High-Gain Observers With Sliding Mode for State and Unknown Input Estimations

3386 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009

High-Gain Observers With Sliding Mode for Stateand Unknown Input Estimations

Kalyana C. Veluvolu, Member, IEEE, and Yeng Chai Soh, Senior Member, IEEE

AbstractTo handle the state estimation of a nonlinear systemperturbed by a scalar disturbance distributed by a known nonlin-ear vector, a sliding-mode term is incorporated into the nonlinearhigh-gain observer (HGO) to realize a robust HGO. By imposing astructural assumption on the unknown input distribution vector,the observability of the disturbance with respect to the outputis safeguarded, and the disturbance can be estimated from thesliding surface. Under a Lipschitz condition for the nonlinearpart, the nonlinear observers are designed under the structuralassumption that the system is observable with respect to anyinput. In the sliding mode, the disturbance under an equivalentcontrol becomes an increment of Lipschitzian function, and theconvergence of the estimation error dynamics can be provensimilar to the analysis of HGOs. The proposed technique can beapplied for fault detection and isolation. The simulation resultsfor the bioreactor application demonstrate the effectiveness of theproposed method.

Index TermsHigh-gain observers (HGOs), nonlinear trans-formation, sliding-mode observers (SMOs), unknown inputestimations.

I. INTRODUCTION

THE STATE estimation of nonlinear systems has been anactive field of research in the last few decades. The worksin [1][3] presented some fundamental results on state estima-tion of systems via state transformation and nonlinear observer.Through a nonlinear change of coordinates, linearization isachieved by output and output derivative injections. High-gainobservers (HGOs) [1], [3] have been proposed for a generalclass of single output systems that is uniformly observable. Theapproach was generalized to a more general class of nonlinearsystems in [4] and [5]. In [5], a constant gain observer isproposed for a special class of nonlinear systems that doesnot require the nonlinear transformation. Furthermore, [6] im-proved the existing design of the HGO by incorporating thenonlinearity of the system into the gain design strategy. The de-sign of other exponential observers such as the GauthierKupkaObserver [2] and the Kalman-Like Observer [3] also follows asimilar framework. Khalil et al. [7][10] further explored theuse of an HGO for feedback control, numerical differentiation,and sampled-data control.

Manuscript received September 12, 2008; revised November 12, 2008. Firstpublished June 5, 2009; current version published August 12, 2009.

K. C. Veluvolu is with the School of Electrical Engineering and Com-puter Science, Kyungpook National University, Daegu 702701, Korea (e-mail:[email protected]).

Y. C. Soh is with the School of Electrical and Electronics Engineer-ing, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2009.2023636

HGOs with sliding-mode control have been used in thedesign of output feedback controllers due to their ability torobustly estimate the unmeasured states and to asymptoticallyattenuate the disturbances [7], [9]. The works in [7] alsoproved a nonlinear separation principle for the stabilizationof nonlinear systems employing the HGO. In [8], discrete-time implementation of HGOs was also explored. The workin [11] designed a discrete-time controller using HGO withthe separation principle. However, HGOs are designed for thenominal systems and the performance degrades in the presenceof uncertainty/disturbances. In this paper, a robust HGO isdeveloped by appending a sliding-mode term to the HGO.

Sliding-mode control is a well-established method for han-dling disturbances and modeling uncertainties through the con-cepts of sliding surface design and equivalent control [12].Based on the same concept, sliding-mode observers (SMOs)have been developed to robustly estimate the system states[12][21]. The Lyapunov-based approach of Walcott and Zak[13] considered the problems of state observation in the pres-ence of bounded uncertainties/unknown inputs based on amatching condition. The approach in [16] and [22] extendedthe design of [12] to linear systems such that the states affectedby the unknown inputs are dealt with by the switching terms.The reconstruction of unknown inputs/faults from equivalentcontrol was also discussed in the same work. The work in[18] extended the SMO design of [13] to Lipschitz nonlinearsystems based on matching conditions [12]. In [23] and [24],estimation in the presence of unknown inputs for mechani-cal systems has been studied using high-order sliding-modetechniques. In addition, there has been tremendous interest inthe application of these techniques in real time for industrialapplications in both continuous and discrete time [25][32].

The equivalent control-based SMO was first proposed byUtkin [12], and its extensions to nonlinear systems were ad-dressed in [14], [15], and [17]. These methods chose the outputestimation error and its higher order derivatives as the slidingsurfaces, which are, in general, not measurable. These meth-ods require structural assumptions for the design of switchingterms for all the states, which are, in general, conservative.In practical applications, as only the output estimation erroris available, and so the higher order derivatives have to beobtained through low-pass filtering [12]. Furthermore, most ofthe works to date are mainly limited to linear systems withconstant disturbance distribution matrix and rely on either thestructural conditions or matching assumptions such that theerror system is stable and free from unknown inputs. Recently,SMOs [19], [20] are designed for a class of nonlinear uncertaincontinuous and discrete-time systems. The gain design depends

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VELUVOLU AND SOH: HGOs WITH SLIDING MODE FOR STATE AND UNKNOWN INPUT ESTIMATIONS 3387

on the Lipschitz constants, and the method may not be feasiblefor systems with large Lipschitz constants. In this paper, theproblem is solved by employing a HGO. In addition, theproposed method only requires the output measurement erroras the sliding surface, and it is directly measurable.

Proposed SMO is integrated into the nonlinear HGO designso that in the sliding mode, the disturbance under an equivalentcontrol becomes an increment of Lipschitzian function, and theconvergence of the error dynamics of the state estimation isproven. The proposed method does not require the matchingcondition on the unknown input distribution vector. A properdesign of the estimation feedback gain ensures asymptoticconvergence of the estimation residual. The observer can bedesigned directly for the original system.

The remainder of this paper is organized as follows.Section II presents the system description and backgroundresults. Section III presents the design of robust nonlinearobserver that incorporates an SMO. In Section IV, the con-vergence analysis and the design of sliding-mode gain arediscussed. Section V presents the estimation of the unknowninput from the sliding mode. In Section VI, the robust nonlinearobserver is applied for estimation of states and uncertainties inbioreactor. Section VII concludes this paper.

Throughout this paper, max(A) denotes the maximumeigenvalue of a matrix A, A denotes the two-norm

max(ATA) of a matrix A, and (A) denotes the conditionnumber (

max(A)/

min(A)) of the matrix.

II. SYSTEM DESCRIPTION AND BACKGROUND RESULTS

A. System Description

In this paper, the following class of uniformly observablesystems are considered for design of the robust SMO:{

x = Ax +(x) + (x)u + p(x)d(x, t)y = Cx

(1)

where x = [x1, x2, . . . , xn]T M, a C connected manifoldof dimension n

A =[0 I(n1)(n1)

01n

]C = [ 1 0 0 ]

are constant matrices; the nonlinear functions (x), (x), andp(x) are smooth vector fields on M; the control input u isupper bounded with umax; the unknown input or disturbanced(x, t) is upper bounded with d. y is the measurable output.

In the following analysis, local coordinates are generallyused. When global properties are considered, notions are sim-plified by assuming thatM accepts a global coordinate system.

Assumption 1: The system (1) satisfies

(x) = [1(x1) 2(x1, x2) n(x1, x2, . . . , xn)]T

(2)(x) = [1(x1) 2(x1, x2) n(x1, x2, . . . , xn)]T

(3)

p(x) = [1 p2(x1, x2) p3(x1, x2, x3)

pn(x1, x2, . . . , xn)]T .(4)

Assumption 2: The functions i(x1, x2, . . . , xi), i(x1, x2,. . . , xi), pi(x1, x2, . . . , xi); i = 1, . . . , n, are Lipschitz func-tions w.r.t. x.

Assumption 3: The distribution vector p(x) with functionspi(x1, x2, . . . , xi); i = 1, . . . , n are bounded with respect totheir arguments on M.

Remark 1: The structure in (4) can be obtained by normaliz-ing p(x) with p1(x), its first component. If p1(x) = 0, then theunknown input is not detectable from the output measurement,and the estimation will be inaccurate due to the effect of largedisturbance on the system states. In general, the matchingcondition [12] is applicable for a constant distribution matrix.For systems with state-dependent nonlinear entries p(x), thedesign of a constant matrix to satisfy the matching condition isnot feasible.

Assumptions 1 and 2 can be conservative, but they charac-terize a system that is uniformly observable for any boundedinput. It has been proven in [1] that Assumption 1 is a sufficientcondition, but not necessary, to ensure uniform observability forany input. The triangular structure in Assumption 1 is necessaryfor the equivalent control-based SMOs [15], [17] to facilitatesuccessive evaluation of higher order derivative terms from themeasurable estimation error. Assumption 3 is necessary for thedevelopment of robust SMO to deal with the unknown input.

The class of systems considered is applicable to electro-mechanical systems such as single-link robotic arm [6], [33],[34] and linear motors. Some examples of uncertain chemi-cal and biochemical processes [35] can also be expressed ortransformed into the form of (1). The uniform observabilityassumptions and Lipschitz assumptions are satisfied due to theboundedness of states in chemical and biochemical processes.

Remark 2: Consider a class of uncertain nonlinear systemsdescribed by{

x(t) = f(x) + b(x)u(t) + p(x)d(x, t)y(t) = h(x)

(5)

where x Rn; f(x), b(x), p(x), and h(x) are nonlinearfunctions; u, d(x, t) are the bounded control input and theunknown input; y is the measurable output. Define a nonlineartransformation

x x = [x1 x2 xn]T

=(x) =[h(x) Lfh(x) L(n1)f h(x)

]T

where the Lie derivative is defined as Lfh(x)= [h(x)/x]f ,

and assuming that the mapping (x) is a diffeomorphism(x). The nonlinear functions in (5) can then be trans-formed into the form of (1): x = [0, 0, . . . , 0, Lnf h(x))]T,x = [1(x),2(x), . . . ,n(x)]T, p(x)

= [p1(x), p2(x),



. . . , pn(x)]T with i(x) = LbL(i1)f h(x), and pi(x) = Lp

L(i1)f h(x) for all i = 1, . . . , n.

B. Background Results

Consider the nonlinear system (1) but without the distur-bance, i.e., d(x, t) = 0{

x = Ax +(x) + (x)uy = Cx.

(6)

For the system in the form of (6) satisfying Assumptions 1and 2, the results in [1][3] proved the estimation convergenceof the estimator

x = Ax +(x) + (x)u + L(y Cx) (7)where x is the estimate of x and L is a properly chosenestimation gain.

For a gain design that is based on HGO [1], [3], L is

L = S1 CT (8)

where S is the unique solution of the Lyapunov equation

S + ATS + SACTC = 0 (9)and is a positive parameter which can be chosen to overcomesystem constants and bounds. The explicit solution of (9) canbe obtained as

S(i, j) =(1)i+jCj1i+j2

i+j1, 1 < i, j < n

where Crn =n!

(n r!)r! . (10)

Furthermore, S is symmetric positive definite for every > 0 (see [11]).

In the following section, the observer design for systemswith uncertainties/disturbances that are not necessarily random/structural is explored.

III. ROBUST NONLINEAR STATE ESTIMATOR

For system (1) satisfying Assumptions 13, the robust non-linear estimator with a sliding-mode term is of the form

x = Ax +(x) + (x)u + L(y Cx) + p(x)ur(t). (11)In the aforementioned equation

L = [l1 l2 ln]T = S1 CT (12)is a properly chosen constant feedback estimation gain basedon HGO, and the design of the scalar-valued robust term ur isbased on sliding-mode theory, and is given as

ur(t) = sign(e1) = sign(y Cx). (13)The sliding-mode estimation gain will be discussed in

Lemma 2.

It is clear that the proposed robust observer copies the form ofthe nonlinear system with appended feedback correction termand switching term (robust term) to deal with the unknowninput/disturbances. The switching term serves as a trackingelement for the unknown input which can be reconstructedfrom the sliding mode. The convergence analysis and the designof feedback gain and sliding-mode gain will be discussed later.

IV. CONVERGENCE ANALYSIS

In the following, the boundedness of the estimation error isfirst proved in Lemma 1. With the established boundedness oferror, the design of sliding-mode gain and the error conver-gence to the sliding surface are then addressed in Lemma 2.With the sliding mode established, the asymptotic stability ofthe estimation error in the sliding mode is then proved inTheorem 1. These are organized as three separate sections.

A. Boundedness of Error DynamicsFor the error dynamics e = [e1, e2, . . . , en]T

= x x, it canbe obtained from (1) and (11) that

e =(A S1 CTC

)e +(x)(x)

+(x)u (x)u + p(x)ur p(x)d(x, t). (14)

For ease of analysis, define as a diagonal matrix anddeduce the following equalities:

=diag(

1,1, . . . ,

1n1

), S =

1S1

A1 = A, C = C1 = C (15)

where S1 is the solution of (9) for = 1.Lemma 1: Consider system (1) satisfying Assumptions 13.

For the estimator (11), there exists 0 > 0 such that > 0, xand e remain bounded.

Proof: Set = e, then =

(A S1 CTC

)1

+ [(x)(x) + [(x) (x)]u]+ [p(x)ur p(x)d(x, t)] . (16)

Using the equalities in (15), it can be obtained that

= (A S11 CTC

)

+ [(x)(x) + [(x) (x)]u]+ [p(x)ur p(x)d(x, t)] . (17)

Using the aforementioned results and with the Lyapunovfunction V = TS1, differentiating w.r.t. time, it can evalu-ated as

V =2TS1

=2TS1(A S11 CTC

)

+ 2TS1 [(x)(x) + [(x) (x)]u]+ 2TS1 [p(x)ur p(x)d(x, t)]



Furthermore, with (9), it can be deduced that 2TS1A =TS1 + TCTC. Hence,

V = TS1 C2

+ 2TS1 [(x)(x) + [(x) (x)]u]+ 2TS1 [p(x)ur p(x)d(x, t)]

V + 2TS1 [(x)(x) + [(x) (x)]u]+ 2TS1 [p(x)ur p(x)ur]+ 2TS1 [p(x)ur p(x)d(x, t)] . (18)

Together with the Lipschitz assumption in Assumption 2 andtriangular structure in Assumption 1, and assuming 1, it canbe evaluated that

[(x)(x) + ((x) (x))u]

n

i=1

1i1 [i(x) i(x) + (i(x) i(x))u]

n

i=1

(li + liumax) eii1

n(l + lumax)e n(l + lumax) (19)

where ei = (e1, . . . , ei), and l = supi |li |, l = supi |li |are the largest Lipschitz constants of and .

Under Assumptions 2 and 3, p(x) is a Lipschitz functionand bounded for some upper bound bp. Similar to (19), dueto the triangular structure of p(x) in Assumption 1, it can beobtained as

(p(x) p(x)) nlp (20)p(x)ur p(x)d(xt) bp(d + ) (21)

where lp is the largest Lipschitz constant of p, |d(x, t)| d and|ur| = .

Using the aforementioned inequalities, the derivative ofLyapunov function can be evaluated as

V V + 2n(l + lumax)S1+ 2nlpS1 + 2S1bp(d + )

[ 2n(l + lumax)(S1) 2nlp(S1)]V+ 2max(S1)bp(d + ) (22)

= c12 + c2 (23)

where c1= 2n(l + lumax)(S1) 2nlp(S1), c2 =

2max(P)bp(d + ). Here, () and max() denote the con-dition number

max()/

min() and maximum eigenvalue

of the matrix. With selection of > 2n(l + lumax)(S1) +2nlp(S1) = 0, it can be shown that c1 > 0. Hence, e isbounded such that e c2/c1. In addition, the bound of ecan be made small by selecting a large .

B. Sliding-Mode Gain Design

For modeling uncertainties from the scalar disturbanced(x, t) that is not necessarily Lipschitzian, the normal estimatorcan only achieve a bounded e instead of e 0.

In the proposed design, SMO with the term p(x)ur is em-ployed in (11) to improve the estimation accuracy. The rationaleof this solution is twofold.

1) With the sliding surfacee1 = 0 (24)

the aim is to design the sliding-mode estimation as (13)to reach and maintain in the sliding mode.

2) To ensure that the term [ur d(x, t)] of the estimationerror dynamics (14) in the sliding mode e1 = 0, i.e.,a zero dynamics, can be substituted by an incrementof Lipschitzian function through an equivalent controlsignal, so that the asymptotic convergence of e can beproved.

The following Lemma 2 and Theorem 1 are devoted to theaforementioned points 1) and 2), respectively.

Lemma 2: For the system (1) satisfying Assumptions 13and the estimator (11), the sliding-mode estimation (13) ensuresthat the sliding surface e1 = 0 can be reached and maintainedprovided there exists 1 > 0 such that > 1 and the sliding-mode gain satisfies

> e2max + d (25)where e2 e2max and l1 , l1 are the Lipschitz constants of1() and 1().

Proof: The first dynamics e1 from (14) can be obtained asfollows:

e1 = e2 l1e1 + 1(x1) 1(x1)+ [1(x1) 1(x1)]u + ur d(x, t). (26)

For the Lyapunov function V1 = 1/2e21, using the above andthe sliding-mode estimation (13), it can be evaluated asV1 = e1e1

= l1e21 + e1 [1(x1) 1(x1)]+ e1 [1(x1) 1(x1)]u + [e2(t) d(x, t)] e1 |e1|.

Under the condition in Lemma 1 that ensures boundedness ofe2, there exists a finite constant gain satisfying (25) such that,if e1 = 0, one hasV1 < l1e21 + e1 [1(x1) 1(x1)] + e1 [1(x1) 1(x1)]u.

Under the Lipschitzian condition in Assumption 2 and theboundedness of the input, it can be obtained as

V1 < l1e21 + l1e21 + l1umaxe21.From the design of high gain, l1 of L from (8) can be

evaluated as l1 = n, where n is the order of the system. Byselecting such that

>l1 + l1umax

n= 1



ensures

l1 > l1 + l1umax.

Hence, it can be shown that

V1 < 0 if e1 = 0.

Hence, the robust term (13) using the gain (25) ensures thatthe sliding surface e1 = 0 can be reached in a finite time andmaintained thereafter.

Remark 3: The boundedness of e is established in Lemma 1and is dependent on . As is an independent parameter, onecan choose arbitrarily to reduce the bound of e. This bound canbe used for the calculation of sliding-mode gain .

C. Error Dynamics in the Sliding Mode

Since the estimator design (11) using the robust term (13)ensures the sliding mode, it is only required to examine theconvergence of the dynamics of e during the sliding mode.In the sliding mode when e1 = 0 and e1 = 0, x1 = x1, theequivalent control of ur can be obtained from (26) as in [12]

ueq = d(x, t) e2,d (27)

where the subscript d denotes the estimated x-related variablesin the sliding mode, i.e., ed = [e1,d, e2,d, . . . , en,d]T

= xd x.Substituting the aforementioned equivalent control (27) into(14), the estimation error dynamics in the sliding mode ofe1 = 0 can be obtained as

ed =(A S1 CTC

)ed + (xd, u) (x, u)

+ (p(xd) p(x))ueq + p(x) [ueq d(x, t)]

=(A S1 CTC

)ed +(xd)(x) + (xd)u

(x)u + (p(xd) p(x))ueq p(x)e2,d. (28)

The equivalent control in the sliding mode clearly cancelsthe disturbance effect in the state estimation. The followingtheorem proves the asymptotic stability of the estimation error.

Theorem 1: Assume that system (1) satisfies Assumptions13. For the estimator (11) with the robust term (13) and thesliding-mode gain (25), there exists 2 > 0 such that > 2,the estimation error is asymptotically stable in the sliding modeof e1 = 0.

Proof: The proof follows the similar pattern as inLemma 1. By selecting d = 0ed and Lyapunov functionV2 = Td S1d, it can be evaluated with (28) similar to (18)

V2 V2 + 2Td S1 [(xd)(x) + [(xd) (x)]u]

+ 2Td S1 [p(xd) p(xd)]ueq 2Td S1p(x)e2,d. (29)

Since e2 and d(x, t) are bounded, then according to (27),ueq ueq for some upper bound ueq. With (21), p(x) bp

and similar to (22), it can be deduced to the form (29)V2 V2 + 2n(l + lumax)S1dd

+ 2nueqlpS1dd+ 2bpS1ded

V2 +[2n(l + lumax)(S1) + 2nlp(S1)ueq

+ 2bp(S1)]V2.

By selecting > 2n(l + lumax)(S1) + 2nlp(S1)ueq+ 2bp(S1) = 2, it can be shown that V2 < 0. Hence, the ro-bust estimator guarantees asymptotic stability of the estimationerror.

Obviously, the equilibrium point of the error dynamics (28) ised = 0, i.e., xd = x. When the error e1 is in the sliding mode,the robust term can be viewed as tracking element for theunknown input or disturbance input. The robust term in thesliding mode can be viewed as an estimate of the disturbanced(x, t). The reconstruction of the disturbance from the slidingmode is discussed in the next section.

Remark 4: To avoid the switching caused by the sign()function, the sign function sign() is approximated by a satu-ration function sat(, ) =

{ /, if | | sign(), if | |> , where >0

is the threshold. By replacing the sign() function with sat(),the error dynamics finally settle to a bound inside the boundarylayer rather than converging to zero in the sliding plane. De-creasing the boundary layer thickness in sat() increases theaccuracy in estimation.

Remark 5: If the nonlinear system (1) is obtained via thetransformation of (5), then the robust nonlinear estimatorfor (5) can be obtained by means of inverse transformationto give

x = f(x) + b(x)u +[(x)x

]1x=x

L(y h(x)

)

+ p(x)sign(y h(x)

). (30)

V. UNKNOWN INPUT ESTIMATION FROM SLIDING MODE

Once the trajectory reaches the sliding mode, all the statesconverge to the true states, i.e., xd x. Therefore,

e2,d 0.The equivalent control ueq information can then be used

to reconstruct the unknown input. From (27), the equivalentcontrol can be approximated as

ueq d(x, t). (31)The use of a low-pass filter for recovering the equivalent

control signal was given by [12]. Continuous approximation ofequivalent injection signal by using a small positive scalar was also implemented in [16]. Similar to the analysis of [16],



in the proposed approach, the unknown input can be estimatedfrom equivalent control as follows:

d(x, t) (sign(e1))eq e1

(|e1|+ ) . (32)

The accuracy of the disturbance estimation will dependon . The estimation only depends on measurement error e1and, hence, can be performed online with state estimation.

VI. APPLICATION TO BIOREACTOR

Consider the following bioreactor, which has been discussedin many works [1], [19], [36]:{

X = mXSkcX+S XDS = 1Y mXSkcX+S + (Sf S)D

(33)

where X and S represents the cell mass concentration, andsubstrate concentrations, respectively. X is measured online.Sf and Y are known positive constants. D > 0 is the dilutionrate, and also the plants control input. In the growth function,the uncertain parameters m > 0 and kc > 0 are modeled asm = 0m + d1(t), kc = k

0c + d2(t), where 0m and k0c are the

known nominal parameters, and d1(t) and d2(t) model the un-certainties. The bioreactor system satisfies all the assumptions.For a detailed analysis of the bioreactor system and the statetransformation, please see [19].

A. Robust Estimator

In the following, the gain design is carried using the high-gain approach. From (10), S, S1 and feedback gain L can becomputed as

S =[

1 12 12 23

]S1 =

[2 2

2 3

]

L =S1 CT =

[22

].

A proper sliding-mode estimation gain is also required.Since Lph(x) = 1, the proposed robust estimator is designed as

X =

0mXS

k0cX + S XD + 2(y X) + sat(y X, )

(34)S = 1

Y

0mXS

k0cX + S+ (Sf S)D

+

[(0mX + S

)22

0mk0cX

2 2S

2

k0cX2

](y X)

Y sat(y X, ). (35)

The unknown input or disturbance can be estimated accord-ing to (32) as

d(x, t) (X X)(X X)+ . (36)

Fig. 1. X(t) and X(t) with the nonlinear estimator in the absence of therobust term.

Fig. 2. S(t) and S(t) with the nonlinear estimator in the absence of therobust term.

B. Simulation Results

The following parameters were chosen for the simulation:0m = 1 1/h, k0c = 1, sf = 0.1 g/L, Y = 1, D = 0.5 1/h. Theadditive parametric uncertainties of m and kc are sin(1.5t)and 0.6 cos(t), respectively. The initial conditions for plantand estimator are set as x(0) = [0.02 0.02]T and x(0) =[0.01 0]T, respectively.

By choosing = 4, the estimation gain can be computedas L = [8 16]T. The gain l1 of L also should satisfy thecondition in Lemma 2. In addition, = 2 and = 0.001. First,the nonlinear estimator is applied without the robust termssat(y X, ) and (/Y ) sat(y X, ) in accordance with(30), and the estimation performance is shown in Figs. 1 and 2.Although the estimator tracks the actual state, the accuracyis not good. Then, the proposed robust nonlinear estimatordescribed by (34) and (35) is employed, and the improvedestimation performance is clearly shown in Figs. 3 and 4. Toestimate the unknown input according to (36), is chosen to be0.001. The unknown input reconstructed from the sliding modeis shown in Fig. 5. The effect of chattering in the absence of



Fig. 3. X(t) and X(t) with the nonlinear estimator with the sliding-mode term.

Fig. 4. S(t) and S(t) with the nonlinear estimator with the sliding-mode term.

Fig. 5. d(x, t) and d(x, t) with sat() function.

sat() function is shown in Fig. 6. For clarity, a small portionof Fig. 6 is shown in Fig. 7. The estimated unknown inputconverges to the actual disturbance after all the states haveconverged.

Fig. 6. d(x, t) and d(x, t) without sat() function.

Fig. 7. d(x, t) and d(x, t) without sat() function.

VII. CONCLUSION

A robust HGO design is developed for a class of single-inputsingle-output nonlinear systems by incorporating asliding-mode term into the nonlinear observer to improve esti-mation accuracy. The sliding surface uses only the measurableoutput estimation error, and the unknown input can be estimatedfrom the measurable sliding surface. The gain design is basedon HGO that guarantees exponential convergence of the esti-mation error. The proposed estimator is applied to the state andunknown input estimation of a bioreactor, and the simulationresults demonstrate the effectiveness of the proposed approach.

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Kalyana C. Veluvolu (M06) received the B.Tech.degree in electrical and electronic engineering fromAcharya Nagarjuna University, Guntur, India, in2002, and the Ph.D. degree in electrical engineeringfrom Nanyang Technological University, Singapore,in 2006.

From 2006 to 2009, he was a Research Fellowwith the Biorobotics Group, Robotics Research Cen-ter, Nanyang Technological University. Since 2009,he has been an Assistant Professor with the Schoolof Electrical Engineering and Computer Science,

Kyungpook National University, Daegu, Korea. His current research interestsare in the areas of nonlinear estimation and filtering, sliding-mode control,braincomputer interface, biomedical signal processing, and surgical robotics.

Yeng Chai Soh (M87SM06) received the B.Eng.degree in electrical and electronic engineering fromthe University of Canterbury, Christchurch, NewZealand, in 1983, and the Ph.D. degree in electri-cal engineering from the University of Newcastle,Callaghan, Australia, in 1987.

Since 1987, he has been with Nanyang Techno-logical University, Singapore, where he is currentlya Professor with the School of Electrical and Elec-tronic Engineering. From 1995 to 2005, he wasconcurrently the Head of the Control and Instrumen-

tation Division. He is currently the Associate Dean (Research) of the Collegeof Engineering. He serves on several project evaluation and review committeesof national research and funding agencies. His current research interests arein the areas of robust system theory and applications, estimation and filtering,information and signal processing, hybrid and switched systems, and opticalfield control and signal processing, and he has published more than 200 refereedjournal articles in these areas.


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High-Gain Observers With Sliding Mode for State and Unknown Input Estimations

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