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A variable structure model reference robust control without a prior knowledge of high frequency gain sign

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A variable structure model reference robust control without a prior knowledge of high frequency gain sign
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  Automatica 44 (2008) 1036–1044www.elsevier.com/locate/automatica Brief paper Avariablestructuremodelreferencerobustcontrolwithoutapriorknowledgeofhighfrequencygainsign  LinYan a , ∗ , Liu Hsu b , Ramon R. Costa b , Fernando Lizarralde b a School of Automation, Beijing University of Aeronautics and Astronautics, Beijing 100083, China b  Department 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 of the model following schemes, one of the basic assumptions of theVS-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 back stepping 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 is 2 y  =  G p (s) [ u  +  d  ] =  k p n p (s)d  p (s) [ u  +  d  ] , (1)where  y  and  u  are the system output and input, respectively, G p (s)  is the plant transfer function,  d  p (s)  and  n p (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 )  G p (s)  is strictly proper and minimum phase. The parameters of   G p (s)  are unknown but belong to a knowncompact set   S  .  The degree n and the relative degree  n ∗ of   G p (s)  are known constants .( A2 )  The sign of the HFG  k p ( =  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) |  d  u (t),  ∀ t   0, (2) where  d  u  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  y M   of a stable reference model given by y M   =  M(s) [ r ] =  k M  1 d  M  (s) [ r ] , k M   > 0 (3)for any piecewise continuous and uniformly bounded referencesignal  r  , where  d  M  (s)  is a monic  Hurwitz  polynomial withdeg (d  M  (s))  =  n ∗ .In this paper, the control is of the following form: u  = ˆ  T   +  u vs , (4)where  u vs  is a variable structure control term to be designed,the constant vector  ˆ   ∈  R 2 n will be defined below and   , theregressor vector, is defined as   := [  T1  y   T2  r ] T ∈ R 2 n , (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  ∈  R n − 1 . It is well known that un-der the above assumptions, there exits a unique constantvector   ∗ = [  ∗ T1   ∗ 0   ∗ T2  k ∗ ] T ∈  R 2 n with   ∗ 1  ∈  R n − 1 ,  ∗ 0  ∈  R ,   ∗ 2  ∈  R n − 1 and  k ∗ =  k M  /k p  ∈  R , such that, mod-ulo exponentially decaying terms due to initial conditions, y  =  G p (s) [  ∗ T  ] =  M(s) [ r ] =  y M   (Narendra & Annaswamy,1989). Since the plant parameters are assumed to be uncertain,the constant vector  ˆ   in (4) is then defined as ˆ   = [ˆ  T1  ˆ  0  ˆ  T2  ˆ k ] T :=  ˆ  + = [ ( ˆ  + 1  ) T  ˆ  + 0  ( ˆ  + 2  ) T  ˆ k + ] T if   k p  > 0 , ˆ  − = [ ( ˆ  − 1  ) T  ˆ  − 0  ( ˆ  − 2  ) T  ˆ k − ] T if   k p  < 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   +  d  f   +  u vs ] +  ǫ , (8)where  ˜   := ˆ  −  ∗ =[˜  T1  ˜  0  ˜  T2  ˜ k ] T ,  d  f   :=  ( 1 − d  1 (s)) [ d  ]  with d  1 (s)  := ˆ  T1 adj ( sI   −   )b , and  ǫ  decays to zero exponentiallydue to initial conditions.When  n ∗ =  1, (3) implies that  M(s)  =  k M  /(s  +   ) ,   > 0.Therefore, from (8) we have ˙ e  = −  e  +  k p ( ˜  T   +  d  f   +  u vs )  +  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 ¯   + ¯ d  f   + ¯ u vs ] +  ǫ , (10)where the polynomial L(s)  := n ∗ − 1  i = 1 L i (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  y a  and the auxiliaryerrors  e 0 ,...,e n ∗ − 1  as y a  :=  M(s)L(s) [ˆ k − 1 ( ¯ u vs  −  u 0 ) ] , (12) e 0  :=  e  −  y a  =  M(s)L(s) [ˆ k − 1 (   +  u 0 ) ] +  ǫ , (13) e i  :=  L − 1 i  (s) [ u i ] −  F  − 1 (s) [ u i − 1 ]= L − 1 i  (s) [ F  − i (s)L 1 ,i (s) [  ] +  u i ] +  ε i , i  =  1 , 2 ,...,n ∗ −  2, (14) e n ∗ − 1  :=  L − 1 n ∗ − 1 (s) [ u n ∗ − 1 ] −  F  − 1 (s) [ u n ∗ − 2 ]=  L − 1 n ∗ − 1 (s) [ ( ˆ k/k ∗ )(F  − (n ∗ − 1 ) (s) [˜  T   +  d  f  ] +  u n ∗ − 1 ) ]+    +  ε n ∗ − 1  (u vs  =  u n ∗ − 1 ) , (15)where  u 0 ,...,u n ∗ − 1  are the VS control signals to be designed, ˆ k( = k M  / ˆ k p )  is given in (7),   :=  ( ˆ k/k ∗ ) ˜  T ¯   +  ( ˜ k/k ∗ ) ¯ u vs  +  ( ˆ k/k ∗ )  ¯ d  f   (16)with  ˜ k  = ˆ k  −  k ∗ (see (8)), the averaging filter F  − 1 (s)  :=  (  s  +  1 ) − 1 (17)with    a small positive constant, and L 1 ,i (s)  = i  k = 1 L k (s) , (18) ε 1  =  F  − 1 (s) [ ( ˆ k(M(s)L(s)) − 1 [ ǫ  −  e 0 ] ) ] , (19) ε j   =  L j  − 1 (s)F  − 1 (s) [ ε j  − 1  −  e j  − 1 ] , j   =  2 ,...,n ∗ −  1, (20)   =  ( ˜ k/k ∗ )L − 1 n ∗ − 1 (s) [ (F  − (n ∗ − 1 ) (s) [ u n ∗ − 1 ] −  u n ∗ − 1 ) ] . (21)The final control signal  u vs  is thus obtained recursively by(13)–(15) as  u vs  =  u n ∗ − 1 .The following lemma summarizes the main results when thesign of   k p  is known. Lemma 1.  Let the plant   (1)  satisfy  (A1), (A3),  and the sign of  k p  be known. The VS-MRRC has the following properties :(1)  If   n ∗ =  1,  let the VS control signal in  (8)  be u vs  =  − f  sgn (e),  if   sgn (k p )> 0 ,f  sgn (e),  if   sgn (k p )< 0 ,  (22) where the modulation function is chosen such that  f > |˜  T   +  d  f  | .  Then ,  the tracking error converges to zero 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 u 0  =  − f  0  sgn (e 0 ),  if   sgn (k p )  =  sgn ( ˆ k)> 0 ,f  0  sgn (e 0 ),  if   sgn (k p )  =  sgn ( ˆ k)< 0 , (23) u i  = − f  i  sgn (e i ), i  =  1 ,...,n ∗ −  1, (24) where the modulation functions are chosen such that  f  0  > |  | ,  f  i  > | F  − i (s)L 1 ,i (s) [  ]| ,  for   i  =  1 ,...,n ∗ −  2, and   f  n ∗ − 1  > | F  − (n ∗ − 1 ) (s) [˜  T  + d  f  ]| ,  respectively. Then ,  for sufficiently small   ,  all signals of the closed-loopsystem are in  L ∞  and the auxiliary errors  e i ,  for  i = 0 ,...,n ∗ − 2,  converge to zero in finite time ,  while thelast auxiliary error   e n ∗ − 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  ˆ k  :=  k M  / ˆ k p , where  k M   > 0 (see (3)), wehave sgn ( ˆ k) = sgn ( ˆ k p ) . In particular, if the sign of   k p  is known,sgn (k p )  =  sgn ( ˆ k)  =  sgn ( ˆ k p ) . Remark 2.2.  The modulation function  f   or  f  j   in Lemma 1 canbe obtained by usingthe assumption (A1) and (2), e.g.,  f   canbe chosen as  f   =  (     +  d  fu  +   )  with    an upper boundof   ˜   ,  d  fu  an upper bound of   | d  f  |  and    an arbitrarily smallpositive constant. 3. Main results 3.1. Signals to be switched  Because the sign of   k p  is  unknown , we have to redefine thecontrol signals (22)  (n ∗ =  1 )  and (23)  (n ∗ > 1 )  as u vs (t)  :=  u + vs (t)  = − f  sgn (e),  if   t   ∈ T + ,u − vs (t)  =  f  sgn (e),  if   t   ∈ T − ,  (25)and u 0 (t)  :=  u + 0  (t)  = − f  0  sgn (e 0 ),  if   t   ∈ T + ,u − 0  (t)  =  f  0  sgn (e 0 ),  if   t   ∈ T − ,  (26)respectively, the vector  ˆ   in (7) as ˆ   =  ˆ  + ,  if   t   ∈ T + , ˆ  − ,  if   t   ∈ T − ,  (27)and design a monitoring function to decide when  (u vs ,  ˆ  )((u 0 ,  ˆ  ))  will be switched from  (u + vs ,  ˆ  + )  to  (u − vs ,  ˆ  − )  ( (u + 0  ,  ˆ  + ) to  (u − 0  ,  ˆ  − ) ) 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 ,  ˆ  + )  and  (u − vs ,  ˆ  − )  ( (u + 0  ,  ˆ  + )  and (u − 0  ,  ˆ  − ) ) are applied, respectively, and satisfy T + ∪ T − = [ 0 , ∞ ),  T + ∩ T − =   ; (28)both T + and T − have the form [ t  k ,t  k + 1 )  ∪ [ t  k + 3 ,t  k + 4 ) ··· ∪ [ t  j  ,t  j  + 1 ) , (29)   L. Yan et al. / Automatica 44 (2008) 1036–1044  1039 where  t  k  or  t  j   denotes switching time that will be definedlater. The term  [ t  k ,t  k + 1 )  ∪ [ t  k + 3 ,t  k + 4 )  indicates that  (u vs ,  ˆ  )((u 0 ,  ˆ  ))  switches between  (u + vs ,  ˆ  + )  and  (u − vs ,  ˆ  − )  ( (u + 0  ,  ˆ  + ) and  (u − 0  ,  ˆ  − ) )  alternately . Comparing (22), (23), (7) with (25),(26), (27) it is clear that if the sign of   k p  is known, we need onlyone  u vs  (u 0 )  and one  ˆ  , while if the sign of   k p  is unknown, both (u + vs ,  ˆ  + )  and  (u − vs ,  ˆ  − )  ( (u + 0  ,  ˆ  + )  and  (u − 0  ,  ˆ  − ) ) are needed. 3.2. Monitoring function and switching law Our purpose is to construct a monitoring function to deter-mine when  (u vs ,  ˆ  )  or  (u 0 ,  ˆ  )  will be switched. n ∗ =  1: We consider the following first-order differentialequation: ˙   = −   +  ε, t   ¯ t  0 ,   ( ¯ t  0 )  =  e( ¯ t  0 ) , (30)which is obtained by ignoring the terms related to  k p  in (9),i.e.,  k p ( ˜  T   +  d  f   +  u vs ) , by intuition that  if the sign of   k p  isestimated correctly ,  e  will converge faster than   . Indeed, let z  :=  e  −   , then it follows from (9) and (30) that ˙ z  = −  z  +  k p ( ˜  T   +  d  f   +  u vs ) . (31)If we  correctly estimate the sign of   k p  for all  t   ¯ t  0 , substituting(22) in (31) and noting that   ( ¯ t  0 )  =  e( ¯ t  0 )  it follows that   =  e  −  k p    t  ¯ t  0 exp [−  (t   −   ) ][ ( ˜  T   +  d  f   +  u vs )(  ) ] d  = e  +    t  ¯ t  0 exp [−  (t   −   ) ]× [| k p | f  sgn (e)  −  k p ( ˜  T   +  d  f  ) ] (  ) d  . (32)Since  f(f > |˜  T  + d  f  | )  in (22) can completely dominate theterm  ˜  T   +  d  f  , it is easy to check that for all  t   ¯ t  0 , | 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   ¯ t  0 , |  (t) |  exp [−  (t   − ¯ t  0 ) ]| e( ¯ t  0 ) |+    t  ¯ t  0 exp [−  (t  −  ) ]| ε(  ) | d   exp [−  (t  −¯ t  0 ) ]| e( ¯ t  0 ) |+  c m  exp [−  m (t   − ¯ t  0 ) ] exp ( −  ¯ t  0 ) , (35)where c m  =  2 c/ |   −   | ,   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   − ¯ t  0 ) ]| e( ¯ t  0 ) | +  c m  exp ( −  t), t   ¯ t  0 . (37)Recalling that the inequality (33) holds once the sign of   k p is correctly estimated, it seems natural to use    as a benchmark to decide whether a switching of   (u vs ,  ˆ  )  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  c m  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   −  t  k ) ]| e(t  k ) | +  c k  exp ( −  k t) , t   ∈ [ t  k , t  k + 1 ), k  =  0 , 1 , ···;  t  0  :=  0, (38)where  t  k  is the switching time,  c k  is any monotonically increas-ing positive sequence satisfying c k  → ∞  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  c m  by   k  and c k , respectively, and by introducing the switching of    k  and c k . Note that  c k  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  (u 0 ,  ˆ  ) , it is invalid to supervise the behavior of   e 0  because thesign of   ˆ k( sgn ( ˆ k) = sgn ( ˆ k p ),  see Remark 2 . 1 )  is known. How-ever, it is clear that an  incorrect   sign of   ˆ k  results in  ˆ k/k ∗ = k p / ˆ k p  < 0, where  ˆ k/k ∗ is the HFG of (15). In that case,  e n ∗ − 1 should diverge under the control (24). Note that from (7), the switching of   ˆ   includes the switching of   ˆ k  and therefore ˆ k/k ∗ > 0 whenever the sign of   ˆ   is chosen  correctly . Hence, todetermine the switching of   (u 0 ,  ˆ  ) , a monitoring function su-pervising  e n ∗ − 1  will suffice. Again, on the assumption that thesign of   k p  has been  correctly estimated for all  t   ¯ t  0 , in a sim-ilar way to obtain (30), multiplying  L n ∗ − 1 (s)  =  s  +   n ∗ − 1  onboth sides of (15) and ignoring the terms related to  ˆ k/k ∗ , wecan construct the following differential equation: ˙   = −   n ∗ − 1   +  ( ˜ k/k ∗ )(   −  u n ∗ − 1 ) +  ( ˙ ε n ∗ − 1  +   n ∗ − 1 ε n ∗ − 1 ),   ( ¯ t  0 )  =  e n ∗ − 1 ( ¯ t  0 ) , (41)where (21) has been used and    :=   t  ¯ t  0 F (t   −   )u n ∗ − 1 (  ) d  with F (t)  the inverse  Laplace  transform of   F  − (n ∗ − 1 ) (s) . Then,similar to the analysis of (32), | e n ∗ − 1 (t) |  |  (t) | , t   ¯ t  0 . (42)Solving (41) and noting that   ( ¯ t  0 )  =  e n ∗ − 1 ( ¯ t  0 ) , it follows that |  (t) |  exp [−  n ∗ − 1 (t   − ¯ t  0 ) ]| e n ∗ − 1 ( ¯ t  0 ) | + |  (t) | + |¯ ε n ∗ − 1 | , t   ¯ t  0 , (43)where  (t)  =  ( ˜ k/k ∗ )    t  ¯ t  0 exp [−  n ∗ − 1 (t   −   ) ] (   −  u n ∗ − 1 )(  ) d  , (44) ¯ ε n ∗ − 1  =  ε n ∗ − 1  −  exp [−  n ∗ − 1 (t   − ¯ t  0 ) ] ε n ∗ − 1 ( ¯ t  0 ) . (45)  1040  L. Yan et al. / Automatica 44 (2008) 1036–1044  0  1 et t  1 0Fig. 1. The trajectories of    k (t)  and  e(t)  intersect at  t  1  and therefore, aswitching of   (u vs ,  ˆ  )  from  (u + vs ,  ˆ  + )  to  (u − vs ,  ˆ  − )  (or  (u − vs ,  ˆ  − )  to  (u + vs ,  ˆ  + ) )occurs; meanwhile,   k (t)  switches from   0 (t)  to   1 (t)  according to (38). As will be shown later, both  ε n ∗ − 1  and  ¯ ε n ∗ − 1  decay to zeroexponentially and are unknown due to the unknown  ǫ  in (10).Hence, by using the way to obtain the term  c m  exp ( −  t)  in(37),  |¯ ε n ∗ − 1 |  can be expressed as  |¯ ε n ∗ − 1 |  c n ∗ − 1  exp ( −  n ∗ − 1 t) ,where  c n ∗ − 1 and   n ∗ − 1  are positive constants. The same as (38),a monitoring function is constructed based on (43):  k  =  exp [−  n ∗ − 1 (t   −  t  k ) ]| e n ∗ − 1 (t  k ) | +  c k  exp [−  k t  ] +   , t   ∈ [ t  k ,t  k + 1 ), k  =  0 , 1 , ···;  t  0  :=  0, (46)where  c k  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,  e n ∗ − 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   ˆ k p  is chosen, we have | e n ∗ − 1 (t) |  EXP  +   C , where it can be proved that  u n ∗ − 1  ∈ L ∞ and  |  (t) |   C  (u n ∗ − 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(e n ∗ − 1 ) (Filippov, 1964), we always have  | e(t  k ) | <  k (t  k ) ( | e n ∗ − 1 (t  k ) | <  k (t  k ))  at  t   =  t  k . Hence, the next switching time  t  k + 1  for (u vs ,  ˆ  )  and   k  ( (u 0 ,  ˆ  )  and   k ) is well-defined: t  k + 1 =  min { t   :  t >t  k ,  | e(t) |=  k (t),  for  n ∗ = 1 ;| e n ∗ − 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) ( | e n ∗ − 1 (t  k ) | <  k (t  k ))  is violated at some finitepoint  t   =  t  k + 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   (u vs ,  ˆ  )  or   (u 0 ,  ˆ  )  be defined by  (47)  with u vs ,u 0  and   ˆ   given by  (25)–(27).  Then the switching algorithmguarantees that  (1)  If   n ∗ = 1,  only finite number of switching of   (u vs ,  ˆ  )  occurs , all signals of the system are in L ∞  and the tracking error converges 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   (u 0 ,  ˆ  )  occurs , all signals of the system are in L ∞  and the tracking error converges exponentially to a residual set whose size is proportional to   . Proof.  (1) The proof is achieved by contradiction. Suppose (u vs ,  ˆ  )  switches between  (u + vs ,  ˆ  + )  and  (u − vs ,  ˆ  − )  without stop-ping. Because  c m  and    given by (36) and (34), respectively,are constant, and  (u vs ,  ˆ  )  has only two choices,  (u + vs ,  ˆ  + ) and (u − vs ,  ˆ  − ) , and changes  alternately , the following conditionsmust hold after some  k  th switching:  (u vs ,  ˆ  )  has a correct sign( (u vs ,  ˆ  ) = (u + vs ,  ˆ  + )  if   k p  > 0 or  (u vs ,  ˆ  ) = (u − vs ,  ˆ  − ) if   k p  < 0),and c m  =  2 c/ |   −   | <c k , (48)exp ( −  t)< exp ( −  k t),  ∀ t >t  k , (49)where  c k  and   k  are defined by (39) and (40), respectively, and t  k  is the switching time. The above conditions, together with(37), imply that  |  (t) | <  k (t) ,  ∀ t >t  k , where we have replaced ¯ t  0  by  t  k . Since, however, for a correct choice of the sign of   k p ,  e satisfies (33), the inequality  |  (t) | <  k (t) ,  ∀ t >t  k  implies that | e(t) | <  k (t),  ∀ t >t  k . (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   (u 0 ,  ˆ  ) . For a properlychosen   > 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  ˆ  + ,  ˆ  − are con-stant vectors, from (15) and (24) with  f  n ∗ − 1  chosen inthe same way as  f   in Remark 2.2, we have   (u vs ) t   ∞  = (u n ∗ − 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);
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