
∂
O (t − δ, t) = −A0 (t)O (t − δ, t) − O (t − δ, t)A(t)
∂t
−C 0 (t)C(t) + Φ0 (t + δ, t)C 0 (t + δ)C(t + δ)Φ(t + δ, t)
Definition: A time variant system is called uniform if there
exists a constant α, depending on δ but not on t such that
R (t − δ, t) ≥ α(δ)I > 0 and O (t, t + δ) ≥ α(δ)I > 0.
Equivalently, the system is uniform if there exists a k > 0
such that kR∞ k > k and kO∞ k > k.
SIB(δ) balancing is performed by simultaneously diagonalizing the reachability and observability gramians. In the
balanced coordinates, any direction has the same measure for
its reachability and observability in time δ.
For infinitesimal balancing, we let δ approach zero. In
the limit, the gramians will obviously be singular, which is
undesirable. Hence we retain sufficiently many powers of δ
to ensure the nonsingularity of R (t) and O (t). The matrix
functions, which are of order δ, need to be expanded to the
(n + 1)-st degree in δ. Consequently, in the expansions (15)
and (17) the instantaneous reachability and observability matrices R∞ (t) and O∞ (t) are respectively replaced by their
n-column truncation Rn (t) and On (t), and ∆(δ) is likewise
truncated to (block)-size n × n.
By the interpretation of the gramians and the existence of
the balancing transformation, the SIB(δ) realization is the
realization for which the coordinate directions provide the
same reachability and observability energies.
4.2
Explicit Form of Balancing Transformation
The balancing transformation T satisfies:
T R∆R0 T
T −T O0 ∆OT −1
= Λ
= Λ
(19)
(20)
This linearized system is time variant, unless x is an equilibrium of the nominal system. Therefore, the results from
section 4 need to be applied. The instantaneous reachability
matrix R∞ contains the columns (A − D)i B, where D is
the derivative in the direction of the nominal flow, i.e., the
covariant derivative [6]. Hence
Consequently, there exists an orthogonal matrix W such that
T R∆
1/2
=T
−T
0
1/2
O∆
W,
(21)
(23)
where W = Ũ Ṽ 0 and Ũ ΣṼ 0 is the singular value decomposition of Z = ∆T /2 H∆1/2 .
5 Balancing Nonlinear Systems
For notational simplicity, we restrict the discussion to single
input single output systems. The general case proceeds analogously. Consider the n-th order nonlinear system, given by
the equations (1) and (2), assumed to hold in some open set
D of IRn . Let f and g be smooth vector functions defined on
the domain D.
5.1 Local Reachability and Observability
(A − D)b =
=
=
f (x)
h(x).
(24)
(25)
Along these trajectories the perturbed system (1) and (2)is
the linearized system given by
x̃˙
ỹ
=
=
Ax x̃ + bx u
cx x̃,
(26)
(27)
where
∂f
Ax =
|x ,
∂x
∂h
cx =
|x .
∂x
(A − D)k+1 b =
(28)
(30)
∂(A − D)k b
∂f
· (A − D)k b −
· f (31)
∂x
∂x
from which it readily follows that
R∞ (f, g) = [g, ad−f g, ad2−f g, · · · ].
(32)
As usual, “ad” denotes the iterated Lie bracket:
g = [f, adkf g], and adf0 g = g.
adk+1
f
Likewise the instantaneous observability matrix O∞ is obtained as follows: Its first row is the gradient dh of h. Its
second row, the transpose of (A0 + D)c0 is the gradient
d(Lf h) of the Lie-derivative Lf h. Iterating, it follows that
(A0 + D)k c0 = dLfk h, and thus (see also [33])



O∞ = 

dh
dLf h
dLf2 h
...



.

(33)
Consequently, the truncated infinitesimal time gramians associated with the linearized system moving along the nominal flow are given by
R δ (f, h) = R(f, g)∆(δ)R0 (f, g)
O δ (f, h) = O0 (f, h)∆(δ)O(f, h).
(34)
(35)
The gramians satisfy the functional equations given in
Section 4.
’ “0
n
X
∂R
∂f
∂f
fi −
R −R
=
∂x
∂x
∂x
i
i=1
0
(f, g)
gg 0 − R∞ (f, g)T (δ)T 0 (δ)R∞
and
n
X
∂O
bx = g(x)
∂g
∂f
g−
f = −[f, g]
∂x
∂x
where [f, g] is the Lie bracket. Iterating, we get
Consider the undriven system, u ≡ 0, as the nominal dynamics. Any other case is easily reduced to this case. Denoting
the state of the nominal system by x, it satisfies
ẋ
y
(29)
(22)
Let Ũ ΣṼ 0 be the singular value decomposition of Z =
∆T /2 H∆1/2 . It is easily verified that the orthogonal matrix W = Ũ Ṽ 0 makes the right hand side of (22) symmetric
and positive definite. Hence a real square root matrix exists, and the balancing transformation is determined modulo
an orthogonal transformation on the left. Hence an essentially balanced realization is readily determined, the (strict)
balanced realization is obtained by an additional orthogonal
realization. We summarize:
Theorem 3: An essentially SIB(δ)-realization is obtained by
the transformation
T = ∆T /2 H0 ∆1/2 W
∂ψ
|x f (x)
∂x
It follows that the second column of the instantaneous reachability is given by
from which in turn
T 0 T = O0 ∆1/2 W ∆−1/2 R−1
Dψx =
’
∂f
∂x
“0
∂f
fi +
O +O
=
∂x
∂x
i
i=1
’ “0
∂h ∂h
0
−
+ O∞
(f, h)T (δ)T 0 (δ)O∞ (f, h)
∂x ∂x
The input normalizing transformation is
reachability and observability gramian in the (local) balanced
form is
T1 = ∆−1/2 (δ)R−1 (f, g),
so that the elements of the canonical gramian, Λδ , are the
square roots of the eigenvalues of O 1 = T1−T O δ T1−1 =
∆T /2 (δ)R0 (f, g)O0 (f, h)∆(δ)O(f, h)R(f, g)∆1/2 (δ).
Note that with the nominal flow, one can associate a Hankel
matrix function
H(f, g, h) = O(f, h)R(f, g).
(36)
This Hankel matrix function, together with the fixed matrix
∆ easily determines the local balanced realization. Indeed,
a second transformation T2 satisfying O 1 = T20 ΛT2 and
T2 T20 = Λ, brings the input normalized realization into
the balanced form. Note that the combined local balancing
transformation T = T2 T1 , depends on the nominal x, but is
otherwise not an explicit function of time for a given time
invariant nonlinear system.
Locally at x, the canonical gramian elements are
the square roots of the eigenvalues of O 1 , or by
cyclic permutation, also the eigenvalues of the matrix
Whereas for scalar
∆(δ)H0 (f, g, h)x ∆(δ)H(f, g, h)x .
linear equation the Hankel matrix is symmetric, this property
is lost in the nonlinear case.
A nonlinear realization with nonsingular Hankel matrix
H(f, g, h) will be called minimal. The following is obvious:
Theorem 4: The linearization of a minimal system (f,g,h)
along its autonomous (or undriven) trajectories, is jointly
reachable and observable.
5.2
Global balancing
The problem is now to extend these local balancing transformations T (x) to a transformation on at least some open set
contained in D. To this effect, the equation
∂ξ
= T (x)
∂x
(37)
needs to be solved. This is a set of n partial differential equations of first order in n variables. It is a special case of a
Mayer-Lie system [1, 2, 5]. It is known that such a system
of equations is not generically solvable. The necessary and
sufficient conditions for solvability are
∂Tij (x) ∂Tik (x)
−
= 0.
∂xk
∂xj
(38)
for all i, j, k = 1, . . . , n. Hence, for scalar systems, flow
balancing can always be performed. For second order systems, n = 2, there may already be an obstruction to balancing. However, one can always determine integrating factors,
si (x), i = 1, 2 for which the Jacobian matrix ST (x) is integrable, where S = diag [s1 , s2 ] (e.g., see [34]). The effect
of the additional non-uniform scaling transformation on the
R s (x) = S(x)R (x)S(x) = S 2 (x)Λ(x)
O s (x) = S −1 (x)O (x)S −1 (x) = S −2 (x)Λ(x)
(39)
(40)
Hence, the uncorrelatedness (diagonality) is retained, as well
as the fact that the product of the two gramians correctly
specifies the canonical gramian. The information about the
relative importance of each coordinate direction is retained
as in the original notion of a balanced realization. This is the
information that is important in model reduction. For this
reason, the scaled balanced realization is still useful, and we
shall define
Definition: A realization for which the reachability and
observability gramians are both diagonal is called an uncorrelated realization.
Thus balanced realizations are special cases of uncorrelated
realizations.
For n ≥ 3 it is not always possible to find a set of
integrating factors [23, p. 30]. We summarize the above
into:
Theorem 5: i) A first order minimal system can be balanced.
ii) A second order minimal system can be brought to
uncorrelated form.
iii) A higher order system can be uncorrelated if and only if
integrating factors exist for which ST is integrable.
Remark: The condition for existence of an uncorrelated
realization is a condition for “flatness” in some particular
sense. However, we refrained from using this term as it has
another meaning. No connection has been established between the integrability conditions above and the established
notion of flatness.
5.3
General scalar system
The balanced realization for the general scalar system
ẋ
y
=
=
f (x) + g(x)u
h(x)
valid in some open interval I is now readily obtained. The instantaneous reachability and observability ‘matrices’ are re2
spectively R = g and O = dh
dx . The gramians are R = g δ
€ dh 2
and O = δ dx . Hence
sŒ
Œ
’ “1/4
Œ dh/dx Œ
O
Œ,
Œ
T =
=± Œ
R
g Œ
qŒ
Œ
√
Œ
and the canonical gramian is Λ = R O = δ Œg dh
dx . The
global balancing in subintervals of I follows by integrating:
r
dh
dξ
= δ |g |.
dx
dx
The transformation T1 = ∆T /2 brings the system in output
normal form (observability gramian is the identity), while the
reachability gramian is then
Example: scalar bilinear system
The bilinear realization for x > 0
ẋ =
y =
ax + nxu
cx
R1
ŒnŒ 2
Œ ξ , giving the
is balanced by the transformation x = Π4c
balanced realization
a
n
ξ + ξu
2
2
|n|
sgn(c)ξ 2 .
4
ξ˙ =
y
=
Note that the elements of the canonical gramian are the
singular values of ∆T /2 R∆1/2 . The short time gramians depend on the function ζ(x1 ) = δ(1 − x1 )2 . For ζ œ 1, the
singular values are approximately the singular values of
√ •
”
δ2
13
2 3
√
R1 =
(44)
1
12 2 3
for ξ > 0.
6
Van der Pol Oscillator
As an example we condider in this section the balanced
model reduction for the Van der Pol oscillator In state space
form, the system is given by
ẋ
=
y
=
”
•
x2
(1 − x21 )x2 − x1
x1 ,
+
”
0
1
•
u
(41)
(42)
∂h
= [1
∂x
0]
dLf h = [0
1].
∂f
∂g
g−
f
∂x
∂x
•”
•
”
0
0
1
=
−1 − 2x1 x2 (1 − x1 )2
1
”
•
1
=
.
(1 − x1 )2
=
The reachability matrix is
”
0
1
1
(1 − x1 )2
•
(43)
The short time Gramians are
O
=
R
=
∆
”
Λ =δ
2
”
1.16
.006
•
(45)
For ζ  1, i.e., for x1  1, we find the asymptotic expression
√ •
”
1 2 4 4
3
3
.
(46)
 δ x1 √
R1 =
3 1
36
R1
The observability matrix is therefore the identity matrix. One
also computes
R=
2
If δ = 1, then
and Lf h = x2 , so that
ad−f g
giving
which is singular. Reachability (along the flow) is lost. The
corresponding canonical gramian is
”
•
2 δ 4 x14 1
Λ2 =
(47)
0
9
where  is a small parameter. One readily finds
dh =
= T1 R T10 = ∆T /2 R0 ∆1/2 ∆T /2 R∆1/2
” 13
7
1 2
+ 12
(1 − x21 )2 δ + 12
 (1 − x12 )2 δ 2
2
√
√12
√
= δ
3
5 3
3 2
2 2
2
6 + 36 (1 − x1 ) δ + 36  (1 − x1 ) δ
√
√
√
•
3
5 3
3 2
2 2
2
6 + 36 (1 − x1 ) δ + 36  (1 − x1 )δ
1
1
1 2
2 2 2
2
12 + 12 (1 − x1 ) δ + 36  (1 − x1 ) δ
δ 2 /3
δ/2 + δ 2 /3(1 − x1 )2
δ/2 + δ 2 /3(1 − x1 )2
1 + δ(1 − x1 )2 + δ 2 /32 (1 − x21 )2
•
= δ2
”
13
7
1
+ 12
(1 − x1 )2 + 12
(1 − x21 )2
√
√
√12
5 3
3
3
2 2
2
6 + 36 (1 − x1 ) + 36 (1 − x1 )
√
√
√
3
3
5 3
2 2
2
6 + 36 (1 − x1 ) + 36 (1 − x1 )
1
1
1
2
2 2
)
x
)
x
+
(1
−
+
(1
−
1
1
12
12
36
•
Now the squared singular values are
”
4
1
2
2 35
− x2 + x4 ±
λ± = δ
36 9 1 18 1
•
q
1
2
4
6
8
304 − 280x1 + 99x1 − 16x1 + x1
18
The diagonal elements cross over at x1 = 2. We see that for
small ζ, i.e. when x1 is sufficiently small, the local balancing
transformation is independent of x1 and x2 . For extremely
large values of x1 however, the canonical gramian is nearly
singular, and the Van der Pol oscillator can be reduced to a
first order system. Near x1 = 2, the canonical gramians are
of the same order of magnitude, and as expected for an oscillator, its behavior cannot be captured by a first order system.
We carry out the balancing transformation in these cases.
When ζ œ 1, The balancing transformation is
"
√ #
α
√
β δ
δ
√
T =
α
√
−β δ
δ
where α = 0.9306 and β = 0.5373. As this is (approximately) independent of the state in this domain, the system
is represented in the given domain by the balanced representation
ξ˙1
=
ξ˙2
=
y
=
1
[4β[α + (β − α)δ]ξ1 + 4β[α + (β + α)δ]ξ2
8αβδ
i
−(ξ1 + ξ2 )(ξ1 − ξ2 )2 + 8αβ 2 δ 3/2 u
1
[4β[α − (β − α)δ]ξ1 + 4β[α − (β + α)δ]ξ2
8αβδ
i
+(ξ1 + ξ2 )(ξ1 − ξ2 )2 − 8αβ 2 δ 3/2 u
1
√ (δξ1 + ξ2 ).
2α δ
Note that the domain ζ œ 1 corresponds to δξ1 + ξ2 sufficiently small. Hence since ξ2 has order δξ1 (in δ), the ξ1 -term
is dominant. The balanced approximation corresponds to the
projection of dynamics to the dominant subsystem. In this
case, neglecting ξ2 gives:
ξ˙1
=
y
=
i
1 h
4β[α + (β − α)δ]ξ1 − ξ13 + 8αβ 2 δ 3/2 u
8αβδ
√
δ
ξ1 .
2α
The case ζ  1 gives the local balancing transformation
(because of the singularity of R in the limit case, the generalized balancing, as pointed out in [30], must be used):
T =
”
3√1
2 δx1
1
2
√1
x1
0
•
”
1 2 2
3 δ x1
0
•
;
O =
”
1 2 2
3 δ x1
1
•
This local balancing cannot be globalized, as the equation
∂ξ
∂x = T (x) is not integrable. However, the additional scaling
by
•
” √
δx1
S=
1
gives the transformed realization
ξ˙1
ξ˙2
y
’
“
3
− 2δ ξ2 − 4ξ22 ξ1 + 4ξ23 + δu
2
1
3
=
ξ1 − ξ2
2δ
2δ
= 2ξ2 .
=
An alternative form of balancing nonlinear systems has been
developed. The key ideas are the use of mobile frames,
entrained with the nominal flow, and the commutation of
balancing and linearization. This method generalizes the
balancing for linear systems, were this property is trivially
satisfied and perhaps therefore has been overlooked for
decades. It was shown that generically, smooth scalar
systems can be balanced. For second order systems, the
classical notion of balancedness seemed too strict. The
notion of uncorrelated realizations was introduced to relax
this. It was shown that second order systems can be brought
to uncorrelated form, which essentially carries the same
information as the balanced form. For higher order systems,
local balancing can be defined, but the patches cannot be
integrated to a global balanced coordinate frame in general.
If you think of the local patches as “floor tiles”, then these
tiles together may not form a surface, unless the Jacobian
field is integrable. It is an open problem to characterize this
integrability, and therefore the ability to globally balance
a nonlinear system, in terms of the given realization. This
of course leaves the possibility of defining balancedness
via alternative methods. However, the method we proposed
is computationally simple up to the point of defining the
local balancing transformations. Even in the integrable
case, it may be a complex problem to determine the global
transformation and thus the global balanced realization
explicitly. But then, these were only the beginnings. . .
Acknowledgement The first author thanks Peter J. Olver
from the University of Minnesota for helpful clarifications.
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rendering the gramians
R =
7 Concluding Remarks
Note that ||ξ̇ξ̇2 ||  1 except in the neighborhood of ξ1 = 3ξ2 .
1
Thus the family ξ1 = constant approximates the local flow.
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