Feedback Equivalence of 1-dimensional Control Systems of the 1-st Order
The problem of local feedback equivalence for 1-dimensional control systems of the 1-st order is considered. The algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found.
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irk-123456789-63092010-02-24T12:01:08Z Feedback Equivalence of 1-dimensional Control Systems of the 1-st Order Lychagin, V.V. Геометрія, топологія та їх застосування Праці міжнародної конференції "Геометрія в Одесі - 2008" The problem of local feedback equivalence for 1-dimensional control systems of the 1-st order is considered. The algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found. 2009 Article Feedback Equivalence of 1-dimensional Control Systems of the 1-st Order / V.V. Lychagin // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 288-302. — Бібліогр.: 9 назв. — англ. 1815-2910 http://dspace.nbuv.gov.ua/handle/123456789/6309 en Інститут математики НАН України |
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Геометрія, топологія та їх застосування Праці міжнародної конференції "Геометрія в Одесі - 2008" Геометрія, топологія та їх застосування Праці міжнародної конференції "Геометрія в Одесі - 2008" |
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Геометрія, топологія та їх застосування Праці міжнародної конференції "Геометрія в Одесі - 2008" Геометрія, топологія та їх застосування Праці міжнародної конференції "Геометрія в Одесі - 2008" Lychagin, V.V. Feedback Equivalence of 1-dimensional Control Systems of the 1-st Order |
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The problem of local feedback equivalence for 1-dimensional control systems of the 1-st order is considered. The algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found. |
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Article |
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Lychagin, V.V. |
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Lychagin, V.V. |
| author_sort |
Lychagin, V.V. |
| title |
Feedback Equivalence of 1-dimensional Control Systems of the 1-st Order |
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Feedback Equivalence of 1-dimensional Control Systems of the 1-st Order |
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Feedback Equivalence of 1-dimensional Control Systems of the 1-st Order |
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Feedback Equivalence of 1-dimensional Control Systems of the 1-st Order |
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Feedback Equivalence of 1-dimensional Control Systems of the 1-st Order |
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feedback equivalence of 1-dimensional control systems of the 1-st order |
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Інститут математики НАН України |
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2009 |
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Геометрія, топологія та їх застосування Праці міжнародної конференції "Геометрія в Одесі - 2008" |
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Feedback Equivalence of 1-dimensional Control Systems of the 1-st Order / V.V. Lychagin // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 288-302. — Бібліогр.: 9 назв. — англ. |
| work_keys_str_mv |
AT lychaginvv feedbackequivalenceof1dimensionalcontrolsystemsofthe1storder |
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2025-07-02T09:14:36Z |
| last_indexed |
2025-07-02T09:14:36Z |
| _version_ |
1836525998939570176 |
| fulltext |
Çáiðíèê ïðàöüIí-òó ìàòåìàòèêè ÍÀÍ Óêðà¨íè2009, ò.6, �2, 288-302
V.V. Lychagin
Department of Mathematics, University of Tromso (Tromso,
Norway) and Institute of Control Science, Russian Academy of
Science (Moscow, Russia)
E-mail: lychagin@yahoo.com
Feedback Equivalence of
1-dimensional Control Systems of
the 1-st Order
The problem of local feedback equivalence for 1-dimensional control sys-
tems of the 1-st order is considered. The algebra of differential invariants
and criteria for the feedback equivalence for regular control systems are
found.
Keywords: differential invariants, invariant differentiation
1. Introduction
In this paper we study the problem of local feedback equivalence
for 1-dimensional control systems of 1-st order.
As in paper ( [8]) we use the method of differential invariants.
To this end we consider control systems as underdetermined ordi-
nary differential equations. This gives a representation of feedback
transformations as a special type of Lie transformations, and we
study and find differential invariants of these representation.
Remark also that from the EDS point of view the case of control
systems considered here is equivalent to the case of second order
systems considered in ( [8]), but from ODE point of view they
have different algebras of feedback differential invariants.
To find a structure of the algebra of feedback differential in-
variants we first find 3 feedback invariant derivations. Then the
differential invariants algebra is generated by two basic differential
c© V. V. Lychagin, 2009
Feedback Equivalence of 1-dimensional Control Systems 289
invariants J and K of orders 2 and 3 respectively and by all their
invariant derivations.
This description allows us to find invariants for the formal feed-
back equivalence problem.
To get a local feedback equivalence we introduce a notion of reg-
ular control system and connect with such a system a 3-dimensional
submanifold Σ in R14.
The main result of the paper states that two regular control
systems are locally feedback equivalent if and only if the corre-
sponding 3-dimensional submanifolds Σ coincide.
2. Representation of Feedback Pseudogroup
Let
(1)
·
x = F (x, u,
·
u),
be an autonomous 1-dimensional control system of the 1-st order.
Here the function x = x (t) describes a dynamic of the state of
the system, and u = u (t) is a scalar control parameter.
We shall consider this system as an undetermined ordinary dif-
ferential equation of the first order on sections of 2-dimensional
bundle π : R3 → R , where π : (x, u, t) 7−→ t.
Let E ⊂ J1 (π) be the corresponding submanifold. In the canon-
ical jet coordinates (t, x, u, x1, u1, . . . ) this submanifold is given by
the equation:
x1 = F (x, u, u1) .
It is known (see, for example, [6]) that Lie transformations in
jet bundles Jk (π) for 2-dimensional bundle π are prolongations of
point transformations, that is, prolongations of diffeomorphisms
of the total space of the bundle π.
We shall restrict ourselves by point transformations which are
automorphisms of the bundle π.
Moreover, if these transformations preserve the class of systems
(1) then they should have the form
(2) Φ : (x, u, t) → (X (x) , U (x, u) , t) .
290 V. V. Lychagin
Diffeomorphisms of form (2) is called feedback transformations.
The corresponding infinitesimal version of this notion is a feedback
vector field, i.e. a plane vector field of the form
Xa,b = a (x) ∂x + b (x, u) ∂u.
The feedback transformations in a natural way act on the con-
trol systems of type (1):
E 7−→ Φ(1) (E) ,
where Φ(1) : J1 (π) → J1 (π) is the first prolongation of the point
transformation Φ.
Passing to functions F, defining the systems, we get the follow-
ing action on these functions:Φ̂ : F 7−→ G, where the function G
is a solution of the equation
(3) Xx G = F (X,U,UxG+ Uuu1) .
The infinitesimal version of this action leads us to the following
representation Xa,b 7−→ X̂a,b of feedback vector fields:
(4) X̂a,b = a ∂x + b ∂u + (u1bu + f bx) ∂u1 + ax f ∂f .
In this formula X̂a,b is a vector field on the 4-dimensional space
R4 with coordinates (u, u, u1, f) , and this field corresponds to the
above action in the following sense.
Each control system (1) determines a 3-dimensional submani-
fold LF ⊂ R4, the graph of F :
LF = {f = F (x, u, u1)} .
Let At be the 1-parameter group of shifts along vector field Xa,b
and let Bt : R4 → R4 be the corresponding 1-parameter group of
shifts along X̂a,b, then these two actions related as follows
L
cAt(F )
= Bt (LF ) .
In other words, if we consider an 1-dimensional bundle
κ : R4 → R3,
Feedback Equivalence of 1-dimensional Control Systems 291
where κ((u, u, u1, f)) = (u, u, u1), then formula (4) defines the
representation X 7−→ X̂ of the Lie algebra of feedback vector
fields into the Lie algebra of Lie vector fields on J0 (κ) , and the
action of Lie vector fields X̂ on sections of bundle κ corresponds
to the action of feedback vector fields on right hand sides of (1)
3. Feedback Differential Invariants
By a feedback differential invariant of order ≤ k we understand
a function I ∈ C∞ (Jkκ
)
on the space of k-jets Jk(κ), which is in-
variant under of the prolonged action of feedback transformations.
Namely,
X̂a,b
(k)
(I) = 0,
for all feedback vector fields Xa,b.
In what follows we shall omit subscript of order of jet spaces, and
say that a function I on the space of infinite jets I ∈ C∞ (J∞κ)
is a feedback differential invariant if
X̂a,b
(·)
(I) = 0,
where X̂a,b
(·)
is the prolongation of the vector field Xa,b in the
space of infinite jets J∞κ.
In a similar way one defines a feedback invariant derivations as
combinations of total derivatives
∇ = A
d
dx
+B
d
du
+ C
d
du1
,
A,B,C ∈ C∞ (J∞κ) ,
which are invariant with respect to prolongations of feedback trans-
formations, that is,
[X̂a,b
(·)
,∇] = 0
for all feedback vector fields Xa,b.
Remark that for these derivations functions ∇ (I) are differen-
tial invariants ( of order, as a rule, higher then order of I) for
any feedback differential invariant I. This observation allows us to
292 V. V. Lychagin
construct new differential invariants from known ones only by the
differentiations.
Recall the construction of the Tresse derivations in our case.
Let J1, J2, J3 ∈ C∞ (Jkκ
)
be three feedback differential invariants,
and let
d̂Ji =
dJi
dx
dx+
dJi
du
du+
dJi
du1
du1
be their total derivatives.
Assume that we are in a domain D in Jkκ, where
d̂J1 ∧ d̂J2 ∧ d̂J3 6= 0.
Then, for any function V ∈ C∞ (J lκ
)
over domain D, one has
decomposition
d̂V = λ1d̂J1 + λ2d̂J2 + λ3d̂J3.
Coefficients λ1, λ2 and λ3 of this decomposition are called the
Tresse derivatives of V and are denoted by
λi =
DV
DJi
.
The remarkable property of these derivatives is the fact that they
are feedback differential invariants (of higher, as a rule, order then
V ) each time when V is a feedback differential invariant.
In other words, the Tresse derivatives
D
DJ1
,
D
DJ2
and
D
DJ3
are feedback invariant derivations.
4. Dimensions of Orbits
First of all, we remark that the submanifold {f = 0} is a sin-
gular orbit for the feedback action in the space of 0-jets J0κ. The
generating function of the feedback vector field X̂a,b has the form:
φa,b = axf − afx − bfu − (u1bu + fbx) fz,
Feedback Equivalence of 1-dimensional Control Systems 293
and the formula for prolongations of vector fields ( [6]) shows that
in the space of 1-jets J1κ, in addition, we have one more singular
orbit {fu1 = 0}. In similar way, we have one more singular orbit
{fu1u1 = 0} in the space of 2-jets. There are no more additional
singular orbits in the spaces of k-jets, when k ≥ 3.
We say that a point xk ∈ Jkκ is regular, if f 6= 0, fu1 6=
0, fu1u1 6= 0 at this point.
In what follows we shall consider orbits of regular points only.
It is easy to see, that the k−th prolongation of the feedback
vector field X̂a,b depends on (k + 1)-jet of function a (x) and (k+
1)-jet of function b (x, u) .
Denote by V k
i and W k
ij the components of the decomposition
X̂a,b
(k)
=
∑
0≤i≤k+1
a(i) (x)V k
i +
∑
0≤i+j≤k+1
∂i+jb
∂xi∂uj
W k
ij.
Then, by the construction, the vector fields V k
i , 0 ≤ i ≤ k+1, and
W k
ij, 0 ≤ i+j ≤ k+1, generate a completely integrable distribution
on the space of k-jets, integral manifolds of which are orbits of the
feedback action in Jkκ.
Straightforward computations show that there are no non trivial
feedback differential invariants of the 1-st order.
Let Ok+1 be a feedback orbit in Jk+1κ, then the projection Ok =
κk+1,k (Ok+1) ⊂ Jkκ is an orbit too, and to determine dimensions
of the orbits one should find dimensions of the bundles: κk+1,k :
Ok+1 → Ok. To do this we should find conditions on functions a
and b under which X̂a,b
(k)
= 0 at a point xk ∈ Jkκ.
Assume that X̂a,b
(k−1)
= 0 at the point xk−1 ∈ Jk−1κ . Then
the vector field X̂a,b
(k)
is a κk,k−1-vertical over this point.
Components
dkφ
dxiduj
∂
∂fσij
294 V. V. Lychagin
of this vector field, where σij = (x, . . . , x︸ ︷︷ ︸,
i-times
u, . . . , u︸ ︷︷ ︸
j-times
), i + j = k, and
components
dkφ
dxidujdu1
∂
∂fτij
,
where τij = (x, . . . , x︸ ︷︷ ︸,
i-times
u, . . . , u︸ ︷︷ ︸
j-times
), i+ j = k − 1 depend on
∂k+1b
∂xi∂uj
,
and
dk+1a
dxk+1
respectively.
All others components
dkφ
dxrdusdut1
∂
∂fσ
are expressed in terms of k-jet of b (x, u) and k-jet of function
a (x) .
It shows that the bundles: κk,k−1 : Ok → Ok−1 are (k + 3)-
dimensional, when k > 1.
Feedback orbits in the space of 2-jets can be found by direct
integration of 12-dimensional completely integrable distribution
generating by the vector fields V 1
i , 0 ≤ i ≤ 3, and W 1
ij, 0 ≤ i+ j ≤
2. Summarizing, we get the following result.
Theorem 1. (1) The first non-trivial differential invariants
of feedback transformations appear in order 2 and they are
functions of the basic invariant
J =
f2 fu1u1
(u1fu1 − f) f2
u1
.
(2) There are
k (k + 1)
2
− 2
Feedback Equivalence of 1-dimensional Control Systems 295
independent differential invariants of pure order k.
(3) Dimension of the algebra of differential feedback invariants
of order k ≥ 2, is equal to
k3
6
+
k2
2
− 5k
3
+ 1.
(4) Dimension of the regular feedback orbits in the space of
k-jets, k ≥ 2, is equal to
(k + 1)2
2
+
23k
3
+
5
2
.
5. Invariant Derivations
We’ll need the following result which allows us to compute in-
variant derivations.
Assume that an infinitesimal Lie pseudogroup g is represented
in the Lie algebra of contact vector fields on the manifold of 1-jets
J1 (Rn) .
Moreover, we will identify elements g with the corresponding
contact vector fields , i.e. we assume that elements of g have the
form Xf (see [6]), where f is the generating function.
Lemma 1. Let x1, . . . , xn be coordinates in Rn, and let
(x1, . . . , xn, u, p1, . . . , pn)
be the corresponding canonical coordinates in the 1-jet space
J1 (Rn) . Then a derivation
∇ =
n∑
i=1
Ai
d
dxi
is g-invariant if and only if functions Ai ∈ C∞ (J∞Rn) , j =
1, . . . , n, are solutions of the following PDE system:
(5) Xf (Ai) +
n∑
j=1
d
dxj
(
∂f
∂pi
)
Aj = 0,
for all i = 1, . . . , n, and Xf ∈ g.
296 V. V. Lychagin
Proof. We have, [6]:
X•
f = Ef −
n∑
i=1
∂f
∂pi
d
dxi
,
where
Ef =
∑
σ
d|σ|f
dxσ
∂
∂pσ
is the evolutionary derivation, σ is a multi index and {pσ} are the
canonical coordinates in J∞Rn.
Using the fact that evolutionary derivations commute with the
total ones and the relation
[∇,X•
f ] = 0,
we get
0 =
n∑
j=1
Aj
d
dxj
,Ef −
n∑
i=1
∂f
∂pi
d
dxi
= −
∑
j
Ef (Aj)
d
dxj
+
∑
i,j
(
−Aj
d
dxj
(
∂f
∂pi
)
d
dxi
+
∂f
∂pi
dAj
dxi
d
dxj
)
= −
∑
s
X•
f (As) +
∑
j
Aj
d
dxj
(
∂f
∂ps
)
d
dxs
.
�
In our case we expect three linear independent feedback invari-
ant derivations. To solve PDE system (5) we first assume that the
unknown functions are functions on the 1-jet space J1R3. Then
collect terms in (5) with a, a
′
, a′′ and b, bx, bu, bxx, bxu and bu u we
get the system of 8 differential equations for 3 unknown functions.
Solving the system we found two independent invariant deriva-
tions. The last one we get in a similar way by assuming that the
unknown functions are functions on the 2-jet space J2R3.
Feedback Equivalence of 1-dimensional Control Systems 297
Finally, we have 3 feedback invariant derivations:
∇1 =
u1fu1 − f
fu1
d
du
+
f − u1fu1
f2
u1
fu
d
du1
,
∇2 =
f
fu1
d
du1
,
∇3 = f
d
dx
+
f
fu1
d
du
+
(
fxfu1 + fu − zfu u1 − fxu1
fu1u1
+
u1fu1 − f
f2
u1
fu
)
d
du1
.
These derivations obey the following commutation relations
[∇2,∇1] = J ∇1
[∇3,∇1] = K ∇2
[∇3,∇2] = −∇3 + J ∇1 + L ∇2
where K and L are some differential invariants of the 3rd order
(see below).
6. Differential Invariants of the 3-rd Order
Theorem 1 shows that there are four independent differential
invariants of the 3-rd order. We get three of them simply by
invariant differentiations:
∇1 (J) ,∇2 (J) ,∇3 (J) .
The symbols of these invariants contain:
• symbol of ∇2 (J) depends on fu1u1u1 ,
• symbol of ∇1 (J) depends on fu1u1u1 and fuu1u1,
• symbol of ∇3 (J) depends on fu1u1u1 ,fuu1u1 and fxu1u1 .
It shows that these differential invariants are independent.
The similar observation shows that the differential invariant
L, which appears in the commutation relations, is a function of
298 V. V. Lychagin
J,∇1 (J) ,∇2 (J) ,∇3 (J) , and the differential invariant K is the
forth independent invariant. It has the following form:
K=−u1fxu + 2u1
f2
u
ffu1
− 2
f2
u
fu1
2 +
+
f uuu1 − 2 fufx + ffxu
fu1
− u1
(f uuu1 − 2 fufx)
f
+
+
c1
fu1fu1u1
2 +
c2
ffu1u1
2 +
c3
fu1u1
2 +
c4
ffu1u1
+
c5
fu1fu1u1
+
c6
fu1u1
,
where
c1 =−ffufxu1fu1u1u1 − u1fufuu1fu1u1u1 + f2
ufu1u1u1 ,
c2 =u1
(
fufu1fuu1u1−f2
ufu1u1u1−fxfufu1fu1u1u1+fxfu1
2fuu1u1
)
+u1
2fuu1 (−fu1fuu1u1 + fufu1u1u1) ,
c3 =ffxu1fuu1u1 + fxfufu1u1u1 − fufuu1u1 − fxfu1fuu1u1
+u1 (fufxu1fu1u1u1 − fu1fxu1fuu1u1 + fuu1fuu1u1) ,
c4 =−u1
(
2 fu1fxfuu1 − fu1fufxu1 + fufuu1 + fu1f uu + fu1
2fxu
)
+u1
2
(
fu1f uuu1 − fufuu1u1 + fuu1
2
)
,
c5 =ffufxu1u1 − ffxu1fuu1 + fufuu1 + u1
(
fufuu1u1 − fuu1
2
)
,
c6 =fuu − fufxu1 + 2 fxfuu1 + fu1fxu − ff xuu1
+u1 (fu1f xuu1 − f uuu1 + fxu1fuu1 − fufxu1u1) .
7. Algebra of Feedback Differential Invariants
By regular orbits we mean feedback orbits of regular points.
Counting the dimensions of regular feedback orbits shows that
the following result is valid.
Theorem 2. Algebra of feedback differential invariants in a neigh-
borhood of a regular orbit is generated by differential invariant J
of the 2-nd order, differential invariant K of the 3-rd order and
all their invariant derivatives.
Feedback Equivalence of 1-dimensional Control Systems 299
8. The Feedback Equivalence Problem
Consider two control systems given by functions F andG. Then,
to establish feedback equivalence, we should solve the differential
equation
(6) F (X,U,UxG (x, u, u1) + Uuu1) −Xx G (x, u, u1) = 0
with respect to unknown functions X (x) and U (x, u) .
Let us denote the left hand side of (6) by H. Then assuming
the general position one can find functions X,Xx, U, Ux, Uu from
the equations
H = Hu1 = H(2)
u1
= H(3)
u1
= H(4)
u1
= 0.
Remark, that the above general conditions are feedback invari-
ant, depends on finite jet of the system and holds in a dense open
domain of the jet space. Therefore, it holds in regular points.
Assume that we get
U = A (x, u, u1) , Ux = B (x, u, u1) ,
Uu = C (x, u, u1) ,X = D (x, u, u1) ,
X ′ = E (x, u, u1)
Then the conditions
Au1 = Bu1 = Cu1 = Du1 = Eu1 = 0,
Du = Eu = 0
and
B = Ax, C = Au, E = Dx
show that if (6) has a formal solution at each point (x, u, u1) in a
domain then this equation has a local smooth solution.
On the other hand if system F at a point p = (x0, u0, u0
1) and
system G at a point p̃ = (x̃0, ũ0, ũ0
1) has the same differential in-
variants then, by the definition, there is a formal feedback trans-
formation which send the infinite jet of F at the point p to the
infinite jet of G at the point p̃.
300 V. V. Lychagin
Keeping in mind these observations and results of theorem 2 we
consider the space R3with coordinates (x, u, u1) and the space R14
with coordinates (j, j1, j2, j3, j11, j12, j13, j22, j23, j33k, k1, k2, k3) .
Then any control system, given by function F (x, u, u1), defines
a map
σF : R3 → R14,
by
j = JF , k = KF ,
ji = (∇i(J))F , ki = (∇i(K))F ,
jij = (∇i∇j(J))F ,
where i, j = 1, 2, 3, and the subscript F means that the differential
invariants are evaluated due to the system.
Let
Φ : R3 → R3
be a feedback transformation.
Then from the definition of the feedback differential invariants
it follows that
σF ◦ Φ = σ
bΦ(F )
.
Therefore, the geometrical image
ΣF = Im (σF ) ⊂ R14
does depend on the feedback equivalence class of F only.
We say that a system F is regular in a domain D ⊂ R3 if
(1) 4-jets of F belong to regular orbits,
(2) σF (D) is a smooth 3-dimensional submanifold in R14, and
(3) three of five functions j, j1, j2, j3, k are coordinates on ΣF .
Assume, for example, that functions j1, j2, j3 are coordinates
on ΣF . The following lemma gives a relation between the Tresse
derivatives and invariant differentiations ∇1,∇2,∇3.
Lemma 2. Let
D
DJ1
,
D
DJ2
,
D
DJ3
Feedback Equivalence of 1-dimensional Control Systems 301
be the Tresse derivatives with respect to differential invariants Ji =
∇i (J).
Then the following decomposition
(7) ∇i =
∑
j
Rij
D
DJj
with feedback differential invariants Rij of order ≤ 4 is valid.
Proof. Applying both parts of (7) to invariant Jk we get
∇i (Jk) = Rik
which is a feedback differential invariant of order ≤ 4. �
Theorem 3. Two regular systems F and G are locally feedback
equivalent if and only if
(8) ΣF = ΣG.
Proof. Let us show that the condition 8 implies a local feedback
equivalence.
Assume that
JF = jF (J1, J2, J3) , J
F
ij = jFij (J1, J2, J3) ,
KF = kF (J1, J2, J3) ,K
F
i = kFi (J1, J2, J3)
on ΣF , and
JG = jG (J1, J2, J3) , J
G
ij = jGij (J1, J2, J3) ,
KG = kG (J1, J2, J3) ,K
G
i = kGi (J1, J2, J3)
on ΣG.
Then condition 8 shows that jF = jG, jFij = jGij , k
F
i = kGi and
kF = kG.
Moreover,as we have seen the invariant derivations ∇1,∇2,∇3
are linear combinations of the Tresse derivatives with coefficients
which are feedback differential invariants of order ≤ 4.
In other words, the above functions jF , kF , jFij , k
F
i and their
partial derivatives in j1, j2, j3 determine the restrictions of all dif-
ferential invariants.
302 V. V. Lychagin
Therefore, condition 8 equalize restrictions of differential invari-
ants not only to order ≤ 4 but in all orders, and provides formal
and therefore local feedback equivalence between F and G. �
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