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Linearization by Output Injection under Approximate Sampling
European Journal of Control (2009)2:205–217 # 2009 EUCA DOI:10.3166/EJC.15.205–217
Linearization by Output Injection under Approximate Sampling Salvatore Monaco1, and Dorothe´e Normand-Cyrot2 1 2
Dipartimento di Informatica e Sistemistica ‘Antonio Ruberti’ Università La Sapienza via Ariosto 25, 00185 Roma, Italy Laboratoire des Signaux et Systèmes, CNRS-ESE, Plateau de Moulon, 91190, Gif-sur-Yvette, France
Linearization by output injection has been studied in both continuous time and discrete-time contexts. In this paper we discuss the possible preservation of such a property under sampling. It is shown that it can be maintained under approximate sampling up to the order of the system itself. Keywords: Observer forms, nonlinear sampled-data systems, discrete-time systems, nonlinear observer design.
1. Introduction Linearization by coordinates change and output injection has been firstly studied in a continuous-time context in [16], [17], and further extended and developed as a basic tool for observer design (we refer to [13] for survey on the subject). Even less popular, the same problem has been investigated in discrete time pursuing different approaches (see [7, 11, 15, 19, 20, 29]). While techniques and results are similar when dealing with maps or vector fields, that is for continuous-time or discrete-time uncontrolled dynamics, specialized studies are necessary when controlled equations are investigated. In this paper linearization by output injection is revised as the dual problem of linearization under state feedback ([5, 9, 14, 18, 21]). The geometric conditions ensuring linearization through coordinates
Correspondence to: D. Normand-Cyrot, E-mail: cyrot@lss. supelec.fr E-mail: [email protected]
change and feedback or output injection admit similar formulations for both continuous-time and discretetime systems. Such a similarity is possible making use of an alternate representation of controlled discrete-time dynamics as two coupled differential/difference equations rather than a map parameterized by the control. In this context the authors showed how results on structural and control properties admit similar formulations [23]. In this formalism, feedback linearization was studied in [24], starting from the geometric conditions to the computations of structural invariants and controller normal forms, possibly through successive approximations of increasing degree. Following the same lines, normal forms associated with linearization under coordinates change and output injection, have been recently computed making use of successive transformations of increasing degree in [3, 26]. These analogies in the results’ formulation legitimate the question of their preservation under sampling, i. e. when applied to the discrete-time model issued from the sampling of the continuous system: a problem which has been widely investigated in the literature. Different sampling procedures give rise to sampled equivalent models which may exhibit different characteristics w. r. to the preservation of some continuous time properties. For, multirate or higher order holding procedures have been introduced to maintain properties under sampling ([10], [27]). As far as feedback linearization is concerned, the problem has been firstly addressed starting from [1] and then studied in an approximate context in [2] and [25]. In these papers, more strictly linked to the present Received 27 April 2008; Accepted 18 December 2008 Recommended by H. Shim and A. Isidori
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contribution, we showed that feedback linearization can be preserved under sampling up to the system order making use of a new filtered output mapping. Moreover, multirate control strategies or multirate sampling procedures have been proposed to enlarge the degree of approximation or to achieve exact solutions [10]. On these bases, we show in this paper that equivalence through coordinates change to the observer form can be preserved under sampling up to the system order, say n, the result is constructive for the coordinates change. It relies on the existence of a fictitious vector field with respect to which the given output mapping has relative degree equal to n. It is immediately understood that this result is the dual of the one in [2] where feedback linearization is achieved through the existence of a fictitious output mapping which has relative degree equal to n. The effects of sampling over the observer design have been investigated in [8] and [28]; in particular in [8], the authors proved that for systems over R2, linearization by output injection under sampling is in fact equivalent to linear equivalence. The present work shows that preservation under time-sampling is in fact possible over Rn when considering the approximate sampled model at the order n in the sampling time . This result suggests to use the approximate model of order n for designing the sampled-data observer since increasing the accuracy of the sampled model by considering higher order terms would only bring to additional computational complexity even if linearization by output injection could be assured. The present work is based on [6] regarding geometric necessary and sufficient conditions for equivalence to the linear observer form through coordinates change and to [25] where a complete description of the sampled equivalent to an input-affine system is given. This paper represents an extended version of [27]. The paper is organized as follows. Section 2 sets the problem and recalls the geometric conditions ensuring linearization through coordinates change and output injection in the continuous-time and discrete-time contexts. Linearization by output injection under sampling is set in the differential/difference context in section 3 where the conditions of section 2 are specified on the sampled equivalent model. In section 4 the main result is given and a constructive procedure for computing the coordinates change which solves the problem is detailed for unforced and forced dynamics with extra conditions on the output injection. The algorithm is worked out on an example: an elementary application in state estimation of the duffing oscillator. Some conclusions end the paper.
S. Monaco and D. Normand-Cyrot
Some standard notations : Given , a vector field over Rn, L denotes the associated formal Lie derivative which acts over real valued functions h : Rn ! R as a first order differential operator L h :¼ dh; the Lie bracket ad 1 2 : ¼ [ 1, 2] between two vector fields over Rn is described by the non commuting product L½1 ;2 ¼ L1 L2 L2 L1 ; the Lie series associated with L is defined P Li by its exponential expansion eL :¼ 1 þ i1 i! , usually denoted as e , which gives e hðxÞ ¼ hðe xÞ; given a diffeomorphism on Rn, of along satisfying Ad denotes the transport 1 . All maps and Ad :¼ ð½Jx Þ1 ¼ d dx vector fields, possibly parameterized, are assumed analytic. The manipulations performed over the asymptotic series expansions are formal ones, no convergence studies are performed.
2. Linearization by Output Injection 2.1. In Continuous Time Let c denote a SISO input-affine system _ ¼ fðxðtÞÞ þ gðxðtÞÞuðtÞ xðtÞ yðtÞ ¼ hðxðtÞÞ
ð1Þ
where x 2 Rn, y 2 R, f and g are analytic vector fields and h an analytic mapping. Assuming, without any loss of generality, that (0, 0) is an equilibrium pair, f(0) ¼ 0, h(0) ¼ 0, linearization by output injection stands for the equivalence under coordinates change to the observer canonical form COF _ ¼ AO zðtÞ þ ðyðtÞ; uðtÞÞ zðtÞ yðtÞ ¼ CO zðtÞ
ð2Þ
with (AO, CO) in the Brunovsky observability form 0 1 0 0 B .. C B 1 ... .C B C B .. .. .. C AO ¼ B . . .C B0 C; B. . C B. . . . . . . . . ... C @. A 0 0 1 0 CO ¼ ð 0
0 1Þ
and ðy; uÞ : R R!Rn , an analytic map characterizing the input-output injection: i.e. the injection on the dynamics which should render linear the dynamics itself.
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Observer Form Under Sampling
The problem, referred to as linearization by output injection as opposed to feedback linearization, is of interest in the design of nonlinear observers. It results from (2) that
an alternate representation as two differential/ difference equations - DDR xþ ¼ F0 ðxÞ
ð5Þ
þ
dx ðuÞ ¼ Gðxþ ðuÞ; uÞ du
z^_ ðtÞ ¼ AO z^ðtÞ þ ðyðtÞ; uðtÞÞ þ Kðy CO z^Þ
with
xþ ð0Þ ¼ xþ ð6Þ
estimates the state with a linear error dynamics: _ ¼ ðAO KCO ÞeðtÞ: eðtÞ Conditions ensuring local equivalence of c to (2) are well known. Theorem 2.1: [16, 17] c is locally equivalent to the observer canonical form (2) if and only if n1 Ac1 : dh; dL h . . . ; dL h ð0Þ ¼ n; f f Ac2 : rci ; rcj ¼ 0 for i þ j 2 ½2; 2n 1; Ac3 : ½g; rci ¼ 0
for
i 2 ½1; n 1;
where rc1 is the vector field solution of 0 1 0 1 dh 0 .. .. C B C B . B C .C dHrc1 :¼ B C rc1 ¼ B @0A @ dLf h A h dLn1 1 f i1
...
@Fðx; uÞ : @u
:¼
ð7Þ
Indeed, the integration of (6) with respect to u between 0 and uk with initial condition specified by (5), xþ ð0Þ ¼ F0 ðxk Þ :¼ Fðxk ; 0Þ, returns xkþ1 in (4), i.e. þ
Z
uk
x ðuk Þ ¼ x ð0Þþ
Gðxþ ðÞ; Þd ¼ Fðxk ; uk Þ:
0
ð3Þ
adfi1 rc1
rcn Þ ¼ Idn
In the same way w.r. to an output mapping y ¼ hðxÞ, one has þ
Z
ykþ1 ¼ hðx ð0ÞÞ þ
uk
0
LGð:; Þ hðxþ ðÞÞd
¼ hðxþ ðuk ÞÞ ¼ hðFðxk ; uk ÞÞ: The Taylor expansion of Gðx; uÞ in powers of u
with Idn the n n-identity matrix. Remark: Applying the Jacobian identity, condition Ac2 is recognized to be equivalent to [rc1, rci] ¼ 0 for i 2 [2, 2n 2]. Remark: In the uncontrolled case (g ¼ 0) in (1), under the stronger condition A0c2 : ½rci ; rcj ¼ 0 for i þ j 2 ½2; 2n þ 1 (equivalently ½rc1 ; rci ¼ 0 for i 2 ½2; 2n), one gets linear equivalence under linear output injection. Remark: The ‘‘fictitious’’ controlled dynamics _ ¼ fðxÞþ urc1 ðxÞ with output mapping y ¼ h(x) xðtÞ exhibits a relative degree equal to n, ðLrc1 h ¼ 0; Lrc2 h ¼ 0; . . . ; Lrcn1 h ¼ 0Þ with moreover Lrcn h ¼ 1. 2.2. In Discrete Time A discrete-time dynamics is usually represented as a map xk ! xkþ1 ¼ Fðxk ; uk Þ
GðFðx; uÞ; uÞ
þ
for i 2 ½2; n: and rci :¼ adf rci1 ¼ ð1Þ Moreover, the coordinates change (x) satisfies ½ Jx ðrc1
where xþ ðuÞ indicates a curve in Rn parameterized by u. (4) and (5–6) describe the same discrete-time dynamics provided that G(x,u) satisfies
ð4Þ
where x 2 Rn ; u 2 R and Fð:; uÞ is an analytic map, analytically parameterized by u. In [22], we proposed
Gðx; uÞ ¼ G0 ðxÞ þ
X ui i1
i!
Gi ðxÞ
ð8Þ
gets the analytic vector fields Gi ðxÞ, i 0, which play a fundamental role in the geometric characterization of the properties under study. Let d denote the DDR (5-6) with output map y ¼ hðxÞ, and define the canonical observer differential/difference representation (CO-DDR) as the DDR of the COF (2). Denoting by i the i – th component of , and assuming, without loss of generality, that @ 1 ðy; uÞ 6¼ 0, which can be achieved through a ð0;0Þ @y possible preliminary linear output injection. One has Proposition 2.1: The discrete-time observer canonical form zkþ1 ¼ AO zk þ ðyk ; uk Þ y ¼ CO z
ð9Þ
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Remark: The fictitious controlled dynamics
admits the DDR z
þ
¼ AO z þ ðyÞ
dzþ ðuÞ ¼ Gðzþ 1 ðuÞ; uÞ; du y ¼ CO z
ð10Þ zþ ð0Þ ¼ zþ
ð11Þ
with (AO, CO) in the Brunovsky form, ðyÞ ¼ ðy; 0Þ and @ ð:; uÞ j 1 ðzþ ðuÞ;uÞ Gðzþ 1 ðuÞ; uÞ ¼ 1 1 @u 1 where 1 ð:; uÞ indicates the inverse of 1 ð:; uÞ, the first component of ð:; uÞ. Proof: The result is an immediate consequence of the definition of the differential equation and the particular form of the matrix AO so getting with zþ 1 ð0Þ ¼ 1 ðyÞ Z u Gðzþ ðy; uÞ ¼ ðyÞ þ 1 ðvÞ; vÞdv: 0 3 Remark: The input-output injection term (11) takes the form ðy; uÞ ¼
@ ðy; uÞ ¼ Gðzþ 1 ðuÞ; uÞ ¼ Gð @u
1 ðy; uÞ; uÞ:
Theorem 2.2: [6] d is locally equivalent to the discrete-time observer canonical form if and only if Ad1 : dh; dðh F0 Þ; ; dðh Fn1 ð0Þ ¼ n; 0 i 2 ½2; n Ad2 : ½rd1 ; rdi ¼ 0; p 0; Ad3 : ½Gp ; rdi ¼ 0 for i 2 ½2; n; where the vector field rd1 satisfies 0 1 0 1 dh 0 .. .. C B B C . Crd1 ¼ B . C dHd rd1 :¼ B @0A @ dh Fn2 A 0 n1 dh F0 1
ð12Þ
and for i 2 ½2; n, rdi ’s are given by rdi ¼ AdF0 rdi1 ¼ AdFi1 rd1 :
xþ ð0Þ ¼ F0 ðxÞ dxþ ðuÞ ¼ rd1 ðxþ ðuÞÞ du with output mapping y ¼ hðxÞ has relative degree equal to n; i.e. using the DDR formalism ðLrd1 h ¼ 0; Lrd2 h ¼ 6 0. 0; . . . ; Lrdn1 h ¼ 0Þ with moreover Lrdn h ¼ 1 ¼
3. Linearization by Output Injection Under Sampling Assuming the control variable u(t) constant over small time intervals of amplitude , with value uk over ½k; ðk þ 1Þ½ for k 0, the sampled equivalent system, s , is the discrete-time system which reproduces the state and output evolutions of c at the sampling instants for any x0 ¼ xðt ¼ 0Þ. As well known the discrete-time dynamics is drift invertible for small , i.e. F0 ðxÞ ¼ ef ðxÞ and ðF0 Þ1 ðxÞ ¼ ef ðxÞ. As a consequence G ðx; uÞ satisfying (7) exists and it is uniquely @F ðx; uÞ defined by G ðx; uÞ :¼ with @u x¼F ðx;uÞ fþug ðxÞ. It is a matter of computations to F ðx; uÞ ¼ e verify that the DDR xþ ¼ F0 ðxÞ dx ðuÞ ¼ G ðxþ ðuÞ; uÞ du y ¼ hðxÞ
½Jx d ð rd1 . . . rdn Þ ¼ Idn : Condition Ad1 is the observability rank condition which ensures the existence and uniqueness of rd1 satisfying (12); Ad2 requires first order nilpotency of the distribution generated by the vector fields ðrd1 ; ; rdn Þ and guarantees the existence of a coordinates change as well as the specific structure of (10); Ad3 guarantees the specific structure (11). It is worthy to note the strict analogy between statements of Theorem 2.1 and Theorem 2.2.
with
xþ ð0Þ ¼ xþ ð14Þ ð15Þ
describes the sampled equivalent s of c. Its inputstate or output-state evolutions match, at the sampling instants t ¼ k those of c. Combinatoric relations between the continuous-time model and its sampled equivalent are detailed in [25], we just recall what is necessary in the present context. First, G ð:; uÞ is described by its asymptotic expansion in Z
G ð:; uÞ ¼
0
Moreover the coordinates change, d , satisfies
ð13Þ
þ
esðadf þuadg Þ gds ¼ G0 þ
i1
0
as well as each Gi ¼ for the first ones G0
Z ¼
P
p piþ1 p! Gi;p
esadf gds :¼
p1 X ð1Þp1 pþ1 X p2
ðp þ 1Þ!
i!
Gi ð16Þ
for i 0 so getting
X ð1Þp1 p p1
0
G1 ¼
X ui
p!
adp1 g f
ð17Þ
Ckp ½adkf g;adfpk1 g ð18Þ
k¼0
p! denotes the binomial. where Ckp :¼ k!ðpkÞ!
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Observer Form Under Sampling
We note that the term in p in G0 described by (17) is g which contains the vector field the Lie bracket adp1 f g one-time and the vector field f, (p 1)-times. G1 in (18) involves Lie brackets of the first order in the adjf g of the form ½adkf g; adfpk1 g in pþ1 for p 1. Such a term contains g, two-times and f, (p 1)-times. More in general, Gi can be iteratively deduced according to a combinatoric rule; it involves Lie brackets of order i in the adjf g and it contains g, (i þ 1)-times. The transport of any Gi along F0 is denoted by AdF Gi and its 0 transport along F0 , q-times is defined as AdFq Gi 0 because F0 . . . F0 ¼ Fq 0 . Theorem 3.1: [27] s is locally equivalent to the discrete-time observer canonical form (10-11) if and only if there exists T > 0 such that for any 2 ½0; T, the conditions below hold true As1 : ðdh; def h; ; deðn1Þf hÞð0Þ ¼ n; As2 : ½rs1 ; rsi ¼ 0 for i 2 ½2; n; As3 : ½Gp ; rsi ¼ 0 for i 2 ½2; n; p 0; where the vector field rs1 satisfies: 0 0 1 1 dh 0 . . B B .. C C .. B C C dH rs1 :¼ B @ deðn2Þf h Ars1 ¼ @ 0 A n1 deðn1Þf h
ð19Þ
and rsi is the transport of rsi1 along F0 , rsi :¼ eadf rsi1 ¼ eði1Þadf rs1 for i 2 ½2; n. Moreover, the coordinates change satisfies ½Jx ð rs1
...
rsn Þ ¼ n1 Idn :
When it is referred to a sampled model, the observer canonical form maintains the discrete-time structure (9-10-11) with -dependent output injection and vector field G . Assuming c locally equivalent to the observer canonical form (2), the present paper discusses the possible preservation of such a property under approximated sampling. The loss of linearization by output injection under sampling, as noted in [8], is the natural counterpart of what occurs when investigating the dual problem of feedback linearization under sampling. As a matter of fact, preservation of linearization by feedback under sampling should imply linear equivalence as conjectured in [1] and proved for n ¼ 2. However, some approximate result can be proven. As proposed in [2], feedback linearization can be maintained till an approximation order with respect to the sampling period equal to the state dimension. The same idea is here developed with respect to linearization by output injection. Our result
stands in proving that the conditions above can be maintained in an approximate meaning. More precisely, these conditions, reformulated as equalities between asymptotic expansions in , hold true up to a certain degree of approximation. With this in mind the following definition is mandatory. Definition 3.1: Assuming c linearizable by outputinjection, the property is maintained under sampling if there exists T > 0 such that for any 2 ½0; T the sampled equivalent model satisfies the property. Approximate solution at order p stands for the existence of a solution on the approximate sampled model at order p in ðerror in Oðpþ1 ÞÞ. The problem has been approached in [27] where we showed that linearization by output injection could be maintained under sampling thanks to the computation of a ‘‘fictitious’’ -dependent vector field with respect to which the relative degree is maintained equal to n at order n 1 (i.e. up to an error in Oðn Þ). As a matter of fact, a stronger result can be proven. Theorem 3.2: For c, locally linear equivalent by output injection, there exists T > 0 such that for any 2 ½0; T a vector field rs1 can be computed to satisfy (19) at any degree of approximation in . Proof: The proof works out showing first that dH ¼ HðÞdH þ Oðn Þ
ð20Þ
with HðÞ 2 Rnn , invertible by construction for any 20; T 0 1 1 0 0 0 . .. B C B1 C . 2 =2 . . B C . B C . 2 HðÞ ¼ B 1 C: . 2 2 B. C .. .. .. .. B. C . .2 2 . . @. A n1 n1 ðn1Þ ðn1Þ 1 ðn 1Þ 2 ðn1Þ! Then, given r1 satisfying (3), we set in (19) 0 1 10 B 20 C n1 B C X B C .. rs1 ¼ i1 i0 rci ¼ ðrc1 . . . rcn ÞB C: . B C i¼1 @ n2 n10 A 0 Since rs1 must solve (19) in Oðn Þ, due to (20), we have 0 1 0 1 10 0 B .. C @ ... A ¼ ðM1 Þ ð21Þ @ . A ¼ M1 0 0 lc 1 n10
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S. Monaco and D. Normand-Cyrot
ðn1Þðn1Þ where ðM1 , 0 Þlc is the last column of M0 2 R invertible by construction; i.e.
1 0 10 1 ... 1 B C .. C .. CB . C B 1 . AB n2 C @ n10 A ... 0 0
1 0 00 1 0 C B 10 C HðÞ B . B B CC B . B M0 B .. C C ¼ n1 @ . @ . AA @ 1 n10 0
so getting 0 B B M0 :¼ B B @
1 ðn1Þ! n1 2 ðn1Þ!
1 ðn2Þ! n2 2 ðn2Þ!
ðn1Þ ðn1Þ!
ðn1Þ ðn2Þ!
.. . n1
.. . n2
Setting now in (19), rs1 ¼
...
1 2
...
4 .. .
... ...
Pn1 i¼1
ðn1Þ2 2
1
1
2 C C C .. C: . A n1
i1 ði0 þ i1 Þrci , i. e.
0 rs1 ¼ ð rc1
...
10 þ 11 20 þ 2 21 .. .
1
B C B C B C rcn ÞB C B n2 C @ n10 þ n1 n11 A 0
and solving equality (19) in Oðnþ1 Þ, 0 B B 10 n B B HðÞdH:rs1 þ n! B B @
0 1 n
2 .. .
ðn 1Þ
1
0 1 0 C B .. C C B C C n CdL h:rc1 ¼ B . C B C C f @ 0 A C A n1 n
leads to 0 1 1 1 11 ðxÞ n C 2 dn ðxÞ 1 B B C .. C .. M0 B @ A¼ . @ A n! . n n11 ðxÞ ðn þ 1Þ 0
with dn ðxÞ :¼ dLnf hðxÞ:rc1 and 10 ¼ 1 computed from P (21). Iterating the procedure and setting 2 i1 ði0 þ i1 þ 2! i2 þ . . .Þrci , a solution rs1 ¼ n1 i¼1 to (19) up to increasing approximation order in
exists thanks to the invertibility of M0 . The proof is constructive for the successive ip which are functions of x for p 1. 3 Starting from the so computed vector field rs1 , truncated at a certain approximation in , special assumptions on the output injection must be taken into account to satisfy conditions As2 , As3 in Theorem 3.1 so maintaining the observer canonical structure under approximate sampling. This is the case in [27] where the output injection is assumed piecewise constant. A more general result is proposed in the sequel where we describe a procedure for computing a -dependent coordinates transformation under which the approximate sampled model exhibits the sampled observer canonical form - sampled COF.
4. Approximate Sampled COF, an Algorithm We first describe an algorithm for computing a coordinates transformation under which the COF structure is recovered under n-th order approximate sampling; i.e. starting from a continuous-time COF, an approximate sampled COF is described. The algorithm is proposed below for unforced systems c ð0Þ, and then extended to forced ones c ðuÞ under some extra conditions on the output injection. 4.1. The Algorithm for c ð0Þ Starting from the continuous-time COF over Rn x_ 1 ¼ 1 ðxn Þ x_ 2 ¼ x1 þ 2 ðxn Þ x_ n ¼ xn1 þ n ðxn Þ y ¼ xn
ð22Þ
we look for a coordinates change z ¼ T ðxÞ such that in the new coordinates, the sampled equivalent model, truncated at order n in exhibits a sampled COF. The proof is constructive for T . Initialization - set zn :¼ xn ¼ y. Step 1 – compute the evolution of y, truncated at order n in (n-th order approximation) yðk þ 1Þ ¼ y þ y_ þ
2 n ðnÞ y€ þ þ y 2 n!
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Observer Form Under Sampling
and put in evidence in the r.h.s. the terms which depend on y only, denoted by n ðy; Þ, from the terms which depend on the state variables xi ; i 2 ½1; n 1 denoted by n ðx; Þ. One gets yðk þ 1Þ ¼ zn ðk þ 1Þ :¼ n ðzn ; Þ þ n ðx; Þ ð23Þ with
Step (i þ 1) – compute the n-th order approximate evolution of zni zni ðk þ 1Þ ¼
p¼0
þ
n n;n ðxÞ n! and by construction n;1 ðxÞ ¼ xn1 . Set
þ ð24Þ
zn ðk þ 1Þ ¼ zn1 þ C1n1 zn þ n ðzn ; Þ n ðyÞ
Step 2 – compute the n-th order approximate evolution of zn1 . From (23 – 24) and dropping the (x, )-dependency for simplicity, one gets n1 ðn1Þ zn1 ðk þ 1Þ ¼ n þ _n þ þ ðn 1Þ! n þ ð1Þ1 C1n1 ðn ðzn ; Þ þ n Þ ¼ ð1
þ n1 þ n1 ðzn ; Þ
ð25Þ
n n1;n ðn 1Þ!
ð1Þp Cpni ni1 þ ni ðx; Þ þ ni ðzn ; Þ
where ni regroups terms in the r.h.s. which depend on zn only while n and n1 ; . . . ; ni regroups terms which do not depend on zn only. One gets n ni; n ðn iÞ!
with ni; iþ1 ¼ xni1 . Set zni1 ¼
i X
ð1Þp Cpn1 n þ
p¼0
þ
1 X
i1 X ð1Þp Cpn2 n1 þ... p¼0
ð1Þp Cpni ni1 þni þð1Þi Cin1 zn
p¼0
and verify that the zni -dynamics takes the form ni ðyÞ
so describing the output-injection term ...
ni ðyÞ.
Step (n – 1) – compute the approximate z2 evolution at order n z2 ðkþ1Þ¼ð1þð1Þ1 C1n1 þ...þð1Þn2 Cn2 n1 Þn þð1þð1Þ1 C1n2 þ...þð1Þn3 Cn3 n2 Þn1 þ...þð1þð1Þ1 C12 Þ3 ðx;Þþ2 þ2 ðzn ;Þ
with n1;2 ¼ xn2 . Set
¼ð1Þn2 n þð1Þðn3Þ n1 þ...þ2 þ2 ðzn ;Þ
zn2 ¼ ð1 C1n1 Þn þ n1 þ C2n1 zn
ð26Þ
so that, from (25–26), the zn1 -dynamics takes the form zn1 ðk þ 1Þ :¼ zn2 C2n1 zn þ n1 ðzn ; Þ ¼ zn2 þ
p¼0
zni ðk þ 1Þ ¼ zni1 ðkÞ þ
where n1 ðzn ; Þ contains the terms in the r.h.s. which depend onzn only; n1 ðx; Þ is a new term containing 2 all n1the terms in _n ðx; Þ þ 2! €n ðx; Þ þ þ ðn1Þ ðx; Þ which do not depend on zn only, ðn 1Þ! n so getting n1 ðx; Þ ¼ 2 n1;2 þ . . . þ
1 X
i2 X ð1Þp Cpn3 n2 þ
ni ðx; Þ ¼ iþ1 ni; iþ1 þ . . . þ
so recovering the desired structure over the n-th equation where n ðyÞ :¼ C1n1 y þ n ðy; Þ specifies the n-th component of the output injection.
C1n1 Þn
ð1Þp Cpn2 n1 þ
p¼0
n! . By construction, the zn -dynamics with Cpn :¼ p!ðnpÞ! takes the form
¼ zn1 þ
i1 X p¼0
n ðx; Þ ¼ n;1 ðxÞ þ . . . þ
zn1 :¼ n ðx; Þ þ ð1Þ1 C1n1 zn
i X ð1Þp Cpn1 n
n1 ðyÞ
so recovering the desired structure with output injection term, n1 ðyÞ :¼ C2n1 y þ n1 ðy; Þ. . . . Iterate the steps so characterizing the (i þ 1)-th step as follows.
where 2 ðzn ; Þ regroups terms in the r.h.s. which depend on zn only and 2 ðx; Þ those which do not depend on zn only so getting 2 ¼ n1 2; n1 þ . . . þ
n 2; n 2!
with 2;n1 ¼ x1 . Set z1 ¼ ð1Þn2 n þ ð1Þðn3Þ n1 þ . . . 3 þ 2 þ ð1Þn1 Cn1 n1 zn
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S. Monaco and D. Normand-Cyrot
Example: Let us illustrate the computation over R2. Consider the continuous-time uncontrolled COF
so computing the z1 -dynamics n1 ðn1Þ Þ z1 ðk þ 1Þ ¼ ð1Þn2 ðn þ _n þ ... þ ðn 1Þ! n n2 ðn2Þ Þ þ ð1Þðn3Þ ðn1 þ _n1 þ ... þ ðn 2Þ! n1 2 þ ... ð3 þ _3 þ €3 Þ þ ð2 þ _2 Þ 2! þ ð1Þn1 ðn ðzn Þ þ n ðx;ÞÞ ¼ 1 ðzn Þ þ _2
x_ 1 ¼ 1 ðx2 Þ x_ 2 ¼ x1 þ 2 ðx2 Þ y ¼ x2 and its approximate sampled equivalent at order 2 in x1 ðk þ 1Þ ¼ x1 þ
1þ
2 2!
x2 ðk þ 1Þ ¼ x2 þ ðx1 þ rewritten as z1 ðk þ 1Þ ¼ 1 ðyÞ with 1 ðyÞ :¼ 1 ðzn Þ þ n n _ _2 because Pn 2 ¼ p x_p1 ¼ 1 ðyÞ is a function of y only and p¼0 ð1Þ Cn ¼ 0.
4.2. Approximate Sampled COF for Unforced Dynamics The result of the previous algorithm is summarized below.
ð27Þ
1 ðx1 þ 2Þ þ
ð 2! 2
2Þ 1þ
0
2 ðx1 þ
2 ÞÞ
y ¼ x2 which clearly does not preserve the observer structure; 0 00 ð:Þ ; ð:Þ ; . . . indicate the successive derivatives w.r.t. its arguments of the function into the parentheses. According to the algorithm, one sets z2 ¼ x2 and puts in evidence in z2 ðk þ 1Þ ¼ yðk þ 1Þ the parts which depend on z2 only from the remaining ones, so getting z2 ðk þ 1Þ ¼ 2 ðz2 Þ þ 2 ðx; Þ ¼ x2 þ
Theorem 4.1: Linearization by output injection of c ð0Þ is maintained under approximate sampling at order n through the coordinates transformation z ¼ ðxÞ ¼ T ððxÞÞ
0
þ x1 þ
2
2 2!
þ
2 ð 2!
1
0
þ
2 2Þ
0
2 x1 :
Setting z1 ¼ 2 ðx; Þ z2 , one recovers 2 ðyÞ
z2 ðk þ 1Þ ¼ z1 ðkÞ þ z2 ðkÞ þ 2 ðz2 ðkÞÞ ¼ z1 þ where denotes the coordinates change which transforms c ð0Þ into its COF (22) and T is the coordinates change computed through the algorithm. Proof: Taking in mind that the algorithm has been worked out on the approximate sampled equivalent, Sððc0 ÞÞ of the continuous-time COF, ðc0 Þ and that the procedures of applying a coordinates transformation and sampling commute S ¼ S, we can deduce from the composition (27). More precisely, setting ¼ T , one has S ¼ T S ¼ T S so proving that transforms directly the approximate sampled equivalent model Sðc ð0ÞÞ of c ð0Þ into the sampled canonical normal form issued from the algorithm, T ðSððc0 ÞÞ provided homogeneous approximations in up to order n are performed. 3 Remark: In [8], it was conjectured (proven for n ¼ 2) that preservation under sampling of linear equivalence through output-injection should imply linear equivalence of c0 . Our result shows that preservation holds true for any n but up to order n in .
so defining z1 -dynamics
2
as
2
z1 ðk þ 1Þ ¼ ðx1 þ ¼ x2
þ 2! ð 2
:¼ z2 þ
2
1Þ þ þ 2þ
2 2!
2 2!
1
0
þ
2 2 Þ.
The
0
2 x1 z2 ðk þ 1Þ
1
2 2!
0
2 2
:¼ 1 ðyÞ
depends on z2 ¼ x2 ¼ y only as 2 depends on x2 only. In conclusion, the coordinates change, z ¼ T ðxÞ described by 2 0 x1 x2 ; z2 ¼ x2 ð28Þ 2! 2 transforms the approximate sampled equivalent to the COF into the 2nd order sampled COF over R2 z1 ¼ x1 þ
z1 ðk þ 1Þ ¼ z2
2
þ
z2 ðk þ 1Þ ¼ z1 þ 2z2 þ y ¼ z2 :
2 ð 2! 2
þ
1
0
2 ð 2!
2
1
þ
2Þ 0
2 2Þ
213
Observer Form Under Sampling
To complete the discussion, it is a matter of computations to verify that the Jacobian of T satisfies the condition set in Theorem 2.3; i.e. ½Jx T r1
r2 ¼ Id2 þ Oð3 Þ
where the vector field r1 is computed to satisfy (19) up to an error in Oð3 Þ and r2 :¼ eadf r1 so getting up to 2 r1 ¼ rc1 d2 ðxÞrc1 ; r2 ¼ rc1 d2 ðxÞrc1 þ rc2 2 2 with d2 ðxÞ ¼ Lrc3 hðxÞ. In the present case, because 0 rc1 ¼ ð1; 0ÞT ; rc2 ¼ ð1; 0ÞT and d2 ðxÞ ¼ 2 , one gets in Oð3 Þ
þ 2! 2
0
0
2
1 1
1 2 0
0
2
1 2
0
2
0 ¼ : 0
The specific structure of the n-th order sampled COF over Rn can be described in terms of the continuoustime COF according to the same procedure.
Remark: An interesting case is represented by controlled system c ðuÞ having relative degree n, h 6¼ 0). (Lg Lkf h ¼ 0; k ¼ 0; n 2; Lg Ln1 f Example: Let us illustrate the procedure on the controlled COF over R3 with input-output injection satisfying (29) x_ 1 ¼ 1 ðx3 ; uÞ x_ 2 ¼ x1 þ 2 ðx3 Þ x_ 3 ¼ x2 þ 3 ðx3 Þ y ¼ x3 with approximate sampled equivalent at order 3 in x1 ðk þ 1Þ ¼ x1 þ
1 ðuÞ
þ
@ 2 i ðy; uÞ ¼0 @y@u
for
i 2 ½2; n
ð29Þ
which is equivalent to assume that the derivatives 0 0 i ðyÞ do not depend on u except 1 ðy; uÞ. The following result holds true. Theorem 4.2: Linearization by output injection of c ðuÞ is maintained under approximate sampling at order n through the coordinates transformation (27) if the conditions (29) are satisfied. Proof: It is readily understood that under (29), the output injection in (2) takes the form ðy; uÞ ¼ ð
1 ðy; uÞ; b2 u
þ
2 ðyÞ; ; bn u
þ
n ðyÞÞ
T
:
Linearity in u of the input-output injection i 2 ðy; uÞ guarantees that the algorithm proposed above works out identically yielding to a coordinates change which does not depend on the control variable which is assumed constant over the sampling intervals. Relaxing (29) yields to a coordinates change which depends on u. 3
1 ðuÞðx2
þ
3Þ
3 0 ð ðuÞðx1 þ 2 Þ 3! 1 00 þ 1 ðuÞðx2 þ 3 Þ2
þ
þ
The result stated can be extended to controlled systems c ðuÞ admitting a COF (2) with output injection satisfying the condition
0
þ
0
0
1 ðuÞ 3 ðx2
þ
x2 ðk þ 1ÞÞ ¼ x2 þ ðx1 þ 4.3. Approximate Sampled COF for Controlled Dynamics
2 2!
þ
0
2 ðx2 3
ð 3! 0
þ
3 ÞÞ 2Þ
þ
2 ð 2!
3Þ
þ
1 ðuÞ
3 ÞÞ
0
1 ðuÞðx2
þ
0
0
2 ðx1
00
þ 2 3 ðx2 þ 3 Þ þ x3 ðk þ 1Þ ¼ x3 þ ðx2 þ 3 Þ
2 ðx2
þ
þ 3Þ
2Þ 2
Þ
2 0 ðx1 þ 2 þ 3 ðx2 þ 3 ÞÞ 2! 3 0 þ ð 1 ðuÞ þ 2 ðx2 þ 3 Þ 3! 00 þ 3 ðx2 þ 3 Þ2 þ
0
þ 3 ðx1 þ y ¼ x3
2Þ
where the u-dependency is in one deduces from z3 ðk þ 1Þ 3 ðz3 ; ; uÞ :¼ z3 þ þ þ
3 ð 3! 0
3 2
1
3
þ
1 ðuÞ
þ
þ
3 ðx; Þ :¼ x2 þ
þ
0
2 3 ðx2
þ
3 ÞÞ
only. Setting z3 :¼ x3 , 2 ð 2! 0
2 3
2
0
þ
3 3Þ 00
þ
2 3 3
0
2 3 3Þ
2 ðx1 þ 2!
0
3 x2 Þ
3 0 00 ð x2 þ 3 x22 3! 2 0 00 þ 3 x1 þ 2 3 3 x2 þ
þ
0
2 3 x2 Þ:
214
S. Monaco and D. Normand-Cyrot
In conclusion, the coordinates change, z ¼ T ðxÞ
Setting z2 :¼ 3 ðx; Þ 2z3 , one computes z3 ðk þ 1Þ ¼ z2 þ 2z3 þ 3 ðz3 ; Þ
z1 ¼ x2 þ
3 ðz3 ; uÞ:
¼ z2 þ
þ _3 ðz3 ; Þ þ
2 € 3 ðz3 ; Þ 2!
z2 ¼ x2 þ
2 ðx1 þ 2!
3 0 ð x2 þ 3! 2 00 þ 2 3 3 x2 þ z3 ¼ x3 ¼ y þ
with ð 1 ðuÞ 2 00 0 þ 3 x2 ðx2 þ 3 Þ þ 3 ðx1 þ
2 € 3 ¼ 3 2ð 2!
2Þ
1 ðuÞ
3
þ
þ
0
2 ðx2
þ
2 ÞÞ
3 ÞÞ
z1 ðk þ 1Þ ¼ z2 ðk þ 1Þ ¼
z2 ðk þ 1Þ ¼ 2 ðz3 ; ; uðkÞÞ
with the
with
0
2
3
þ
0
3 3
2
þ
3 ð2 2 00
3 3 x2
2 3 x2 Þ
3 x1
2x3
1 ðz3 ðkÞ; uðkÞÞ z1 ðkÞ þ 2 ðz3 ðkÞ; uðkÞÞ
i ðz3 ; uÞ
3 ðz3 ðkÞ; uðkÞÞ
described above for i ¼ [1,3].
with 1 ¼ x2 x32 and 2 ¼ 0. Under approximate sampling at order 2, the COF structure is lost
2 ðz3 ; uÞ
x1 ðk þ 1Þ ¼ x1 þ ðx2 x32 Þ
:¼ z3 þ 2 ðz3 ; ; uÞ and also
2 ðx1 3x22 x1 Þ 2 2 x2 ðk þ 1Þ ¼ x2 þ x1 þ ðx2 x32 Þ 2 y ¼ x2 : þ
z1 ðk þ 1Þ ¼ 3 þ 3 3 _3 2 € 3 þ 2 þ _2 2! 2 0 ¼ z3 þ 3 þ ð 2 þ 2 3 0 þ ð 1 ðuÞ 2 2 3 2 3! :¼ 1 ðz3 ; uÞ:
0
0
x_ 1 ¼ x2 x32 x_ 2 ¼ x1 y ¼ x2
z2 ðk þ 1Þ ¼ z1 z3 þ 2 ðz3 ; ; uÞ
2 ðz3 ; uÞ
þ
A very simple example is the duffing oscillator which admits the COF
Setting z1 :¼ 3 þ 2 þ z3 , one has
¼ z1 þ
3 x2 Þ
4.4. The Duffing Oscillator
1 ðuÞ
2Þ
00 ð x2 þ 2 3 2 0 0 þ 2 x2 þ 3 x1 Þ:
2 ðx; Þ :¼ 2 x1 þ
2 3 x2
2 x2
0
z3 ðk þ 1Þ ¼ z2 ðkÞ þ y ¼ z3
3 ðz3 ; Þ þ 2 ðz3 ; Þ
2 ðz3 ; ; uÞ :¼ 23 þ 2
00
0
transforms the controlled continuous-time COF over R3 into the controlled approximate sampled COF at order 3 in
we get
þ
3 x2 Þ
þ
z2 ðk þ 1Þ ¼ 23 ðz3 ; ; uÞ 3 ðz3 ; Þ
_3 ¼ 2 ðx1 þ
0
3 00 00 ð2 3 x22 þ 3 3 x2 þ 3 3! 0 0 þ 2 3 x1 32 x2 Þ þ x3
Moreover, by computing
with
2 ðx1 2!
0
3 3Þ 0
3 2þ
It is straightforward to verify that the coordinates change (28) takes the very simple linear form 00
2 3 3þ
0
2 3 3Þ
z1 ¼ x1 x2 z2 ¼ x2 ¼ y
215
Observer Form Under Sampling + continuous; * sampled; o emulated
+ continuous; * sampled; o emulated 1
8
0.8
6
0.6 4 0.4 2
0.2 0
0
−0.2
−2
−0.4 −4
−0.6
−6
−0.8 −1
0
1
2
3
4
5
6
Fig. 1. ¼ 0:05:
−8
0
2
4
6
8
10
4
5
Fig. 3. ¼ 0:1:
+ continuous; * sampled; o emulated
+ continuous; * sampled; o emulated 6
8 6
4
4 2 2 0
0 −2
−2
−4 −4
−6 −8
0
2
4
6
8
10
−6
0
1
2
3
Fig. 2. ¼ 0:05:
Fig. 4. ¼ 0:1:
so getting under transformation and up to an error in Oð3 Þ the Sampled COF
In Figs. 1–4, the behavior of the approximate sampled observer (30) with eigenvalues at e1 is compared with the sampled values of the continuoustime observer, which has linear error dynamics with eigenvalues 1 ¼ 2 ¼ 1, and with the approximate sampled observer at the first order in the output injection with eigenvalues at zero, named emulated, i. e. given by
2 ðz2 z32 Þ 2 2 z2 ðk þ 1Þ ¼ z1 þ 2z2 þ ðz2 z32 Þ 2 y ¼ z2 :
z1 ðk þ 1Þ ¼ z2 þ
It is a simple exercise to verify that the discrete-time observer 2 ðy y3 Þ þ k1 ðy z^2 Þ ð30Þ 2 2 z^2 ðk þ 1Þ ¼ z^1 þ 2y þ ðy y3 Þ þ k2 ðy z^2 Þ 2 y ¼ z2
z^1 ðk þ 1Þ ¼ y þ
yields to a linear error dynamics e1 ðk þ 2Þ þ k1 e1 ðkÞ þ k2 e1 ðk þ 1Þ ¼ 0 for e1 ¼ z1 z^1 .
z~1 ðk þ 1Þ ¼ z~1 y y3 k~1 ðy z~2 Þ z~2 ðk þ 1Þ ¼ z~2 þ ~ z1 k~2 ðy z~2 Þ
ð31Þ
y ¼ z2 In all the simulations the observer dynamics is initialized at zero and the evolution of x1 is represented together with its estimates. It results from the figures that to a fast convergence, which is typical of a discretetime device, the proposed observer associates very good steady state performances. Figs 1 and 2 show how the
216
performances of the emulated observer degraded with the initial error: the system evolves starting from x0 ¼ ð0:5; 0:5ÞT and x0 ¼ ð1; 1ÞT , respectively. Figs. 3 and 4 put in light the performances of the proposed sampled observer also when increasing the sampling interval and the initial error x0 ¼ ð0:5; 2ÞT , while the convergence of the emulated observer is lost even by reducing the sampling interval.
5. Conclusions In this paper we have shown that linearization through coordinates change and input-output injection of uncontrolled dynamics can be preserved under sampling up to approximations at order n, the state dimension. The proof is constructive for the coordinates change so exhibiting n-th order approximate sampled counterparts of COF over Rn. The result is extended to controlled dynamics with reference to a specific structure of the input-output injection. A global version of these results, with respect to the x-dependency, could be given assuming that the conditions hold globally and assuming completeness of the vector fields describing the continuous-time dynamics. Work is progressing to relax the extra conditions set on the output injection by considering multi-output injections; i.e. depending also on time instants internal to the sampling intervals, a concept some how similar to multirate control.
Acknowledgment This work was supported by the University ItalyFrance/France-Italy-UIF/UFI under the programme Galileo/Galile´e.
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S. Monaco and D. Normand-Cyrot
6. Califano C, Monaco S, Normand-Cyrot D. On the observer design for discrete-time nonlinear systems. Syst Control Lett 2003; 49:255–265. 7. Ciccarella G, Dalla Mora M, Germani A. Observers for discrete-time nonlinear systems. Syst Control Lett 1993; 20:373–382. 8. Chung ST, Grizzle JW. Sampled-data observer error linearization. Automatica 1990; 26:997–1007. 9. Grizzle JW. Feedback linearization of discrete-time systems. Lecture Notes in Control and Information. Science 1986; 83: 273–281. 10. Grizzle JW, Kokotovic PV. Feedback linearization for sampled-data systems. IEEE Trans Autom Control 1988; 33: 857–859. 11. Huijberts HJC. On existence of extended observers for nonlinear discrete-time systems, Lecture Notes in Control and Information. Science 1999; 244:73–92. 12. Isidori A. Nonlinear Control Systems, 3rd edn. Springer Verlag, 1995. 13. Jouan Ph. Immersion of nonlinear systems into linear systems modulo output injection. Siam J Control Optim 2003; 41: 1756–1778. 14. Jakubczyk B. Feedback linearization of discrete-time systems. Syst Control Lett 1987; 9: 441–446. 15. Kazantzis N, Kravaris C. Discrete-time nonlinear observer design using functional equations. Syst Control Lett 2001; 42: 1–94. 16. Krener AJ, Isidori A. Linearization by output injection and nonlinear observers. Syst Control Lett 1983; 3, 47–52. 17. Krener AJ, Respondek W. Nonlinear observers with linearizable error dynamics. SIAM J Control and Optim 1985; 23: 197–216. 18. Lee HG, Arapostathis A, Marcus SI. On the linearization of discrete-time systems. Int J Control 1987; 45: 1783–1785. 19. Lee W, Nam K. Observer design for autonomous discrete-time nonlinear systems. Syst Control Lett 1991; 17: 49–58. 20. Lin W, Byrnes CI. Remarks on linearization of discretetime autonomous systems and nonlinear observer design. Syst Control Lett 1995; 25: 31–40. 21. Monaco S, Normand-Cyrot D. The immersion under feedback of a multidimensional discrete time nonlinear system into a linear one. Int J Control 1983; 38: 245– 261. 22. Monaco S, Normand-Cyrot D. A unifying representation for nonlinear discrete-time and sampled dynamics. J Math Syst, Estimation Control 1995; 4: 477–503. 23. Monaco S, Normand-Cyrot D. On the differential/ difference representation of sampled dynamics. Proceedings of IEEE-CDC-EUCA-ECC-05, 2005, Se´ville, pp. 6597–6602. 24. Monaco S, Normand-Cyrot D. Normal forms and approximated feedback linearization in discrete time. Syst Control Lett 2006; 55: 71–80. 25. Monaco S, Normand-Cyrot D. Some comments about normal forms under sampling. Proceedings of ACC-06, 2006, Minneapolis, Minnesota, USA, pp. 1487–1492. 26. Monaco S, Normand-Cyrot D. Controller and observer normal forms in discrete time. In: Astofi A, Marconi L (eds), Analysis and Design of Nonlinear Control Systems in Honor of A Isidori, Springer, 2008, pp. 377–397.
Observer Form Under Sampling
27. Monaco S, Normand-Cyrot D. Linearization by outputinjection under sampling, Proceedings of ECC, Kos, Gre`ce, pp. 5270–5273. 28. Moraal PE, Grizzle JW. Observer design for nonlinear systems with discrete-time measurement. IEEE Trans Autom Control 1995; 40:395–404.
217 29. Song Y, Grizzle JW. The extended Kalman filter as a local asymptotic observer for discrete-time nonlinear systems. J Math Syst Est Contr 1995; 5: 59–78. 30. Xia X-H, Gao W-B. Nonlinear observer design by observer error linearization. SIAM J Control Optim 1989; 27: 199–216.
Linearization by Output Injection under Approximate Sampling Salvatore Monaco1, and Dorothe´e Normand-Cyrot2 1 2
Dipartimento di Informatica e Sistemistica ‘Antonio Ruberti’ Università La Sapienza via Ariosto 25, 00185 Roma, Italy Laboratoire des Signaux et Systèmes, CNRS-ESE, Plateau de Moulon, 91190, Gif-sur-Yvette, France
Linearization by output injection has been studied in both continuous time and discrete-time contexts. In this paper we discuss the possible preservation of such a property under sampling. It is shown that it can be maintained under approximate sampling up to the order of the system itself. Keywords: Observer forms, nonlinear sampled-data systems, discrete-time systems, nonlinear observer design.
1. Introduction Linearization by coordinates change and output injection has been firstly studied in a continuous-time context in [16], [17], and further extended and developed as a basic tool for observer design (we refer to [13] for survey on the subject). Even less popular, the same problem has been investigated in discrete time pursuing different approaches (see [7, 11, 15, 19, 20, 29]). While techniques and results are similar when dealing with maps or vector fields, that is for continuous-time or discrete-time uncontrolled dynamics, specialized studies are necessary when controlled equations are investigated. In this paper linearization by output injection is revised as the dual problem of linearization under state feedback ([5, 9, 14, 18, 21]). The geometric conditions ensuring linearization through coordinates
Correspondence to: D. Normand-Cyrot, E-mail: cyrot@lss. supelec.fr E-mail: [email protected]
change and feedback or output injection admit similar formulations for both continuous-time and discretetime systems. Such a similarity is possible making use of an alternate representation of controlled discrete-time dynamics as two coupled differential/difference equations rather than a map parameterized by the control. In this context the authors showed how results on structural and control properties admit similar formulations [23]. In this formalism, feedback linearization was studied in [24], starting from the geometric conditions to the computations of structural invariants and controller normal forms, possibly through successive approximations of increasing degree. Following the same lines, normal forms associated with linearization under coordinates change and output injection, have been recently computed making use of successive transformations of increasing degree in [3, 26]. These analogies in the results’ formulation legitimate the question of their preservation under sampling, i. e. when applied to the discrete-time model issued from the sampling of the continuous system: a problem which has been widely investigated in the literature. Different sampling procedures give rise to sampled equivalent models which may exhibit different characteristics w. r. to the preservation of some continuous time properties. For, multirate or higher order holding procedures have been introduced to maintain properties under sampling ([10], [27]). As far as feedback linearization is concerned, the problem has been firstly addressed starting from [1] and then studied in an approximate context in [2] and [25]. In these papers, more strictly linked to the present Received 27 April 2008; Accepted 18 December 2008 Recommended by H. Shim and A. Isidori
206
contribution, we showed that feedback linearization can be preserved under sampling up to the system order making use of a new filtered output mapping. Moreover, multirate control strategies or multirate sampling procedures have been proposed to enlarge the degree of approximation or to achieve exact solutions [10]. On these bases, we show in this paper that equivalence through coordinates change to the observer form can be preserved under sampling up to the system order, say n, the result is constructive for the coordinates change. It relies on the existence of a fictitious vector field with respect to which the given output mapping has relative degree equal to n. It is immediately understood that this result is the dual of the one in [2] where feedback linearization is achieved through the existence of a fictitious output mapping which has relative degree equal to n. The effects of sampling over the observer design have been investigated in [8] and [28]; in particular in [8], the authors proved that for systems over R2, linearization by output injection under sampling is in fact equivalent to linear equivalence. The present work shows that preservation under time-sampling is in fact possible over Rn when considering the approximate sampled model at the order n in the sampling time . This result suggests to use the approximate model of order n for designing the sampled-data observer since increasing the accuracy of the sampled model by considering higher order terms would only bring to additional computational complexity even if linearization by output injection could be assured. The present work is based on [6] regarding geometric necessary and sufficient conditions for equivalence to the linear observer form through coordinates change and to [25] where a complete description of the sampled equivalent to an input-affine system is given. This paper represents an extended version of [27]. The paper is organized as follows. Section 2 sets the problem and recalls the geometric conditions ensuring linearization through coordinates change and output injection in the continuous-time and discrete-time contexts. Linearization by output injection under sampling is set in the differential/difference context in section 3 where the conditions of section 2 are specified on the sampled equivalent model. In section 4 the main result is given and a constructive procedure for computing the coordinates change which solves the problem is detailed for unforced and forced dynamics with extra conditions on the output injection. The algorithm is worked out on an example: an elementary application in state estimation of the duffing oscillator. Some conclusions end the paper.
S. Monaco and D. Normand-Cyrot
Some standard notations : Given , a vector field over Rn, L denotes the associated formal Lie derivative which acts over real valued functions h : Rn ! R as a first order differential operator L h :¼ dh; the Lie bracket ad 1 2 : ¼ [ 1, 2] between two vector fields over Rn is described by the non commuting product L½1 ;2 ¼ L1 L2 L2 L1 ; the Lie series associated with L is defined P Li by its exponential expansion eL :¼ 1 þ i1 i! , usually denoted as e , which gives e hðxÞ ¼ hðe xÞ; given a diffeomorphism on Rn, of along satisfying Ad denotes the transport 1 . All maps and Ad :¼ ð½Jx Þ1 ¼ d dx vector fields, possibly parameterized, are assumed analytic. The manipulations performed over the asymptotic series expansions are formal ones, no convergence studies are performed.
2. Linearization by Output Injection 2.1. In Continuous Time Let c denote a SISO input-affine system _ ¼ fðxðtÞÞ þ gðxðtÞÞuðtÞ xðtÞ yðtÞ ¼ hðxðtÞÞ
ð1Þ
where x 2 Rn, y 2 R, f and g are analytic vector fields and h an analytic mapping. Assuming, without any loss of generality, that (0, 0) is an equilibrium pair, f(0) ¼ 0, h(0) ¼ 0, linearization by output injection stands for the equivalence under coordinates change to the observer canonical form COF _ ¼ AO zðtÞ þ ðyðtÞ; uðtÞÞ zðtÞ yðtÞ ¼ CO zðtÞ
ð2Þ
with (AO, CO) in the Brunovsky observability form 0 1 0 0 B .. C B 1 ... .C B C B .. .. .. C AO ¼ B . . .C B0 C; B. . C B. . . . . . . . . ... C @. A 0 0 1 0 CO ¼ ð 0
0 1Þ
and ðy; uÞ : R R!Rn , an analytic map characterizing the input-output injection: i.e. the injection on the dynamics which should render linear the dynamics itself.
207
Observer Form Under Sampling
The problem, referred to as linearization by output injection as opposed to feedback linearization, is of interest in the design of nonlinear observers. It results from (2) that
an alternate representation as two differential/ difference equations - DDR xþ ¼ F0 ðxÞ
ð5Þ
þ
dx ðuÞ ¼ Gðxþ ðuÞ; uÞ du
z^_ ðtÞ ¼ AO z^ðtÞ þ ðyðtÞ; uðtÞÞ þ Kðy CO z^Þ
with
xþ ð0Þ ¼ xþ ð6Þ
estimates the state with a linear error dynamics: _ ¼ ðAO KCO ÞeðtÞ: eðtÞ Conditions ensuring local equivalence of c to (2) are well known. Theorem 2.1: [16, 17] c is locally equivalent to the observer canonical form (2) if and only if n1 Ac1 : dh; dL h . . . ; dL h ð0Þ ¼ n; f f Ac2 : rci ; rcj ¼ 0 for i þ j 2 ½2; 2n 1; Ac3 : ½g; rci ¼ 0
for
i 2 ½1; n 1;
where rc1 is the vector field solution of 0 1 0 1 dh 0 .. .. C B C B . B C .C dHrc1 :¼ B C rc1 ¼ B @0A @ dLf h A h dLn1 1 f i1
...
@Fðx; uÞ : @u
:¼
ð7Þ
Indeed, the integration of (6) with respect to u between 0 and uk with initial condition specified by (5), xþ ð0Þ ¼ F0 ðxk Þ :¼ Fðxk ; 0Þ, returns xkþ1 in (4), i.e. þ
Z
uk
x ðuk Þ ¼ x ð0Þþ
Gðxþ ðÞ; Þd ¼ Fðxk ; uk Þ:
0
ð3Þ
adfi1 rc1
rcn Þ ¼ Idn
In the same way w.r. to an output mapping y ¼ hðxÞ, one has þ
Z
ykþ1 ¼ hðx ð0ÞÞ þ
uk
0
LGð:; Þ hðxþ ðÞÞd
¼ hðxþ ðuk ÞÞ ¼ hðFðxk ; uk ÞÞ: The Taylor expansion of Gðx; uÞ in powers of u
with Idn the n n-identity matrix. Remark: Applying the Jacobian identity, condition Ac2 is recognized to be equivalent to [rc1, rci] ¼ 0 for i 2 [2, 2n 2]. Remark: In the uncontrolled case (g ¼ 0) in (1), under the stronger condition A0c2 : ½rci ; rcj ¼ 0 for i þ j 2 ½2; 2n þ 1 (equivalently ½rc1 ; rci ¼ 0 for i 2 ½2; 2n), one gets linear equivalence under linear output injection. Remark: The ‘‘fictitious’’ controlled dynamics _ ¼ fðxÞþ urc1 ðxÞ with output mapping y ¼ h(x) xðtÞ exhibits a relative degree equal to n, ðLrc1 h ¼ 0; Lrc2 h ¼ 0; . . . ; Lrcn1 h ¼ 0Þ with moreover Lrcn h ¼ 1. 2.2. In Discrete Time A discrete-time dynamics is usually represented as a map xk ! xkþ1 ¼ Fðxk ; uk Þ
GðFðx; uÞ; uÞ
þ
for i 2 ½2; n: and rci :¼ adf rci1 ¼ ð1Þ Moreover, the coordinates change (x) satisfies ½ Jx ðrc1
where xþ ðuÞ indicates a curve in Rn parameterized by u. (4) and (5–6) describe the same discrete-time dynamics provided that G(x,u) satisfies
ð4Þ
where x 2 Rn ; u 2 R and Fð:; uÞ is an analytic map, analytically parameterized by u. In [22], we proposed
Gðx; uÞ ¼ G0 ðxÞ þ
X ui i1
i!
Gi ðxÞ
ð8Þ
gets the analytic vector fields Gi ðxÞ, i 0, which play a fundamental role in the geometric characterization of the properties under study. Let d denote the DDR (5-6) with output map y ¼ hðxÞ, and define the canonical observer differential/difference representation (CO-DDR) as the DDR of the COF (2). Denoting by i the i – th component of , and assuming, without loss of generality, that @ 1 ðy; uÞ 6¼ 0, which can be achieved through a ð0;0Þ @y possible preliminary linear output injection. One has Proposition 2.1: The discrete-time observer canonical form zkþ1 ¼ AO zk þ ðyk ; uk Þ y ¼ CO z
ð9Þ
208
S. Monaco and D. Normand-Cyrot
Remark: The fictitious controlled dynamics
admits the DDR z
þ
¼ AO z þ ðyÞ
dzþ ðuÞ ¼ Gðzþ 1 ðuÞ; uÞ; du y ¼ CO z
ð10Þ zþ ð0Þ ¼ zþ
ð11Þ
with (AO, CO) in the Brunovsky form, ðyÞ ¼ ðy; 0Þ and @ ð:; uÞ j 1 ðzþ ðuÞ;uÞ Gðzþ 1 ðuÞ; uÞ ¼ 1 1 @u 1 where 1 ð:; uÞ indicates the inverse of 1 ð:; uÞ, the first component of ð:; uÞ. Proof: The result is an immediate consequence of the definition of the differential equation and the particular form of the matrix AO so getting with zþ 1 ð0Þ ¼ 1 ðyÞ Z u Gðzþ ðy; uÞ ¼ ðyÞ þ 1 ðvÞ; vÞdv: 0 3 Remark: The input-output injection term (11) takes the form ðy; uÞ ¼
@ ðy; uÞ ¼ Gðzþ 1 ðuÞ; uÞ ¼ Gð @u
1 ðy; uÞ; uÞ:
Theorem 2.2: [6] d is locally equivalent to the discrete-time observer canonical form if and only if Ad1 : dh; dðh F0 Þ; ; dðh Fn1 ð0Þ ¼ n; 0 i 2 ½2; n Ad2 : ½rd1 ; rdi ¼ 0; p 0; Ad3 : ½Gp ; rdi ¼ 0 for i 2 ½2; n; where the vector field rd1 satisfies 0 1 0 1 dh 0 .. .. C B B C . Crd1 ¼ B . C dHd rd1 :¼ B @0A @ dh Fn2 A 0 n1 dh F0 1
ð12Þ
and for i 2 ½2; n, rdi ’s are given by rdi ¼ AdF0 rdi1 ¼ AdFi1 rd1 :
xþ ð0Þ ¼ F0 ðxÞ dxþ ðuÞ ¼ rd1 ðxþ ðuÞÞ du with output mapping y ¼ hðxÞ has relative degree equal to n; i.e. using the DDR formalism ðLrd1 h ¼ 0; Lrd2 h ¼ 6 0. 0; . . . ; Lrdn1 h ¼ 0Þ with moreover Lrdn h ¼ 1 ¼
3. Linearization by Output Injection Under Sampling Assuming the control variable u(t) constant over small time intervals of amplitude , with value uk over ½k; ðk þ 1Þ½ for k 0, the sampled equivalent system, s , is the discrete-time system which reproduces the state and output evolutions of c at the sampling instants for any x0 ¼ xðt ¼ 0Þ. As well known the discrete-time dynamics is drift invertible for small , i.e. F0 ðxÞ ¼ ef ðxÞ and ðF0 Þ1 ðxÞ ¼ ef ðxÞ. As a consequence G ðx; uÞ satisfying (7) exists and it is uniquely @F ðx; uÞ defined by G ðx; uÞ :¼ with @u x¼F ðx;uÞ fþug ðxÞ. It is a matter of computations to F ðx; uÞ ¼ e verify that the DDR xþ ¼ F0 ðxÞ dx ðuÞ ¼ G ðxþ ðuÞ; uÞ du y ¼ hðxÞ
½Jx d ð rd1 . . . rdn Þ ¼ Idn : Condition Ad1 is the observability rank condition which ensures the existence and uniqueness of rd1 satisfying (12); Ad2 requires first order nilpotency of the distribution generated by the vector fields ðrd1 ; ; rdn Þ and guarantees the existence of a coordinates change as well as the specific structure of (10); Ad3 guarantees the specific structure (11). It is worthy to note the strict analogy between statements of Theorem 2.1 and Theorem 2.2.
with
xþ ð0Þ ¼ xþ ð14Þ ð15Þ
describes the sampled equivalent s of c. Its inputstate or output-state evolutions match, at the sampling instants t ¼ k those of c. Combinatoric relations between the continuous-time model and its sampled equivalent are detailed in [25], we just recall what is necessary in the present context. First, G ð:; uÞ is described by its asymptotic expansion in Z
G ð:; uÞ ¼
0
Moreover the coordinates change, d , satisfies
ð13Þ
þ
esðadf þuadg Þ gds ¼ G0 þ
i1
0
as well as each Gi ¼ for the first ones G0
Z ¼
P
p piþ1 p! Gi;p
esadf gds :¼
p1 X ð1Þp1 pþ1 X p2
ðp þ 1Þ!
i!
Gi ð16Þ
for i 0 so getting
X ð1Þp1 p p1
0
G1 ¼
X ui
p!
adp1 g f
ð17Þ
Ckp ½adkf g;adfpk1 g ð18Þ
k¼0
p! denotes the binomial. where Ckp :¼ k!ðpkÞ!
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Observer Form Under Sampling
We note that the term in p in G0 described by (17) is g which contains the vector field the Lie bracket adp1 f g one-time and the vector field f, (p 1)-times. G1 in (18) involves Lie brackets of the first order in the adjf g of the form ½adkf g; adfpk1 g in pþ1 for p 1. Such a term contains g, two-times and f, (p 1)-times. More in general, Gi can be iteratively deduced according to a combinatoric rule; it involves Lie brackets of order i in the adjf g and it contains g, (i þ 1)-times. The transport of any Gi along F0 is denoted by AdF Gi and its 0 transport along F0 , q-times is defined as AdFq Gi 0 because F0 . . . F0 ¼ Fq 0 . Theorem 3.1: [27] s is locally equivalent to the discrete-time observer canonical form (10-11) if and only if there exists T > 0 such that for any 2 ½0; T, the conditions below hold true As1 : ðdh; def h; ; deðn1Þf hÞð0Þ ¼ n; As2 : ½rs1 ; rsi ¼ 0 for i 2 ½2; n; As3 : ½Gp ; rsi ¼ 0 for i 2 ½2; n; p 0; where the vector field rs1 satisfies: 0 0 1 1 dh 0 . . B B .. C C .. B C C dH rs1 :¼ B @ deðn2Þf h Ars1 ¼ @ 0 A n1 deðn1Þf h
ð19Þ
and rsi is the transport of rsi1 along F0 , rsi :¼ eadf rsi1 ¼ eði1Þadf rs1 for i 2 ½2; n. Moreover, the coordinates change satisfies ½Jx ð rs1
...
rsn Þ ¼ n1 Idn :
When it is referred to a sampled model, the observer canonical form maintains the discrete-time structure (9-10-11) with -dependent output injection and vector field G . Assuming c locally equivalent to the observer canonical form (2), the present paper discusses the possible preservation of such a property under approximated sampling. The loss of linearization by output injection under sampling, as noted in [8], is the natural counterpart of what occurs when investigating the dual problem of feedback linearization under sampling. As a matter of fact, preservation of linearization by feedback under sampling should imply linear equivalence as conjectured in [1] and proved for n ¼ 2. However, some approximate result can be proven. As proposed in [2], feedback linearization can be maintained till an approximation order with respect to the sampling period equal to the state dimension. The same idea is here developed with respect to linearization by output injection. Our result
stands in proving that the conditions above can be maintained in an approximate meaning. More precisely, these conditions, reformulated as equalities between asymptotic expansions in , hold true up to a certain degree of approximation. With this in mind the following definition is mandatory. Definition 3.1: Assuming c linearizable by outputinjection, the property is maintained under sampling if there exists T > 0 such that for any 2 ½0; T the sampled equivalent model satisfies the property. Approximate solution at order p stands for the existence of a solution on the approximate sampled model at order p in ðerror in Oðpþ1 ÞÞ. The problem has been approached in [27] where we showed that linearization by output injection could be maintained under sampling thanks to the computation of a ‘‘fictitious’’ -dependent vector field with respect to which the relative degree is maintained equal to n at order n 1 (i.e. up to an error in Oðn Þ). As a matter of fact, a stronger result can be proven. Theorem 3.2: For c, locally linear equivalent by output injection, there exists T > 0 such that for any 2 ½0; T a vector field rs1 can be computed to satisfy (19) at any degree of approximation in . Proof: The proof works out showing first that dH ¼ HðÞdH þ Oðn Þ
ð20Þ
with HðÞ 2 Rnn , invertible by construction for any 20; T 0 1 1 0 0 0 . .. B C B1 C . 2 =2 . . B C . B C . 2 HðÞ ¼ B 1 C: . 2 2 B. C .. .. .. .. B. C . .2 2 . . @. A n1 n1 ðn1Þ ðn1Þ 1 ðn 1Þ 2 ðn1Þ! Then, given r1 satisfying (3), we set in (19) 0 1 10 B 20 C n1 B C X B C .. rs1 ¼ i1 i0 rci ¼ ðrc1 . . . rcn ÞB C: . B C i¼1 @ n2 n10 A 0 Since rs1 must solve (19) in Oðn Þ, due to (20), we have 0 1 0 1 10 0 B .. C @ ... A ¼ ðM1 Þ ð21Þ @ . A ¼ M1 0 0 lc 1 n10
210
S. Monaco and D. Normand-Cyrot
ðn1Þðn1Þ where ðM1 , 0 Þlc is the last column of M0 2 R invertible by construction; i.e.
1 0 10 1 ... 1 B C .. C .. CB . C B 1 . AB n2 C @ n10 A ... 0 0
1 0 00 1 0 C B 10 C HðÞ B . B B CC B . B M0 B .. C C ¼ n1 @ . @ . AA @ 1 n10 0
so getting 0 B B M0 :¼ B B @
1 ðn1Þ! n1 2 ðn1Þ!
1 ðn2Þ! n2 2 ðn2Þ!
ðn1Þ ðn1Þ!
ðn1Þ ðn2Þ!
.. . n1
.. . n2
Setting now in (19), rs1 ¼
...
1 2
...
4 .. .
... ...
Pn1 i¼1
ðn1Þ2 2
1
1
2 C C C .. C: . A n1
i1 ði0 þ i1 Þrci , i. e.
0 rs1 ¼ ð rc1
...
10 þ 11 20 þ 2 21 .. .
1
B C B C B C rcn ÞB C B n2 C @ n10 þ n1 n11 A 0
and solving equality (19) in Oðnþ1 Þ, 0 B B 10 n B B HðÞdH:rs1 þ n! B B @
0 1 n
2 .. .
ðn 1Þ
1
0 1 0 C B .. C C B C C n CdL h:rc1 ¼ B . C B C C f @ 0 A C A n1 n
leads to 0 1 1 1 11 ðxÞ n C 2 dn ðxÞ 1 B B C .. C .. M0 B @ A¼ . @ A n! . n n11 ðxÞ ðn þ 1Þ 0
with dn ðxÞ :¼ dLnf hðxÞ:rc1 and 10 ¼ 1 computed from P (21). Iterating the procedure and setting 2 i1 ði0 þ i1 þ 2! i2 þ . . .Þrci , a solution rs1 ¼ n1 i¼1 to (19) up to increasing approximation order in
exists thanks to the invertibility of M0 . The proof is constructive for the successive ip which are functions of x for p 1. 3 Starting from the so computed vector field rs1 , truncated at a certain approximation in , special assumptions on the output injection must be taken into account to satisfy conditions As2 , As3 in Theorem 3.1 so maintaining the observer canonical structure under approximate sampling. This is the case in [27] where the output injection is assumed piecewise constant. A more general result is proposed in the sequel where we describe a procedure for computing a -dependent coordinates transformation under which the approximate sampled model exhibits the sampled observer canonical form - sampled COF.
4. Approximate Sampled COF, an Algorithm We first describe an algorithm for computing a coordinates transformation under which the COF structure is recovered under n-th order approximate sampling; i.e. starting from a continuous-time COF, an approximate sampled COF is described. The algorithm is proposed below for unforced systems c ð0Þ, and then extended to forced ones c ðuÞ under some extra conditions on the output injection. 4.1. The Algorithm for c ð0Þ Starting from the continuous-time COF over Rn x_ 1 ¼ 1 ðxn Þ x_ 2 ¼ x1 þ 2 ðxn Þ x_ n ¼ xn1 þ n ðxn Þ y ¼ xn
ð22Þ
we look for a coordinates change z ¼ T ðxÞ such that in the new coordinates, the sampled equivalent model, truncated at order n in exhibits a sampled COF. The proof is constructive for T . Initialization - set zn :¼ xn ¼ y. Step 1 – compute the evolution of y, truncated at order n in (n-th order approximation) yðk þ 1Þ ¼ y þ y_ þ
2 n ðnÞ y€ þ þ y 2 n!
211
Observer Form Under Sampling
and put in evidence in the r.h.s. the terms which depend on y only, denoted by n ðy; Þ, from the terms which depend on the state variables xi ; i 2 ½1; n 1 denoted by n ðx; Þ. One gets yðk þ 1Þ ¼ zn ðk þ 1Þ :¼ n ðzn ; Þ þ n ðx; Þ ð23Þ with
Step (i þ 1) – compute the n-th order approximate evolution of zni zni ðk þ 1Þ ¼
p¼0
þ
n n;n ðxÞ n! and by construction n;1 ðxÞ ¼ xn1 . Set
þ ð24Þ
zn ðk þ 1Þ ¼ zn1 þ C1n1 zn þ n ðzn ; Þ n ðyÞ
Step 2 – compute the n-th order approximate evolution of zn1 . From (23 – 24) and dropping the (x, )-dependency for simplicity, one gets n1 ðn1Þ zn1 ðk þ 1Þ ¼ n þ _n þ þ ðn 1Þ! n þ ð1Þ1 C1n1 ðn ðzn ; Þ þ n Þ ¼ ð1
þ n1 þ n1 ðzn ; Þ
ð25Þ
n n1;n ðn 1Þ!
ð1Þp Cpni ni1 þ ni ðx; Þ þ ni ðzn ; Þ
where ni regroups terms in the r.h.s. which depend on zn only while n and n1 ; . . . ; ni regroups terms which do not depend on zn only. One gets n ni; n ðn iÞ!
with ni; iþ1 ¼ xni1 . Set zni1 ¼
i X
ð1Þp Cpn1 n þ
p¼0
þ
1 X
i1 X ð1Þp Cpn2 n1 þ... p¼0
ð1Þp Cpni ni1 þni þð1Þi Cin1 zn
p¼0
and verify that the zni -dynamics takes the form ni ðyÞ
so describing the output-injection term ...
ni ðyÞ.
Step (n – 1) – compute the approximate z2 evolution at order n z2 ðkþ1Þ¼ð1þð1Þ1 C1n1 þ...þð1Þn2 Cn2 n1 Þn þð1þð1Þ1 C1n2 þ...þð1Þn3 Cn3 n2 Þn1 þ...þð1þð1Þ1 C12 Þ3 ðx;Þþ2 þ2 ðzn ;Þ
with n1;2 ¼ xn2 . Set
¼ð1Þn2 n þð1Þðn3Þ n1 þ...þ2 þ2 ðzn ;Þ
zn2 ¼ ð1 C1n1 Þn þ n1 þ C2n1 zn
ð26Þ
so that, from (25–26), the zn1 -dynamics takes the form zn1 ðk þ 1Þ :¼ zn2 C2n1 zn þ n1 ðzn ; Þ ¼ zn2 þ
p¼0
zni ðk þ 1Þ ¼ zni1 ðkÞ þ
where n1 ðzn ; Þ contains the terms in the r.h.s. which depend onzn only; n1 ðx; Þ is a new term containing 2 all n1the terms in _n ðx; Þ þ 2! €n ðx; Þ þ þ ðn1Þ ðx; Þ which do not depend on zn only, ðn 1Þ! n so getting n1 ðx; Þ ¼ 2 n1;2 þ . . . þ
1 X
i2 X ð1Þp Cpn3 n2 þ
ni ðx; Þ ¼ iþ1 ni; iþ1 þ . . . þ
so recovering the desired structure over the n-th equation where n ðyÞ :¼ C1n1 y þ n ðy; Þ specifies the n-th component of the output injection.
C1n1 Þn
ð1Þp Cpn2 n1 þ
p¼0
n! . By construction, the zn -dynamics with Cpn :¼ p!ðnpÞ! takes the form
¼ zn1 þ
i1 X p¼0
n ðx; Þ ¼ n;1 ðxÞ þ . . . þ
zn1 :¼ n ðx; Þ þ ð1Þ1 C1n1 zn
i X ð1Þp Cpn1 n
n1 ðyÞ
so recovering the desired structure with output injection term, n1 ðyÞ :¼ C2n1 y þ n1 ðy; Þ. . . . Iterate the steps so characterizing the (i þ 1)-th step as follows.
where 2 ðzn ; Þ regroups terms in the r.h.s. which depend on zn only and 2 ðx; Þ those which do not depend on zn only so getting 2 ¼ n1 2; n1 þ . . . þ
n 2; n 2!
with 2;n1 ¼ x1 . Set z1 ¼ ð1Þn2 n þ ð1Þðn3Þ n1 þ . . . 3 þ 2 þ ð1Þn1 Cn1 n1 zn
212
S. Monaco and D. Normand-Cyrot
Example: Let us illustrate the computation over R2. Consider the continuous-time uncontrolled COF
so computing the z1 -dynamics n1 ðn1Þ Þ z1 ðk þ 1Þ ¼ ð1Þn2 ðn þ _n þ ... þ ðn 1Þ! n n2 ðn2Þ Þ þ ð1Þðn3Þ ðn1 þ _n1 þ ... þ ðn 2Þ! n1 2 þ ... ð3 þ _3 þ €3 Þ þ ð2 þ _2 Þ 2! þ ð1Þn1 ðn ðzn Þ þ n ðx;ÞÞ ¼ 1 ðzn Þ þ _2
x_ 1 ¼ 1 ðx2 Þ x_ 2 ¼ x1 þ 2 ðx2 Þ y ¼ x2 and its approximate sampled equivalent at order 2 in x1 ðk þ 1Þ ¼ x1 þ
1þ
2 2!
x2 ðk þ 1Þ ¼ x2 þ ðx1 þ rewritten as z1 ðk þ 1Þ ¼ 1 ðyÞ with 1 ðyÞ :¼ 1 ðzn Þ þ n n _ _2 because Pn 2 ¼ p x_p1 ¼ 1 ðyÞ is a function of y only and p¼0 ð1Þ Cn ¼ 0.
4.2. Approximate Sampled COF for Unforced Dynamics The result of the previous algorithm is summarized below.
ð27Þ
1 ðx1 þ 2Þ þ
ð 2! 2
2Þ 1þ
0
2 ðx1 þ
2 ÞÞ
y ¼ x2 which clearly does not preserve the observer structure; 0 00 ð:Þ ; ð:Þ ; . . . indicate the successive derivatives w.r.t. its arguments of the function into the parentheses. According to the algorithm, one sets z2 ¼ x2 and puts in evidence in z2 ðk þ 1Þ ¼ yðk þ 1Þ the parts which depend on z2 only from the remaining ones, so getting z2 ðk þ 1Þ ¼ 2 ðz2 Þ þ 2 ðx; Þ ¼ x2 þ
Theorem 4.1: Linearization by output injection of c ð0Þ is maintained under approximate sampling at order n through the coordinates transformation z ¼ ðxÞ ¼ T ððxÞÞ
0
þ x1 þ
2
2 2!
þ
2 ð 2!
1
0
þ
2 2Þ
0
2 x1 :
Setting z1 ¼ 2 ðx; Þ z2 , one recovers 2 ðyÞ
z2 ðk þ 1Þ ¼ z1 ðkÞ þ z2 ðkÞ þ 2 ðz2 ðkÞÞ ¼ z1 þ where denotes the coordinates change which transforms c ð0Þ into its COF (22) and T is the coordinates change computed through the algorithm. Proof: Taking in mind that the algorithm has been worked out on the approximate sampled equivalent, Sððc0 ÞÞ of the continuous-time COF, ðc0 Þ and that the procedures of applying a coordinates transformation and sampling commute S ¼ S, we can deduce from the composition (27). More precisely, setting ¼ T , one has S ¼ T S ¼ T S so proving that transforms directly the approximate sampled equivalent model Sðc ð0ÞÞ of c ð0Þ into the sampled canonical normal form issued from the algorithm, T ðSððc0 ÞÞ provided homogeneous approximations in up to order n are performed. 3 Remark: In [8], it was conjectured (proven for n ¼ 2) that preservation under sampling of linear equivalence through output-injection should imply linear equivalence of c0 . Our result shows that preservation holds true for any n but up to order n in .
so defining z1 -dynamics
2
as
2
z1 ðk þ 1Þ ¼ ðx1 þ ¼ x2
þ 2! ð 2
:¼ z2 þ
2
1Þ þ þ 2þ
2 2!
2 2!
1
0
þ
2 2 Þ.
The
0
2 x1 z2 ðk þ 1Þ
1
2 2!
0
2 2
:¼ 1 ðyÞ
depends on z2 ¼ x2 ¼ y only as 2 depends on x2 only. In conclusion, the coordinates change, z ¼ T ðxÞ described by 2 0 x1 x2 ; z2 ¼ x2 ð28Þ 2! 2 transforms the approximate sampled equivalent to the COF into the 2nd order sampled COF over R2 z1 ¼ x1 þ
z1 ðk þ 1Þ ¼ z2
2
þ
z2 ðk þ 1Þ ¼ z1 þ 2z2 þ y ¼ z2 :
2 ð 2! 2
þ
1
0
2 ð 2!
2
1
þ
2Þ 0
2 2Þ
213
Observer Form Under Sampling
To complete the discussion, it is a matter of computations to verify that the Jacobian of T satisfies the condition set in Theorem 2.3; i.e. ½Jx T r1
r2 ¼ Id2 þ Oð3 Þ
where the vector field r1 is computed to satisfy (19) up to an error in Oð3 Þ and r2 :¼ eadf r1 so getting up to 2 r1 ¼ rc1 d2 ðxÞrc1 ; r2 ¼ rc1 d2 ðxÞrc1 þ rc2 2 2 with d2 ðxÞ ¼ Lrc3 hðxÞ. In the present case, because 0 rc1 ¼ ð1; 0ÞT ; rc2 ¼ ð1; 0ÞT and d2 ðxÞ ¼ 2 , one gets in Oð3 Þ
þ 2! 2
0
0
2
1 1
1 2 0
0
2
1 2
0
2
0 ¼ : 0
The specific structure of the n-th order sampled COF over Rn can be described in terms of the continuoustime COF according to the same procedure.
Remark: An interesting case is represented by controlled system c ðuÞ having relative degree n, h 6¼ 0). (Lg Lkf h ¼ 0; k ¼ 0; n 2; Lg Ln1 f Example: Let us illustrate the procedure on the controlled COF over R3 with input-output injection satisfying (29) x_ 1 ¼ 1 ðx3 ; uÞ x_ 2 ¼ x1 þ 2 ðx3 Þ x_ 3 ¼ x2 þ 3 ðx3 Þ y ¼ x3 with approximate sampled equivalent at order 3 in x1 ðk þ 1Þ ¼ x1 þ
1 ðuÞ
þ
@ 2 i ðy; uÞ ¼0 @y@u
for
i 2 ½2; n
ð29Þ
which is equivalent to assume that the derivatives 0 0 i ðyÞ do not depend on u except 1 ðy; uÞ. The following result holds true. Theorem 4.2: Linearization by output injection of c ðuÞ is maintained under approximate sampling at order n through the coordinates transformation (27) if the conditions (29) are satisfied. Proof: It is readily understood that under (29), the output injection in (2) takes the form ðy; uÞ ¼ ð
1 ðy; uÞ; b2 u
þ
2 ðyÞ; ; bn u
þ
n ðyÞÞ
T
:
Linearity in u of the input-output injection i 2 ðy; uÞ guarantees that the algorithm proposed above works out identically yielding to a coordinates change which does not depend on the control variable which is assumed constant over the sampling intervals. Relaxing (29) yields to a coordinates change which depends on u. 3
1 ðuÞðx2
þ
3Þ
3 0 ð ðuÞðx1 þ 2 Þ 3! 1 00 þ 1 ðuÞðx2 þ 3 Þ2
þ
þ
The result stated can be extended to controlled systems c ðuÞ admitting a COF (2) with output injection satisfying the condition
0
þ
0
0
1 ðuÞ 3 ðx2
þ
x2 ðk þ 1ÞÞ ¼ x2 þ ðx1 þ 4.3. Approximate Sampled COF for Controlled Dynamics
2 2!
þ
0
2 ðx2 3
ð 3! 0
þ
3 ÞÞ 2Þ
þ
2 ð 2!
3Þ
þ
1 ðuÞ
3 ÞÞ
0
1 ðuÞðx2
þ
0
0
2 ðx1
00
þ 2 3 ðx2 þ 3 Þ þ x3 ðk þ 1Þ ¼ x3 þ ðx2 þ 3 Þ
2 ðx2
þ
þ 3Þ
2Þ 2
Þ
2 0 ðx1 þ 2 þ 3 ðx2 þ 3 ÞÞ 2! 3 0 þ ð 1 ðuÞ þ 2 ðx2 þ 3 Þ 3! 00 þ 3 ðx2 þ 3 Þ2 þ
0
þ 3 ðx1 þ y ¼ x3
2Þ
where the u-dependency is in one deduces from z3 ðk þ 1Þ 3 ðz3 ; ; uÞ :¼ z3 þ þ þ
3 ð 3! 0
3 2
1
3
þ
1 ðuÞ
þ
þ
3 ðx; Þ :¼ x2 þ
þ
0
2 3 ðx2
þ
3 ÞÞ
only. Setting z3 :¼ x3 , 2 ð 2! 0
2 3
2
0
þ
3 3Þ 00
þ
2 3 3
0
2 3 3Þ
2 ðx1 þ 2!
0
3 x2 Þ
3 0 00 ð x2 þ 3 x22 3! 2 0 00 þ 3 x1 þ 2 3 3 x2 þ
þ
0
2 3 x2 Þ:
214
S. Monaco and D. Normand-Cyrot
In conclusion, the coordinates change, z ¼ T ðxÞ
Setting z2 :¼ 3 ðx; Þ 2z3 , one computes z3 ðk þ 1Þ ¼ z2 þ 2z3 þ 3 ðz3 ; Þ
z1 ¼ x2 þ
3 ðz3 ; uÞ:
¼ z2 þ
þ _3 ðz3 ; Þ þ
2 € 3 ðz3 ; Þ 2!
z2 ¼ x2 þ
2 ðx1 þ 2!
3 0 ð x2 þ 3! 2 00 þ 2 3 3 x2 þ z3 ¼ x3 ¼ y þ
with ð 1 ðuÞ 2 00 0 þ 3 x2 ðx2 þ 3 Þ þ 3 ðx1 þ
2 € 3 ¼ 3 2ð 2!
2Þ
1 ðuÞ
3
þ
þ
0
2 ðx2
þ
2 ÞÞ
3 ÞÞ
z1 ðk þ 1Þ ¼ z2 ðk þ 1Þ ¼
z2 ðk þ 1Þ ¼ 2 ðz3 ; ; uðkÞÞ
with the
with
0
2
3
þ
0
3 3
2
þ
3 ð2 2 00
3 3 x2
2 3 x2 Þ
3 x1
2x3
1 ðz3 ðkÞ; uðkÞÞ z1 ðkÞ þ 2 ðz3 ðkÞ; uðkÞÞ
i ðz3 ; uÞ
3 ðz3 ðkÞ; uðkÞÞ
described above for i ¼ [1,3].
with 1 ¼ x2 x32 and 2 ¼ 0. Under approximate sampling at order 2, the COF structure is lost
2 ðz3 ; uÞ
x1 ðk þ 1Þ ¼ x1 þ ðx2 x32 Þ
:¼ z3 þ 2 ðz3 ; ; uÞ and also
2 ðx1 3x22 x1 Þ 2 2 x2 ðk þ 1Þ ¼ x2 þ x1 þ ðx2 x32 Þ 2 y ¼ x2 : þ
z1 ðk þ 1Þ ¼ 3 þ 3 3 _3 2 € 3 þ 2 þ _2 2! 2 0 ¼ z3 þ 3 þ ð 2 þ 2 3 0 þ ð 1 ðuÞ 2 2 3 2 3! :¼ 1 ðz3 ; uÞ:
0
0
x_ 1 ¼ x2 x32 x_ 2 ¼ x1 y ¼ x2
z2 ðk þ 1Þ ¼ z1 z3 þ 2 ðz3 ; ; uÞ
2 ðz3 ; uÞ
þ
A very simple example is the duffing oscillator which admits the COF
Setting z1 :¼ 3 þ 2 þ z3 , one has
¼ z1 þ
3 x2 Þ
4.4. The Duffing Oscillator
1 ðuÞ
2Þ
00 ð x2 þ 2 3 2 0 0 þ 2 x2 þ 3 x1 Þ:
2 ðx; Þ :¼ 2 x1 þ
2 3 x2
2 x2
0
z3 ðk þ 1Þ ¼ z2 ðkÞ þ y ¼ z3
3 ðz3 ; Þ þ 2 ðz3 ; Þ
2 ðz3 ; ; uÞ :¼ 23 þ 2
00
0
transforms the controlled continuous-time COF over R3 into the controlled approximate sampled COF at order 3 in
we get
þ
3 x2 Þ
þ
z2 ðk þ 1Þ ¼ 23 ðz3 ; ; uÞ 3 ðz3 ; Þ
_3 ¼ 2 ðx1 þ
0
3 00 00 ð2 3 x22 þ 3 3 x2 þ 3 3! 0 0 þ 2 3 x1 32 x2 Þ þ x3
Moreover, by computing
with
2 ðx1 2!
0
3 3Þ 0
3 2þ
It is straightforward to verify that the coordinates change (28) takes the very simple linear form 00
2 3 3þ
0
2 3 3Þ
z1 ¼ x1 x2 z2 ¼ x2 ¼ y
215
Observer Form Under Sampling + continuous; * sampled; o emulated
+ continuous; * sampled; o emulated 1
8
0.8
6
0.6 4 0.4 2
0.2 0
0
−0.2
−2
−0.4 −4
−0.6
−6
−0.8 −1
0
1
2
3
4
5
6
Fig. 1. ¼ 0:05:
−8
0
2
4
6
8
10
4
5
Fig. 3. ¼ 0:1:
+ continuous; * sampled; o emulated
+ continuous; * sampled; o emulated 6
8 6
4
4 2 2 0
0 −2
−2
−4 −4
−6 −8
0
2
4
6
8
10
−6
0
1
2
3
Fig. 2. ¼ 0:05:
Fig. 4. ¼ 0:1:
so getting under transformation and up to an error in Oð3 Þ the Sampled COF
In Figs. 1–4, the behavior of the approximate sampled observer (30) with eigenvalues at e1 is compared with the sampled values of the continuoustime observer, which has linear error dynamics with eigenvalues 1 ¼ 2 ¼ 1, and with the approximate sampled observer at the first order in the output injection with eigenvalues at zero, named emulated, i. e. given by
2 ðz2 z32 Þ 2 2 z2 ðk þ 1Þ ¼ z1 þ 2z2 þ ðz2 z32 Þ 2 y ¼ z2 :
z1 ðk þ 1Þ ¼ z2 þ
It is a simple exercise to verify that the discrete-time observer 2 ðy y3 Þ þ k1 ðy z^2 Þ ð30Þ 2 2 z^2 ðk þ 1Þ ¼ z^1 þ 2y þ ðy y3 Þ þ k2 ðy z^2 Þ 2 y ¼ z2
z^1 ðk þ 1Þ ¼ y þ
yields to a linear error dynamics e1 ðk þ 2Þ þ k1 e1 ðkÞ þ k2 e1 ðk þ 1Þ ¼ 0 for e1 ¼ z1 z^1 .
z~1 ðk þ 1Þ ¼ z~1 y y3 k~1 ðy z~2 Þ z~2 ðk þ 1Þ ¼ z~2 þ ~ z1 k~2 ðy z~2 Þ
ð31Þ
y ¼ z2 In all the simulations the observer dynamics is initialized at zero and the evolution of x1 is represented together with its estimates. It results from the figures that to a fast convergence, which is typical of a discretetime device, the proposed observer associates very good steady state performances. Figs 1 and 2 show how the
216
performances of the emulated observer degraded with the initial error: the system evolves starting from x0 ¼ ð0:5; 0:5ÞT and x0 ¼ ð1; 1ÞT , respectively. Figs. 3 and 4 put in light the performances of the proposed sampled observer also when increasing the sampling interval and the initial error x0 ¼ ð0:5; 2ÞT , while the convergence of the emulated observer is lost even by reducing the sampling interval.
5. Conclusions In this paper we have shown that linearization through coordinates change and input-output injection of uncontrolled dynamics can be preserved under sampling up to approximations at order n, the state dimension. The proof is constructive for the coordinates change so exhibiting n-th order approximate sampled counterparts of COF over Rn. The result is extended to controlled dynamics with reference to a specific structure of the input-output injection. A global version of these results, with respect to the x-dependency, could be given assuming that the conditions hold globally and assuming completeness of the vector fields describing the continuous-time dynamics. Work is progressing to relax the extra conditions set on the output injection by considering multi-output injections; i.e. depending also on time instants internal to the sampling intervals, a concept some how similar to multirate control.
Acknowledgment This work was supported by the University ItalyFrance/France-Italy-UIF/UFI under the programme Galileo/Galile´e.
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