Peaking free variable structure control of uncertain linear systems based on a high-gain observer

Automatica 45 (2009) 1156–1164

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Peaking free variable structure control of uncertain linear systems based on a high-gain observerI José Paulo V.S. Cunha a,∗ , Ramon R. Costa b , Fernando Lizarralde b , Liu Hsu b a

State University of Rio de Janeiro, 20550-900, Rio de Janeiro, Brazil

b

COPPE/Federal University of Rio de Janeiro, 21945-970, Rio de Janeiro, Brazil

article

info

Article history: Received 15 December 2006 Received in revised form 13 November 2008 Accepted 20 December 2008 Available online 28 February 2009 Keywords: Variable-structure control Sliding mode High-gain observer Model reference control Peaking phenomena Output feedback

a b s t r a c t An output-feedback model-reference variable structure controller based on a high-gain observer (HGO) is proposed and analyzed. For single-input–single-output (SISO) linear plants with relative degree greater than one, the control law is generated using the HGO signals only to drive the sign function of the variable structure control component while the sign function gain, also called modulation, as well as the other components of the control signal are generated using signals from state variable filters which do not require high gain and are free of peaking. This scheme achieves global exponential stability with respect to a small residual set and does not generate peaking in the control signal. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction This paper is concerned with the design of output feedback control laws for uncertain linear systems using variable structure control (VSC) and a high-gain observer (HGO). Here, the generation of large fast transients in the control signal, known as peaking phenomena, is avoided while preserving global stability. VSC is an efficient tool to design controllers for plants under significant uncertainty conditions. Owing to the practical difficulty of measuring all states, as required in early works, output-feedback strategies for VSC were proposed (e.g. Emelyanov, Korovin, Nersisian, and Nisenzon (1992), Esfandiari and Khalil (1992) and Walcott and Żak (1988)). Recently, higher order sliding modes for plants of arbitrary relative degree have been also considered by Levant (1998) using robust exact differentiators. Theoretically, controllers based on such differentiators may lead to exact output

I The material in this paper was partially presented at 16th IFAC World Congress, Prague, July 4-8, 2005. This work was partially supported by CNPq and FAPERJ, Brazil. This paper was recommended for publication in revised form by Associate Editor Murat Arcak, under the direction of Editor Hassan K. Khalil. ∗ Corresponding address: Department of Electronics and Telecommunication Engineering, State University of Rio de Janeiro, 20550-900, Rio de Janeiro, Brazil. Tel.: +55 21 2587 7633; fax: +55 21 2577 1373. E-mail addresses: [email protected], [email protected] (J.P.V.S. Cunha), [email protected] (R.R. Costa), [email protected] (F. Lizarralde), [email protected] (L. Hsu).

0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2008.12.018

tracking. However, stability and/or convergence of the overall control system was guaranteed only locally. While it is well known from the seminal paper (Bondarev, Bondarev, Kostyleva, & Utkin, 1985) that output feedback sliding mode control is possible with the use of asymptotic observers, a good knowledge of the plant model is needed. For uncertain plants, a solution for state estimation is the HGO, which is robust to model uncertainties (Emelyanov et al., 1992; Esfandiari & Khalil, 1992). A drawback of the HGO is the peaking phenomenon, which may be destabilizing and even provoke finite-time escape in closed-loop nonlinear systems (Atassi & Khalil, 2000; Sussmann & Kokotović, 1991). The relevance of peaking elimination is explained as follows. If the system satisfies a global Lipschitz condition (e.g., if the system is linear), global asymptotic stability can be obtained with HGOs (Busawon, El Assoudi, & Hammouri, 1993; Gauthier, Hammouri, & Othman, 1992), at the expense of unacceptable transient responses (Atassi & Khalil, 2000). In linear plants with actuator constraints, peaking can lead to saturation of the control signal and, consequently, to performance degradation, as pointed out by Méndez-Acosta, Femat, and Campos-Delgado (2004). Moreover, peaking may cause undesirable mechanical wear and energy loss. For example, in flow control systems operating at high flow rates, large peaks in the control signal can abruptly close a valve, causing water hammer which can severely damage pipes, valves and other mechanical parts. These facts are well known in industrial process control when derivative action is used.

J.P.V.S. Cunha et al. / Automatica 45 (2009) 1156–1164

Consequently, peaking avoidance has motivated the development of several control design remedies (Li, Ang, & Chong, 2006). Several previous works give some alternatives for peaking alleviation, for instance: (1) The amplitude of the control signal can be globally bounded through saturation (Oh & Khalil, 1995, 1997). This may restrict stability to become local or semi-global and precludes global stabilization of unstable linear systems; (2) The HGO free of peaking proposed by Chitour (2002) is based on a timevarying observer gain. However, this algorithm may fail in actual systems since disturbances may excite peaking when the observer gain is large. (3) The semi-high gain observer (Lu & Spurgeon, 1998), which is a HGO with a non-conservative value for the observer gain, is computed such that closed-loop stability is guaranteed. However, this procedure, which was developed for stabilization purposes, seems inadequate for tracking applications where the observer gain must be large enough to keep the residual output error small. This paper presents an output-feedback model-reference variable structure controller that uses simultaneously a HGO and state variable filters to implement the control law. The main contributions are: (i) To introduce a HGO-based controller for uncertain linear plants which is free of peaking and yet guarantees global exponential stability with respect to a small residual set. To the best of our knowledge, such a result is original. The importance of the absence of peaking is illustrated by experimental results in Section 6.2. (ii) To offer a significantly more simple controller than the earlier variable structure model-reference robust controller (VS-MRRC)1 (Hsu, Araújo, & Costa, 1994; Hsu, Lizarralde, & Araújo, 1997). The VS-MRRC also solves the output-feedback tracking control problem, however the number of required state variables significantly exceeds the HGO by 2(n + 1)(n∗ − 1) − 1, where n is the order and n∗ is the relative degree of the plant. For example, if n = 5 and n∗ = 3 the HGO-based controller requires 14 state variables, while the VS-MRRC requires 37 state variables. This is due to the more involved structure of the VS-MRRC which requires several filters for the computation of the gains of n∗ modulated relays, instead of a single modulated relay in the new HGO-based controller. (iii) The HGO structure allows a more natural extension to nonlinear plants with nonlinearities depending on unmeasured states (e.g. Oliveira, Peixoto, & Hsu, 2008). Such an extension is not trivial for the VS-MRRC. Notations: The L∞e norm of the signal x(t ) ∈ R is defined as kxt ,t0 k∞ := supt0 ≤τ ≤t kx(τ )k. A mixed time-domain and Laplace transform domain is adopted, i. e., s denotes either the Laplace variable or the differential operator, according to the context. The output signal y of a linear time-invariant system with transfer function matrix H (s) and input u is denoted by H (s)u. The timedomain convolution is denoted by h(t ) ∗ u(t ). n

1157

phase; (A2) G(s) is strictly proper; (A3) the order of the system (n) is known; (A4) the relative degree of G(s), n∗ , is known; (A5) the sign of kp is known and kp > 0 for simplicity. Two additional assumptions are needed in the design of the modulation function of the control law: (A6) the disturbance d(t ) is piecewise continuous and a bound d¯ (t ) is known such that |d(t )| ≤ d¯ (t ) ≤ d¯ sup < +∞, ∀t ≥ 0; (A7) the parameters of G(s) are uncertain but the coefficients of Dp (s) and Np (s) belong to known bounded sets and, a bound kp is known such that 0 < kp ≤ kp . To simplify the analysis and design of the controller, the reference model is defined by yM = WM (s)r ,

WM (s) =

kM L(s) (s + γ )

Let a linear, time-invariant, observable and controllable plant be described by the input–output model y = G(s)[u + d(t )],

G(s) = kp

Np (s) Dp (s)

,

(1)

where u ∈ R is the input, y ∈ R is the output, d ∈ R is an unmeasured input disturbance, kp ∈ R is the high frequency gain, Np (s) and Dp (s) are monic polynomials. The following assumptions regarding the plant are usual in model-reference adaptive control (Ioannou & Sun, 1996): (A1) G(s) is minimum

1 Originally, it was named variable structure model-reference adaptive controller (VS-MRAC) due to its close relation with MRAC.

(2)

where yM (t ) is the output signal, r (t ) is a piecewise continuous and uniformly bounded reference signal, kM > 0 is the high frequency gain of the model, L(s) is a monic Hurwitz polynomial of degree N := n∗ − 1 and γ > 0. The objective is to design an output-feedback controller to achieve asymptotic convergence of the output error e(t ) := y(t ) − yM (t )

(3)

to zero, or to some small residual neighborhood of zero. If the plant and the disturbance d(t ) are perfectly known, a control law which achieves matching between the closed-loop transfer function and WM (s) is given by (Cunha, Hsu, Costa, & Lizarralde, 2003) u∗ = θ ∗T ω − wd (t ) ∗ d(t ),

(4)

where wd (t ) is the impulse response of a system with transfer function Wd (s) = 1 − θ1∗T A(s)/Λ(s), where A(s) = [sn−2 , sn−3 , . . . , s, 1]T , and Λ(s) is an arbitrary monic Hurwitz polynomial of degree n − 1. The signal wd (t ) ∗ d(t ) cancels the input parameter vector is given by θ ∗T =  ∗T disturbance  d(t ). The ∗T ∗ ∗ ∗ θ1 , θ2 , θ3 , θ4 , with θ1 , θ2∗ ∈ R(n−1) , θ3∗ , θ4∗ ∈ R and the regressor vector is ω = ω1T , ω2T , y, r



ω1 =

A(s) u, Λ(s)

ω2 =

T

with state variable filters:

A(s) y. Λ(s)

(5)

1 The matching parameters θ ∗ can be computed from θ4∗ = k− p kM and a Diophantine equation (Ioannou & Sun, 1996, eq. (6.3.13)). Consider the system state X := [xTp , ω1T , ω2T ]T , where xp ∈ Rn is the plant state, and a non-minimal realization {Ac , Bc , Co } of WM (s) with state vector XM and Ac Hurwitz. Then, the state error Xe := X − XM and the output error e(t ) satisfy

X˙ e = Ac Xe + Bc k u − θ ∗T ω + wd (t ) ∗ d(t ) ,

(6)

e = C o Xe ,

(7)



2. Problem statement and preliminaries

,



where k := (θ4 ) = kM kp (Hsu et al., 1994). For the HGO design and overall stability analysis, a reduced order error model is advantageous. To this end, consider a Kalman decomposition (Kailath, 1980, pp. 132–134) for the system (6) and (7) with partial observable states xoc (controllable) and xoc¯ (uncontrollable) satisfying: −1

∗ −1

x˙ oc = A11 xoc + A12 xoc¯ + B1 k u − u∗ ,



x˙ oc¯ = A22 xoc¯ , e = C1 xoc + C2 xoc¯ ,



(8) (9) (10)

where {A11 , B1 , C1 } is a minimal realization of WM (s). The characteristic polynomial of A11 is DM (s) = L(s)(s + γ ) = ∗ ∗ −1 sn + an∗ −1 sn −1 + · · · + a1 s + a0 . Noting that C1 Ai11 B1 = 0

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J.P.V.S. Cunha et al. / Automatica 45 (2009) 1156–1164 n ∗ −1

(i = 1, . . . , n∗ − 1) and C1 A11 (i)

B1 = kM , then by successive (i)

−1 differentiation xoc¯ = Ai22 xoc¯ and C1 xoc = C1 Ai11 xoc + C1 Ai11 B1 k[u − i ∗ u ] + C1 Ei−1 xoc¯ , where Ei = A11 A12 + Ei−1 A22 (i = 1, . . . , n∗ ) with E0 = A12 . Then, one can write:

DM (s) e = C1 DM (A11 ) xoc + kp u − u∗ + E xoc¯ ,





(11)

Pn∗

where E = C2 DM (A22 ) + C1 i=1 ai Ei−1 . Noting that DM (A11 ) = 0 from the Cayley–Hamilton Theorem, Eq. (11) can be rewritten as e = WM (s)k u − u + kp E xoc¯ . ∗



−1

such that {AM , BM , S } is a realization of the transfer function WM (s)L(s), which has relative degree one. After some algebraic manipulations using the controllability canonical form in (Kailath, 1980, pp. 39–45), S is given by



(12)

0 0 . . S = .   T

0 1

Consider the observer canonical form realization

 −a n ∗ − 1 1 · · · ..  .. . . AM =   −a 1 0 −a 0 0 ···   CM = 1 0 · · · 0 ,

0

0





..  . , 

 ..  .  BM =   ,

1 0



(13)

of the model WM (s). Then a realization of the error Eq. (12) is given by x˙ e = AM xe + BM k u − θ ∗T ω + wd (t ) ∗ d(t ) + πe ,

(14)

e = CM xe ,

(15)



where xe is a new error state and πe = Pπ xoc¯ = Tπ Xe for some constant matrices Pπ and Tπ . Since A22 is Hurwitz, πe (t ) satisfies |πe (t )| ≤ ke e−λe t kXe (0)k with some appropriate constants ke , λe > 0. 3. Variable structure control For plants of relative degree n∗ = 1, the VS-MRRC in (Hsu et al., 1994, Sec. IV.A) is based on output-feedback and requires no state observer. In this case, N = n∗ − 1 = 0, L(s) = 1 and the control law is given by u = −ρ sgn(e).

(16)

If the modulation function ρ ∈ R satisfies the inequality ρ ≥ δ + θ ∗T ω − wd (t ) ∗ d(t ) , ∀t ≥ 0, where δ ≥ 0 is an arbitrary constant, then the output error e(t ) converges exponentially to zero, as can be concluded by applying Lemma 1 of (Hsu et al., 1997). Moreover, if δ > 0, then the output error e(t ) becomes zero after some finite time. If n∗ > 1, the control law (16) cannot guarantee stability and convergence properties. In the present paper a control strategy based on a HGO is proposed. A natural idea would be to estimate the plant state, as in the robust stabilization controller of Oh and Khalil (1995). Here, however, the error state is estimated instead. In this way, the reference model (stable and perfectly known) is used instead of the plant model (uncertain and possibly unstable) for the estimator. Chang and Lee (1996) recognize that the design of an observer for the reference model is easier than the design of an observer for the plant. A similar approach was proposed by Oh and Khalil (1997) for trajectory tracking in nonlinear systems. Oh and Khalil (1995, 1997) apply VSC laws where the control signal is globally bounded through saturation in order to avoid the peaking phenomena. However, as a result, only semi-global stability properties can be guaranteed. Here, a HGO is employed only to generate the switching law. The modulation function in the control law is synthesized using signals from the state filters. This will lead us to obtain global stability without peaking phenomena in the plant and control signals. ∗ It is possible to design a matrix S ∈ R1×n , which defines the ideal sliding surface σ (xe ) = Sxe = 0 with xe from (14), +

. .. ..

. an∗ −1

0 0

0 1

1

1 . ..

an∗ −1

an ∗ − 2

···

−1 

an∗ −1

   an∗ −2  ..   . a1

1



lN −1   .   . ,  .   l 

(17)

1

l0

where li are the coefficients of the polynomial L(s) = sN + lN −1 sN −1 + · · · + l1 s + l0 . At this point the proposed control law is introduced

0 kM



··· ···

u = unom + U , u

nom

=θ

nomT

U = −ρ sgn(S xˆ e ),

(18)

ω,

(19)

where xˆ e is some available estimate for xe and θ nom is the nominal value of θ ∗ . Particularly, if xˆ e (t ) ≡ xe (t ) and the modulation function satisfies the inequality

ρ ≥ δ + (θ nom − θ ∗ )T ω + wd (t ) ∗ d(t ) ,

∀t ≥ 0,

(20)

where δ ≥ 0 is an arbitrary constant, then the sliding surface σ (ˆxe ) = 0 will be reached at least exponentially and, kxe k and the output error e(t ) converge exponentially to zero, as can be concluded by applying Lemma 1 of (Hsu et al., 1997). Finite time escape in the system signals can be prevented if the modulation function is designed to satisfy the inequality

ρ ≤ kω kωk + krd ,

∀t ≥ 0 ,

(21)

for some constants kω , krd > 0. Indeed, this implies that the state error in (6) can grow at most exponentially (see Appendix). A modulation function which satisfies inequalities (20) and (21) is given by

ρ = δ + θ¯ T |ω| + dˆ (t ),

(22)

with dˆ (t ) = d¯ (t ) + c1 e−γd t ∗ d¯ (t )

(≥ |wd (t ) ∗ d(t )|) ,

(23)

where θ¯ T =

  θ¯1 , . . . , θ¯2n , |ω| = [|ω1 |, . . . , |ω2n |]T and θ¯j ≥ ∗ θ − θ nom (j = 1, . . . , 2n). The nominal parameter vector θ nom j j is usually computed such that the closed-loop transfer function is matched to the reference model WM (s) when the plant parameters are nominal, u = unom and d(t ) ≡ 0. The nominal control unom is of practical importance, since it allows the reduction of the modulation function ρ and control signal amplitudes if the parameter uncertainty kθ ∗ − θ nom k is small. Since assumption (A6) is valid, the upper bound (23) can be obtained through the application of (Cunha, Costa, & Hsu, 2008, Theorem 1), which requires that c1 ≥ 0 be an appropriate constant and γd satisfies γ0 > γd > 0, with γ0 := minj {−Re(pj )} being the stability degree (Boyd & Barratt, 1991, p. 138), where {pj } are the poles of the transfer function matrix A(s)Λ−1 (s). Some methods for the computation of the coefficients c1 and γd of the filter (23) were proposed by Cunha et al. (2008).

J.P.V.S. Cunha et al. / Automatica 45 (2009) 1156–1164 Table 1 Proposed control algorithm for linear systems with n∗ > 1.

equation results x˙˜ e = Ae (ε −1 )˜xe + BM knom U¯ − k πe (t ) ,



Reference model

yM = WM (s)r.

Output error State filters

e = y − yM . ω1 = ΛA((ss)) u, ω2 =

Regressor vector Control law Nominal control Modulation function High-gain observer

T 2

T



(31)

where

A(s) y. Λ(s)

ω = [ω ω y r ] . u=u + U , U = −ρ sgn (σ ), σ = S xˆ e . unom = (θ nom )T ω. ρ = δ + θ¯ T |ω| + dˆ (t ), dˆ (t ) = d¯ (t ) + c1 e−γd t ∗ d¯ (t ).   knom x˙ˆ e = AM xˆ e + BM kp U − α(ε −1 ) − aM (CM xˆ e − e). M T 1 nom

1159

Ae (ε −1 ) := AM − α(ε −1 ) − aM CM ,



U¯ :=



1 − (k

×

h

(32)

) k U + (k ) k i
T θ − θ nom ω − wd (t ) ∗ d(t ) . nom −1



nom −1



∗

(33)

Then, applying the scaling transformation (Emelyanov et al., 1992) 4. High-gain observer Since the state xe is not measured, the control law will use the state (xˆ e ) estimated by the HGO x˙ˆ e = AM xˆ e + BM knom U − α(ε −1 ) − aM e˜ ,

(24)

e˜ = CM xˆ e − e,

(25)





1 nom where e˜ is the observer output error, knom := k− , knom is p M kp

the nominal value of kp , aM = [an∗ −1 , . . . , a1 , a0 ]T and, U is given in (18). The coefficients αi in the observer feedback vector (Lu & Spurgeon, 1998)

hα

n ∗ −1

α(ε −1 ) =

α1

···

ε

ε

α0 iT ε n∗

n∗ −1

(26)

must be chosen such that the characteristic polynomial of the ∗ closed-loop observer is Hurwitz, which holds if Nα (s) = sn + n∗ −1 αn∗ −1 s + · · · + α0 is Hurwitz and ε > 0. Since it is desired that the uncertainties and disturbances have negligible effects in the estimated state xˆ e , the norm of the observer feedback vector (kα(ε −1 )k) shall be large, which requires ε to be sufficiently small. The proposed controller is summarized in Table 1. 4.1. Upper bound for the estimation error Lemma 1 below gives upper bounds for the state estimation error x˜ e (t ) := xˆ e (t ) − xe (t ) of the HGO. These bounds are required in the stability proof of the closed-loop control system. In order to simplify Lemma 1, a scalar λα which satisfies the inequality ¯ α is introduced, where λ¯ α := minj {−Re(zj )} 0 < λe < λα < λ is the stability degree of the polynomial Nα (s), where {zj } are the roots of Nα (zj ) = 0. Lemma 1. Consider the observer (24) and (25) and the error Eqs. (14) and (15). If the control signal satisfies the inequality (21), the signals r (t ) and d(t ) are uniformly bounded, the polynomial Nα (s) is Hurwitz and the parameter ε ∈ (0, 1], then ∃k1 , . . . , k6 > 0 such that the state estimation error (x˜ e ) and the observer output error (e˜ ) satisfy the inequalities

k˜xe (t )k ≤

k˜xe (t )k ≤

k1 xe n∗ −1 λα e− ε t

ε +

k˜ (0)ke−

k˜xe (0)ke n ∗ −1

|˜e(t )| ≤ k1 k˜xe (0)k +ε

+ k2 εkXe (0)ke−λe t

∗ [k3 kω(t )k + k4 ] ,

k1

ε

λα ε t

λα e− ε t

λ n∗ −1 − εα t e

− λεα t

+ k2 εkXe (0)ke

(27) −λe t

+ ε C (t , 0), (28)

∗

+ k2 ε n kXe (0)ke−λe t

∗ [k3 kω(t )k + k4 ] ,

(29)

C (t , t0 ) = k5 kωt ,t0 k∞ + k6 .

(30)

Proof. The observer output error Eq. (25) is rewritten as e˜ = CM x˜ e . Subtracting (14) from (24), the state estimation error dynamic

(34)

the error Eq. (31) is rewritten as

  ε x˙¯ e = A¯ e x¯ e + ε B¯ M (ε) knom U¯ − kπe (t ) ,

(35)

where the companion matrix A¯ e has the characteristic polyno∗ mial Nα (s), B¯ M (ε) = [0, . . . , 0, ε n −1 ]T kM and, consequently, ∗ kB¯ M (ε)k = ε n −1 kM . Therefore, the norm of the state x¯ e is bounded by

k¯xe (t )k ≤ k1 k¯xe (0)ke− + k7 ε n

λα ε t

∗ − 1 − λα t ε

e

∗

+ k2 ε n kXe (0)ke−λe t ∗ kU¯ (t )k,

∀t ≥ 0.

(36)

Since the modulation function satisfies the inequality (21) and the signals r (t ) and d(t ) are uniformly bounded, it can be concluded from (33) that ∃kU¯ ω , kU¯ ≥ 0 such that kU¯ t ,0 k∞ ≤ kU¯ ω kωt ,0 k∞ + kU¯ , ∀t ≥ 0. On the other hand, from the definition of the ∗ transformation (34), one has kT (ε)k = 1 and kT −1 (ε)k = ε 1−n , since ε ∈ (0, 1]. Considering these facts, the upper bounds (27) and (28) for k˜xe (t )k are obtained from (36). The upper bound (29) is obtained from the inequalities |˜e| = |CM T −1 (ε)¯xe | = |CM x¯ e | ≤ k¯xe k and (36).  4.2. Peaking phenomena Peaking phenomena are well known to exist in high-gain observers, as can be seen in (Khalil, 2002, Sec. 14.5.1). The parameter ε should be chosen sufficiently small in order to reduce the residual estimation error and to speed up the response of the observer. However, the term p(t , ε −1 ) :=

k1

ε

n ∗ −1

k˜xe (0)ke−

λα ε t

,

(37)

present in the upper bounds for the state estimation error (27) and (28), indicates the possible occurrence of large peak amplitudes of ∗ order 1/ε n −1 in the estimation error during the initial transient. The peak extinction time of an HGO is introduced below in view of its application in the proof of the stability of the closed-loop control system (Appendix). This concept is based on the dynamics of the state estimation error of an HGO with U¯ (t ) ≡ 0 and πe (t ) ≡ 0 given by x˙˜ e (t ) = Ae (ε −1 )˜xe (t ),

t ≥ 0,

where the Hurwitz matrix Ae (ε

−1

(38)

) is defined in (32).

Definition 1. The peak extinction time (te ) of the HGO is the smallest time value such that inequality

k˜xe (t )k ≤ k˜xe (0)k,

∀t ≥ 0, where

o

n

∗ T (ε) = diag 1, ε, . . . , ε n −1 ,

x¯ e = T (ε)˜xe ,

∀t ≥ te ≥ 0,

∀˜xe (0),

(39)

holds for a fixed value of the parameter ε ∈ (0, 1]. The precise computation of the peak extinction time of a HGO may be difficult. However, a convenient upper bound t¯e can be obtained from inequalities (27) and (39) (U¯ (t ) ≡ 0 and πe (t ) ≡ 0),

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J.P.V.S. Cunha et al. / Automatica 45 (2009) 1156–1164

which gives2 k1

ε n∗ −1

λα e− ε t

≤ 1,

∀t ≥ t¯e ≥ 0,

(40)

where t¯e ≥ te , which leads to 1 ∗ −1 t¯e = (n∗ − 1)λ− α ε (n − 1) ln(k1 ) − ln(ε) ,





(41)

which is similar to Eq. (31) of (Atassi & Khalil, 1999). It can be concluded that t¯e is uniformly bounded with respect to the parameter ε ∈ (0, 1] and tends to zero as ε → +0, since k1 ≥ 1, λα > 0 and n∗ ≥ 2 are constants. 5. Closed-loop stability analysis The controller defined by the variable structure control law (18) and the HGO (24) and (25) is analyzed. To avoid peaking in the control signal and in the plant signals, the state filter signals (5) are used to generate the modulation function. However, the peaking is still present in the estimated state, since this phenomenon is intrinsic in the HGO (24) and (25). For closed-loop stability analysis



T

purpose, the state vector is defined as z := XeT x˜ Te . Note that, similarly to ω, y can be expressed in terms of the state z and the reference model signals (y = [Co 0]z + yM from (3) and (7)) and since πe = Tπ Xe (see below (15)), then πe can also be expressed in terms of the state. Theorem 2 below states the stability properties of the error system. Theorem 2. For n∗ > 1, consider the plant (1), the control law (18) and (19), the state filters (5) and the observer (24) and (25). If the assumptions (A1)–(A7) are satisfied and the modulation function ρ satisfies inequalities (20) and (21) then, for ε > 0 sufficiently small, the system composed by the error Eqs. (6), (7) and (31) with state z will be globally exponentially stable with respect to a residual set of radius KX (ε)e−λz t + O(ε) which converges exponentially to O(ε), i.e., there exist constants kz , λz > 0 and a class K function KX (ε) such that

kz (t )k ≤

kz

e kz (0)k + KX (ε)e ε n ∗ −1 |e(t )| and kXe (t )k ≤ kz e−λz t kz (0)k −λz t

−λz t

+ O(ε),

+ KX (ε)e−λz t + O(ε),

(42)

(43)

∀z (0), ∀t ≥ 0. Moreover, if the amplitude of the modulation function ρ is large enough, then the sliding mode surface σ (ˆxe ) = 0 is reached in finite time and thereafter the sliding mode takes place. Proof. See Appendix.



The trajectory-tracking controller proposed by Oh and Khalil (1997) is also based on VSC and an HGO. This controller prevents the peaking phenomena in the plant through the global saturation of the control signal. However, their results guarantee only semiglobal stability and the finite time convergence of√ the state of the tracking error equation to a residual set of order ε . In contrast, the results in Theorem 2 are stronger, i.e., global stability and exponential convergence of the error states to a residual set of order ε hold. However, it should be stressed that our results hold for linear systems, in contrast to Oh and Khalil (1997) who considered nonlinear systems. The controller presented in Table 1 avoids peaking in the control signal u and in the plant signals, as can be concluded from the upper bound for the error signals (43) given in Theorem 2. However, peaking is still present in the state estimated by the HGO (24) and (25), as can be seen in inequality (42) which bounds the state estimation error (x˜ e ). If needed, this problem can be circumvented

through magnitude scaling of the error state of (14) and (15), i.e., ζ = T (ε)xe . The peaking-free HGO is obtained applying the transformation ζˆ = T (ε)ˆxe in the HGO (24) and (25), as described by Cunha, Hsu, Costa, and Lizarralde (2005). It can be shown that the resulting HGO is free of peaking since the scaled state estimation error given by x¯ e (t ) = T (ε)˜xe (t ) is bounded by (36). 6. Application example The proposed control scheme is applied to the position control of a car driven by a permanent magnet DC motor, via a rack-andpinion mechanism. The system is a Linear Position Servo module IP01 supplied by Quanser Consulting. This simple servomechanism allows a clear description of the controller design and the evaluation of closed-loop performance in actual experimental conditions with disturbances (e.g., dry friction), measurement noise, unmodeled dynamics and large uncertainties. The control algorithms are implemented in a digital computer using the Euler integration method with 0.2 ms step size. The car position (y) is measured by a potentiometer connected to a 13 bit A/D converter through a signal conditioner, which reduces aliasing and keeps measurement noise amplitude <0.4 mm peakto-peak. The control signal (u) is a voltage generated by a 12 bit D/A converter connected to a linear power operational amplifier which drives the DC motor. The servomechanism transfer function is given by G(s) ≈ kp /[s(s − p1 )], where the uncertain parameters are in the ranges: p1 ∈ [−15.0, −9.5] and kp ∈ [1.59, 2.99]. Friction is represented by an input disturbance uniformly bounded by d¯ sup = 0.3 V, which agrees with open-loop experiments. 6.1. Controller design The chosen reference model is WM (s) = 50/[(s + 5)L(s)] (L(s) = (s + 10)), which has unity DC gain. The state filters have Λ(s) = s + 10. The chosen HGO feedback vector coefficients are α1 = 30 and α0 = 225. The nominal control parameter vector is computed such that the nominal plant transfer function Gnom (s) = 2.41/[s(s + 13.4)] becomes matched to the reference model transfer function, which results in the solution of the Diophantine equation (Ioannou & Sun, 1996, eq. (6.3.13)) given by θ nom = [−1.6, −22.6, −18.5, 20.7]T . The modulation function (22) should be designed such that the upper bound (20) be satisfied for any values of the uncertain parameters. Moreover, it is desired to keep the control signal amplitude at moderate levels, which can be obtained if the constants in the modulation function are small as possible. Therefore, δ = 0 and the modulation function coefficients vector θ¯ T = [3.9, 39.9, 14.7, 10.8] has been computed such that the inequalities

θ¯j ≥

sup kp ∈[1.59,2.99] p1 ∈[−15.0,−9.5]

∗ θ (kp , p1 ) − θ nom , j j

are satisfied ∀j ∈ {1, . . . , 4}, where θ ∗ (kp , p1 ) is the solution of the Diophantine equation (Ioannou & Sun, 1996, eq. (6.3.13)) which depends on the plant uncertain parameters. Since friction is uniformly bounded, the upper bound dˆ (t ) ≡ 0.47 allows compensation of this disturbance. This is an upper bound for the steady state response of the filter Wd (s) to a step with amplitude d¯ sup , which satisfies

 2 The peak extinction time is analogous to the return time defined in (Boyd, El Ghaoui, Feron, & Balakrishnan, 1994, Sec. 5.2.3), where the problem of finding upper bounds for the return time is formulated.

(44)

dˆ ≥ 

sup kp ∈[1.59,2.99] p1 ∈[−15.0,−9.5]

 A(0) ∗ T 1 − θ (kp , p1 )  d¯ sup . 1 Λ(0)

(45)

J.P.V.S. Cunha et al. / Automatica 45 (2009) 1156–1164

Fig. 1. Control signal (u) obtained with the modulation function generated by: (a) the high-gain observer (48); (b) the state filters (22). The reference is r (t ) ≡ 0 mm and ε = 0.1.

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Fig. 2. Measured car position (y). Results with modulation function generated by: (a) the high-gain observer (48), ε = 0.1; (b) the high-gain observer, ε = 0.03; (c) the state filters (22), ε = 0.1; (d) the state filters, ε = 0.03. The reference signal is r (t ) ≡ 0 mm.

To allow the evaluation of the effects of peaking in the control of linear plants, the control law proposed by Oh and Khalil (1997) has also been implemented. In this case, the control law can be expressed by

 u = Su sat

unom + U



Su

,

U = −ρ sgn(S xˆ e ),

(46)

unom = Kfnom xˆ p + knom r, c

(47)

ρ = δ + K¯ f |ˆxp | + k¯ c |r | + d¯ (t ),

(48)

where xˆ p = xˆ e + xM is an estimate of the plant state, xˆ e is estimated by the HGO (24) and (25) and, xM is the reference model state. The parameters Kfnom = [−10.8, −0.664] and knom = c 20.7 of the nominal state feedback unom are computed such that the transfer function of the closed-loop system matches WM (s) in nominal conditions. The modulation function parameters are computed following (Devaud & Caron, 1975) to keep the control signal amplitude at moderate levels after the peaking: δ = 0.1, K¯ f = [31.3, 2.8], k¯ c = 10.8 and, d¯ (t ) ≡ d¯ sup = 0.3 V. The saturation (sat(·)) limits the control signal peak amplitude to Su = 10 V. Lower values of Su can impair the tracking when large voltages in the motor are demanded. 6.2. Experimental results In the experimental results presented in Figs. 1 and 2, the initial estimated state is xˆ e (0) = [0.1, 0]T , all the other initial conditions are zero and, the reference signal is r (t ) ≡ 0 mm. Since the modulation function (22) and nominal control (19) are based on the state filters (5), while Eqs. (47) and (48) apply the state estimated by the HGO (24) and (25), their control signals are different as can be seen in Fig. 1. In Fig. 1(a), the peaking in the control signal without saturation has a large amplitude (≈19.3 V) since the peaking in the state estimated by the HGO propagates into the modulation function and nominal control. The amplitude of the actual control signal is limited by a saturation in (46). Moreover, it has been verified through simulations that the amplitude of the modulation function (48) can be undesirably increased due to the HGO’s sensitivity to high frequency noise. On the other hand, the amplitude of the modulation function (22) is free of peaking as can be seen in Fig. 1(b) and, is noiseimmune by virtue of the low-pass characteristics of the state filters. The effect of peaking in the output of the system can be seen

Fig. 3. Measured car position (y) and the output signal of the reference model (yM ) for ε = 0.03.

in Fig. 2. The car position deviates more than 6 mm from the reference (0 mm) when the modulation function is generated by the HGO, while this deviation is less than 0.12 mm for the proposed modulation function. The transient performance is improved when the parameter ε is reduced from 0.1 to 0.03, since the peaking extinction time is smaller for ε = 0.03. However, the controllers are more noise sensitive for higher gains in the observer. The discrete-time implementation of the controllers and unmodeled dynamics induce the chattering signal seen in the control signals (Fig. 1). Being a high enough frequency phenomenon, it could be tolerated since it did not significantly annoy the controller performance. To evaluate the tracking performance of the proposed controller in the following experiments, the reference signal is a square wave with amplitude 50 mm and frequency 0.5 Hz. The actual car mass is 0.665 kg, which is 46% larger than the nominal mass (0.455 kg). The initial car position is y(0) = −40.0 mm, while all the other initial conditions are zero. Fig. 3 shows the remarkable tracking performance of the proposed control scheme. The amplitude of the tracking error for ε = 0.03 is about 10 times smaller than is obtained for ε = 0.3, as can be seen in Fig. 4 after t = 0.5 s. This can be predicted from Theorem 2, since the residual tracking error is of the order of O(ε). Fig. 4 also shows the large tracking error that results when only the nominal control is applied (i.e., u(t ) = (θ nom )T ω(t )), which leads to the conclusion that the performance of this nominal linear controller is impaired by disturbances and parameter uncertainties.

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From these inequalities, it is possible to show that

  kxe (t )k and kXe (t )k ≤ ke1 eλe1 t kXe (0)k + ke2 eλe1 t − 1 , (A.2) holds ∀t ≥ 0. There exists some class K function Kte (ε) which is larger than the peaking extinction time, i.e., te ≤ t¯e ≤ Kte (ε), as can be concluded from (41). Then, an upper bound for the righthand side of (A.2) valid ∀t ∈ [0, te ] is given by

kxe (t )k and kXe (t )k ≤ (ke3 + ke4 ε)kXe (0)k + Ke6 (ε),

(A.3)

which combined with the upper bound (28) for the state estimation error leads to the inequality (∀t ∈ [0, te ])

k˜xe (t )k ≤

Fig. 4. Output error (e): proposed peaking-free controller (ε = 0.3 and ε = 0.03) and nominal controller (u = unom ).

It has been verified experimentally that, after the initial transient, the performances of both controllers are similar since the modulation functions (22) and (48) become of similar magnitudes and the switching laws are the same. 7. Conclusion An output-feedback sliding-mode controller, for uncertain linear SISO systems, based on a high-gain observer (HGO) was proposed. In contrast to controllers of this kind found in the literature, an important property of the new scheme is that the control signal is free of peaking, without the need for globally bounding it with saturation. The relevance of avoiding the peaking effect for linear systems resides mainly on the deterioration of transient performance it may cause. Here, control peaking avoidance was achieved by using the HGO estimated state only to compute the switching function, while the modulation of the control signal was generated by state filters without peaking. The proposed scheme is shown to be theoretically globally exponentially stable with respect to a small residual set. Hence, both peaking avoidance and global stabilization can be achieved. To the best of our knowledge, this result is original. Experiments carried out with a positioning servomechanism have shown the robustness of the proposed scheme with respect to nonlinear input disturbances, unmodeled dynamics, measurement noise and significantly large parameter uncertainties. Experimental results show that peaking can severely affect the transient behavior, which was also verified by simulations not presented here due to the lack of space. The proposed controller resulted in better transient performance showing that it can circumvent the peaking problem.

k1

ε

n ∗ −1

k˜xe (0)ke−

λα ε t

+ ε [ke9 kXe (0)k + ke10 ] .

Now, an upper bound for the state z valid for t > t1 := te is developed. The instant t1 is taken as a new initial time after the extinction of the peaking in the HGO. Thereafter,

  k˜xe (t )k ≤ k˜xe (0)k + ε (ke13 kXe (0)k + ke12 ) e−λe (t −t1 ) + ε C (t , t1 ),

∀t ≥ t 1 .

(A.5)

The error equation (14) is rewritten as x˙ e = AM xe + BM k [U + dU + πe (t )] ,

(A.6)

where dU := (θ nom − θ ∗ )T ω + wd (t ) ∗ d(t ). Since the estimated state (xˆ e (t ) = xe (t ) + x˜ e (t )) is applied, the control law (18) can be rewritten as U = −ρ sgn σ (ˆxe ) ,

(A.7)

σ (ˆxe ) = Sxe + S x˜ e .

(A.8)




Remembering that {AM , BM , S } is a controllable nonminimal kM 1 realization of WM (s)L(s) = s+γ and k = k− M kp then, from the dynamic Eq. (A.6) and the algebraic Eq. (A.8), the signal σ (ˆxe ) can be represented as

σ (ˆxe ) =

kp

[U + dU (t )] + π1 (t ) + β(t ),

s+γ

(A.9)

where π1 (t ) + β(t ) = kp e−γ t ∗ πe (t ) + S x˜ e (t ). Through the application of the upper bound (A.5) for k˜xe k, one has

  |π1 (t )| ≤ ke14 k˜xe (0)k + kXe (0)k + ε ke12 e−λe (t −t1 ) , kβt ,t1 k∞ ≤ ε kβ C (t , t1 ),

∀t ≥ t 1 .

(A.10)

Since kp is assumed positive and the modulation function satisfies ρ(t ) ≥ |dU (t )| (∀t ≥ 0), the Lemma 2 of (Hsu et al., 1997) can be applied to the system composed of (A.9) and the control law (A.7), which results in the upper bound

|σ (ˆxe )| and |σˆ (ˆxe )| ≤ ke15 [kz (0)k + ε ke12 ] e−λe (t −t1 ) + 2ε kβ C (t , t1 ), ∀t ≥ t1 , where σˆ := σ − π1 (t ) − β(t ). Reminding that u = u

Appendix. Proof of Theorem 2

(A.4)

nom

(A.11)

+ U, one

k

In what follows, all the k’s and λ’s, not yet defined in the main text, denote appropriate positive constants, K ’s are class K functions and the operator norm (k·k) is induced by the norm L∞e . Since the regressor vector can be given by ω = Ω1 X + Ω2 r, where Ω1 and Ω2 are appropriate matrices (Hsu et al., 1994, Sec. II.B) and r (t ) is uniformly bounded, then kωk ≤ kΩ kXe k + kM1 , ∀t ≥ 0. Moreover, since the control law is given by (18) and the modulation function satisfies (21), the control signal is bounded by |u| ≤ kω1 kωk + krd ≤ kΩ 1 kXe k + kM2 , ∀t ≥ 0. From the error Eq. (6) and the upper bound kwd ≥ |wd (t ) ∗ d(t )|, ∀t ≥ 0, the following inequalities can be obtained

  kX˙ e k ≤ kAc kkXe k + kBc kk |u| + kθ ∗T kkωk + kwd ≤ kΩ 2 kXe k + kM3 ,

∀t ≥ 0.

(A.1)

p can note that s+γ in (A.9) operates in the same signal U + dU in (6). 1 ˙ From (A.9) it can be concluded that U + dU = k− ˆ + γ σˆ ]. Then, p [σ the tracking error can be rewritten from (6) as

h

i

X˙ e = Ac Xe + Bc k σ˙ˆ + γ σˆ .

(A.12)

To eliminate the derivative term σ˙ˆ , a variable transformation X¯ e := Xe − Bc kσˆ is performed yielding X˙¯ e = Ac X¯ e + (Ac + γ I )Bc kσˆ .

(A.13)

Since Ac is Hurwitz and the signal σˆ satisfies the upper bound (A.11), it can be verified that, ∀t ≥ t1 ,

¯ (t , t1 ). kX¯ e (t )k ≤ ke16 [kz (0)k + ε ke12 ] e−λe (t −t1 ) + εkC

J.P.V.S. Cunha et al. / Automatica 45 (2009) 1156–1164

Moreover, as described in what follows, ∀t ≥ t1 , and ke(t )k ≤ ke17 [kz (0)k + ε ke12 ] e−λe (t −t1 )

kXe (t )k + ε ke18 C (t , t1 ), kωt ,t1 k∞ ≤ ε ke19 C (t , t1 ) + ke20 kz (0)k + km , C (t , t1 ) ≤

k0red + ke21 kz (0)k 1 − ε ke22

(A.14) (A.15)

.

(A.16)

The inequalities in (A.14) are developed from Xe = X¯ e + Bc kσˆ . Inequality (A.15) is obtained from (A.14), since kωk ≤ kM + kΩ kXe k. Now, from (A.15) and (30), it can be concluded that C (t , t1 ) ≤ ε ke22 C (t , t1 ) + ke21 kz (0)k + k0red , from which the upper 1 bound (A.16) (valid for ε < k− e22 ) can be obtained. The upper bound for the complete state, ∀t ≥ t1 ,

kz (t )k ≤ [ke23 kz (0)k + ε ke24 ] e−λe (t −t1 ) + ε ke25 C (t , t1 ), is obtained through the combination of the upper bounds (A.5) for k˜xe k and (A.14) for kXe k. Then, the application of the upper bound (A.16) results in

kz (t )k ≤ [ke23 kz (0)k + ε ke24 ] e−λe (t −t1 ) +ε

ke26 + ke27 kz (0)k 1 − ε ke22

,

(A.17)

∀t ≥ t1 , which can be rewritten as kz (t )k ≤ [ke23 kz (0)k + ε ke24 ] e−λe (t −t1 ) +ε [ke28 + ke29 kz (0)k] ,

(A.18)

∀t ≥ t1 , valid for 0 < ε ≤ kε < min(1, ke22 ). Therefore,   kz (t )k ≤ ke23 e−λe (t −t1 ) + ε ke29 kz (0)k + O(ε), −1

holds ∀t ≥ t1 , where the residual term O(ε) is independent from the initial conditions. Noting that the initial time is irrelevant in the development of the above expressions, the inequality

  kz (t )k ≤ ke30 e−λe (t −ti ) + ε ke31 kz (ti )k + O(ε), holds for any t ≥ ti ≥ t1 (i = 1, 2, 3, . . .). This leads to the recursive linear inequality

kz (ti+1 )k ≤ λkz (ti )k + O(ε),

(A.19)

with λ = ke30 exp(−λe T1 )+ε ke31 and some period T1 = ti+1 − ti > 1 0. For 0 < ε ≤ ε ∗ < k− e31 and choosing T1 > 0 large enough, λ < 1 is obtained. Thus, for ε > 0 sufficiently small, the recursion (A.19) converges exponentially to a residual set of order ε . The upper bounds (42) and (43) for the norms of the error signals, valid ∀t ≥ 0, are finally obtained, since after t1 the state z (t ) converges exponentially to a residual set of order ε and, the upper bounds (A.3) for kXe k and (A.4) for k˜xe k hold for 0 ≤ t ≤ t1 . To prove the existence of the sliding mode, the following transformation is applied to the estimated state [¯xT1 , σ ]T = P xˆ e , ∗ ∗ P = [P1T , S T ]T , with P1 ∈ Rn −1×n being chosen such that P1 BM = 0 and det(P ) 6= 0. Thereafter, the observer state Eq. (24) can be rewritten in the regular form x˙¯ 1 = A¯ 11 x¯ 1 + A¯ 12 σ − P1 α(ε −1 ) − aM e˜ ,





σ˙ = A¯ 21 x¯ 1 + A¯ 22 σ + SBM k

nom

(A.20)

  U − S α(ε −1 ) − aM e˜ ,

(A.21)

where A¯ ij are appropriate partitions of PAM P −1 and, U is given by M (A.7). Since {AM , BM , S } is a controllable realization of s+γ , its only observable eigenvalue is −γ . Moreover, since the last row of P is the only linearly independent row (S) of the observability matrix, it follows that A¯ 21 = 0 and A¯ 22 = −γ . The high-frequency gain of the reference model is given by kM = SBM . Therefore, the state equation (A.21) can be rewritten as

k

  σ˙ = −γ σ + knom U − S α(ε −1 ) − aM e˜ . p

(A.22)

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From (A.22) and the upper bound for the signal e˜ given by (29), it can be concluded that, if the modulation function satisfies inequality (21) as well as

  ∗ ρ ≥ δ + (knom )−1 S α(ε −1 ) − aM ε n −1 p × e−

λα ε t

∗ [k3 kω(t )k + k4 ] ,

(A.23)

∀t ≥ ts , then the condition for the existence of sliding mode σ σ˙ < 0 is verified ∀t > ts , with ts ≥ 0 being the instant when the exponentially decaying terms due to (29) become smaller than the constant δ > 0.  References Atassi, A. N., & Khalil, H. K. (1999). A separation principle for the stabilization of a class of nonlinear systems. IEEE Transactions on Automatic Control, 44(9), 1672–1687. Atassi, A. N., & Khalil, H. K. (2000). Separation results for the stabilization of nonlinear systems using different high-gain observer designs. Systems & Control Letters, 39(3), 183–191. Bondarev, A. G., Bondarev, S. A., Kostyleva, N. E., & Utkin, V. I. (1985). Sliding modes in systems with asymptotic state observers. Automation and Remote Control, 46(6), 679–684. Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. SIAM. Boyd, S. P., & Barratt, C. H. (1991). Linear controller design: Limits of performance. Prentice-Hall. Busawon, K., El Assoudi, A., & Hammouri, H. (1993). Dynamical output feedback stabilization of a class of nonlinear systems. In Proc. IEEE conf. on decision and control (pp. 1966–1971). Chang, P. H., & Lee, J. W. (1996). A model reference observer for time-delay control and its application to robot trajectory control. IEEE Transactions on Control Systems Technology, 4(1), 2–10. Chitour, Y. (2002). Time-varying high-gain observers for numerical differentiation. IEEE Transactions on Automatic Control, 47(9), 1565–1569. Cunha, J. P. V. S., Costa, R. R., & Hsu, L. (2008). Design of first-order approximation filters for sliding-mode control of uncertain systems. IEEE Transactions on Industrial Electronics, 55(11), 4037–4046. Cunha, J. P. V. S., Hsu, L., Costa, R. R., & Lizarralde, F. (2003). Output-feedback model-reference sliding mode control of uncertain multivariable systems. IEEE Transactions on Automatic Control, 48(12), 2245–2250. Cunha, J.P.V.S., Hsu, L., Costa, R.R., & Lizarralde, F. (2005). Sliding mode control of uncertain linear systems based on a high gain observer free of peaking. In Proc. of the 16th IFAC world congress (pp. 717–722). Devaud, F. M., & Caron, J. Y. (1975). Asymptotic stability of model reference systems with bang–bang control. IEEE Transactions on Automatic Control, 20(5), 694–696. Emelyanov, S. V., Korovin, S. K., Nersisian, A. L., & Nisenzon, Y. Y. (1992). Output feedback stabilization of uncertain plants: A variable structure systems approach. International Journal of Control, 55(1), 61–81. Esfandiari, F., & Khalil, H. K. (1992). Output feedback stabilization of fully linearizable systems. International Journal of Control, 56(5), 1007–1037. Gauthier, J. P., Hammouri, H., & Othman, S. (1992). A simple observer for nonlinear systems applications to bioreactors. IEEE Transactions on Automatic Control, 37(6), 875–880. Hsu, L., Araújo, A. D., & Costa, R. R. (1994). Analysis and design of I/O based variable structure adaptive control. IEEE Transactions on Automatic Control, 39(1), 4–21. Hsu, L., Lizarralde, F., & Araújo, A. D. (1997). New results on output-feedback variable structure model-reference adaptive control: Design and stability analysis. IEEE Transactions on Automatic Control, 42(3), 386–393. Ioannou, P. A., & Sun, J. (1996). Robust adaptive control. Prentice-Hall. Kailath, T. (1980). Linear systems. Prentice Hall. Khalil, H. K. (2002). Nonlinear systems. Prentice Hall. Levant, A. (1998). Robust exact differentiation via sliding mode technique. Automatica, 34(3), 379–384. Li, Y., Ang, K. H., & Chong, G. C. Y. (2006). PID control system analysis and design. IEEE Control Systems Magazine, 26(1), 32–41. Lu, X. Y., & Spurgeon, S. K. (1998). Output feedback stabilization of SISO nonlinear systems via dynamic sliding modes. International Journal of Control, 70(5), 735–759. Méndez-Acosta, H. O., Femat, R., & Campos-Delgado, D. U. (2004). Improving the performance on the chemical oxygen demand regulation in anaerobic digestion. Industrial & Engineering Chemistry Research, 43(1), 95–104. Oh, S., & Khalil, H. K. (1995). Output feedback stabilization using variable structure control. International Journal of Control, 62(4), 831–848. Oh, S., & Khalil, H. K. (1997). Nonlinear output-feedback tracking using high-gain observer and variable structure control. Automatica, 33(10), 1845–1856. Oliveira, T.R., Peixoto, A.J., & Hsu, L. (2008). Peaking free output-feedback sliding mode control of uncertain nonlinear systems. In Proc. American contr. conf. (pp. 389–394).

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Sussmann, H. J., & Kokotović, P. V. (1991). The peaking phenomenon and the global stabilization of nonlinear systems. IEEE Transactions on Automatic Control, 36(4), 424–440. Walcott, B. L., & Żak, S. (1988). Combined observer-controller synthesis for uncertain dynamical systems with applications. IEEE Transactions on Systems, Man and Cybernetics, 18(1), 88–104.

José Paulo V.S. Cunha was born in Rio de Janeiro, Brazil, on March 9, 1965. He received the B.Sc. degree in electrical engineering from the State University of Rio de Janeiro in 1988, and the M.Sc. and D.Sc. degrees in electrical engineering from the Federal University of Rio de Janeiro in 1992 and 2004, respectively. From 1992 to 1996, he was a teacher at the Centro Federal de Educação Tecnológica do Rio de Janeiro. Since 1997, he has been a Professor at the Department of Electronics and Telecommunication Engineering of the State University of Rio de Janeiro. His research interests include sliding mode control, control of electromechanical systems, robotics, marine systems, underwater vehicles and the development of instrumentation and measurement systems. He is a member of the Sociedade Brasileira de Automática and of the IEEE Control Systems Society. Ramon R. Costa was born in São Paulo, Brazil in 1956. He received the B.Sc. and M.Sc. degrees in electrical engineering from the Federal Engineering School of Itajubá, Brazil, in 1979 and 1982, respectively, and the D.Sc. degree in electrical engineering from the Federal University of Rio de Janeiro, Brazil, in 1990. Currently, he is an Associate Professor in the Department of Electrical Engineering of COPPE at the Federal University of Rio de Janeiro. In the period 1999–2000 he was a visiting scholar in the Center for Control Engineering and Computation at the University of California, Santa Barbara (UCSB). His research interests include adaptive control, sliding mode control, and robotics. He is a member of the Sociedade Brasileira de Automática.

Fernando Lizarralde was born in Bell Ville, Argentina. He received the Ingeniero Electricista degree from Universidad Nacional del Sur, Bahia Blanca, Argentina, in 1989, the M.Sc. and Ph.D. degrees in Electrical Engineering from the Federal University of Rio de Janeiro, Brazil, in 1992 and 1998, respectively. He spent 1989 as a research fellow at the Universidad Nacional del Sur. In 1994-1995 he was a visiting scholar at the Rensselaer Polytechnic Institute, Troy (NY). Since 1996, he has been with the Department of Electrical Engineering at the Federal University of Rio de Janeiro, where he is currently an Associate Professsor. His current research interests include nonlinear control systems, adaptive control, variable structure control, visual servoing, stability and oscillations of nonlinear systems and their applications to industrial process control, industrial robotics and underwater robotics. Prof. Lizarralde is a member of the Sociedade Brasileira de Automática and of the IEEE Control Systems Society. Currently, he is an associate editor for the IEEE Control Systems Society Conference Editorial Board. Liu Hsu was born in Hunan, China. He received the B.Sc. and M.Sc. degrees in electrical engineering from the Instituto Tecnológico de Aeronáutica (ITA), São José dos Campos, Brazil, in 1968 and 1970, respectively, and the Docteur d’Etat ès Sciences Physiques degree from the Université Paul Sabatier, Toulouse, France in 1974. His doctoral research was developed at the Laboratorie d’Automatique et d’Analyse des Systèmes — CNRS, Toulouse. In 1975 he joined the Graduate School and Research in Engineering of the Federal University of Rio de Janeiro. His current research interests include adaptive control systems, variable structure systems, nonlinear systems dynamics and their applications to industrial process control, nanopositioning, industrial robotics, and terrestrial and underwater robotics. He is a member of the Sociedade Brasileira de Automática (SBA) and of the IEEE Control Systems Society. Liu Hsu is also a member of the Brazilian Academy of Sciences. He is an associate editor of the scientific journal of the SBA ‘‘Controle & Automação’’ and of the Proceedings of the Brazilian Academy of Sciences. In 2005, he was nominated to the Brazilian National Order of Scientific Merit.