On the Convergence of Extended State Observer for Nonlinear Systems with Uncertainty

Transcription

1 On the Convergence of Extended State Observer for Nonlinear Systems with Uncertainty Bao-Zhu Guo a,b,c,d, Zhi-liang Zhao a a Department of Mathematics, University of Science and Technology of China, Hefei 36, Anhui, PR China b Academy of Mathematics and Systems Science, Academia Sinica, Beijing 8, PR China c School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 5, Johannesburg, South Africa d School of Mathematical Science, Shanxi University, Taiyan, 36, PRChina Abstract The extended state observer first proposed by Jingqing Han in [Control and Decision, ()(995), 85-88] is the key link toward the active disturbance rejection control that is taking off as a technology after numerous successful applications in engineering Unfortunately, there is no a rigorous proof of convergence to date In this paper, we attempt to tackle this long unsolved extraordinary problem The main idea is to transform the error equation of objective system with its extended state observer into a asymptotical stable system with small disturbation, for which the effect of total disturbance error is eliminated by the high-gain Key words: Extended state observer, nonlinear systems, uncertainty Introduction The observer design is all along a major issue in nonlinear systems control The Luenberger observer-based approach is the main method They are many other approaches like slide mode based approach Some of them are of advantage in robustness, but most of them come up against the problems like uncertainty estimation, adaptability, and anti-chattering We refer to [3,4,7,7,8, 4] and references therein, and the recent book [] A particular attention is paid to [] where the comparison of the approach discussed in this work with other existing approaches were presented in details This work was carried out with the support of National Natural Science Foundation of China, the National Basic Research Program of China(CB88), and the National Research of South Afric The corresponding author bzguo@issaccn Preprint submitted to Elsevier 3 March

2 In his seminal work [9], Han proposed the following extended state observer (ESO): ˆx (t) = ˆx (t) α g (ˆx (t) y(t)), ˆx (t) = ˆx 3 (t) α g (ˆx (t) y(t)), ˆx n (t) = ˆx n+ (t) α n g n (ˆx (t) y(t)) + u(t), () ˆx n+ (t) = α n+ g n+ (ˆx (t) y(t)), for an n-dimensional SISO nonlinear system x (n) (t) = f(t, x(t), ẋ(t),, x (n ) (t)) + w(t) + u(t), y(t) = x(t), which can be written as ẋ (t) = x (t), x () = x, ẋ (t) = x 3 (t), x () = x, ẋ n (t) = f(t, x (t), x (t),, x n (t)) +w(t) + u(t), x n () = x n, y(t) = x (t), () where u C(R, R) is the input (control), y the output (measurement), f C(R n, R) a possibly unknown system function, and w C(R, R) the uncertain external disturbance f + w is called total disturbance in the elaborated expository paper [] (x, x,, x n ) is the initial state, and α i, i =,,, n + are regulable gain constants The main idea of the extended state observer is that for the appropriately chosen functions g i C(R, R), the state of the observer ˆx i, i =,,, n and ˆx n+ can be, through regulating α i, considered as the approximations of the corresponding state x i for i =,,, n, and the total disturbance f + w, respectively The last remarkable fact is the source in which the external state observer is rooted This idea can be found in high-gain observer with augmented state variable used in [5] (see also an expository survey in [4]) The numerical studies (eg, [9]) and many other studies the years followed have shown that for some nonlinear functions g i and parameter α i, the observer () performs a very satisfactory adaptability, robustness, and anti-chattering For other perspectives of this remarkable series researches, we refer the reader to [] Unfortunately, although huge applications have been carried out in engineering applications since then (see eg, [,9,7 3]), the choice of functions g i is essentially experiential In order to apply conveniently in practice, it is proposed in [6] the following linear extended state observer (LESO

3 in short in what follows), a special case of (): ˆx (t) = ˆx (t) + α ε (y(t) ˆx (t)), ˆx (t) = ˆx 3 (t) + α ε (y(t) ˆx (t)), (3) ˆx n (t) = ˆx n+ (t) + α n (y(t) ˆx (t)) + u(t), ˆx n+ (t) = α n+ + (y(t) ˆx (t)), where α i, i =,,, n + are pertinent constants, and ε the constant gain The LESO (3) is essentially similar with the extended high-gain observer used in [5] to stabilize a little bit more sophisticated system than () The convergence of the linear high-gain observer for determined systems without augmented variable state is studied in [5] and Theorem 6 of [] on page 8 (see also Chapter 4 of [3] on page 6) A nice survey of high-gain observer with or without augmented state variable can be found in [4] However, the convergence of nonlinear high-gain observer with large uncertainties, particularly, with augmented state variable is, to the best of our knowledge, not available in literature The main part of ESO is not only to estimate the state (x, x,, x n ), but also the total disturbance f + w in which f is also not known in many cases Through ESO (state plus augmented state observer), we are able to cancel the total disturbance in the design of control From this point of view, the study of ESO becomes significant both theoretically and practically For LESO (3), [6] presents some convergence results where either f +w is completely known or f +w is completely unknown The convergence results for some special LESO are also available in [5] In this paper, we are concerned with the convergence of nonlinear extended state observe (NLESO in short in what follows) of the following: y(t) ˆx (t) = ˆx (t) + ˆx (t) g, y(t) ˆx (t) = ˆx 3 (t) + ˆx (t) g, (4) y(t) ˆx (t) ˆx n (t) = ˆx n+ (t) + g n + u(t), ˆx n+ (t) = y ε g ˆx (t) n+, which is a special case of () and an nonlinear generalization of LESO (3) for gain ε and pertinent chosen functions g i, i =,,, n + Notice that the solution of (4) depends on parameter ε, but here we miss ε by abuse of notation without confusion The following assumptions are presumed: Assumption (H) The possibly unknown functions f, w are continuously differentiable with 3

4 respect to their variables, and f u + f + ẇ + t + f n x i c + c j x j k (5) j= for some positive constants c j, j =,,, n and positive integer k We proceed as follows In next section, Section, we show that the NLESO (4) is convergent under additional assumptions Various special NLESO are presented in Section 3 Meanwhile, some examples and numerical simulations are presented from time to time for illustrations The convergence of extended state observer In this section, we consider the convergence of the NLESO (4) To this end, we make additional assumptions Assumption (H) w and the solution x i of () satisfy w + x i (t) B for some constant B > and all i =,, n, and t Assumption (H3) There exist constants λ i (i =,, 3, 4), α, β and positive definite, continuous differentiable functions V, W : R n+ R such that y V (y) λ y, λ 3 y W (y) λ 4 y n V (y i+ g i (y )) V g n+ (y ) W (y), i= y i y n+ V β y, y n+ where y = (y, y,, y n+ ), denotes the Euclid norm of R n+ Theorem Suppose that Assumption (H)-(H3) are satisfied Then (i) For every positive constant a, lim x i(t) ˆx i (t) = uniformly in t [a, ) ε (ii) lim x i (t) ˆx i (t) O (+ i ), t where x i, ˆx i denote the solutions of () and (4) respectively, i =,,, n +, x n+ = f + w is the extended state variable for system () 4

5 Proof By extra state variable x n+ = f + w ([]), system () can be written as ẋ (t) = x (t), x () = x, ẋ (t) = x 3 (t), x () = x, ẋ n (t) = x n+ (t) + u(t), x n () = x n, ẋ n+ (t) = L(t), x n+ () = L(), y(t) = x (t), where L(t) = f(t, x (t), x (t),, x n (t)) + w(t) We first notice that (t) = d ds f(s, x (s),, x n (s)) + ẇ(εt) s=εt = t f(εt, x (εt),, x n (εt)) + n i= +u(εt) x n f(εt, x (εt),, x n (εt)) + ẇ(εt) x i+ (εt) x i f(εt, x (εt),, x n (εt)) () () From Assumptions (H)-(H), there is a positive constant M > such that (t) M for all t Set e i (t) = x i (t) ˆx i (t), η i (t) = e i(εt), i =,,, n + (3) εn+ i Then a direct computation shows that η = (η, η,, η n+ ) satisfies η (t) = η (t) g (η (t)), η () = e (), η (t) = η 3 (t) g (η (t)), η () = e (), η(t) n = η n+ g n (η (t)), η n () = e n(), ε η n+ (t) = g n+ (η (t)) + ε (t), η n+ () = e n+ () (4) By Assumption (H3), finding the derivative of V (η(t)) with respect to t along the solution η(t) of system (4) gives d n dt V (η(t)) = V (η i+ g i (η )) η i i= V g n+ (η ) + η n+ W (η) + εmβ η λ 3 λ V (η) + λ V η n+ ε εmβ V (η) (5) It follows that d V (η(t)) λ 3 λ εmβ V (η(t)) + (6) dt λ 5

6 By Assumption (H3) again, we have η(t) V (η(t)) V (η()) e λ 3 λ t + εmβ t e λ 3 λ (t s) ds (7) This together with (3) yields ( t t e i (t) = + i η i ε ε) n+ i η ε λ + i V (η()) e λ 3 t λ ε + εmβ t ε e λ 3 λ (t/ε s) ds (8) uniformly in t [a, ) as ε Both (i) and (ii) of Theorem then follow from (8) The proof is complete The result of Theorem enables us to deduce the convergence of LESO (3) immediately Actually, if the matrix following is Hurwitz: α α E =, (9) α n α n+ let P be the positive definite matrix solution of the Lyapunov equation P E + E P = I for (n + )-dimensional identity matrix I Define the Lyapunov function V, W : R n+ R by V (η) = P η, η, W (η) = η, η, η R n+ () Then λ min (P ) η V (η) λ max (P ) η, () n V (η i+ α i η ) V α n+ η = η η = η = W (η), i= η i η n+ and V η n+ V η = η P P η = λ max (P ) η where λ max (P ) and λ min (P ) are the maximal and minimal eigenvalues of P, respectively So V, W defined by () satisfies Assumption (H3) Therefore, as a consequence of Theorem, we have improved the stability of LESO (3) presented in [6] where the a in the following Corollary below is required to be large Corollary If the matrix E defined by (9) is Hurwitz and Assumptions (H)-(H) are satisfied, then (i) For every positive constant a, lim x i(t) ˆx i (t) = uniformly in t [a, ) ε 6

7 (ii) lim x i (t) ˆx i (t) O (+ i ), t where x i, ˆx i denote the solutions of (), (3) respectively, i =,, n +, x n+ = f + w is the extended state variable for system () Example For the system following ẋ (t) = x (t), ẋ (t) = x (t) x (t) + w(t) + u(t), () y(t) = x (t), we design the LESO following: ˆx (t) = ˆx (t) + 3 ε (y(t) ˆx (t)), ˆx (t) = ˆx 3 (t) + 3 ε (y(t) ˆx (t)) + u(t), ˆx 3 (t) = ε 3 (y(t) ˆx (t)) (3) For this example, the corresponding matrix 3 E = 3, (4) for which all eigenvalues are equal to, so it is Hurwitz For any bounded control u and bounded disturbance w, ẇ (for instance the finite superposition of sinusoidal disturbance w(t) = p i= a i sin b i t), the solution of () is bounded Figure below gives numerical simulations for Example where we take x () = x () =, ˆx () = ˆx () = ˆx 3 () =, u(t) = sin t, w(t) = cos t, ε = (5) (a) x (green), ˆx (red), x ˆx (blue) (b) x (green), ˆx (red), x ˆx (blue) (c) x 3 (green), ˆx 3 (red), x 3 ˆx 3 (blue) Fig Linear ESO (3) for system () It is seen from Figure that the LESO (3) is very effectively in tracking the system () not only for the state (x, x ) but also for the extended state (uncertainty) x 3 7

8 For system (), we can also design an NLESO following: ˆx (t) = ˆx (t) + 3 y(t) ε (y(t) ˆx ˆx (t) (t)) + εϕ, ε ˆx (t) = ˆx 3 (t) + 3 ε (y(t) ˆx (t)) + u(t), (6) ˆx 3 (t) = ε 3 (y(t) ˆx (t)) where ϕ : R R is defined as ( 4, r, π ), ϕ(r) = ( 4 sin r, r π, π ), π 4, r, (7) In this case, g i in (4) can be specified as g (y ) = 3y + ϕ(y ), g (y ) = 3y, g 3 (y ) = y (8) The Lyapunov function V : R 3 R for this case is given by where V (y) = P y, y + y P = 4 ϕ(s)ds, (9) is the positive definite solution of Lyapunov equation P E + E P = I for E given by (4) A direct computation shows that i= V (y i+ g i (y )) V g 3 (y ) y i y 3 = y y y3 (y y y 3 + ϕ(y ))ϕ(y ) + (y 3y )ϕ(y ) y 8 + 7y 8 + 3y 3 W (y, y, y 3 ) 4 () So all conditions of Assumption (H3) are satisfied Therefore, (6) serves as a well defined NLESO for () according to Theorem Now take the same data as (5) The simulations for NLESO (6) are plotted as Figure below It is seen from Figure that the NLESO (6) is at least as good as LESO (3) in tracking the state and the extended state of the system () In what follows, we relax the conditions of Assumption (H3) by Assumption (H4) following 8

9 (a) x (green), ˆx ( red), x ˆx (blue) (b) x (green), ˆx (red), x ˆx (blue) (c) x 3 (green), ˆx 3 ( red), x 3 ˆx 3 (blue) Fig NLESO (6) for system () Assumption (H4) There exist constants R, α > and positive definite, continuous differentiable functions V, W : R n+ R such that {y V (y) d} is bounded for any d >, n V (y i+ g i (y )) V g n+ (y ) W (y), i= y i y n+ V αw (y) for y > R y n+ We can obtain the weak convergence result stated in Theorem below Theorem Under Assumptions (H),(H) and (H4), the nonlinear extended state observer (4) is convergent in the sense that for any σ (, ), there exists ε σ (, ) such that for any ε (, ε σ ), x i (t) ˆx i (t) < σ, t (T ε, ), where T ε > depends on ε, x i, ˆx i denote the solutions of (), (4) respectively, i =,, n+, x n+ = f + w is the extended state variable for system () Proof For positive definite functions V, W, it follows from Lemma 43 of [3] on page 45 that there exist wedge functions K i : [, ) [, ), i =,, 3, 4 such that K ( (y, y,, y n+ ) ) V (y, y,, y n+ ) K ( (y, y,, y n+ ) ), K 3 ( (y, y,, y n+ ) ) W (y, y,, y n+ ) K 4 ( (y, y,, y n+ ) ) Denote by η(t; η ) the solution of (4), where η = be split into several claims ( e () ε, e ) () n ε,, e n+() The proof will n Claim : There exists an ε (, ) such that for any ε (, ε ), there exists a t ε > such that {η(t; η ) t [t ε, )} {η V (η) C}, () where C = max y R V (y) < 9

10 It is easily shown that there exists an ε (, Mα ) such that for each ε (, ε ), there is a constant t ε > satisfying η(t ε ; η ) R This is because if for all t >, η(t; η ) > R, then finding the derivative of V with respect to t along the solution of (4) leads to a contradiction to the positivity of V : dv dt n = along the system (4) i= V η i (η i+ g i (η )) ( αεm)w (η) K 3(R) V g n+ (η ) + ε V η n+ η n+ < Now we prove Claim by contradiction Firstly, since V (y) C + } is bounded, we have A = V y n+ V (y) sup Y {y C V (y) C+} y n+ is continuous and the set {y C < Secondly, W (η) K 3 ( η ) K 3 K (V (η)) K 3 K (C) >, η {y C V (y) C + } () Suppose that Claim is false Since η(t ε ; η ) R, it has V (η(t ε ; η )) C Let ε = min {, K 3K (C) AM Then there exist an ε < ε and t ε, t ε (t ε, ), t ε < t ε such that and Combining () and (5) yields } (3) η (t ε ; η ) {η V (η) = C}, η (t ε ; η ) {η V (η) > C}, (4) {η (t; η ) t [t ε, t ε ]} {y C V (y) C + } (5) inf W (η(t; η )) K 3 K t [t ε,tε ] (C) (6) Therefore, for t [t ε, t ε ], dv (η(t; η )) dt = dv dt along the system (4) W (η(t; η )) + AMε K 3 K (C) + AM K 3K (C) AM which shows that V (η(t; η )) is non-increasing in [t ε, t ε ], and hence =, V (η(t ε ; η )) V (η(t ε ; η )) = C

11 This contradicts the second inequality of (4) The Claim follows Claim [ : There ] is an ε σ (, ε ) such that for any ε (, ε σ ), there exists a T ε t ε, t ε + such that η(t ε ; η ) < δ c K 3 K (δ) Actually, for any given σ >, since V is continuous, there exists a δ (, σ) such that V (η) K (σ), η δ (7) Now, for each η {η V (η) δ}, W (η) K 3 ( η ) K 3 K (V (η)) K 3 K (δ) > (8) By Claim, for any ε (, ε ), {η(t; η ) t [t ε, )} {η V (η) C}, and hence H = sup t [t ε, ) V (η(t; η )) η n+ sup η {η V (η) C} Suppose that Claim is false Then for ε σ = min {ε, K 3K (δ) HM there exists an ε < ε σ such that η(t; η ) δ for any t (8) concludes that for any ε (, ε σ ) and all t Integral above inequality over V dv (η(t; η )) dt [ t ε, t ε + [ t ε, t ε + V (η) η n+ < }, (9) [ t ε, t ε + C K 3 K (δ) C K 3 K ] (δ), = dv dt along the system (4) V (η(t; η )) W (η(t; η )) + Mε η n+ K 3K (δ) C K 3 K (δ) < ], to give ( ( )) C tε+ C η K 3 K (δ) ; η K = 3 K dv (η(t;, η (δ) )) dt + V (η(t ε ; η )) t ε dt This is a contradiction since for each t also K 3K (δ) C K 3 K (δ) + V (η(t ε; η )) ] This together with [ ] t ε, t ε + C, η(t; η K 3 K (δ) ) δ The Claim follows Claim 3: For each ε (, ε σ ), if there exists a T ε [t ε, ) such that η(t ε ; η ) {η η δ},

12 then { η (t; η ) t (T ε, ] } {η η σ} (3) Suppose Claim 3 is not valid Then there exist t ε > t ε T ε such that η(t ε ; η ) = δ, η(t ε ; η ) > σ, {η(t; η ) t [t ε, t ε ]} {η η δ} (3) This together with (8) concludes that for t [t ε, t ε ], K ( η (t ε ; η ) ) V (η (t ε ; η )) = t ε t ε t ε t ε dv (η(t; η )) dt + V (η (t ε dt ; η )) K 3K (δ) dt + V (η (t ε ; η )) V (η (t ε ; η )) (3) By (7) and the fact η(t ε ; η ) = δ, we have This together with (3) gives V (η(t ε ; η )) K (σ) K ( η (t ε ; η ) ) K (σ) Since the wedge function K is increasing, the above inequality implies that η(t ε ; η ) σ, which contradicts the middle inequality of (3) The Claim 3 is verified Theorem then follows by combing Claims -3 It should be pointed out that Theorem is obtained based on Assumption (H4) rather than Assumption (H3), which is more less restrictive than Assumption (H3) This is because in Assumption (H3), positive definite functions V, W should satisfy conditions y V (y) λ y, λ 3 y W (y) λ 4 y that are not required in Assumption (H4) So under the assumptions of Theorem, it is more flexible to construct examples than under the assumptions of Theorem Now we construct an example that satisfies Assumption (H4) but it is hard to verify whether or not it satisfies Assumption (H3) In order to construct the example, we need the definition following Definition A function V : R n R is called homogeneous of degree d with respect to weights {r i > } n i=, if V (λ r x, λ r x,, λ rn x n ) = λ d V (x, x,, x n ) (33) for all λ > and all (x, x,, x n ) R n A vector field F : R n R n is called homogeneous of degree d with respect to weights {r i > } n i=, if for any i =,,, n, F i (λ r x, λ r x,, λ rn x n ) = λ d+r i F i (x, x,, x n ) (34) for all λ > and all (x, x,, x n ) R n, where F i is the i-th component of F

13 If V is satisfies (33) and is differentiable with respect to x n, then the partial derivative of V in x n satisfies λ rn V (λ r x, λ r x,, λ rn x n ) = λ d V (x, x,, x n ) (35) x n x n The above equality is very convenient to be used for checking the homogeneity of V that we have known the homogeneity of V x n provided Let n =, g (y ) = 3[y ] α, g (y ) = 3[y ] α, g 3 (y ) = [y ] 3α in Theorem Let vector field y g (y ) F (y) = y 3 g (y ), (36) g 3 (y ) where [y ] = sign(y ) y It is easy to verify that the vector field F in (36) is homogeneous of degree α with respect to the weights {, α, α } Since the matrix E given in (4) is Hurwitz, it follows from [] that for some α (, 3 ), the system ẏ(t) = F (y(t)) is finite-time stable From theorem of [] and theorem 6 of [], we get that there exists positive definite, radially unbounded function V : R 3 R such that V is homogeneous of degree γ with respect to the weights {, α, α }, and V (y) (y 3 g (y )) y V (y) y 3 V (y) y (y g (y )) + g 3 (y ) is negative definite and homogeneous of degree γ +α From V (y) (35) and the homogeneity of V, we can get that is homogeneous of degree γ + α y 3 By Lemma 4 of [], there exists positive constants b, b, b 3 > such that V (y) y 3 b (V (y)) γ (α ) γ (37) and b (V (y)) γ ( α) γ V (y) V (y) (y g (y )) + (y 3 g (y )) y y V (y) g 3 (y ) b 3 (V (y)) γ ( α) γ y 3 (38) Let W (y) = c (V (y)) γ ( α) γ Since V is radially unbounded positive definite function, we have, for each d >, that {y V (y) d} is bounded, and lim y V (y) = This together with (37) yields that for α (, 3 ) W (y), lim y V (y) = So there is a B > such that for y 3 y B, V y 3 W (y) Therefore, Assumption (H4) is satisfied 3

14 By Theorem, we can then construct an NLESO following: [ ] α y(t) ˆx (t) ˆx (t) = ˆx (t) + 3ε, ε [ ] α y(t) ˆx (t) ˆx (t) = ˆx 3 (t) u(t), ε ˆx 3 (t) = [ ] 3α y(t) ˆx (t) ε ε (39) Set α = 8, ε = 5 and other parameters as that in (5) We plot the numerical results for LESO (3) in Figure 3 and the NLESO (39) in Figure 4, both for system () (a) x (green), ˆx (red), x ˆx (blue) (b) x (green), ˆx (red), x ˆx (blue) (c) x 3 (green), ˆx 3 (red), x 3 ˆx 3 (blue) (d) Magnification of Figure 3(c) Fig 3 Linear ESO (3) for system () (a) x (green), ˆx (red), x ˆx (blue) (b) x (green), ˆx (red), x ˆx (blue) (c) x 3 (green), ˆx 3 (red), x 3 ˆx 3 (blue) Fig 4 Nonlinear ESO (39) for system () 4

15 The numerical results show that for the same tuning parameter ε, the NLESO (39) is more accurate with small peak value compared with the LESO (3) In Figure 3(c), the peak value of ˆx 3 is between [, ], while in Figure 4(c), the peak value of ˆx 3 is between [ 5, 5] To end this section, we point out that if we only estimate the state rather than extended state (uncertainty), Assumptions (H) and (H) can be replaced by the following Assumptions (A) or (A) where the boundedness of derivative of disturbance is removed Assumption (A) u, w C(R, R) f C(R n+, R) are bounded Assumption (A) The solution x i of () and u, w C(R, R) are bounded, f C(R n+, R) satisfies n f(t, x, x,, x n ) c + c j x j k j Under Assumption (A) or (A), the state observer can be designed as (4) below to estimate the state of () but usually not for unknown f j= y(t) ˆx (t) = ˆx (t) + ˆx (t) g, y(t) ˆx (t) = ˆx 3 (t) + 3 ˆx (t) g, ˆx n (t) = y(t) ε g ˆx (t) n + u(t), (4) Also, we need Assumption (A3) below, which is similar to Assumption (H3) Assumption (A3) There exist constants λ i (i =,, 3, 4), α, β and positive definite, continuous differentiable functions V, W : R n R such that y V (y) λ y, λ 3 y W (y) λ 4 y n V (y i+ g i (y )) V g n (y ) W (y), i= y i y n V y n β y, where y = (y, y,, y n ), denotes the Euclid norm of R n Proposition Suppose that Assumption (A) or (A) is satisfied If Assumption (A3) is satisfied, then (i) For every positive constant a, lim x i(t) ˆx i (t) = uniformly in t [a, ) ε 5

16 (ii) lim x i (t) ˆx i (t) O (+ i ), t where x i, ˆx i denote the solutions of () and (4) respectively, i =,,, n Proof Let η i (t) = x i(εt) ˆx i (εt) i, i =,,, n, where x i, ˆx i are solutions of () and (4) respectively A straightforward computation shows that η (t) = η (t) g (η (t)), η (t) = η 3 (t) g (η (t)), η n (t) = g n (η (t)) + ε (t), (4) where (t) = f(εt, x (εt),, x n (εt)) + w(εt) It is seen that (4) is similar to (4) So the proof is also similar Actually, From Assumption (A) or (A) we get that sup t [, ) (t) M for some M > By Assumption (A3), finding the derivative of V (η(t)) with respect to t along the solution η(t) of system (4) gives n d dt V (η(t)) = V (η i+ g i (η )) V g n (η ) + V ε η i η n η n i= W (η) + εmβ η λ 3 λ V (η) + λ εmβ V (η) (4) It follows that d V (η(t)) λ 3 λ εmβ V (η(t)) + (43) dt λ By Assumption (H3) again, we have η(t) V (η(t)) V (η()) e λ 3 λ t + εmβ t e λ 3 λ (t s) ds (44) It yields ( t t x i (t) ˆx i (t) = i η i ε ε) n i η ε λ i V (η()) e λ 3 t λ ε + εmβ t ε e λ 3 λ (t/ε s) ds (45) uniformly in t [a, ) as ε Both (i) and (ii) of Proposition then follow from (45) 3 Some special extended observer In this section, we state some special applications of the extended state observer The first special case is what so called the output tracking differentiator, Differentiator is another big issue that 6

17 has been extensively studied in literature, see for instance [8,6] Suppose that v is the tracked signal Let x i (t) = v (i ) (t) Then x i, i =,,, n satisfy ẋ (t) = x (t), ẋ (t) = x 3 (t), ẋ n (t) = v (n ) (t), y(t) = x (t) = v(t) (3) The corresponding NLESO (4) becomes v(t) ˆx (t) = ˆx (t) + ˆx (t) g, v(t) ˆx (t) = ˆx 3 (t) + ˆx (t) g, v(t) ˆx (t) ˆx n (t) = ˆx n+ (t) + g n, ˆx n+ (t) = v(t) ε g ˆx (t) n+, (3) The convergence result in this special case is stated as Proposition 3 below, which is actually a consequence of Theorem Proposition 3 [Output tracking differentiator] Suppose that Assumption (H3) holds and v (n) is bounded Then (i) For every positive constant a, lim ε v(i ) (t) ˆx i (t) = uniformly in t [a, ) (ii) lim v (i ) (t) ˆx i (t) O (+ i ), t where ˆx i is the solution of (3), i =,, n + The next example is different to system () and those discussed in [6], in which the system function f is known The unknown part is only the external disturbance w In this case, we try 7

18 to utilize known information as much as possible Our NLESO in this case can be modified as ˆx (t) = ˆx (t) + x (t) ˆx (t) g, ˆx (t) = ˆx 3 (t) + x (t) ˆx (t) g, (33) x (t) ˆx (t) ˆx n (t) = ˆx n+ (t) + g n + f(t, ˆx (t), ˆx (t),, ˆx n (t)) + u(t), ˆx n+ (t) = ε g x (t) ˆx (t) n+, which is used to estimate not only the state (x, x,, x n ) but also the extended state w Using the same notation as in (3) and setting x n+ = w in this case, we get η (t) = η (t) g (η (t)), η () = e (), η (t) = η 3 (t) g (η (t)), η () = e (), η n (t) = η n+ (t) g n (η (t)) + δ (t), η n () = e n(), ε η n+ (t) = g n+ (η (t)) + εδ (t), η n+ () = e n+ (), (34) where δ (t) = f(t, x (εt),, x n (εt)) f(t, ˆx (εt),, ˆx n (εt)), δ (t) = ẇ(εt) (35) Proposition 3 [Modified extended state observer] In addition to the conditions in Assumption (H3), we assume that V y n α y, αρ < λ3, where ρ is the Lipschitz constant of f: f(t, x,, x n ) f(t, y,, y n ) ρ x y, If ẇ is bounded in (), then (i) For every positive constant a, t, x = (x, x,, x n ), y = (y, y,, y n ) R n lim x i(t) ˆx i (t) = uniformly in t [a, ) ε (36) (ii) lim x i (t) ˆx i (t) O (+ i ), t where x i, ˆx i denote the solutions of (), (33) respectively, i =,,, n +, x n+ = w 8

19 Proof Finding the derivative of V (η(t)) with respect to t along the solution η(t) of system (34) yields d n dt V (η(t)) = V (η i+ g i (η )) V g n+ (η ) + V δ + V εδ η i η n+ η n η n+ i= W (η) + αρ η + εmβ η λ 3 αρ λ V (η) + λ It follows that d V (η(t)) λ 3 αρ dt λ This together with Assumption (H3) gives η(t) By (3), it follows that V (η(t)) εmβ V (η) (37) λ εmβ V (η(t)) + (38) V (η()) e λ 3 αρ λ t + εmβ t e λ 3 αρ λ (t s) ds (39) ( t t e i (t) = + i η i ε ε) n+ i η ε λ + i V (η()) e (λ 3 αρ)t λ ε + εmβ t ε e λ 3 αρ λ (t/ε s) ds (3) uniformly in t [a, ) as ε The (ii) of Theorem 3 also follows from (3) The proof is complete Remark 3 Proposition 3 is a special case of Theorem The only difference is that in Proposition 3, f is known while in Theorem, f is not This results in the difference in designing the observer: in (33), we are able to utilize the known information of f, while in (4), this information is lacking Nevertheless, The proof of Proposition 3 is similar to that of Theorem, we put here just for the sake of completeness Example 3 For the system following ẋ (t) = x (t), ẋ (t) = sin(x (t)) + sin(x (t)) + w(t) + u(t), 4π y(t) = x (t), (3) where w is the external disturbance, we design the corresponding modified linear extended state observer as: ˆx (t) = ˆx (t) + 6 ε (y(t) ˆx (t)), ˆx (t) = ˆx 3 (t) + ε (y(t) ˆx (t)) + sin(ˆx (t)) + sin(ˆx (t)) + u(t), 4π ˆx 3 (t) = 6 ε 3 (y(t) ˆx (t)) (3) 9

20 6 For this example, the associated matrix E = 6 it is Hurwitz has three eigenvalues {,, 3}, so Use Matlab to solve the Lyapunov equation P E + E P = I, to find that the eigenvalues of P satisfying λ max (P ) 33 < π Let V, W : R n+ R be defined as V (η) = P η, η, W (η) = η, η Then and λ min (P ) η V (η) λ max (P ) η V, η λ max(p ) η, V (η 6η ) + V V (η 3 η ) 6η = η (P E + E P )η = W (η) (33) η η η Now f(x, x ) = sin(x (t))+sin(x (t)), and we find that L = Hence Lλ 4π π max(p ) < Therefore, for any bounded control u and bounded disturbance w, ẇ (for instance the finite superposition of sinusoidal disturbance w(t) = p i= a i sin b i t), by Theorem 3, for any a > lim x i(t) ˆx i (t) = uniformly in t [a, ) ε and lim x i(t) ˆx i (t) O ( + i), t where x 3 = w(t) and x i, ˆx i are solutions of (3) and (3) respectively We use the data x () = x () =, ˆx () = ˆx () = ˆx 3 () =, u(t) = cos t, w(t) = sin t to simulate Example 3 The results are plotted in Figure 5 below (a) x (green), ˆx (red), x ˆx (blue) (b) x (green), ˆx (red), x ˆx (blue) (c) x 3 (green), ˆx 3 (red), x 3 ˆx 3 (blue) Fig 5 The modified extended state observer (3) for system (3) It is seen from Figure 5 that the modified extended state observer (3) tracks very satisfactorily both the state and extended state of system (3), respectively The last case is more special that the system function f is known and the external w = In this case, system () becomes a special deterministic nonlinear system To design the observer, we make the following assumption

21 Assumption (H5) f is locally Lipschitz continuous, and h(t, x) = t f(t, x) + n i= x i+ (t) f(t, x) f(t, x) + u(t) (34) x i x n is globally Lipschitz continuous in x = (x, x,, x n ) uniformly for t (, ), where x n+ (t) = f(t, x(t)) The state observer for () now is designed as following: y(t) ˆx (t) = ˆx (t) + ˆx (t) g, y(t) ˆx (t) = ˆx 3 (t) + ˆx (t) g, y(t) ˆx (t) ˆx n (t) = ˆx n+ (t) + g n + u(t), ˆx n+ (t) = h(t, ˆx) + y ε g ˆx (t) n+ (35) Theorem 3 Under assumptions (H3) and (H5), it has (i) There exists an ε > such that for any ε (, ε ), lim x i(t) ˆx i (t) = t (ii) For any a >, lim x i(t) ˆx i (t) = uniformly in t [a, ), ε where x i, ˆx i are the solutions of (), (35) respectively, i =,,, n+, x n+ (t) = f(t, x (t),, x n (t)) Proof Using the same notation in (3), since x n+ = f(t, x,, x n ), we have the error system following: η = η g (η ), η () = e () ε, n η = η 3 g (η ), η () = e (), η n = η n+ g n (η ), η n () = e n(), ε η n+ = g n+ (η ) + εδ 4 (t), η n+ () = e n+ (), where δ 4 (t) = h(εt, x(εt)) h(εt, ˆx(εt)), x = (x, x,, x n ), ˆx = (ˆx, ˆx,, ˆx n ) (36)

22 A direct computation shows that the derivative of V with respect to t along the solution of system (36) satisfies d dt V (η(t)) = n i= V η i (η i+ g i (η )) V η n+ g n+ (η ) + W (η) + εβρ η λ 3 εβρ V (η), λ V η n+ εδ 4 (37) for ε (, λ 3 βρ ), where ρ is the Lipschitz constant of h(t, x) It follows that d dt V (η(t)) λ 3 εβρ λ V (η(t)) (38) By Assumption (H3), we have V (η(t)) η(t) V (η()) e λ 3 εβρ λ t (39) Return back to e by (3), to get finally that ( t t e i (t) = + i η i ε ε) n+ i η ε V (η()) e (λ 3 εβρ )t λ ε, (3) uniformly in t [a, ) as ε, or for any ε (, ε ), ε = λ 3 βρ, as t The proof is complete Remark 3 If no extended state variable x n+ is taken into account, and the NLESO observer is taken as LESO, Theorem 3 is just a special case of Theorem 6 of [] on page 8 4 Concluding remark In this paper, the convergence of various high gain nonlinear extended state observers for a kind of nonlinear systems with uncertainty both from internal and external is proved The high gain is used to eliminate the influence of uncertainties Examples and numerical analysis are followed to illustrate the effectiveness of the corresponding observers Due to high gain and the difference between the initial states of the tracked system and the observer, the peaking phenomena observed in [5] in the beginning of the time interval is also observed from the numerical analysis One of the problems in high gain observer perhaps is the robustness to time delay In order to observe the robustness to time delay, we simulate the Example by replacing y(t) = x (t) with y(t) = x (t + τ) for τ = 3 The parameters are taken as the same as that in (5) The result is plotted in Figure 6 below It is seen that the extended state observer can tolerate the small output time delay Acknowledgment The authors would like to thank antonymous referees for their constructive suggestions and comments that improve substantially the original manuscript

23 (a) x (green), ˆx (red), x ˆx (blue) References (b) x (green), ˆx (red), x ˆx (blue) Fig 6 Linear ESO (3) for system () with time delay (c) x 3 (green), ˆx 3 (red), x 3 ˆx 3 (blue) [] G Besancon, Nonlinear Observers and Applications, Springer Verlag, New York, 7 [] SP Bhat and DS Bernstein, Geometric homogeneity with applications to finite-time stability, Math Control Signals Systems, 7(5), -7 [3] MJ Corless, J Tu, State/input estimation for a class of uncertain systems, Automatica, 34(998), [4] M Darouach, M Zasadzinski, JS Xu, Full-order observers for linear systems with uncertain inputs, IEEE Trans Auto Contr, 39(994), [5] LB Freidovich, H K Khalil, Performance recovery of feedback-linearization-based Designs, IEEE Trans Auto Contr, 53(8), [6] Z Gao, Scaling and bandwith-parameterization based controller tuning, American Control Conference, 3, [7] V Gourshankar, P Kudva, K Ramar, Reduced order observer for multivariable systems with inaccessible disturbance inputs, Int J Contr, 5(977), 3-39 [8] BZ Guo and ZL Zhao, On convergence of tracking differentiator, Int J Contr, in press [9] JQ Han, A class of extended state observers for uncertain systems, Control & Decision, ()(995), (in Chinese) [] JQ Han, From PID to active disturbance rejection control, IEEE Trans Ind Electron, 56(9), 9-96 [] Y Hou, Z Gao, F Jiang, and BT Boulter, Active disturbance rejection control for web tension regulation, IEEE Conference on Decision and Control,, [] Y Huang and JQ Han, A new synthesis method for uncertain systems-the self-stable region approach, International Journal of Systems Science, 3(999), [3] HK Khalil, Nonlinear Systems, Prentice Hall, New Jersey, [4] HK Khalil, High-gain observers in nonlinear feedback control, Int Conf Control, Automation and Systems, Oct4-7, Korea, 8 [5] H Khalil, High-gain observer in nonlinear feedback control, in New Direction in Nonlinear Observer Design, Lecture Notes in Control and Inform Sci, 44, Springer, London, 999 3

24 [6] A H Dabroom and H K Khalll, Discrete-time implementation of high-gain observers for numerical differentiation, Int J Control, 7(999), [7] A Koshkouei, A Zinober, Sliding mode controller observer design for SISO linear systems, Int J Sys Sci, 9(998), [8] P Kudva, N Viswanadham, A Ramakrishna, Observers for linear systems with unknown inputs, IEEE Trans Auto Contr, 5(98), 3-5 [9] R Miklosovic and Z Gao, A dynamic decoupling method for controlling high performance turbofan engines, Proc of the 6th IFAC World Congress, July 4-8, 5, Czech Republic [] W Perruquetti, T Floquet and E Moulay, Finite-time observers: application to secure communication, IEEE Trans Automatic Control, 53(8), [] L Rosier, Homogeneous Lyapunov function for homogeneous continuous vector field, Sys & Control Letters, 9(99), [] J Slotine, J Hedrick, E Misawa, On sliding observers for nonlinear systems, ASME J Dynam Sys Measur & Contr, 9(987),45-5 [3] BL Walcott, MJ Corless, SH Zak, Comparative study of non-linear state-observation technique, Int J Contr, 45(987), 9-3 [4] B L Walcott, SH Zak, State observation of nonlinear uncertain dynamical systems, IEEE Trans Auto Contr, 3(987), 66-7 [5] X X Yang, Y Huang, Capability of extended state observer for estimating uncertainties, American Control Conferencce, 9, [6] Q Zheng, LGao, ZGao, On stability analysis of active disturbance rejection control for nonlinear time-varying plants with unknow dynamics, IEEE Conference on Decision and Control, 7, [7] Q Zheng, L Gong, DH Lee and Z Gao, Active disturbance rejection control for MEMS gyroscopes, American Control Conference, 8, [8] Q Zheng, LL Dong and Z Gao, Control and rotation rate estimation of vibrational MEMS gyroscopes, IEEE Multi-Conference on Systems and Control, 7, 8-3 [9] Q Zheng, ZGao, On applications of active disturbance rejection control, Chinese Control Conference,, [3] Q Zheng, Z Chen, Z Gao, A dynamic decoupling control approach and tts applications to chemical processes, American Control Conference, 7, [3] 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, 7,