Dynamic Equivalence of Control Systems and Infinite Permutation Matrices
To each dynamic equivalence of two control systems is associated an infinite permutation matrix. We investigate how such matrices are related to the existence of dynamic equivalences.
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| citation_txt | Dynamic Equivalence of Control Systems and Infinite Permutation Matrices / J.N. Clelland, Y. Hu, M.W. Stackpole // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 14 назв. — англ. |
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| description | To each dynamic equivalence of two control systems is associated an infinite permutation matrix. We investigate how such matrices are related to the existence of dynamic equivalences.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 063, 16 pages
Dynamic Equivalence of Control Systems
and Infinite Permutation Matrices
Jeanne N. CLELLAND †, Yuhao HU † and Matthew W. STACKPOLE ‡
† Department of Mathematics, 395 UCB, University of Colorado, Boulder, CO 80309-0395, USA
E-mail: Jeanne.Clelland@colorado.edu, Yuhao.Hu@colorado.edu
‡ Maxar Technologies, 1300 W. 120th Ave, Westminster, CO 80234, USA
E-mail: Matt.Stackpole@maxar.com
Received February 05, 2019, in final form August 20, 2019; Published online August 26, 2019
https://doi.org/10.3842/SIGMA.2019.063
Abstract. To each dynamic equivalence of two control systems is associated an infinite per-
mutation matrix. We investigate how such matrices are related to the existence of dynamic
equivalences.
Key words: dynamic equivalence; control systems
2010 Mathematics Subject Classification: 34H05; 58A15; 58A17
1 Introduction
A control system is an underdetermined ODE system of the form
ẋ = f(t,x,u),
where x = (xi) are called the state variables and u = (uα) the control variables. The meaning
of “control” is clear: Under suitable regularity conditions, specifying a control function u(t) and
an initial state x(t0) uniquely determines a local “trajectory” x(t) that satisfies the ODE system
and the initial state. When f does not explicitly depend on t, the control system is said to be
autonomous, which we will assume throughout this paper.
Let ẋ = f(x,u) and ẏ = g(y,v) be two control systems. Suppose that there exist mappings
φ = φ
(
x,u, u̇, ü, . . . ,u(p)
)
and ψ = ψ
(
y,v, v̇, v̈, . . . ,v(q)
)
such that,
• for any solution (x(t),u(t)) of the first system, the function (y,v) = φ
(
x,u, u̇, ü, . . . ,u(p)
)
is a solution of the second;
• for any solution (y(t),v(t)) of the second system, the function (x,u) = ψ
(
y,v, v̇, v̈, . . . ,
v(q)
)
is a solution of the first;
• moreover, applying φ and ψ successively to a solution (x(t),u(t)) of the first system yields
the same solution (x(t),u(t)), and, similarly, applying ψ and φ successively to a solution
(y(t),v(t)) of the second system yields the same solution (y(t),v(t)).
If all these conditions are satisfied, we say that the pair of maps (φ, ψ) establishes a dynamic
equivalence between the two control systems. Intuitively, a dynamic equivalence provides a one-
to-one correspondence between the solutions of one control system with those of the other.
Fixing a dynamic equivalence (φ, ψ), one can always find the smallest p, q ≥ 0 so that φ =
φ
(
x,u, u̇, ü, . . . ,u(p)
)
and ψ = ψ
(
y,v, v̇, v̈, . . . ,v(q)
)
. We call such a pair (p, q) the height of
the corresponding dynamic equivalence. A dynamic equivalence with height (0, 0) is known as
a static (feedback) equivalence, in which case φ, ψ are inverses of each other as diffeomorphisms.
mailto:Jeanne.Clelland@colorado.edu
mailto:Yuhao.Hu@colorado.edu
mailto:Matt.Stackpole@maxar.com
https://doi.org/10.3842/SIGMA.2019.063
2 J.N. Clelland, Y. Hu and M.W. Stackpole
An immediate question is: How much more general is the notion of dynamic equivalence than
that of static equivalence? Classical results (see [10, 12]) suggest that the answer depends on the
number of control variables. In particular, a dynamic equivalence between two control systems
with a single control variable is necessarily static. It is also well known that the number of
control variables is invariant under a dynamic equivalence. However, in the cases of 2 or more
controls, a precise answer to the question above remains largely unknown.
In [14], the author considered all control-affine systems with 3 states and 2 controls, proving
that three statically non-equivalent systems are pairwise dynamically equivalent at height (1, 1).
In addition, he introduced a new method of studying dynamic equivalences of two control sys-
tems. He found that to each dynamic equivalence is associated an infinite permutation matrix.
Intuitively, such a matrix tells us how the ‘generating 1-forms’ of certain prolongations of the two
control systems (viewed as Pfaffian systems), when chosen appropriately, relate under a dynamic
equivalence.
In the current work, we present further properties of dynamic equivalences that can be derived
using the associated infinite permutation matrices. First we prove that there is a rank matrix
(Definition 3.6) associated to a dynamic equivalence, which has a more ‘invariant’ nature than
an associated infinite permutation matrix (Proposition 3.7). Then we prove several inequalities
and equalities (Propositions 4.1 and 4.2) satisfied by the rank matrix. Using these results, we
prove an inequality satisfied by the height (p, q) of a dynamic equivalence (Theorem 4.3). In
particular, this inequality implies the
Theorem 4.5. The height (p, q) of a dynamic equivalence between two control systems with n1
and n2 states, respectively, and 2 controls must satisfy n1 + p = n2 + q.
2 Control systems and dynamic equivalence
2.1 Control systems
Definition 2.1. A control system with n states and m controls is an underdetermined ODE
system
ẋ = f(x,u), (2.1)
where
x = (xi) ∈ Rn, u = (uα) ∈ Rm,
and f = (fi) : Rn+m → Rn is a smooth function satisfying rank
( ∂fi
∂uα
)
= m on some open domain1
in Rn+m. Here, xi are called the state variables, uα the control variables.
For a control system, there is an equivalent geometric characterization. Let D ⊂ Rn be the
domain of the state variables x = (xi). The admissible t-derivatives of xi, as imposed by the
control system, are given by specifying a submanifold Σ ⊂ TD that submerses onto D with
rank-m fibers. Each fiber is precisely parametrized by the control variables u = (uα). The
submanifold Σ induces an embedding
ι : R× Σ ↪→ R× TD,
which is the identity in the R-factor (with coordinate t) and t-independent in the Σ-factor. In
coordinates, this embedding may be written as
ι(t,x,u) = (t,x, f(x,u)),
1Since our study is local, we henceforth assume that such a domain is the entire Rn+m.
Dynamic Equivalence of Control Systems and Infinite Permutation Matrices 3
which satisfies
ι∗(dxi − ẋidt) = dxi − fi(x,u)dt.
In other words, the system (2.1) corresponds to the Pfaffian system (M, CM ), where M := R×Σ,
and CM is the restriction to M of the standard contact system C = 〈dxi − ẋidt〉ni=1 on the jet
bundle J1(R,D) ∼= R × TD. Conversely, let Σ ⊂ TD be a submanifold that submerses onto
a domain D ⊂ Rn with rank-m fibers. The Pfaffian system (M, CM ) corresponds to a control
system with n states and m controls.
2.2 Prolongations of a control system
Definition 2.2. Let (M, CM ) be a control system with n states and m controls. Let (t,x,u)
be coordinates on M . Suppose that CM is generated by dxi − fi(x,u)dt (i = 1, . . . , n). By
definition, the first total prolongation2 of (M, CM ) is the Pfaffian system
(
M (1), C(1)
)
, where
M (1) = M × Rm with the coordinates
(
t,x,u,u(1)
)
; C(1) is the Pfaffian system generated by
dxi − fi(x,u)dt, duα − u(1)
α dt, i = 1, . . . , n, α = 1, . . . ,m.
Let k be a positive integer. One can start from a control system (M, CM ) with n states
and m controls and generate total prolongations successively for k times. The result is called
the k-th total prolongation of (M, C), denoted as
(
M (k), C(k)
)
, where M (k) has the coordinates(
t,x,u,u(1), . . . ,u(k)
)
and C(k) is generated by the 1-forms
dxi − fi(x,u)dt, duα − u(1)
α dt, du(`)
α − u(`+1)
α dt,
i = 1, . . . , n, α = 1, . . . ,m, ` = 1, . . . , k − 1.
When k = 0, we simply let
(
M (0), C(0)
)
denote (M, CM ).
It is clear that
(
M (k), C(k)
)
is a control system with n+ km states and m controls.
2.3 Dynamic equivalence
Given two control systems, it is natural to regard them as equivalent if one can establish a one-
to-one correspondence between their solutions.
Of course, two control systems ẋ = f(x,u) and ẏ = g(y,v) are equivalent in the sense above
when they can be transformed into each other by a change of variables of the form y = φ(x),
v = ξ(x,u) and x = φ−1(y), u = ρ(y,v). This notion of equivalence is called static equivalence,
which, in particular, requires the two equivalent control systems to have same number of states
and the same number of controls. However, it is possible for two systems with different numbers
of states to have a one-to-one correspondence between their solutions, as is indicated by the
following standard property of jet bundles.
Proposition 2.3. Let (M, CM ) be a control system. Let π : M (k) →M be the canonical projec-
tion from its k-th total prolongation. Any integral curve τ : R → M of (M, CM ) has a unique
lifting τ (k) : R → M (k) (i.e., satisfying π ◦ τ (k) = τ) to an integral curve of
(
M (k), C(k)
)
. In
addition, for each integral curve σ : R→M (k) of
(
M (k), C(k)
)
, its projection π ◦ σ is an integral
curve of (M, CM ).
In other words, given two control systems, a one-to-one correspondence between their solu-
tions may involve differentiation. This motivates the following notion of equivalence.
2Geometrically, M (1) is known as the space of integral line elements of (M, I). See [3]. In particular, one can
show that this definition of
(
M (1), C(1)
)
is independent of the choice of coordinates on M .
4 J.N. Clelland, Y. Hu and M.W. Stackpole
Definition 2.4. Two control systems (M, CM ) and (N, CN ) are said to be dynamically equivalent
if there exist integers p, q ≥ 0 and submersions Φ: M (p) → N and Ψ: N (q) →M that satisfy3
(i) Φ, Ψ preserve the t-variable and are t-independent in the state and control components;
(ii) Φ cannot factor through any M (k) for k < p; Ψ cannot factor through any N (`) for ` < q;
(iii) for each integral curve τ : R→M of (M, CM ), Φ ◦ τ (p) is an integral curve of (N, CN ); for
each integral curve σ : R→ N of (N, CN ), Ψ ◦ σ(q) is an integral curve of (M, CM );
(iv) letting τ and σ be as in (iii), we have
τ = Ψ ◦
(
Φ ◦ τ (p)
)(q)
, σ = Φ ◦
(
Ψ ◦ σ(q)
)(p)
.
For the convenience of the reader, we present the commutative diagram:
M (p) N (q)
R M N R.
Φπ
Ψ
πτ (p)
τ
σ(q)
σ
Remark 2.5. We observe the following:
(a) It is easy to verify that Definition 2.4 defines an equivalence relation.
(b) By this definition, a control system (M, CM ) is dynamically equivalent to each of its total
prolongations
(
M (k), C(k)
)
.
(c) A dynamic equivalence with p = q = 0 is a static equivalence. To see this, let (t,x,u)
and (t,y,v) be coordinates on M and N , respectively. Represent Φ in local coordinates
as (t,y,v) =
(
t, φs(x,u), φc(x,u)
)
. (Here the superscripts ‘s’ and ‘c’ of φ stand for ‘state’
and ‘control’, respectively.) Since Φ maps integral curves of (M, CM ) to integral curves of
(N, CN ), it is necessary that, for each dyi − gi(y,v)dt ∈ CN , its pull-back
Φ∗(dyi − gi(y,v)dt) = d
(
φsi (x,u)
)
− gi
(
φs(x,u), φc(x,u)
)
dt
is contained in CM . It follows that φs(x,u) is independent of u. A similar argument applies
to Ψ. Finally, Condition (iv) in Definition 2.4 implies that Φ and Ψ are inverses of each
other.
(d) This definition of dynamic equivalence corresponds to the notion of endogenous transfor-
mation in the control literature (see [10]). In broader contexts, it is related to the notion
of Lie–Bäcklund equivalences (see [1, 6, 9]), that of C-transformations (see [5, 8]) and
equivalences between differential algebras (see [7]).
Definition 2.6. We call the pair of integers (p, q) in Definition 2.4 the height of the correspond-
ing dynamic equivalence.
Remark 2.7. Keeping the notations from the above, p is the highest derivative of u that Φ
depends on, and q is the highest derivative of v that Ψ depends on. On the other hand, one
may find the highest derivative of each vβ that Ψ depends on and sum over β. Of course, this
depends on the choice of coordinates on N . Such a sum is related to the notion of differential
weight (of Ψ) defined in [11], where its relation with differential flatness is investigated.
3A more careful definition would set the domains of Φ and Ψ to be open subsets of M (p) and N (q), respectively.
See, for example, [12]. Since our results are local, for the economy of notations, we will be content with the
definition presented here.
Dynamic Equivalence of Control Systems and Infinite Permutation Matrices 5
Given two dynamically equivalent control systems (M, CM ) and (N, CN ), if needed, one could
always apply a partial prolongation (for details, see [14]) to one of them such that the resulting
systems have the same number of states and are still dynamically equivalent. When this is
achieved, the proposition below would become applicable.
Proposition 2.8. Let (M, CM ) and (N, CN ) be control systems with the same number of states.
The height (p, q) of a dynamic equivalence between them must satisfy either p = q = 0 or p, q > 0.
Proof. Suppose that the following commutative diagram represents a dynamic equivalence of
height (p, 0) (p > 0) between (M, CM ) and (N, CN ):(
M (p), C(p)
)
(M, CM ) (N, CN ).
π Φ
Ψ
By the assumption, rank(CM ) = rank(CN ). Since
π∗CM = (Ψ ◦ Φ)∗CM = Φ∗(Ψ∗CM ) ⊆ Φ∗CN ,
and since π, Φ, Ψ are all submersions, it is necessary that 〈π∗CM 〉 = 〈Φ∗CN 〉. Now, Φ is constant
along the Cauchy characteristics of Φ∗CN , since the Cartan system (see [3]) of CN generates the
entire cotangent bundle of N . On the other hand, the Cauchy characteristics of π∗CM are
precisely the fibres of π. This proves that Φ factors through M , a contradiction to the choice
of p. The case when p = 0, q > 0 is similar. �
3 Infinite permutation matrices associated
to a dynamic equivalence
In this section, we assume p, q > 0 unless otherwise noted.
Given a control system (M, CM ), one automatically obtains a system of projections
πk,j : M (k) →M (j), k ≥ j.
The inverse limit of this projective system is denoted as
M (∞) := lim←−
k
M (k).
Let πk : M (∞) →M (k) be the canonical projections. Since π∗k,jC(j) ⊆ C(k) for all k ≥ j ≥ 0, one
can define C(∞) to be the differential system generated by
⋃
k≥0 π
∗
kC(k).
In coordinates, if (M, CM ) corresponds to the system ẋ = f(x,u), then M (∞) has the standard
coordinates
(
t,x,u,u(1), . . .
)
; C(∞) is generated by the 1-forms
dxi − fi(x,u)dt, duα − u(1)
α dt, du(`)
α − u(`+1)
α dt,
i = 1, . . . , n, α = 1, . . . ,m, ` ≥ 1.
The pair
(
M (∞), C(∞)
)
is called the infinite prolongation of (M, CM ).
Now suppose that (M, CM ) (ẋ = f(x,u)) and (N, CN ) (ẏ = g(y,v)) are two control systems
with n1, n2 states and m1, m2 controls, respectively. (Note that, at this point, we do not
assume that either the number of states or the number of controls is the same for both systems).
6 J.N. Clelland, Y. Hu and M.W. Stackpole
Furthermore, suppose that a dynamic equivalence of height (p, q) between (M, CM ) and (N, CN )
is given by
Φ: M (p) → N, Ψ: N (q) →M.
It can be shown that Φ and Ψ induce, respectively, maps
Φ(∞) : M (∞) → N (∞), Ψ(∞) : N (∞) →M (∞),
which satisfy
Φ(∞) ◦Ψ(∞) = Id, Ψ(∞) ◦ Φ(∞) = Id.
Moreover, if we let
ω0 = dx− f(x,u)dt, ωk = du(k−1) − u(k)dt, k ≥ 1,
η0 = dy − g(y,v)dt, ηk = dv(k−1) − v(k)dt, k ≥ 1,
then there exist matrices Aij , B
i
j (i, j ≥ 0) satisfying4
P1. for k ≥ 0, we have
Φ(∞)∗ηk = Ak0ω
0 +Ak1ω
1 + · · ·+Akp+kω
p+k, (3.1)
Ψ(∞)∗ωk = Bk
0η
0 +Bk
1η
1 + · · ·+Bk
q+kη
q+k, (3.2)
where Akp+k, B
k
q+k 6= 0 for all k ≥ 0.
P2. All Akp+k are equal for k ≥ 1; similarly for Bk
q+k. Hence, we denote
A∞ := Akp+k, B∞ := Bk
q+k, k ≥ 1.
P3. rank
(
A0
p
)
= rank(A∞) > 0, rank
(
B0
q
)
= rank(B∞) > 0.
In particular, P1 follows from the definition of a dynamic equivalence of height (p, q); P2 can
be seen by taking the exterior derivative of (3.1) on both sides for k ≥ 1 then reducing modulo
ω0, . . . ,ωp+k and similarly for (3.2). For more details of these proofs, see [14].5
To prove P3, note that there exists an n2 ×m2 matrix F such that
dη0 ≡ Fη1 ∧ dt mod η0
≡ FA1
p+1ω
p+1 ∧ dt mod ω0,ω1, . . . ,ωp.
On the other hand,
dη0 = d
(
A0
0ω
0 + · · ·+A0
pω
p
)
≡ −A0
pω
p+1 ∧ dt mod ω0,ω1, . . . ,ωp.
Consequently,
FA1
p+1 = −A0
p.
By Definition 2.1, F has full rank. Therefore, rank
(
A1
p+1
)
= rank
(
A0
p
)
. The case for Bi
j is
similar. Property P3 follows.
4Here we have assumed p, q > 0, otherwise, these properties may not hold. For instance, consider the standard
dynamic equivalence between an arbitrary (M, CM ) and
(
M (k), C(k)
)
.
5The proofs in [14] were made under the assumptions: n1 = n2 and m1 = m2, but they can be easily modified
to work for the situation being considered here.
Dynamic Equivalence of Control Systems and Infinite Permutation Matrices 7
3.1 Two classical theorems
Concerning dynamic equivalences between two control systems, the following two classical the-
orems are of fundamental importance.
Theorem 3.1 ([10, 12]). Two dynamically equivalent control systems must have the same num-
ber of control variables.
Theorem 3.2 ([4, 13]). A dynamic equivalence between two control systems with n states and
single control can only have height (p, q) = (0, 0) (i.e., the equivalence is static).
Remark 3.3. The original result of Cartan takes a different form. In particular, the notion of
equivalence considered in [4] is absolute equivalence. On the other hand, it is a consequence of
Corollary 26 in [13] that two control systems are absolutely equivalent (in an appropriate sense
as t-independence of maps is assumed) if and only if they are dynamically equivalent. Thus,
Theorem 3.2 may be regarded as a version of Cartan’s result (see also Corollary 12 in [13]).
It turns out that these two theorems are consequences of the fact that the following matrices
are inverses of each other, after taking into account the maps Φ(∞) and Ψ(∞):
A =
A
0
0 · · · A0
p 0 · · ·
A1
0 · · · A1
p A∞ · · ·
...
...
...
...
. . .
, B =
B
0
0 · · · B0
q 0 · · ·
B1
0 · · · B1
p B∞ · · ·
...
...
...
...
. . .
. (3.3)
In fact, suppose that (M, CM ) and (N, CN ) have n1, n2 states and m1, m2 controls, respectively.
We can arrange that n1 = n2 =: n by partially prolonging one of the systems, if necessary.
Moreover, since partial prolongations preserve the number of controls, this will have no effect
on the values of m1 or m2. Then by Proposition 2.8, there are two possible cases: p = q = 0
or p, q > 0. In the former case, we have static equivalence. In the latter case, the form of
matrix A implies that the n+ rm2 linearly independent components of η0, . . . ,ηr are all linear
combinations of the n+(r+p)m1 components of ω0, . . . ,ωr+p. When m2 > m1, this is impossible
because n+ (r + p)m1 < n+ rm2 as long as
r >
m1p
m2 −m1
.
Theorem 3.1 follows. Furthermore, when p, q > 0, A and B being inverses of each other requires
that either A∞B∞ = 0 or B∞A∞ = 0. This is impossible when m1 = m2 = 1, in which case A∞
and B∞ are just nonvanishing functions. Theorem 3.2 is an immediate consequence.
More generally, we have the
Lemma 3.4. Suppose that E is a dynamic equivalence of height (p, q) (p, q > 0) between two
control systems with m controls and not necessarily the same number of states. The associated
matrices A∞ and B∞ must satisfy
2 ≤ rank(A∞) + rank(B∞) ≤ m. (3.4)
Proof. This is because p, q > 0 implies A∞B∞ = 0. �
3.2 The infinite permutation matrix S
Let (M, CM ), (N, CN ), ωi, ηi, A, B be as above. In [14], it is proved that there exist transfor-
mationsω̄
0
ω̄1
...
=
g
0
0 0 . . .
g1
0 g1
1 . . .
...
...
. . .
ω
0
ω1
...
,
η̄
0
η̄1
...
=
h
0
0 0 . . .
h1
0 h1
1 . . .
...
...
. . .
η
0
η1
...
, (3.5)
where gii = gi+1
i+1, hii = hi+1
i+1 for all i ≥ 1, such that, pointwise,
8 J.N. Clelland, Y. Hu and M.W. Stackpole
C1.
{
span
{
ω0, . . . ,ωk
}
= span
{
ω̄0, . . . , ω̄k
}
,
span
{
η0, . . . ,ηk
}
= span
{
η̄0, . . . , η̄k
}
,
k ≥ 0;
C2.
{
dω̄` = −ω̄`+1 ∧ dt mod ω̄0, . . . , ω̄`,
dη̄` = −η̄`+1 ∧ dt mod η̄0, . . . , η̄`,
` ≥ 1;
C3. Φ(∞)∗η̄ = Āω̄, Ψ(∞)∗ω̄ = B̄η̄, where Ā, B̄ take the same form as (3.3) (in particular,
all Ākp+k are equal for k ≥ 1, etc.), and both Ā and B̄ are infinite permutation matrices,
that is, each row/column of Ā and B̄ contains a single 1 with the rest of the entries being
all 0.
In addition, we have
Proposition 3.5. Assume that p, q > 0. The matrices Ā and B̄ satisfy Ā = B̄
T
. In particular,
the infinite permutation matrix Ā must be of the form
S := Ā =
Ā0
0 Ā0
1 · · · Ā0
p 0 · · ·
Ā1
0 Ā1
1 · · · Ā1
p Ā∞
. . .
...
...
. . .
...
. . .
Āq0 Āq1 · · · Āqp
0 B̄T
∞
...
. . .
. . .
; (3.6)
in other words, Āq+kk = B̄T
∞ (k ≥ 1), and Āq+`k = 0 for all ` > k.
Proof. The fact Ā = B̄
T
follows from ĀB̄ = B̄Ā = diag(1, 1, . . . ) and Property C3 above.
Consequently, Āq+kk =
(
B̄k
q+k
)T
= B̄T
∞ for k ≥ 1. For a similar reason, Āq+`k = 0 for all ` > k. �
Definition 3.6. Let S, taking the form of (3.6), be an infinite permutation matrix obtained
from a dynamic equivalence with height (p, q) (p, q > 0) between two control systems. Let
rij = rank
(
Āij
)
. We define the rank matrix associated to S to be
R(S) :=
(
rij
)
.
Given a dynamic equivalence, an associated matrix S may depend on the choice of the
transformations ω̄ = Gω and η̄ = Hη. However, we have
Proposition 3.7. If S1 and S2 are two infinite permutation matrices obtained from the same
dynamic equivalence, then their rank matrices satisfy
R(S1) = R(S2).
Proof. Suppose that the underlying dynamic equivalence has height (p, q) (p, q > 0). One can
write S1 and S2 in block forms
S1 =
U0
0 · · · U0
p 0 · · ·
...
. . .
...
. . .
U q0 · · · U qp
0
. . .
...
. . .
, S2 =
V 0
0 · · · V 0
p 0 · · ·
...
. . .
...
. . .
V q
0 · · · V q
p
0
. . .
...
. . .
.
Dynamic Equivalence of Control Systems and Infinite Permutation Matrices 9
Let uij := rank
(
U ij
)
and vij := rank
(
V i
j
)
. Since S1, S2 arise from the same dynamic equivalence,
there exist invertible block lower triangular matrices6
K =
k
0
0 0 . . .
k1
0 k1
1 . . .
...
...
. . .
, L =
`
0
0 0 . . .
`1
0 `1
1 . . .
...
...
. . .
,
where kii = ki+1
i+1, `ii = `i+1
i+1 for all i ≥ 1, such that
S1K = LS2 =
W 0
0 · · · W 0
p 0 · · · · · · · · · · · ·
W 1
0 · · · · · · W 1
p+1
. . .
...
. . .
. . .
W q
0 · · · · · · · · · · · · W q
p+q 0 · · ·
...
. . .
. . .
.
As results of the forms of K and L, we have
(i) uip+i = vip+i for all i ≥ 0. This is because W i
p+i = U ip+ik
p+i
p+i = `iiV
i
p+i, where both kp+ip+i
and `ii are invertible.
(ii) u0
i = v0
i for all 0 ≤ i < p. To see why this is true, consider the submatrix
(
W 0
i W
0
i+1 · · ·W 0
p
)
.
For each i < p, its row rank equals to v0
i + v0
i+1 + · · · + v0
p, which must be equal to its
column rank u0
i + u0
i+1 + · · ·+ u0
p.
(iii) u1
i = v1
i for all 0 ≤ i < p + 1. To see this, consider the submatrix
(
W 0
i ··· W 0
p 0
W 1
i ··· W 1
p W 1
p+1
)
. Its
row rank
(
v0
i + · · ·+ v0
p
)
+
(
v1
i + · · ·+ v1
p+1
)
must be equal to its column rank
(
u0
i + · · ·+
u0
p
)
+
(
u1
i + · · ·+ u1
p+1
)
. Let i decrease from p and use (ii). The desired result follows.
(iv) uji = vji for j ≥ 2, 0 ≤ i < p+ j. This can be verified by a similar comparison between the
column and row ranks of the submatrices
W 0
i · · · W 0
p 0 · · · 0
W 1
i · · · W 1
p W 1
p+1
. . .
...
...
. . . 0
W j
i · · · W j
p W j
p+1 · · · W j
p+j
, i < p+ j.
This completes the proof. �
As a consequence of Proposition 3.7, we have
Corollary 3.8. If S1 and S2 are two infinite permutation matrices obtained from a dynamic
equivalence with height (p, q) (p, q > 0), then there exist block diagonal matrices
K = diag
(
kii
)
i≥0
, L = diag
(
`ii
)
i≥0
,
where kii and `ii are usual permutation matrices of appropriate sizes,7 such that
S1 = LS2K.
6Respectively, the block sizes of K and L are the same as those of G and H.
7That is, the sizes of kii and `ii are consistent with the product of block matrices LS2K.
10 J.N. Clelland, Y. Hu and M.W. Stackpole
Proof. This is because S1 and S2 (i) are permutation matrices; and (ii) have the same rank in
each pair of corresponding blocks. �
Corollary 3.9. Let S be an infinite permutation matrix obtained from a dynamic equivalence
of height (p, q) (p, q > 0). Using the notations in (3.3), the associated rank matrix R(S) =
(
rij
)
satisfies
rkp+k = rank(A∞), rq+kk = rank(B∞), k ≥ 0.
Proof. To see why this is true, first notice that a transformation (3.5) preserves the ranks
of A∞ and B∞; then use Property P3. �
Remark 3.10. A dynamic equivalence may be viewed as an invertible differential operator [2,
5, 9]. In particular, one may, according to [5], define the associated d-scheme of squares. It is
interesting to observe how the relations (3)–(5) in [5] resemble the conditions that rij satisfy for
our rank matrix, but we will not pursue this relation further in the current article.
4 The height of a dynamic equivalence
Given two control systems (M, CM ) (n1 states, m controls) and (N, CN ) (n2 states, m controls)
that are dynamically equivalent, it is interesting to ask: What are the possible heights of a dy-
namic equivalence? A particular instance is Theorem 3.2, which tells us that the height suggests
how control systems with m = 1 and m > 1 are qualitatively different. The current section will
present some new results in this direction.
4.1 Some rank equalities and inequalities
Throughout this section, let (M, CM ) and (N, CN ) be as above. Suppose that a dynamic equiva-
lence between them has height (p, q) with p, q > 0. Let S be an associated infinite permutation
matrix (equation (3.6)), obtained from a choice of coframes
(
ω̄0, ω̄1, . . .
)
and
(
η̄0, η̄1, . . .
)
. Let
R(S) =
(
rij
)
be the corresponding rank matrix (Definition 3.6).
Proposition 4.1. rij satisfy the following equalities∑
i≥0
ri0 = n1,
∑
j≥0
r0
j = n2,
∑
i≥0
rik =
∑
j≥0
rkj = m, k = 1, 2, . . . . (4.1)
Proof. This is because the matrix S is an infinite permutation matrix. �
Proposition 4.2. rij satisfy the following inequalities:
(i) for i, j 6= 0, then
rij ≤ min
{
j+1∑
k=0
ri+1
k ,
i+1∑
k=0
rkj+1
}
;
(ii) for j 6= 0,
r0
j ≤ min
{
r0
j+1 + r1
j+1, (n2 −m) +
j+1∑
k=0
r1
k
}
;
Dynamic Equivalence of Control Systems and Infinite Permutation Matrices 11
(iii) for i 6= 0,
ri0 ≤ min
{
ri+1
0 + ri+1
1 , (n1 −m) +
i+1∑
k=0
rk1
}
;
(iv) r0
0 ≤ min
{
(n1 −m) + r0
1 + r1
1, (n2 −m) + r1
0 + r1
1
}
.
Proof. To prove (i), the case when rij = 0 is trivial. Otherwise, suppose that the submatrix Āij
of S has 1’s precisely at positions (ak, bk), 1 ≤ ak, bk ≤ m, 1 ≤ k ≤ rij . Dropping pullback
symbols, we have
η̄iak = ω̄jbk .
Condition C2 demands
dη̄iak ≡ −η̄
i+1
ak
∧ dt mod η0, . . . ,ηi (4.2)
and
dω̄jbk ≡ −ω̄
j+1
bk
∧ dt mod ω0, . . . ,ωj . (4.3)
At most
j∑
k=0
ri+1
k congruences in (4.2) are reduced to the following form once the congruence is
taken modulo η0, . . . ,ηi,ω0, . . . ,ωj :
dη̄iak ≡ 0 mod η0, . . . ,ηi,ω0, . . . ,ωj ;
similarly, at most
i∑
k=0
rkj+1 congruences in (4.3) are reduced to the following form once the
congruence is taken modulo ω0, . . . ,ωj ,η0, . . . ,ηi:
dω̄jbk ≡ 0 mod ω0, . . . ,ωj ,η0, . . . ,ηi.
The remaining congruences in (4.2) and (4.3), reduced modulo η0, . . . ,ηi,ω0, . . . ,ωj , must
match up as identical congruences; in particular, the corresponding η̄i+1
ak
and ω̄j+1
bk
must be
equal. Since the equalities η̄i+1
ak
= ω̄j+1
bk
are at most ri+1
j+1 in number, we obtain the inequalities
rij ≤
j+1∑
k=0
ri+1
k , rij ≤
i+1∑
k=0
rij+1,
which justifies (i).
To prove (iv), suppose that r0
0 > 0 and that the submatrix Ā0
0 has 1’s precisely at positions
(ak, bk), 1 ≤ ak ≤ n2, 1 ≤ bk ≤ n1, 1 ≤ k ≤ r0
0. We have
η̄0
ak
= ω̄0
bk
.
There exist functions C1
k , . . . , C
s
k, D1
k, . . . , D
s
k such that
dη̄0
ak
≡ −
(
C1
k η̄
1
1 + · · ·+ Cskη̄
1
s
)
∧ dt mod η0, (4.4)
dω̄0
bk
≡ −
(
D1
kω̄
1
1 + · · ·+Ds
kω̄
1
s
)
∧ dt mod ω0. (4.5)
12 J.N. Clelland, Y. Hu and M.W. Stackpole
Since the rank of dη0 (modulo η0) is m, we have
rank
(
Cαk
)
≥ m−
(
n2 − r0
0
)
; (4.6)
similarly, we have
rank
(
Dα
k
)
≥ m−
(
n1 − r0
0
)
. (4.7)
On the other hand, by taking the congruences (4.4) and (4.5) reducing modulo both ω0 and η0,
it is not hard to see that
rank
(
Cαk
)
≤ r1
0 + r1
1, rank
(
Dα
k
)
≤ r0
1 + r1
1. (4.8)
Combining (4.6), (4.7) and (4.8), we obtain the inequalities in (iv).
The proofs of (ii) and (iii) are similar; we leave them to the reader. �
4.2 Admissible heights
Theorem 4.3. Let E be a dynamic equivalence between two control systems with n1, n2 states,
respectively, and m controls. The height (p, q) of E must satisfy: if p, q > 0, then
min{(p− 1)δ + r1p+ n1, (q − 1)δ + r2q + n2} ≥ max{r1p+ n1, r2q + n2}, (4.9)
where r1 = rank(A∞), r2 = rank(B∞), δ = m− r1 − r2.
Proof. The rank matrix associated to E takes the form
R =
r0
0 · · · r0
p 0 · · ·
...
. . .
...
. . .
rq0 · · · rqp
0
. . .
...
. . .
,
where rkp+k = rank(A∞) = r1, rq+kk = rank(B∞) = r2 for all k ≥ 0.
Let
C :=
q−1∑
i=0
p+i∑
j=p
rij −
q−1∑
i=0
rip+i, D :=
p−1∑
j=0
q+j∑
i=q
rij −
p−1∑
j=0
rq+jj .
By Proposition 4.1, we have
0 ≤ C ≤ (q − 1)δ, 0 ≤ D ≤ (p− 1)δ. (4.10)
Furthermore, we have
E := (q − 1)(m− r1)−
(
n1 − r0
0 − r2
)
− C = (p− 1)(m− r2)−
(
n2 − r0
0 − r1
)
−D.
Using (4.10), we obtain
max{r1p+ n1, r2q + n2} ≤ E − r0
0 + n1 + n2
≤ min{(p− 1)δ + r1p+ n1, (q − 1)δ + r2q + n2}.
The conclusion follows. �
Dynamic Equivalence of Control Systems and Infinite Permutation Matrices 13
Corollary 4.4. Let E be a dynamic equivalence between two control systems with n1, n2 states,
respectively, and m controls. If rank(A∞) + rank(B∞) = m, then the height (p, q) of E must
satisfy: when p, q > 0, we have
n1 + rank(A∞) · p = n2 + rank(B∞) · q. (4.11)
Proof. This is an immediate consequence of setting δ = 0 in (4.9). �
Theorem 4.5. The height (p, q) of a dynamic equivalence between two control systems with n1
and n2 states, respectively, and 2 controls must satisfy n1 + p = n2 + q.
Proof. Let (M, CM ) and (N, CN ) be the two control systems in question with a dynamic equiv-
alence given by submersions Φ: M (p) → N and Ψ: N (q) →M .
First consider the case when n1 = n2. By Proposition 2.8, we have either p = q = 0 or p, q > 0.
In the former case, there is nothing to prove. In the latter case, when m = 2, Lemma 3.4 implies
that the only possibility is rank(A∞) = rank(B∞) = 1; then apply Corollary 4.4.
It remains to consider the case when n1 6= n2. Assume that n1 > n2. The case when p, q > 0
follows from a similar argument as the above. Thus, it suffices to consider the case when p = 0.
Suppose that, under a choice of coordinates (t,y,v) on N , Ψ depends nontrivially on v
(q)
α for
some α. Fixing an index β 6= α, we can construct a partial prolongation
(
N̄ , C̄N
)
by letting
N̄ = N × Rn1−n2 with coordinates
(
t,y,v, v
(1)
β , . . . , v
(n1−n2)
β
)
and C̄N being generated by
CN , dvβ − v
(1)
β dt, . . . , dv
(n1−n2−1)
β − v(n1−n2)
β dt.
Let Φ̄ : M (n1−n2) → N̄ (indicated by the dashed arrow in the diagram below) denote the submer-
sion induced by Φ. Let π̄(q) : N̄ (q) → N (q) be the submersion induced by the natural projection
π̄ : N̄ → N
M (n1−n2) N̄ (q)
N (q)
M N̄
N.
π̄(q)
Ψ
π
Φ π̄
It is easy to see that the pair of submersions Φ̄ and Ψ◦ π̄(q) establishes a dynamic equivalence
between (M, CM ) and
(
N̄ , C̄N
)
of height (n1−n2, q). Since M and N̄ have the same dimension,
we conclude that n1 − n2 = q. This completes the proof. �
Remark 4.6. Theorem 4.5 suggests a qualitative distinction between control systems with
m = 2 and those with m > 2. In fact, when m > 2, it may be, for a dynamic equivalence, that
rank(A∞) + rank(B∞) < m, and (4.11) does not need to hold. For details, see Example 4.7
below.
4.3 Examples
Example 4.7. Let (M, CM ) be a control system with 4 states and 3 controls, where M has
coordinates (t,x,u) and CM is generated by
dx1 − u1dt, dx2 − x1dt, dx3 − u2dt, dx4 − u3dt. (4.12)
14 J.N. Clelland, Y. Hu and M.W. Stackpole
A partial prolongation (M, CM ) can be obtained by adjoining equations of the form u̇
(k)
α = u
(k+1)
α
to the original system. For example, consider N1 = M × R3 with the coordinates(
t,
(
x, u1, u2, u
(1)
2
)
,
(
u
(1)
1 , u
(2)
2 , u3
))
,
where u
(1)
1 , u
(1)
2 and u
(2)
2 are coordinates on the R3 component; let C1 be the Pfaffian system
generated by (4.12) and the 1-forms
du1 − u(1)
1 dt, du2 − u(1)
2 dt, du
(1)
2 − u
(2)
2 dt.
Alternatively, consider N2 = M × R3 with the coordinates(
t,
(
x, u3, u
(1)
3 , u
(2)
3
)
,
(
u1, u2, u
(3)
3
))
,
where u
(1)
3 , u
(2)
3 and u
(3)
3 are coordinates on the R3 component; let C2 be the Pfaffian system
generated by (4.12) and the 1-forms
du3 − u(1)
3 dt, du
(1)
3 − u
(2)
3 dt, du
(2)
3 − u
(3)
3 dt.
The standard submersions Φ: N
(3)
1 → N2, Ψ: N
(2)
2 → N1 give rise to a dynamic equivalence
between (N1, C1) and (N1, C2) (both having 7 states and 3 controls) with height (p, q) = (3, 2).
The associated rank matrix is
(rij) =
4 1 1 1
2 0 0 0 1
1 1 0 0 0 1
1 1 0 0 0 1
1 1 0 0 0 1
. . .
. . .
.
In this example, (4.9) becomes an equality.
Example 4.8. Let (M, CM ) (ẋ = f(x,u)) and (N, CN ) (ẏ = f(y,v)) be two copies of a same
control system with 3 states and 2 controls:
ẋ1 = u1,
ẋ2 = u2,
ẋ3 = f(x2, x3, u2),
ẏ1 = v1,
ẏ2 = v2,
ẏ3 = f(y2, y3, v2).
For any p > 1, the following pair of submersions Φ: M (p) → N and Ψ: N (p) → M define
a dynamic equivalence with height (p, p) between (M, CM ) and (N, CN ):
(y,v) = Φ
(
x,u, . . . ,u(p)
)
=
(
u
(p−1)
2 − x1, x2, x3;u
(p)
2 − u1, u2
)
,
(x,u) = Ψ
(
y,v, . . . ,v(p)
)
=
(
v
(p−1)
2 − y1, y2, y3; v
(p)
2 − v1, v2
)
.
This shows that there is no a priori upper bound of dynamic equivalences relating a fixed pair
of control systems; it would only be meaningful to ask whether a ‘minimum height’ exists among
all possible dynamic equivalences between such a fixed pair.
Example 4.9. Assume n1 > n2 in Theorem 4.5. This theorem tells us that even though
dimN (q) ≥ dimM (in order for N (q) to submerse onto M) as long as
q ≥ n1 − n2
2
,
Dynamic Equivalence of Control Systems and Infinite Permutation Matrices 15
an actual dynamic equivalence could only exist with q ≥ n1 − n2. One may compare this fact
with Theorem 52 in [13].
As an example, we consider the PVTOL system (see [10]).
Using the coordinates (t, (x, z, θ, ẋ, ż, θ̇), (u1, u2)) on M , let CM be generated by the six 1-
forms:
dx− ẋdt,
dz − żdt,
dθ − θ̇dt,
dẋ− (−u1 sin θ + εu2 cos θ)dt,
dż − (u1 cos θ + εu2 sin θ − 1)dt,
dθ̇ − u2dt,
where ε is a constant.
Using the coordinates (t, (y1, y2), (ẏ1, ẏ2)) on N , let CN be the trivial system generated by
the two 1-forms:
dy1 − ẏ1dt,
dy2 − ẏ2dt.
By the argument above, if there exists a dynamic equivalence between (M, CM ) and (N, CN )
(aka. (M, CM ) being ‘differentially flat’), then such a dynamic equivalence must satisfy q ≥
n1 − n2 = 6− 2 = 4.
In fact, one can verify that the relation
(y1, y2) = (x− ε sin θ, z + ε cos θ)
induces a dynamic equivalence between (M, CM ) and (N, CN ) of height (p, q) = (0, 4). For more
details, see [10].
Acknowledgements
We would like to thank our referees for their careful reading of the manuscript and helpful
comments that led to considerable improvement of this article. The first and third authors
were supported in part by NSF grant DMS-1206272. The first author was supported in part by
a Collaboration Grant for Mathematicians from the Simons Foundation.
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https://doi.org/10.1007/s00200-010-0137-x
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https://arxiv.org/abs/1106.5437
1 Introduction
2 Control systems and dynamic equivalence
2.1 Control systems
2.2 Prolongations of a control system
2.3 Dynamic equivalence
3 Infinite permutation matrices associated to a dynamic equivalence
3.1 Two classical theorems
3.2 The infinite permutation matrix S
4 The height of a dynamic equivalence
4.1 Some rank equalities and inequalities
4.2 Admissible heights
4.3 Examples
References
|
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| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Clelland, J.N. Hu, Y. Stackpole, M.W. 2025-12-04T13:04:46Z 2019 Dynamic Equivalence of Control Systems and Infinite Permutation Matrices / J.N. Clelland, Y. Hu, M.W. Stackpole // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 14 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34H05; 58A15; 58A17 arXiv: 1902.00598 https://nasplib.isofts.kiev.ua/handle/123456789/210232 https://doi.org/10.3842/SIGMA.2019.063 To each dynamic equivalence of two control systems is associated an infinite permutation matrix. We investigate how such matrices are related to the existence of dynamic equivalences. We would like to thank our referees for their careful reading of the manuscript and helpful comments that led to considerable improvement of this article. The first and third authors were supported in part by NSF grant DMS-1206272. The first author was supported in part by a Collaboration Grant for Mathematicians from the Simons Foundation. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Dynamic Equivalence of Control Systems and Infinite Permutation Matrices Article published earlier |
| spellingShingle | Dynamic Equivalence of Control Systems and Infinite Permutation Matrices Clelland, J.N. Hu, Y. Stackpole, M.W. |
| title | Dynamic Equivalence of Control Systems and Infinite Permutation Matrices |
| title_full | Dynamic Equivalence of Control Systems and Infinite Permutation Matrices |
| title_fullStr | Dynamic Equivalence of Control Systems and Infinite Permutation Matrices |
| title_full_unstemmed | Dynamic Equivalence of Control Systems and Infinite Permutation Matrices |
| title_short | Dynamic Equivalence of Control Systems and Infinite Permutation Matrices |
| title_sort | dynamic equivalence of control systems and infinite permutation matrices |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210232 |
| work_keys_str_mv | AT clellandjn dynamicequivalenceofcontrolsystemsandinfinitepermutationmatrices AT huy dynamicequivalenceofcontrolsystemsandinfinitepermutationmatrices AT stackpolemw dynamicequivalenceofcontrolsystemsandinfinitepermutationmatrices |