On Convergence of Nonlinear Active Disturbance Rejection for SISO Systems

Transcription

1 On Convergence of Nonlinear Active Disturbance Rejection for SISO Systems Bao-Zhu Guo 1, Zhi-Liang Zhao 2, 1 Academy of Mathematics and Systems Science, Academia Sinica, Beijing, , China bzguo@issaccn 2 School of Mathematical Science, University of Science and Technology, Hefei, , China gsdxzzl@mailustceducn Abstract: The active disturbance rejection control (ADRC) was proposed by Jingqing Han in the late 1990s, which offers a new and inherently robust controller building block that requires very little information of the plant Originally, the proposal was based largely on experiments with numerous simulations on various systems of different nature Later, the effective of the control strategy has also been demonstrated in many engineering applications such as motion control, web tension regulation, chemical processes However, many theoretical issues, including its applicability in stabilization, output regulation remain unanswered In this paper, we consider the ADRC for general single input single output nonlinear systems subject to dynamical and external uncertainties We establish conditions that guarantee the ADRC achieving closed-loop system stability, disturbance rejection, and reference tracking Key Words: Nonlinear system, Lyapunov stability, feedback stabilization, robust design 1 Introduction The active disturbance rejection control (ADRC), as an unconventional design strategy, was first proposed by Han in his pioneer work [6] It has been now acknowledged to be an effective control strategy in the absence of proper models and in the presence of model uncertainty Its power was originally demonstrated by numerical simulations [5, 6], and later by many engineering practices such as motion control, tension control in web transport and strip precessing systems, DC-DC power converts in power electronics, continuous stirred tank reactor in chemical and process control, micro-electro-mechanical systems gyroscope For more details on practical perspectives, we refer to a nice recent summary paper [17] Other concrete examples can also be found in [8, 15, 16, 18] and the references therein Like some other important ideas in control, the theoretical study on ADRC lags far behind the applications Apart from some early preliminary arguments in [2] for reference tracking of linear systems, and some efforts on state observer in [14] for linear ADRC, and linear ADRC [1] also, there is still no, up to date, a rigorous convergence proof for nonlinear ADRC Let us first briefly recall the main idea of ADRC The details can be found in [8] For a n-dimensional SISO nonlinear system x (n) (t) =f(x(t), ẋ(t),,x (n 1) (t),w(t)) + bu(t), y(t) =x(t), This work was carried out with the support of the National Natural Science Foundation of China, the National Basic Research Program of China (2011CB808002) which can be written as ẋ 1 (t) =x 2 (t), ẋ 2 (t) =x 3 (t), ẋ n (t) =f(x 1 (t),,x n (t),w(t)) + bu(t), y(t) =x 1 (t), (11) where y is the output (observation), u is the input (control), w is the external disturbance, f represents the nonlinear dynamic function of the plant which is possibly unknown, and b>0 is a parameter which is also unknown in some circumstances The objective of ADRC is to design an observer-based output feedback control so that the output y tracks a given reference signal v, and at the same time x i (t) tracks v (i 1) for every i =2, 3,,nprovided that the latter exist in some sense We shall seen later that this general formulation covers not only the special output regulation problem, but also the output feedback stabilization by setting v 0 To begin with, we are given a reference system: ẋ 1(t) =x 2(t), ẋ 2(t) =x 3(t), ẋ n(t) =ϕ(x 1(t),,x n(t)),ϕ(0, 0,,0) = 0, (12) Assume throughout this paper, the system (12) is globally asymptotically stable The ADRC is composed of three components The first one is to recover all v (i 1),i =2,,n+1through the reference v itself This is realized by the so-called tracking /12/$2600 c 2012 IEEE 3524

2 differentiator (TD): ż 1R (t) =z 2R (t), ż 2R (t) =z 3R (t), TD : ż nr (t) =z (n+1)r (t), ( ż (n+1)r (t) =R n ψ z 2R (t) R,, z (n+1)r(t) R n z 1R (t) v(t), ) (13) The control objective of ADRC is to make x i z ir x i That is, x i converges to z ir or v (i 1) in the desirable way of reference system x i converging to zero The TD was first proposed in [7], but its (correct) convergence becomes available only recently It is proved in [4] that under the global asymptotic stability assumption of free system (corresponding to R =1,v =0in (13)), it is indeed true that z ir v (i 1) as R for all i =1, 2,,n+1, provided that v (i 1) exist in the classical sense or in the sense of distribution The second component of the ADRC is the extended state observer (ESO) for system (11): ˆx 1 (t) =ˆx 2 (t)+ n 1 g 1 (θ(t)), ˆx 2 (t) =ˆx 3 (t)+ n 2 g 2 (θ(t)), ESO : ˆx n (t) =ˆx n+1 (t)+g n (θ(t)) + u(t), ˆx n+1 (t) = 1 g n+1(θ(t)), (14) where θ(t) =(y(t) ˆx 1 (t))/ n The above is a special form of the general ESO proposed in [5] in estimating, in real time, both the state x i (i =1, 2,,n) of (11), and the extended state x n+1 = f +(b )u, where > 0 is a nominal parameter of b, and >0is the regulable high gain It is pointed out that it is this remarkable component that the ADRC is rooted The convergence of the extended state observer is presented in [3] where it is shown that (14) is convergent in the sense that ˆx i x i as 0 and t for all i =1, 2,,n+ 1 The third and the last link of ADRC is to design an observer-based output feedback control: u(t) = 1 [ ϕ(ˆx(t) zr (t)) + z (n+1)r (t) ˆx n+1 (t) ], (15) where (ˆx =(ˆx 1, ˆx 2,,ˆx n ), ˆx n+1 ) is the solution of (14) and (z R =(z 1R,z 2R,,z nr ),z (n+1)r ) is the solution of (13) It is observed from the expression of feedback controller (15) that the third term ˆx n+1 is used to cancel, in real time, the effect of total disturbance x n+1 = f +(b )u, and the first two terms ϕ(ˆx z R )+z (n+1)r is used, after recovering the differentials v (i 1) by z R, to achieve the reference stable system (12) with difference x i z ir x i From these observations, we see that the ADRC, unlike the traditional design like internal mode principle (IMP), adopt a entirely new strategy in dealing with the uncertainty Definition 11 Let x i (1 i n) and ˆx i (1 i n +1) be the solutions of the closed-loop system (11) under the feedback (15) with ESO (14), coupling TD (13) and reference system (12) Let x n+1 = f +(b )u be the extended state variable We say that the ADRC is convergent, if for any given initial values of (11), (13), (14), there exits a constant R 0 > 0 such that for any R>R 0, lim 0,t [x i (t) ˆx i (t)] = 0, (16) lim 0,t [x i (t) z ir (t)] = 0 Moreover, for any given a>0, lim R z 1R (t) v(t) = 0 uniformly for t [a, ) Note that the convergence of the closed-loop system (11) under the feedback (15) is first presented in [14] for linear ADRC, that is, all functions g i in (14) and ϕ in (12) and (15) are linear ones In addition, it is assumed in [14] that the following function h(t) = d [f(x(t),w(t)) + (b )u(t)] (11) = 1 i=1 x i+1(t) f (x(t),w(t)) x i +[f(x(t),w(t)) + bu(t)] f (17) (x(t),w(t)) +ẇ(t) f w (x(t),w(t)) + (b ) u(t), is bounded, where x =(x 1,x 2,,x n ) It is actually a priori condition and hence very hard to verify, because h in (17) is not only a function of x, w but also a function of ˆx, z R in the closed-loop A similar idea like linear ADR- C can also be found in [11], in particular, [1] has actually proved the convergence of the stabilization for linear ADR- C In the next section, Section 2, we give a rigorous convergence proof results of the nonlinear ADRC which is formed by the closed-loop system of (11) under the feedback (15) coupling with system (12)-(14) under some checkable conditions The linear ADRC is hence deduced as a consequence 2 Convergence of ADRC with total disturbance Due to the relative independency of (13) and (12) with the other components of ADRC, we write the closed-loop system of (11) under the feedback (15) coupling with (14) as follows: ẋ 1 (t) =x 2 (t), ẋ 2 (t) =x 3 (t), ẋ n (t) =f(x(t),w(t)) + (b )u(t) + u(t), ˆx 1 (t) =ˆx 2 (t)+ n 1 g 1 (θ 1 (t)), (21) ˆx n (t) =ˆx n+1 (t)+g n (θ 1 (t)) + u(t), ˆx n+1 (t) = 1 g n+1(θ 1 (t)), u(t) = 1 [ϕ(ˆx(t) z R (t)) + z (n+1)r (t) ˆx n+1 (t)], th Chinese Control and Decision Conference (CCDC) 3525

3 where z R =(z 1R,z 2R,,z nr ), (z R,z (n+1)r ) is the solution of (13), x = (x 1,x 2,,x n ), ˆx = (ˆx 1, ˆx 2,, ˆx n ), and θ 1 (t) =(x 1 (t) ˆx 1 (t))/ n The following assumptions about f,w,ϕ,b are needed in the establishment of the convergence The assumption A1 is made for system (11) itself and external disturbance, Assumption A2 is for ESO (14) and unknown parameter b, Assumption A3 is for reference system (12), and Assumption A4 is for TD (13) Assumption A1 f C 1 (R n+1 ), w C 1 (R), both w, ẇ are bounded on R, and all partial derivatives of f with respect to its independent variables are bounded over R n+1 Assumption A2 g i (r) k i r for some positive constants k i for all i =1, 2,,n+1 There exist constants λ 1i (i =1, 2, 3, 4), β 1, and positive definite continuous differentiable functions V 1,W 1 : R n+1 R such that 1) λ 11 y 2 V 1 (y) λ 12 y 2, λ 13 y 2 W 1 (y) λ 14 y 2, y R n+1, 2) i=1 (y i+1 g i (y 1 )) V 1 (y) g n+1 (y 1 ) V 1 (y) W 1 (y), y R n+1, y n+1 3) V 1 (y) y n+1 β 1 y, y R n+1 Moreover, the parameter b satisfies b b0 k n+1 < λ13 β 1, where is the nominal parameter Here and throughout the paper, we always use to denote the corresponding Euclidian norm Assumption A3 ϕ is globally Lipschitz continuous with Lipschitz constant L: ϕ(x) ϕ(y) L x y for all x, y R n There exist constants λ 2i (i=1,2,3,4), β 2, and positive definite continuous differentiable functions V 2,W 2 : R n R such that 1) λ 21 y 2 V 2 (y) λ 22 y 2, λ 23 y 2 W 2 (y) λ 24 y 2, 1 2) i=1 y V 2 i+1 (y) +ϕ(y 1,y 2,,y n ) V 2 (y) W 2 (y), y n 3) V 2 y n β 2 y, y R n Assumption A4 Both v and v are bounded over [0, ), ψ is locally Lipschitz continuous, and the system (13) with v =0,R=1is globally asymptotically stable Theorem 21 Let x i (1 i n) and ˆx i (1 i n +1) be the solutions of the closed-loop system (21), x n+1 = f(x, w)+(b )u be the extended state, and z 1R be the solution of (13) Under Assumptions A1 A4, the following statements hold true for any given initial values of (13) and the closed-loop system (21) (i) For any σ>0and τ>0, there exists a constant R 0 > 0 such that z 1R (t) v(t) <σuniformly in t [τ, ) for all R>R 0 (ii) For every R>R 0, there is a R-dependent constant 0 > 0 (specified by (216) later) such that for any (0, 0 ), there exists a t > 0 such that for all R>R 0, (0, 0 ), t>t, and x i (t) ˆx i (t) Γ 1 n+2 i,i=1, 2,,n+1, (22) x i (t) z ir (t) Γ 2, i =1, 2,,n (23) Γ 1 and Γ 2 are R-dependent positive constants only (iii) For any σ>0, there exist R 1 >R 0, 1 (0, 0 ) such that for any R>R 1 and (0, 1 ), there exists a t R > 0 such that for all R>R 1, (0, 1 ), and t>t R, it has x 1 (t) v(t) <σ Proof Statement (i) follows directly from Theorem 21 of [4], which concludes the convergence of tracking differentiator (13) under Assumption A4 Moreover, from the proof of Theorem 21 of [4], we know that for any R>R 0, there exists an M R > 0 such that (z 1R (t),z 2R (t),,z (n+1)r (t), ż (n+1)r (t)) M R (24) for all t 0 Note that statement (iii) is the direct consequence of (i) and (ii) We list it here to indicate the output tracking function of ADRC only It suffices to prove (ii) Let 1 e i (t) = n+1 i [x i(t) ˆx i (t)], i=1, 2,,n+1 (25) Then we have ėi (t) ==e i+1 (t) g i (e 1 (t)), i n, (26) ė n+1 (t) =h(t) g n+1 (e 1 (t)), where x n+1 (t) =f(x(t),w(t)) + (b )u(t) = f(x(t),w(t)) + b b0 [ϕ(ˆx(t) z R (t)) +z (n+1)r (t) ˆx n+1 (t)], (27) is the extended state of system (21) and h(t) is defined by (17) Under feedback law (15), since x n+1 = f +(b )u, (f + bu) f f = x n+1 + u f, we can compute h as h(t) = d [f(x(t),w(t)) + (b )u(t)] (21) = i=1 x i+1(t) f (x(t),w(t)) x i +ẇ(t) f w (x(t),w(t)) + [ϕ(ˆx(t) z R(t)) +z (n+1)r (t) ˆx n+1 (t)] f (x(t),w(t)) + b b 0 [ˆx i+1 (t)+ n i g i (e 1 (t)) i=1 z (i+1)r (t)] ϕ (ˆx(t) z R (t)) +ż (n+1)r (t) 1 } g n+1(e 1 (t)), (28) th Chinese Control and Decision Conference (CCDC)

4 where f x i (x(t),w(t)) denotes the value of the i-th partial derivative of f C 1 (R n+1 ) at (x(t),w(t)) R n+1 ; similarly for ϕ (ˆx(t) z R (t)) Set η i (t) =x i (t) z ir (t), i=1, 2,,n, η(t) =(η 1 (t),η 2 (t),,η n (t)) It follows from (21) and (26) that η 1 (t) =η 2 (t), η n (t) =ϕ(η 1 (t),η 2 (t),,η n (t)) + e n+1 (t) +[ϕ(ˆx(t) z R (t)) ϕ(x(t) z R (t))], ė 1 (t) =e 2 (t) g 1 (e 1 (t)), ė n (t) =e n+1 (t) g n (e 1 (t)), ė n+1 (t) =h(t) g n+1 (e 1 (t)) Furthermore, set ẽ(t) =(e 1 (t),e 2 (t),,e n (t)), e(t) =(e 1 (t),e 2 (t),,e n+1 (t)) By Assumption A3, we have (0, 1) (29) ϕ(ˆx(t) z R (t)) ϕ(x(t) z R (t)) L ẽ(t) (210) From the boundedness of the partial derivatives of f, z R (see (24)) and w, it follows that for some R-dependent positive numbers M,N 0,N 1,N > 0 (by mean value theorem with f(0, 0)) f(x(t),w(t)) M η(t) + N (211) Notice that by (27), we have ˆx n+1 (t) = b [ e n+1 (t)+f(x(t),w(t)) + b (ϕ(ˆx(t) z R (t)) + z (n+1)r (t) ] (212) By (24), (211) (212), Assumptions A1 and A3, the function h given by (28) satisfies h(t) B 0 + B 1 e(t) + B 2 η(t) + B e(t), (213) where B = k n+1 b /,B 0,B 1,B 2 are R-dependent positive numbers We proceed the proof in three steps as follows Step 1 Show that for every R>R 0, there exists a R- dependent 0 > 0 such that for any (0, 0 ), there exist t 1 and r>0such that the solution of (29) satisfies (e(t),η(t)) r for all t>t 1, where r is a R- dependent constant Consider the positive definite function V : R 2n+1 R given by V (e 1,,e n+1,η 1,,η n ) = V 1 (e 1,,e n+1 )+V 2 (η 1,,η n ), (214) where V 1 and V 2 are positive definite functions specified in Assumptions A2 and A3 respectively Computing the derivative of V along the solution of (29), owing to Assumptions A2, A3, and the inequalities (210) and (213), we obtain (29) = 1 [ (e i+1 (t) g i (e 1 (t))) V 1 e i i=1 g n+1 (e 1 (t)) V 1 +h(t) V ] i=1 η i+1(t) V 2 η i +ϕ + e n+1 (t) +[ϕ(ˆx(t) z R (t)) ϕ(x(t) z R (t))]} V 2 1 ( W 1 + B 0 + B 1 e(t) } +B 2 η(t) + B )β e(t) 1 e(t) W 2 + (L +1)β 2 e(t) η(t) W 1 + B 0 β 1 e(t) + B 1 β 1 e(t) 2 W 2 + (B 2 β 1 +(L +1)β 2 ) e(t) η(t) + β 1B [ ] e(t) 2 β 1 B 1 e(t) 2 +β 1 B 0 e(t) λ 23 η(t) 2 + e(t) λ 13 β 1 B (B 2β 1 +(L +1)β 2 ) η(t) [ ] β 1 B 1 e(t) 2 +β 1 B 0 e(t) λ 23 η(t) 2 + λ 13 β 1 B e(t) (B 2 β 1 +(L +1)β 2 ) 2 η(t) 2 2(λ 13 β 1 B) (215) th Chinese Control and Decision Conference (CCDC) 3527

5 [ 2 +β 1 B 0 e(t) β 1 B 1 ] e(t) 2 [ λ 23 (B 2β 1 +(L +1)β 2 ) 2 ] η(t) 2, 2(λ 13 β 1 B) where we used Assumption B<λ 13 /β 1 Set r =max 0 =min 1, 2, 4(1+B0β1), } λ 13 β 1B 2β, (λ 13 β 1B)λ 23 1(B 0+B 1) (B 2β 1+(L+1)β 2) 2 λ 23 } (216) For any (0, 0 ) and (e(t),η(t)) r, we consider the derivative of V along the solution of (29) in two cases Case 1: e(t) r/2 In this case, e(t) 1 and hence e(t) 2 e(t) By the definition 0 of (216), it has, λ 23 (B2β1+(L+1)β2)2 2(λ 13 β 1B) >0, thus it follows from (215) that (29) ( ) β 1 B 1 e(t) 2 + β 1 B 0 e(t) 2 2 ( ) β 1 B 1 β 1 B 0 e(t) 2 2 = λ 13 β 1 B 2β 1 (B 1 + B 0 ) e(t) 2 < 0, 2 where we used again the definition 0 of (216) which gives λ 13 β 1 B 2β 1 (B 1 + B 0 ) > 0 Case 2: e(t) <r/2 In this case, from η(t) + e(t) (e(t),η(t)), it has η(t) r/2 By the definition 0 of (216), λ 13 β 1 B 2β 1 B 1 > 0 Thus it follows from (215) that (29) β 1 B 0 e(t) (λ 23 (B 2β 1 +(L +1)β 2 ) 2 2(λ 13 β 1 B) ) η(t) 2 λ 23 2 η(t) 2 + β 1 B 0 e(t) λ ( 23 r ) 2 ( r + B0 β ) = r ( ) rλ23 4B 0 β 1 < 0, 2 4 where t>t 1, and we used the definition r of (216) which gives rλ 23 4B 0 β 1 > 0 Combining the above two cases yields that for any (0, 0 ),if (e(t),η(t)) r then < 0 (29) Therefor, there exists a t 1 such that (e(t),η(t)) r for all t>t 1 Step 2 Establish the convergence of x i (t) ˆx i (t) Consider the following subsystem which is composed of the last n +1equations in system (29): ė 1 (t) =e 2 (t) g 1 (e 1 (t)), (217) ė n (t) =e n+1 (t) g n (e 1 (t)), ė n+1 (t) =h(t) g n+1 (e 1 (t)) Since (e(t),η(t) r for all t>t 1, we get, together with (213), that h(t) M 0 + B e(t) / for all t>t 1 and some R-dependent constant M 0 > 0 Under Assumption A2, we can compute the derivative of V 1 alone the solution of (217) as follows dv 1 (217) = 1 ( (e i+1 (t) g i (e 1 (t))) V 1 e i i=1 g n+1 (e 1 (t)) V 1 +h(t) V ) 1 λ13 Bβ1 e(t) 2 + M 0 β 1 e(t) λ13 Bβ1 λ 12 + M0β1 λ11 λ 11 V 1 V1 (218) In the last step above, we used again Assumption B λ 13 /β 1 It follows that d V 1 + M 0β 1 λ11 2λ 11, t>t 1, λ 13 Bβ 1 V1 2λ 12 and hence V1 V 1 (e(t 1 ))e λ 13 Bβ 1 2λ (t t 12 1) + M0β1 λ11 λ 11 t t 1 e λ13 Bβ1 2λ 12 (t s) ds, t>t 1 (219) (220) This together with (25) implies that there exist t 2 >t 1 and R-dependent constant Γ 1 > 0 such that x i (t) ˆx i (t) = n+1 i e i (t) n+1 i e(t) n+1 i V1 Γ 1 n+2 i, t>t 2, λ 11 (221) where we used the facts xe x < 1 for all x > 0 and (e(t),η(t)) r for all t>t 1, which was proved in Step 1 Step 3 Establish the convergence of x(t) z R (t) Consider the following system which is composed of the th Chinese Control and Decision Conference (CCDC)

6 first n equations in system (29): η 1 (t) =η 2 (t), η 2 (t) =η 3 (t), η n (t) =ϕ + e n+1 (t) +[ϕ(ˆx(t) z R (t)) ϕ(x(t) z R (t))] (222) Under Assumption A3, and using (210), we can compute the derivative of V 2 along the solution of (222) as follows: dv 2 (11) = 1 i=1 η i+1(t) V 2 η i +ϕ + e n+1 (t) +[ϕ(ˆx(t) z R (t)) ϕ(x(t) z R (t))]} V 2 η n W 2 + (L +1)β 2 e(t) η(t) λ 23 λ 22 V 2 + N 0 V 2, (223) where t>t 2, N 0 is some R-dependent positive constant and we used the fact e(t) (n+1)b 1 proved in (221) It then follows that d V 2 λ 23 λ 22 V2 + N 0, t>t 2 (224) This together with Assumption A3 implies that for all t> t 2, that λ21 η(t) V2 λ 21 ( λ21 e λ 23 λ (t t 22 2) V 2 (η(t 2 )) λ 21 +N 0 ) t t 2 e λ23 λ (t s) 22 ds (225) Since the first term of the right side of (225) tends to zero as t goes to infinity; and the second term is bounded by multiplied by an -independent constant, it follows that there exist t >t 2 and Γ 2 > 0 such that x(t) z R (t) Γ 2 for all t>t Thus (23) follows This completes the proof Remark 21 Equations (22) and (23) are stronger than (16) in Definition 11 for the convergence of ADRC We point out that in Theorem 21, 0, 1, Γ 1, Γ 2,t,t R are all R-dependent; 0, 1, Γ 1, Γ 2 are independent of initial value of (21); t,t R dependent on initial value of (21) Remark 22 We remark that it is not necessary to assume that the functions g i (1 i n +1)are bounded by linear functions (as required in Assumption A2) if we want to conclude the convergence result of [3] for ESO (14) This additional assumption for g i can be considered as a sufficient condition that guarantees the separation principe For linear g i, we conclude the results of [1] REFERENCES [1] LB Freidovich, HK Khalil, Performance recovery of feedback-linearization-based Designs, IEEE Trans Auto Contr, 53(2008), [2] BZ Guo, JQ Han and FB Xi, Linear trackingdifferentiator and application to online estimation of the frequency of a sinusoidal signal with random noise perturbation, International Journal of Systems Science, 33(2002), [3] BZ Guo and ZL Zhao, On the convergence of extended state observer for nonlinear systems with uncertainty, Sys & Contr Lett, 60(2011), [4] BZ Guo and ZL Zhao, On convergence of nonlinear tracking differentiator, International Journal of Control, 84(2011), [5] JQ Han, A class of extended state observers for uncertain systems, Control & Decision, 10(1)(1995), (in Chinese) [6] JQ Han, Auto-disturbance rejection control and applications, Control & Decision, 13(1)(1998), 19-23, (in Chinese) [7] JQ Han, Control theory: Model approach or control approach, J Sys Sci & Math Scis, 9(4)(1989), (in Chinese) [8] JQ Han, From PID to active disturbance rejection control, IEEE Trans Ind Electron, 56(2009), [9] JQ Han and W Wang, Nonlinear tracking-differentiator, J Sys Sci & Math Scis, 14(2)(1994), (in Chinese) [10] T Hu and Z Lin, Output regulation of linear systems with bounded continuous feedback, IEEE Trans Auto Contr, 49(2004), [11] HX Li and PPJVan Den Bosch, A robust disturbancebased control and its applications, Internaltional Journal of Control, 58(1993), [12] X Wang, Z Chen and G Yang, Finite-time-convergent differentiator based on singular perturbation technique, IEEE Trans Auto Contr, 52(2007), [13] XX Yang, Y Huang, Capability of extended state observer for estimating uncertainties, American Control Conferencce, 2009, [14] Q Zheng, L Gao and ZGao, On stability analysis of active disturbance rejection control for nonlinear time-varying plants with unknow dynamics, IEEE Conference on Decision and Control, 2007, [15] Q Zheng, L Gong, DH Lee and Z Gao, Active disturbance rejection control for MEMS gyroscopes, American Control Conference, 2008, [16] Q Zheng, LL Dong and Z Gao, Control and rotation rate estimation of vibrational MEMS gyroscopes, IEEE Multi- Conference on Systems and Control, 2007, [17] Q Zheng, ZGao, On pratical applications of active disturbance rejection control, Chinese Control Conference, 2010, [18] W Zhou and Z Gao, An active disturbance rejection approach to tension and velocity regulations in Web processing lines, IEEE Multi-conference on Systems and Control, 2007, th Chinese Control and Decision Conference (CCDC) 3529