Some Comments about Normal Forms under Sampling

Proceedings of the 2006 American Control Conference Minneapolis, Minnesota, USA, June 14-16, 2006

WeC03.1

Some comments about normal forms under sampling S. Monaco ∗

∗

and D. Normand-Cyrot

∗∗

Dipartimento di Informatica e Sistemistica, Universit`a di Roma “La Sapienza”, Via Eudossiana 18, 00184 Rome, Italy. ∗∗ Laboratoire des Signaux et Syst` emes, CNRS-Sup´elec, Plateau de Moulon, 91192 Gif-sur-Yvette, France.

Abstract : The paper is a first attempt to charaterize how continuous-time normal form structures are transformed under sampling. Two aspects are pointed out: generically with respect to the sampling period, the degree of the normal form cannot increase under sampling; under sampling, an homogeneous continuous-time normal can be transformed under coordinates change into a discrete-time homogeneous normal form. The study is set in the context of differential/difference representations of discrete-time dynamics. Keywords : Normal forms, nonlinear sampled systems. I. I NTRODUCTION The theory of normal forms is widely investigated in the recent literature in both the continuous-time and discretetime contexts through specific studies. How normal forms are transformed under sampling is the problem addressed in the present paper. The study is limited to single-input-affine continuous-time dynamics which are linearly controllable (controllable in first approximation). Having in mind that the normal form degree of a given controlled dynamics can be related to its ”residual” complexity (i.e. the degree of the nonlinear terms which cannot be eliminated through coordinates change and feedback), the present study clarifies what is occuring on the equivalent sampled model. Does this complexity increase or not ?, in which sense ? For major simplicity, these aspects are detailed on an example on R3 in section III and then enounced in the general case omitting the proofs. The study confirms the well known fact that complexity increases under sampling, which results in the present case in the existence of a normal form of degree necessarily less or equal to that of the continuous-time dynamics. However, when sampling a system already transformed into its normal form then, the sampled equivalent exhibits, modulo coordinates change only and generically w.r. to the sampling period, a discrete-time normal form of the same degree. Sampling before reduction into normal form increases complexity while sampling after reduction in normal form does not. This is intuitively clear thinking to the procedure of reduction composed with successive coordinates change and feedback and recalling that coordinates change and sampling procedures commute while feedback and sampling procedures do not (see [4],[1],[2]).

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More in details, with respect to the first point, starting from a continuous-time dynamics with quadratic part, whatever be the degree of its normal form, the sampled model generally exhibits a normal form of degree 2. This implies that the degree of the normal form of the sampled dynamics decreases (complexity increases) since it falls down to the minimum order of the given continuous dynamics. Regarding the structures of normal forms under sampling, two contributions are discussed. On one hand, starting from a system admitting a normal form of degree m, the normal form structure of the part of degree < m created under sampling is described; sampling before reduction. On the other hand, starting from a m-th degree homogeneous continuoustime normal form, its sampled equivalent is compared with a m-th degree homogeneous discrete-time normal form. Can this sampled dynamics be referred to as a discrete-time normal form ? More precisely, does it contain resonance terms only or terms which can be simplified under coordinates change and feedback?. The conclusions are that it may contain terms of higher degree than m, but under discretetime coordinates change only, a discrete-time normal form of the same degree can be recovered. Our study is based on ([7],[6],[15]) regarding continous-time normal forms, ([12],[13]) regarding discrete-time normal forms while in ([3],[8],[5]) discrete-time quadratic and cubic normal forms in different formats are proposed. We set the study in the framework of differential/difference representations of discrete-time dynamics ([9],[10]) because such a representation is well defined (without loss of generality) in the sampled context and makes it possible a quite complete study, at least regarding dynamics controllable in first approximation, so providing results, techniques and concepts admitting continuous-time counterparts. To be self contained enough, Section II contains some recalls about the DDR in the sampled context and recent results about normal forms in both continuous-time and discrete-time systems. II. R ECALLS AND PRELIMINARIES Throughout the paper continuous-time input-affine dynamics Σc of the form

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x(t) ˙

= f (x(t)) + u(t)g(x(t))

(1)

with f and g analytic vector fields, are considered. The nonlinear difference equation

takes the form (Aδ , B δ ) ⎛

x(k + 1) = F (x(k), u(k))

⎜ 0 1 δ... ⎜ Aδ = ⎜ .. .. ⎝

δ

1

(2)

denotes its sampled equivalent model in the form of a map; i.e. the state evolutions of (1) and (2) coincide at the sampling instants t = kδ (k ≥ 0 and sampling interval δ), under constant input over [kδ, (k + 1)δ[. As well known, (2) admits the formal exponential representation F δ (x(k), u(k)) = eδ(f +u(k)g) In |x(k) (3) i  L where ef = 1 + i≥1 i!f , indicates the operator Lie series associated with nthe vector∂ field f regarded as the Lie derivative Lf = i=1 fi (x) ∂x , 1 indicates the identity i operator and In the indentity function on Rn .

0

dF δ (.,u) |F −δ (.,u) du

δ Gδ (., u) =

and

e−sadf +ug gds = Gδ1 +

 ui i≥1

0

i!

Gδi+1

(6)

where adf g = [f, g] = Lf ◦Lg − Lg ◦Lf indicates the ususal Lie bracket of vector fields. Integrating (5) between 0 and u(k) for the initial state condition x+ (0) specified by (4), one recovers (2). Combinatoric relations between these two representations are in [14]. Remark. Approximated sampled models of fixed degree in δ are referred to truncation of the series expansions (3) or ((4)-(5)) and finite sampling corresponds to series solutions of finite degree in δ. It is easily verified that the DDR representation associated with a linear dynamics, a pair (A, B), takes the form of a pair (Aδ , B δ ) x+ (0) = Aδ x dx+ (u) = Bδ du δ  with Aδ = eδA ; B δ = esA Bds = δB + i≥1 0

. 0

. ...

1 0

1

δ n−1 (n−1)! δ n−2 (n−2)!

.. . 1

. ...

⎞ ⎟ ⎟ ⎟, ⎠

⎛

δn n!

Bδ = ⎝

⎞

.. ⎠ . . δ

. 0

. ...

1 1

δ i+1 i (i+1)! A B.

(8)

1

Let thus (As , Bs ), be the sampled Brunovsky ⎛ ⎞ 1 δ ... 0 ⎛ .. ⎟ ⎜ ⎜ 0 1 δ . ⎟ As = ⎜ ⎟ , Bs = ⎝ .. .. ⎝ ⎠ 0

. 0

. ...

δ 1

form given by ⎞

0 .. ⎠ . . δ

(10)

The following lemma is instrumental. Lemma 2.1: The coordinates change on Rn defined by

Aδ − 1 

Aδ − 1 n−1 ]T (11) , ..., C δ Tn (δ) = [C δ , C δ δ δ where C δ indicates the last row of the matrix

−1 δ n B δ , (Aδ − I)B δ , ..., (Aδ − I)n−1 B δ transforms (Aδ , B δ ) in (8) into (As , Bs ) in (10). B. Continuous-time normal forms Assuming that Σc in (1) has a linear controllable part in Brunovsky form (7), let us rewrite the dynamics around the linear part as the series expansion   x(t) ˙ = Ax + f [m] (x) + Bu + u g [m−1] (x) m≥2

m≥2

[m]

indicates the homogeneous part of degree m ≥ where (.) 2 of the function into the parentheses. Following the literature, let Σ[m] be the homogeneous polynomial approximation of degree m of Σc around its linear part; i.e. x˙ = Ax + f [m] (x) + Bu + ug [m−1] (x).

Under sampling, the continous-time Brunovsky pair (A, B) ⎛ ⎞ 0 1 ... 0 ⎛ ⎞ 0 .. ⎟ ⎜ ⎜ 0 0 1 . ⎟ . A=⎜ (7) ⎟ , B = ⎝ .. ⎠ .. .. ⎝ ⎠ 0

. 0

0

F δ (x, 0) = eδf In |x ; (F δ )−1 (x, 0) = e−δf In |x = F −δ (x, 0)

with Gδ (., u) :=

...

The procedure for computing the normal forms being strongly dependent from the linear part structure, it is worthy to make reference in the sampled context to what we will refer to as the sampled Brunovsky form with the same structure as (Ad , Bd ) below with normalized sampling period (δ = 1) ⎛ ⎞ 1 1 ... 0 ⎛ ⎞ 0 . ⎟ ⎜ ⎜ 0 1 1 .. ⎟ .. ⎠ . ⎝ Ad = ⎜ (9) ⎟ , Bd = . .. .. ⎝ ⎠

A. Differential/difference representation under sampling For sufficiently small δ ensuring series convergence, (2) (equivalently (3)) is drift invertible and so is F δ (x, u) (with inverse F −δ (x, u)), for u ∈ U, a neighborhood of 0. It follows that (3) admits the equivalent differential/difference representation - DDR  x+ (0) = eδf In x (4) dx+ (u) (5) = Gδ (x+ (u), u) du

δ

(12)

Performing transformations of degree m (coordinates change plus feedback) of the form z u

= x + φ[m] (x) = v+α

[m]

(x) + β

[m−1]

(x)v

(13) (14)

it has been shown ([7],[6],[15]) that, homogeneous normal forms of degree m containing all the terms which cannot be cancelled under homogeneous transformation (the resonance terms), can be computed. Two classes of normal forms have

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been introduced depending if one chooses to fully linearize the drift term or the controlled part. Hereafter, we consider controller normal form with constant control vector field and resonance terms in the drift but the analysis could be performed considering dual normal form with linear drift term and resonance terms in the control vector field. A [m] controller normal form of degree m, Σcnf , is described by (12) with (A, B) as in (7) and g [m−1] (x) = 0,

[m]

[m]

f [m] = (f1 , . . . , fn−2 , 0, 0)T

n [m] [m−2] fp (z) = i=p+2 zi2 fpi (z1 , . . . , zi ); p = (1, ..., n − 2). The computation of normal forms works out solving the homogeneous homological equations of degree m −f [m] (x) −g [m−1] (x)

and for p = (1, ..., n − 2),

[m]

−F [m] (x) = φ[m] (Ad x) − Ad φ[m] (x) + γ0 (x)Bd

F

(x)

i=0 m−1 i

v i=0

Bd +

=

 ui  i≥1

i!

[m−i−1]

x˙ 1 (t) x˙ 2 (t) x˙ 3 (t)

[m]

+

dx (u) du

=

be its approximation

Ad x + F [m] (x)

 ui

m−1

=

Bd +

i=0

(16)

m≥2

with (Ad , Bd ) given by (9) and let Σd of degree m around (Ad , Bd ) x+

(x+ (u))

i!

(17)

[m−i−1]

Gi+1

(x+ (u)).

(18)

Performing transformations of degree m composed with homogeneous coordinates change of the form (13) but generalized homogeneous feedback of the form [m]

u = v + γ [m] (x, v) = v + γ0 (x) +

m  vi i=1

i!

[m−i]

γi

[m−1]

[m−i] Gi

=

= 0,

T [m] [m] F [m] = F1 , ..., Fn−2 , 0, 0 T

[m−i] [m−i] Gi;1 , ..., Gi;n−1 , 0

[m−i−1]

γi+1

(x) = LBd φ[m] (x) +

−1 (A−1 d x − vAd Bd )Bd .

= x2 (t) + 2x2 (t)x23 (t) + ux3 (t) = x3 (t) = u(t) + x2 (t)x3 (t).

Applying the results in ([7],[6],[15]), it can be immediately verified that Σ3c admits a cubic normal form: its quadratic terms only can be simplified under quadratic coordinates change z = x + φ[2] (x) and feedback u = v + α[2] (x) + x2 vβ [1] (x)). Setting (z1 = x1 − 23 , z2 = x2 , z3 = x3 ) together with (u = v − x2 x3 ) yields to (z˙1 (t) = z2 + z2 z32 ; z˙2 (t) = z3 ; z˙3 (t) = v), which does not contain quadratic terms anymore and cannot be further simplified so defining the normal form of degree 3, Σ3cnf , with one resonance term z2 z32 . What does it happen under sampling ?

(x) (19) A. Normal forms under sampling

it has been shown in ([13], [12]) that normal forms of degree m, containing all the resonance terms can be characterized. A controller homogeneous discrete-time normal form of degree [m] m, Σdnf is of the form (17-18) with G1

[m−i−1]

Gi+1

Consider Σ3c on R3 with controllable linear approximation

(15)

Gi+1

i!

i!

III. N ORMAL FORMS UNDER SAMPLING ; AN EXAMPLE

m≥2

dx+ (u) du

 vi

m−1

−

Discrete-time normal forms have been introduced in ([3],[8],[5],[13]) for quadratic and cubic approximations. It has been shown in ([13],[12]) that the study greatly benefits from the differential/difference representations (DDR) setting proposed in ([9],[10]). More precisely, referring to dynamics controllable in first approximation, it has been possible to introduce controller and dual normal forms to define a set of discrete-time homogeneous invariants, to characterize the normal forms in terms of these invariants. Let Σd be the DDR of a single-input discrete-time dynamics controllable in first approximation  [m] + Ad x +

(z1 , . . . , zi ).

Remark. We note the strong analogy between [m] [m−1] = 0) (f [m] , g [m−1] = 0) in Σcnf and (F [m] , G1 [m] in Σdnf but the presence of nonlinearities in the control [m−i] variable associated with the (Gi≥2 )’s which cannot be cancelled even through generalized feedback (19). The computation works out solving the homogeneous dicrete-time homological equations of degree m

C. Discrete-time normal forms

=

[m−2]

zi2 Fpi

i=p+2

= LAx φ[m] (x) − Aφ[m] (x) + α[m] (x)B = LB φ[m] (x) + β [m−1] (x)B.

x

n 

Fp[m] (z) =

Thinking to the procedure for obtaining normal forms as composed with two separate actions: coordinates change and feedback, it is shown hereafter that the capability of simplifying the structure of the equivalent sampled model under coordinates change is maintained while the feedback capability is lost under sampling. [2]

;

i = (2, ..., m)

1) The coordinates change action under sampling: Let Σs [2] be the sampled equivalent to Σc , the quadratic approxima-

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m−1 x + i=2 φ[i] (x), its sampled equivalent Σs is transformed, ¯