a discrete-time observer based on the polynomial...

European Journal of Control (2009)2:143-156 © 2009 EUCA DOl: 1O.31661EJC.15.143-156

European Journal of

Control

A Discrete-time Observer Based on the Polynomial Approximation of the Inverse 0 bservability Map

Alfredo Germani 1,2, * and Costanzo Manes 1,3, **

I Department of Electrical and Infonnation Engineering, University of L' Aquila, Poggio di Roio, 67040 L' Aquila, Italy; 2Universiw. Campus Bio-Medico of Rome, Via Alvaro del Portillo, 21, 00128 Roma Italy; 3Istituto di Analisi dei Sistemi ed Informatica del CNR "A Ruberti" , Viale Manzoni 30. 00185 Roma, Italy

This paper investigates the problem of asymptotic state observation for analytic nonlinear discrete-time systems. A new techniquefor the observer construction. based on the Taylor polynomial approximation of the inverse of the observability map, is presented and discussed. The degree chosen for the approximation is the most important design parameter of the observer, in that it can be chosen such to ensure the convergence of the observation error at any desired exponential rate, in any prescribed convergence region (semiglobal exponential convergence).

Keywords: Observer design, discrete-time nonlinear systems, polynomial approximation.

1. Introduction

The importance of the problem of asymptotic state observation of nonlinear dynamical systems is well known, and a large amount of scientific literature on this subject is available. Although the case of continuous time systems has been more intensively studied in the past, an increasing number of authors is devoting the attention to the state observation problem in the discrete-time framework. Many papers discuss the extension to discrete-time systems of observer design techniques devised and developed for

*Email: alfredo.germani @univaq.it ** Correspondence to: C. Manes, Email: [email protected]

continuous time systems. An approach widely investigated is to find a nonlinear change of coordinates and, if necessary, an output transformation, that transform the system into some canonical form suitable for the observer design using linear methodologies. First papers on the subjcct are [10], [23], [24], where autonomous systems are considered, and quite restrictive conditions are given for the existence of the coordinate transformation that allows the observer design with linearizable error dynamics. Other papers concerned with the problem of finding conditions for the existence of the change of coordinates are [30] and [31] , for autonomous systems, and [3] and [7] for nonautonomous systems. The drawback of this approach is that in general the computation of the coordinate transformation, when existing, is a very difficult task. Also the technique proposed in [22] req uires to solve functional equations that in general do not admit a closed form solution. An interesting approach for the construction of observers with linear error dynamics for systems admitting a differential/ difference representation is reported in [26]. Another approach consists in designing observers in the original coordinates, finding iterative algorithms, typically inspired by the Newton method, that asymptotically solve suitably defined observability maps (see e.g. [12], [13] and [27]). Sufficient conditions of local convergence are provided, in general, under

Received 10 April 2008; Accepted 18 December 2008 Recommended by D. Normand-Cyrol and S. Monaco.

144

the assumption of Lipschitz nonlinearities. The use of the Extended Kalman Filter as a local observer for noise-free systems has been investigated in [29], [4], [5] and [28]. In [28], sufficient conditions of convergence are given in term of existence of a solution for some Linear Matrix Inequalities (LMI). Some authors propose observers for particular classes of systems. Tn [32] an observer with standard Luenberger fOlm and constant gain is proposed for autonomous systems with the state transition function characterized by Lipschitz nonlinearities and with linear output transformation. Sufficient convergence conditions are given in term of LMI. Nonlinear systems with linear measurements are also considered in [I] and [6]. The problem of asymptotic state reconstruction for systems described by bilinear state-transition functions and rational output functions is considered in [17], where, using the tool of the Kronecker algebra, the problem has been solved through the immersion of the nonlinear system into a linear one of larger dimension. Tn [19] the use of polynomial approximations of nonlinear discrete-time systems for solving the state observation problem has been investigated, and conditions for the exponential convergence of the polynomial extended Kalman filter [18], when used as an observer, are studied.

This paper presents a new method for the construction of an observer for nonlinear discrete-time systems of the type

x(t + 1) = fix(t)), tEll, (1)

yet) = h(x(t)) ,

where x(t) E]R11 is the system state, yet) E JR is the measured output, f: JRI1f----+JRI1 is the one-step state transition map, and h : ]R11f----+JR is the output function. System (1) is assumed to be globally analytic, that means that both f and h are analytic functions in all compact sets of JR". The approach employed here can be considered the discrete time equivalent of the approach followed in [11] and [14]. The method is based on the Taylor polynomial approximation of the inverse of the observability map (the function that maps states into output sequences) up to a given degree 1/ 2: 1, and constitute a nontrivial extension of the one presented in [12], where only the first degree Taylor approximation has been considered, and only local convergence is guaranteed. The main novelty of the contribution here presented is that, thanks to the analyticity assumption, the degree of the approximation can be chosen such to guarantee the exponential convergence of the observation error to zero when the initial observation error is inside a ball of prescribed radius, centered at the origin. It follows that

A . Germani and C. Manes

the proposed observer has the important property to be a semiglobal exponential observer, in that both the convergence rate and the radius of the convergence region can be arbitrarily assigned.

The paper is organized as follows. In section IT some preliminary definitions and notations are given, and previous work of the authors on the subject is discussed. Tn section III the new observation algorithm is presented and its convergence properties are discussed. Simulation results and conclusions follows. An extended Appendix is also present, where the theory of the Taylor approximation of the inverse of a nonlinear vector function is explained in some detail.

2. Preliminaries

In this section some definitions, notations and concepts that will be used throughout the paper are presented, and relations with the authors' previous work are discussed. The symbolr(x), with r E N, is recursively defined as:

f o(x) = x,

fr+l(x) = (fofr)(x) = f(fr(x)) , r > 1 - , (2)

(the symbol 0 denotes the composition of functions) , and is used to denote the r-steps state transition, that allows the computation of future system states as a function of past states:

x(t + r) = fr(x(t)). (3)

The output at time t + r can be written as a function of x(t) as follows

yet + r) = h o.rCx(t)). (4)

Definition 2.1: The observability map for the system (1) is a vector fun ction z + iI>(x) , with iI> : JRI1f----+JRI1 , defIned as:

[

h of~-I (x) 1 <p(x) = : .

hofl(x) hex)

(5)

Let Yt denote the stacked output sequence y ( t) in the interval [t,t + n)

Yt

= [y (t + ~ - 1) l. y( t + I)

yet)

(6)

Polynomial Approximation of the Inverse Observability Map

By the definition of the observability map, the following holds:

Y t = <T?(x(t)). (7)

Definition 2.2: The observability matrix for the system (l) is:

d Q(x) = dx <T?(x). (8)

Following the notations introduced in the Appendix, Q(x) will be also written as Vx<T?(x) or Vx (:9 <T?(x) (see (l00) in the Appendix).

Remark 1: Note that for linear systems the above definition of observability matrix coincides with the classical linear one.

Definition 2.3: The nonlinear system (1) is said to be observable in an open subset 0 <;;;; lR,n if its observability map (5) is invertible in 0, i.e., there exists 1) -1(z) such that <T? - ' (<T?(x)) = x,for all x En. If ° = ]Rn, then the system (1) is said to be globally observable.

The Definition 2.3 of observability states the theoretical possibility of the reconstruction of the state of a nonlinear system, at a given time t, exploiting the knowledge of the output sequence in the interval [t,t + n), as

(9)

Such a possibility in most cases remains theoretical because, in general, closedforms for the inverse of the observability map are not available. Note that the observability property in a set D implies that the observability matrix Q (x) is nonsingular in n. Definition 2.4: The nonlinear system (J) is said to be uniformly Lipschitz observable in an open subset D <;;;; ]R11 iff it is observable, and ifboth maps <T? ( x) and <T? - fez}, are uniformly Lipschitz in D and in <T?(D) , re.)pectiveiy. This means that there exist two real numbers I - I and Iif> such that

II<T?(xd - <T?(X2) II :::; r IIXI - x211 , 'v'XI , X2 ED,

!!<T? - I(ZI) - <T?-I(z2)11 :::; 1<1> Illzl - z211,

liz" Z2 E <T?(n).

(10)

If D = lRn, then the system (1) is said to be globally

uniformly Lipschitz observable.

Definition 2.5: Consider a dynamic system of the form

((t + I) = g(((t) , y (t), e) ,

((t) = m(((t), y(t) , e), t E Z, (11 )

145

where ((t) E lRs is the system state, e E ]R<7 is a set of constant parameters, and y(t) E lR is the output of system (1).

• System (11), with a given choice of e, is said to be a local asymptotic observer for (1) if there exist an open set D ~ ]R1l and 8 E jR+ such that, for any pair x(to), ~(to) ED,

Il x (to) -- ~(to)11 :::; 8 =? lim Ilx(t) - ~(t)11 = o. /-+0

( 12)

• The system (1 t) is said to be a semiglobal asymptotic observer if for any D ~ ]R11 and 8 E jR+ there exists a choice e such to guarantee the implication (12) for any pair x(to), ~(to) E D.

• The system (11), with a given choice of e, is said to be a global asymptotic observer for (1) if for any pair x(to) , ~(to) E ]R11 it is limt-->o Ilx(t) -~(t)11 = o.

Remark 2: Assume that a system of the type (11) provides a sequence ~ ( t ) such that for all pairs x(to), ~(to) E ]Rn it is

(13)

If the system ( 1) is uniformly Lipschitz observable, then ( 11) is a global asymptotic observer. This happens because the Lipschitz property implies the following

Ilx(t) - ~(t) 1 1 = 11<T?-I(<T?(x(t))) - <T?-'(<T?(~(t)))11

= II <T? - I ( Yt ) - <T? - I ( <T? ( ~ ( t) ) ) \I

:::; 1<1>- 111 Yt - <T?(~(t))II ,

(14 )

and therefore Ilx(t) - ~(t)11 - O. Consider a globally observable system of the type

( 1). Thanks to the assumption of invertibility of the observability map, this can be used to define a new set of coordinates for system ( 1):

z(t) = <T?(x(t - n + 1)). ( 15)

Note that z(t) is used here to represent the system state at time t - n + 1. Recalling the identity (7), it follows that

z( t) = Yt - n+l. ( 16)

146

Let (Ab,Bb,Cb) be a Brunowsky triple of dimension n, defined as:

0 0 0 0 0

1 0 0 0 0

Ab = 0 1 0 0 0 E }RI1 XI1 ,

0 0 0 0

1

0

Bb 0 E }R11, Cb = [0 0 0 ... I] E }R11.

o (17)

Using these matrices the following equations holds

Yt - I1+2 = Ab Yt- n+ 1 + BbY( t + 1),

y(t-n+ 1) = CbYt - n+l . ( 18)

With the substitution z ( t) = Y, _ n + I, these can be rewritten as

z(t+ 1) = AbZ(t) + Bby(t + 1),

y(t - n + 1) = Chz(t) ( 19)

II' the inverse map - I( .) were available, then the original state x(t) could be easily computedfrom z(t) in two steps: first x(t - n + 1) is computed as

x(t - n + 1) = -I (z(t)) , (20)

and then x(t) is computed as

x(t) =fn-'(x(t-n+ 1)) = f n- 1(-I(z(t)).

(21 )

ltfollows that, using the z-coordinates, system (1) can be rewritten as:

z(t+ I) = AbZ(t) + BbhoP(-l(z(t))) ,

y(t) = h ofl1- 1( - 1 (z(t))). (22)

The problem is that a closedform for the inverse of the observability map of a given system is not available, in general. Numerical iterative methods can be usedfor the computation 01' x(t - n + 1), by finding the (unique) root of

z(t) - (x) = O. (23)

A. Germani and C. Manes

In [12] the following observer structure was proposed

'l/J(t+ 1) = j( 1j;(t))

+(Q(f(1j;(t))))-1 (Bb (y(l+ 1) -hop (1j;(t)) )

+K(y(t-n+ l)-h(1j;(t))))

x(t) = fl1-I( 1j;(t)), t>to=n-l,

(24)

where Q( .) is the observability matrix (8) of system ( 1). For this observer, the following local convergence result holds true (Theorem 2.1 of[12])

Theorem 1: For a system of the type (1) there exists a finite gain vector K E }Rn and a suitable b > 0 such that the system (24) is a local observer, provided that:

• System (I) is globally Lipschitz observable definition 2.4);

• h andf are uniformly Lipschitz in }RD;

• Ilr-I(1j;(to)) -x(to)11 > b.

(see

The proof of this theorem in [12] exploits the representation (22) of system (I). Note that the theorem is here stated using the notations introduced in this paper, and that the Lipschitz observability condition implies the full rank property of the observability matrix Q.

3. The Polynomial Observer

The observer proposed in this paper is based on the idea of using the Taylor approximation of a chosen degree of the inverse map as a numerical method for solving (23). This approach has strong connections with the iterative technique presented in [16], which generalizes the Newton-Rapson approach. The observation algorithm derived is characterized by the classical prediction-correction structure, but in this observer the correction term is a polynomial function of the output prediction error, whereas standard algorithms have a linear correction term.

The main assumption needed for the construction of the proposed observer is that both functionsfand h in (1) are analytic in all }Rll. Well known results of real analysis establish that the composition of analytic functions gives back an analytic function, and that if an analytic function is invertible, then the inverse is analytic too. It follows that the observability map z = (x) defined in (5) is analytic in all IRD

, and that globally observable analytic systems admit an analytic inverse x = - 1 (z) in all IRll.

It is important to stress that the Taylor polynomial approximation of the inverse of the observability map z = <I)(x) can be computed without the explicit knowledge of the inverse function x = <I) - I(Z), by using only the derivatives of <I)(x). The mathematical tools needed for the construction of the polynomial Taylor approximation of inverse functions are described in the Appendix. Using the notations stated in the Appendix, the following derivatives can be defined

Gdz) = V1kJ 0 <I)-I (z), k = 0, 1, ... , (25)

where Go(z) = <I) - I(Z) and G1(z) is the Jacobian '1;;(1) - 1 (z). Let

rk = sup IIGk(z)ll. (26) ZElFtn

The Taylor theorem states that

v 1 <I)-I (z) = ~ k! Gk(z)(z - z)lkJ + rll+1 (z, z),

(27)

where the remainder of the series is infinitesimal of order v + 1 w.r.t. (z - z):

Ilrl/+1 11 ::; (: :+~)! liz - zill/+I. (28)

The exponents [kJ in the Taylor formula (27) denote the Kronecker powers of matrices and vectors (see Appendix).

Proposition 2: If; 'lip > 0, the sequence rk defined in (26) is such that

r k 1· kP lm-----k---->o(k + I)! = 0,

(29)

then <I) - 1 (z) is analytic in alllRf1.

Proof The proof is a straightforward consequence of the inequality (28). •

Let

Gk(X) = Gk 0 <I)(x) = (V1kJ 0 <I)-I (Z)) z= <P(x )

(30)

and let z = <I)(x) and x = <I)-I (z). Observing that v 10J 0 <I)-I (z) = <I)-I (z) = x, i.e. Go(x) = x, the Taylor formula (27) becomes

1/ 1 x = ,Y + t; k! Gk(X)(Z - <I)(x))lkJ + rl/+ I (z, <I)(x)) ,

(31 )

147

where

Ilrv+111 ::; (::+~)! liz - <I)(x)llv+l. (32)

Note that rk = SUpzElFtn IIGk(z)11 = SUPxElFtn IIGk(x)ll. The main result proved in the Appendix is that the

sequence of derivatives Gk(X) of the inverse function <I) - I( . ) can be computed as functions of the matrix coefficients FIl.llx) recursively defined as

Fk,k = (V;/f» lk] = QlkJ(x), k = 1,2, . . .

Fh ,k = V x 0 Fh ,k-I + (Vx<l)) 0 Fh- l ,k-l,

h=l, .. ·,k - l ,

FO,k = 0. (33)

Lemma 9 of the Appendix proves that the sequence of matrices G,.;( x) needed in the Taylor formula (31) can be computed as

Go = x, G1 = (V'x<l))-l,

k = 2,3,···.

(34)

Note that

(35)

What is proposed in this paper is a family of observers, whose construction is based on the truncation of the Taylor series (31). Each observer in the family is characterized by the degree of the polynomial approximation, and recursively provides at time t an estimate of the past state x(t - n + 1), denoted xU). From this, an estimate x(t) of the current state is obtained as x(t) = f- I (X(t)). The observer starts the operations at a given time to, while the output measurements are available starting from time to _ 11 + 1. The observer state is X(t), whose initial value X(to) is an a priori estimate of x(to - n + 1). The recursive equations of the proposed observer are the following:

Polynomial Observer of degree v

91 (t) = (Q(f(x(t)))) -I, (36)

(I, (t) ~ - (~ g,,(t)Fh~ (f(x(t)))) 9\'1, (37)

k = 2, ... , v

148

~x,y( t + 1) = K(y(t - n + 1) - h(X(t)

+ B,,(y(t + 1) - h 0t(X(t)) ),

/1 I X(t+ 1) =f(x(t)) + 8k!9k(t)~~,ly (t+ 1),

x(t) = r-I(x(t)) ,

(38)

(39)

(40)

Equation (40) provides the observed state x(t). In the observer equations, the observability matrix Q(x) defined in (2.2) and the matrix functions Fh,k(X) defined in (33) are needed. Note that Fl,l(x) = Q(x) . The gain vector K E JRIl must be chosen such that all eigenvalues of Ab - KCb are in the open unit circle in the complex plane (Ab and Cb are the Brunowsky pair defined in (17».

Remark 3: Comparing equations (36)-(37) with equations (34) it is clear that the matrices 9dt) in the summation (39) are the coefficients (h of the Taylor approximation of the inverse map evaluated at z =.f(X(t)). Note also that, when // = 1 the algorithm . (36) - (40) coincides with the observer (24) presented in [12].

Remark 4: Note that in the observer equations (36) (40), the only matrix inverse to be numerically computed at each time step is the inverse of the observability matrix Q( . ), evaluated at fix(t» , in (36), independently of the order // of the observer. The matrix functions Fh,

k(X) at each time step must be evaluated at f(x(t )) . Considering the dimension of the matrices in the summations (37) and (39)

(41 )

It IS easy to see how the complexity increases by increasing the observer degree. However, the only operations involved, after the evaluation of the coefficients Fh,k, are simple products and sums of matrices.

The following theorem provides conditions for semiglobal exponential convergence of the observer (36) - (40). For convenience, in the proof the gain vector Kin (38) is chosen such that all eigenvalues of Ab - KC" are real and distinct.

Theorem 3: Consider the system (1) and its observability map defined in (5). Consider system (36) --(40), where K is such that all eigenvalues of Ab - KCb are real and distinct in the interval ( - I,]),

Assume that: Hpl : the system (1) is uniformly globally Lipschitz

observable;


H p2 : The functions f and h are analytic in all JRI1 and uniformly Lipschitz in JR/!, i.e.:

:3l'h >0: Ilh(xl) - h(x2)11 ~ I'hllXI - x211, 'v'XI,X2 E JRIl;

:31'1> 0 : Ilf(xl) - f(x2) II ~ I'flixi - x211, 'v'XI ,X2 E JR/!;

(42)

Hp3 the constants r k defined in equation (26) are such that: r k < 00, 'v'k E N, and

'v'p > 0, (43)

Then, for any pair of real numbers ex and 8, with ex E (0,1) and /5 > 0, there exist an integer //, the degree of the observer (36)-(40), and a positive real f-t such that

Ilx(t) - x(t) 11 ~ f-t ext- to Ilx(to - n + 1) - x(to)II,

(44)

for all pairs X(to), x(to - n + 1) E JRIl such that Ilx(to - n + 1) - x(to)11 ~ 8.

Proof By assumption Hp2, both f and h are analytic functions in all JR'\ and therefore the observability map is analytic in all JR/!. Thanks to the global observability assumption HPi' the function is invertible and analytic in all JRI1 , and therefore acts as a global change of coordinates (note that global analyticity of - I is also ensured by HPJ' see Proposition 2). Let (t) be the -transformed state of the observer (36)-(40), defined by:

i(t) = <T>(X(t)) , (45)

which is an estimate of z(t) = Y(t _ 11 + I) (recall that z (t) = (x(t - n + 1»), and define the error in z-coordinates as e(t) = z(t) - z(t). From the observer equations (36)- (40), considering the equivalence 9k(t) = Gdf(x(t))), the update law of z(t) takes the form

z( t + 1)

~ (fiX(t)) + t ~! Gk(f(X(t) )L'.~1 (t + I)), (46)

Consider now a sequence i( t) defined by the update law

z(t + 1) = (f(X(t))) + ~X,y( t + 1), (47)

where ~X,y(t+ I )=K(y(t-n+ 1 )-h(X(t)) )+Bh(y(t+ I) -hor(X( t))) , as defined in (38). Note that, by definition of the map <T>, it is

(f(x(t))) = Ab(X(t)) + Bb h o.r(X(t)),

and h(X(t)) = CbZ(t).

Thus, recalling (45), it follows

(48)

z(t + 1) = AbZ(t) + Bb h Ofll(X(t)) + ~X,y(t + 1)

= (Ab - KCb)Z(t ) + Kh(X(t))

+Bb h 0 fll(X(t)) + ~X,y(t + 1)

(49)

and, by the definition of ~x,y(t + 1), it is

Now, rewrite (19) for z(t + 1) as follows

z(t + 1) = (A b - KCb)z(t) + Bby(t + 1)

+Ky(t-n+ 1).

Using this and (50) yields

149

(56)

z(t + 1) - z(t + 1) = (Ab - KCb) (z(t) - z(t)),

(57)

and using (54)

e(t+ 1) = (Ab - KCb)e(t ) + r(t+ 1); (58) z(t + 1) = (Ab - KCb) Z(t) + Bby(t+ 1)

+ K y( t - n + 1). (50) where

Now, consider the Taylor expansion of - I(Z(t + 1)) around (f(X(t)) (see eq.(31):

-I(z(t + 1)) =f(x (t)) II 1 _

+ Lk!Gk(f{X(t)))~~,~ (t+ 1) k= 1 .

+ 1'11+1 (t + 1)

= X( t + 1) + r 11+ 1 (t + 1) ,

(51 )

where the remainder rv + I(t + 1) is such that

Ilrll+l(t+ 1)11::; (:~+~)l I I Z(t+ 1) - (f(x (t))) IIII+1

< r ll+1 II~ ,(t+I)II II+l . -(v+l)l X,)

(52)

Rewrite equation (51) as

Then

where

x (t + 1) = - I (Z (t + I)) - r 11+ I (t + 1).

e(t + 1) = z(t+ 1) - z(t+ 1)

= z(t + 1) - (X(t + 1))

= z(t+ I) - (-I(Z(t+ 1) )

- r 11+ I ( t + 1))

= z( t + 1) - z( t + 1)

(53)

+ (11

('Vx (x)) ~(.\') dS' )rll+l(t + 1)

(54)

~(s) = -I(z(t+ 1)) -srll+I(t+ 1) . (55)

r(t+ I) = (11('VX(X)) x=~(S) d))rHI(t+ 1).

(59)

By the Uniform Lipschitz Observability assumption II 'Vx (x) II ::; I q" 'l/x E R Il

, and therefore

(60)

Recalling the bound (52) on r v + I (t + I), and observing that, from definition (38),

II~x,y(t + 1) II::; I h(IIKII + IJ) Ilx(t - n + 1) - X(t) II ::; l it (IIKII +IJ )rq, I Ilz( t) - z(t) II

(61 )

it follows

(62)

where

(63)

By assumption, K is such that all eigenvalues of Ab - KCb are real and distinct. Let KA denote the gain that assigns a set of eigenvalues A = {>.1, ... ,AIl }, and let AMax = maxj= I, .. . ,n{ll-\ill}· Choose A such that AM ax

< a . Let VA denote the Vandermonde matrix

- [~ ~I . ... A7.-1

J VA - : : : : '

1 A . .. ).'1-1 11 /J

(64)

which is nonsingular, being Ai of \' i =1= j , and such that

(65)

150

It is easy to verify that

[All]

K;.., = _V:\I :1 . )..'1

n

(66)

Note that, being AMax < 1, it is v'n ::; II V;.., II < nand 11K;.., II ::; II V:\'IIv'n· Consider the following transformation of the error:

''7(t) = V;..,e(t). (67)

Then

.,,(t + 1) = A.,,(t) + V;..,Nt + 1) (68)

which implies that:

1177(t+ 1)11::; AMax ll77(t)11 + IIV;..,llllr(t+ 1)11 ::; AMaxll77(t) 11

+ II V 11;3( ) II V>:'77(t)llv+'

;.., 1/ (1/+ I)! ::; AMax IIT/( t) II

+11V,1111V~11l "+ Ii1(v) 1!:(21~~:1 ::; (AMax + II V;.., II II v:11(+1

(3(1/) 1177(t)llv

)II77(t)ll. (1/+1)!

(69)

Now, choose the degree of the observer 1/ such that, for the given a E (0, I) and 6 > 0, it is

(: :+~ )! (II V;.., II II V:\III(IIK;..,II + /rhh/'P -- I t+'

::; D( a - AMax) . (70)

This inequality can be satisfied for a sufficiently large 1/, thanks to the assumption HPJ' From this, thanks to the definition of ;3(1/) given in (63), it follows

A + II Villi V- I II /.1+1 (3( ) (II V;..,I!t'PDt < Max ;..,;.., 1/ (1/+ 1)! _ a. (71 )

It is easy to see that if at a given time t it is IIx(t) - x(t - n + 1)11::; D, then

1177(t)11 = II V;..,e(t) II = II V;.., ((x(t - n + 1» - (x(t) II ::; II V;.., I 1t'PD,

(72)

A_ Germani and C Manes

so that

IIT/(t)II V+' = IIT/(t)II V II77(t)11 ::; (II V;..,I!t'PDtll77(t) II· (73)

From this, thanks to inequality (69) it foHows

1177(t + 1)11 ::; (AMax + (II V;.., II II V;l llt+1

(t;p8)" (74) (3(1/) (1/ + 1 )!) 1177(t)ll ,

and thanks to (71)

1177(t + ])11::; all77(I)II· (75)

Thus, being a E (0,1), if at time t It IS 1177(t)ll::; II V;.., IIt'PD, then also at time t + 1 it is 1177(t + 1) II ::; II VAI!t'P8. Therefore, if at to Ilx(to) - x(to - n + 1)11 ::; D, then 1177(to) II ::; II VA I 1t'P 8, and thanks to (75)

(76)

This implies

Ile(t)ll ::; at-toIIVAIIIIV:\llllle(to)II, (77)

and, being e(t) = 11(x(t - n + 1» - (x(t»II , it follows

Ilx(t-n+ l)-x(I)11 ::; at-to/'P/'P_I I IV;.., 1111 V:\ 111 11x( to - n + 1) - X( to) II·

(78)

and finally

Ilx(t) - x(t) 11 ::; at-to,,/;-I/'P/'P III VAil II v:\lll Ilx(to - n + 1) - x(to)ll, (79)

which is the thesis (44), with J..l = /;-I/'P"/ifJ-I II V;.., II II V>:'II · •

4. Simulation Results

The polynomial observer (36)-(40) has been tested on many systems. In general, the theoretical convergence results have been confirmed by the simulation tests. Here, some results concerning a chaos synchronization problem have been presented.

Consider the well known Henon map [20]

(80)

which, for some values of the parameters a and b, defines a discrete time chaotic system. The canonical values of the parameters of the Henon map are a = 1.4 and b = 0.3, which produce a chaotic behavior of the system.

The polynomial observer (36)-(40) has been applied to the problem of estimating the state of the Henon system when the constant parameter b is unknown. Only the first state variable XI(t) is assumed to be measured. This problem is known as a chaos synchronization problem with uncertain parameters (see e.g. [15], [25]).

The unknown parameter b is modeled as a constant state variable, i.e. X3(t + 1) = X3(t) = b. The system equations considered for the observer construction are the following

[

XI(t+ I)] [1-1.4XT(t)+X2(t)] X2(t + 1) = X[ (t)X3(t)

X3(t + 1) X3(t) (81 )

y(t) = X[.

The polynomial observers of orders v = 1,2, 3 and 4 have been applied. The matrix functions Fh,k(X), defined in (33), where k = 1, ... ,4 and h = I,... k, have been computed using the symbolic toolbox of MATLAB(Q, by reproducing exactly the recursive computations in (33).

The map is as follows:

[I - 1.4( I - 1.4X; + X2)' + X,XI ] (x) = 1 - 1.4xT + X2

XI

(82)

[7 .84( I - 1.4X; + X2)XI + Xl

F I, I (x) = Q(x) = -2.8.\:[ 1

[ -32.928X; + 7.84 + 7.84x2 7.84xI F I,2(X) = -2.800 0

0 0

In matrix F1,3(x) the only nonzero elements are [F1,3]1,l = - 65.856x1 and [F1,3]1,2 = [F1,3]I,4 = [F[ 3]1 10 = 7.84. Matrices Fh k for h > 1 are rather big

~ , ,

sparse matrices, and can not be easily re),?orted. Note that matrices Fk,k can be computed as Fj~~ (see (33)).

The gain matrix used in the simulations here reported is the following

K = [-0.0017 0.0428 -0.3600]' (85)

151

1.5 ,--------,-----.----~--------,

0,5

o

--{).S

-1

-1.5 o'------------'-----:-'-----:-':------=-'

Fig. 1. True and observed variable Xl,

and is such to assign eigenvalues Al = 0.1, A2 = 0.12 and A3 = 0.14 to the Brunowsky pair (Ab' Cb) of dimension n = 3. The Figs 1-3 report simulation results where all observers for v = 1,2,3 and 4 provide convergent state estimates. The true and observed initial states are

X(O) = [-0.6 -0.1 0.3]T

x(O) = [-0.3 -0.2 0.2 f'. (86)

It can be seen that higher order observers produce faster transients.

In many cases, the first order observer does not converge, while higher order ones are convergent. For instance, when the initial states are as follows

X(O) = [-0.6 -0.1 0.3]T

x(O)=[O 0 0.2]T, (87)

-2.8 + 3.92xT - 2.8x2 XI]

0 0

10 (83) 0 0

7.840x[ -2.800 0 1 0

~] 0 0 0 0 0 0 0 0 0 0

(84)

the first order observer does not converge.

5. Conclusions

A new technique for the observer construction for analytic discrete-time nonlinear systems has been presented in this paper. The method is based on the Taylor


20. Henon M. A two-dimensional mapping with a strange attractor. Commun Math Phy 1976; 50, 69-77.

21. Graham A. Kronecker Products and Matrix Calculus with Applications, Ellis Horwood: Chichester, 1981.

22. Kazantzis N, Kravaris C. Discrete-time nonlinear observer design using functional equations. Syst Control Lett 2001; 42, 81 - 94.

23 . Lee W, Nam K. Observer design for autonomous discretetime nonlinear systems. Syst Control Letl 1991; 17: 49-58.

24. Lin W, Byrnes CI. Remarks on linearization of discretetime autonomous systems and nonlinear observer design. Syst Control Lett 1995; 25: 31-40.

25. Millerioux G, Daafouz J. Global chaos synchronization and robust filtering in noisy context. IEEE Trans Circuits Syst 1, Fundam Theory App12001; 48(10): 1170- 1176.

26. Monaco S, Normand-Cyrot D, Barbot JP. Observers for differential-difference representations of discretetime systems. Proceedings of the European Control Conference 2007, 5783-5788, Kos, Greece, 2007.

27. Moraal PE, Grizzle lE. Observer design for nonlinear systems with discrete-time measurements. IEEE Trans Autom Control 1995; 40: 395- 404.

28. Reif K, Unbehauen R. The extended Kalman filter as an exponential observer for nonlinear systems. IEEE Trans Signal Process 1999; 47, 2324-2328.

29. Song Y, Grizzle JW. The extended Kalman filter as a local asymptotic observer for discrete-time nonlinear systems. J Math Systems Estimate Control 1995; 5, 59- 78.

30. Xiao M, Kazantzis N, Kravaris C, Krener AJ. Nonlinear discrete-time observer design with linearizable error dynamics. IEEE Trans Au/om Control 2003; 48: 622-626.

31. Xiao M.A direct method for the construction of nonlinear discrete-time observer with linearizable error dynamics. IEEE Trans Autom Control 2006; 51: 128-135.

32. Zemouche A, Boutayeb M. Observer design for lipschitz nonlinear systems: the discrete-time case. IEEE Trans Circuits Syst II, Express Briefs 2006; 53: 777-781. .-

Appendix

Polynomial Approximationof Inverse Mappings

This appendix describes the mathematical tools of the Kronecker algebra, used in this paper to represent the formulas of the Taylor series of vector functions of several variables and of their inverse functions. Here, the symbol I" denotes the identity matrix in JR

I1 XI1.

The Kronecker product of two matrices M and N of dimensions p x q and r x s respectively, is the (p . r)

x (q - s) matrix

[

mllN

M ® N= :

mplN

(88)

153

where the mil are the entries of M. The Kronecker power of a matrix M is recursively defined as

Note that if ME JRp x q , then M [i] E JRp'x q' . Some useful properties of the Kronecker product are the following;

(A + B) ® (C+ D) = A ® C + A ® D + B ® C

+B®D

(90)

A ® (B ® C) = (A ® B) ® C (91)

(A· C) ® (B· D) = (A ® B) . (C ® D) (92)

det(A ® B) = det(B ® A) (93)

Repeated application of the properties (91) and (92) provides the identities

(AB) [i] = A[i]B[i] and (A[k])-l = (A-1)[kl .

(94)

Using (92), given A E JRn x l1 and B E JRm x m

, the following properties can be easily derived

(95)

(96)

From the last, thanks to (93), it follows det(A ® B) = det(A ® 1m) det(In ® B). Using (93) it is det(In ® B) = det(B ® In). Recognizing the block diagonal structure of 1m ® A and of 111 ® B it follows

det(A ® B) = (det(A))m(det(B)Y. (97)

A quick survey on the Kronecker algebra can be found in the Appendix of [9]. See [21] for more properties.

The Kronecker formalism is also useful for representing derivatives of matrix functions of several variables, and provides a compact notation for the Taylor polynomial approximation of nonlinear vector functions. Let M(x) be a smooth matrix function of x E JRI1, i.e.: M: jRl1f-+JRrxc, and let \7x ® M: jRl1f-+jRrxl1-c denote the following matrix of derivatives:

(98)

154

This matrix Jacobian is a formal Kronecker product of the vector \7x = [a/(Jxl ... a/axnl with the matrix M(x). Repeated derivatives can be recursively defined as

\7~l ® M= M,

\7t+1J ® M = \7 x ® (\7tJ ® M) , i? 1.

(99)

Let F : JR.11 ---+ JR.nJ be a smooth nonlinear map. In this case \7 x @ F is the standard Jacobian of F, and therefore throughout the paper the following symbols will be indifferently used

dF , \7 x ® F = dx = \7 x F . (100)

The last symbol provides the most compact notation. Note that for matrix functions the use of \7x@ is necessary.

A useful property of the operator \7x is the following:

\7x @ (A(x) . B(x)) = (\7 x 0 A) (In ® B)

+ A (\7x ® B) ,

where A and B are generic matrix functions.

(101)

Let F(x) and H(z) be functions F : JR.111---*JR.11 and H : JR.11 ---+ JR.rx c. The composi tion H 0 F : JR.n ---+ JR.rx c is defined as

H 0 F(x) = H(F(x)) . (102)

Proposition 4: If H(z) and F(x) are differentiable, then

\7x @ (H 0 F) = ((\72 @H) 0 F) ((\7xF) @Ie).

( 103)

where the term

(\7;: 0 H) 0 F(x). (104)

denotes the derivative of H w.r.t. z computed at z= F(x).

Proof Observe that

V" 0 (H 0 F) = [a( H 0 F) aXI

a(H 0 F)] aXil '

(105)


and that

a(HoF) = taH(z) aFj(x) ox; . 1 azJ· aXi

J= F(x)

aFI1[ OX; C

= ((\7z @ H) 0 F) (~~ @Ie).

(106)

Substitution of this into the right hand side of (105) provides equation (103).

Throughout the paper, the symbol Sx,,(p) will denote the open ball of radius p centered at Xo E JR.I\ i.e. Sxo(p) = {x E JR.1l: Ilx - xoll > p}. The symbol Sxo (p) will denote the closure of Sxo(p).

Using the Kronecker notation, the Taylor Theorem for vector functions of several variables can be written as follows:

Theorem 5: Let F : JR.11 ---+ JR.m be a function defined on the closed ball. S Xo (p) ,for some Xo E JR.n and p > O. If,for a given integer v, all derivatives \71kJ ® F, k = 1, ... , v + 1, exist and are continuous at every point of Sx/ p), then the following identity holds for all x E Sxo(p):

IJ I F(x) = ~ k! (\7~l ® F(x)) Xo (x - xO) [kl

+ Rv+l (x , xo)(x - XO)[IJ+l] ,

(107)

where the remainder sati4ies the inequality

IIRl/+I(x)11 ::; ( 11)' sup II\7t+ll ® F(x)ll. v + . XESxo (p)

(l08)

The function F is said to be analytic in the set Sxo(p) if the Taylor series is convergent for all x E Sxo(p). Defining

'Pk(XO, p) = sup 11\7~l @ F(x) I\, (109) XESxo(p)

a sufficient condition for the analyticity ofF in Sx/ p) is the following

lim ipk(XO , p) = O. (110) k->oo k!

Polynomial Approximation 0/ (he Inverse Ohservahility Map

Theorem 6: Let F: jRll I--)-jRll be a smooth function defined on alljRl1. If, for any choice of Xo E jRlI and of p E jR + the sequence CPk(XO, p) defined in (109) is such that

I· lPk(XO , p)/ - 0 1m k' - ,

k --400 • ( 111 )

then F is analytic in all Rn. Proof The proof is a direct con seq uence of the remainder expression (108).

Let F - I: Rfl -----* Rn denote the inverse map of the function F , i.e.

( 112)

so that F - 10 F is the identity map in RII: F-I(F(x)) = x. Let us define the following notation for the derivatives of the inverse function F - l:

A useful equivalent recursive definition is

90(z) = F- I (z) ,

9k+1 (z) = \7z ® 9k(Z) , k = 0, 1, ... , ( 114)

Lemma 7: The derivatives of the inverse mapping F - ' , i.e. the matrix functions 9k (Z) , k= 1,2, ... , defined in equation (113), satisfy the following chain of identities:

where

k

L(9h 0 F)Fh,k = 0, k = 2,3 , . .. , h=1

rk,k = (\7xF)[k], k = 1,2, ...

Fh,k = \7 x ® Plz ,k- I + (\7 xF ) ® Fh - I ,k-I ,

h= 1, ... , k - l ,

FO ,k = O.

(115)

( 116)

( 117)

Proof The property (91 0 F)Fl.I = In, with Fl,1 = V' x F , it will be proved in Lemma 8. From this

Y\: ® ((91 0 F)FI ,I) = o. (118)

The computation of the derivative, using (101), gives

\7 x®( (91 of)Fl,l) = (920F)(\7 xF ®In)(I,JSl FI,l)

+(9l oF)(\7x®Fl,l)=O.

(119)

155

From the property (A® B)(C® D) = (AC)®(CD) it follows

(\7 xF ® In) (In c>9 FI ,d = \7 xF c>9 FI ,I = (\7xF) [2]

= F2,2 '

(120)

Moreover, from (117), it is F1,2 = V' x® Fl ,l. Therefore equation (119) can be written as

(121 )

so that (116) is proved for k = 2. The identity (116) for k > 2 will be proved by induction. Then, assume that (116) holds for some k '2 2, and prove that it holds also for k + 1. If (116) is true, then

(122)

Differen tiating, and using (10 1), yields

k

L ( (9h+ I 0 F) ((\7xF) ® Inh) (In ® Fh ,k) h= 1 (123)

+ (9h ® F)(\7x ® Fh ,k )) = o.

Reorganizing the subscripts one gets

(9k+ I 0 F) ((\7xF) ® Ink) (In ® Fk,k) k

+ L(9h 0 F)(\7 x ® Fh,k h=1

that can be rewritten as

k+ 1 L(9h 0 F)Fh,k+1 = 0, h=1

where

Fk+l ,k+l = ((\7xF) ® lnk) (In ®Fk,k),

(124)

(125)

Fh,k+1 =\7x ® F/z ,k + ((\7x F)®Inh J)( ln ® Fh-l ,k),

1 ~ h ~ k.

(126)

Exploiting the property (A ® B)(C® D) = (AC)® (BD)

A+I,k+1 = (\7xF) ® Fk,k,

Fh,k+1 = \7 x ® Fh ,k + (\7xF) ® Fh- I,k, (127)

1 ~ h ~ k.

a discrete-time observer based on the polynomial...

Documents