Int. J. Appl. Math. Comput. Sci., 2013, Vol. 23, No. 2, 383–394DOI: 10.2478/amcs-2013-0029

STATE ESTIMATION FOR A CLASS OF NONLINEAR SYSTEMS

BENOÎT SCHWALLER ∗, DENIS ENSMINGER ∗ , BIRGITTA DRESP-LANGLEY ∗∗,JOSÉ RAGOT ∗∗∗

∗ Laboratory of Inventive Design (LGECO), EA 3938 INSA StrasbourgUniversity of Strasbourg, 24 Boulevard de la Victoire, 67000 Strasbourg, France

e-mail: schwaller@convergence.u-strasbg.fr

∗∗Laboratory of Mechanical and Civil Engineering (LMGC)University of Montpellier 2, UMR 5508 CNRS, 860 route de St. Priest, 34090 Montpellier, France

e-mail: dresp@lmgc.univ-montp2.fr

∗∗∗Centre for Automatic Control of NancyUMR 7039, University of Lorraine, CNRS, 2 avenue de Haye, 54516 Vandœuvre lès Nancy, France

e-mail: Jose.Ragot@ensem.inpl-nancy.fr

We propose a new type of Proportional Integral (PI) state observer for a class of nonlinear systems in continuous timewhich ensures an asymptotic stable convergence of the state estimates. Approximations of nonlinearity are not necessary toobtain such results, but the functions must be, at least locally, of the Lipschitz type. The obtained state variables are exactand robust against noise. Naslin’s damping criterion permits synthesizing gains in an algebraically simple and efficientway. Both the speed and damping of the observer response are controlled in this way. Model simulations based on a Sprottstrange attractor are discussed as an example.

Keywords: nonlinear systems, state observers, continuous time.

1. Introduction

State observers have been intensely exploited sincethe work of Luenberger (1966) to model, control, oridentify linear and nonlinear systems, as in the studiesby Krener and Isidori (1983), Zeitz (1987), Bastin andGevers (1988), Boutat et al. (2009), and Zheng et al.(2009), relating to nonlinear systems transformable intoa canonical form. The key idea in such approachesis to produce approximate measures of nonlinearityof the order 1, as in Extended Luenberger Observers(ELOs) (Ciccarella et al., 1993). The approximationof nonlinearities in the canonical form (which resultsin an ELO) has already been suggested (Bestle andZeitz, 1983), and this approach can be extended to higherorder approximations (Röbenack and Lynch, 2004). Anobserver using a Partial nonlinear Observer CanonicalForm (POCF) (Röbenack and Lynch, 2006) has weakerobservability and integrability existence conditions thanthe well-established nonlinear Observer Canonical Form(OCF).

Nonlinear sliding mode observers usea quasi-Newtonian approach, applied afterpseudo-differentiation of the output signal (Huiand Żak, 2005; Veluvolu et al., 2007; Efimov andFridman, 2011). State observers using Extended KalmanFilters (EKFs) provide another method of transformingnonlinear systems (Boutayeb and Aubry, 1999; 2010;Khémiri et al., 2011). In particular, Besançon et al. (2010)exploit an adaptive observer based on Kalman filters forphysical systems to which Liénard transformations canbe applied. Finding an appropriate method for parametersynthesis remains one of the major difficulties withstate observers for nonlinear systems (cf. the works ofTornambè (1992), Besançon et al. (2004), Boizot et al.(2010) and Farza et al. (2011), where high-gain stateobservers to deal with this problem were proposed).High-gain state observers reduce observation errors fora range of predetermined amplitudes or fluctuations bymaking the observations independent from parameters. Aweak point of this method is its sensitivity to noise anduncertainty.

schwaller@convergence.u-strasbg.frdresp@lmgc.univ-montp2.frJose.Ragot@ensem.inpl-nancy.fr

384 B. Schwaller et al.

In network identification and encryption, observerswith delays are used to synchronize chaotic oscillators, asshown in several studies (Feki, 2009; Ibrir, 2009; Ghoshet al., 2010). Noise and uncertainty are not criticalfactors in such a context. This can be very different inthe case of industrial processes, as shown in a recentstudy by Bodizs et al. (2011), where the performancesof observers using ELOs, EKFs or Integrated KalmanFilters (IKFs) are compared. The influence of noiseand uncertainty on these observer types was emphasized,with more reliable results produced by ELOs, whichpermits the exact state reconstruction of highly perturbedsystems. For PI and ELO observer classes, Söffkeret al. (1995) as well as Morales and Ramirez (2002)demonstrated a compensation effect on measurementerrors. Abdessameud and Khelfi (2006) as well as Chenet al. (2011) addressed the problem of uncertainty ofnonlinear models.

In the present study here, we deal with a specific classof nonlinear SISO (Single Input Single Output) systems,described by Fliess (1990), called the generalizedcontroller canonical form. In principle, every uniformlyobservable (Hermann and Krener, 1977; Gauthier andBornard, 1981) sufficiently smooth SISO system withinput u and output y can be transformed into this normalform,

ẋ(t) = A x(t) + f(t), (1a)

y(t) = cT x(t) + Φ [ u(t) ] , (1b)

A =

⎡⎢⎢⎣

0 1 0 . . .0 0 1 . . .

. . . 0 0 1−a1 −a2 . . . −an

⎤⎥⎥⎦ , (1c)

cT =[c1 . . . cn

], (1d)

fT (t) =[0 . . . Ψ [ x(t), u(t)]

], (1e)

with the following definitions.

Definition 1.

n order of differential equationu1(t) input of systemui(t) (i − 1)-th temporal derivative of

u1(t)uT (t) input vector [u1(t), . . . , un(t)]xi(t) (i − 1)-th temporal derivative of

x1(t)xT (t) state vector [x1(t), . . . , xn(t)]ai, ci model parametersc output vector of dimension nΨ [ x(t), u(t)] a nonlinear function of the type C1

Φ [ u(t) ] function of inputs u(t)y(t) output variable

θ ≤ n index of last coefficient ci �= 0ω0 = n

√a1 eigenfrequency of the system.

We introduce a PI observer with a modified high gainobserver, which includes nonlinear functions and linearparameters. This approach has two advantages:

• it permits an asymptotic stable reconstruction of staterepresentations without having to increase gains toinfinity;

• it obtains the limiting conditions of the observerstability in a systematic way, without makingapproximations on the nonlinearities, and withouthaving to employ optimization algorithms (e.g., bysolving Riccati equations).

The proposed approach is completely deterministicand allows controlling the speed and damping of thedynamic response of the observer, according to itsrequirements, without having to search for a minimumto guarantee its stability. The only requirement isthat the nonlinear functions be at least locally of theLipschitz type, which can be satisfied for many practicalapplications (Düffing, Van der Pol, Bernoulli equations,inverted penduli, nonlinear friction models for DC motorsor valve actuators (Shuang et al., 2010)).

To achieve this, the mathematical representation ofthe physical system is normalized with regard to theeigenfrequency ω̃, with r̃ = ω̃/ω0. This has theconsiderable advantage of giving a normalized space thatis independent of the temporal constants of the system,and fulfills an important function in the observer gainsynthesis. Such a normalized representation is possible intime (Gille et al., 1988) as well as in the frequency domain(Gißler and Schmid, 1990) for linear systems.

Definition 2. We define the following scaled state repre-sentation:

ẋ(τ) = Â x(τ) + f̃(τ), (2a)

y(τ) = c̃T x(τ) + Φ [ u(τ) ] , (2b)

 =

⎡⎢⎢⎣

0 1 0 . . .0 0 1 . . .

. . . 0 0 1−ã1 −ã2 . . . −ãn

⎤⎥⎥⎦ , (2c)

c̃T =[c̃1 . . . c̃n

], (2d)

f̃(τ)T

=[0 . . . Ψ̃ [ x(τ), u(τ)]

], (2e)

τ = ω̃t, ω̃ = r̃ ω0 (2f)

ui(t) = ui(τ)ω̃i−1, xi(t) = xi(τ) ω̃

i−1,

ãi = ai/ω̃n−i+1, c̃i = ci ω̃

i−1, (2g)ẋn(τ) = ẋn(t)/ω̃

n, i = 1, . . . , n, (2h)

State estimation for a class of nonlinear systems 385

f̃(τ) being a vector with dimension n.Equations (2f)–(2h) define time dilation of the state

representation and its new parameters, allowing fordilatation or retraction of the temporal scale and theamplitude of derivatives from the order from 1 to nwithout changing the pattern of the signal xi(τ). In thefunction Ψ (1e), the terms ui(t) and xi(t) are replacedby ui(τ) ω̃

i−1 and xi(τ) ω̃i−1, and each expression is

divided by ω̃ n to obtain the normalized function Ψ̃.In the following paragraphs, we will define

the observer structure (Section 2.1), characterize theobservation error (Section 2.2), and prove that theobserver state vector uniformly converges toward thestate vector of the physical system (Section 2.3). Thenthe problem of parameters synthesis will be discussed(Section 2.4) and model simulations for a strange attractorsystem (Sprott, 1994) will be shown (Section 3).

2. Structure and synthesis of the observer

2.1. System and observer definitions. To obtain thevariable x1(τ), if θ = 1, we use

x1(τ) =y(τ) − Φ [ u(τ) ]

c̃1. (3)

When θ > 1, y(τ) − Φ [ u(τ) ] is filtered usingẏ(τ) = K y(τ) + k [ y(τ) − Φ [ u(τ) ] ] , (4a)

K =

⎡⎢⎢⎢⎣

0 1 0 . . .0 0 1 . . .

. . . 0 0 1

− c̃1c̃θ

. . . . . . − c̃θ−1c̃θ

⎤⎥⎥⎥⎦ , (4b)

y(τ)T =[y1(τ) . . . yθ−1(τ)

], y(0) = 0, (4c)

kT =[0 . . . 1/c̃θ

]. (4d)

To analyze the effect, we rewrite (2b) in scalar form,ignoring c̃θ+1, . . . , c̃n, which are all zero:

y(τ) − Φ [ u(τ) ]c̃θ

= xθ(τ) +θ−1∑i=1

c̃ic̃θ

xi(τ). (5)

In (4a) we insert (5) in the place of ẏθ−1(τ) and afterreducing it to the same denominator it becomes

c̃θ ẏθ−1(τ) +θ−1∑i=1

c̃i yi(τ) =θ∑

i=1

c̃i xi(τ) (6a)

y1(s)[c̃1 + . . . + c̃θ sθ−1

]

= x1(s)[c̃1 + . . . + c̃θ sθ−1

](6b)

with (6b), the Laplace transformation of (6a). The transferfunction y1(s)/x1(s) = 1, and xi(τ) = yi(τ), i =1, . . . , θ. In the rest of the study, and without limiting thegenerality, we will consider solely the case where θ = 1.

Definition 3. To generate state estimates for thesystem (2), the following modified high gain observer isdefined:

˙̂x(τ) = Â x̂(τ) + h Δy1(τ) + j(τ) + Â g(τ), (7a)

x̂(τ) =[x̂1(τ) . . . x̂n(τ)

], (7b)

hT =[hn . . . h1

], (7c)

Δy1(τ) = x1(τ) − x̂1(τ) (7d)j(τ)T =

[0 . . . I0(τ) + I1(τ)

], (7e)

g(τ)T =[0 . . . I2(τ)

], (7f)

İ0(τ) = h0 Δy1(τ). (7g)

Figure 1 shows the functional diagram of an observerof the order n = 3. The error estimate Δy1(τ) (7d) is usedwith gains h1, . . . , hn in (7a) to compensate for distancesbetween system and observer states. The matrix  (2c)used in (7a) retains the generalized controller canonicalform in the observer. The vectors j(τ) and g(τ), ofdimension n, serve to compensate the effects of f̃(τ) andof possible perturbations or external noise to reduce thestate distance. The nonlinear functions I1(τ) and I2(τ)used in j(τ) and g(τ) are defined later in (18d)–(18h).The vector x̂(τ) + g(τ) used in feedback by parametersai in the equation x̂n(τ) is also the vector for estimatingx(τ). Apart from controlling the synthesis of gainparameters hi (Section 2.4), the use of (7) requires settingthe state vector x̂(0) at initial conditions 0. This structurecan be readily applied to the reference examples given in(40). For the parameters ãi = 0, i = 1, . . . , n, h0 = 0,and the functions I1(τ) = I2(τ) = 0, (7) becomesan observer with a high gain for SISO systems. Thefunctional difference between the two approaches is thatan observer with a high gain works continually to reducethe error Δy1(τ) to produce state estimations, whilst (7)no longer takes into account Δy1(τ) after convergence ofthe error to zero.

2.2. Characterization of the observer error dynam-ics. In a first step, we rewrite the system given in (7)in terms of a single relation depending only on x̂1(τ)and Δy1(τ) and their successive derivatives. In a secondstep, the differential equation for state distances betweensystem and observer is determined. To achieve this, wededuce from (7a) the recursive relation used to generateestimates for successive state characteristics as functionsof the output errors:

x̂i+1(τ) = ˙̂xi(τ) − hn−i+1 Δy1(τ),i = 1 , . . . , n − 2, (8a)

x̂n(τ) = ˙̂xn−1(τ) − h2 Δy1(τ) + I2(τ). (8b)

386 B. Schwaller et al.

-

1s

1s

1s

1s

-

h3h2h1h0I0

I1 I2

ã1ã3

x̂1x̂2x̂3

x1

x̃1

ã2

Δy1

Fig. 1. Observer structure and functional characteristics.

To simplify the expressions for successive derivativesof the observer state, we introduce

x̃i+1(τ) =d (i) x̂1(τ)

dτ (i), i = 0, . . . , n − 1, (9)

with as many derivatives of x̂1(τ) as needed. The vectorof successive derivatives for x̂1(τ) is thus

x̃(τ)T =[x̃1(τ) . . . x̃n(τ)

], (10)

which leads to defining the output errors given in (7d) interms of

Δy1(τ) = x1(τ) − x̃1(τ). (11)Successive derivatives of Δy1(τ) are written as

Δyi+1(τ) =d(i) Δy1(τ)

dτ (i), i = 1, . . . , n − 1. (12)

From (12) we deduce

Δyi+1(τ) = Δẏi(τ), i = 1, . . . , n − 1. (13)

From (10), (11) and (12) we obtain

Δyi(τ) = xi(τ) − x̃i(τ), (14a)Δy(τ)T =

[Δy1(τ) . . . Δyn(τ)

]. (14b)

Repeatedly applying (8) with this new notation leadsto a new form of the vector x̂(τ):

x̂(τ) = x̃(τ) + G Δy(τ) − g(τ), (15a)

G =

⎡⎢⎢⎢⎣

0 . . .−hn 0 . . .

.... . . 0 . . .

−h2 . . . −hn 0

⎤⎥⎥⎥⎦ . (15b)

Then, we re-introduce (15a) into (7a) to obtain

˙̂x(τ) = Â x̃(τ) + Â G Δy(τ) + h Δy1(τ) + j(τ). (16)

The temporal derivative of (15a) can be written asfollows:

˙̂x(τ) = ˙̃x(τ) + J Δy(τ) − ġ(τ), (17a)

J =

⎡⎢⎢⎢⎢⎢⎣

0 0 . . .0 −hn 0 . . .0

.... . . 0 . . .

0 −h3 . . . −hn 00 −h2 −h3 . . . −hn

⎤⎥⎥⎥⎥⎥⎦

. (17b)

Then, reducing the right-hand side of (16) to (17a),we obtain a new state system, which only depends onx̂1(τ), Δy1(τ) and their successive derivatives:

˙̃x(τ) = Â x̃(τ) + H Δy(τ) + f̂(τ), (18a)

H =

⎡⎢⎣

0 . . . 0 0...

......

h1 + g1 . . . hn−1 + gn−1 hn

⎤⎥⎦ , (18b)

gi =n−1∑j=i

hj+1 ãn+i−j , (18c)

f̂(τ)T

= j(τ)T + ġ(τ)T (18d)

=[0 . . . I0(τ) + I1(τ) + İ2(τ)

], (18e)

İ0(τ) = h0 Δy1(τ), (18f)I1(τ) = f1 ( x̃1(τ), x̃2(τ), u(τ) ) , (18g)I2(τ) = f2 ( x̃1(τ), x̃2(τ) ) . (18h)

The vector f̂(τ), composed of functions I0(τ), I1(τ)and İ2(τ), is of the type C1. The variables x̃1(τ), x̃2(τ) =dx̂1(τ)/dτ used in functions f1, f2 are directlyobtainable. The derivative of I2(τ) allows constructingfunctions by involving [ x̃1(τ), x̃2(τ), x̃3(τ) ]. Thisallows a large number of choices of İ2(τ)+I1(τ). Secondof choices of İ2(τ) + I1(τ) and third order nonlinearsystems are often composed of such a, particularlypolynomial, function (41d). We will now calculate the

State estimation for a class of nonlinear systems 387

difference between (2a) and (18a). This gives

Δẏ(τ) =[Â − H

]Δy(τ) + f̃(τ) − f̂(τ), (19a)

 − H =

⎡⎢⎢⎣

0 1 0 . . .0 0 1 . . .. . . 0 0 1−Ã1 . . . . . . −Ãn

⎤⎥⎥⎦ , (19b)

Ãi ={

ãi + hi, i = n,ãi + hi + gi, i < n,

(19c)

f̃(τ)T

=[0 . . . Ψ̃ [ x(τ), u(τ)]

], (19d)

which enables us to obtain the state representation of thedistances between the physical system (vector x(τ)) andvector x̃(τ) ((9) and (10)) and the observer. Here f̃(τ) iscomposed of the function Ψ̃ of the physical system, whichis of the type C1. The distance with f̂(τ) only takes partin the equation n of (19a). The matrix  − H retains thecanonical form of regulation.

If the distances Δy(τ) tend towards 0, given (15),the variables x̂(τ) + g(τ) tend towards x̃(τ), which tendtowards x(τ). As a consequence, the distance due tononlinearities

ΔΨ̃(τ) = Ψ̃ [ x(τ), u(τ)] − I1(τ) − İ2(τ) (20)

tends towards 0. For the rest of the derivation, it is usefulto define the augmented vectors Δy

a(τ), xa(τ), x̃a(τ) :

Δya(τ) =

[Δy(τ)T , Δyn+1(τ)

]T, (21a)

xa(τ) =[x(τ)T , xn+1(τ)

]T, (21b)

x̃a(τ) =[x̃(τ)T , x̃n+1(τ)

]T, (21c)

Δyn+1(τ) = xn+1(τ) − x̃n+1(τ) = Δẏn(τ) (21d)

xn+1(τ) =d(n) x1(τ)

dτ (n)= ẋn(τ), (21e)

x̃n+1(τ) =d(n) x̃1(τ)

dτ (n)= ˙̃xn(τ). (21f)

For analyzing the integrator effects (7g), wetemporally differentiate (19a) through (13) and (21),and reintroduce (7g) into df̂(τ)/dτ for İ0(τ). Thefunction Ψ(t) (1e) in the vector f̃(t) is also differentiated,and terms ui(t) and xi(t) are replaced by ui(τ) ω̃

i

and xi(τ) ω̃i, with each expression divided by ω̃ n

to obtain the normed functions ˙̃Ψ(τ), İ1(τ) and Ï2(τ)(see (41d), (48) as an example). Hence, we obtain the

following (n + 1) × (n + 1) state representation:

Δẏa(τ) = Ã Δy

a(τ) + Δ ˙̃Ψ(τ), (22a)

à =

⎡⎢⎢⎣

0 1 0 . . .0 0 1 . . .

. . . 0 0 1−Ã0 −Ã1 . . . −Ãn

⎤⎥⎥⎦ , (22b)

Ã0 = h0, (22c)

Δ ˙̃Ψ(τ)T

=[0 . . . Δ ˙̃Ψ(τ)

], (22d)

Δ ˙̃Ψ(τ) = ˙̃Ψ [ xa(τ), u(τ)] − İ1(τ) − Ï2(τ). (22e)

Given (22b) and (22c), the observer gain has becomeunitary.

Considering that ˙̃Ψ [ xa(τ), u(τ)] is a nonlinearfunction Lipschitz in xa(τ) and uniformly bounded inu(τ) in an invariant set, with a Lipschitz constant L, i.e.,

‖Δ ˙̃Ψ(τ)‖ � L ‖Δya(τ)‖ (23)

For many systems, if functions Δ ˙̃Ψ(τ) are not globally ofthe Lipschitz type, they may be locally so.

2.3. Convergence of state observations. The observerconvergence analysis consists in proving the globallyasymptotic evolution of the error estimate for statereconstruction. In other words, regardless of the initialconditions, the observer state is to converge toward thestate of the physical system. This leads to the followingtheorem.

Theorem 1. Let us consider a physical SISO systemdescribed by (2), for which the observer structure (7) maybe used and the distance (22e) exists. If the latter is locallyof the Lipschitz type (23), then the observer (7) will belocally stable if the gains hi controlling the constants Ãi(19c) can be adjusted so that they satisfy the followingconditions:

Ãi ≥ ω̃n+1L2 (17/8) + 2ω̃ n+1

, i = 0, . . . , n − 1,(24a)

Ãn ≥ ω̃n+1L2(

12

+n

32

)+

14

+1

2 ω̃n+1. (24b)

If the distance (22e) is globally of the Lipschitz type andif the gains hi satisfy (24), then the observer (7) will beglobally asymptotically stable.

Proof. It can be achieved by proving the stability of(22a) using an appropriate Lyapunov function, like the

388 B. Schwaller et al.

following quadratic function:

Vn(τ) = Δya(τ)T P Δy

a(τ), (25a)

P =

⎡⎢⎢⎢⎣

1 0 . . . 00 1 . . . 0...

. . ....

1/2 1/2 . . . 1

⎤⎥⎥⎥⎦ , (25b)

where P is an (n + 1) × (n + 1) lower triangular matrixdefined as positive and satisfying the Sylvester criteria.To prove the uniform convergence of state distances, (25)needs to be derived:

V̇n(τ) = Δẏa(τ)T P Δy

a(τ)

+ Δya(τ)T P Δẏ

a(τ).

(26)

We reintroduce (22a) replacing Δẏa(τ) and obtain

V̇n(τ) = Δya(τ)T

Q Δya(τ) + N(τ), (27a)

Q = ÃTP + P Ã, (27b)

N(τ) = Δ ˙̃Ψ(τ)T

P Δya(τ)

+ Δya(τ)T P Δ ˙̃Ψ(τ), (27c)

where N(τ) describes the influence of the nonlinearfunctions on state distances. By factorizing the firstterm of the right-hand side of (27a) through the productsΔyi(τ) Δyj(τ), we obtain a lower triangular matrix Qof dimension (n + 1) × (n + 1). The coefficients of theprincipal diagonal element are written as

qii =

{−Ãi−1/2, i = 1, . . . , n,(1/2)− 2 Ãn, i = n,

(28a)

Ãi > 0, i = 0, . . . , n − 1, Ãn > 1. (28b)If the conditions (28b) hold, the coefficients qii (28a)satisfy the Sylvester criteria to obtain the semi-negativityof Q: all successive minors have opposite signs.

Conditions applying to coefficients Ãi determine theobserver gains hi and follow from (19c) and (22c). Todetermine the sign of V̇n(τ) requires expanding N(τ)(27c) through (22d) and (25b)):

N(τ) = Δya(τ)T S Δ ˙̃Ψ(τ), (29a)

S = P + PT . (29b)

To determine the sign of V̇n(τ), we need to majorize theterm N(τ) in (27a) and (29a). This is possible on thebasis of the following inequality (Raghavan and Hedrick,1994):

Δya(τ)T S Δ ˙̃Ψ(τ) ≤ Δy

a(τ)T R Δy

a(τ), (30a)

R =ω̃n+1 L2

4S S +

I

ω̃ n+1, (30b)

In (30a), factorizing the products Δyi(τ) Δyj(τ)yields a positive lower triangular matrix R (30b) of thedimension (n + 1) × (n + 1), which permits rewriting(29a) as an inequality:

N(τ) ≤ Δya(τ)T R Δy

a(τ), (31a)

rii =

⎧⎪⎨⎪⎩

ω̃n+1L21716

+1

ω̃ n+1, i �= n,

ω̃n+1L2(1 +

n

16

)+

1ω̃n+1

, i = n,(31b)

with rii (31b) as positive coefficients of the principaldiagonal element. The inequality (31a) and the Lipschitzconditions (23) permit majorizing N(τ) in (27a):

V̇n(τ) ≤ Δya(τ)T (

Q + R)Δy

a(τ). (32)

To determine the sign of V̇n(τ), we need

Q + R ≤ 0. (33)

The sum Q + R yields a lower triangular matrix thatsatisfies Sylvester citeria of semi-negativity if the ninequalities (24) are satisfied. Then, if I1(τ) (18g) and

I2(τ) (18h) exist, if Δ˙̃Ψ (22e)) is Lipschitz (23, V̇n(τ)

is semi-negative and (22a) is globally and asymptoticallystable. The observer is locally stable if (23) is locally Lip-schitz.

�

2.4. Synthesis of observer parameters. Theforegoing proof of uniform convergence and overallasymptotic stability of the observer make it possible toconceive a synthesis of parameters hi by applying Lya-punov’s first-order approximation (Lyapunov, 1892; Gilleet al., 1988; Fuller, 1992) to (22a). This implies neglecting

nonlinear terms in functions Δ ˙̃Ψ(τ), which then takesthe form of a classical linear differential equation. Moreprecisely, this approximation deals with singular pointsof a nonlinear system. Lyapunov, in particular, made animportant theoretical claim (first-order method) accordingto which, except in certain so-called “critical” cases, thenature and especially the stability of an singular pointcan be determined neglecting terms with degrees greaterthan 2. This would consist in linearizing a nonlinearsystem by limiting it to the smallest domains of variationin variables (“small movements”); in other words, byassimilating the nonlinear functions of variables to thefirst term of their serial development.

The proof of globally asymptotic stability given inSection 2.3 limits the problem of parameter synthesisnear the point of equilibrium Δy(τ)T → 0. In ourcase here, this leads to a synthesis of gains hi and keepsthe roots of the characteristic equation of the linearizedobserver in conditions where such an approximation is

State estimation for a class of nonlinear systems 389

valid. Lyapunov’s “critical case” is avoided. The“small movements” in our case are the differences instate trajectories between the physical system and itsadapted observer. Since the role of an observer isto constantly reduce or minimize these differences, thetheorem proposed here appears ideally suited. Otherwise,very complex equations, specific to each model of anobserver’s input-output behaviour, would be required.Such an unnecessary complexity seriously complicatesattempts to synthesize the coefficients hi, while thestability of a linearized system conforms to that of a realsystem. On the basis of these considerations, we canre-write (22a) as follows:

Δyn+2(τ) +n∑

i=1

Ãi Δyi+1(τ) + h0 Δy1(τ) = 0. (34)

Observer stability and convergence, as proven in theprevious section here, do not account for error damping.Naslin’s normal damping polynomials (Naslin, 1960;1963, Humbert and Ragot, 1970; Gissler and Schmid,1990; Kim, 2002) define algebraic criteria for a simpleand rapid analysis of the damping associated with theparameter space in (34), allowing a parameter synthesisof gains hi which ensures the convergence of vectors x̃(τ)(10), x̂(τ) (7b), (15a) and allows controlling the transitorystates. A normal polynomial D(s) = a0 + a1 s + . . . +an+1 s

n+1 + sn+2 of degree n + 2 is defined in terms ofa characteristic pulsation ω0 and an associated interval γ.Characteristic pulsations of D(s) as defined by Naslin arewritten in terms of

ωi =ai

ai+1, i = 0, . . . , n. (35)

The relationship between characteristic pulsationsand interval γ is

ωi = ω0 γi, i = 0, . . . , n. (36)

The definition of the first pulsation ω0 determines that ofall others. The coefficient γ is assimilated with the generaldamping of D(s). Given (2g), and by normalizing theparameters of D(s), the relation between characteristicpulsations and intervals is

ω̃i =Ãi

Ãi+1=

γi

r̃. (37)

This condition applies to the characteristic polynomialassociated with (34). The Naslin technique of gainadjustment satisfies the Routh–Hurwitz criteria within anadimensional space. For the observer gain synthesis,coefficient hn is determined for the highest order takinginto account (22d):

Ãn =γn

r̃, hn =

γn

r̃− ãn (38)

For practical reasons, the coefficient γ is oftenchosen arbitrarily between 2 and 5/2, which producesa specific response and damping in the system. Thecoefficients hn−1, . . . , h1, h0 are obtained by using

hi =n∏

j=i

(γj

r̃

)− ãi − gi, i = 1, . . . , n − 1, (39a)

h0 =n∏

j=0

(γj

r̃

). (39b)

Naslin damping polynomials provide a simple andefficient means of observer parameter synthesis fullyrespecting the conditions for convergence (24), whichensure stability. The speed of convergence is controlledby (2) through parameter r̃ and the damping by γ. Giventhat ãi, gi are normalized, the gains obtained satisfy theconditions (24) independently of the time scale of thesystem.

3. Simulations

To illustrate our developments, we consider an observerof a strange attractor of type D (Sprott, 1994). Thiscorresponds to a system of the order n = 3 defined interms of

ż1(t) = − z2(t), (40a)ż2(t) = z1(t) + z3(t), (40b)

ż3(t) = z1(t) z3(t) + 3 z2(t)2, (40c)

y(t) = z1(t). (40d)

It is possible to define the system (40) in terms of itssuccessive derivatives:

ẋ1(t) = x2(t), (41a)ẋ2(t) = x3(t), (41b)

ẋ3(t) = Ψ̃ (x(t)) , (41c)

Ψ̃ (x(t)) = x1(t)2 + x1(t) x3(t)

− x2(t) − 3 x2(t)2, (41d)

and a linking system with (41) and (40):

z1(t) = x1(t), (42a)z2(t) = − x2(t), (42b)z3(t) = − x1(t) − x3(t), (42c)y(t) = z1(t). (42d)

This transformation permits observing the systemthrough (7). Figure 2(a) visualizes the state trajectory inthe two-dimensional plane (x − y coordinates), with theinitial parameters z1(0) = −4, 27z2(0) = −0, 39z3(0) =0, 3. Figure 2(b) permits observing estimates of the

390 B. Schwaller et al.

system’s temporal response, showing a superposition ofseveral eigenfrequencies and random amplitudes. Thehighest is chosen as the norm. We consider ω0 = 2πwith a ratio r̃ = 6. This choice of initial values placesthe observer in a “pessimistic” regime where the assumedstate distance is the largest possible. Normalization of (2)applied to our example here gives

ẋ1(τ) = x2(τ), (43a)ẋ2(τ) = x3(τ), (43b)

ẋ3(τ) = Ψ̃ [ x(τ) ] , (43c)

Ψ̃ [ x(τ) ] =1

ω̃ 3x1(τ)

2 +1ω̃

x1(τ) x3(τ)

− 1ω̃ 2

x2(τ) − 3ω̃

x2(τ)2. (43d)

The linking system for (43) and (40) becomes

z1(t) = x1(τ), (44a)z2(t) = − ω̃ x2(τ) = −37, 7 x2(τ), (44b)z3(t) = − x1(τ) − ω̃ 2 x3(τ)

= − x1(τ) − 1421, 2 x3(τ), (44c)y(τ) = x1(τ). (44d)

The observer of the system (43) illustrated by Fig. 1 iswritten in terms of

˙̂x1(τ) = x̂2(τ) + h3 Δy1(τ), (45a)˙̂x2(τ) = x̂3(τ) + I2(τ) + h2 Δy1(τ), (45b)˙̂x3(τ) = I0(τ) + I1(τ) + h1 Δy1(τ), (45c)

İ0(τ) = h0 Δy1(τ), (45d)Δy1(τ) = x1(τ) − x̂1(τ), (45e)

with the nonlinear functions decomposed into

I2(τ) =1ω̃

x̃1(τ) x̃2(τ), (46a)

İ2(τ) =1ω̃

[x̃1(τ)

2 + x̃1(τ) x̃2(τ)], (46b)

I1(τ) =1

ω̃ 3x̃1(τ)

2 − 1ω̃ 2

x̃2(τ) − 4ω̃

x̃2(τ)2. (46c)

A Lyapunov approximation of the first order (34)through (38) and (39), generates the observer gainsynthesis

h3 =γ3

r̃, h2 =

γ2 γ3

r̃ 2,

h1 =γ1 γ2 γ3

r̃ 3, h0 =

γ0 γ1 γ2 γ3

r̃ 4.

(47)

For three different γ, we obtain the gains given in Table 1.

Table 1. Gains obtained for the numerical example.

γ h3 h2 h1 h0

1,5 0,56 0,89 0,053 0,0092 1,33 0,89 0,3 0,05

The Lipschitz constant L is determined to provethat the observer converges irrespective of the initialconditions of the attractor. The temporal derivative of(41d) is computed and then normalized:

˙̃Ψ (x(t)) = 2 x1(t) x2(t) + x1(t) x4(t)− 5 x2(t) x3(t) − x3(t), (48a)

˙̃Ψ (x(τ)) =2

ω̃ 3x1(τ) x2(τ) +

1ω̃

x1(τ) x4(τ)

− 5ω̃

x2(τ) x3(τ) − 1ω̃ 2

x3(τ). (48b)

To obtain the expression of x4(τ), we differentiate (42c)and replace (42b) by x1(t). Then the whole is normalized(2g), which yields

x4(τ) ω̃3 = z2(t) − ż3(t). (49)

This allows completing (44) to construct the vector[x1(τ) . . . x4(τ)]:

x1(τ) = z1(t), (50a)x2(τ) = − (1/ω̃) z2(t), (50b)x3(τ) = − (1/ω̃ 2) (z1(t) + z3(t)) , (50c)x4(τ) = (1/ω̃

3) (z2(t) − ż3(t)) , (50d)which is used in (48b). It is possible to determine themaximum Lipschitz constant L for vector x̂(τ) = 0 of thesystem on the basis of simulations and through (23). Inour example here, L = 3 · 10−5. The gains determined inTable 1 satisfy the conditions (24), yielding local stability

since Δ ˙̃Ψ(τ) can only be locally Lipschitz.Figure 2(c) illustrates the convergence of observer

state trajectories towards the system state trajectoriesfor the three parameter types. The observer stabilizesasymptotically, with oscillations that are stronger whenγ is weaker. The choice for γ between 2 and 2.5 isexperimentally justified. The speed of convergence is notaffected by this coefficient.

A second simulation allows comparison of theobserver (7) for a γ = 2, 5 with that of a high gainobserver (I1(τ) = I2(τ) = h0 = 0) having a chainof temporarily nonnormalized integrators with gainsh2 = 39, 3h1 = 616, 9h0 = 3875, 8, and with an observerstructure (Gauthier et al., 1992) suggesting parametersθ = 62, 8 to determine gains hi. Figures 2(e) and

State estimation for a class of nonlinear systems 391

−2.5−2−1.5−1−0.5

00.51

1.52

−4.5−4−3.5−3−2.5−2−1.5−1−0.5 0

z 1

z0

z1(t), z2(t)

-4.5-4

-3.5-3

-2.5-2

-1.5-1

-0.50

0 5 10 15 20 25 30 35 40 45 50

seconds

z1

(a) (b)

-5-4-3-2-1012345

0 0.5 1 1.5 2

seconds

γ = 1, 5γ = 2

γ = 2, 5

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10

seconds

x1(τ) x̂1(τ) y(t)

(c) (d)

-5-4-3-2-10123

0 0.2 0.4 0.6 0.8 1

seconds

high gainGauthierγ = 2, 5

-0.006-0.005-0.004-0.003-0.002-0.001

00.0010.0020.003

2 4 6 8 10 12 14 16 18 20

seconds

high gainGauthierγ = 2, 5

(e) (f)

Fig. 2. Observation of the Sprott D strange attractor: state trajectory of the system (a), oscillations of z1(t) (b), influence of dampingγ (c), observer output under the influence of white noise (d), high-gain observer (e), fluctuations of the high-gain observer (f).

2(f) demonstrate the convergence of (7) which tendsconstantly towards zero, while there is still a residual errorfor the high gain and Gauthier observers, which ends byfollowing fluctuations of Ψ [ x(t), u(t)] within the limits

of an amplitude factor.A third simulation is performed with parameters

r̃ = 20, ω0 = 2π, γ = 2. A limited band-pass whitenoise with a signal to noise ratio of 10 is added to the

392 B. Schwaller et al.

measured output y(τ). Reconstruction of the latter x̃1(τ)in Fig. 2(d) perfectly matches x1(τ). This is ensuredby the integration I0(τ), which compensates for anystatic bias due to the noise on the basis of the functionsI1(τ), I2(τ).

4. Conclusions and perspectives

The observer introduced here proposes a new proportionalintegral strategy applied to continuous nonlinear systems.It permits the exact and stable reconstruction of statetrajectories of a physical system. The observer structureallows expressing the convergence dynamics in terms of anonlinear differential equation with constant coefficients.The stability of the observer was demonstrated as longas the nonlinear functions were at least locally of theLipschitz type. In this case, it is possible to determineconditions which systematically guarantee this stability,using a weighted quadratic Lyapunov function.

We proceed to an observer parameter synthesiswhich allows providing its transitory states with temporalspecifications. Naslin’s damping polynomials arewell-adapted and simple to use here.

The reconstructed state vector has the majoradvantage of being directly exploitable for state controlwithout any additional transformations.

The approach introduced here should prove useful inthe context of state estimates for nonlinear systems usinga decoupled multiple model. Further simulations underconditions of parametric uncertainty, minimized by onlineparameter identification, may help increase the robustnessof such estimates, even in the presence of instrumentalnoise and external system perturbation. It may ultimatelybe extended to MIMO systems and to the modelling ofsystems based on Lure decompositions.

ReferencesAbdessameud, A. and Khelfi, M.F. (2006). A variable structure

observer for the control of robot manipulators, Interna-tional Journal of Applied Mathematics and Computer Sci-ence 16(2): 189–196.

Bastin, G. and Gevers, M. (1988). Stable adaptive observersfor nonlinear time-varying systems, IEEE Transactions onAutomatic Control 33(7): 650–658.

Besançon, G., Voda, A. and Jouffroy, G. (2010). A note on stateand parameter estimation in a Van der Pol oscillator, Auto-matica 46(10): 1735–1738.

Besançon, G., Zhang, Q. and Hammouri, H. (2004). High gainobserver based state and parameter estimation in nonlinearsystems, 6th IFAC Symposium on Nonlinear Control Sys-tems, NOLCOS, Stuttgart, Germany, pp. 325–332.

Bestle, D. and Zeitz, M. (1983). Canonical form observer designfor non-linear time-variable systems, International Journalof Control 38(2): 419–431.

Bodizs, L., Srinivasan, B. and Bonvin, D. (2011). On thedesign of integral observers for unbiased output estimationin the presence of uncertainty, Journal of Process Control21(3): 379–390.

Boizot, N., Busvelle, E. and Gauthier, J. (2010). An adaptivehigh-gain observer for nonlinear systems, Automatica46(9): 1483–1488.

Boutat, D., Benali, A., Hammouri, H. and Busawon, K.(2009). New algorithm for observer error linearizationwith a diffeomorphism on the outputs, Automatica45(10): 2187–2193.

Boutayeb, M. and Aubry, D. (1999). A strong trackingextended Kalman observer for nonlinear discrete-timesystems, IEEE Transactions on Automatic Control44(8): 1550–1556.

Chen, W., Khan, A.Q., Abid, M. and Ding, S.X. (2011).Integrated design of observer based fault detection fora class of uncertain nonlinear systems, InternationalJournal of Applied Mathematics and Computer Science21(3): 423–430, DOI: 10.2478/v10006-011-0031-0.

Ciccarella, G., Mora, M. D. and Germani, A. (1993). ALuenberger-like observer for nonlinear systems, Interna-tional Journal of Control 57(3): 537–556.

Efimov, D. and Fridman, L. (2011). Global sliding-modeobserver with adjusted gains for locally Lipschitz systems,Automatica 47(3): 565–570.

Farza, M., M’Saad, M., Triki, M. and Maatoug, T. (2011). Highgain observer for a class of non-triangular systems, Systemsand Control Letters 60(1): 27–35.

Feki, M. (2009). Observer-based synchronization of chaoticsystems, Chaos, Solitons and Fractals 39(3): 981–990.

Fliess, M. (1990). Generalized controller canonical forms forlinear and nonlinear dynamics, IEEE Transactions on Au-tomatic Control 35(9): 994–1001.

Fuller, A. (1992). Guest editorial, International Journal of Con-trol 55(3): 521–527.

Gauthier, J. and Bornard, G. (1981). Observability for any u(t)of a class of nonlinear systems, IEEE Transactions on Au-tomatic Control AC-26(4): 922–926.

Gauthier, J., Hammouri, H. and Othman, S. (1992). Asimple observer for nonlinear systems: Applications tobioreactors, IEEE Transactions on Automatic Control37(6): 875–880.

Ghosh, D., Saha, P. and Chowdhury, A. (2010). Linear observerbased projective synchronization in delay Roessler system,Communications in Nonlinear Science and Numerical Sim-ulation 15(6): 1640–1647.

Gille, J., Decaulne, P. and Pélegrin, M. (1988). Systèmes asservisnon linéaires, 5 ième edn, Dunod, Paris, pp. 110–146.

Gißler, J. and Schmid, M. (1990). Vom Prozeß zur Regelung.Analyse, Entwurf, Realisierung in der Praxis, 1st Edn.,Siemens, Berlin/Munich, pp. 350–399.

Hermann, R. and Krener, A. (1977). Nonlinear controllabilityand observability, IEEE Transactions on Automatic Con-trol AC-22(5): 728–740.

State estimation for a class of nonlinear systems 393

Hui, S. and Żak, S.H. (2005). Observer design for systems withunknown inputs, International Journal of Applied Mathe-matics and Computer Science 15(4): 431–446.

Humbert, C. and Ragot, J. (1970). Comportement asymptotiquedes rapports caractéristiques dans l’optimisationquadratique, Revue A 12(1): 32–34.

Ibrir, S. (2009). Adaptive observers for time-delaynonlinear systems in triangular form, Automatica45(10): 2392–2399.

Kim, Y.C., Keel, L.H. and Manabe. S.M. (2002). Controllerdesign for time domain specifications, Proceedings of the15th Triennial World Congress, Barcelona, Spain, pp.1230–1235.

Khémiri, K., Ben Hmida, F., Ragot, J. and Gossa, M.(2011). Novel optimal recursive filter for state and faultestimation of linear stochastic systems with unknowndisturbances, International Journal of Applied Mathe-matics and Computer Science 21(4): 629–637, DOI:10.2478/v10006-011-0049-3.

Krener, A. and Isidori, A. (1983). Linearization by outputinjection and nonlinear observers, Systems & Control Let-ters 3(1): 47–52.

Luenberger, D. (1966). Observers for multivariablesystems, IEEE Transactions on Automatic ControlAC-11(2): 190–197.

Lyapunov, A. (1892). The General Problem of Motion Sta-bility, Annals of Mathematics Studies, Vol. 17, PrincetonUniversity Press, (translated into English in 1949).

Morales, A. and Ramirez, J. (2002). A PI observerfor a class of nonlinear oscillators, Physics Letters A297(3–4): 205–209.

Naslin, P. (1960). Nouveau critère d’amortissement, Automa-tisme 5(6): 229–236.

Naslin, P. (1963). Polynomes normaux et critère algébriqued’amortissement, Automatisme 8(6): 215–223.

Raghavan, S. and Hedrick, J. (1994). Observer design for aclass of nonlinear systems, International Journal of Con-trol 59(2): 515–528.

Röbenack, K. and Lynch, A.F. (2004). An efficient method forobserver design with approximately linear error dynamics,International Journal of Control 77(7): 607–612.

Röbenack, K. and Lynch, A. (2006). Observer design usinga partial nonlinear observer canonical form, InternationalJournal of Applied Mathematics and Computer Science16(3): 333–343.

Shuang, C., Guodong, L. and Xianyong, F. (WCE 2010).Parameters identification of nonlinear DC motor modelusing compound evolution algorithms, Proceedings of theWorld Congress on Engineering, London, UK, pp. 15–20.

Sprott, J. (1994). Some simple chaotic flows, Physical Review E50(2): 647–650.

Söffker, D., Yu, T. and Müller, P. (1995). State estimationof dynamical systems with nonlinearities by usingproportional-integral observers, International Journal ofSystems Science 26(9): 1571–1582.

Tornambè, A. (1992). High-gain observers for non-linearsystems, International Journal of Systems Science23(9): 1475–1489.

Veluvolu, K., Soh, Y. and Cao, W. (2007). Robust observer withsliding mode estimation for nonlinear uncertain systems,IET Control Theory and Applications 1(5): 1533–1540.

Zeitz, M. (1987). The extended Luenberger observer fornonlinear systems, Systems and Control Letters Archive9(2): 149–156.

Zheng, G., Boutat, D. and Barbot, J. (2009). Multi-outputdependent observability normal form, Nonlinear Analysis:Theory, Methods and Applications 70(1): 404–418.

Benoı̂t Schwaller received his engineer’s de-gree with specialisation in control from Fach-hochschule Offenburg (Germany) in 1990. Thenhe joined the University of Strasbourg (France),where he obtained his Ph.D. in automatic con-trol and biomechanics in 1993. Since 1999,Benoı̂t Schwaller has been a Maı̂tre Assistantat the University Institute of Technology ofSchiltigheim and the University of Strasbourg,and conducts his research at INSA Strasbourg,

Graduate School of Science and Technology (LGECO). His fields of in-terest include real time control, nonlinear observer modelling and iden-tification.

Denis Ensminger began with an informatics andmanagement technological diploma, then passeda master degree in computer science and finallya Ph.D. in informatics and bio-mechanics at theLouis Pasteur University of Strasbourg (France).He worked as an engineer for companies such asINTERTECHNIQUE, SIEMENS or ATRAITS.Since 1992, he has been teaching computer sci-ence at the Louis Pasteur Institute of Technologyat the University of Strasbourg. Presently, his re-

search in INSA Strasbourg, Graduate School of Science and Technology(LGECO), is mostly on signal processing, applied artificial intelligenceand embedded systems. He still works for industry as a consultant.

394 B. Schwaller et al.

Birgitta Dresp-Langley obtained a Ph.D. in cog-nitive sciences in 1992 at University Paris V. Shehas been a tenured research scientist with CNRSsince 1993. Her fields of interest include cog-nitive systems, perceptual processes, modelling,and abstract expressionist paintings.

José Ragot received his engineer’s degree withspecialisation in control from ECN Nantes, Grad-uate School of Science and Technology (France),in 1969. Then, he joined the University of Nancy(France), where he received his master’s degreein 1970. In 1973 he obtained a Ph.D. and in 1980a Diplôme de Doctorat-es-Science. Since 1985,José Ragot has been a full professor at the Na-tional Polytechnic Institute of Lorraine and a re-searcher in the Automatic Control Research Cen-

ter of Nancy (CRAN). His fields of interest include data validation, pro-cess diagnosis, fault detection and isolation, as well as modelling andidentification.

Received: 2 April 2012Revised: 5 August 2012Re-revised: 23 October 2012

IntroductionStructure and synthesis of the observerSystem and observer definitionsCharacterization of the observer error dynamicsConvergence of state observationsSynthesis of observer parameters

SimulationsConclusions and perspectives

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 150 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 150 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 2.00000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName (http://www.color.org?) /PDFXTrapped /False

/CreateJDFFile false /SyntheticBoldness 1.000000 /Description >>> setdistillerparams> setpagedevice