THEORETICAL & APPLIED MECHANICS LETTERS 2, 043006 (2012)

Stability analysis of the extended state observers by Popov criterionChristiav Erazo,a) Fabiola Angulo, and Gerard OlivarUniversidad Nacional de Colombia-Sede Manizales-Facultad de Ingeniera y Arquitectura-Departamentode Ingeniera Electrica, Electronica y Computacion-Percepcion y Control Inteligente-Bloque Q,Campus La Nubia, Manizales 170003, Colombia

(Received 4 June 2012; accepted 20 June 2012; published online 10 July 2012)

Abstract The analysis and design of the extended state observer (ESO) involves a continuousnon-smooth structure, thus the study of the ESO dynamic requires mathematical tools of thenonlinear systems analysis. This paper establishes the sucient conditions for absolute stabilityof the ESO. Based on this study, a methodology to estimate several nonlinear functions in dy-namics systems is proposed. c 2012 The Chinese Society of Theoretical and Applied Mechanics.[doi:10.1063/2.1204306]

Keywords Popov criterion, extended observer, ESO

A special class of state observers, called extendedstate observers (ESO) was proposed by Han in Ref. 1.This observer has the ability to estimate the states andthe whole system, including uncertainties, in an inte-grated form. This technique has been widely used inindustrial applications with excellent results;2,3 in fact,the ESO is the brain of the control technique knownas ADRC (active disturbance rejection control) whichdetects and compensates actively the internal and ex-ternal perturbations of the system.4,5 Although ESOhas been widely used in several applications, the sta-bility analysis is still an open problem since the ESOhas a non-smooth structure which makes the analysis adicult task. However, stability studies for the secondorder ESO have been reported in literature. In Ref. 6 apiecewise smooth Lyapunov function is proposed in or-der to analyze the convergence properties for the secondorder ESO. Similarly, in Ref. 7 a stability study for thesecond order ESO using the self-stable region (SSR) ap-proach is proposed. The trouble is that extending thesestudies to higher-order ESOs is a complicated task andhas not yet been reported in literatures. In this pa-per we propose to perform a convergence analysis forthe high order ESO using the Popov criterion. Basi-cally, we propose represent the observer error systemas a feedback interconnection of a linear system and anonlinearity in order to use the Popov criterion.8 In ad-dition, using the above results we propose a nonlinearestimator functions based on the ESO.

A brief review of this approach will be presented be-low. Consider a nonlinear dynamical system describedby

y(n) = f(y; _y; ; y(n1); w(t)) + bu(t); (1)where f() is an uncertain nonlinear function, w(t) isan unknown disturbance, u(t) is the control signal andy(t) is the system output which can be measured. Thesystem (1) is augmented as

_x1 = x2;

a)Corresponding author. Email: ccerazoo@unal.edu.co.

...

_xn = xn+1 + bu;

_xn+1 = h1; (2)

and y = x1, where f(x1; x2; ; xn; w(t)) is han-dled as an extended state xn+1. Here, h1 =_f(x1; x2; ; xn; w(t)) is assumed unknown butbounded. With this consideration, the nonlinearobserver design for system (2) is described by

_z1 = z2 + 1g1(e1);

...

_zn = zn+1 + ngn(e1) + bu;

_zn+1 = h2 + n+1gn+1(e1); (3)

and y^ = z1, where e1 = y y^ = x1 z1, zn+1 is anestimate of uncertain function f(), 0is are feedbackgains, and each gi() is dened as

gi(ei; i; ) =

8 ei

1i; jeij

; (4)

where 2 (0; 1). In order to improve the transientresponse of the estimation error and decrease the ob-server sensitivity to model and external disturbances,the nonlinear function (4) is incorporated into the ob-server, such as discussed in Ref. 8. Equation (4) is anonlinear gain function where small errors produce highgains, and large error produce small gains. The last is-sue prevents excessive gain which could cause high fre-quency.

Let ei = xi zi, i = 1; 2; ; n + 1. The observerestimation error is dened as

_e1 = e2 1g1(e1; 1; );...

_en = en+1 ngn(e1; n; );_en+1 = n+1gn+1(e1; n+1; ): (5)

To relax notation, in that follows gi(e1; i; ) will benoted as gi. Taking the same parameter values i for all

043006-2 C. Erazo, F. Angulo, and G. Olivar Theor. Appl. Mech. Lett. 2, 043006 (2012)

components and adding and subtracting the term ie1to the right side of Eq. (5), Eq. (5) can be expressed as_e = Ae+ bu, where u = '(e) is a function of error e,and '(e) is dened as '(e) = g1 e1, and A and b arematrix given by

A =

2666641 1 0 02 0 1 0...

......

...n 0 0 1n+1 0 0 0

377775 ;

b =

0BBBB@12...nn+1

1CCCCA : (6)Remark 1: i must be computed such that A is Hur-witz and the par (A; b) is controllable.

With the aim to apply the Popov criterion, it isnecessary that '(e) satises the sector condition. It issaid that the function '(e) belongs to sector [0 k] if

'(e)[ke '(e)] 0; 8 t 2 R+ and 8 e 2 R;(7)

where is the region of the error which guarantees thestability of the observer. In this case, condition is ful-lled if 0 < < 1 and k 1=1 1. Then, is takenfrom this range. According to the Popov criterion, thesystem (5) is absolutely stable if there is 0 (notcorresponding to any eigenvalue of A) such that

1

k+Re[G(jw)] wIm[G(jw)] > 0; 8w 2 R (8)

with a suitable selection of parameters (i; ) and0is, the ESO will have asymptotic stability.

Consider the following dynamical system

_x1 = x2;

_x2 = f(x1; x2; w) + bu; (9)

and y = x1, where w is an external disturbance, jb is aconstant value and f() is the dynamics of the systemwhich is unknown. Assuming that f() is dierentiable,the extended state observer is dened as

_z1 = z2 + 1g1(e1);

_z2 = z3 + 2g2(e1) + bu;

_z3 = 3g3(e1); (10)

and y^ = z1, where z3 is the estimate of the uncertainfunction f() and e1 = y y^. The observer estimationerror is dened as

_e1 = e2 1g1(e1);_e2 = e3 2g2(e1);_e3 = 3g3(e1); (11)

2.0 1.5 1.0 0.5 0 0.5 1.0

0.5

0.4

0.3

0.2

0.1

0

0.1

0.2

ReGjw

wImGjw

1/

(1 ),0)(1-

Fig. 1. Graphical interpretation of 3rd order observer sta-bility.

and ~y = e1. Now, according to the above methodology,system (11) can be rewritten as24 _e1_e2

_e3

35 =24 1 1 02 0 13 0 0

3524 e1e2e3

35+24 123

35u;~y =

1 0 0

24 e1e2e3

35 ; (12)where u = '1(e1) and '1(e1) = g1 e1. The system(11) is absolutely stable if

1 +1w

4 2w4 + 2w2 213w2 + 23(3 1w2)2 + (2w w3)2

w1w5 + 3w3

(3 1w2)2 + (2w w3)2 > 0; (13)

for the sake of simplicity the pole placement method canbe used for the initial design of this observer. In thiscase, suppose the poles of the observer on (S +8)3, thelinear gains are 1 = 24, 2 = 192, and 3 = 512, andthe observe (10) will be globally asymptotically stableif 1 > 0:947 5. Taking = 0:5 and = 0:9 absolutestability is obtained. Figure 1 shows a graphical inter-pretation. The plot of Re[(G(jw))] vs. wIm[(G(jw))] isto the right of the line with slope 1= that intercepts thepoint (1=k; 0) (see Ref. 8, for a complete explanation).The limit 0.947 5 was found as it is explained below. (1)For simplicity we assume = 0; (2) The derivative ofRe[G(jw)] is computed and equaled to zero; (3) It isveried that the condition given by Eq. (8) is satisedfor values obtained in the above (2); (4) Values of and are computed.

Now, taking into account the ESO abilities to esti-mate nonlinear functions, a theorem for the estimationof nonlinear functions of a dynamical system is pro-posed. The number of nonlinear functions to estimatedepends on the number of available outputs. A sim-ilar procedure to the previous case can be developed

043006-3 Stability analysis of the extended state observers by Popov criterion Theor. Appl. Mech. Lett. 2, 043006 (2012)

to prove the stability, using the multivariable Popovcriterion. In this paper, the stability is proven whenLuenberguer observer is used.

Theorem: Consider an nth order nonlinear dy-namical system given by

_x = Ax+Bu+Ef(x);

y = Cx; (14)

where x 2 Rn is the state vector, y 2 Rm is the outputvector, u 2 R is the signal control, f(x) : Rn ! Rmcorresponds to the set of nonlinear functions, which aregoing to estimate, A, B, C and E are nn, n1, mn, and n m matrices, respectively. Therefore in thissystem it can be estimated m nonlinear functions givenby f(x), if the pair (A;C) is observable, the derivativeof f(x) is bounded, and the observer design leads toBIBO stable error dynamics.

Proof : The rst step is to augment the systemorder, taking the nonlinear pieces as extended states,as follows

_x = Ax+Bu+Ex^;_^x = _f(x); (15)

where x^ 2 Rm is the extended state corresponding tothe m nonlinear functions, which are going to estimate,_f(x) are the derivatives of nonlinear functions, are un-known but bounded. Now, the extended state observercan be expressed as

_z = Az +Bu+Ez^ +KCe;_^z = h2 +GCe; (16)

where z is the estimation of the state x, e is the esti-mation error and it is dened as e = x z, z^ is theestimation of nonlinear functions f(x), h2 2 Rm is afunction dened by the user, K and G are n m andm m gain matrices. Now, the dynamics of estima-tion errors are obtained by subtracting Eq. (16) fromEq. (15) as follows

_e_^e

=

AKC EGC 0m

ee^

+

0hh

; (17)

where h = _f(x) h2, 0m is a m m zero matrix,0h 2 Rn is a zero vector and h 2 Rm. Since the sys-tem is observable, the nonlinear functions will have abounded estimation error (due to elements of h) if and

only if all eigenvalues of A^ are placed on the open leftcomplex half plane. Where A^ is a (n +m) (n +m)matrix and it is dened by Eq. (17). The gain termsthat can be chosen arbitrarily are nm+m2, which cor-respond to terms of K and G matrices, respectively.Now, because the system is observable, the order ofcharacteristic polynomial of A^ is n + m and its zeroscan be placed arbitrarily, i.e., the characteristic polyno-mial will have n +m unknowns; therefore, the systemcan be solved if m (n+m) n +m, which is true form 1. If it does not have information about _f(x), agood selection of h2 is h2 = 0.

On the other hand, the exponential gain functions ofthe nonlinear functions observer are bounded by slidingobserver gain and Luenberger observer gain, therefore,these satisfy the sector theorem and it can be used themultivariable Popov criterion to determine the absolutestability as we said previously.

The previous proposed methodology was veriedthrough numerical simulations on a third order non-linear dynamic system with two outputs, as it is shownbelow.

Consider a nonlinear system described by

_x1 = x2 x1 2x12 + u;_x2 = 2x2 + 2x1x2; (18)

and y1 = x1 and y2 = x2. Taking into account themethodology, the rst step is to augment the order ofthe system, considering the nonlinear pieces as extendedstates

_x1 = x2 x1 + x3 + u;_x2 = 2x2 + x4;_x3 = h1;

_x4 = h2; (19)

where f1 and f2 are the nonlinear functions to be es-timated. For the observer design we assume h1;2 = 0.Now, ESO for system (19) is expressed as

_z1 = z2 z1 + z3 + u+ 11g11(e1);_z2 = 2z2 + z4 + 22g22(e2);_z3 = 31g31(e1) + 32g32(e2);

_z4 = 41g41(e1) + 42g42(e2); (20)

where z3 and z4 are the estimation of the nonlinear func-tions and the signal control u corresponds to unit step.Next, the observer error dynamics are obtained by sub-tracting Eq. (19) from Eq. (20)

Ae =

264 1 + 11 1 1 00 2 22 0 131 32 0 041 42 0 0

375 : (21)The pole placement technique is used to place the ob-server poles in 2 and each i in ESO is determined.The parameters of ESO are i = 1; 0:25; 0:25; 0:25; 0:25and = 104. The estimation of nonlinear pieces andtheir errors are displayed in percent in Figs. 2 and 3,respectively.

The error percents of the nonlinear functions con-verge quickly and accurately to zero. It is worth notingthat as the ESO does not assume the knowledge of thesystem dynamics, the transient of the estimation erroris high; however, in steady state it converges to zero.

Consider a nonlinear system described by

_x1 = 3x2 x1 + x3 + u;_x2 = 4x1 2x2 + x32 cosx2;_x3 = x1 3x3 x1x3; (22)

043006-4 C. Erazo, F. Angulo, and G. Olivar Theor. Appl. Mech. Lett. 2, 043006 (2012)

2.5

2.0

1.5

1.0

0.5

0 1 2 3 4 5 6 t/s

Real

Estimated

f/x

0 1 2 3 4 5 6t/s

0

0.5

1.0

1.5

RealEstimated

f/xx

Fig. 2. Estimation of nonlinear pieces.

and y1 = x2 and y2 = x3. The nonlinear parts are con-sidered as extended states, with the nonlinear functionsf1 = x3

2 cosx2 and f2 = x1x3 to be estimated. Forthe observer design we assume h1;2 = 0. Now, the ESOfor the system (22) is

_z1 = 3z2 z1 + z3 + u+ 12g12(e2) + 13g13(e3)_z2 = 2z2 4z1 + z4 + 22g22(e2);_z3 = 3z3 + z1 + z5 + 33g33(e3);_z4 = 42g42(e2) + 43g43(e3);

_z5 = 52g52(e2) + 53g53(e3); (23)

where z4 and z5 are the estimation of the nonlinearfunctions and the signal control u corresponds to unitstep. Following the previous procedure and applyingpole placement technique for placing the poles at 2,we compute each i. The nonlinear functions percent-age error are displayed in Fig. 4. All estimated convergequickly and accurately to zero. As the ESO does notassume the knowledge of the system dynamics the tran-sient of the estimation error is high; however, in steadystate it converges to zero.

Requirements on and were found graphically toguarantee the stability of 2nd order ESO. A methodol-ogy to estimate nonlinear functions on dynamical sys-tems based on ESO approach was proposed. The sim-ulations carried out with dierent kinds of nonlinearsystems have shown that the methodology is eective,assuring that the estimation of states and uncertain-ties can be performed in an integrated way. Neverthe-less, the observer for nonlinear functions is only validfor steady state, because in the transient steady thederivative of the nonlinear functions is dierent fromzero. Thus the estimation fails in transient part. On theother hand, for nonlinear functions whose time deriva-tive is dierent from zero in steady state but bounded,the estimation error will converge to a nite value whichwill depend on the observer gains and the bound of non-linear functions.

0 1 2 3 4 5 60

10

20

30

40

50

0

20

40

60

80

100

0 1 2 3 4 5 6

t/s

0 1 2 3 4 5 6t/s

10

0

Error/(%)

Error/(%)

Fig. 3. Error percent of nonlinear functions.

0 1 2 3 4 5 60

10

20

30

40

50

t/s

Error/(%)

%E4%E5

4.0 4.5 5.0 5.50

0.20.40.60.8

Fig. 4. Error percent of nonlinear functions.

This paper was presented at the 11th Conference on

Dynamical Systems|Theory and Applications, December 5-

8, 2011, Lodz, Poland. This work was supported by Pro-

grama de Jovenes Investigadores e Innovadores COLCIEN-

CIAS (DFIA-0494) and Universidad Nacional de Colombia-

Manizales (12475), Vicerrectora de Investigacion, DIMA.

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