Transforming Nonlinear Single-Input Control Systems to Normal Forms via Feedback Issa Amadou Tall∗, Witold Respondek Institut National des Sciences Appliqu´ees de Rouen Laboratoire de Math´ematiques de l’INSA Pl. Emile Blondel, 76 131 Mont Saint Aignan Cedex, France {tall,wresp}@lmi.insa-rouen.fr Keywords: feedback equivalence, normal forms, canonical forms, invariants.
Abstract We study the problem of bringing a nonlinear single-input system to a normal form. We follow an approach of Kang and Krener based on studying, step by step, the action of homogeneous transformations on homogeneous terms of the system. The first part of the paper deals with systems whose linearization is controllable. We give explicit transformations bringing the system to a normal form proposed by Kang. Then we construct a canonical form. In the second part, we give a normal form for systems whose linearization is not controllable. The normal forms, in the controllable and uncontrollable case, are illustrated by two physical examples.
1
A natural question to ask is whether we can take the system ˜ to be linear, i.e., whether we can linearize the system Σ via Σ feedback. Necessary and sufficient geometric conditions for this to be the case have been given in [7] and [8]. Those conditions turn out to be, except for the planar case, restrictive and a natural problem which arises is to find normal forms for nonlinearizable systems. Although being natural, this problem is very involved and has been extensively studied during the last twenty years (see [3], [5], [9], [11], [12], [13], [14], [15] among others). In our paper we will follow a very fruitful approach proposed by Kang and Krener [13] and then followed by Kang [11], [12]. Their idea, which is closely related with classical Poincar´e’s technique for linearization of dynamical systems (see e.g. [1]), is to analyze the system Σ and the feedback transformation Γ step by step and, as a ˜ also consequence, to produce a simpler equivalent system Σ step by step.
Introduction
The problem of transforming the nonlinear control singleinput system Σ : ξ˙ = f (ξ) + g(ξ)u by a feedback transformation of the form Γ:
x = Φ(ξ) u = α(ξ) + β(ξ)v
to a simpler form has been extensively studied during the last twenty years. The transformation Γ brings Σ to the system ˜ : x˙ = f˜(x) + g˜(x)v, Σ whose dynamics are given by f˜ = Φ∗ (f + gα) g˜ = Φ∗ (gβ), where for any vector field f and any diffeomorphism Φ we denote (Φ∗ f )(x) = dΦ(Φ−1 (x)) · f (Φ−1 (x)).
∗ On leave from the Dep. of Mathematics, University Cheikh Anta Diop, Dakar, Senegal, e-mail:
[email protected]
The paper is organized as follows. In Section 2 we will introduce, following [13] and [11], homogeneous feedback transformations. We give a normal form, obtained by Kang [11] and discuss invariants of homogeneous transformations, also obtained by him. Then we provide an explicit construction of transformations bringing the system to Kang’s normal form. It turns out that a given system can admit different normal forms. In Section 3 we thus construct a canonical form and give one of our main results stating that two control systems are fedback equivalent if and only if their canonical forms coincide. In Section 4 we study systems whose linear part is not controllable. In this case, we give a normal form which generalizes, on one hand, that obtained by Kang [12] for terms of degree 2, and, on the other hand, generalizes those discussed in Section 2 and Section 3. Finally, in Section 5 we provide two physical examples of variable lenght pendulum and Kapitsa’s pendulum. The first one illustrates the introduced notions of normal and canonical forms in the case of controllable linear part. The second one illustrates normal forms for systems with noncontrollable linear part, and, in particular, the role of resonances in obtaining such forms
2 Normal form and m-invariants All objects, i.e., functions, maps, vector fields, control systems, etc., are considered in a neighborhood of 0 ∈ Rn and assumed to be C ∞ -smooth. Let h be a smooth function. By h(x) = h[0] (x) + h[1] (x) + h[2] (x) + · · · =
∞ X
h[k] (x)
k=0
we denote its Taylor expansion around zero, where h[k] (x) stands for a homogeneous polynomial of degree k. Similarly, for a map Φ of an open subset of Rn to Rn (resp. for a vector field f on an open subset of Rn ) we will denote by Φ[k] (resp. by f [k] ) the term of degree k of its Taylor [k] expansion at zero, i.e., each component Φj of Φ[k] (resp. [k]
fj of f [k] ) is a homogeneous polynomial of degree k in x. ¯i = (x1 , . . . , xi ). Denote also x Consider the system Σ : ξ˙ = =
f (ξ) + g(ξ)uP ∞ F ξ + Gu + k=2 (f [k] (ξ) + g [k−1] u), (2.1) and = g(0). We will assume throughwhere F = ∂F (0) G ∂ξ out the paper that f (0) = 0 and g(0) 6= 0. Consider the Taylor developement Γ∞ of the feedback transformation Γ given by P∞ x = Φ(ξ) = T ξ + k=2 Φ[k] (ξ) Γ∞ : u = α(ξ) + β(ξ)vP ∞ = Kξ + Lv + k=2 (α[k] (ξ) + β [k−1] (ξ)v), (2.2) where the matrix T is invertible and L 6= 0. Let us analyze the action of Γ∞ on the system Σ step by step. To start with, consider the linear part ξ˙ = F ξ + Gu of the system Σ given by (2.1). If this linear system is controllable, which we will assume in Sections 2 and 3, then it can be transformed by a linear feedback transformation of the form x = Tξ Γ1 : (2.3) u = Kξ + Lv to the Brunovsky canonical form (A, B). Assuming that the linear part (F, G), of the system Σ given by (2.1) has been transformed to the Brunovsky canonical form (A, B), we follow an idea of Kang and Krener [13], [11] and apply successively a series of transformations
Observe that each transformation Γm , for m ≥ 2, leaves invariant all homogeneous terms of degree smaller than m and we will call Γm a homogeneous feedback transformation of order m. We will study the action of Γm on the following system Σ[m] : ξ˙ = Aξ + Bu + f [m] (ξ) + g [m−1] (ξ)u + O(ξ, u)m+1 . (2.5) The starting point is the following result, proved in [11]. ˜ [m] given by Consider another system Σ ˜ [m] : x˙ = Σ +
Ax + Bv + f˜[m] (x) + g˜[m−1] (x)v O(x, v)m+1 . (2.6)
Proposition 1 The feedback transformation Γm , defined by ˜ [m] , given (2.4), brings the system Σ[m] , given by (2.5), to Σ by (2.6), if and only if the following relations hold for any 1 ≤ j < n: [m] [m] [m] [m] LAξ (φj ) − φj+1 (ξ) = f˜j (ξ) − fj (ξ) [m] [m−1] [m−1] LB (φj ) = g˜j (ξ) − gj (ξ)
[m]
LAξ (φn ) + α[m] (ξ)
[m] [m] f˜n (ξ) − fn (ξ)
=
[m−1]
[m]
LB (φn ) + β [m−1] (ξ) =
g˜n
[m−1]
(ξ) − gn
This proposition represents the essence of the method developped by Kang and Krener and used in our paper. The problem of studying the feedback equivalence of two sys˜ requires, in general, solving a system of 1-st tems Σ and Σ order partial differential equations. On the other hand, if we perform the analysis step by step, then the problem of establishing the feedback equivalence of two systems Σ[m] and ˜ [m] reduces to solving the algebraic system (2.7). Σ Using the above proposition, Kang [11] proved the following result. Theorem 1 The system Σ[m] can be transformed via a homogeneous feedback transformations Γm , into the following [m] normal form ΣN F n
[m−2]
x˙ 1
=
x2 + Σ x2i P1,i i=3 .. .
x˙ j
=
xj+1 +
n
(¯ xi ) + O(x, v)m+1
[m−2]
Σ x2i Pj,i
i=j+2
(¯ xi ) + O(x, v)m+1
(2.4)
x˙ n−2
=
.. . [m−2] xn ) + O(x, v)m+1 xn−1 + x2n Pn−2,n (¯
for m = 2, 3, . . . . A feedback transformation defined as a series of successive compositions of Γm , m = 1, 2, . . . will also be denoted by Γ∞ because, as a formal power series, it is of the form (2.2). We will not address the problem of convergence and we will call such a series of successive compositions a formal feedback transformation.
x˙ n−1
=
xn
Γm :
[m]
x = ξ + Φ (ξ) u = v + α[m] (ξ) + β [m−1] (ξ)v,
(ξ). (2.7)
x˙ n
= v,
(2.8) [m−2] [m−2] (¯ xi ) = Pj,i (x1 , · · · , xi ) are homegeneous where Pj,i polynomials of degree m − 2.
In order to construct invariants of homogeneous feedback transformations let us define Xim−1 (ξ) = adiAξ+f [m] (ξ) (B + g [m−1] (ξ)). [m]t,i
Following Kang [11], we denote by a neous part of degree m − 2 of m−1 m−1 , Xi−2 |W CAt−1 Xi−1
(ξ) the homoge-
n−i+2
{ a[m]t,i (ξ) : 2 ≤ i ≤ n − 1, 1 ≤ t ≤ n − i } and {a ˜[m]t,i (x) : 2 ≤ i ≤ n − 1, 1 ≤ t ≤ n − i } denote, respectively, their m-invariants. Theorem 2 The m-invariants have the following properties: ˜ [m] are equivalent via a homo(i) the systems Σ[m] and Σ geneous feedback of order m, modulo higher order terms, if and only if a[m]t,i = a ˜[m]t,i for any 2 ≤ i ≤ n − 1 and any 1 ≤ t ≤ n − i; [m] (ii) the m-invariants of the system ΣN C , defined by (2.8), are given by [m−2]
x2n−i+2 Pt,n−i+2 (¯ xn−i+2 )
(2.10)
for any 2 ≤ i ≤ n − 1 and any 1 ≤ t ≤ n − i. Our first aim is to find explicitely feedback transforma[m] tions bringing the system Σ[m] to its normal form ΣN F . [m−1] [m−1] Define the functions Ψj,i (ξ) by setting Ψj,0 (ξ) = [m−1]
Ψ1,1
(ξ) = 0 and
[m−1]
Ψj,i
[m−1] × adn−i + Aξ g
n−i−1
t Σ (−1)t+1 adAξ adAn−i−t−1 B f [m] ,
t=0
(2.11)
if 1 ≤ j < i ≤ n, and by setting [m−1]
Ψj,i
[m]
[m−1]
(ξ) = LAn−i B fj−1 (ξ¯i ) + LAξ (Ψj−1,i (ξ¯i ))|Wi
[m−1] +Ψj−1,i−1 (ξ¯i−1 )
+
R ξi
[m−1] LAn−i+1 B Ψj−1,i (ξ¯i )dξi , 0
(2.12)
[m−1]
Ψj,i
[m]
(ξ¯i )dξi ,
1 ≤ j < n,
[m]
=
fn−1 (ξ) + LAξ φn−1 (ξ),
α[m] (ξ)
=
[m] [m] − fn (ξ) + LAξ φn (ξ) ,
β [m−1] (ξ)
=
[m] [m−1] (ξ) + LB φn (ξ) . − gn
(2.13)
We have the following result Theorem 3 The feedback transformation x = ξ + φ[m] (ξ) m Γ : u = v + α[m] (ξ) + β [m−1] (ξ)v,
(2.14)
where (φ[m] , α[m] , β [m−1] ) are defined by (2.13), brings the [m] system Σ[m] to a normal form ΣN F given by (2.8). We would like to discuss the interest of Theorem 3. Notice that the transformations (2.13) are easily computable via differentiation and integration of homogeneous polynomials. Therefore, for any given control systems, Theorem 3 gives explicit transformations bringing the homogeneous part of the system to its normal form. For example, if the system is feedback linearizable, up to order m0 (see [14]) then a diffeomorphism and a feedback compensating all nonlinearities of order lower than m0 can be calculated explicitely without solving partial differential equations. More generaly, by a successive applications of transformations given by (2.13) we can bring the system, without solving partial differential equations, to its normal form given in Theorem 4 below. Proof of Theorem 3. From the expressions of α[m] (ξ) and β [m−1] (ξ) given by (2.13) and the last two equations of (2.7) we get f˜n[m] (ξ) = 0
and
g˜n[m−1] (ξ) = 0.
[m]
Plugging φj , defined by (2.13), to the second equation of (2.7), gives [m−1]
(ξ) = (−1)n−i+1 CAj−1
is the restriction of
φn (ξ)
(2.9)
The functions a[m]t,i (ξ), for 2 ≤ i ≤ n − 1, 1 ≤ t ≤ n − i, will be called m-invariants of Σ[m] . The following result of Kang [11] asserts that m-invariants a[m]t,r (ξ) are complete invariants of homogeneous feedback and, moreover, illustrates their meaning for the normal form [m] ˜ [m] , ΣN F . Consider two systems Σ[m] , given by (2.5), and Σ given by (2.6). Let
∂x2n−i+2
n Rξ Σ i i=1 0
φj (ξ) =
,
Wi = {ξ ∈ Rn | ξi+1 = · · · = ξn = 0 } .
(ξ¯i )
to the submanifold Wi . Define
[m]
defined as follows:
∂2
[m−1] Ψj,i (ξ) [m]
where C = (1, 0, · · · , 0), and let the submanifolds Wi be
a[m]t,i =
[m−1]
if 1 ≤ i ≤ j, where Ψj,i
Ψj,n
[m−1]
(ξ) = g˜j
[m−1]
(ξ) − gj
(ξ),
[m−1]
which, by (2.7), implies g˜j (ξ) = 0, for j = 1, · · · , n − 1. Now we consider the first equation of (2.7). From the [m] [m] expression of φn we get f˜n−1 (ξ) = 0 and for any i = 1, · · · , n we obtain by derivation [m]
[m]
[m]
[m]
[m]
∂fj (ξ) ∂φj ∂φj ∂φj+1 ∂ f˜j (ξ) = + LAξ ( )+ − . ∂ξi ∂ξi ∂ξi ∂ξi−1 ∂ξi (2.15)
If i ≥ j + 2, we obtain [m] ∂ f˜j (ξ) = ∂ξi |Wi−1
=
[m]
=
[m]
[m]
∂fj (ξ) ∂ξi
∂fj (ξ¯i ) ∂ξi
+
∂φ LAξ ( ∂ξji
[m]
)+
∂φj ∂ξi−1
[m−1] + LAξ (Ψj,i (ξ¯i ))
[m]
−
∂φj+1 ∂ξi
|Wi−1
|Wi−1
[m−1] [m−1] +Ψj,i−1 (ξ¯i−1 ) − Ψj+1,i (ξ¯i ).
Hence, by an induction argument, we obtain, compare also [16], [m]
∂fj (ξ) ∂ξi
[m−1]
+ LAξ (Ψj,i
(ξ¯i ))
[m−1] [m−1] +Ψj,i−1 (ξ¯i−1 ) − Ψj+1,i (ξ¯i ) = 0
and, finally, we get [m]
∂ f˜j (ξ) = 0, ∂ξi |Wi−1
j + 2 ≤ i ≤ n.
(2.16)
If 1 ≤ i ≤ j + 1, then, using (2.15) and (2.13), we get [m] ∂ f˜j (ξ) = ∂ξi =
= +
|Wi
[m] ∂fj (ξ)
∂ξi
[m] ∂fj (ξ¯i ) ∂ξi
R ξi 0
+ LAξ (
[m] ∂φj
∂ξi
)+
[m−1]
+ LAξ (Ψj,i
[m−1] ¯ (ξi ) ∂Ψj,i dξi ∂ξi−1
[m] ∂φj
∂ξi−1
−
[m] ∂φj+1
∂ξi
|Wi
[m−1] (ξ¯i ))|Wi + Ψj,i−1 (ξ¯i−1 )
From the expression (2.12) we deduce that: [m]
1 ≤ i ≤ j + 1.
(2.17)
From relations (2.16) and (2.17) we conclude that [m] f˜j (ξ) =
n
Σ ξi2 Pj,i (ξ¯i )
i=j+2
and thus the system Σ[m] given by (2.5) is feedback equiva[m] lent to the system ΣN F given by (2.8). To illustrate results of this Section, we consider the system Σ[m] , given by (2.5) on R3 . Theorem 1 implies that the system Σ[m] is equivalent, via the feedback Γm defined by [m] (2.13), to its normal form ΣN F (see (2.8)) x˙ 1 x˙ 2 x˙ 3
Theorem 4 Consider the system Σ given by (2.1). There exists a formal feedback transformation Γ∞ which brings the ∞ system Σ to a normal form ΣN F given by n x˙ 1 = x2 + Σ xi2 P1,i (x1 , · · · , xi ) i=3 .. . n x˙ j = xj+1 + Σ x2i Pj,i (x1 , · · · , xi ) i=j+2 . ∞ . : ΣN F . = xn−1 + x2n Pn−2,n (x1 , · · · , xn ) x ˙ n−2 x˙ n−1 = xn x˙ n = v , (2.18) where Pj,i (x1 , · · · , xi ) are formal power series.
3 Canonical form
[m−1] − Ψj+1,i (ξ¯i ).
∂ f˜j (ξ) = 0, ∂ξi |Wi
where P [m−2] is a homogeneous polynomial of degree m−2. Consider the system Σ of the form (2.1) and assume that its linear part (F, G) is controllable. Apply successively to it a series of transformations Γm , m = 1, 2, . . . such that each [m] Γm brings Σ[m] to its normal form ΣN F , for instance we can take a series of transformations defined by (2.13). Successive repeating of Theorem 1 gives
= x2 + x23 P [m−2] (x1 , x2 , x3 ) + O(x, v)m+1 = x3 = v,
A natural and fundamental question which arises is whether the system Σ can admit two different normal forms, that is, whether the normal forms given by Theorem 4 are in fact canonical forms. It turns out that a given system can admit different normal forms, see [11], and the aim of this Section is to construct a canonical form for Σ. Consider the system Σ of the form (2.1). Since its linear part (F, G) is assumed to be controllable, we bring it, via a linear transformation and linear feedback, to the Brunovsky canonical form (A, B). Let the first homogeneous term of Σ which cannot be anihilated by a feedback transformation be of order m0 . As proved by Krener [14], the degree m0 is given by the largest integer j + 1 such that all distributions Dk = span {g, . . . , adk−1 g}, for 1 ≤ k ≤ n − 1, are f involutive modulo terms of order j − 1. We can thus, due to Theorems 1 and 2, assume that, after applying a suitable feedback, Σ takes the form Σ : ξ˙ = Aξ + Bu + f˜[m0 ] (ξ) ∞ [k] P f (ξ) + g [k−1] (ξ)u , +
(3.1)
k=m0 +1
where (A, B) is in Brunovsky canonical form and the first nonvanishing homogeneous vector field f˜[m0 ] is of the form n P 2 [m0 −2] ¯ (ξi ), 1≤j ≤n−2 ξi Pj,i [m0 ] i=j+2 ˜ fj (ξ) = 0, n − 1 ≤ j ≤ n. (3.2)
Let (i1 , . . . , in−s ), where i1 + · · · + in−s = m0 , be the largest, in the lexicographic ordering, (n − s)-tuple of nonnegative integers such that for some 1 ≤ j ≤ n − 2 we have [m0 ]
∂ m0 f˜j
Σ : ξ˙ = f (ξ) + g(ξ)u,
(ξ)
6= 0.
in−s · · · ∂ξn−s
∂ξ1i1
) ∂ m0 f˜[m0 ] (ξ) j t0 = sup j = 1, · · · , n − 2, i 6 0 . = in−s ∂ξ11 · · · ∂ξn−s (3.3) We have the following result (
Theorem 5 The system Σ given by (2.1) is equivalent by a formal feedback Γ∞ to a system of the form Σ∞ ˙ = Ax + Bv + CF : x
∞ X
f˜[k] (x),
(3.4)
k=m0
where, for any k ≥ m0 , n P 2 [k−2] xi Pj,i (¯ xi ), [k] i=j+2 f˜j (x) = 0,
∂xi11
(x) i
n−s · · · ∂xn−s
i
n−s ∂xi11 · · · ∂xn−s
= x2 + x23 P (x1 , x2 , x3 ) = x3 = v,
where P (x1 , x2 , x3 ) is a formal power series. Assume, for simplicity, that m0 = 2 which is equivalent to the following generic condition: g, adf g, and [g, adf g] are linearly independent at 0 ∈ R3 . This implies that we can express P = P (x1 , x2 , x3 ) as P = c + P1 (x1 ) + x2 P2 (x1 , x2 ) + x3 P3 (x1 , x2 , x3 ),
1≤j ≤n−2 n − 1 ≤ j ≤ n,
x ˜ = Φ(x) v = α(x) + β(x)˜ v
(3.5)
= ±1
(3.6)
(x1 , 0, . . . , 0) = 0.
which normalizes the constant c and anihilates the formal power series P1 (x1 ). More precisely, Γ∞ transforms Σ to its canonical form Σ∞ CF x ˜˙ 1 x ˜˙ 2 x ˜˙ 3
and, moreover, for any k ≥ m0 + 1, [k] ∂ m0 f˜t0 (x)
x˙ 1 x˙ 2 x˙ 3
where P1 (0) = 0. Observe that any P (x1 , x2 , x3 ), of the above form, gives a normal form Σ∞ N F . In order to get the canonical form Σ∞ CF , we use Theorem 5 which assures the existence of a feedback transformation Γ∞ of the form
additionally, we have
[m0 ]
(3.8)
where ξ(·) ∈ R3 , is equivalent, via a formal feedback Γ∞ , to its normal form Σ∞ N F (see (2.18))
Define
∂ m0 f˜t0
To illustrate results of this Section, we consider the system Σ on R3 whose linear part is controllable. Theorem 4 implies that the system
(3.7)
The form Σ∞ CF satisfying (3.5), (3.6) and (3.7) will be called the canonical form of Σ. The name is justified by the following Theorem 6 Two systems Σ1 and Σ2 are formally feedback equivalent if and only if their canonical forms Σ∞ 1,CF and Σ∞ 2,CF coincide. Kang [11], generalizing [13], proved that any system Σ can be brought by a formal feedback to the normal form (3.4) for which (3.5) is satisfied. He also observed that his normal forms are not unique. Our results, Theorems 5 and 6, complete his study. We show that for each degree m of homogenity we can use a 1-dimensional subgroup of feedback transformations which preserves the “triangular” structure of (3.5) and at the same time allows us to normalize one term. The form of (3.6) and (3.7) is a result of this normalization. Proofs of Theorems 5 and 6 are given in [17] and [16], and contain an explicit construction of feedback transformations bringing Σ to its canonical form Σ∞ CF .
x1 , x = x ˜2 + x ˜32 P˜ (˜ ˜2 , x ˜3 ) = x ˜3 = v˜,
x1 , x where the formal power series P˜ (˜ ˜2 , x ˜3 ) is of the form x1 , x ˜2 ) + x ˜3 P˜3 (˜ ˜3 ). ˜3 ) = 1 + x ˜2 P˜2 (˜ x1 , x ˜2 , x ˜2 , x P˜ (˜ x1 , x An example of constructing the canonical form for a physical model of variable lenght pendulum will be given in Section 5.
4 Systems with Noncontrollable Linear Part Consider a system of the form ∞ Σ : ξ˙ = F ξ + Gu + Σ f [k] (ξ) + g [k−1] (ξ)u k=2
(4.1)
and assume that its linear part (F, G) is not controllable. If the rank of Kalman controllability matrix equals n − r, then, see e.g. [10], we can apply a linear feedback transformation Γ1 , of the form (2.3) such that x = (x1 , x2 ) ∈ Rr × Rn−r and in (x1 , x2 )-coordinates we have 0 A1 0 , B= , (4.2) A= B2 0 A2
where (A2 , B2 ) is in Brunovsky form of dimension n − r. In what follows, we assume, for simplicity, that all eigenvalues {λ1 , . . . , λr } of the matrix A1 are real and distinct. Put λ = (λ1 , . . . , λr ). For any integer m ≥ 2, the set of r-tuples of nonnegative integers α = (α1 , . . . , αr ), such that α1 + · · · + αr = m and < λ, α > = λ1 α1 + · · · + λr αr = λj for some λj , forms the so called “resonances” of order m. A classical result of Poincar´e (see e.g. [1]) states that resonances are the only obstruction for, at least formal, linearization of vector fields. In fact, we can anihilate all terms except, perhaps, for those monomials cξ1α1 · · · ξrαr for which the multiindex α = (α1 , . . . , αr ) is resonant. It is thus not surprising that we rediscover resonant terms in the two following results which provide normal forms for systems with noncontrollable linear part. A particular case of Theorem 7, for m = 2, was obtained by Kang [12]. Theorem 7 Consider the system Σm : ξ˙ = Aξ + Bu + f [m] (ξ) + g [m−1] (ξ)u + O(ξ, u)m+1 , where (A, B) is of the form (4.2), with A1 being diagonal and (A2 , B2 ) being a Brunovsky pair of dimension n − r. (i) There exists a transformation Γm which transforms Σm into the following system x˙ = Ax + Bv + f˜[m] (x) + O(x, v)m+1 , such that
[m] f˜j (x) =
m P P [m−k] (¯ xr ) γj,α xα + xkr+1 Qj −λj =0 k=1 n P 2 [m−2] + xi ), 1 ≤ j ≤ r i=r+2xi Rj,i (¯
n P
i=j+2
[m−2]
x2i Pj,i
(¯ xi ),
r+1≤j ≤n−2
n−1≤j ≤n (4.3) where < λ, α > = λ1 α1 + · · · + λr αr , α1 + · · · + αr = m, αr 1 xα = xα 1 · · · xr and γj,α ∈ R. 0,
[m−2]
(ii) Moreover, the polynomials Pj,i
[m−k] Qj
,
[m−2]
Rj,i
are invariant under any transformation Γ form (2.4).
m
, and
of the
A successive application of this result gives the following formal normal form of nonlinear single-input systems. Theorem 8 Consider the control system Σ of the form (4.1) and let the rank of the Kalman controllability matrix of (F, G) be n − r. Assume that the r × r-matrix A1 , defined by (4.2), is diagonalizable. The system Σ is equivalent, by a formal feedback transformation Γ∞ , given by (2.2), to a
system of the form x˙ j
= +
n P
i=r+2
x˙ j
=
P
λj xj +
−λj =0
x2i Rj,i (¯ xi ),
xj+1 +
n P
i=j+2
x˙ n−1
=
xn
x˙ n
=
v,
γj,α xα +
∞ P
k=1
xkr+1 Qkj (¯ xr )
1 ≤ j ≤ r,
x2i Pj,i (¯ xi ), r + 1 ≤ j ≤ n − 2,
(4.4) r where < λ, α > = λ1 α1 + · · · + λr αr , xα = x1α1 · · · xα r , γj,α ∈ R and the first sum is taken, for each λj , over all r-tuples of nonnegative integers α = (α1 , . . . , αr ) such that < λ, α > = λj and α1 + · · · + αr ≥ 2. Above, Qkj , Pj,i , and Rj,i are formal power series depending on the indicated variables. Proofs of Theorems 7 and 8 are given in [16]. Notice that the form of the linearly controllable part of (4.4) reminds the normal form Σ∞ N F given by (2.18), the only difference being that now the formal power series Pj,i are parametrized by the state vector of the uncontrollable part. In particular, we can bring the series Pj,i to a canonical form generalizing, in an obvious way, that described by Theorem 5. The more difficult problem of normalizing the uncontrollable part will be discussed in another paper. In the next Section we will provide an example of a fourdimensional system, the Kapitsa pendulum, illustrating the normal form (4.4). In fact, the linearization of Kapitsa pendulum possesses an uncontrollable mode of dimension two whose eigenvalues are in resonance of order two and its normal forms contain infinitely many resonant terms (see (5.5) and (5.6) below).
5
Examples
In this Section we give physical examples illustrating normal and canonical forms defined and studied in the paper. We start with the case of the controllable linear part. Consider the variable lenght pendulum of Bressan and Rampazzo [4] (see also [6]). We denote by ξ1 the lenght of the pendulum, by ξ2 its velocity, by ξ3 the angle with respect to the horizontal, and by ξ4 the angular velocity. The control u = ξ˙4 = ξ¨3 is the angular acceleration. The mass is normalized to 1. The equations are (compare [4] and [6]) ξ˙1 ξ˙2 ξ˙3 ξ˙4
= = = =
ξ2 −g sin ξ3 + ξ1 ξ42 ξ4 u,
where g denotes the gravity. Notice that if we suppose to control the angular velocity ξ4 = ξ˙3 , which is the case of [4]
and [6], then the system is three-dimensional but the control enters nonlinearly. Observe that the linear part of the system is controllable at any equilibrium point ξ0 = (ξ10 , ξ20 , ξ30 , ξ40 )t =(ξ10 , 0, 0, 0)t . Our goal is to produce, for variable lenght pendulum, a normal form and the canonical form as well as to answer the question whether the systems corresponding to various values of the gravity constant g are feedback equivalent. In order to get a normal form, put x1 x2 x3 x4 v
= ξ1 = ξ2 = −g sin ξ3 = −gξ4 cos ξ3 = gξ42 sin ξ3 − gu cos ξ3 . = x2 1 = x3 + x42 g2x−x 2 3 = x4 = v,
which gives a normal form. Indeed, we rediscover Σ∞ NF , given by (2.18), with P1,3 = 0, P1,4 = 0, and P2,4 =
g2
[3]
= µxi , 1 ≤ i ≤ 4, = µv,
where µ = 1/g. We get the following canonical form for the variable lenght pendulum x ˜˙ 1 x ˜˙ 2 ˜˙ 3 x x ˜˙ 4
= ξ2 + ξ1 ξ4 + ξ4 P1 (ξ1 ) = g0 ξ1 − ξ2 ξ4 + ξ2 ξ4 P2 (ξ1 ) + ξ42 Q2 (ξ1 ) + R2 (ξ1 ) = ξ4 = v, (5.2) where g0 = g /l , the case = 1 corresponds to α0 = 2nπ while = −1 to α0 = (2n + 1)π, and P1 , P2 , Q2 , R2 are analytic functions such that
0
0
0
P1 (0) = P2 (0) = R2 (0) = 0. One can easily check that the quadratic transformation y1 y Γ2 : 2 y3 y4
= ξ1 − ξ1 ξ3 = ξ2 + ξ2 ξ3 = ξ3 = ξ4
brings the system (5.2) into the following one ˜ 1 (y1 , y3 ) = y2 + y2 y3 P˜1 (y3 ) + y4 Q ˜ ˜ 2 (y1 , y2 , y3 ) = g0 y1 + y1 y3 P2 (y1 , y3 ) + y4 Q 2˜ + y4 R2 (y1 , y3 ) + S˜2 (y1 ) y˙ 3 = y4 y˙ 4 = v (5.3) ˜ 1 , P˜2 , Q ˜2, R ˜ 2 , and S˜2 are analytic functions. where P˜1 , Q Since the vector field defined in R3 by y˙ 1 y˙ 2
6= 0,
is (i1 , i2 , i3 , i4 ) = (1, 0, 0, 2). In order to normalize f2 , put x ˜i v˜
(5.1)
where α denotes the angle of the pendulum with respect to the vertical axis z, by u we denote the vertical velocity of the suspension point, p is proportionnal to the generalized impulsion, g is the gravity constant and l is the lenght of the pendulum. We assume to control the acceleration w = u. ˙ Introduce the following coordinates (ξ1 , ξ2 , ξ3 , ξ4 )t = (α, p, z/l, u/l)t and take the control v = w/l. We will consider the system (5.1) around the equilibrium point (α0 , p0 , z0 , u0 )T = (kπ, 0, 0, 0)T . It thus rewrites as follows:
and
∞ , In order to bring the system to its canonical form ΣCF 2 x1 firstly observe that m0 = 3. Indeed, the function x4 g2 −x2 3 starts with third order terms, which corresponds to the fact that the invariants a[2]t,i vanish for any i = 2, 3 and any [3] 1 ≤ t ≤ 4− i. The only non zero component of f [3] is f2 = [1] x24 P2,4 . Hence t0 = 2 and the only, and thus the largest, quadruplet (i1 , i2 , i3 , i4 ) of nonnegative integers, satisfying i1 + i2 + i3 + i4 = 3 and such that
∂xi11 · · · ∂xi44
= p + ul sin α 2 = ( gl − ul2 cos α) sin α − ul p cos α = u,
P1 (0) = P2 (0) = Q2 (0) = R2 (0) = 0
x1 . − x23
[3] ∂ 3 f2
α˙ p˙ z˙
ξ˙1 ξ˙2 ξ˙3 ξ˙4
The system becomes x˙ 1 x˙ 2 x˙ 3 x˙ 4
The equations of the Kapitsa pendulum are given by (see [2] and [6])
= x ˜2 x ˜1 = x ˜3 + x ˜24 1−˜ x23 = x ˜4 = v˜.
Independently of the value of the gravity constant g, all systems are feedback equivalent to each other. Now we consider the Kapitsa pendulum in order to illustrate normal forms in the case of noncontrollable linear part.
˜ 1 (y1 , y3 )∂/∂y1 + Q ˜ 2 (y1 , y2 , y3 )∂/∂y2 + ∂/∂y3 f =Q does not vanish at 0 ∈ R3 , there exists an analytic transformation x = Φ(y) of the form x1 x2 x3
= φ1 (y1 , y2 , y3 ) = φ2 (y1 , y2 , y3 ) = y3
such that (Φ∗ f )(x) = ∂/∂x3 .
This transformation, completed with x4 = y4 , brings the system (5.3) to x˙ 1 x˙ 2 x˙ 3 x˙ 4
= + = + = =
¯ 1 (x1 , x2 ) x2 + R ¯ 1 (x1 , x2 , x3 ) x3 P¯1 (x1 , x2 , x3 ) + x24 Q ¯ 2 (x1 , x2 ) g0 x1 + R ¯ 2 (x1 , x2 , x3 ) x3 P¯2 (x1 , x2 , x3 ) + x24 Q x4 v.
[3] B. Bonnard, Quadratic control systems, Mathematics of Control, Signals, and Systems, 4 (1991), pp. 139–160. (5.4)
Let us remark that the dimension of the controllable part of the system (5.2), and thus of (5.4), is 2. In the case = 1, the √ eigenvalues of the uncontrolable part are λ1 = −λ2 = g0 √ while in the case = −1, they are λ1 = −λ2 = i g0 . In both cases they are resonant with the resonance relations which, in both cases, are λ1 = mλ1 + (m − 1)λ2 for m ≥ 2 and λ2 = (m − 1)λ1 + mλ2 for m ≥ 2. Then applying Poincar´e’s method (see [1]), we get rid, by a formal diffeomorphism in the (x1 , x2 )-space, of ¯1, R ¯ 2 )t and all nonresonant terms of the dynamical part (R transform the system, respectively for = 1 and = −1 to one of the following normal forms (compare (4.4)). Put λ = √ g0 . Then for = 1, which corresponds to real eigenvalues of the uncontrollable mode, the normal form is P x˙ 1 = λx1 + m≥2 am x1 (x1 x2 )m−1 ¯ 1 (x1 , x2 , x3 ) + x3 P¯1 (x1 , x2 , x3 ) + x24 Q x˙ 2 x˙ 3 x˙ 4
P = −λx2 + m≥2 bm x2 (x1 x2 )m−1 ¯ 2 (x1 , x2 , x3 ) + x3 P¯2 (x1 , x2 , x3 ) + x24 Q = x4 = v.
[2] V. Bogaevski and A. Povzner, Algebraic Methods in Nonlinear Physics, Springer-Verlag, New York, 1991.
(5.5)
For = −1, which corresponds to complex eigenvalues of the uncontrollable mode, the normal form is P x˙ 1 = λx2 + m≥2 (cm x1 + dm x2 )(x12 + x22 )m−1 ¯ 1 (x1 , x2 , x3 ) + x3 P¯1 (x1 , x2 , x3 ) + x24 Q
P = −λx1 + m≥2 (−dm x1 + cm x2 )(x12 + x22 )m−1 ¯ 2 (x1 , x2 , x3 ) + x3 P¯2 (x1 , x2 , x3 ) + x42 Q x˙ 3 = x4 x˙ 4 = v. (5.6) Notice that we transform the original system to the form (5.4) using an analytic diffeomorphism and only the passage from (5.4) to (5.5) or (5.6) is done via a formal transformation. x˙ 2
References [1] V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Second Edition, Springer-Verlag, 1988.
[4] A. Bressan and F. Rampazzo, On differential systems with quadratic impulses and their applications to Lagrangian mechanics, SIAM J. Control and Optim, 31 (1993), pp.1205-1230. [5] S. Celikovsky and H. Nijmeijer, Equivalence of nonlinear systems to triangular form: the singular case, Systems and Control Letters, 27 (1996), pp. 135-144. [6] M. Fliess, J. L´evine, P. Martin, and P. Rouchon, Flatness and defect of nonlinear systems: Introductory theory and examples, International Journal of Control, 6 (1995), pp. 1327-1361. [7] L. R. Hunt and R. Su, Linear equivalents of nonlinear time varying systems, in: Proc. MTNS, Santa Monica, CA, 1981, pp. 119-123. [8] B. Jakubczyk and W. Respondek, On linearization of control systems, Bull. Acad. Polon. Sci. Ser. Math., 28 (1980), pp. 517-522. [9] B. Jakubczyk and W. Respondek, Feedback classification of analytic control systems, in: Analysis of Controlled Dynamical Systems, B. Bonnard et al. (eds.), Birkh¨auser, Boston, 1991, pp. 262–273. [10] T. Kailath, Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980. [11] W. Kang , Extended controller form and invariants of nonlinear control systems with single input, J. of Mathem. System, Estimation and Control, (4) 1994, pp.253-256. [12] W. Kang , Quadratic normal forms of nonlinear control systems with uncontrollable linearization, Proc. 34th CDC, New Orleans, 1995. [13] W. Kang and A.J. Krener, Extended quadratic controller normal form and dynamic feedback linearization of nonlinear systems, SIAM J. Control and Optim, 30 (1992), pp.1319-1337. [14] A.J. Krener, Approximate linearization by state feedback and coordinate change, Systems Control Letters, 5 (1984), pp. 181-185. [15] W. Respondek, Feedback classification of nonlinear control systems in R2 and R3 , in: Geometry of Feddback and Optimal Control, Marcel Dekker, New York, 1998, 347-382. [16] I.A. Tall, Classification des syst`emes de contrˆoles non lin´eaires a` une entr´ee, Th`ese de 3eme cycle, University of Dakar, 1999.
[17] I.A. Tall and W. Respondek, Nonlinear single-input control systems with controllable linearization: normal forms, canonical forms, and invariants, in preparation. [18] I.A. Tall and W. Respondek, Normal forms, canonical forms, and invariants of single single-input control systems under feedback, submitted to 39th CDC, Sydney, 2000.
