IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 61, NO. 6, JUNE

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1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 6, NO 6, JUNE Active Disturbance Rejection Control Approach to Output-Feedback Stabilization of a Class of Uncertain Nonlinear Systems Subject to Stochastic Disturbance Bao-Zhu Guo, Ze-Hao Wu, and Hua-Cheng Zhou Abstract The active disturbance rejection control ADRC is now considered as a powerful control strategy in dealing with large uncertainty covering unknown dynamics, external disturbance, and unknown part in coefficient of the control However, all theoretical works up to present are limited to deterministic uncertainty In this technical note, we generalize the ADRC to uncertain nonlinear systems subject to external bounded stochastic disturbance described by an uncertain stochastic differential equation driven by white noise We first design an extended state observer ESO that is used to estimate both state, and total disturbance which includes the internal uncertain nonlinear part and the external uncertain stochastic disturbance It is shown that the resulting closed-loop system is practically stable in the mean-square topology The numerical experiments are carried out to illustrate effectiveness of the proposed approach Index Terms Disturbance, output-feedback, stabilization, uncertain stochastic systems I INTRODUCTION The idea of active disturbance rejection control ADRC initiated by Han [5] has been attracting more attention by the industry practitioners as presented in latest survey-type paper [3] The numerous concrete applications in different fields include control of flexible joint manipulator [], control of a model-scale helicopter [0], control of omnidirectional mobile robot [4], control of synchronous motors [5], and control of delta parallel robot [3] On the other hand, the progress has also been made in theoretical foundations, see [4], [7] In addition, the ADRC has been successfully applied to stabilization for systems described by partial differential equations However, in these works, the external disturbance is always assumed to be deterministic This is not very realistic because many real control systems are most often subject to the influence of randomness, which is a source of instability of the control systems Actually, output-feedback stabilization for stochastic nonlinear systems driven by white noise has been receiving increasing attention Manuscript received May 3, 05; revised June, 05 and August, 05; accepted August 6, 05 Date of publication August 4, 05; date of current version May 5, 06 This work supported by the National Natural Science Foundation of China, the National Basic Research Program of China 0CB80800, and the National Research Foundation of South Africa Recommended by Associate Editor N Kazantzis B-Z Guo is with School of Mathematical Science, Shanxi University, China, Key Laboratory of System and Control, Academy of Mathematics and Systems Science, The Chinese Academy of Sciences, Academia Sinica, Beijing 0090, China, and the School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, Wits 050, South Africa bzguo@issaccn Z-H Wu and H-C Zhou are with the Key Laboratory of System and Control, Academy of Mathematics and Systems Science, Academia Sinica, Beijing 0090, China zehaowu@amssaccn; hczhou@amssaccn Color versions of one or more of the figures in this paper are available online at Digital Object Identifier 009/TAC in last two decades By employing a quartic Lyapunov function, the paper [] presents a backstepping design to achieve a first result on global output-feedback stabilization for stochastic nonlinear systems A problem of stochastic disturbance attenuation is discussed in [] for the systems driven by white noise of unknown covariance Several different output-feedback controllers are developed for stochastic nonlinear systems with unmeasured states, such as tracking control [8] and decentralized control [] However, in these studies, the system functions are supposed to be known or the system uncertainties are linearly parameterized with respect to known nonlinear functions To overcome this obstacle, output-feedback control approaches such as adaptive neural network [9] are investigated for uncertain stochastic systems with unmeasured states However, much less results are available on output-feedback stabilization problem for stochastic nonlinear systems with both uncertain system functions and stochastic non-white exogenous disturbance In this technical note, we consider a class of nonlinear systems disturbed by some bounded colored noise which satisfies an uncertain Itô-type stochastic differential equation Actually, such kind of exogenous disturbance exists in many practical dynamical systems A typical example is the stochastic nonlinear reactor systems, in which the external reactivity noise may be more realistic to be assumed to be non-white because of various noise mechanisms and time constants involved in the various feedback paths But the fundamental noise sources that bring about the reactivity noise through various feedback mechanisms may be regarded as white, ie, the external reactivity noise can be produced by passing the white noise through a filter, described by an Itô-type stochastic differential equation [6] This motivates us to consider the ADRC approach to the uncertain nonlinear systems with bounded uncertain non-white stochastic disturbance In contrast to most of the available results, the main contributions of this technical note lie in: an extended state observer ESO is designed, for the first time, not only to estimate the unmeasured states, but also the internal uncertain nonlinear part and the bounded external uncertain stochastic disturbance, and an ESO-based output-feedback control is designed to stabilize the system In addition, most of the available output-feedback controllers are used to guarantee the global asymptotic stability in probability [] for the case where the noise vector field is vanishing at the origin or the noise-to-state or inputto-state stability in probability [], [8] otherwise In this technical note, however, we address a kind of practical stability for the stochastic nonlinear systems with nonvanishing noise vector field Precisely, we consider the following uncertain stochastic nonlinear system: dx t =x tdt dx t =x 3 tdt dx n t =[f t, x t,,x n t ϕ t, x t,,x n t wtut] dt yt =x t where x t,,x n t R n, ut R, yt R are the state, the control input, and the output measurement of system, respectively IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information

2 64 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 6, NO 6, JUNE 06 f,ϕ :[0, R n R are possibly unknown measurable system functions, wt R is the uncertain external disturbance which is assumed to satisfy an Itô-type stochastic differential equation as follows: dwt =φ t, wt dt ψ t, wt db t,w0 = w 0 where {B t } t 0 is a one-dimensional standard Brownian motion defined on a complete probability space Ω, F, {F t } t 0,P with a natural filtration {F t } t 0 ie, F t = σ{b s, 0 s t}, and unknown F 0 -measurable initial value w 0 which is independent of the σ- algebra generated by B t,t 0 The functions φ, ψ :[0, R R are possibly unknown measurable functions We proceed as follows In the next section, Section II, we design a nonlinear extended state observer to estimate both state and total disturbance, the most original feature of the ADRC approach compared with other approaches aforementioned The mean-square convergence is proved under additional assumptions As a direct consequence, we conclude the convergence of linear extended state observer Section III is devoted to practical mean-square convergence for the closed-loop system The convergence of closed-loop system under the linear extended state observer based feedback is also concluded In Section IV, some numerical experiments for a physical motivated example are carried out to illustrate effectiveness of the proposed approach II EXTENDED STATE OBSERVER For a given vector X R n,weusex to denote its transpose, and X the Euclidean norm The Assumption H is a prior assumption about the unknown functions f,ϕ, φ, ψ, and control ut Assumption H: The possibly unknown functions f and ϕ are continuously differentiable with respect to their variables, and there exist known constants c i >0,i=0,,,n and C>0 such that ut ft, x,,x n ft, x,,x n t c 0 c i x i 3 ϕt, x,,x n ϕt, x,,x n t ft, x,,x n x i ϕt, x,,x n x i c n 4 φt, w ψt, w C w 5 Motivated by [4] for deterministic systems, we design a nonlinear extended state observer NLESO in shorthand for system as follows: dˆx t =ˆx tdt ε n g yt ˆx t ε dt n dˆx n t =ˆx n tdt g yt ˆx t n ε dt utdt n dˆx n t = g ε n dt yt ˆx t ε n Notice that the solution of 6 depends on parameter ε but we drop ε by abuse of notation without confusion The following Assumption H is a prior assumption about the solutions of and the external stochastic disturbance wt Assumption H: There exists a known constant B>0 such that: i The solution of x i t of satisfies sup t 0 E x i t B i =,,n ii wt B almost surely for all t 0 The following Assumption H3 is on the nonlinear functions g i on NLESO 6 6 Assumption H3: There exist constants λ i i =,, 3, 4 and twice continuously differentiable function V : R n R which is positive definite and radially unbounded such that λ y V y λ y,λ 3 y W y λ 4 y V y y i y i g i y V y y n α y, V y y β n y =y,y,,y n R n V y y n g n y W y for some nonnegative continuous function W : R n R and constants α, β > 0 Theorem II: Under Assumptions H H3, the nonlinear extended state observer 6 is convergent in the sense that for any initial values x 0 R n, ˆx 0 R n and any positive constant T>0 where [x i t ˆx i t] 0 uniformly in [T, as ε 0 n E x n t Δ =ft, x t,,x n tϕ t, x t,,x n t wt 8 is the total disturbance or the extra state variable Proof: Set Δt = d dt f t, x t,,x n t wt d dt ϕ t, x t,,x n t ϕ t, x t,,x n t φ t, wt 9 Σt =ϕ t, x t,,x n t ψ t, wt 0 System can be written as dx t =x tdt dx n t =x n tdt utdt dx n t =Δtdt ΣtdB t 7 Notice that for any ε>0, ˆBt =/ εb εt is also a standard Brownian motion Set x i t =x i t ˆx i t, η i t = x iεt,,,,n εn i A direct computation shows that ηt =η t,η t,,η n t satisfies dη t =[η t g η t] dt dη n t =[η n t g n η t] dt dη n t =[εδεt g n η t] dt εσεtd ˆB t 3 By Assumption H3, we apply Itô s formula to V ηt with respect to t along the solution ηt of system 3 to obtain dv ηt [ V η = η i η i g i η ] V η g n η dt η n V η εδεt dt V η εσεt d η n η ˆB t n εσ εt V η dt 4 ηn

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 6, NO 6, JUNE By Assumptions H and H, there exists M>0 such that E Δεt M, E Σεt M for all t 0 We also notice that there exists ε 0 > 0 such that γ := λ 3 /λ ε / 0 / > 0 Suppose that ε 0,ε 0 Wethenhave dev ηt dt EW ηt αε E Δεt V ηt βε λ E Σεt λ 3 EV ηt λ ε α 3 ε EV ηt E Δεt λ βε E Σεt γev ηt Mα ε 5 λ So, for every T>0 t EV η e ε γt ε EV η0 Mα λ ε t ε 0 e γ t ε s ds 0 6 uniformly in t [T, as ε 0 Thus for all i =,,,n E x i t =ε n i t E η i ε ε n i t E η ε εn i t EV η 0 7 ε uniformly in t [T, as ε 0 This completes the proof of the theorem Remark II: Compared with the deterministic case, we can see from the proof of Theorem II that since the total disturbance contains the stochastic term, the Itô differential of the total disturbance has diffusion term So the Itô differential of the Lyapunov function for the error equation contains not only the gradient, but also the higher order Hessian term This makes the estimation for total disturbance much more difficult than for the deterministic case The simplest ESO is the linear extended state observer LESO that is a special case of 6 dˆx t =ˆx tdt a ε yt ˆx t dt 8 dˆx n t =ˆx n tdt an yt ˆx ε n t dt utdt dˆx n t = a n yt ˆx ε n t dt where a i,,,,nare chosen so that the following matrix is Hurwitz: a 0 0 a 0 0 E = 9 a n 0 0 a n 0 0 λ n n Corollary II: Under Assumptions H H, the linear extended state observer 8 is convergent in the sense that for any initial values x 0 R n, ˆx 0 R n and any T>0 [x i t ˆx i t] 0 uniformly in [T, as ε 0 n E where x n t defined by 8 is the total disturbance or the extra state Proof: From Theorem II, it is suffice to show that there exist two Lyapunov functions satisfying Assumption H3 Actually, let P be the positive definite matrix solution of the Lyapunov equation PE E P = I for n -dimensional identity matrix I Define the Lyapunov functions V,W : R n R by V η =η Pη, W η =η η for η R n Then λ min P η V η λ max P η V η V η η i a i η a n η η i η n = η = W η V η η n V η η = η P P η =λ max P η V η ηn V η η P λ maxp η R n 0 where λ max P and λ min P are the maximal and minimal eigenvalues of P, respectively So V η and W η satisfy Assumption H3 The result then follows from Theorem II III ESO BASED OUTPUT FEEDBACK The next step is to design an ESO-based feedback control First, we notice that when there is no total disturbance, the following state feedback control: ut = k x t k x t k n x n t where k,k,,k n are constants to be chosen so that the matrix following: A = k k k n k n n n is Hurwitz can stabilize system Since we have obtained estimations for all states and total disturbance claimed by Theorem II, an ESObased output feedback control is naturally designed as ut = k ˆx t k ˆx t k nˆx n t ˆx n t where the last term ˆx n t is obviously used to compensate the effect of the total disturbance f ϕ w because of ˆx n f ϕ wthis is just the estimation/cancellation nature of ADRC Let x i t,η i t, i =,,,nbe variables defined in Under feedback, the closed-loop system of and 6 is equivalent to dx t =x tdt dx n t =[ k x t k n x n tk x t k n x n t x n t] dt dx n t =Δtdt ΣtdB t 3 dη t =[η t g η t] dt dη n t =[η n t g n η t] dt dη n t =[εδεt g n η t] dt εσεtd ˆB t where Δt and Σt are given by 9 and 0, respectively The main result on the practical mean-square stability of the closed-loop system of, 6, and is summarized in the succeeding theorem Theorem III: Under Assumptions H, ii of H and H3, for any initial values x 0 R n, ˆx 0 R n, the closed-loop system of, 6, and, which is equivalent to 3, admits a solution and is practically stable in the sense that lim E [ x i tˆx i t ] =0

4 66 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 6, NO 6, JUNE 06 Proof: Let Then 3 is equivalent to dθ t =εθ tdt θ i t =x i εt,,,,n dθ n t =[ εk θ t εk n θ n t] dt [ε n k η t ε k n η n tεη n t] dt dθ n t =εδεtdt εσεtd ˆB t dη t =[η t g η t] dt dη n t =[η n t g n η t] dt dη n t =[εδεt g n η t] dt εσεtd ˆB t 4 Since the matrix A defined by is Hurwitz, there exists a positive definite matrix solution Q to the Lyapunov equation QA A Q = I for n-dimensional identity matrix I Define the Lyapunov function K : R n R by Kθ =θ Qθ, whereθ =θ,,θ n It is easy to see that there exist m>0, ξ > 0,andξ > 0 such that Kθ θ n m θ, ξ θ Kθ ξ θ, θ =θ,,θ n 5 and from Assumptions H and ii of H that there exist N > 0,N > 0 such that E Δεt N E θt N 6 and there exists ε > 0 such that γ 0 := m k εn λ m kn ε λ m ε λ ε λ 3 λ > 0 nε α 0 := ε ε α N λ > 0 7 Suppose that ε<min{ε, γ 0 ξ /α 0 } Apply Itô s formula to Kθt and V ηt with respect to t along the solution of system 4, respectively to obtain, from 5 7 and 5 that dekθtev ηt dt εe θt E Kθ [ ε n k η tε k n η n tεη n t ] θ n ε λ 3 EV ηt ε 3 α E Δεt Mβ λ λ ε m εe θt k ε n m kn ε m ε λ λ λ EV ηt nε ε 3 E θt ε λ 3 EV ηt λ ε 3 α N E θt α N λ λ ε γ 0 EV ηt εα 0 EK θt α N ξ λ ε εα 0 [EV ηt EK θt] α N ξ λ ε Hence there exists t ε = b> such that for all t t ε b ε, EK θt EV ηt e εα 0 ξ α N λ t [EK θ0 EV η0] t ε 0 e εα 0 ξ t s ds EK θ0e εα 0 t ξ ε EV η0 α N ξ ε Mβξ λ α 0 α 0 8 We thus obtain that θt is mean-square bounded, ie, there exists B> 0 such that sup t tε E θt B So i of Assumption H holds By Theorem II, x i t 0 9 n E in t [t ε, as ε 0 Letx =x,,x n Apply Itô s formula to Kxt with respect to t along the solution of system 3 to obtain dek xt dt E xt E Kx x n [k x t k n x n t x n t] E xt n μ E xt m [ k μ E x t kne x n t E x n t ] n μ EK xt m k ξ μ E n x i t where Kx/ x n m x by virtue of 5 and 0 <μ< /n, k =max{k,,k n, } Hence E xt = E x i t e nμ t t ε ξ E xt ε ξ m k μξ t It then follows from 9 that: lim E x i t =0 lim E ˆx i t lim E t ε e nμ n ξ t s E x i sds 30 x i te x i t =0 This completes the proof of the theorem Like Corollary II, under the LESO 8 based feedback control, we have immediately the Corollary III Corollary III: Under Assumptions H and ii of H, for any initial values x 0 R n, ˆx 0 R n, the closed-loop system of, 8, and admits a solution and is practically stable in the sense that lim E [ x i tˆx i t ] =0 IV NUMERICAL SIMULATIONS In this section, we present some numerical simulations on stabilization for a linear harmonic oscillator subject to external bounded stochastic disturbance presented in [7], which is described by the following equation: ẍtα xt = βẋth xt, ẋt wt 3

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 6, NO 6, JUNE where α is the frequency of degenerated linear oscillator, β is the coefficient of linear damping, h is the amplitude of bounded noise, wt is the bounded external stochastic disturbance that contains the parameter noise and the influence of a random environment For this oscillator, the total energy is [7] Et = ẋ t α x t 3 As pointed out in [7], the bounded stochastic disturbance wt of the oscillator can be of the form wt :=cosλt ξb t ν 33 where λ and ξ are constants representing center frequency and strength of frequency perturbation, respectively, and ν is random phase uniformly distributed in [0, π] Here we consider the case of ν =0 for simplicity In this case, it is easy to compute by Itô s formula that wt is a bounded disturbance which satisfies an Itô-type stochastic differential equation where the corresponding functions φ,ψ in satisfy the condition 5 Set x t =xt, x t =ẋt 34 Then the control system of the motion of the oscillator 3 with the input control ut and the output measurement yt =xt can be rewritten as follows: { dx t =x tdt dx t =[ α x t βx thx t,x t wtut] dt yt =x t 35 which is of the form We design a NLESO 6 for system 35 as dˆx t=ˆx tdt 3 yt ˆx ε t dtεϕ yt ˆx t ε dt dˆx t=ˆx 3 tdt 3 yt ˆx ε t dtutdt 36 dˆx 3 t= yt ˆx ε 3 t dt where ϕ :R R is defined as {, s, ] π πs ϕs = sin, s, 37 π, s [, π Similar to [4], Assumption H3 in 7 is satisfied for this example Hence 36 serves as a well-defined NLESO for 35 from Theorem II For numerical computations, the Milstein approximation method [6] is used to discretize systems 35 and 36 Figs and display the numerical results for 35 and 36, where we take { h x t,x t = 6 cos x 5 tx t wt =cos t B t and the initial values are taken as,α=,β = 3,ε=00 38 x 0 =,x 0 =, ˆx 0 = ˆx 0 = ˆx 3 0 = 0 39 For the open-loop system, we suppose without loss of generality that the bounded input ut is also disturbed by a stochastic noise ut =sint B t 40 It is seen from Fig that the NLESO 36 is very effective in tracking the system 35 not only for the state x t,x t but also for the extended state total disturbance x 3 t defined by x 3 t = x t 3 x t 6 5 cos x tx t cos t B t Fig Open-loop for the state x t,x t, and total disturbance x 3 t, and their estimations ˆx t, ˆx t, andˆx 3 t under the control ut = sint B t by NLESO 36 For the closed-loop system, the NLESO based feedback control is designed as follows: ut = ˆx t 3ˆx t ˆx 3 t 4 The maximal real part of the eigenvalues of matrix 0 A = 3 4 is From Theorem III, we know that the closed-loop system is practically stable under the output feedback control 4 in the sense

6 68 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 6, NO 6, JUNE 06 V C ONCLUDING REMARKS In this technical note, the ADRC is extended to a class of nonlinear systems with large uncertainties that come from both internal unknown dynamics and external stochastic disturbance The key idea is to use an ESO to estimate, in real time, not only the state but also the uncertainties, and then compensate uncertainties in the feedback loop It is shown that the resulting closed-loop system is practically stable in the mean-square topology Compared to the deterministic case, the ADRC approach for stochastic nonlinear systems is far from fully understanding It is found for deterministic systems in [3] and [5] that the ADRC approach can be established based on generalized proportional integral observer GPI instead of ESO to deal with more complicated perturbed differentially flat nonlinear systems with measurable flat outputs It would be interesting to generalize the present result to stochastic counterpart of systems considered in [3] and [5] Fig Closed-loop for the state x t,x t, and total disturbance x 3 t, and their estimations ˆx t, ˆx t, andˆx 3 t under the control ut = ˆx t 3ˆx t ˆx 3 t by NLESO 36 that [ α lim EEt = lim Ex t ] Ex t =0 43 In addition, Fig shows the stability of each component of the trajectory x t,x t This is because 43 implies almost sure convergence of each x i t,, The peaking values are observed in both Figs and, which can be overcome by time-varying gain approach presented in recent paper [7] REFERENCES [] H Deng and M Krstić, Output-feedback stochastic nonlinear stabilization, IEEE Trans Autom Control, vol 44,no,pp ,Feb 999 [] H Deng and M Krstić, Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance, Syst Control Lett, vol 39, pp 73 8, 000 [3] Z Gao, On the centrality of disturbance rejection in automatic control, ISA Trans, vol 53, pp , 04 [4] B Z Guo and Z L Zhao, On the convergence of extended state observer for nonlinear systems with uncertainty, Syst Control Lett,vol60, pp , 0 [5] J Q Han, From PID to active disturbance rejection control, IEEE Trans Ind Electron, vol 56, no 3, pp , Mar 009 [6] D J Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev, vol 43, pp , 00 [7] Z Huang, W Zhu, Y Ni, and J Ko, Stochastic averaging of strongly nonlinear oscillators under bounded noise excitation, J Sound and Vibrat, vol 54, pp 45 67, 00 [8] H Ji and H Xi, Adaptive output-feedback tracking of stochastic nonlinear systems, IEEE Trans Autom Control, vol 5, no, pp , Feb 006 [9] J Li, W Chen, J Li, and Y Fang, Adaptive NN output-feedback stabilization for a class of stochastic nonlinear strict-feedback systems, ISA Trans, vol 48, pp , 009 [0] F Leonard, A Martini, and G Abba, Robust nonlinear controls of model scale helicopter sunder lateral and vertical windgusts, Trans Control Syst Technol, vol 0, pp 54 63, 0 [] S Liu, J Zhang, and Z Jiang, Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems, Automatica, vol 43, pp 38 5, 007 [] R Madoński, M Kordasz, and P Sauer, Application of a disturbance rejection controller for robotic-enhanced limbre habilitation trainings, ISA Trans, vol 53, pp , 04 [3] M Ramírez-Neria, H Sira-Ramírez, A Luviano-Juárez, and A Rodríguez-Ángeles, Active disturbance rejection control applied to a deltaparallel robot in trajectory tracking tasks, Asian J Control, vol 7, pp , 05 [4] H Sira-Ramírez, C López-Uribe, and M Velasco-Villa, Linear observer-based active disturbance rejection control of the omnidirectional mobile robot, Asian J Control, vol 5, pp 5 63, 0 [5] H Sira-Ramírez, J Linares-Flores, C García-Rodríguez, and M A Contreras-Ordaz, On the control of the permanent magnet synchronous motor: An active disturbance rejection control approach, IEEE Trans Control Syst Technol, vol, no 5, pp , Sep 04 [6] O Sako, Dynamical behavior of fluctuations in a non-linear reactor system perturbed by non-white reactivity noise, Proc Faculty Eng Tokai Univ, vol, pp 3 34, 985 [7] Z L Zhao and B Z Guo, On active disturbance rejection control for nonlinear systems using time-varying gain, Eur J Control, vol 3, pp 6 70, 05