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Observer design for nonlinear discrete-time systems.pdf

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INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust Nonlinear Control 2010; 20:504–514 Published online 14 April 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.1443 Observer design for nonlinear discrete-time systems: Immersion and dynamic observer error linearization techniques Jian Zhang 1, ∗, † Gang Feng 2 and Hongbing Xu 1 1 School of Automation Engineering, University of Electronic Science and Technology
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  INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL  Int. J. Robust Nonlinear Control  2010;  20 :504–514Published online 14 April 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.1443 Observer design for nonlinear discrete-time systems: Immersion and dynamicobserver error linearization techniques Jian Zhang 1 , ∗ , † Gang Feng 2 and Hongbing Xu 1 1 School of Automation Engineering ,  University of Electronic Science and Technology of China ,  Chengdu , Sichuan 610054 ,  People’s Republic of China 2  Department of Manufacturing Engineering and Engineering Management  ,  The City University of Hong Kong ,  Hong Kong SUMMARYThis paper focuses on the observer design for nonlinear discrete-time systems by means of nonlinear observer canonicalform. At first, sufficient and necessary conditions are obtained for a class of autonomous nonlinear discrete-time systems tobe immersible into higher dimensional observer canonical form. Then a method called dynamic observer error linearizationis developed. By introducing a dynamic auxiliary system, the augmented system is shown to be locally equivalent to thegeneralized observer form, whose nonlinear terms contain auxiliary states and output of the system. A constructive algorithmis also provided to obtain the state coordinate transformation. These results are an extension of their counterparts of nonlinearcontinuous-time systems to nonlinear discrete-time systems ( Syst. Control Lett.  1986;  7 :133–142;  SIAM. J. Control Optim. 2003;  41 :1756–1778;  Int. J. Control  2004;  77 :723–734;  Automatica  2006;  42 :321–328;  IEEE Trans. Automat. Control  2007; 52 :83–88;  IEEE Trans. Automat. Control  2004;  49 :1746–1750;  Automatica  2006;  42 :2195–2200;  IEEE Trans. Automat.Control  1996;  41 :598–603;  Syst. Control Lett.  1997;  31 :115–128). Copyright  q  2009 John Wiley & Sons, Ltd. Received 31 July 2008; Revised 11 December 2008; Accepted 5 January 2009KEY WORDS : nonlinear; discrete-time; observer 1. INTRODUCTIONObserver design for nonlinear systems has been anactive research topic for decades. In general, thereare two main approaches. In the first approach  [ 1–3 ] , ∗ Correspondence to: Jian Zhang, School of Automation Engi-neering, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, People’s Republic of China. † E-mail: zhangj@uestc.edu.cnContract / grant sponsor: Research Grants Council of the HongKong Special Administrative Region of China; contract / grantnumber: CityU-112907 nonlinear systems are brought into a so-called observerform by means of a nonlinear state transformation.This special form yields a completely linear errordynamics of state such that linear observer theorycan be applied. However, for this approach, theconditions for achieving the desired state coordi-nate transformation are in general difficult to satisfy,which requires the solution of a set of partial differ-ential equations. More recently, significant researchefforts have been directed to alleviate the difficultyby developing more generalized state transformationprocedures and more generalized canonical forms sothat the larger class of nonlinear systems can be dealtwith  [ 4,5 ] . Copyright  q  2009 John Wiley & Sons, Ltd.  OBSERVER DESIGN FOR NONLINEAR DISCRETE-TIME SYSTEMS  505On the other hand, the second approach is concernedwith designing observers for nonlinear systems withoutthe need of state transformation. The high-gainobservers in  [ 6–8 ]  are attractive because of theirrobustness. Available design techniques include pole-placement approach  [ 6 ] , Riccati equation approach  [ 7 ] and Lyapunov equation approach  [ 8 ] . However, theirapplication is quite limited due to the strict conditionsthat the system must be in semi-triangular structure.An observer for Lipschitz nonlinear systems has beenstudied in  [ 9,10 ] , where the system was split into alocally or globally Lipschitz part, and a linear partwas assumed to be observable from the output. Theobserver can be designed by solving a Lyapunov equa-tion or a Riccati equation. Unfortunately, the approachsuffers from a limitation that the Lipschitz constanthas to be small.The above results have been obtained for the contin-uous case. Later on, the observer design for nonlineardiscrete-time systems has also attracted much interestof many researchers particularly since digital tech-nology is increasingly used at the implementationstage of model-based control law. Based on geometriccontrol theory, sufficient and necessary conditions weregiven in  [ 11,12 ]  under which a nonlinear discrete-time systems may be transformed into a system inobserver form via a state transformation. In contrast tothese approaches, some of the restrictive assumptionswere relaxed by using past values of the output in [ 13 ] . Another nonlinear observer design method wasproposed in  [ 14 ]  that can be viewed as the discrete-timeversion of the continuous-time observer presented in [ 15 ] . Unfortunately, the restrictive global Lipschitzconditions inevitably arise in the discrete-time versionas well. The paper  [ 16 ]  presented a robust observerfor nonlinear discrete-time systems, which is basedon inverting the map from a past system state to theoutput sequence. An actual estimation is calculatedby repeatedly applying the system function to theestimation of the past system state. In  [ 17 ] , the authorsproposed a nonlinear discrete-time observer based onthe Newton algorithm for the simultaneous solution of a set of nonlinear equations, which yields an asymp-totic observer for a broad class of systems. However,this approach requires time-wise iterative solution of a nonlinear algebraic equation, and the convergenceconditions derived can not be easily checked in prac-tice. In addition, a simple and useful reduced-orderobserver for a large class of nonlinear discrete-timesystems can be found in  [ 18 ] , where a set of asymp-totic convergence conditions for observer errors wasestablished, which appeared to be nonrestrictive in thepresence of mild nonlinearities. The observer designproblem was translated into the problem of solving asystem of first order linear nonhomogeneous functionalequations in  [ 19 ] , and a set of necessary and sufficientconditions for the solvability of these equations wasderived based on functional equations theory.More recently, immersions rather than diffeo-morphism techniques have been investigated in theobserver design for nonlinear continuous-time systems,which is motivated by the fact that there exists aclass of nonlinear systems that cannot be transformedinto observer form via diffeomorphism, but can beimmersed into a higher dimensional observer canonicalform. Several constructive algorithms for immersionare given in  [ 20–24 ] . Similar to the idea of dynamicfeedback, a new framework called dynamic observererror linearization has been introduced for observerdesign recently  [ 25,26 ] . They provide sufficient andnecessary conditions under which nonlinear systemscan be transformed into higher-dimensional observerforms by adding auxiliary dynamic systems such as achain of integrators or a stable linear system. Theseapproaches are recognized as efficient methods torelax the structural restriction in the observer errorlinearization.In this paper, we extend the immersion and dynamicobserver error linearization techniques to nonlineardiscrete-time systems. A number of novel results forobserver design of nonlinear discrete-time system arepresented. Motivated by  [ 27,28 ] , we also provide aconstructive algorithm for dynamic observer errorlinearizationbasedoninput–outputdifferenceequation.The paper is organized as follows. Some prelimi-naries are provided in Section 2. In Section 3, a numberof novel results of observer design for nonlineardiscrete-time systems as well as a constructive algo-rithm for dynamic observer error linearization arepresented. Some illustrative examples are also includedto demonstrate the proposed methods. Conclusions aregiven in Section 4. Copyright  q  2009 John Wiley & Sons, Ltd.  Int. J. Robust Nonlinear Control  2010;  20 :504–514DOI: 10.1002/rnc  506  J. ZHANG, G. FENG AND H. XU 2. PRELIMINARIESConsider a class of nonlinear discrete-time systemsdescribed by the following form  x ( k  + 1 )  =  f   (  x ( k  ))  y ( k  )  =  h (  x ( k  ))  (1)where  x ( k  ) ∈  R n ,  f   :  R n →  R n and  h :  R n →  R , aresmooth functions, with  f   ( 0 ) = 0 and  h ( 0 ) = 0.  Definition 1 Define the observability map   :  R n →  R n by  (  x ) =  dhd  ( h ◦  f   )... d  ( h ◦  f   n − 1 )  (2)where  f   k  denotes the  k  -times composite function  f   ◦···◦  f   ( k   times),andassumetheorigintobeequilibriumpoint of system (1). Then we call system (1) locallyobservable on an open subset  U   ∈  R n containing thesrcin if rank  (  (  x )) = n , ∀  x  ∈ U  .  Definition 2 The following canonical structure is called nonlineardiscrete-time observer canonical form, or in shortobserver form:  z ( k  + 1 )  =  Az ( k  ) +  (  y ( k  ))  y ( k  )  =  Cz ( k  ) (3)where  A  =  0 1  ···  0 ...... ............  10  ··· ···  0  C   = [ 1 0  ···  0 ] where   T =[  1   2  ···   n ]  is a vector of scalar func-tions of   y ( k  ) .If the system (1) can be transformed to the observerform (3) by means of state and output transformations,then an observer can be constructed easily so that theobserver error satisfies a linear dynamical system equa-tion. A necessary and sufficient condition on the exis-tence of state transformations is given in the followinglemma.  Lemma 1  (  Lin and Byrnes  [ 12 ] )Suppose that the nonlinear discrete-time autonomoussystem of the form (1) is observable on an open subset U   ∈  R n containing the srcin. Then system (1) is locallyequivalent to a system in the nonlinear discrete-timeobserver form (3) via a state transformation if and onlyif the Hessian matrix of the function  h ◦  f   n ◦  − 1 ( s )  isdiagonal, where  x  =  − 1 ( s )  is the inverse map of  s =[ h (  x ),  h ◦  f   (  x ), ···  h ◦  f   n − 1 (  x ) ] T (4)  Remark 1 The condition of Lemma 1 is equivalent to  2 ( h ◦  f   n (  x ))  ( h ◦  f   i (  x ))  ( h ◦  f   j (  x )) = 0  ( i ,  j = 0 ,..., n − 1 , i =  j ) It should be noted that a nonlinear discrete-time systemmight not be able to be transformed to the observerform with its present output by a state transformation,but might be transformable with a transformed output,as shown by the following example.  Example 1 Consider a nonlinear discrete-time system defined onan open subset  U  0 ={ (  x 1 ,  x 2 ) :|  x 1 | < 1 , |  x 2 | < 1 }  x 1 ( k  + 1 )  =  x 2 ( k  )  x 2 ( k  + 1 )  =  1 / 2 (  x 1 ( k  ) +  x 22 ( k  ))  y ( k  )  =  sin (  x 1 ( k  )) (5)By means of Lemma 1, the system is not transformableinto an observer form (3) under state transformation,because of   2 (  y ( k  + 2 ))  (  y ( k  + 1 ))  (  y ( k  )) = 0 Copyright  q  2009 John Wiley & Sons, Ltd.  Int. J. Robust Nonlinear Control  2010;  20 :504–514DOI: 10.1002/rnc  OBSERVER DESIGN FOR NONLINEAR DISCRETE-TIME SYSTEMS  507Yet, applying both local output and state transformation  z 1 ( k  )  =  x 1 ( k  )  z 2 ( k  )  =  x 2 ( k  ) − 1 / 2  x 21 ( k  ) ¯  y ( k  )  =  arcsin  y ( k  ) (6)system (5) is equivalent to  z 1 ( k  + 1 )  =  z 2 ( k  ) + 1 / 2 ¯  y 2 ( k  )  z 2 ( k  + 1 )  =  1 / 2 ¯  y ( k  ) ¯  y ( k  )  =  z 1 ( k  ) (7)which is exactly in the observer form (3).3. MAIN RESULTSIn this section, we give two main results of this work. 3.1. An immersion approach to nonlinear discrete-time observer design Let us first consider a motivating example.  Example 2 Consider the discrete-time system  x 1 ( k  + 1 )  = − 1 / 4  x 1 ( k  ) +  x 2 ( k  ) +  x 22 ( k  )  x 2 ( k  + 1 )  =  1 / 2  x 2 ( k  )  y ( k  )  =  x 1 ( k  ) (8)It can be checked that it is not transformable intotwo-dimensional observer form even if output trans-formation and state diffeomorphism are all employed.However, this system is immersible into three-dimensional observer form. In fact, we define animmersion transformation  z =  (  x ) :  R 2 →  R 3 as  z 1 ( k  )  =  x 1 ( k  )  z 2 ( k  )  = − 34  x 1 ( k  ) +  x 2 ( k  ) +  x 22 ( k  )  z 3 ( k  )  =  x 1 ( k  ) 8 − 14  x 2 ( k  ) − 12  x 22 ( k  ) (9)Then the system is immersible into the observer form  z 1 ( k  + 1 )  =  z 2 ( k  ) + 1 / 2  y ( k  )  z 2 ( k  + 1 )  =  z 3 ( k  ) + 1 / 16  y ( k  )  z 3 ( k  + 1 )  = − 1 / 32  y ( k  )  y ( k  )  =  z 1 ( k  ) (10)In this case, the two-dimensional discrete-time systemadmits the following three-dimensional observer: ˆ  z ( k  + 1 )  =  (  A − KC  ) ˆ  z ( k  ) +  (  y ( k  )) ˆ  x 1 ( k  )  = ˆ  z 1 ( k  ),  ˆ  x 2 ( k  ) =ˆ  z 1 ( k  ) + 2 ˆ  z 2 ( k  ) + 4 ˆ  z 3 ( k  ) (11)where  A  =  0 1 00 0 10 0 0  ,  C  =[ 1 0 0 ]   =  12  y ( k  ), 116  y ( k  ), −  132  y ( k  )  T and  K   is a design parameter column vector such that  A − KC   have desired prescribed eigenvalues.Now we introduce a definition on immersion analo-gous to continuous-time systems.  Definition 3 System (1) is said to be immersible into  n + d  dimensional observer form (3) if there exists a  C  ∞ mapping   : D →  ( D ), D ⊂  R n ,  ( D ) ⊂  R n + d  and alocal diffeomorphism   :  I  o →  (  I  o ),  I  o ⊂  R ,  (  I  o ) ⊂  R such that for every initial condition  x 0  and  z 0  with  z 0 =  (  x 0 )  then   ◦ h (  x k  (  x 0 )) = Cz k  (  z 0 )  for every  k  ,0  k   K   x . Here  K   x  = sup { k   0 |  x k  (  x 0 ) ∈ D } .  Remark 2 Definition 3 is the counterpart of the continuous-timeversion for discrete-time systems. For the detaileddiscussion, refer to Definition 2.2 and Theorem 2.3in  [ 21 ] . Theorem 1 System (1) can be immersed into the observer form (3)if and only if there exist a diffeomorphism   , an integer Copyright  q  2009 John Wiley & Sons, Ltd.  Int. J. Robust Nonlinear Control  2010;  20 :504–514DOI: 10.1002/rnc
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