Extended state observer for uncertain lower triangular nonlinear ...

Control Theory Tech, Vol. 14, No. 3, pp. 179–188, August 2016

Control Theory and Technology http://link.springer.com/journal/11768

Extended state observer for uncertain lower triangular nonlinear systems subject to stochastic disturbance Zehao WU† , Baozhu GUO Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China Received 26 January 2016; revised 5 May 2016; accepted 1 June 2016

Abstract The extended state observer (ESO) is the most important part of an emerging control technology known as active disturbance rejection control to this day, aiming at estimating “total disturbance” from observable measured output. In this paper, we construct a nonlinear ESO for a class of uncertain lower triangular nonlinear systems with stochastic disturbance and show its convergence, where the total disturbance includes internal uncertain nonlinear part and external stochastic disturbance. The numerical experiments are carried out to illustrate effectiveness of the proposed approach. Keywords: Extended state observer, stochastic disturbance, lower triangular nonlinear systems DOI 10.1007/s11768-016-6019-4

1

Introduction The extended state observer (ESO) is the most impor-

tant component of active disturbance rejection control (ADRC), an emerging control technology proposed by Han in his pioneer work [1]. ADRC is now acknowledged to be an effective control strategy in dealing with systematically so called “total disturbance” which can

part of control input, in a large scale. The most remarkable feature of ADRC lies in its estimation/cancellation nature, where the total disturbance is considered as an extended state and is estimated, in real time, through ESO. The total disturbance is finally cancelled (compensated) in the feedback loop by its estimation. This estimation/cancellation nature of ADRC makes it capable of eliminating the uncertainty before it causes negative

include the coupling between unknown system dynam-

effect to control plant and the control energy can there-

ics, external disturbance, and the superadded unknown

fore be saved significantly in engineering applications.

† Corresponding author. E-mail: [email protected]. Tel.: +86-15311202539. This work was supported by the National Natural Science Foundation of China (No. 61273129).

© 2016 South China University of Technology, Academy of Mathematics and Systems Science, CAS, and Springer-Verlag Berlin Heidelberg

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The idea of ADRC has been attracting more attention by the industry practitioners as presented in an up-to-date survey paper [2]. The numerous concrete applications in different fields include flexible joint manipulator control [3], control of a model-scale helicopter [4], omnidirectional mobile robot control [5], vibrational control in MEMS gyroscopes [6], or control system in superconducting RF cavities [7], as reviewed, among many others, in [8]. On the other hand, some progresses have also been made in theoretical foundations, see [9–16], name just a few. The first ESO is designed in [17] as follows: ⎧ ⎪ xˆ˙ (t) = xˆ2 (t) − α1 g1 (xˆ1 (t) − y(t)), ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ xˆ˙ 2 (t) = xˆ3 (t) − α2 g2 (xˆ1 (t) − y(t)), ⎪ ⎪ ⎪ ⎪ ⎨ .. ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ xˆ˙ n (t) = xˆn+1 (t) − αn gn (xˆ1 (t) − y(t)) + u(t), ⎪ ⎪ ⎪ ⎪ ⎩ xˆ˙ n+1 (t) = −αn+1 gn+1 (xˆ1 (t) − y(t)),

(1)

for the following n-dimensional single input and single output (SISO) nonlinear system ⎧ ⎪ x˙ 1 (t) = x2 (t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x˙ 2 (t) = x3 (t), ⎪ ⎪ ⎪ ⎪ ⎨ .. ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˙ ⎪ ⎪ xn (t) = f (t, x1 (t), . . . , xn (t)) + w(t) + u(t), ⎪ ⎪ ⎩ y(t) = x1 (t),

(2)

where f : [0, ∞) × Rn → R is possibly an unknown function, w(t) the external disturbance, y(t) the measured output, and u(t) is the control input. The main idea of ESO is to choose some appropriate gi ( · )’s so that the xˆi (t) approaches xi (t) for all i = 1, 2, . . . , n + 1 through regulating gain constants αi , where xn+1 (t)  f (t, x1 (t), . . . , xn (t)) + w(t) is called the “total disturbance” which is seen including internal uncertain nonlinear part and external disturbance. It is demonstrated by numerical simulations that for some nonlinear functions gi ( · )’s and tuning parameters αi ’s, ESO (1) takes its advantage of satisfactory adaptability, robustness, and anti-chattering [17]. The multiple choice of tuning parameters has been changed in [18], 1 in terms of bandwidth, by constant gain with linear ε

gain functions gi ( · )’s as follows: ⎧ a1 ⎪ xˆ˙ 1 (t) = xˆ2 (t) + (y(t) − xˆ1 (t)), ⎪ ⎪ ⎪ ε ⎪ ⎪ ⎪ a2 ⎪ ˙ ⎪ ⎪ xˆ2 (t) = xˆ3 (t) + 2 (y(t) − xˆ1 (t)), ⎪ ⎪ ε ⎪ ⎪ ⎨ .. ⎪ . ⎪ ⎪ ⎪ ⎪ an ⎪ ⎪ ⎪ xˆ˙ n (t) = xˆn+1 (t) + n (y(t) − xˆ1 (t)) + u(t), ⎪ ⎪ ε ⎪ ⎪ ⎪ ⎪ xˆ˙ (t) = an+1 (y(t) − xˆ (t)). ⎩ n+1 1 εn+1

(3)

The ESO (3) (with linear gain functions) is thus said to be linear ESO (LESO). The convergence of LESO (3) for SISO systems is presented in [19, 20]. As a special case of ESO (1) and a nonlinear generalization of LESO (3), a nonlinear ESO (NLESO in short) of the following: ⎧ y(t) − xˆ1 (t) ⎪ ⎪ ⎪ xˆ˙ 1 (t) = xˆ2 (t) + εn−1 g1 ( ), ⎪ ⎪ ⎪ εn ⎪ ⎪ ⎪ ⎪ y(t) − xˆ1 (t) ⎪ ⎪ ), xˆ˙ 2 (t) = xˆ3 (t) + εn−2 g2 ( ⎪ ⎪ ⎪ εn ⎪ ⎪ ⎨ .. (4) ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ y(t) − xˆ1 (t) ⎪ ˙ ⎪ ) + u(t), ⎪ ⎪ xˆn (t) = xˆn+1 (t) + gn ( ⎪ εn ⎪ ⎪ ⎪ ⎪ y(t) − xˆ1 (t) 1 ⎪ ⎪ ), ⎩ xˆ˙ n+1 (t) = gn+1 ( ε εn is proposed in [9], where convergence of NLESO (4) for SISO systems is concluded. Shortly afterwards, a series of convergence results on NLESO for more general deterministic systems are developed, see, for instance, [10–13]. Although great progress has been achieved, most of the literatures including the aforementioned ones, however, are focused mainly on deterministic systems, and little attention is paid to stochastic counterparts. This motivates us, in this paper, to consider ESO for a class of uncertain stochastic nonlinear systems where the exterˆ nal stochastic disturbance satisfying an Ito-type stochastic differential equation. A typical example of such kind of exogenous disturbance is the “colored noise” whose fundamental noise sources through various feedback mechanisms may be regarded as white so that it can be produced by passing the white noise through a filter, ˆ described by an Ito-type stochastic differential equation, see, for instance, [21, 22]. Actually, “colored noise” exists in many practical systems such as physical model systems [23, 24] and chemical model systems [25]. The paper [16] focuses on the filtering problem of general discrete nonlinear uncertain systems with nonlinear un-

Z. Wu, B. Guo / Control Theory Tech, Vol. 14, No. 3, pp. 179–188, August 2016

known dynamics, stochastic process and measurement noises, where the extended state filter (ESF) is constructed to estimate timely the uncertainties of the system. In this paper, however, the bounded continuous time noise, considered as part of the stochastic total disturbance, is estimated by ESO. Precisely, the system that we consider is an uncertain lower triangular nonlinear system with external stochastic disturbance as ⎧ ⎪ x˙ 1 (t) = x2 (t) + h1 (u(t), x1 (t)), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x˙ 2 (t) = x3 (t) + h2 (u(t), x1 (t), x2 (t)), ⎪ ⎪ ⎨ .. ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x˙ n (t) = f (t, x(t), w(t)) + hn (u(t), x(t)), ⎪ ⎪ ⎪ ⎪ ⎩ y(t) = x1 (t),

(5)

where x(t) = (x1 (t), . . . , xn (t))T ∈ Rn , u(t) ∈ Rm , and y(t) = x1 (t) ∈ R are the state, control (input), and output (measurement) of system, respectively; The functions hi : Rm+i → R (i = 1, 2, . . . , n) are known, whereas f : [0, ∞) × Rn+1 → R is possible unknown measurable; The w(t) ∈ R is used to describe the external stochastic disturbance which is assumed to satisfy the following uncertain stochastic differential equation: dw(t) = φ(t, w(t))dt + ψ(t, w(t))dBt ,

(6)

where w(0) = w0 and {Bt }t0 is an one-dimensional standard Brownian motion defined on a complete probability space (Ω, F , {Ft }t0 , P) with Ω being a sample space, F a σ-field, {Ft }t0 a filtration, and P the probability measure; The functions φ : [0, ∞) × R → R, ψ : [0, ∞) × R → R are unknown measurable functions. We proceed as follows. In the next section, Section 2, we design a NLESO to estimate both state and stochastic total disturbance. The mean-square convergence is proved. As a direct consequence, we conclude convergence for LESO. In Section 3, a special kind of ESO is applied to system (5) where the stochastic total disturbance includes external stochastic disturbance only. The corresponding convergence is developed. In Section 4, some numerical experiments are carried out to illustrate effectiveness of the proposed approach.

2

Convergence of extended state observer

Motivated from deterministic case [13], we introduce a NLESO proposed in [13] with constant high gain tuning

181

parameter for system (5) as follows: ⎧ ⎪ ⎪ xˆ˙ 1 (t) = xˆ2 (t) + εn−1 g1 (η1 (t)) + h1 (u(t), xˆ1 (t)), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ xˆ˙ 2 (t) = xˆ3 (t) + εn−2 g2 (η1 (t)) + h2 (u(t), xˆ1 (t), xˆ2 (t)), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ... ⎪ ⎪ ⎪ ⎪ ⎪ ˆ xˆ˙ n (t) = xˆn+1 (t) + gn (η1 (t)) + hn (u(t), x(t)), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y(t) − xˆ1 (t) 1 ⎪ ⎪ , ⎩ xˆ˙ n+1 (t) = gn+1 (η1 (t)), η1 (t) = n ε ε (7) where gi ∈ C(R; R) (i = 1, 2, . . . , n + 1) are designed functions to be specified later, ε > 0 is the tuning parameter, and xˆn+1 (t) is used to estimate the stochastic total disturbance xn+1 (t)  f (t, x(t), w(t)). Here and throughout the paper, we always drop ε for the solution to (7) by abuse of notation without confusion. To obtain convergence of NLESO (7) for system (5), we need some assumptions. Assumption 1 is a prior assumption about the functions hi ( · ), f ( · ), φ( · ), and ψ( · ). Assumption 1 f ( · ) is twice continuously differentiable with respect to its arguments, and there exist (known) constants Ci > 0 (i = 1, 2) and non-negative functions ζ1 ∈ C(Rm ; R), ζ2 ∈ C(Rn ; R), ζ3 , ζ4 ∈ C(R; R) such that for all t  0, x ∈ Rn , w ∈ R, |hi (u, x1 , . . . , xi ) − hi (u, xˆ1 , . . . , xˆi )|  ζ1 (u)((x − xˆ1 ), . . . , (xi − xˆi )), hi (0, . . . , 0) = 0, i = 1, 2, . . . , n;

(8)

n ∂ f (t, x, w) ∂ f (t, x, w)  | |+ | ∂t ∂xi i=1 ∂ f (t, x, w) ∂ f 2 (t, x, w) | +| |+| ∂w ∂w2  C1 + ζ2 (x) + ζ3 (w);

(9)

| f (t, x, w)| + |

|φ(t, w)| + |ψ(t, w)|  C2 + ζ4 (w).

(10)

Assumption 2 is a prior assumption about the control input u(t) and the external stochastic disturbance w(t). Assumption 2 There exists a (known) constant M  0 such that |u(t)| + |w(t)|  M almost surely for all t  0. Remark 1 Since the stochastic disturbance w(t) is regarded as part of an extended state variable of system (5) to be estimated by ESO, it is reasonable to assume that the w(t) itself and its “variation” (or differential) are bounded. It is also easy to verify that many practical deterministic disturbances such as “cos(a + bt)”, “sin(a + bt)” and stochastic disturbances such as “cos(at + bBt )”, “sin(at+bBt )” satisfy the assumption, where a, b are con-

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stants and the deterministic disturbances are covered by letting ψ( · ) ≡ 0 and φ( · ) be independent of w in (6). Assumption 3 is on the designed functions gi ( · )’s in ESO (7). Assumption 3 There exist constants λi (i = 1, . . . , 4) and twice continuously differentiable function V : Rn+1 → R which is positive definite and radially unbounded such that ⎧ ⎪ 2 2 ⎪ ⎪ ⎪ λ1 y  V(y)  λ2 y , ⎪ ⎪ ⎪ ⎪ λ3 y2  W(y)  λ4 y2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n ∂V(y) ∂V(y) ⎪ ⎪ ⎪ (yi+1 − gi (y1 )) − gn+1 (y1 ) ⎪ ⎪ ∂yi ∂yn+1 ⎪ ⎪ ⎨ i=1 (11) ⎪  −W(y), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂V(y) ⎪ ⎪ | |  αy, i = 1, 2, . . . , n + 1, ⎪ ⎪ ⎪ ∂yi ⎪ ⎪ ⎪ ⎪ ∂2 V(y) ⎪ T n+1 ⎪ ⎪ | ⎪ ⎩ ∂y2 |  β, ∀y = (y1 , y2 , . . . , yn+1 ) ∈ R , n+1

for some nonnegative continuous function W : Rn+1 → R and constants α, β > 0. Assumption 3 guarantees that the zero equilibrium of the following system: ˙ = (y2 (t) − g1 (y1 (t)), . . . , yn+1 (t) − gn (y1 (t)), y(t) −gn+1 (y1 (t)))T

(12)

is asymptotically stable. Theorem 1 Suppose that the solution to system (5) satisfies: sup x(t)  N almost surely for some cont0

stant N > 0. Then under Assumptions 1–3, there exists a constant ε∗ > 0 (specified by (21) later) such that the ESO (7) is convergent in the sense that for any ε ∈ (0, ε∗ ), ˆ xˆn+1 (0)) ∈ Rn+1 , and any initial values x(0) ∈ Rn , (x(0), any positive constant a > 0, E|xi (t) − xˆi (t)|  Γε i = 1, 2, . . . , n + 1, 2

2n+3−2i

uniformly in [a, +∞), (13)

where Γ > 0 is an ε-independent constant. Proof Set ⎧ xi − xˆi ⎪ ⎪ ⎪ ⎪ ηi = εn+1−i , i = 1, 2, . . . , n + 1, ⎪ ⎪ ⎪ ⎪ ⎨ Δi = hi (u, x1 , . . . , xi ) − hi (u, xˆ1 , . . . , xˆi ), ⎪ ⎪ ⎪ ⎪ i = 1, 2, . . . , n, ⎪ ⎪ ⎪ ⎪ ⎩ η = (η , . . . , η )T . 1

(14)

n+1

By Assumptions 1 and 2, we suppose without loss of

generality that sup |ζ1 (u(t))|  C3 for some constant t0

C3 > 0. Let 0 < ε < 1. It then follows from Assumption 1 that |Δi (t)|2  C23 [(x1 (t) − xˆ1 (t))2 + . . . + (xi (t) − xˆi (t))2 ] = C23 [ε2n |η1 (t)|2 + . . . + ε2(n+1−i) |ηi (t)|2 ]  C23 ε2(n+1−i) η(t)2 , i = 1, 2, . . . , n.

(15)

ˆ formula gives Applying Ito’s  d f (t, x(t), w(t))along(5),(6) ∂ n−1  = f (t, x(t), w(t)) + [xi+1 (t) ∂t i=1 ∂ f (t, x(t), w(t)) +hi (u(t), x1 (t), . . . , xi (t))] ∂xi ∂ f (t, x(t), w(t)) +[ f (t, x(t), w(t)) + hn (u(t), x(t))] ∂xn ∂ f (t, x(t), w(t)) 1 ∂2 f (t, x(t), w(t)) + φ(t, w(t)) + × 2 ∂w ∂w2 ∂ f (t, x(t), w(t)) ×ψ2 (t, w(t)) dt + ψ(t, w(t))dBt ∂w  Λ1 (t)dt + Λ2 (t)dBt . (16) By Assumptions 1 and 2, there exists a constant C4 > 0 such that |Λ1 (t)| + |Λ2 (t)|  C4 almost surely ∀t  0.

(17)

System (5) can be then written as ⎧ ⎪ dx1 (t) = [x2 (t) + h1 (u(t), x1 (t))]dt, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dx2 (t) = [x3 (t) + h2 (u(t), x1 (t), x2 (t))]dt, ⎪ ⎪ ⎪ ⎪ ⎨ .. ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dxn (t) = [xn+1 (t) + hn (u(t), x(t))]dt, ⎪ ⎪ ⎪ ⎪ ⎩ dxn+1 (t) = Λ1 (t)dt + Λ2 (t)dBt .

(18)

A straightforward computation shows that η(t) satisfies ⎧ 1 1 ⎪ ⎪ ⎪ ⎪ dη1 (t) = ε [η2 (t) − g1 (η1 (t))]dt + εn Δ1 (t)dt, ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ dη2 (t) = [η3 (t) − g2 (η1 (t))]dt + n−1 Δ2 (t)dt, ⎪ ⎪ ⎪ ε ε ⎪ ⎪ ⎨ .. ⎪ . ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ dηn (t) = [ηn+1 (t) − gn (η1 (t))] + Δn (t)dt, ⎪ ⎪ ε ε ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ dηn+1 (t) = − gn+1 (η1 (t))dt + Λ1 (t)dt + Λ2 (t)dBt . ε (19) ˆ formula to V(η(t)) with respect to t along We apply Ito’s

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the solution η(t) of system (19) to obtain dV(η(t)) n ∂V(η(t)) 1  = [ (ηi+1 (t) − gi (η1 (t))) ε i=1 ∂ηi ∂V(η(t)) − gn+1 (η1 (t))]dt ∂ηn+1 n ∂V(η(t))  1 ∂V(η(t)) + Δi (t)dt + Λ1 (t)dt n+1−i ∂ηi ∂ηn+1 i=1 ε ∂V(η(t)) 1 ∂2 V(η(t)) 2 Λ2 (t)dBt + Λ2 (t)dt. (20) + ∂ηn+1 2 ∂η2n+1 Suppose

λ3 1 − nαC3 − > 0 for some ε0 > 0, and 2ε0 4 0 < ε < ε∗  min{1, ε0 }.

(21)

By (15), (17), (20), and Assumption 3, it follows that dEV(η(t)) dt 1 1  − EW(η(t)) + nαC3 Eη(t)2 + αC4 Eη(t) + βC24 ε 2 λ3 1 1  − Eη(t)2 + (nαC3 + )Eη(t)2 + α2 C24 + βC24 ε 4 2 λ3 1 2 2 2 − EV(η(t)) + α C4 + βC4 . (22) 2ελ2 2 Therefore, for every a > 0 and all t  a,

where gi ( · ) s (i = 1, . . . , n + 1) in ESO (7) are linear functions: gi (r) = ai r, r ∈ R. Define the matrix as follows: ⎡ ⎢⎢ −a1 1 ⎢⎢ ⎢⎢ ⎢⎢ −a2 0 ⎢⎢ ⎢ . . E = ⎢⎢⎢⎢ .. .. ⎢⎢ ⎢⎢⎢ −an 0 ⎢⎢ ⎢⎣ −an+1 0

⎤ 0 · · · 0 ⎥⎥⎥ ⎥⎥ 1 · · · 0 ⎥⎥⎥ ⎥ .. .. ⎥⎥⎥⎥ . . ⎥⎥⎥ ⎥⎥ 0 · · · 1 ⎥⎥⎥ ⎥⎥ ⎦ 0 ··· 0

.

(27)

(n+1)×(n+1)

t0

λ

− 2ελ3 t

EV(η(0)) t λ 1 − 3 (t−s) ds + (α2 C24 + βC24 ) e 2ελ2 0 2 λ3 a ελ2 − (4α2 C24 + 2βC24 ).  e 2ελ2 EV(η(0)) + λ3 2

(23)

Since both the first term and the second one of the righthand side of (23) are bounded by ε multiplied by an ε-independent constant, there exists an ε-independent constant γ > 0 such that for all t ∈ [a, ∞), EV(η(t)  γε.

(24)

Thus for all i = 1, 2, . . . , n + 1 and t ∈ [a, +∞), E|xi (t) − xˆi (t)|2 = ε2n+2−2i E|ηi (t)|2  ε2n+2−2i Eη(t)2 

⎧ a1 ⎪ xˆ˙ 1 (t) = xˆ2 (t) + (y(t) − xˆ1 (t)) + h1 (u(t), xˆ1 (t)), ⎪ ⎪ ⎪ ε ⎪ ⎪ ⎪ ⎪ ⎪ a2 ⎪ ⎪ xˆ˙ 2 (t) = xˆ3 (t) + 2 (y(t) − xˆ1 (t)) + h2 (u(t), xˆ1 (t), xˆ2 (t)), ⎪ ⎪ ⎪ ε ⎪ ⎪ ⎪ ⎨ .. ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ an ⎪ ⎪ ˆ ⎪ xˆ˙ n (t) = xˆn+1 (t) + n (y(t) − xˆ1 (t)) + hn (u(t), x(t)), ⎪ ⎪ ε ⎪ ⎪ ⎪ ⎪ ⎪ a ⎪ ⎩ xˆ˙ n+1 (t) = n+1 (y(t) − xˆ1 (t)), εn+1 (26)

Corollary 1 Suppose that the solution to system (5) satisfies: sup x(t)  N almost surely for some constant

EV(η(t)) e

γ is an ε-independent constant. This comλ1 pletes the proof of the theorem.  The simplest ESO is certainly the LESO which is a special case of (7): where Γ 

ε2n+2−2i EV(η(t))  Γε2n+3−2i , λ1

(25)

N > 0 and the matrix E is Hurwitz. Then under Assumptions 1 and 2, there exists a constant ε∗ > 0 such that the LESO (26) is convergent in the sense that for any ˆ xˆn+1 (0)) ∈ ε ∈ (0, ε∗ ), any initial values x(0) ∈ Rn , (x(0), Rn+1 , and any positive constant a > 0, E|xi (t) − xˆi (t)|2  Γε2n+3−2i uniformly in [a, +∞), i = 1, 2, . . . , n + 1, (28) where Γ > 0 is an ε-independent constant. Proof Let Q be the unique positive definite matrix solution to the Lyapunov equation QE + ET Q = −I(n+1)×(n+1) for (n + 1)-dimensional identity matrix I(n+1)×(n+1) . Define the Lyapunov functions V, W : Rn+1 → R by V(y) = yT Qy, W(y) = yT y for y ∈ Rn+1 . It is easy to verify that all conditions of Assumption 3 are satisfied. The result then follows directly from Theorem 1.  Remark 2 When hi ( · ) ≡ 0 (i = 1, 2, . . . , n), system

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(5) is of the form: ⎧ ⎪ x˙ 1 (t) = x2 (t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x˙ 2 (t) = x3 (t), ⎪ ⎪ ⎪ ⎪ ⎨ .. ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˙ x ⎪ ⎪ n (t) = f (t, x1 (t), . . . , xn (t), w(t)) + u(t), ⎪ ⎪ ⎩ y(t) = x1 (t).

|φ(t, w)| + |ψ(t, w)|  D4 + D5 ζ6 (w).

(33)

Theorem 2 Suppose that the solution to system (5) satisfies: sup x(t)  N almost surely for some constant (29)

We thus conclude the results of [26] by Theorem 1 and Corollary 1.

t0

N > 0. Then under Assumptions 1∗ , 2 and 3, there exists a constant ε∗ > 0 (specified by (42)) such that the NLESO (30) is convergent in the sense that for any ε ∈ (0, ε∗ ), ˆ any initial values x(0) ∈ Rn , (x(0), xˆn+1 (0)) ∈ Rn+1 , and any positive constant a > 0, E|xi (t) − xˆi (t)|2  Γε2n+3−2i uniformly in [a, +∞),

3 Extended state observer utilizing known part Although theoretically, ESO spans the concept of disturbance which can include even the parts which are hardly to be treated by practitioner, ESO should utilizes the known information of the plant as much as possible to improves its performance. In this section, a special ESO is considered, where the system function f (t, x, w) = f1 (t, x) + f2 (w), with f1 ( · ) being known. In other words, the total disturbance comes from external stochastic disturbance only. In this case, the NLESO in this case can be modified as ⎧ ⎪ ⎪ xˆ˙ 1 (t) = xˆ2 (t) + εn−1 g1 (η1 (t)) + h1 (u(t), xˆ1 (t)), ⎪ ⎪ ⎪ ⎪ n−2 ⎪ ˙ ⎪ ⎪ xˆ2 (t) = xˆ3 (t) + ε g2 (η1 (t)) + h2 (u(t), xˆ1 (t), xˆ2 (t)), ⎪ ⎪ ⎪ ⎨ .. . ⎪ ⎪ ⎪ ⎪ ⎪ xˆ˙ (t) = xˆ (t) + g (η (t)) + h (u(t), x(t)), ⎪ ˆ ⎪ n n+1 n 1 n ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ xˆ˙ n+1 (t) = gn+1 (η1 (t)) + f1 (t, xˆ1 (t), . . . , xˆn (t)), ε (30)

i = 1, 2, . . . , n + 1, where Γ > 0 is an ε-independent constant. ˆ formula, Proof By Ito’s  d f2 (w(t))along(6)

∂ f2 (w(t)) 1 ∂2 f2 (w(t)) 2 ψ (t, w(t))dt φ(t, w(t))dt + 2 ∂w2 ∂w ∂ f2 (w(t)) + ψ(t, w(t))dBt ∂w  Λ3 (t)dt + Λ4 (t)dBt . (35) =

By Assumptions 1∗ and 2, there exists a constant D6 > 0 such that |Λ3 (t)| + |Λ4 (t)|  D6 almost surely ∀t  0.

| f1 (t, x1 , . . . , xn ) − f1 (t, xˆ1 , . . . , xˆn )|

|

 D1 ((x − xˆ1 ), . . . , (xn − xˆn ));

(31)

∂ f 2 (w) ∂ f2 (w) | + | 2 2 |  D2 + D3 ζ5 (w); ∂w ∂w

(32)

(36)

System (5) can then be written as ⎧ ⎪ dx1 (t) = [x2 (t) + h1 (u(t), x1 (t))]dt, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dx2 (t) = [x3 (t) + h2 (u(t), x1 (t), x2 (t))]dt, ⎪ ⎪ ⎪ ⎪ ⎨ .. ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dxn (t) = [xn+1 (t) + f1 (t, x(t)) + hn (u(t), x(t))]dt, ⎪ ⎪ ⎩ dxn+1 (t) = Λ3 (t)dt + Λ4 (t)dBt .

where xˆn+1 (t) is again used to estimate the stochastic total disturbance xn+1 (t)  f2 (w(t)). To have convergence of NLESO (30), we replace Assumption 1 by Assumption 1∗ . Assumption 1∗ f2 ( · ) is twice continuously differentiable with respect to its argument and there exist (known) constants Di > 0 (i = 1, . . . , 5) and nonnegative functions ζ5 , ζ6 ∈ C(R; R) such that for all t  0, x ∈ Rn , w ∈ R,

(34)

(37) Set ˆ Ξ(t) = f (t, x(t)) − f (t, x(t)).

(38)

By Assumption 1∗ , |Ξ(t)|  εD1 η(t).

(39)

Z. Wu, B. Guo / Control Theory Tech, Vol. 14, No. 3, pp. 179–188, August 2016

t λ 1 − 3 (t−s) ds + (α2 D26 + βD26 ) e 2ελ2 0 2 λ3 a ελ2 − (4α2 D26 + 2βD26 ).  e 2ελ2 EV(η(0)) + λ3 (44)

A straightforward computation shows that η(t) satisfies ⎧ 1 1 ⎪ ⎪ ⎪ ⎪ dη1 (t) = ε [η2 (t) − g1 (η1 (t))]dt + εn Δ1 (t)dt, ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ dη2 (t) = [η3 (t) − g2 (η1 (t))]dt + n−1 Δ2 (t)dt, ⎪ ⎪ ⎪ ε ε ⎪ ⎪ ⎪ ⎨ .. ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 1 ⎪ ⎪ ⎪ dηn (t) = [ηn+1 (t)− gn (η1 (t))]dt+[ Δn (t)+ Ξ(t)]dt, ⎪ ⎪ ε ε ε ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ dηn+1 (t) = − gn+1 (η1 (t))dt + Λ3 (t)dt + Λ4 (t)dBt , ε (40) where ηi (t) (i = 1, 2, . . . , n + 1) and Δi (t) (i = 1, 2, . . . , n) are defined as that in (14). By Assumption 3, we apˆ formula to V(η(t)) with respect to t along the ply Ito’s solution η(t) of system (40) to obtain dV(η(t)) n ∂V(η(t)) 1  (ηi+1 (t) − gi (η1 (t))) = [ ε i=1 ∂ηi ∂V(η(t)) 1 ∂V(η(t)) gn+1 (η1 (t))]dt + Ξ(t) − ∂ηn+1 ε ∂ηn n ∂V(η(t))  1 ∂V(η(t)) + Δi (t)dt + Λ3 (t)dt n+1−i ∂ηi ∂ηn+1 i=1 ε ∂V(η(t)) 1 ∂2 V(η(t)) 2 Λ4 (t)dBt + Λ4 (t)dt. (41) + 2 ∂η2n+1 ∂ηn+1 Suppose once again that some ε1 > 0 and

1 λ3 − αD1 − nαC3 − > 0 for 2ε1 4

0 < ε < ε∗  min{1, ε1 }.

(42)

Then it follows from (15), (36), (39), and Assumption 3 that dEV(η(t)) dt 1  − EW(η(t)) + αD1 Eη(t)2 + nαC3 Eη(t)2 ε 1 + αD6 Eη(t) + βD26 2 λ3 1 2  − Eη(t) + (αD1 + nαC3 + )Eη(t)2 ε 4 1 + α2 D26 + βD26 2 λ3 1 − EV(η(t)) + α2 D26 + βD26 . (43) 2ελ2 2 Therefore, for every a > 0 and for all t  a, λ

− 2ελ3 t

EV(η(t))  e

2

EV(η(0))

185

Since both the first term and the second one of the righthand side of (44) are bounded by ε multiplied by an ε-independent constant, there exists an ε-independent constant ξ > 0 such that for all t ∈ [a, ∞), EV(η(t)  ξε.

(45)

Thus for all i = 1, 2, . . . , n + 1 and t ∈ [a, +∞), E|xi (t) − xˆi (t)|2 = ε2n+2−2i E|ηi (t)|2  ε2n+2−2i Eη(t)2 ε2n+2−2i  EV(η(t))  Γε2n+3−2i , λ1

(46)

ξ is an ε-independent constant. This comλ1 pletes the proof of the theorem.  Similarly, the corresponding LESO of (30) is as follows: where Γ 

⎧ a1 ⎪ ⎪ xˆ˙ (t) = xˆ2 (t) + (y(t) − xˆ1 (t)) + h1 (u(t), xˆ1 (t)), ⎪ ⎪ 1 ε ⎪ ⎪ ⎪ a2 ⎪ ⎪ ˙ ⎪ xˆ2 (t) = xˆ3 (t) + 2 (y(t) − xˆ1 (t)) + h2 (u(t), xˆ1 (t), xˆ2 (t)), ⎪ ⎪ ε ⎪ ⎪ ⎪ ⎨ .. ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ an ⎪ ⎪ ˆ xˆ˙ n (t) = xˆn+1 (t) + n (y(t) − xˆ1 (t)) + hn (u(t), x(t)), ⎪ ⎪ ⎪ ε ⎪ ⎪ a ⎪ ⎪ ⎩ xˆ˙ n+1 (t) = n+1 (y(t) − xˆ1 (t)) + f1 (t, xˆ1 (t), . . . , xˆn (t)). εn+1 (47) Similarly to the proof of Corollary 1, we have immediately Corollary 2. Corollary 2 Suppose that the solution to system (5) satisfies: sup x(t)  N almost surely for some constant t0

N > 0 and the matrix E is Hurwitz. Then under Assumptions 1∗ and 2, there exists a constant ε∗ > 0 such that the LESO (47) is convergent in the sense that for any ˆ xˆn+1 (0)) ∈ ε ∈ (0, ε∗ ), any initial values x(0) ∈ Rn , (x(0), n+1 R , and any positive constant a > 0, E|xi (t) − xˆi (t)|2  Γε2n+3−2i uniformly in [a, +∞), i = 1, 2, . . . , n + 1, (48) where Γ > 0 is an ε-independent constant.

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4 Numerical simulation In this section, we present an example to illustrate the effectiveness of the proposed ESO to estimate both state and stochastic total disturbance. Consider the following second order uncertain lower triangular nonlinear system with exogenous stochastic disturbance: ⎧ ⎪ ⎪ ⎪ ⎪ x˙ 1 (t) = x2 (t) + h1 (u(t), x1 (t)), ⎪ ⎨ x˙ 2 (t) = f (t, x1 (t), x2 (t), w(t)) + h2 (u(t), x1 (t), x2 (t)), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y(t) = x1 (t), (49)

is Hurwitz with all eigenvalues being identical to −1. In this case, gi ( · ) in (7) can be specified as g1 (y1 ) = 3y1 + Ψ (y1 ), g2 (y1 ) = 3y1 , g3 (y1 ) = y1 . (55) The Lyapunov function V : R3 → R for this case is given by V(y) = yT Py +

put u(t) is also disturbed by stochastic noise: u(t) = sin(t + Bt ).

(51)

We design a NLESO (52) for system (49) as follows: ⎧ ⎪ ˙ ⎪ ⎪ ⎪ xˆ1 (t) = xˆ2 (t) + sin(xˆ1 (t)) + ⎪ ⎪ ⎪ y(t) − xˆ1 (t) ⎪ ⎪ ⎪ +εΨ ( ), ⎪ ⎨ ε2 ⎪ ⎪ ⎪ ⎪ xˆ˙ 2 (t) = xˆ3 (t) + sin(xˆ2 (t)) + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ xˆ˙ 3 (t) = (y(t) − xˆ1 (t)), ε3

⎛ ⎞ ⎜⎜ 1 − 1 −1 ⎟⎟ ⎜⎜ ⎟⎟ 2 ⎜⎜ ⎟⎟ ⎜⎜ 1 ⎟ 1 P = ⎜⎜⎜− 1 − ⎟⎟⎟⎟ ⎜⎜ 2 ⎟⎟ 2 ⎜⎜ ⎟ ⎝ −1 − 1 4 ⎟⎠ 2

⎧ 1 ⎪ ⎪ − , ⎪ ⎪ ⎪ π ⎪ ⎪ ⎪ ⎪ πs ⎨1 Ψ (s) = ⎪ sin , ⎪ ⎪ π 2 ⎪ ⎪ ⎪ ⎪ ⎪1 ⎪ ⎩ , π

to discretize systems (49) and (52). Figs. 1–3 display the numerical results for (49) and (52) where we take ⎧ 1 1 ⎪ ⎪ w(t) = cos( t + Bt ), ⎪ ⎪ ⎪ 3 3 ⎪ ⎪ ⎪ ⎨ f (t, x1 (t), x2 (t), w(t)) ⎪ ⎪ ⎪ ⎪ ⎪ = −2x1 (t) − 3x2 (t) + w(t) + cos t, ⎪ ⎪ ⎪ ⎩ ε = 0.01,

3 (y(t) − xˆ1 (t)) + u(t), ε2

where Ψ ( · ) : R → R is defined as s ∈ (−∞, −1], s ∈ (−1, 1),

the initial values: ⎧ ⎪ ⎪ ⎨ x1 (0) = 1, x2 (0) = −1, ⎪ ⎪ ⎩ xˆ1 (0) = xˆ2 (0) = xˆ3 (0) = 0,

(58)

(59)

and the time discrete step as (53)

s ∈ [1, +∞).

Δt = 0.001.

(60)

We notice that the matrix of linear main part of system

First, we notice that the corresponding matrix in (27) for the linear part of (52) ⎛ ⎞ ⎜⎜ −3 1 0 ⎟⎟ ⎜⎜ ⎟⎟ ⎜ ⎟ E = ⎜⎜⎜ −3 0 1 ⎟⎟⎟ ⎜⎜ ⎟⎟ ⎝ ⎠ −1 0 0

(57)

is the positive definite solution to the Lyapunov equation PE + ET P = −I for E given by (54). Similar to [9], Assumption 3 in (11) is satisfied for this example. Hence (52) serves as a well-defined NLESO for (49) from Theorem 1. The Milstein approximation method [27] is used

3 (y(t) − xˆ 1 (t)) ε

(52)

Ψ (s)ds, ∀y = (y1 , y2 , y3 )T ∈ R3 ,

where

(50)

are known functions, whereas the stochastic total disturbance x3  f (t, x1 , x2 , w) is completely unknown. We suppose without loss of generality that the bounded in-

0

(56)

where h1 (u, x1 ) = sin x1 , h2 (u, x1 , x2 ) = sin x2 + u,

 y1

(54)

(49) ⎛ ⎞ ⎜⎜ 0 1 ⎟⎟ ⎜⎜ ⎟⎟⎟ ⎜⎝ −2 −3⎠

(61)

is also Hurwitz. In addition, since the control input and external stochastic disturbance are uniformly bounded

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187

almost surely, the solution to system (49) is also uniformly bounded almost surely. Therefore, all the Assumptions in Theorem 1 are satisfied. It is seen from Figs. 1–3 that the NLESO (52) is very effective in tracking system (49) not only for the state (x1 (t), x2 (t)) but also for the extended state (total disturbance) x3 (t) defined by 1 1 x3 (t) = −2x1 (t) − 3x2 (t) + cos( t + Bt ) + cos t. (62) 3 3 It is observed from Fig. 1 that the tracking effect for x1 (t) is the best, and x2 (t) the second from Fig. 2, and x3 (t) the third from Fig. 3. These are coincident with theoretical estimation (13) that the estimation errors for x1 (t), x2 (t), and x3 (t) are bounded by O(ε5 ), O(ε3 ), and O(ε) in practical mean square sense, respectively. In addition, the peaking values near the initial stages are observed in both Figs. 2 and 3, which can be reduced by time-varying gain approach presented in the recent paper [28].

Fig. 3 The tracking effect for stochastic total disturbance x3 .

5 Conclusions In this paper, an extended state observer (ESO) is designed for a class of lower triangular nonlinear systems with large stochastic total disturbance which comes from both internal unknown dynamics and external stochastic disturbance. The ESO is used to estimate, in real time, not only the state but also the stochastic total disturbance by the measured output. The meansquare convergence of both nonlinear ESO and linear ESO are presented. In addition, a special kind of ESO is constructed for a class of lower triangular nonlinear systems where the stochastic total disturbance includes external stochastic disturbance only. The corresponding mean-square convergence is also concluded. The numerical simulations validate the theoretical results. References

Fig. 1 The tracking effect for state x1 .

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Zehao WU was born in Guangdong, China,

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in 1988. He received his B.Sc. degree in Mathematics from Guangdong Polytechnic Normal University, Guangzhou, China, in 2011, and M.Sc. degree in Mathematics from Xiamen University, Xiamen, China, in 2014. He is currently pursuing the Ph.D. degree at the Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing. His research interests include stochastic systems control and distributed parameter systems control. E-mail: [email protected]. Baozhu GUO received the Ph.D. degree from the Chinese University of Hong Kong in applied mathematics in 1991. From 1985 to 1987, he was a Research Assistant at Beijing Institute of Information and Control,

[20] Q. Zheng, L. Gao, Z. Gao. On stability analysis of active disturbance rejection control for nonlinear time-varying plants with unknown dynamics. Proceedings of the 46th IEEE Conference on Decision and Control, 2007: 4090 – 4095.

China. During the period 1993–2000, he was with the Beijing Institute of Technology,

[21] S. Faetti, P. Grigolini. Unitary point of view on the puzzling problem of nonlinear systems driven by colored noise. Physical Review A, 1987, 36(1): 441 – 444.

Since 2000, he has been with the Academy of Mathematics and Systems Science, the Chinese Academy of Sciences, where he is a research

[22] Y. Jia, X. Zheng, X. Hu, et al. Effects of colored noise on stochastic resonance in a bistable system subject to multiplicative

first as an associate professor (1993–1998) and subsequently a professor (1998–2000).

professor in Mathematical System Theory. His research interests include the theory of control and application of infinite-dimensional systems. E-mail: [email protected].