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A PI observer for a class of nonlinear oscillators
13 May 2002
Physics Letters A 297 (2002) 205–209 www.elsevier.com/locate/pla
A PI observer for a class of nonlinear oscillators America Morales a , Jose Alvarez-Ramirez b,∗ a Programa de Investigacion en Matematicas Aplicadas, IMP, Mexico b Division de Ciencias Basicas e Ingenieria, Universidad Autónoma Metropolitana-Iztapalapa, Mexico
Received 15 March 2000; received in revised form 27 November 2001; accepted 25 February 2002 Communicated by A.R. Bishop
Abstract The aim of this Letter is to propose a robust observer for a class of uncertain second-order nonlinear system. The observer is endowed with proportional and integral observation error corrections to compensate for modeling errors. Sufficient conditions for the stability of the observation error dynamics are provided. 2002 Elsevier Science B.V. All rights reserved. Keywords: PI observer; Modeling error
Observer schemes are widely used for the reconstruction of nonmeasured states of dynamical systems. Among the most successful schemes is the Luenberger observer [1], which comprises a copy of the system plus a correction that is proportional to the observation error. Several extensions of the Luenberger observer have been proposed to cope with uncertainties in the system model. Adaptive and variable-structure versions of the Luenberger observer have been reported in the literature [2–4]. The basic idea is to endow the classical proportional structure with an additional observation error function to compensate for model uncertainties. Most of these compensation configurations (i) are of nonlinear nature and require some structural information of the model uncertainties, (ii) are restricted to special classes of nonlinearities, (iii) and the implementation may require the solution of partial differential equations (see, for instance, [1]). * Corresponding author.
E-mail address: [email protected] (J. Alvarez-Ramirez).
In some applications, it is not an easy task to provide the structural information required for observer construction. Such is the case of nonparametric uncertainties, such that traditional adaptive observers [1] can lead to erroneous observations. To overcome this drawback, proportional-plus-integral (PI) observers have been proposed [6]. The idea is to incorporate an additional correction term that is proportional to the integral of the observation error, so that, at steadystate conditions, the integral contains the steady-state modeling error. In this way, the natural applications of PI observers have proven to be successful in the case of linear system with constant disturbances [5]. However, PI observers fail when there exist poorly knowledge in the systems and unknown inputs due to nonlinearities. The aim of this Letter is to show that the PI observer scheme can deal with nonconstant model uncertainties. To this end, a PI observer configuration is proposed for a class of second-order systems with uncertain nonlinearities. Sufficient conditions for the stability of the observation error dynamics are provided.
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 1 9 1 - 3
206
A. Morales, J. Alvarez-Ramirez / Physics Letters A 297 (2002) 205–209
Consider a second-order nonlinear oscillator of the form y¨ − a2 y˙ − a1 y = g(y, ˙ y, t, Π), where y is available for measurements, g(x, t, Π) is a continuously differentiable nonlinear function and Π is a set of ˙ one can physical parameters. If x1 = y and x2 = y, write x˙1 = x2 ,
x˙2 = a1 x1 + a2 x2 + g(x, t, Π),
(1)
where x = (x1 , x2 Notice that denotes the transpose of x. The class of systems (1) includes the van der Pol and the Duffing oscillators. The following assumptions are taken: )T .
xT
A.1. Only the system output y = x1 is available for measurements. A.2. The function g(x, t, Π) is uncertain. An esti¯ is available. Both g(x, t, Π) and mate g(x, ¯ t, Π) ¯ are continuously differentiable Lipg(x, ¯ t, Π) schitz functions of their arguments. A.3. The system state x(t) evolves in a compact, possibly chaotic, attractor A. Essentially, system (1) is a mechanical system where the measured state y = x1 is the system position, and the nonmeasured state x2 is the system velocity. In practice, velocity measurements are rarely made because of high noise amplification rate and commonly expensive devices. On the other hand, the nonlinear function g(x, t, Π) is poorly known because of uncertainties in models and physical parameters. Let us rewrite the system (1) as follows: x˙1 = x2 ,
x˙2 = a1 x1 + a2 x2 + g¯ x, t, Π¯ + η x, t, Π, Π¯ ,
(2)
¯ = g(x, t, Π) − g(x, ¯ is the where η(x, t, Π, Π) ¯ t, Π) modeling error function. One can see the modeling erdef ¯ as an additional ror trajectory z(t) = η(x(t), t, Π, Π) system state that is not available for measurements. In this way, the idea is to construct an observer to estimate simultaneously the nonmeasured states x2 (t) and z(t) from measurements of the system output y = x1 . For the simultaneous estimation of x2 (t) and z(t), the following dynamical system is proposed: ¯ t, Π¯ + z¯ + 2τe−1 y˙ − x¯2 , x˙¯ 2 = a1 y + a2 x¯2 + g¯ x, z˙¯ = τe−2 y˙ − x¯2 , (3)
where x¯2 and z¯ are, respectively, estimates of x2 and z, x¯ = (x1 , x¯2 ) and τe > 0 is the estimation timeconstant. The dynamical system (3) plays the role of a reduced-order Luenberger observer for the nonmeasured states x2 and z. Notice that the estimation of x2 comprises a copy of the x2 -dynamics with a correction term that is proportional to the velocity observation error y˙ − x¯2 . On the other hand, the estimation of the modeling error dynamics z(t) comprises only a correction term τe−2 (y˙ − x¯2 ). This is because there is not knowledge of the modeling error dynamics. It is noted that the time-derivative y˙ of the measured position y = x1 appears in the right-hand side of the system (3). In principle, one could use numerical differentiators to obtain an approximate value of y, ˙ which can be subsequently used in (3) to obtain x¯ 2 (t) and z¯ (t). However, this procedure can yield excessive amplification of measurement noise. A more systematic approach to implement (3) is to introduce the new def
def
coordinates w1 = x¯ 2 − 2τe−1 y and w2 = z¯ − τe−2 y, so that the dynamical system (3) is equivalent to the following one: ¯ t, Π¯ + z¯ − 2τe−1 x¯2 , w˙ 1 = a1 y + a2 x¯2 + g¯ x, w˙ 2 = −τe−2 x¯2 ,
(4)
where the estimates x¯2 (t) and z¯ (t) are computed by x¯2 = w1 + 2τe−1 y,
z¯ = w2 + τe−2 y.
(5)
In this way, the initial conditions for the dynamical system (4) are w1 (0) = x¯2 (0) − 2τe−1 y(0) and w2 (0) = z¯ (0) − τe−2 y(0). It is interesting to note that the system (3) has the structure of a PI observer velocity x2 . In fact, t for the since z¯ (t) = z¯ (0) + τe−2 0 (y(t ˙ ) − x¯2 (t )) dt , one has that x˙¯ 2 = a1 y + a2 x¯2 + g¯ x, ¯ t, Π¯ + z¯ (0) + 2τe−1 y˙ − x¯2 + τe−2
t
y˙ t − x¯2 t dt .
0
The term in the brackets is actually a PI correction term acting on the velocity estimation error y˙ − x¯ 2 . The aim of the integral correction term is to compensate for uncertainties in the nonlinear function g(x, t, Π).
A. Morales, J. Alvarez-Ramirez / Physics Letters A 297 (2002) 205–209
To establish the stability properties of the proposed PI observer (3), introduce the following estimation errors: −1 e τ (x2 − x¯ 2 ) e= 1 = e (6) . z − z¯ e2 Then, e˙1 = −τe−1 (2e1 − e2 ) + a2 e1 ¯ t, Π¯ , + τe−1 g¯ x, t, Π¯ − g¯ x, e˙2 = −τe−1 e1 + γ x, t, Π, Π¯ ,
e˙
¯ is Lipschitz (assumption A.2), (i) since g(x, ¯ t, Π) there exists a positive constant kg such that ¯ − g( ¯ < kg e1 , τe−1 |g(x, ¯ t, Π) ¯ x, ¯ t, Π)| (ii) since x(t) evolves in a compact set (assump¯ , for all t 0, tion A.3), B2 γ (x(t), t, Π, Π) 2 (iii) λmin (P ) e V (e) λmax (P ) e 2 . Let τemax be defined by
(7)
¯ = z˙ is given by where γ (x, t, Π, Π) ∂η ¯ γ x, t, Π, Π = x2 ∂x1 ∂η a1 x1 + a2 x2 + g(x, t, Π) + ∂x2 ∂η + . ∂t In compact notation, one can write the system (7) as follows: = τe−1 Ae + τe−1 B1 τe a2 e1
207
τemax =
1 . 2λmax (P )(|a2 | + kg )
(10)
For 0 < τe < τemax , the inequality in (9) is stable in the sense that V (e(t)) is bounded for all t 0. Such inequality can be integrated to give
V e(t) exp − τe−1 − α1 λmin (P )t V (0) +
2λmax (P )
[τe−1 − α1 ]λmin (P )
× 1 − exp − τe−1 − α1 λmin (P )t . In this way, in the asymptotic limit (i.e., as t → ∞), def
+ g¯ x, t, Π¯ − g¯ x, ¯ t, Π¯ + B2 γ x, t, Π, Π¯ ,
V∞ = limt →∞ V (e(t)) is given by
(8)
where B1 = (1, 0)T , B2 = (0, 1)T , and −2 1 A= . −1 0 Notice that A is a stable matrix with its two eigenvalues located at −1. Since A is a stable matrix, there exists a positive-definite matrix P such that P A + AT P = −I. Consider the Lyapunov function V (e) = eT P e. The time-derivative of V (e) along the dynamics (8) is given by V˙ (e) = −τe−1 e 2 + 2eT P τe−1 B1 τe a2 e1 + g¯ x, t, Π¯
− g¯ x, ¯ t, Π¯ + B2 γ x, t, Π, Π¯
− τe−1 − α1 λmin (P )V (e) + 2λmax (P ) , (9) where α1 = 2λmax (P )(|a2 | + kg ), and λmax (P ) and λmin (P ) denote, respectively, the maximum and minimum eigenvalues of P . To obtain the above inequality, the following three facts have been used:
V∞
2λmax (P ) [τe−1
− α1 ]λmin (P )
,
for all 0 < τe < τemax . This shows that V (e(t)) is globally bounded and converges to a neighborhood of the origin. Since V∞ is of the order of τe , the asymptotic size V∞ of V (e(t)) can be made as smaller as desired by taking sufficiently small values of the estimation time-constant. Considering that e 2 V (e)/λmin (P ), the above implies that the estimation error e can be made as smaller as desired by adjusting the value of the estimation time-constant τe > 0. Although we have restricted ourselves to the case of second-order system of the form y¨ − a2 y˙ − a1 y = g(y, ˙ y, t, Π), where y is the measured output, the observer construction and stability analysis can be easily extended to the n-dimensional case y (n) − n (n−j ) = g(y (n−1) , . . . , y, t, Π), where j =1 an−j y y (j ) = d j y/dt j . This can be done by writing the system as a set of n first-order differential equations as follows: x˙j = xj +1 ,
j = 1, . . . , n − 1,
x˙n = a1 x1 + · · · + an xn + g(x, t, Π).
208
A. Morales, J. Alvarez-Ramirez / Physics Letters A 297 (2002) 205–209
Similarly to the second-order system (1), the reduced order observer becomes x˙¯ j = x¯j +1
βj −1 y˙ − x¯2 , j = 2, . . . , n − 1, ¯ t, Π x˙¯ n = a1 y + · · · + an x¯n + g¯ x, + τe−(n−1) βn−1 y˙ − x¯2 , z˙¯ = τe−n βn y˙ − x¯ 2 , (11) −(j −1)
+ τe
where the constants βj ’s are chosen such that the polynomial s n + βn s n−1 + · · · + β1 = 0 has all its roots in the open left-hand side of the complex plane. −j By defining the new variables wj = x¯j +1 − τe βj y, −n j = 1, . . . , n − 1, and wn = z¯ − τe βn y, the observer can be rewritten as −j
w˙ j = x¯j +2 − τe βj −1 x¯2 ,
j = 1, . . . , n − 2, w˙ j −1 = a1 y + · · · + an x¯n + g¯ x, ¯ t, Π − τe−(n−1) βn−1 x¯2 , w˙ n = −τe−n βn x¯2 ,
(12)
where the estimated states and modeling error are −(j −1) βj −1 y, j = 2, . . . , n, given by x¯j = wj −1 + τe and z¯ = wn + τe−n βn y. As for the second-order case, it can be shown that the observation error can be made as small as desired by taking sufficiently small values of the estimation time-constant τe . To illustrate the performance and stability of the proposed controller, the Duffing oscillator will be used. In such case, g(x, t, Π) = −x13 + ζ cos(ωt). The Duffing oscillator displays chaotic behavior for the following set of parameters: a1 = 1.0, a2 = 0.15, ζ = 0.3 and ω = 1.0 [6]. Fig. 1 presents the performance of the PI observer for three different values in the estimation time-constant τe . It is observed that the estimation error becomes smaller as τe takes smaller values. Fig. 2 presents the observer performance under a +10% step change in the parameter a1 at t = 10. As expected from the stability analysis, the observer is able to reject moderate disturbances in the system parameters without serious performance degradation. Summarizing, a PI observer to estimate nonmeasured states and uncertainties has been presented. The observer is simple in structure and its tuning procedure depends only on one parameter with a well defined directionality. That is, the smaller the value of the estimation time-constant, the smaller the estimation error.
Fig. 1. Performance of the PI observer in the reconstruction of the velocity for three different values of the estimation time-constant τe .
Fig. 2. Performance of the PI observer when a +10% step disturbance in the Duffing parameter a2 occurs at t = 10.
A. Morales, J. Alvarez-Ramirez / Physics Letters A 297 (2002) 205–209
References [1] H. Nijmeijer, A.J. van der Schaft, Nonlinear Dynamical Control Systems, Springer, New York, 1991. [2] M. Hou, P.C. Müller, IEEE Trans. Automat. Control 37 (1992) 96. [3] F. Yang, R.W. Wilde, IEEE Trans. Automat. Control 33 (1988) 87.
209
[4] E.A. Misawa, J.K. Hedrick, J. Dynamic Systems 3 (1989) 123. [5] D. Söftker, T.J. Yu, P.C. Müller, Int. J. Systems Sci. 26 (1995) 545. [6] S. Wiggings, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, Berlin, 1981.
Physics Letters A 297 (2002) 205–209 www.elsevier.com/locate/pla
A PI observer for a class of nonlinear oscillators America Morales a , Jose Alvarez-Ramirez b,∗ a Programa de Investigacion en Matematicas Aplicadas, IMP, Mexico b Division de Ciencias Basicas e Ingenieria, Universidad Autónoma Metropolitana-Iztapalapa, Mexico
Received 15 March 2000; received in revised form 27 November 2001; accepted 25 February 2002 Communicated by A.R. Bishop
Abstract The aim of this Letter is to propose a robust observer for a class of uncertain second-order nonlinear system. The observer is endowed with proportional and integral observation error corrections to compensate for modeling errors. Sufficient conditions for the stability of the observation error dynamics are provided. 2002 Elsevier Science B.V. All rights reserved. Keywords: PI observer; Modeling error
Observer schemes are widely used for the reconstruction of nonmeasured states of dynamical systems. Among the most successful schemes is the Luenberger observer [1], which comprises a copy of the system plus a correction that is proportional to the observation error. Several extensions of the Luenberger observer have been proposed to cope with uncertainties in the system model. Adaptive and variable-structure versions of the Luenberger observer have been reported in the literature [2–4]. The basic idea is to endow the classical proportional structure with an additional observation error function to compensate for model uncertainties. Most of these compensation configurations (i) are of nonlinear nature and require some structural information of the model uncertainties, (ii) are restricted to special classes of nonlinearities, (iii) and the implementation may require the solution of partial differential equations (see, for instance, [1]). * Corresponding author.
E-mail address: [email protected] (J. Alvarez-Ramirez).
In some applications, it is not an easy task to provide the structural information required for observer construction. Such is the case of nonparametric uncertainties, such that traditional adaptive observers [1] can lead to erroneous observations. To overcome this drawback, proportional-plus-integral (PI) observers have been proposed [6]. The idea is to incorporate an additional correction term that is proportional to the integral of the observation error, so that, at steadystate conditions, the integral contains the steady-state modeling error. In this way, the natural applications of PI observers have proven to be successful in the case of linear system with constant disturbances [5]. However, PI observers fail when there exist poorly knowledge in the systems and unknown inputs due to nonlinearities. The aim of this Letter is to show that the PI observer scheme can deal with nonconstant model uncertainties. To this end, a PI observer configuration is proposed for a class of second-order systems with uncertain nonlinearities. Sufficient conditions for the stability of the observation error dynamics are provided.
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 1 9 1 - 3
206
A. Morales, J. Alvarez-Ramirez / Physics Letters A 297 (2002) 205–209
Consider a second-order nonlinear oscillator of the form y¨ − a2 y˙ − a1 y = g(y, ˙ y, t, Π), where y is available for measurements, g(x, t, Π) is a continuously differentiable nonlinear function and Π is a set of ˙ one can physical parameters. If x1 = y and x2 = y, write x˙1 = x2 ,
x˙2 = a1 x1 + a2 x2 + g(x, t, Π),
(1)
where x = (x1 , x2 Notice that denotes the transpose of x. The class of systems (1) includes the van der Pol and the Duffing oscillators. The following assumptions are taken: )T .
xT
A.1. Only the system output y = x1 is available for measurements. A.2. The function g(x, t, Π) is uncertain. An esti¯ is available. Both g(x, t, Π) and mate g(x, ¯ t, Π) ¯ are continuously differentiable Lipg(x, ¯ t, Π) schitz functions of their arguments. A.3. The system state x(t) evolves in a compact, possibly chaotic, attractor A. Essentially, system (1) is a mechanical system where the measured state y = x1 is the system position, and the nonmeasured state x2 is the system velocity. In practice, velocity measurements are rarely made because of high noise amplification rate and commonly expensive devices. On the other hand, the nonlinear function g(x, t, Π) is poorly known because of uncertainties in models and physical parameters. Let us rewrite the system (1) as follows: x˙1 = x2 ,
x˙2 = a1 x1 + a2 x2 + g¯ x, t, Π¯ + η x, t, Π, Π¯ ,
(2)
¯ = g(x, t, Π) − g(x, ¯ is the where η(x, t, Π, Π) ¯ t, Π) modeling error function. One can see the modeling erdef ¯ as an additional ror trajectory z(t) = η(x(t), t, Π, Π) system state that is not available for measurements. In this way, the idea is to construct an observer to estimate simultaneously the nonmeasured states x2 (t) and z(t) from measurements of the system output y = x1 . For the simultaneous estimation of x2 (t) and z(t), the following dynamical system is proposed: ¯ t, Π¯ + z¯ + 2τe−1 y˙ − x¯2 , x˙¯ 2 = a1 y + a2 x¯2 + g¯ x, z˙¯ = τe−2 y˙ − x¯2 , (3)
where x¯2 and z¯ are, respectively, estimates of x2 and z, x¯ = (x1 , x¯2 ) and τe > 0 is the estimation timeconstant. The dynamical system (3) plays the role of a reduced-order Luenberger observer for the nonmeasured states x2 and z. Notice that the estimation of x2 comprises a copy of the x2 -dynamics with a correction term that is proportional to the velocity observation error y˙ − x¯2 . On the other hand, the estimation of the modeling error dynamics z(t) comprises only a correction term τe−2 (y˙ − x¯2 ). This is because there is not knowledge of the modeling error dynamics. It is noted that the time-derivative y˙ of the measured position y = x1 appears in the right-hand side of the system (3). In principle, one could use numerical differentiators to obtain an approximate value of y, ˙ which can be subsequently used in (3) to obtain x¯ 2 (t) and z¯ (t). However, this procedure can yield excessive amplification of measurement noise. A more systematic approach to implement (3) is to introduce the new def
def
coordinates w1 = x¯ 2 − 2τe−1 y and w2 = z¯ − τe−2 y, so that the dynamical system (3) is equivalent to the following one: ¯ t, Π¯ + z¯ − 2τe−1 x¯2 , w˙ 1 = a1 y + a2 x¯2 + g¯ x, w˙ 2 = −τe−2 x¯2 ,
(4)
where the estimates x¯2 (t) and z¯ (t) are computed by x¯2 = w1 + 2τe−1 y,
z¯ = w2 + τe−2 y.
(5)
In this way, the initial conditions for the dynamical system (4) are w1 (0) = x¯2 (0) − 2τe−1 y(0) and w2 (0) = z¯ (0) − τe−2 y(0). It is interesting to note that the system (3) has the structure of a PI observer velocity x2 . In fact, t for the since z¯ (t) = z¯ (0) + τe−2 0 (y(t ˙ ) − x¯2 (t )) dt , one has that x˙¯ 2 = a1 y + a2 x¯2 + g¯ x, ¯ t, Π¯ + z¯ (0) + 2τe−1 y˙ − x¯2 + τe−2
t
y˙ t − x¯2 t dt .
0
The term in the brackets is actually a PI correction term acting on the velocity estimation error y˙ − x¯ 2 . The aim of the integral correction term is to compensate for uncertainties in the nonlinear function g(x, t, Π).
A. Morales, J. Alvarez-Ramirez / Physics Letters A 297 (2002) 205–209
To establish the stability properties of the proposed PI observer (3), introduce the following estimation errors: −1 e τ (x2 − x¯ 2 ) e= 1 = e (6) . z − z¯ e2 Then, e˙1 = −τe−1 (2e1 − e2 ) + a2 e1 ¯ t, Π¯ , + τe−1 g¯ x, t, Π¯ − g¯ x, e˙2 = −τe−1 e1 + γ x, t, Π, Π¯ ,
e˙
¯ is Lipschitz (assumption A.2), (i) since g(x, ¯ t, Π) there exists a positive constant kg such that ¯ − g( ¯ < kg e1 , τe−1 |g(x, ¯ t, Π) ¯ x, ¯ t, Π)| (ii) since x(t) evolves in a compact set (assump¯ , for all t 0, tion A.3), B2 γ (x(t), t, Π, Π) 2 (iii) λmin (P ) e V (e) λmax (P ) e 2 . Let τemax be defined by
(7)
¯ = z˙ is given by where γ (x, t, Π, Π) ∂η ¯ γ x, t, Π, Π = x2 ∂x1 ∂η a1 x1 + a2 x2 + g(x, t, Π) + ∂x2 ∂η + . ∂t In compact notation, one can write the system (7) as follows: = τe−1 Ae + τe−1 B1 τe a2 e1
207
τemax =
1 . 2λmax (P )(|a2 | + kg )
(10)
For 0 < τe < τemax , the inequality in (9) is stable in the sense that V (e(t)) is bounded for all t 0. Such inequality can be integrated to give
V e(t) exp − τe−1 − α1 λmin (P )t V (0) +
2λmax (P )
[τe−1 − α1 ]λmin (P )
× 1 − exp − τe−1 − α1 λmin (P )t . In this way, in the asymptotic limit (i.e., as t → ∞), def
+ g¯ x, t, Π¯ − g¯ x, ¯ t, Π¯ + B2 γ x, t, Π, Π¯ ,
V∞ = limt →∞ V (e(t)) is given by
(8)
where B1 = (1, 0)T , B2 = (0, 1)T , and −2 1 A= . −1 0 Notice that A is a stable matrix with its two eigenvalues located at −1. Since A is a stable matrix, there exists a positive-definite matrix P such that P A + AT P = −I. Consider the Lyapunov function V (e) = eT P e. The time-derivative of V (e) along the dynamics (8) is given by V˙ (e) = −τe−1 e 2 + 2eT P τe−1 B1 τe a2 e1 + g¯ x, t, Π¯
− g¯ x, ¯ t, Π¯ + B2 γ x, t, Π, Π¯
− τe−1 − α1 λmin (P )V (e) + 2λmax (P ) , (9) where α1 = 2λmax (P )(|a2 | + kg ), and λmax (P ) and λmin (P ) denote, respectively, the maximum and minimum eigenvalues of P . To obtain the above inequality, the following three facts have been used:
V∞
2λmax (P ) [τe−1
− α1 ]λmin (P )
,
for all 0 < τe < τemax . This shows that V (e(t)) is globally bounded and converges to a neighborhood of the origin. Since V∞ is of the order of τe , the asymptotic size V∞ of V (e(t)) can be made as smaller as desired by taking sufficiently small values of the estimation time-constant. Considering that e 2 V (e)/λmin (P ), the above implies that the estimation error e can be made as smaller as desired by adjusting the value of the estimation time-constant τe > 0. Although we have restricted ourselves to the case of second-order system of the form y¨ − a2 y˙ − a1 y = g(y, ˙ y, t, Π), where y is the measured output, the observer construction and stability analysis can be easily extended to the n-dimensional case y (n) − n (n−j ) = g(y (n−1) , . . . , y, t, Π), where j =1 an−j y y (j ) = d j y/dt j . This can be done by writing the system as a set of n first-order differential equations as follows: x˙j = xj +1 ,
j = 1, . . . , n − 1,
x˙n = a1 x1 + · · · + an xn + g(x, t, Π).
208
A. Morales, J. Alvarez-Ramirez / Physics Letters A 297 (2002) 205–209
Similarly to the second-order system (1), the reduced order observer becomes x˙¯ j = x¯j +1
βj −1 y˙ − x¯2 , j = 2, . . . , n − 1, ¯ t, Π x˙¯ n = a1 y + · · · + an x¯n + g¯ x, + τe−(n−1) βn−1 y˙ − x¯2 , z˙¯ = τe−n βn y˙ − x¯ 2 , (11) −(j −1)
+ τe
where the constants βj ’s are chosen such that the polynomial s n + βn s n−1 + · · · + β1 = 0 has all its roots in the open left-hand side of the complex plane. −j By defining the new variables wj = x¯j +1 − τe βj y, −n j = 1, . . . , n − 1, and wn = z¯ − τe βn y, the observer can be rewritten as −j
w˙ j = x¯j +2 − τe βj −1 x¯2 ,
j = 1, . . . , n − 2, w˙ j −1 = a1 y + · · · + an x¯n + g¯ x, ¯ t, Π − τe−(n−1) βn−1 x¯2 , w˙ n = −τe−n βn x¯2 ,
(12)
where the estimated states and modeling error are −(j −1) βj −1 y, j = 2, . . . , n, given by x¯j = wj −1 + τe and z¯ = wn + τe−n βn y. As for the second-order case, it can be shown that the observation error can be made as small as desired by taking sufficiently small values of the estimation time-constant τe . To illustrate the performance and stability of the proposed controller, the Duffing oscillator will be used. In such case, g(x, t, Π) = −x13 + ζ cos(ωt). The Duffing oscillator displays chaotic behavior for the following set of parameters: a1 = 1.0, a2 = 0.15, ζ = 0.3 and ω = 1.0 [6]. Fig. 1 presents the performance of the PI observer for three different values in the estimation time-constant τe . It is observed that the estimation error becomes smaller as τe takes smaller values. Fig. 2 presents the observer performance under a +10% step change in the parameter a1 at t = 10. As expected from the stability analysis, the observer is able to reject moderate disturbances in the system parameters without serious performance degradation. Summarizing, a PI observer to estimate nonmeasured states and uncertainties has been presented. The observer is simple in structure and its tuning procedure depends only on one parameter with a well defined directionality. That is, the smaller the value of the estimation time-constant, the smaller the estimation error.
Fig. 1. Performance of the PI observer in the reconstruction of the velocity for three different values of the estimation time-constant τe .
Fig. 2. Performance of the PI observer when a +10% step disturbance in the Duffing parameter a2 occurs at t = 10.
A. Morales, J. Alvarez-Ramirez / Physics Letters A 297 (2002) 205–209
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