Automatica 44 (2008) 1036–1044www.elsevier.com/locate/automatica

Brief paper

A variable structure model reference robust control without a priorknowledge of high frequency gain sign�

Lin Yana,∗, Liu Hsub, Ramon R. Costab, Fernando Lizarraldeb

aSchool of Automation, Beijing University of Aeronautics and Astronautics, Beijing 100083, ChinabDepartment of Electrical Engineering, COPPE/UFRJ. P.O. Box 68540, Rio de Janeiro, Brazil

Received 9 March 2006; received in revised form 1 June 2007; accepted 19 August 2007Available online 20 December 2007

Abstract

The design of a variable structure model reference robust control without a prior knowledge of high frequency gain sign is presented. Based onan appropriate monitoring function, a switching scheme for some control signals is proposed. It is shown that after a finite number of switching,the tracking error converges to zero at least exponentially for plants with relative degree one or converges exponentially to a small residual setfor plants with higher relative degree, and the input disturbance can be completely rejected without affecting the tracking performance.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Variable structure control; High frequency gain; Model following; Switching control; Nussbaum function

1. Introduction

Variable structure model reference robust control (VS-MRRC)1 was introduced by Hsu and Costa (1989), Hsu,Araujo, and Costa (1994) and Hsu, Lizarralde, and Araujo(1997), as a new means of I/O based model following con-troller design technique for linear time invariant plants withinput disturbance and has been extended to MIMO systems(Cunha, Hsu, Costa, & Lizarralde, 2003). The main interest inthe VS-MRRC relies on its strong stability, disturbance rejec-tion and nice tracking performance properties, as compared toparameter adaptive and robust linear controllers. Like most ofthe model following schemes, one of the basic assumptions ofthe VS-MRRC is that the plant high frequency gain (HFG) sign

� This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor ChangyunWen under the direction of Editor Miroslav Krstic. This work was supportedby NSF of China (60174001) and FAPERJ of Brazil (No. E-26/152.058/2001).

∗ Corresponding author.E-mail address: linyanee2@yahoo.com.cn (L. Yan).

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

0005-1098/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2007.08.011

is known a priori. The objective of this paper is to generalizethe VS-MRRC to the case of unknown HFG sign.

The relaxation of the assumption of known HFG sign haslong been an attractive topic in control community and can betraced back to the paper by Morse (1982). Several approacheshave been proposed so far and most of them, however, arebased on Nussbaum function (NF) (Mudgett & Morse, 1985;Nussbaum, 1983). Related work can also be found in backstepping design (Zhang, Wen, & Soh, 2000) and nonlinearsystems with unknown control directions (Ge & Wang, 2003;Ye & Jiang, 1998). However, as many authors pointed out,Nussbaum-based controllers may always cause a poor transientperformance and much higher control amplitude, and may failto maintain system stability if input disturbance or unmodelleddynamics exists. Therefore, it is of limited practical use.

An alternative way is switching. Switching was firstintroduced by Martensson (1985) to stabilize a system witha set of candidate controllers that switch directly accordingto plant output and a predetermined switching logic, and wasextended to more general cases by Fu and Barmish (1986)and Miller and Davison (1989) with the objective to achieveLyapunov stability with minimum prior information. However,the Lyapunov stability will lose if input disturbance exists.

L. Yan et al. / Automatica 44 (2008) 1036–1044 1037

In Li, Wen, and Soh (2001), by applying a state transformationto a class of nonlinear switched systems, switching laws wereproposed based on a Lyapunov function. In Miller and Davison(1991), a switching scheme for the gain of control signal wasproposed for unity feedback systems so that the tracking errormay have an arbitrarily good transient and steady-state perfor-mance specifications given by designer in advance even whenthe plant HFG sign is unknown. However, the price of thissolution is that the amplitude of the control signal will beextremely high if a higher tracking precision is needed since thetracking error is reciprocally proportional to the controller gain.Another kind of switching, say, indirect switching, in whichmultiple models are used to determine both when and to whichcontroller one should switch, was introduced by Middleton,Goodwin, and Mayne (1988). In Narendra and Balakrishnan(1997), an adaptive control using multiple models was givento improve transient performance, but how to cope with plantswith input disturbance or unmodelled dynamics is unknown.A detailed discussion about indirect switching can be foundin Hespanha, Liberzon, and Morse (2003) and the referencescontained therein.

In this paper, for plants without a prior knowledge of HFGsign, a switching scheme is proposed for the VS-MRRC. Themain idea is to construct a monitoring function based on aspecially chosen differential equation to supervise the behav-ior of the tracking error or one of the auxiliary errors andthen a switching law for some control signals is presented. Weshow that under the supervision of our proposed monitoringfunction, only a finite number of switching is needed and theinput disturbance can be completely rejected without affectingthe tracking performance. A scheme for the VS-MRRC sys-tem based on NF is also given and shows that in contrast tothe proposed switching scheme, its transient performance andamplitude of control signal are unacceptable.

2. Problem formulation and preliminary results

The plant to be controlled is a single input/single output LTIsystem, whose I/O form is2

y = Gp(s)[u + d] = kp

np(s)

dp(s)[u + d], (1)

where y and u are the system output and input, respectively,Gp(s) is the plant transfer function, dp(s) and np(s) aremonic polynomials of degree n and m, respectively, and d is anunknown disturbance. With respect to the controlled plant, wemake the following assumptions.

(A1) Gp(s) is strictly proper and minimum phase. Theparameters of Gp(s) are unknown but belong to a knowncompact set S. The degree n and the relative degree n∗of Gp(s) are known constants.

(A2) The sign of the HFG kp(�= 0) is unknown.

2 The representation of convolution operation h(t) ∗ z(t) = H(s)[z] willbe used throughout this paper.

(A3) The disturbance d satisfies

|d(t)|�du(t), ∀t �0, (2)

where du is a known, piece-wise continuous and uni-formly bounded function.

Our goal is to design the control input u so that y tracks theoutput yM of a stable reference model given by

yM = M(s)[r] = kM

1

dM(s)[r], kM > 0 (3)

for any piecewise continuous and uniformly bounded referencesignal r, where dM(s) is a monic Hurwitz polynomial withdeg(dM(s)) = n∗.

In this paper, the control is of the following form:

u = �T� + uvs, (4)

where uvs is a variable structure control term to be designed,the constant vector � ∈ R2n will be defined below and �, theregressor vector, is defined as

� := [�T1 y �T

2 r]T ∈ R2n, (5)

where �1 and �2 are generated by input/output filters accordingto

�1 = ��1 + bu, �1(0) = 0,

�2 = ��2 + by, �2(0) = 0, (6)

where � is Hurwitz, (�, b) is a controllable pair and� ∈ R(n−1)×(n−1), b ∈ Rn−1. It is well known that un-der the above assumptions, there exits a unique constantvector �∗ = [�∗T

1 �∗0 �∗T

2 k∗]T ∈ R2n with �∗1 ∈ Rn−1,

�∗0 ∈ R, �∗

2 ∈ Rn−1 and k∗ = kM/kp ∈ R, such that, mod-ulo exponentially decaying terms due to initial conditions,y = Gp(s)[�∗T�] = M(s)[r] = yM (Narendra & Annaswamy,1989). Since the plant parameters are assumed to be uncertain,the constant vector � in (4) is then defined as

� = [�T1 �0 �T

2 k]T

:={

�+ = [(�+1 )T �+

0 (�+2 )T k+]T if kp > 0,

�− = [(�−1 )T �−

0 (�−2 )T k−]T if kp < 0,

(7)

which is an estimate of �∗ and is obtained from nominal plant.From (1)–(7), the tracking error can be written as

e = M(s)k∗−1[�T� + df + uvs] + ε, (8)

where � := �−�∗ =[�T1 �0 �T

2 k]T, df := (1−d1(s))[d] with

d1(s) := �T1 adj(sI − �)b, and ε decays to zero exponentially

due to initial conditions.When n∗ = 1, (3) implies that M(s) = kM/(s + �), � > 0.

Therefore, from (8) we have

e = −�e + kp(�T� + df + uvs) + ε, (9)

where ε exponentially decays.

1038 L. Yan et al. / Automatica 44 (2008) 1036–1044

When n∗ > 1, we can write (8) as

e = M(s)L(s)k∗−1[�T� + df + uvs] + ε, (10)

where the polynomial

L(s) :=n∗−1∏i=1

Li(s) =n∗−1∏i=1

(s + �i ), �i > 0 (11)

is chosen such that M(s)L(s) is an SPR (strictly positive real)function, and x denotes filtered version of a signal x by L−1(s).Further, we introduce the predicted error ya and the auxiliaryerrors e0, . . . , en∗−1 as

ya := M(s)L(s)[k−1(uvs − u0)], (12)

e0 := e − ya = M(s)L(s)[k−1(� + u0)] + ε, (13)

ei := L−1i (s)[ui] − F−1(s)[ui−1]

= L−1i (s)[F−i (s)L1,i (s)[�] + ui] + εi ,

i = 1, 2, . . . , n∗ − 2, (14)

en∗−1 := L−1n∗−1(s)[un∗−1] − F−1(s)[un∗−2]

= L−1n∗−1(s)[(k/k∗)(F−(n∗−1)(s)[�T� + df ] + un∗−1)]

+ + εn∗−1 (uvs = un∗−1), (15)

where u0, . . . , un∗−1 are the VS control signals to be designed,k(=kM/kp) is given in (7),

� := (k/k∗)�T� + (k/k∗)uvs + (k/k∗)df (16)

with k = k − k∗ (see (8)), the averaging filter

F−1(s) := (s + 1)−1 (17)

with a small positive constant, and

L1,i (s) =i∏

k=1

Lk(s), (18)

ε1 = F−1(s)[(k(M(s)L(s))−1[ε − e0])], (19)

εj = Lj−1(s)F−1(s)[εj−1 − ej−1], j = 2, . . . , n∗ − 1, (20)

= (k/k∗)L−1n∗−1(s)[(F−(n∗−1)(s)[un∗−1] − un∗−1)]. (21)

The final control signal uvs is thus obtained recursively by(13)–(15) as uvs = un∗−1.

The following lemma summarizes the main results when thesign of kp is known.

Lemma 1. Let the plant (1) satisfy (A1), (A3), and the sign ofkp be known. The VS-MRRC has the following properties:

(1) If n∗ = 1, let the VS control signal in (8) be

uvs ={−f sgn(e), if sgn(kp) > 0,

f sgn(e), if sgn(kp) < 0,(22)

where the modulation function is chosen such thatf > |�T� + df |. Then, the tracking error converges tozero in some finite time t = t and remains zero ∀t � t asa sliding mode on the surface e = 0.

(2) If n∗ > 1, let the VS control signals in (13)–(15) be

u0 ={−f0 sgn(e0), if sgn(kp) = sgn(k) > 0,

f0 sgn(e0), if sgn(kp) = sgn(k) < 0,(23)

ui = −fi sgn(ei), i = 1, . . . , n∗ − 1, (24)

where the modulation functions are chosen such thatf0 > |�|, fi > |F−i (s)L1,i (s)[�]|, for i = 1, . . . , n∗ − 2,and fn∗−1 > |F−(n∗−1)(s)[�T�+df ]|, respectively. Then,for sufficiently small , all signals of the closed-loopsystem are in L∞ and the auxiliary errors ei , fori =0, . . . , n∗ −2, converge to zero in finite time, while thelast auxiliary error en∗−1 and the tracking error e expo-nentially converge to some residual sets whose radiusesare proportional to .

Proof. See Hsu and Costa (1989) and Hsu et al. (1997). �

Remark 2.1. Since k := kM/kp, where kM > 0 (see (3)), wehave sgn(k)= sgn(kp). In particular, if the sign of kp is known,sgn(kp) = sgn(k) = sgn(kp).

Remark 2.2. The modulation function f or fj in Lemma 1 canbe obtained by usingthe assumption (A1) and (2), e.g., f canbe chosen as f = (�‖�‖ + df u + �) with � an upper boundof ‖�‖, df u an upper bound of |df | and � an arbitrarily smallpositive constant.

3. Main results

3.1. Signals to be switched

Because the sign of kp is unknown, we have to redefine thecontrol signals (22) (n∗ = 1) and (23) (n∗ > 1) as

uvs(t) :={

u+vs(t) = −f sgn(e), if t ∈ T+,

u−vs(t) = f sgn(e), if t ∈ T−,

(25)

and

u0(t) :={

u+0 (t) = −f0 sgn(e0), if t ∈ T+,

u−0 (t) = f0 sgn(e0), if t ∈ T−,

(26)

respectively, the vector � in (7) as

� ={

�+, if t ∈ T+,

�−, if t ∈ T−,(27)

and design a monitoring function to decide when (uvs, �)

((u0, �)) will be switched from (u+vs, �

+) to (u−vs, �

−) ((u+0 , �+)

to (u−0 , �−)) and vice versa, where for both n∗ = 1 and n∗ > 1,

the sets T+ and T− generically denote the union of timeintervals on which (u+

vs, �+) and (u−

vs, �−) ((u+

0 , �+) and

(u−0 , �−)) are applied, respectively, and satisfy

T+ ∪ T− = [0, ∞), T+ ∩ T− = ; (28)

both T+ and T− have the form

[tk, tk+1) ∪ [tk+3, tk+4) · · · ∪ [tj , tj+1), (29)

L. Yan et al. / Automatica 44 (2008) 1036–1044 1039

where tk or tj denotes switching time that will be defined

later. The term [tk, tk+1) ∪ [tk+3, tk+4) indicates that (uvs, �)

((u0, �)) switches between (u+vs, �

+) and (u−vs, �

−) ((u+0 , �+)

and (u−0 , �−)) alternately. Comparing (22), (23), (7) with (25),

(26), (27) it is clear that if the sign of kp is known, we need only

one uvs (u0) and one �, while if the sign of kp is unknown, both

(u+vs, �

+) and (u−vs, �

−) ((u+0 , �+) and (u−

0 , �−)) are needed.

3.2. Monitoring function and switching law

Our purpose is to construct a monitoring function to deter-mine when (uvs, �) or (u0, �) will be switched.

n∗ = 1: We consider the following first-order differentialequation:

� = −�� + ε, t � t0, �(t0) = e(t0), (30)

which is obtained by ignoring the terms related to kp in (9),i.e., kp(�T� + df + uvs), by intuition that if the sign of kp isestimated correctly, e will converge faster than �. Indeed, letz := e − �, then it follows from (9) and (30) that

z = −�z + kp(�T� + df + uvs). (31)

If we correctly estimate the sign of kp for all t � t0, substituting(22) in (31) and noting that �(t0) = e(t0) it follows that

� = e − kp

∫ t

t0

exp[−�(t − �)][(�T� + df + uvs)(�)] d�

= e +∫ t

t0

exp[−�(t − �)]× [|kp|f sgn(e) − kp(�T� + df )](�) d�. (32)

Since f (f > |�T� + df |) in (22) can completely dominate theterm �T� + df , it is easy to check that for all t � t0,

|e(t)|� |�(t)|. (33)

We then consider the solution of (30). Because ε decaysexponentially, there exist positive constants � and c, such that

|ε(t)|�c exp(−�t), t �0. (34)

Therefore, for all t � t0,

|�(t)|� exp[−�(t − t0)]|e(t0)|+

∫ t

t0

exp[−�(t−�)]|ε(�)| d�� exp[−�(t−t0)]|e(t0)|+ cm exp[−�m(t − t0)] exp(−�t0), (35)

where

cm = 2c/|� − �|, �m = min{�, �}. (36)

Since a less � can only make the estimate of ε more conserva-tive, we can choose, for the sake of simplicity, � < �, so that�m = �. Therefore, the inequality (35) can be rewritten as

|�(t)|� exp[−�(t − t0)]|e(t0)| + cm exp(−�t), t � t0. (37)

Recalling that the inequality (33) holds once the sign of kp

is correctly estimated, it seems natural to use � as a benchmark

to decide whether a switching of (uvs, �) is needed, i.e., theswitching occurs only when (33) is violated. However, since ε

is not available for measurement or in other words, c, � as wellas cm given by (34) and (36), respectively, are unknown, wehave to use a monitoring function k to replace � and invokethe switching of k:

k(t) = exp[−�(t − tk)]|e(tk)| + ck exp(−�kt),

t ∈ [tk, tk+1), k = 0, 1, · · · ; t0 := 0, (38)

where tk is the switching time, ck is any monotonically increas-ing positive sequence satisfying

ck → ∞ as k → ∞, (39)

and �k is any monotonically decreasing positive sequence sat-isfying

�k → 0 as k → ∞ (40)

with �0 < �. Comparing (37) with (38), it is clear that we obtain k from (37) mainly by replacing both � and cm by �k andck , respectively, and by introducing the switching of �k andck . Note that ck increases as the switching proceeds while �k

satisfies (40) and therefore, can explain why the condition � < �is reasonable.

n∗ > 1: Note that from (13), to determine the switching of(u0, �), it is invalid to supervise the behavior of e0 because thesign of k(sgn(k) = sgn(kp), see Remark 2.1) is known. How-ever, it is clear that an incorrect sign of k results in k/k∗ = kp/

kp < 0, where k/k∗ is the HFG of (15). In that case, en∗−1should diverge under the control (24). Note that from (7),the switching of � includes the switching of k and thereforek/k∗ > 0 whenever the sign of � is chosen correctly. Hence, todetermine the switching of (u0, �), a monitoring function su-pervising en∗−1 will suffice. Again, on the assumption that thesign of kp has been correctly estimated for all t � t0, in a sim-ilar way to obtain (30), multiplying Ln∗−1(s) = s + �n∗−1 onboth sides of (15) and ignoring the terms related to k/k∗, wecan construct the following differential equation:

� = − �n∗−1� + (k/k∗)(� − un∗−1)

+ (εn∗−1 + �n∗−1εn∗−1), �(t0) = en∗−1(t0), (41)

where (21) has been used and � := ∫ t

t0F(t − �)un∗−1(�) d�

with F(t) the inverse Laplace transform of F−(n∗−1)(s). Then,similar to the analysis of (32),

|en∗−1(t)|� |�(t)|, t � t0. (42)

Solving (41) and noting that �(t0) = en∗−1(t0), it follows that

|�(t)|� exp[−�n∗−1(t − t0)]|en∗−1(t0)| + |�(t)| + |εn∗−1|,t � t0, (43)

where

�(t) = (k/k∗)∫ t

t0

exp[−�n∗−1(t − �)](� − un∗−1)(�) d�, (44)

εn∗−1 = εn∗−1 − exp[−�n∗−1(t − t0)]εn∗−1(t0). (45)

1040 L. Yan et al. / Automatica 44 (2008) 1036–1044

0

1

e

tt10

Fig. 1. The trajectories of k(t) and e(t) intersect at t1 and therefore, a

switching of (uvs, �) from (u+vs, �

+) to (u−

vs, �−

) (or (u−vs, �

−) to (u+

vs, �+

))occurs; meanwhile, k(t) switches from 0(t) to 1(t) according to (38).

As will be shown later, both εn∗−1 and εn∗−1 decay to zeroexponentially and are unknown due to the unknown ε in (10).Hence, by using the way to obtain the term cm exp(−�t) in(37), |εn∗−1| can be expressed as |εn∗−1|�cn∗−1 exp(−�n∗−1t),where cn∗−1and �n∗−1 are positive constants. The same as (38),a monitoring function is constructed based on (43):

�k = exp[−�n∗−1(t − tk)]|en∗−1(tk)| + ck exp[−�kt] + �,

t ∈ [tk, tk+1), k = 0, 1, · · · ; t0 := 0, (46)

where ck and �k are given by (39) and (40), respectively, � isa positive design parameter and the time constant is given in(17) .

Remark 3.1. The derivation of the design parameter � is basedon the fact that unlike the case of n∗ = 1, en∗−1 does not con-verge to zero due to the term �(t) in (44), which is derivedfrom in (15) that can be regarded as an unmodelled dy-namics. Indeed, if the correct sign of kp is chosen, we have|en∗−1(t)|�EXP + C, where it can be proved that un∗−1 ∈L∞and |�(t)|�C‖(un∗−1)t‖∞; hence, |�(t)| must be smallwith a sufficiently small . Here and throughout, ‖(x)t‖∞ :=sup�� t |x(�)| and, EXP and C generically denote an exponen-tially decaying term and a positive constant independent of ,respectively.

From (38) ((46)) and the absolute continuity of e(en∗−1)

(Filippov, 1964), we always have |e(tk)| < k(tk) (|en∗−1(tk)|< �k(tk)) at t = tk . Hence, the next switching time tk+1 for(uvs, �) and k ((u0, �) and �k) is well-defined:

tk+1=

⎧⎪⎨⎪⎩

min{t : t > tk, |e(t)|= k(t), for n∗=1;|en∗−1(t)|=�k(t), for n∗>1},if the minimum exists,

+∞, otherwise.

(47)

That is, the switching occurs only when the condition|e(t)| < k(t) (|en∗−1(tk)| < �k(tk)) is violated at some finitepoint t = tk+1. Fig. 1 illustrates the switching when n∗ = 1.

3.3. Main theorem

We are now ready to establish the following main results.

Theorem 1. Assume that the plant (1) satisfies the assump-tions (A1)–(A3). Let the VS-MRRC system be given by (9) and(10)–(15) for both n∗ = 1 and n∗ > 1, respectively, and theswitching time for (uvs, �) or (u0, �) be defined by (47) withuvs, u0 and � given by (25)–(27). Then the switching algorithmguarantees that

(1) If n∗=1, only finite number of switching of (uvs, �) occurs,all signals of the system are in L∞ and the tracking errorconverges to zero at least exponentially;

(2) If n∗ > 1, there exist a � > 0 and a sufficiently small ,such that only finite number of switching of (u0, �) occurs,all signals of the system are in L∞ and the tracking errorconverges exponentially to a residual set whose size isproportional to .

Proof. (1) The proof is achieved by contradiction. Suppose(uvs, �) switches between (u+

vs, �+) and (u−

vs, �−) without stop-

ping. Because cm and � given by (36) and (34), respectively,are constant, and (uvs, �) has only two choices, (u+

vs, �+)and

(u−vs, �

−), and changes alternately, the following conditionsmust hold after some kth switching: (uvs, �) has a correct sign((uvs, �) = (u+

vs, �+) if kp > 0 or (uvs, �) = (u−

vs, �−)if kp < 0),

and

cm = 2c/|� − �| < ck , (48)

exp(−�t) < exp(−�kt), ∀t > tk , (49)

where ck and �k are defined by (39) and (40), respectively, andtk is the switching time. The above conditions, together with(37), imply that |�(t)| < k(t), ∀t > tk , where we have replacedt0 by tk . Since, however, for a correct choice of the sign of kp, esatisfies (33), the inequality |�(t)| < k(t), ∀t > tk implies that

|e(t)| < k(t), ∀t > tk . (50)

Hence, from (47), no switching will occur again, a contradic-tion. Because k converges to zero exponentially, (50) showsthat e converges to zero at least exponentially. Finally, r, e ∈L∞, together with the relative degree one and minimum phaseassumptions, imply that � ∈ L∞ and therefore, all the closed-loop signals are uniformly bounded.

(2) The proof includes two steps.Step 1: Finite number of switching of (u0, �). For a properly

chosen � > 0, the proof is quite similar to the case of n∗ = 1and is omitted. Also, note that the following facts hold:

• No finite time escape occurs. Because �+, �− are con-stant vectors, from (15) and (24) with fn∗−1 chosen inthe same way as f in Remark 2.2, we have ‖(uvs)t‖∞ =‖(un∗−1)t‖∞ �C‖(�)t‖∞ + C and therefore, from (4),‖(u)t‖∞ �C‖(�)t‖∞ + C, which guarantees that all sig-nals of the system are in L∞e (Hsu et al., 1997, p. 388);

L. Yan et al. / Automatica 44 (2008) 1036–1044 1041

• |εn∗−1|��1 exp(−�2t), ∀t �0 for some constants �1,�2 > 0. According to Lemma 1, ei , for i=0, 1, . . . , n∗−2,converge to zero in finite time because the HFG signof each system in (13) or (14) is known. Hence, us-ing (19) and (20), one can recursively show that εi , fori = 1, . . . , n∗ − 1, decay exponentially due to ε. Note thatk is piece-wise constant (k+ or k−) before it stops switch-ing and therefore, does not affect the convergence of ε1.

Step 2:All the closed-loop signals are inL∞ and |e(t)|�EXP+C. From (47), the finite number of switching implies thatthere exists a finite tk such that |en∗−1(t)| < �k(t)= EXP + �,t � tk , which together with the convergence property of ei

(i = 0, 1, . . . , n∗ − 2), guarantees that for sufficiently small ,using the same technique of Theorem 2 of Hsu et al. (1997),the tracking error |e(t)|�EXP + C and � ∈ L∞. That is,all the closed-loop signals are in L∞.See Theorem 2 of Hsuet al. (1997) for a complete proof. �

The following corollary shows a more interesting fact for theVS-MRRC system.

Corollary 1. If all the initial conditions of (8) are zero, thenat most one switching of (uvs, �) or (u0, �) is needed.

Proof. First, we consider the case of n∗ = 1. Because ε = 0implies that ε in (9) is zero, it follows from (34) and (36) thatc = cm = 0 and thus (37) can be rewritten as

|�(t)|� exp[−�(t − t0)]|e(t0)|, t � t0. (51)

Therefore, once we correctly estimate the sign of kp, from (38)and replacing t0 by tk in (51), the inequality |�(t)| < k(t) holdsfor any k, ck and �k which, together with (33), implies that

|e(t)| < k(t), ∀t > tk . (52)

In view of (47), if we correctly estimate the sign of kp at t0 =0

(k=0), no switching of (uvs, �)occurs; whereas, one switching(k = 1) is enough.

We then consider the case of n∗ > 1. From (13), zero initialconditions, together with f0 > |�|, imply that e0 = 0, ∀t �0,which from (19) implies that ε1 =0, ∀t �0. By the same token,using (14) and (20) we have that εn∗−1 = 0, ∀t �0. Now, from(43) with a proper � > 0 the same conclusion for n∗ = 1 can bereached for the switching of (u0, �). �

3.4. Control law design based on NF

To show the effectiveness of the above switching scheme,the control law design based on NF is discussed in this section.To this end, let the VS control (25) be replaced by

uvs = N(x)f sgn(e), (53)

where f = (�‖�‖ + df u + �) (see Remark 2.2) and the NF ischosen as (Narendra & Annaswamy, 1989)

N(x) = x2 cos x. (54)

We let

x = f |e|, x(0) = x0. (55)

Since the adaptive law can guarantee that uvs finally dominatesthe uncertainty, the switching of � given by (27) is not consid-ered.

Remark 3.2. Here, we only consider the case of n∗ = 1. Weemphasize that n∗ = 1 is enough to compare the two schemesbecause for n∗ > 1, our switching scheme actually copes witha set of relative degree one subsystems.

Theorem 2. Assume that the plant (1) satisfies the assumptions(A1)–(A3) with n∗ = 1. Let the VS-MRRC system be given by(9) with the control signal uvs defined by (53). Then all signalsof the system are in L∞ and the tracking error e converges tozero asymptotically.

Proof. Let

V = e2/2, (56)

whose time derivative along the solution of (9) is given by

V = ee = −�e2 + kp(�T� + df )e + kpuvse + εe

= − �e2 + kp(�T� + df )e + kpN(x)f e sgn(e) + εe

= − �e2 + kp(�T� + df )e + kpN(x)x + εe. (57)

Integrating both sides of (57) it follows that

V (t) = V (0) − �∫ t

0e2 d� + kp

∫ t

0(�T� + df )e d�

+ kp

∫ t

0N(x)xd� +

∫ t

0εe d�. (58)

Using the triangle inequality εe�(�/2)e2 + (1/2�)ε2 with

(�/2) < � where � is a positive constant, and noting that |(�T�+

df )e|�f |e| := x, we obtain that

V (t) + (� − �/2)

∫ t

0e2 d�

�V (0) + |kp|∫ t

0x d� + kp

∫ x(t)

x0

N(x) dx + 1

2�

∫ t

0ε2 d�

�V (0) + |kp|x + kp

∫ x(t)

x0

N(x) dx + c, (59)

where 1/2�∫ t

0 ε2 d� − |kp|x0 < c. Replacing (54) into (59) itis clear that the term x2 sin x, which appears in the integral∫ x(t)

x0N(x) dx and oscillates between −x2 and x2, dominates

the right-hand side of the above inequality when x is large andconsequently, V (t) + (� − �/2)

∫ t

0 e2 d��0 implies that x hasto be bounded in spite of the sign of kp. Hence, from (59) andin view of (56), e ∈ L∞ ∩ L2, which together with x ∈ L∞implies that �, uvs ∈ L∞ and therefore, from (9) e ∈ L∞. Ac-cording to Barbalat’s Lemma (Narendra & Annaswamy, 1989),e → 0 as t → ∞. �

1042 L. Yan et al. / Automatica 44 (2008) 1036–1044

3.5. Comparison of the switching scheme with NF schemes

For the switching scheme, whenever the condition |e(t)| = k(t) in (47) is satisfied, (uvs, �) switches immediately, whichresults in a nice transient performance that can be furtherstrengthened by properly choosing � (f = (�‖�‖+df u +�)),ck and �k: A larger � implies that the convergence of e isfaster than k if the correct control direction is chosen. Con-sequently, e and k may not intersect. This is why by usingthe proposed switching algorithm, only one or fewer switchingcan be observed. However, a larger � may cause higher am-plitude. Moreover, from (47) and (38), a less ck and a larger�k may improve the tracking performance but may cause newswitching. Note that the effect of the increase of the switchingnumber on the tracking performance is not significant dueto the fast switching and small increment (ck+1 − ck) of themonitoring function.

The monitoring function given by (38) satisfies |e(tk)|< k(tk)(|e(0)| < 0(0)) for each switching instant as illus-trated by Fig. 1. Whether the next switching will occur isdetermined only by the control direction and the design pa-rameters �, ck , �k , i.e., the initial conditions have no effect on k and the switching is insensitive to the initial conditions.

For the NF scheme given by Theorem 2, from (53), to changethe control direction, one has to wait until cos x changes its signduring which the amplitude of the control signal may becomeextremely large due to x = ∫ t

0 f |e| d and the term x2 in (54),where x increases monotonically before e converges to zero;moreover, if e(0) is large, the term cos x implies that the controlsignal and the corresponding tracking error will oscillate manytimes. Hence, if the initial conditions are large, peaking mayoccur by using the NF scheme, the transient performance maybe very poor and the amplitude of control signal may be muchhigher than that of the proposed switching scheme.

In comparison with those NF-based parameter adaptive con-trol schemes(Mudgett & Morse, 1985; Zhang et al., 2000),our switching scheme can guarantee nice tracking performanceeven with the existence of input disturbance. However, the pricewe have to pay is that the bounds for parametric uncertain-ties and input disturbance must be known (see the assumptions(A1), (A3)), which, instead of using adaptive law, enables usto use variable structure control to dominate the uncertaintiesand the disturbance. Since the controller gain of the switchingscheme is proportional to the bounds, it may be large if thebounds are large, which is the main drawback of our swithchingscheme. Also, note that by using NF-based parameter adaptivecontrol, the controller gain may still be much larger than thatof the switching scheme when initial conditions are large.

4. Simulation results

Example 1. We compare the switching scheme and the NFscheme given by Theorem 2. The relative degree one plant is

y = Gp(s)[u + d] = kp(s + 1)

(s2 − 3s + a)[u + d], (60)

0 0.5 1-4

-2

0

2

4

6

8

Tracking error

Monitoring function

0 1-80

-60

-40

-20

0

20

40

60

80

Time (s)

Time (s)

1.5

1.50.5

Fig. 2. (a) Tracking error and monitoring function. (b) Control signal byusing the switching scheme.

whose uncertainty concerning the plant parameters is

S = {kp, a: 0.5�kp �2 or − 2�kp � − 0.5, −1�a�3}.The disturbance d(t) = sin(t). The reference model is

yM = M(s)[r] = 1

s + 2[r], (61)

where r is a square wave with unit amplitude and frequency1 rad/s. The parameters of the input/output filters are chosen as� = −2 and b = 1. To fairly compare the two schemes, only�+ = [�T

1 �0 �T2 k]T = [1 − 5.6 8.8 0.8]T is applied, which

corresponds to the nominal plant parameters kp=1.25 and a=1.We choose |k|=1.25, |�0|=8.4, |�1|=0 and |�2|=17.2 so thatuvs can dominate the uncertainties even when � is not switched.The price to be paid is that the amplitude of the control signalu in (4) may be larger than that using the switching of �. Thedesign parameters �k and ck of the monitoring function k

in (38) are chosen (somewhat arbitrarily) as �k = 1/(k + 1)

and ck = k + 1, respectively. For both schemes, we choosef = (�‖�‖ + df u + �) with � = 0.1 (see Remark 2.2). In thesimulation the plant parameters are chosen as a=3 and kp=0.5but the control signal given by (27) is chosen, at t = 0, to beu−

vs, i.e., an incorrect control signal is given at the beginningof the simulation since sgn(kp) > 0.The simulations are donewith the initial conditions xp(0) = [3 3]T, where xp denotesthe state vector of the plant. Fig. 2 shows that for the switchingscheme, only one peak is present and one switching occurs.However, Fig. 3 shows that by using the NF scheme, peakingoccurs and the amplitude of the control signal is about 3000!

L. Yan et al. / Automatica 44 (2008) 1036–1044 1043

0 1-2

0

2

4

6

8

10

-3000

-2000

-1000

0

1000

2000

3000

Time (s)

0.5 1.5

0 1

Time (s)

0.5 1.5

Fig. 3. (a) Tracking error using the NF scheme. (b) Control signal using theNF scheme.

To avoid chattering, sgn(e) is replaced by e/(|e| + 0.001)

(Utkin, Guldner, & Shi, 1999) for the two schemes.

Example 2. The relative degree two plant is

y = Gp(s)[u + d] = kp

s2 − 0.6s + a[u + d] (62)

with initial state xp(0) = [−0.5 − 0.5]T. The uncertainty con-cerning the plant parameters is

S = {kp, a: 0.5�kp �4 or − 4�kp � − 0.5, −3�a� − 1}.The disturbance d = sin(5t). The reference model is

yM = M(s)[r] = 1

(s + 1)2 [r], (63)

where r is a square wave with its amplitude 1 and frequency2 rad/s. The parameters of input/output filters are � = −1 andb = 1. We let L(s) = s + 1 in (11) and the nominal plantparameters in (62) be k±

p =±2 and a=−2, from which we have

�+=[−2.6 −3.58 −0.52 0.5]T and �−=[−2.6 3.58 0.52 −0.5]T, respectively; further, in view of the set S, we have |k| =1.75, |�0|=14.78, |�1|=0 and |�2|=10.4. We design u0 and u1=uvs with f0 and f1 satisfying f0 > |�| and f1 > |F−1[�T� +df ]|(see (26), (24)), respectively. We choose = 0.01 in (17).The monitoring function �k in (46) is chosen such that �k =1/(k + 1), ck = k + 1 and � = 5. The true plant parameters arechosen as kp=k+

p =1 and a=−1 but in the simulation, (u−vs, �

−)

is used at t = 0; that is, an incorrect estimate of the sign of kp

is applied at the beginning of the simulation. Fig. 4(b) showsthat the unique switching occurs at about 0.2 s and the system

0 1 2 3 4 5

0

y

ym

-3

-2

-1

0

1

2

3

Monitoring function

Auxiliary error e1

-10

-8

-6

-4

-2

0

2

4

6

8

10

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.1

0.2

Time (s)

0 1 2 3 4 5

Time (s)

0 1 2 3 4 5

Time (s)

Fig. 4. (a) Tracking performance. (b) Monitoring function and auxiliary errore1. (c) Control signal.

output y henceforth tracks the model output yM as shown inFig. 4(a). To alleviate chattering due to the sign function, thecontrol signal uvs is filtered by 1/(0.001s + 1) as suggested byUtkin et al. (1999) and is shown in Fig. 4(c).

5. Conclusion

In this paper, a switching scheme has been introduced for thecontroller design of the VS-MRRC without a prior knowledgeof high frequency gain sign. The main idea is to construct amonitoring function to supervise the behavior of the trackingerror (for n∗ = 1) or the last auxiliary error (for n∗ > 1). Thena switching algorithm is proposed based on the monitoringfunction. We have shown that our scheme can guarantee a nicetracking performance, while maintaining a reasonable controleffort. An algorithm based on NF has also been introduced; theanalysis and simulation show that, from a practical point ofview, its performance and control effort are unacceptable.

1044 L. Yan et al. / Automatica 44 (2008) 1036–1044

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Lin Yan received the M.Sc. and Ph.D degrees inelectrical engineering from the Beijing Univer-sity of Aeronautics and Astronautics (BUAA),Beijing, China, in 1988 and 1999, respectively.Since 1999, he has been a faculty memberin the School of Automation, BUAA. During2002–2003, he held a postdoctoral position atGraduate School and Research in Engineeringof the Federal University of Rio de Janeiro(COPPE/UFRJ), Brazil. His research interestsinclude robust control and adaptive control.

Liu Hsu received the B.Sc. and M.Sc. degreesin electrical engineering from the Instituto Tec-nológico de Aeronáutica (ITA), Brazil, and theDocteur d’Etat degree from the Université PaulSabatier, Toulouse, France, in 1974. In 1975,he joined the Graduate School and Research inEngineering of the Federal University of Riode Janeiro (COPPE/UFRJ). His current researchinterests include adaptive control systems, vari-able structure systems, stability and oscillationsof nonlinear systems and their applications to in-dustrial process control, industrial robotics and

underwater robotics. Prof. Liu Hsu is a member of the Sociedade Brasileirade Automática and of the IEEE Control Systems Society. He is a memberof the Brazilian Academy of Sciences. In 2005, he was awarded the Brazil’sNational Order of Scientific Merit.

Ramon R. Costa received the B.Sc. and M.Sc.degrees in electrical engineering from the Fed-eral Engineering School of Itajubá, Brazil, in1979 and 1982, respectively and the D.Sc de-gree from the Federal University of Rio deJaneiro, Brazil, in 1990. Currently, he is a pro-fessor in the Department of Electrical Engineer-ing of COPPE/UFRJ. His research interests in-clude adaptive and variable structure control andits applications in computer-controlled systems.

Fernando C. Lizarralde received the B.Sc.and M.Sc. degrees in electrical engineeringfrom the Universidad Nacional del Sur, BahiaBlanca, Argentina and the Federal Universityof Rio de Janeiro, Brazil, in 1989 and 1992,respectively, and the D.Sc degree from theFederal University of Rio de Janeiro, Brazil,in 1998. Currently, he is an associate professorin the Department of Electrical Engineering ofCOPPE/UFRJ. His research interests includeadaptive and variable structure control systems,control of rigid/flexible robot manipulators,

attitude control and control of nonholonomic systems.