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Improving observer design for discrete-time TS descriptor models under the quadratic framework
Proceedings of Intelligence the 2nd IFAC Conference oninEmbedded Computational and Telematics Control Systems, Computational and Telematics Control Systems, Proceedings of the 2nd IFAC Conference on Embedded Proceedings of Intelligence theMaribor, 2nd IFAC IFAC Conference oninEmbedded Embedded Systems, June 22-24, 2015. Slovenia Proceedings of the 2nd Conference on Systems, Proceedings of Intelligence theMaribor, 2nd IFAC Conference onin June 22-24, 2015. Slovenia Computational and Telematics Control Available online at Systems, www.sciencedirect.com Computational Intelligence and Telematics inEmbedded Control Computational Intelligence and Telematics in Control Computational Intelligence and Telematics in Control June 22-24, 22-24, 2015. 2015. Maribor, Slovenia Slovenia June 22-24, 2015. Maribor, Slovenia June Maribor, June 22-24, 2015. Maribor, Slovenia
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Improving observer design for discrete-time TS descriptor models under the IFAC-PapersOnLine 48-10 (2015) 276–281 Improving observer design for discrete-time TS descriptor models under the quadratic framework Improving for TS framework Improving observer observer design design quadratic for discrete-time discrete-time TS descriptor descriptor models models under under the the quadratic framework quadratic Víctor Estrada-Manzo*, Zsófiaframework Lendek**, Thierry Marie Guerra*
Víctor Estrada-Manzo*, Zsófia Lendek**, Thierry Marie Guerra* Víctor Estrada-Manzo*, Estrada-Manzo*, Zsófia Lendek**, Lendek**, Thierry Marie Marie Guerra* Víctor Víctor Estrada-Manzo*, Zsófia Zsófia Lendek**, Thierry Thierry Marie Guerra* Guerra* *University of Valenciennes and Hainaut-Cambrésis, LAMIH UMR CNRS 8201, *University ofLeValenciennes and Hainaut-Cambrésis, LAMIH UMR. CNRS 8201, Mont Houy, 59313, Valenciennes Cedex 9, France *University of ofLeValenciennes Valenciennes and Hainaut-Cambrésis, Hainaut-Cambrésis, LAMIH UMR. CNRS CNRS 8201, Mont Houy, 59313, Valenciennes Cedex 9, France *University and LAMIH UMR (e-mail:{victor.estradamanzo, guerra}@univ-valenciennes.fr) *University ofLeValenciennes and Hainaut-Cambrésis, LAMIH UMR. CNRS 8201, 8201, Mont Houy, Houy, 59313, 59313, Valenciennes Cedex 9, France France (e-mail:{victor.estradamanzo, guerra}@univ-valenciennes.fr) Le Mont Valenciennes Cedex 9, . **Department ofHouy, Automotion, Technical University ofFrance Cluj-Napoca, Le Mont 59313, Valenciennes Cedex 9, . (e-mail:{victor.estradamanzo, guerra}@univ-valenciennes.fr) **Department of Automotion, Technical University Romania. of Cluj-Napoca, (e-mail:{victor.estradamanzo, guerra}@univ-valenciennes.fr) Memorandumului 28, 400114, Cluj-Napoca, (e-mail:{victor.estradamanzo, guerra}@univ-valenciennes.fr) **Department of Automotion, Automotion, Technical University Romania. of Cluj-Napoca, Cluj-Napoca, Memorandumului 28, 400114, Cluj-Napoca, **Department of Technical University of [email protected]) **Department of(e-mail: Automotion, Technical University Romania. of Cluj-Napoca, Memorandumului 28, 400114, 400114, Cluj-Napoca, (e-mail: [email protected]) Memorandumului 28, Cluj-Napoca, Romania. Memorandumului 28, 400114, Cluj-Napoca, Romania. (e-mail: [email protected]) [email protected]) (e-mail: (e-mail: [email protected]) Abstract: The paper presents a new quadratic Lyapunov function for observer design for discrete-time Abstract: The paper systems. presents The a new quadratic function for observer design for as discrete-time nonlinear descriptor main idea isLyapunov to represent the original nonlinear model a TakagiAbstract: The paper paper systems. presents The a new new quadratic Lyapunov function for observer observer design for as discrete-time nonlinear descriptor main idea is to represent the original nonlinear model a Finsler’s TakagiAbstract: The presents a quadratic Lyapunov function for design for discrete-time Sugeno one and then use Lyapunov’s direct method to design the observer. The well-known Abstract: The paper presents a new quadratic function for observer design for as discrete-time nonlinear descriptor systems. The main idea method isLyapunov to represent represent thethe original nonlinear model a Finsler’s TakagiSugeno one and then use Lyapunov’s direct to design observer. The well-known nonlinear descriptor systems. The main idea is to the original nonlinear model as a TakagiLemma is descriptor used to design a non-Parallel-Distributed-Compensator-like observer together with a quadratic nonlinear systems. The maindirect idea method is to represent thethe original nonlinear model TakagiSugeno oneused andtothen then useaLyapunov’s Lyapunov’s to design design observer. The well-known Finsler’s Lemma is design non-Parallel-Distributed-Compensator-like observer together withasa aquadratic Sugeno one and use direct method to observer. well-known Finsler’s Lyapunov function. This procedure yields design conditions in the terms of linearThe matrix inequalities. The Sugeno one and then use Lyapunov’s direct method to design the observer. The well-known Finsler’s Lemma is used used to design design non-Parallel-Distributed-Compensator-like observer together with aa quadratic quadratic Lyapunov function. This procedure yields design conditions in terms of lineartogether matrix inequalities. The Lemma is to aaa non-Parallel-Distributed-Compensator-like observer with effectiveness of the proposed approaches is illustrated via numerical examples. Lemma is used to design non-Parallel-Distributed-Compensator-like observer together with a quadratic Lyapunov function. This procedure procedure yields design conditions conditions in terms terms of linear linear matrix matrix inequalities. inequalities. The The effectiveness of the proposed approaches is illustrated via numerical examples. Lyapunov function. This yields design in of Lyapunov function. This procedure yields design Control) conditions in terms of linear inequalities. effectiveness of the proposed proposed approaches is illustrated illustrated via numerical numerical examples. © 2015, IFAC (International Federation of Automatic Hosting by Elsevier Ltd.matrix All rights reserved. The effectiveness of the approaches is via examples. effectiveness of the proposed approaches is illustrated examples. via wasnumerical first introduced in (Taniguchi et al., 1999) to represent 1. INTRODUCTION was first introduced inmodels (Taniguchi al., 1999) represent nonlinear descriptor whichet appear in to mechanical 1. INTRODUCTION was first introduced in (Taniguchi et al., 1999) to represent nonlinear descriptor models which appear in mechanical was first introduced in (Taniguchi et al., 1999) to represent 1. INTRODUCTION INTRODUCTION systems (Luenberger, 1977). An exact TS representation of a 1. Takagi-Sugeno models (Takagi and Sugeno, 1985) have nonlinear was first(Luenberger, introduced in1977). (Taniguchi et appear al., 1999) to represent descriptor models which in mechanical systems An exact TS representation 1. INTRODUCTION nonlinear descriptor models which appear in mechanical Takagi-Sugeno models (Takagi and Sugeno, 1985) have numberofofa model withdescriptor several pmodels nonlinear termsappear gives inr mechanical become an interesting alternative for the analysis and systems nonlinear which (Luenberger, An exact representation r numberof ofa model with several p1977). nonlinear termsTS gives Takagi-Sugeno models (Takagi (Takagi and for Sugeno, 1985) have have p systems 1977). exact TS representation of become an interesting alternative the analysis and Takagi-Sugeno models and Sugeno, 1985) rules (with r(Luenberger, 2 several it canAn reach computational controller/observer synthesis for nonlinear models; this ishave due model systems (Luenberger, Aneasily exact TS representation ofofaa p ), thusp1977). Takagi-Sugeno models (Takagi and Sugeno, 1985) r number nonlinear terms gives r number of model with several p nonlinear terms gives become an an interesting interesting alternative for models; the analysis analysis and rules ( r 2 Since ), thusits itTS can easily structure reach computational controller/observer synthesis nonlinear this is and due model become alternative for the intractability. descriptor to their convex structure thatforallows using theanalysis Lyapunov’s rseparates number the of with several p nonlinear terms reach gives computational pp become an interesting alternative for the and p rules ( r 2 ), thus it can easily controller/observer synthesis for nonlinear models; this is due intractability. Since itsthe TS two descriptor structure separates the to theirmethod convex structure that allows using the Lyapunov’s rules (( rr 22 pterms ), thus it can easily reach computational controller/observer synthesis for nonlinear models; this is due non-constant in sides of the system (Estradadirect (Tanaka and Wang 2001). Moreover, when the rules ), thus it can easily reach computational controller/observer synthesis forallows nonlinear models; this is due intractability. terms Since in itsthe TS two descriptor structure separates the to their theirmethod convex structure that using the Lyapunov’s Lyapunov’s sides the system (Estradadirect (Tanaka andthat Wang 2001). Moreover, the non-constant intractability. TS descriptor structure separates the to convex structure allows using the Manzo et al., Since 2014b;its Guelton et al., of 2008; Taniguchi et al., sector nonlinearity is used, the resulting TS model iswhen an exact intractability. Since itsGuelton TS two descriptor structure separates the to their convex structure that allows using the Lyapunov’s non-constant terms in the sides of the system (Estradadirect method (Tanaka and Wang 2001). Moreover, when the Manzo et al., 2014b; et al., 2008; Taniguchi et al., non-constant terms in the two sides of the system (Estradasector nonlinearity is used, the resulting TS model iswhen anetexact direct method (Tanaka and Wang 2001). Moreover, the keeping the descriptor form reduce the computational representation of the original nonlinear one (Ohtake al., 2000), non-constant terms in the two sides of the system (Estradadirect method (Tanaka and Wang 2001). Moreover, when the Manzo keeping et al., al., 2014b; 2014b; Gueltonform et al., al.,reduce 2008;the Taniguchi et al., al., sector nonlinearity nonlinearity is used, used, the resulting resulting TS model is an anetexact the descriptor computational Manzo et Guelton et 2008; Taniguchi et representation of the original nonlinear one (Ohtake al., 2000), sector is the TS model is burden. 2001). nonlinearity A TS model isused, a blending of local linear models and Manzo et al., 2014b; Gueltonform et al.,reduce 2008;the Taniguchi et al., sector isis the resulting TS model is anetexact exact 2000), keeping keeping the descriptor descriptor computational representation of the the original nonlinear one (Ohtake al., burden. 2000), the form reduce the computational 2001). A TS model a blending of local linear models and representation of original nonlinear one (Ohtake et al., nonlinear membership functions (MFs) (Lendek et al., et 2010; keeping the descriptor form reduce the computational representation of theis original nonlinear one (Ohtake al., 2000), burden. 2001). A A TS TS model blending of local local linear models and burden. nonlinear membership (MFs) (Lendek et al., 2010; In most applications not all the states are available for control 2001). model is aafunctions blending of linear models and Tanaka and Wang, 2001). The main design goal is to get In burden. 2001). A TS model is a blending of local linear models and most applications not all the statestoareestimate availablethe formissing control nonlinearand membership functions (MFs) (Lendek et al., al., 2010; Tanaka The matrix main design goal is (LMIs) to get purposes; an observer is needed nonlinear membership functions (MFs) (Lendek et 2010; conditions inWang, terms2001). of linear inequalities In most applications not all the states are available for control nonlinear membership functions (MFs) (Lendek et al., 2010; purposes; an observer is needed to estimate the missing In most applications not all the states are available for control Tanaka and andinWang, Wang, 2001). The matrix main design design goal is is (LMIs) to get get states. conditions terms of linear inequalities Inapplications this paper, the main idea is toare design an observer via Tanaka 2001). The main goal to In most not all the states available for control (Boyd et al., 1994; Scherer and Weiland, 2005). purposes; an observer is needed to estimate the missing Tanaka Wang, 2001). TheWeiland, main design goal is (LMIs) to get states. In this paper, thesimilar main idea to design observer purposes; an observer is needed to estimate the missing conditions in1994; terms of linear linear matrix inequalities (Boyd etand al.,in Scherer and 2005). a Lyapunov function as is Case 1 in an (Lendek et via al. conditions terms of matrix inequalities (LMIs) purposes; an observer is needed to estimate the missing states. In this this paper, paper, thesimilar main idea idea is to design design an observer via conditions in1994; terms of linear matrix 2005). inequalities (LMIs) astates. Lyapunov function as is Case 1 in an (Lendek etwith al. In the main to observer via (Boyd et al., al.,et Scherer andso-called Weiland, 2015). Using the well-known Finsler’s Lemma together In (Wang al., Scherer 1996) the Parallel-Distributed(Boyd et 1994; and Weiland, 2005). states. In this paper, the main idea is to design an observer via a Lyapunov Lyapunov function similar Finsler’s as Case Case Lemma in (Lendek (Lendek etwith al. (Boyd et al.,et1994; andso-called Weiland, Parallel-Distributed2005). 2015). the well-known together In (Wang al., Scherer 1996) the function similar 111 in et al. aUsing non-PDC-like observeras can be designed. Although it Compensator (PDC) together with a quadratic Lyapunov aaQLFs, Lyapunov function similar ascan Case in (Lendek etwith al. 2015). Using the well-known Finsler’s Lemma together In (Wang et al., 1996) the so-called Parallel-DistributedQLFs, a non-PDC-like observer be designed. Although it 2015). Using the well-known Finsler’s Lemma together with Compensator (PDC) together with quadratic Lyapunov In (Wang et al., 1996) the so-called Parallel-Distributedwell know that NQLFs are more relaxed than QLFs, wewith are function (QLF) were introduced for acontrol purposes. The is 2015). Using the well-known Finsler’s Lemma together In (Wang et al., 1996) the so-called Parallel-DistributedQLFs, non-PDC-like observer canrelaxed be designed. designed. Although it Compensator (PDC) together with quadratic Lyapunov well aaknow areQLFs more than are QLFs, non-PDC-like can be Although it function (QLF)(PDC) weretreated introduced for aacontrol purposes. The is Compensator together with quadratic Lyapunov interested in that the NQLFs studyobserver of because theyQLFs, give we a coobserver design was in (Bergsten and Driankov, 2002; QLFs, non-PDC-like observer canrelaxed be designed. Although it Compensator together with quadratic Lyapunov is well well aknow know that NQLFs areQLFs more than QLFs, we are function design (QLF)(PDC) weretreated introduced for acontrol control purposes. The interested in the study of because they give a cois that NQLFs are more relaxed than QLFs, we are observer was in (Bergsten and Driankov, 2002; function (QLF) were introduced for purposes. The negativity problem of 3 sums, which is less computationally Palm and(QLF) Driankov, 1999). The use of purposes. non-quadratic is well know that NQLFs are more relaxed than QLFs, we are function were introduced for control The interested problem in the the study study of QLFs QLFs because they give aa cocoobserver design was treated treated in (Bergsten (Bergsten andofDriankov, Driankov, 2002; negativity of 3 sums, which is less they computationally in of because give Palm anddesign Driankov, 1999). Theallows use non-quadratic observer was in and 2002; complex than NQLF. Lyapunov functions (NQLF) reducing the interested interestedthan in NQLF. the study of QLFs because give a coobserver was treated in (Bergsten andofDriankov, 2002; negativity problem of 33 sums, sums, which is less less they computationally Palm and anddesign Driankov, 1999). Theallows use non-quadratic complex Lyapunov functions (NQLF) reducing the Palm Driankov, 1999). The use of non-quadratic negativity problem of which is computationally conservativeness of QLF; however, in the continuous-time negativitythan problem of 3 sums, which is less computationally Palm and Driankov, 1999). Theallows non-quadratic complex NQLF. Lyapunov functions (NQLF) reducing the complex conservativeness ofmust QLF; however, inusetheofcontinuous-time Summarizing, the aims of this work are: given a discrete-time Lyapunov functions (NQLF) allows reducing the case, researchers face the difficulty of the timecomplex than than NQLF. NQLF. Lyapunov functions (NQLF) allows reducing the Summarizing, the aims this work are: given a discrete-time conservativeness of QLF; however, in the continuous-time case, researchers must face the Guerra difficulty ofBlanco the timeTS descriptor model, 1)ofdesign non-PDC-like observer for conservativeness of QLF; however, in the continuous-time derivatives of the MFs (Bernal and 2010; et al. TS Summarizing, the aims1)of ofdesign this work work are: given given aa discrete-time discrete-time conservativeness of QLF; however, in the continuous-time descriptor model, non-PDC-like observer for Summarizing, the aims this are: case, researchers must face the difficulty of the timederivatives of the MFs (Bernal and Guerra 2010; Blanco et al. such a TS model, 2) introduce a new structure on the case, researchers must face the difficulty of the timeSummarizing, the aims of this work are: given a discrete-time 2001; Tanaka et al. must 2003).face However, in the discrete-time case such TS descriptor descriptor model,2)1) 1) introduce design non-PDC-like non-PDC-like observer for case, researchers the difficulty of the timea TS model, a new structure on the TS model, design observer for derivatives of the MFs (Bernal and Guerra 2010; Blanco et al. 2001; Tanaka etyielded al. 2003). However, in the2010; discrete-time case Lyapunov function to design perform the observer design derivatives of MFs (Bernal and Guerra Blanco et al. TS descriptor model,2)1) non-PDC-like observer for the NQLF hasthe successful results (Ding et al. 2006; such a TS model, introduce a new structure on the derivatives of the MFs (Bernal and Guerra 2010; Blanco et al. Lyapunov function to perform the observer design such a TS model, 2) introduce a new structure on the 2001; Tanaka etyielded al. 2003). 2003). However, in the the(Ding discrete-time case conditions, the NQLF has successful results et al. 2006; 3) illustrate the advantages via numerical 2001; Tanaka et al. However, in discrete-time case such a TS model, 2) introduce a new structure on the GuerraTanaka and Vermeiren 2004; Kruszewski etdiscrete-time al. 2008; Lendek Lyapunov function function to perform perform the observer observer design 2001; etyielded al. 2003). However, in theet case conditions, 3) illustrate the advantages via numerical Lyapunov to the design the NQLF NQLF has successful results (Ding et al. al.Lendek 2006; Guerra and has Vermeiren Kruszewski al. 2008; examples. the yielded2004; successful results (Ding et 2006; Lyapunov function to perform the observer design et al.NQLF 2014). conditions, 3) illustrate the advantages via numerical the has yielded successful results (Ding et al. 2006; examples. conditions, 3) illustrate the advantages via numerical Guerra and Vermeiren Vermeiren 2004; 2004; Kruszewski Kruszewski et et al. al. 2008; 2008; Lendek Lendek conditions, 3) illustrate the advantages via numerical et al. 2014). Guerra and examples. Guerra and Vermeiren 2004; Kruszewski et al. 2008; Lendek examples. rest of paper is organized as follows: Section 2 provides et al. al. 2014). 2014). et Recently, via some matrix manipulations (de Oliveira and The examples. rest of paper is organized as follows: 2 provides et al. 2014).via some matrix manipulations (de Oliveira and The Recently, some useful notation and lemmas used Section along the paper; Skelton 2001; Shaked 2001) some works have obtained some The rest of paper is organized as follows: Section 2 provides provides useful notation and lemmas used along the paper; The rest of paper is organized as follows: Section 2 Recently,2001; via some some matrix manipulations (dehave Oliveira and Section Skelton Shaked 2001) some works obtained 3 introduces the discrete-time TS descriptor model Recently, via matrix manipulations (de Oliveira and The paper is organized as follows: Section 2 provides advantages for some the quadratic case altogether(de withOliveira a non-PDC somerest useful notation anddiscrete-time lemmas used along the paper; Recently,2001; via matrix manipulations and some Section 3ofintroduces the TS descriptor model useful notation and lemmas used along the paper; Skelton Shaked 2001) some workswith havea non-PDC obtained advantages for the quadratic case altogether and motives the study of them; Section 4 presents previous Skelton 2001; Shaked 2001) some works have obtained some useful notation and lemmas used along the paper; controller/observer (Jaadari et al.some 2012;works Marquez et al. 2013; Section 3 introduces the discrete-time TS descriptor model Skelton 2001; Shaked 2001) have obtained motives studythe ofand them; Section 4 presents Section discrete-time descriptor model advantages for the the quadratic case altogether with non-PDC controller/observer (Jaadari etcase al. 2012; Marquez et al. 2013; and results in33 introduces thethe literature gives the TS main resultsprevious on the advantages quadratic altogether with aaa non-PDC Section introduces the discrete-time TS descriptor model Marquez, etfor al. the 2014). and motives the study of them; Section 4 presents previous advantages for quadratic case altogether with non-PDC results in the literature and gives the main results on motives the study of them; Section 44 presents previous controller/observer (Jaadari et et al. al. 2012; 2012; Marquez Marquez et et al. al. 2013; 2013; and Marquez, et al. 2014). observer design; Section 5 shows the performance of the the controller/observer (Jaadari and motives the study of them; Section presents previous results in in design; the literature literature and gives the the main results on on the controller/observer (Jaadari et al. 2012; Marquez et al. 2013; observer Section 5 shows the performance of the the gives main Marquez, etthe al. 2014). 2014). proposed approaches via and examples. Section 6 results concludes the Despite allet work mentioned above, there are few results results Marquez, al. results in the literature and gives the main results on the observer design; design; Section shows the the performance of the Marquez,alletthe al. 2014). approaches via examples. Section 6 concludes observer Section 55 shows performance of Despite work mentioned above, areTS few results proposed the paper. referring to TS descriptor systems. This there type of rewriting observer Section 5 shows the performance of the the proposed design; approaches via examples. examples. Section concludes Despite all all the work mentioned mentioned above, there areTS few results paper. proposed approaches via Section 66 concludes the referring to the TS descriptor systems. This there type of rewriting Despite work above, are few results proposed approaches via examples. Section 6 concludes the Despite all the work mentioned above, there are few results paper. referring to TS descriptor descriptor systems. This This type type of of TS TS rewriting rewriting 276 paper. referring TS systems. Copyrightto © 2015 IFAC referring TS descriptor systems. This type of TS rewriting 276 paper. Copyrightto © 2015 IFAC
2405-8963 Copyright©© ©2015, 2015IFAC IFAC (International Federation of Automatic Control) 276 Hosting by Elsevier Ltd. All rights reserved. Copyright 2015 IFAC 276 Copyright 2015 IFAC 276 Peer review© of International Federation of Automatic Copyright ©under 2015responsibility IFAC 276Control. 10.1016/j.ifacol.2015.08.144
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x respectively. Arguments will be omitted when their
2. NOTATION AND TOOLS
meaning is clear.
Throughout the paper the following shorthand notation is used to represent convex sums of matrix expressions: re
r
, v z h z
h i i v k 1 i 1
k
k
When using the sector nonlinearity approach (Ohtake et al. 2001), the p nonlinear terms in the right-hand side of (2) are
,
captured via convex MFs hi z 0 , i 1, 2, , 2 p , 1
h h z 1 , hi z i , r
r 1 l l h i 1 i 1
r
h z 1 . Proceeding similarly with the
i 1 i
vk z 0 , k 1, 2, , 2 pe ,
re
r
v z E x h z A x B u
k k k ij k 1 i 1
Lemma 1 (Relaxation Lemma) (Tuan et al. 2001): Let
i, j 1, 2, , r , k 1, 2, , re be matrices of appropriate
i
(3)
i 1
where matrices Ai , Bi , Ci , i 1, 2, , r , represent the i-th
hi z h j z vk z ijk 0, holds if
linear right-hand side model (3) and Ek , k 1, 2, , re ,
k 1 i 1 j 1
represent the k-th linear left-hand side model of the TS descriptor model. In this work, the MFs depend on the premise variables grouped in the vector z which are
iik 0, i, k ,
(1) 2 k ii ijk ijk 0, i j , k . r 1 Lemma 2 (Finsler’s Lemma) (de Oliveira and Skelton 2001): T nn , and mn such that Let n , rank n ; the following expressions are equivalent:
a) T 0, n : 0, 0 .
assumed to be known. The following example illustrates why it is important to keep the nonlinear descriptor form instead of calculating the standard state space: x A x x B x u , (4) with A x E 1 x A x and B x E 1 x B x .
b) n m : T T 0.
Example 1. Consider a nonlinear descriptor model:
Property 1: Let 0 and be matrices of appropriated size. The following expression holds:
E x x A x x Bu ,
with
T
i
y hi z Ci x ,
re
1
i
r
dimensions. Then
T
v z 1 . This
k 1 k
Using the above methodology, an exact representation of (2) in x is given by the following TS descriptor model:
When double convex sums appear, the following relaxation lemma is employed to drop off the MFs.
r
re
method allows obtaining an exact TS descriptor model of the nonlinear descriptor one (more details are given in the pioneering work (Taniguchi et al. 2000)).
transpose of the symmetric element; for in-line expressions it represents the transpose of the terms on its left-hand side.
r
pe
nonlinear terms in the left-hand side, the MFs are
where i , k , and l are matrices of appropriate dimensions. The subscript h and v denote the associated MF. In matrix expressions, an asterisk denotes the
vhh
277
0
T
1
T
.
T
sin x1 x1 C x , 0
0 B , and 1
Consider a nonlinear descriptor system in discrete-time: y C x x ,
cos x1 1 A x , 1.1 0.5
(5)
1 1 x12 2 . A TS E x 1 1 x12 1 representation in the form (3) gives re 2 due to the
3. PROBLEM STATEMENT E x x 1 A x x B x u
y C x x ,
(2)
nonlinear term 1 1 x12 and r 4 due to the number of nonlinearities cos x1 and sin x1 x1 . Note that in (5) the
where x n is the state vector, u m is the control input vector, y o is the output vector, and is the current
matrix B is constant. Since E x is non-singular, one can
sample. Matrices A x , B x , and C x are
calculate its inverse and then construct a standard TS model: 2 1 1 1 x22 1 x22 1 ; from here, E x 3 4 x22 2 x24 1 1 x22 2
assumed to be bounded and smooth in a compact set of the state space x . This paper considers the descriptor matrix E x as a non-singular matrix at least in a compact set
one can see that at least four different nonlinearities have to be considered, which results in r 16 . Moreover, the
x . In what follows, x and x will stand for x 1 and 277
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V ek ekT ek ekT ek 0.
standard representation contains nonlinear terms in B x E 1 x B which forces the use of sum relaxations for
In order to use Finsler’s Lemma the inequality (10) is expressed as follows:
the controller design. Table I compares the observer design conditions for the standard TS and TS descriptor forms. Approach
No. of sums
No. of LMI conditions
Q Standard TS1 + Lemma 1
3
r 2 1 257
NQ Standard TS2 + Lemma 1
4
r3 r 4112
Q TS Descriptor3 + Lemma 1
3
re r 2 1 33
NQ TS Descriptor3 + Lemma 1
4
T
e 0 e V e (11) 0. Feasible e 0 e solution Thus, using Finsler’s Lemma the inequality constraint (11) together with the equality one (8) gives No 0 (12) 0 Ah Ch Ev 0, No where 2 n n will be defined later on.
3
re r r 132
No
4.1 Previous results
Yes
The quadratic approach in (Estrada-Manzo et al. 2014a) is summarized in the following Lemma :
Table I. Computational Complexity for different results in Example 1.
Lemma 3: (Estrada-Manzo et al. 2014a) The estimation error e is asymptotically stable if there exist matrices P PT 0
Via Table I, one can see that better results may be obtained with TS descriptor models. Therefore, the observer design is done for the TS descriptor models via the direct Lyapunov method. The next section recalls some previous results and gives new LMI conditions for observer design.
and Fhv , such that the following inequality holds P vhh PAh Fhv Ch
An observer for the descriptor model (3) is given by: yˆ Ch xˆ ,
0. PEv E P P T v
(13)
The observer gains are recovered with Lhv P 1 Fhv . The final observer structure is
4. RESULTS Ev xˆ Ah xˆ Bh u y yˆ
(10)
Ev xˆ Ah xˆ Bh u Lhv y yˆ yˆ Ch xˆ .
(6)
(14)
Proof: see Theorem 1 in (Estrada-Manzo et al. 2014b).
where the observer gain may change according to the approach under study.
The non-quadratic case presented in (Estrada-Manzo et al. 2014a) is summarized in the following Lemma.
Defining the estimation error e x xˆ , its dynamics are as follows:
Lemma 4: (Estrada-Manzo et al. 2014a) The estimation error e is asymptotically stable if there exist matrices
Ev e Ah Ch e ,
Ph PhT 0 , H h , Lhv , such that the next inequality holds
(7)
Ph hhhh 0. T T H h Ah Lhv Ch H h Ev Ev H h Ph The final observer structure is
which can be expressed as the following equality constraint Ah Ch
e Ev 0. e
(8)
Ev xˆ Ah xˆ Bh u H h1 Lhv y yˆ
For design purposes consider the following Lyapunov function candidate: V e eT e 0, T 0,
yˆ Ch xˆ .
(15)
(16)
Proof: Recall (12). Choosing the observer gain as H h1 Lhv , the Lyapunov matrix as Ph , and
(9)
where may be constant (quadratic approach) or depend on MFs (non-quadratic approach). The variation of the Lyapunov function (9) writes
T
0 H hT yields (15), thus the proof is ended.
Remark 1: The inequality conditions in Lemma 3 and 4 are easily transformed into LMIs once the MFs are removed. To this end, Lemma 1 could be applied, and many other relaxation lemmas are available (Kim and Lee 2000; Sala and Ariño 2007; Wang et al. 1996).
1
(Lendek et al. 2010) (Guerra et al. 2012) 3 (Estrada-Manzo et al. 2014a) 2
278
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4.2 New quadratic Lyapunov function
G T P 1G H h Ah Lhv Ch
In this subsection a new Lyapunov function is presented. Recall (10) and select G T P 1G , therefore a new Lyapunov function yields V e eT G T P 1Ge 0,
Theorem 1: The estimation error e is asymptotically stable
if there exist matrices P PT 0 , G , and Fhv , such that the following inequality holds
Approach Lemma 3
(19)
Proof: Recall (12). Choosing the observer gain as Lhv ,
Lemma 4
the Lyapunov matrix as G P G , and 0 P T
T
1
yields G T P 1G PAh Fhv Ch
0, PEv EvT P G T P 1G
Theorem 1
(20)
with the change of variable Fhv PLhv . Finally, using Property 1 on the position (1,1) and the Schur complement on the position (2,2) renders (18); thus concluding the proof.
Theorem 2
Since Finsler’s Lemma allows “separating” — in a sense — the observer gain and the Lyapunov matrix (Marquez et al. 2013), a way to take advantage of the classical non-PDC-like observer is obtained by choosing H h1 Lhv ; this result is
H h Ev EvT H hT G
The final observer structure is
(23)
No. of LMI conditions
No. of decision variables 0.5n n 1
re r 2 1
r re n o
0.5r n n 1
re r 3 r
r re n o r n 2 0.5n n 1
re r 2 1
n 2 r re n o 0.5n n 1 n 2 r n 2
re r 2 1
r re n o
5. EXAMPLES The following example is adopted from (Estrada-Manzo et al. 2014a).
Theorem 2: The estimation error e is asymptotically stable
G G T P v hh H h Ah Lhv Ch 0
1
Table II. Computational complexity of the various approaches.
summarized in the following: if there exist matrices P PT 0 , G , H h , and that the following inequality holds
T
The complexity in terms of number of decision variable and LMI conditions is summarized in Table II.
(18)
The observer gains are recovered with Lhv P 1 Fhv . The final observer structure is yˆ Ch xˆ .
0. H h Ev E H G P G T h
Remark 2: Note that the PDC-like observer structure in Theorem 1 is the same as in Lemma 3, while the non-PDClike observer in Theorem 2 is the same as in Lemma 4; but the design procedure is done via different Lyapunov functions. The new Lyapunov function may be less conservative since it naturally introduces more slack variables; it could be seen as the dual of the one presented in (Lendek et al. 2015) for control purposes.
P PT 0 , thus it is non-singular. The following where result can be stated:
Ev xˆ Ah xˆ Bh u Lhv y yˆ
T v
Using Property 1 on the position (1,1) and the Schur complement on the position (2,2) gives (21); thus concluding the proof.
(17)
G GT P v T hh PAh Fhv Ch PEv Ev P 0. P G 0
279
Example 2. (Estrada-Manzo et al. 2014a) Consider a discrete-time TS descriptor model as in (3) with r re 2 ,
Lhv such
0.1 a 0.1 a 0.9 0.9 , E2 , 1.1 1.1 0.4 b 0.4 b 1 1 a 1 1 a , A2 A1 , 1.5 0.5 1.5 0.5
E1 (21) 0.
P
T
T
0 0 C1 , C2 1 b . b 1
Ev xˆ Ah xˆ Bh u H h1 Lhv y yˆ
(22) yˆ Ch xˆ . Proof: Recall (12). Choosing the observer gain as H h1 Lhv , the Lyapunov matrix as G T P 1G , and
The parameters are defined as a 1,1 and b 1,1 . Fig. 1 shows the feasible regions for the proposed approaches. Results obtained via Lemma 3 ( ), via Lemma 4 ( ), and via Theorem 1 ( ).
T
0 H hT yields 279
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Example 2 illustrates the improvements made by the new L12 Lyapunov function. In Fig. 1, one can see that the new Lyapunov function together with a PDC-like observer overcomes the quadratic approach in (Estrada-Manzo et al. 2014a).
0.5
The TS descriptor exactly represents the nonlinear model in the compact set x 2 , thus the designed observer allows the asymptotic converge of the estimation error for any initial condition of the original nonlinear model (see Fig. 2).
b
0 -0.5 -0.5
0 a
0.5
e1
5
-1 -1
1
Using results in Theorem 2, it is possible to obtain the same feasibility set as in Lemma 4 for this specific Example 2. Both conditions allow designing the same observer structure (16). Table II shows that conditions in Theorem 2 are less demanding than Lemma 4.
h4 ;
cos x 1 1
h3 , 2,
2 0
1 1
2 1
2
4
6 8 Sample
10
12
e2
Acknowledgements
h1 0102 ,
This work is supported by the Ministry of Higher Education and Research, the CNRS, the Nord-Pas-de-Calais Region (grant No. 2013_11067), a grant of the Technical University of Cluj-Napoca. The authors gratefully acknowledge the support of these institutions.
with
sin x1 / x1 0.2167 1.2167 , 2 0
11 1 01 , and 12 1 02 . The state variable x1 is
available. Conditions in Lemma 3 and Theorem 1 are not feasible. Feasible results are obtained via Theorem 2 (nonPDC-like observer and the new quadratic Lyapunov function); for brevity, only some of the resulting gains are given: P
12
In this paper, LMI conditions are established for the observer design of TS descriptor models. Using Finsler’s Lemma together with a quadratic Lyapunov function it is possible to design a non-PDC-like observer; thus relaxing previous results. Numerical examples illustrate the advantages of the presented approaches.
T
h2 ,
1 1
10
6. CONCLUSIONS
0 1 0.2167 B , C C C4 , and C 1 3 2 . 1 0 0 2 1
6 8 Sample
Fig. 2. Evolution of the estimation error for several initial conditions for Example 3.
1 1 1 1 r 4, A1 A2 , A3 A4 , 0.5 1.1 0.5 1.1
v2 1 v1 ,
4
0 -5 0
2 1 2 0 , and E2 re 2, E1 ; 1 1 0 1 on the right-hand side:
v1 1 1 x12 , The MFs are:
2
5
Example 3. Recall the nonlinear descriptor in Example 1. Considering the compact set x 2 and using the sector nonlinearity approach a TS descriptor model as in (3) , we obtain: on the left-hand side:
T
0 -5 0
Fig. 1. Feasibility sets for Example 2.
1 0
0.05 0.52 0.04 , L32 0.26 , L41 0.02 , 0.01 0.19 0.68 0.02 , H3 . 1.06 0.32 0.99
For this example, conditions in Lemma 4 are also feasible. However, the number of LMIs in Theorem 2 is 33 while for Lemma 4 is 132 (see Table II).
1
1 0
0.28 , L22 0.11 0.69 H1 0.32
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Improving observer design for discrete-time TS descriptor models under the IFAC-PapersOnLine 48-10 (2015) 276–281 Improving observer design for discrete-time TS descriptor models under the quadratic framework Improving for TS framework Improving observer observer design design quadratic for discrete-time discrete-time TS descriptor descriptor models models under under the the quadratic framework quadratic Víctor Estrada-Manzo*, Zsófiaframework Lendek**, Thierry Marie Guerra*
Víctor Estrada-Manzo*, Zsófia Lendek**, Thierry Marie Guerra* Víctor Estrada-Manzo*, Estrada-Manzo*, Zsófia Lendek**, Lendek**, Thierry Marie Marie Guerra* Víctor Víctor Estrada-Manzo*, Zsófia Zsófia Lendek**, Thierry Thierry Marie Guerra* Guerra* *University of Valenciennes and Hainaut-Cambrésis, LAMIH UMR CNRS 8201, *University ofLeValenciennes and Hainaut-Cambrésis, LAMIH UMR. CNRS 8201, Mont Houy, 59313, Valenciennes Cedex 9, France *University of ofLeValenciennes Valenciennes and Hainaut-Cambrésis, Hainaut-Cambrésis, LAMIH UMR. CNRS CNRS 8201, Mont Houy, 59313, Valenciennes Cedex 9, France *University and LAMIH UMR (e-mail:{victor.estradamanzo, guerra}@univ-valenciennes.fr) *University ofLeValenciennes and Hainaut-Cambrésis, LAMIH UMR. CNRS 8201, 8201, Mont Houy, Houy, 59313, 59313, Valenciennes Cedex 9, France France (e-mail:{victor.estradamanzo, guerra}@univ-valenciennes.fr) Le Mont Valenciennes Cedex 9, . **Department ofHouy, Automotion, Technical University ofFrance Cluj-Napoca, Le Mont 59313, Valenciennes Cedex 9, . (e-mail:{victor.estradamanzo, guerra}@univ-valenciennes.fr) **Department of Automotion, Technical University Romania. of Cluj-Napoca, (e-mail:{victor.estradamanzo, guerra}@univ-valenciennes.fr) Memorandumului 28, 400114, Cluj-Napoca, (e-mail:{victor.estradamanzo, guerra}@univ-valenciennes.fr) **Department of Automotion, Automotion, Technical University Romania. of Cluj-Napoca, Cluj-Napoca, Memorandumului 28, 400114, Cluj-Napoca, **Department of Technical University of [email protected]) **Department of(e-mail: Automotion, Technical University Romania. of Cluj-Napoca, Memorandumului 28, 400114, 400114, Cluj-Napoca, (e-mail: [email protected]) Memorandumului 28, Cluj-Napoca, Romania. Memorandumului 28, 400114, Cluj-Napoca, Romania. (e-mail: [email protected]) [email protected]) (e-mail: (e-mail: [email protected]) Abstract: The paper presents a new quadratic Lyapunov function for observer design for discrete-time Abstract: The paper systems. presents The a new quadratic function for observer design for as discrete-time nonlinear descriptor main idea isLyapunov to represent the original nonlinear model a TakagiAbstract: The paper paper systems. presents The a new new quadratic Lyapunov function for observer observer design for as discrete-time nonlinear descriptor main idea is to represent the original nonlinear model a Finsler’s TakagiAbstract: The presents a quadratic Lyapunov function for design for discrete-time Sugeno one and then use Lyapunov’s direct method to design the observer. The well-known Abstract: The paper presents a new quadratic function for observer design for as discrete-time nonlinear descriptor systems. The main idea method isLyapunov to represent represent thethe original nonlinear model a Finsler’s TakagiSugeno one and then use Lyapunov’s direct to design observer. The well-known nonlinear descriptor systems. The main idea is to the original nonlinear model as a TakagiLemma is descriptor used to design a non-Parallel-Distributed-Compensator-like observer together with a quadratic nonlinear systems. The maindirect idea method is to represent thethe original nonlinear model TakagiSugeno oneused andtothen then useaLyapunov’s Lyapunov’s to design design observer. The well-known Finsler’s Lemma is design non-Parallel-Distributed-Compensator-like observer together withasa aquadratic Sugeno one and use direct method to observer. well-known Finsler’s Lyapunov function. This procedure yields design conditions in the terms of linearThe matrix inequalities. The Sugeno one and then use Lyapunov’s direct method to design the observer. The well-known Finsler’s Lemma is used used to design design non-Parallel-Distributed-Compensator-like observer together with aa quadratic quadratic Lyapunov function. This procedure yields design conditions in terms of lineartogether matrix inequalities. The Lemma is to aaa non-Parallel-Distributed-Compensator-like observer with effectiveness of the proposed approaches is illustrated via numerical examples. Lemma is used to design non-Parallel-Distributed-Compensator-like observer together with a quadratic Lyapunov function. This procedure procedure yields design conditions conditions in terms terms of linear linear matrix matrix inequalities. inequalities. The The effectiveness of the proposed approaches is illustrated via numerical examples. Lyapunov function. This yields design in of Lyapunov function. This procedure yields design Control) conditions in terms of linear inequalities. effectiveness of the proposed proposed approaches is illustrated illustrated via numerical numerical examples. © 2015, IFAC (International Federation of Automatic Hosting by Elsevier Ltd.matrix All rights reserved. The effectiveness of the approaches is via examples. effectiveness of the proposed approaches is illustrated examples. via wasnumerical first introduced in (Taniguchi et al., 1999) to represent 1. INTRODUCTION was first introduced inmodels (Taniguchi al., 1999) represent nonlinear descriptor whichet appear in to mechanical 1. INTRODUCTION was first introduced in (Taniguchi et al., 1999) to represent nonlinear descriptor models which appear in mechanical was first introduced in (Taniguchi et al., 1999) to represent 1. INTRODUCTION INTRODUCTION systems (Luenberger, 1977). An exact TS representation of a 1. Takagi-Sugeno models (Takagi and Sugeno, 1985) have nonlinear was first(Luenberger, introduced in1977). (Taniguchi et appear al., 1999) to represent descriptor models which in mechanical systems An exact TS representation 1. INTRODUCTION nonlinear descriptor models which appear in mechanical Takagi-Sugeno models (Takagi and Sugeno, 1985) have numberofofa model withdescriptor several pmodels nonlinear termsappear gives inr mechanical become an interesting alternative for the analysis and systems nonlinear which (Luenberger, An exact representation r numberof ofa model with several p1977). nonlinear termsTS gives Takagi-Sugeno models (Takagi (Takagi and for Sugeno, 1985) have have p systems 1977). exact TS representation of become an interesting alternative the analysis and Takagi-Sugeno models and Sugeno, 1985) rules (with r(Luenberger, 2 several it canAn reach computational controller/observer synthesis for nonlinear models; this ishave due model systems (Luenberger, Aneasily exact TS representation ofofaa p ), thusp1977). Takagi-Sugeno models (Takagi and Sugeno, 1985) r number nonlinear terms gives r number of model with several p nonlinear terms gives become an an interesting interesting alternative for models; the analysis analysis and rules ( r 2 Since ), thusits itTS can easily structure reach computational controller/observer synthesis nonlinear this is and due model become alternative for the intractability. descriptor to their convex structure thatforallows using theanalysis Lyapunov’s rseparates number the of with several p nonlinear terms reach gives computational pp become an interesting alternative for the and p rules ( r 2 ), thus it can easily controller/observer synthesis for nonlinear models; this is due intractability. Since itsthe TS two descriptor structure separates the to theirmethod convex structure that allows using the Lyapunov’s rules (( rr 22 pterms ), thus it can easily reach computational controller/observer synthesis for nonlinear models; this is due non-constant in sides of the system (Estradadirect (Tanaka and Wang 2001). Moreover, when the rules ), thus it can easily reach computational controller/observer synthesis forallows nonlinear models; this is due intractability. terms Since in itsthe TS two descriptor structure separates the to their theirmethod convex structure that using the Lyapunov’s Lyapunov’s sides the system (Estradadirect (Tanaka andthat Wang 2001). Moreover, the non-constant intractability. TS descriptor structure separates the to convex structure allows using the Manzo et al., Since 2014b;its Guelton et al., of 2008; Taniguchi et al., sector nonlinearity is used, the resulting TS model iswhen an exact intractability. Since itsGuelton TS two descriptor structure separates the to their convex structure that allows using the Lyapunov’s non-constant terms in the sides of the system (Estradadirect method (Tanaka and Wang 2001). Moreover, when the Manzo et al., 2014b; et al., 2008; Taniguchi et al., non-constant terms in the two sides of the system (Estradasector nonlinearity is used, the resulting TS model iswhen anetexact direct method (Tanaka and Wang 2001). Moreover, the keeping the descriptor form reduce the computational representation of the original nonlinear one (Ohtake al., 2000), non-constant terms in the two sides of the system (Estradadirect method (Tanaka and Wang 2001). Moreover, when the Manzo keeping et al., al., 2014b; 2014b; Gueltonform et al., al.,reduce 2008;the Taniguchi et al., al., sector nonlinearity nonlinearity is used, used, the resulting resulting TS model is an anetexact the descriptor computational Manzo et Guelton et 2008; Taniguchi et representation of the original nonlinear one (Ohtake al., 2000), sector is the TS model is burden. 2001). nonlinearity A TS model isused, a blending of local linear models and Manzo et al., 2014b; Gueltonform et al.,reduce 2008;the Taniguchi et al., sector isis the resulting TS model is anetexact exact 2000), keeping keeping the descriptor descriptor computational representation of the the original nonlinear one (Ohtake al., burden. 2000), the form reduce the computational 2001). A TS model a blending of local linear models and representation of original nonlinear one (Ohtake et al., nonlinear membership functions (MFs) (Lendek et al., et 2010; keeping the descriptor form reduce the computational representation of theis original nonlinear one (Ohtake al., 2000), burden. 2001). A A TS TS model blending of local local linear models and burden. nonlinear membership (MFs) (Lendek et al., 2010; In most applications not all the states are available for control 2001). model is aafunctions blending of linear models and Tanaka and Wang, 2001). The main design goal is to get In burden. 2001). A TS model is a blending of local linear models and most applications not all the statestoareestimate availablethe formissing control nonlinearand membership functions (MFs) (Lendek et al., al., 2010; Tanaka The matrix main design goal is (LMIs) to get purposes; an observer is needed nonlinear membership functions (MFs) (Lendek et 2010; conditions inWang, terms2001). of linear inequalities In most applications not all the states are available for control nonlinear membership functions (MFs) (Lendek et al., 2010; purposes; an observer is needed to estimate the missing In most applications not all the states are available for control Tanaka and andinWang, Wang, 2001). The matrix main design design goal is is (LMIs) to get get states. conditions terms of linear inequalities Inapplications this paper, the main idea is toare design an observer via Tanaka 2001). The main goal to In most not all the states available for control (Boyd et al., 1994; Scherer and Weiland, 2005). purposes; an observer is needed to estimate the missing Tanaka Wang, 2001). TheWeiland, main design goal is (LMIs) to get states. In this paper, thesimilar main idea to design observer purposes; an observer is needed to estimate the missing conditions in1994; terms of linear linear matrix inequalities (Boyd etand al.,in Scherer and 2005). a Lyapunov function as is Case 1 in an (Lendek et via al. conditions terms of matrix inequalities (LMIs) purposes; an observer is needed to estimate the missing states. In this this paper, paper, thesimilar main idea idea is to design design an observer via conditions in1994; terms of linear matrix 2005). inequalities (LMIs) astates. Lyapunov function as is Case 1 in an (Lendek etwith al. In the main to observer via (Boyd et al., al.,et Scherer andso-called Weiland, 2015). Using the well-known Finsler’s Lemma together In (Wang al., Scherer 1996) the Parallel-Distributed(Boyd et 1994; and Weiland, 2005). states. In this paper, the main idea is to design an observer via a Lyapunov Lyapunov function similar Finsler’s as Case Case Lemma in (Lendek (Lendek etwith al. (Boyd et al.,et1994; andso-called Weiland, Parallel-Distributed2005). 2015). the well-known together In (Wang al., Scherer 1996) the function similar 111 in et al. aUsing non-PDC-like observeras can be designed. Although it Compensator (PDC) together with a quadratic Lyapunov aaQLFs, Lyapunov function similar ascan Case in (Lendek etwith al. 2015). Using the well-known Finsler’s Lemma together In (Wang et al., 1996) the so-called Parallel-DistributedQLFs, a non-PDC-like observer be designed. Although it 2015). Using the well-known Finsler’s Lemma together with Compensator (PDC) together with quadratic Lyapunov In (Wang et al., 1996) the so-called Parallel-Distributedwell know that NQLFs are more relaxed than QLFs, wewith are function (QLF) were introduced for acontrol purposes. The is 2015). Using the well-known Finsler’s Lemma together In (Wang et al., 1996) the so-called Parallel-DistributedQLFs, non-PDC-like observer canrelaxed be designed. designed. Although it Compensator (PDC) together with quadratic Lyapunov well aaknow areQLFs more than are QLFs, non-PDC-like can be Although it function (QLF)(PDC) weretreated introduced for aacontrol purposes. The is Compensator together with quadratic Lyapunov interested in that the NQLFs studyobserver of because theyQLFs, give we a coobserver design was in (Bergsten and Driankov, 2002; QLFs, non-PDC-like observer canrelaxed be designed. Although it Compensator together with quadratic Lyapunov is well well aknow know that NQLFs areQLFs more than QLFs, we are function design (QLF)(PDC) weretreated introduced for acontrol control purposes. The interested in the study of because they give a cois that NQLFs are more relaxed than QLFs, we are observer was in (Bergsten and Driankov, 2002; function (QLF) were introduced for purposes. The negativity problem of 3 sums, which is less computationally Palm and(QLF) Driankov, 1999). The use of purposes. non-quadratic is well know that NQLFs are more relaxed than QLFs, we are function were introduced for control The interested problem in the the study study of QLFs QLFs because they give aa cocoobserver design was treated treated in (Bergsten (Bergsten andofDriankov, Driankov, 2002; negativity of 3 sums, which is less they computationally in of because give Palm anddesign Driankov, 1999). Theallows use non-quadratic observer was in and 2002; complex than NQLF. Lyapunov functions (NQLF) reducing the interested interestedthan in NQLF. the study of QLFs because give a coobserver was treated in (Bergsten andofDriankov, 2002; negativity problem of 33 sums, sums, which is less less they computationally Palm and anddesign Driankov, 1999). Theallows use non-quadratic complex Lyapunov functions (NQLF) reducing the Palm Driankov, 1999). The use of non-quadratic negativity problem of which is computationally conservativeness of QLF; however, in the continuous-time negativitythan problem of 3 sums, which is less computationally Palm and Driankov, 1999). Theallows non-quadratic complex NQLF. Lyapunov functions (NQLF) reducing the complex conservativeness ofmust QLF; however, inusetheofcontinuous-time Summarizing, the aims of this work are: given a discrete-time Lyapunov functions (NQLF) allows reducing the case, researchers face the difficulty of the timecomplex than than NQLF. NQLF. Lyapunov functions (NQLF) allows reducing the Summarizing, the aims this work are: given a discrete-time conservativeness of QLF; however, in the continuous-time case, researchers must face the Guerra difficulty ofBlanco the timeTS descriptor model, 1)ofdesign non-PDC-like observer for conservativeness of QLF; however, in the continuous-time derivatives of the MFs (Bernal and 2010; et al. TS Summarizing, the aims1)of ofdesign this work work are: given given aa discrete-time discrete-time conservativeness of QLF; however, in the continuous-time descriptor model, non-PDC-like observer for Summarizing, the aims this are: case, researchers must face the difficulty of the timederivatives of the MFs (Bernal and Guerra 2010; Blanco et al. such a TS model, 2) introduce a new structure on the case, researchers must face the difficulty of the timeSummarizing, the aims of this work are: given a discrete-time 2001; Tanaka et al. must 2003).face However, in the discrete-time case such TS descriptor descriptor model,2)1) 1) introduce design non-PDC-like non-PDC-like observer for case, researchers the difficulty of the timea TS model, a new structure on the TS model, design observer for derivatives of the MFs (Bernal and Guerra 2010; Blanco et al. 2001; Tanaka etyielded al. 2003). However, in the2010; discrete-time case Lyapunov function to design perform the observer design derivatives of MFs (Bernal and Guerra Blanco et al. TS descriptor model,2)1) non-PDC-like observer for the NQLF hasthe successful results (Ding et al. 2006; such a TS model, introduce a new structure on the derivatives of the MFs (Bernal and Guerra 2010; Blanco et al. Lyapunov function to perform the observer design such a TS model, 2) introduce a new structure on the 2001; Tanaka etyielded al. 2003). 2003). However, in the the(Ding discrete-time case conditions, the NQLF has successful results et al. 2006; 3) illustrate the advantages via numerical 2001; Tanaka et al. However, in discrete-time case such a TS model, 2) introduce a new structure on the GuerraTanaka and Vermeiren 2004; Kruszewski etdiscrete-time al. 2008; Lendek Lyapunov function function to perform perform the observer observer design 2001; etyielded al. 2003). However, in theet case conditions, 3) illustrate the advantages via numerical Lyapunov to the design the NQLF NQLF has successful results (Ding et al. al.Lendek 2006; Guerra and has Vermeiren Kruszewski al. 2008; examples. the yielded2004; successful results (Ding et 2006; Lyapunov function to perform the observer design et al.NQLF 2014). conditions, 3) illustrate the advantages via numerical the has yielded successful results (Ding et al. 2006; examples. conditions, 3) illustrate the advantages via numerical Guerra and Vermeiren Vermeiren 2004; 2004; Kruszewski Kruszewski et et al. al. 2008; 2008; Lendek Lendek conditions, 3) illustrate the advantages via numerical et al. 2014). Guerra and examples. Guerra and Vermeiren 2004; Kruszewski et al. 2008; Lendek examples. rest of paper is organized as follows: Section 2 provides et al. al. 2014). 2014). et Recently, via some matrix manipulations (de Oliveira and The examples. rest of paper is organized as follows: 2 provides et al. 2014).via some matrix manipulations (de Oliveira and The Recently, some useful notation and lemmas used Section along the paper; Skelton 2001; Shaked 2001) some works have obtained some The rest of paper is organized as follows: Section 2 provides provides useful notation and lemmas used along the paper; The rest of paper is organized as follows: Section 2 Recently,2001; via some some matrix manipulations (dehave Oliveira and Section Skelton Shaked 2001) some works obtained 3 introduces the discrete-time TS descriptor model Recently, via matrix manipulations (de Oliveira and The paper is organized as follows: Section 2 provides advantages for some the quadratic case altogether(de withOliveira a non-PDC somerest useful notation anddiscrete-time lemmas used along the paper; Recently,2001; via matrix manipulations and some Section 3ofintroduces the TS descriptor model useful notation and lemmas used along the paper; Skelton Shaked 2001) some workswith havea non-PDC obtained advantages for the quadratic case altogether and motives the study of them; Section 4 presents previous Skelton 2001; Shaked 2001) some works have obtained some useful notation and lemmas used along the paper; controller/observer (Jaadari et al.some 2012;works Marquez et al. 2013; Section 3 introduces the discrete-time TS descriptor model Skelton 2001; Shaked 2001) have obtained motives studythe ofand them; Section 4 presents Section discrete-time descriptor model advantages for the the quadratic case altogether with non-PDC controller/observer (Jaadari etcase al. 2012; Marquez et al. 2013; and results in33 introduces thethe literature gives the TS main resultsprevious on the advantages quadratic altogether with aaa non-PDC Section introduces the discrete-time TS descriptor model Marquez, etfor al. the 2014). and motives the study of them; Section 4 presents previous advantages for quadratic case altogether with non-PDC results in the literature and gives the main results on motives the study of them; Section 44 presents previous controller/observer (Jaadari et et al. al. 2012; 2012; Marquez Marquez et et al. al. 2013; 2013; and Marquez, et al. 2014). observer design; Section 5 shows the performance of the the controller/observer (Jaadari and motives the study of them; Section presents previous results in in design; the literature literature and gives the the main results on on the controller/observer (Jaadari et al. 2012; Marquez et al. 2013; observer Section 5 shows the performance of the the gives main Marquez, etthe al. 2014). 2014). proposed approaches via and examples. Section 6 results concludes the Despite allet work mentioned above, there are few results results Marquez, al. results in the literature and gives the main results on the observer design; design; Section shows the the performance of the Marquez,alletthe al. 2014). approaches via examples. Section 6 concludes observer Section 55 shows performance of Despite work mentioned above, areTS few results proposed the paper. referring to TS descriptor systems. This there type of rewriting observer Section 5 shows the performance of the the proposed design; approaches via examples. examples. Section concludes Despite all all the work mentioned mentioned above, there areTS few results paper. proposed approaches via Section 66 concludes the referring to the TS descriptor systems. This there type of rewriting Despite work above, are few results proposed approaches via examples. Section 6 concludes the Despite all the work mentioned above, there are few results paper. referring to TS descriptor descriptor systems. This This type type of of TS TS rewriting rewriting 276 paper. referring TS systems. Copyrightto © 2015 IFAC referring TS descriptor systems. This type of TS rewriting 276 paper. Copyrightto © 2015 IFAC
2405-8963 Copyright©© ©2015, 2015IFAC IFAC (International Federation of Automatic Control) 276 Hosting by Elsevier Ltd. All rights reserved. Copyright 2015 IFAC 276 Copyright 2015 IFAC 276 Peer review© of International Federation of Automatic Copyright ©under 2015responsibility IFAC 276Control. 10.1016/j.ifacol.2015.08.144
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x respectively. Arguments will be omitted when their
2. NOTATION AND TOOLS
meaning is clear.
Throughout the paper the following shorthand notation is used to represent convex sums of matrix expressions: re
r
, v z h z
h i i v k 1 i 1
k
k
When using the sector nonlinearity approach (Ohtake et al. 2001), the p nonlinear terms in the right-hand side of (2) are
,
captured via convex MFs hi z 0 , i 1, 2, , 2 p , 1
h h z 1 , hi z i , r
r 1 l l h i 1 i 1
r
h z 1 . Proceeding similarly with the
i 1 i
vk z 0 , k 1, 2, , 2 pe ,
re
r
v z E x h z A x B u
k k k ij k 1 i 1
Lemma 1 (Relaxation Lemma) (Tuan et al. 2001): Let
i, j 1, 2, , r , k 1, 2, , re be matrices of appropriate
i
(3)
i 1
where matrices Ai , Bi , Ci , i 1, 2, , r , represent the i-th
hi z h j z vk z ijk 0, holds if
linear right-hand side model (3) and Ek , k 1, 2, , re ,
k 1 i 1 j 1
represent the k-th linear left-hand side model of the TS descriptor model. In this work, the MFs depend on the premise variables grouped in the vector z which are
iik 0, i, k ,
(1) 2 k ii ijk ijk 0, i j , k . r 1 Lemma 2 (Finsler’s Lemma) (de Oliveira and Skelton 2001): T nn , and mn such that Let n , rank n ; the following expressions are equivalent:
a) T 0, n : 0, 0 .
assumed to be known. The following example illustrates why it is important to keep the nonlinear descriptor form instead of calculating the standard state space: x A x x B x u , (4) with A x E 1 x A x and B x E 1 x B x .
b) n m : T T 0.
Example 1. Consider a nonlinear descriptor model:
Property 1: Let 0 and be matrices of appropriated size. The following expression holds:
E x x A x x Bu ,
with
T
i
y hi z Ci x ,
re
1
i
r
dimensions. Then
T
v z 1 . This
k 1 k
Using the above methodology, an exact representation of (2) in x is given by the following TS descriptor model:
When double convex sums appear, the following relaxation lemma is employed to drop off the MFs.
r
re
method allows obtaining an exact TS descriptor model of the nonlinear descriptor one (more details are given in the pioneering work (Taniguchi et al. 2000)).
transpose of the symmetric element; for in-line expressions it represents the transpose of the terms on its left-hand side.
r
pe
nonlinear terms in the left-hand side, the MFs are
where i , k , and l are matrices of appropriate dimensions. The subscript h and v denote the associated MF. In matrix expressions, an asterisk denotes the
vhh
277
0
T
1
T
.
T
sin x1 x1 C x , 0
0 B , and 1
Consider a nonlinear descriptor system in discrete-time: y C x x ,
cos x1 1 A x , 1.1 0.5
(5)
1 1 x12 2 . A TS E x 1 1 x12 1 representation in the form (3) gives re 2 due to the
3. PROBLEM STATEMENT E x x 1 A x x B x u
y C x x ,
(2)
nonlinear term 1 1 x12 and r 4 due to the number of nonlinearities cos x1 and sin x1 x1 . Note that in (5) the
where x n is the state vector, u m is the control input vector, y o is the output vector, and is the current
matrix B is constant. Since E x is non-singular, one can
sample. Matrices A x , B x , and C x are
calculate its inverse and then construct a standard TS model: 2 1 1 1 x22 1 x22 1 ; from here, E x 3 4 x22 2 x24 1 1 x22 2
assumed to be bounded and smooth in a compact set of the state space x . This paper considers the descriptor matrix E x as a non-singular matrix at least in a compact set
one can see that at least four different nonlinearities have to be considered, which results in r 16 . Moreover, the
x . In what follows, x and x will stand for x 1 and 277
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V ek ekT ek ekT ek 0.
standard representation contains nonlinear terms in B x E 1 x B which forces the use of sum relaxations for
In order to use Finsler’s Lemma the inequality (10) is expressed as follows:
the controller design. Table I compares the observer design conditions for the standard TS and TS descriptor forms. Approach
No. of sums
No. of LMI conditions
Q Standard TS1 + Lemma 1
3
r 2 1 257
NQ Standard TS2 + Lemma 1
4
r3 r 4112
Q TS Descriptor3 + Lemma 1
3
re r 2 1 33
NQ TS Descriptor3 + Lemma 1
4
T
e 0 e V e (11) 0. Feasible e 0 e solution Thus, using Finsler’s Lemma the inequality constraint (11) together with the equality one (8) gives No 0 (12) 0 Ah Ch Ev 0, No where 2 n n will be defined later on.
3
re r r 132
No
4.1 Previous results
Yes
The quadratic approach in (Estrada-Manzo et al. 2014a) is summarized in the following Lemma :
Table I. Computational Complexity for different results in Example 1.
Lemma 3: (Estrada-Manzo et al. 2014a) The estimation error e is asymptotically stable if there exist matrices P PT 0
Via Table I, one can see that better results may be obtained with TS descriptor models. Therefore, the observer design is done for the TS descriptor models via the direct Lyapunov method. The next section recalls some previous results and gives new LMI conditions for observer design.
and Fhv , such that the following inequality holds P vhh PAh Fhv Ch
An observer for the descriptor model (3) is given by: yˆ Ch xˆ ,
0. PEv E P P T v
(13)
The observer gains are recovered with Lhv P 1 Fhv . The final observer structure is
4. RESULTS Ev xˆ Ah xˆ Bh u y yˆ
(10)
Ev xˆ Ah xˆ Bh u Lhv y yˆ yˆ Ch xˆ .
(6)
(14)
Proof: see Theorem 1 in (Estrada-Manzo et al. 2014b).
where the observer gain may change according to the approach under study.
The non-quadratic case presented in (Estrada-Manzo et al. 2014a) is summarized in the following Lemma.
Defining the estimation error e x xˆ , its dynamics are as follows:
Lemma 4: (Estrada-Manzo et al. 2014a) The estimation error e is asymptotically stable if there exist matrices
Ev e Ah Ch e ,
Ph PhT 0 , H h , Lhv , such that the next inequality holds
(7)
Ph hhhh 0. T T H h Ah Lhv Ch H h Ev Ev H h Ph The final observer structure is
which can be expressed as the following equality constraint Ah Ch
e Ev 0. e
(8)
Ev xˆ Ah xˆ Bh u H h1 Lhv y yˆ
For design purposes consider the following Lyapunov function candidate: V e eT e 0, T 0,
yˆ Ch xˆ .
(15)
(16)
Proof: Recall (12). Choosing the observer gain as H h1 Lhv , the Lyapunov matrix as Ph , and
(9)
where may be constant (quadratic approach) or depend on MFs (non-quadratic approach). The variation of the Lyapunov function (9) writes
T
0 H hT yields (15), thus the proof is ended.
Remark 1: The inequality conditions in Lemma 3 and 4 are easily transformed into LMIs once the MFs are removed. To this end, Lemma 1 could be applied, and many other relaxation lemmas are available (Kim and Lee 2000; Sala and Ariño 2007; Wang et al. 1996).
1
(Lendek et al. 2010) (Guerra et al. 2012) 3 (Estrada-Manzo et al. 2014a) 2
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4.2 New quadratic Lyapunov function
G T P 1G H h Ah Lhv Ch
In this subsection a new Lyapunov function is presented. Recall (10) and select G T P 1G , therefore a new Lyapunov function yields V e eT G T P 1Ge 0,
Theorem 1: The estimation error e is asymptotically stable
if there exist matrices P PT 0 , G , and Fhv , such that the following inequality holds
Approach Lemma 3
(19)
Proof: Recall (12). Choosing the observer gain as Lhv ,
Lemma 4
the Lyapunov matrix as G P G , and 0 P T
T
1
yields G T P 1G PAh Fhv Ch
0, PEv EvT P G T P 1G
Theorem 1
(20)
with the change of variable Fhv PLhv . Finally, using Property 1 on the position (1,1) and the Schur complement on the position (2,2) renders (18); thus concluding the proof.
Theorem 2
Since Finsler’s Lemma allows “separating” — in a sense — the observer gain and the Lyapunov matrix (Marquez et al. 2013), a way to take advantage of the classical non-PDC-like observer is obtained by choosing H h1 Lhv ; this result is
H h Ev EvT H hT G
The final observer structure is
(23)
No. of LMI conditions
No. of decision variables 0.5n n 1
re r 2 1
r re n o
0.5r n n 1
re r 3 r
r re n o r n 2 0.5n n 1
re r 2 1
n 2 r re n o 0.5n n 1 n 2 r n 2
re r 2 1
r re n o
5. EXAMPLES The following example is adopted from (Estrada-Manzo et al. 2014a).
Theorem 2: The estimation error e is asymptotically stable
G G T P v hh H h Ah Lhv Ch 0
1
Table II. Computational complexity of the various approaches.
summarized in the following: if there exist matrices P PT 0 , G , H h , and that the following inequality holds
T
The complexity in terms of number of decision variable and LMI conditions is summarized in Table II.
(18)
The observer gains are recovered with Lhv P 1 Fhv . The final observer structure is yˆ Ch xˆ .
0. H h Ev E H G P G T h
Remark 2: Note that the PDC-like observer structure in Theorem 1 is the same as in Lemma 3, while the non-PDClike observer in Theorem 2 is the same as in Lemma 4; but the design procedure is done via different Lyapunov functions. The new Lyapunov function may be less conservative since it naturally introduces more slack variables; it could be seen as the dual of the one presented in (Lendek et al. 2015) for control purposes.
P PT 0 , thus it is non-singular. The following where result can be stated:
Ev xˆ Ah xˆ Bh u Lhv y yˆ
T v
Using Property 1 on the position (1,1) and the Schur complement on the position (2,2) gives (21); thus concluding the proof.
(17)
G GT P v T hh PAh Fhv Ch PEv Ev P 0. P G 0
279
Example 2. (Estrada-Manzo et al. 2014a) Consider a discrete-time TS descriptor model as in (3) with r re 2 ,
Lhv such
0.1 a 0.1 a 0.9 0.9 , E2 , 1.1 1.1 0.4 b 0.4 b 1 1 a 1 1 a , A2 A1 , 1.5 0.5 1.5 0.5
E1 (21) 0.
P
T
T
0 0 C1 , C2 1 b . b 1
Ev xˆ Ah xˆ Bh u H h1 Lhv y yˆ
(22) yˆ Ch xˆ . Proof: Recall (12). Choosing the observer gain as H h1 Lhv , the Lyapunov matrix as G T P 1G , and
The parameters are defined as a 1,1 and b 1,1 . Fig. 1 shows the feasible regions for the proposed approaches. Results obtained via Lemma 3 ( ), via Lemma 4 ( ), and via Theorem 1 ( ).
T
0 H hT yields 279
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Example 2 illustrates the improvements made by the new L12 Lyapunov function. In Fig. 1, one can see that the new Lyapunov function together with a PDC-like observer overcomes the quadratic approach in (Estrada-Manzo et al. 2014a).
0.5
The TS descriptor exactly represents the nonlinear model in the compact set x 2 , thus the designed observer allows the asymptotic converge of the estimation error for any initial condition of the original nonlinear model (see Fig. 2).
b
0 -0.5 -0.5
0 a
0.5
e1
5
-1 -1
1
Using results in Theorem 2, it is possible to obtain the same feasibility set as in Lemma 4 for this specific Example 2. Both conditions allow designing the same observer structure (16). Table II shows that conditions in Theorem 2 are less demanding than Lemma 4.
h4 ;
cos x 1 1
h3 , 2,
2 0
1 1
2 1
2
4
6 8 Sample
10
12
e2
Acknowledgements
h1 0102 ,
This work is supported by the Ministry of Higher Education and Research, the CNRS, the Nord-Pas-de-Calais Region (grant No. 2013_11067), a grant of the Technical University of Cluj-Napoca. The authors gratefully acknowledge the support of these institutions.
with
sin x1 / x1 0.2167 1.2167 , 2 0
11 1 01 , and 12 1 02 . The state variable x1 is
available. Conditions in Lemma 3 and Theorem 1 are not feasible. Feasible results are obtained via Theorem 2 (nonPDC-like observer and the new quadratic Lyapunov function); for brevity, only some of the resulting gains are given: P
12
In this paper, LMI conditions are established for the observer design of TS descriptor models. Using Finsler’s Lemma together with a quadratic Lyapunov function it is possible to design a non-PDC-like observer; thus relaxing previous results. Numerical examples illustrate the advantages of the presented approaches.
T
h2 ,
1 1
10
6. CONCLUSIONS
0 1 0.2167 B , C C C4 , and C 1 3 2 . 1 0 0 2 1
6 8 Sample
Fig. 2. Evolution of the estimation error for several initial conditions for Example 3.
1 1 1 1 r 4, A1 A2 , A3 A4 , 0.5 1.1 0.5 1.1
v2 1 v1 ,
4
0 -5 0
2 1 2 0 , and E2 re 2, E1 ; 1 1 0 1 on the right-hand side:
v1 1 1 x12 , The MFs are:
2
5
Example 3. Recall the nonlinear descriptor in Example 1. Considering the compact set x 2 and using the sector nonlinearity approach a TS descriptor model as in (3) , we obtain: on the left-hand side:
T
0 -5 0
Fig. 1. Feasibility sets for Example 2.
1 0
0.05 0.52 0.04 , L32 0.26 , L41 0.02 , 0.01 0.19 0.68 0.02 , H3 . 1.06 0.32 0.99
For this example, conditions in Lemma 4 are also feasible. However, the number of LMIs in Theorem 2 is 33 while for Lemma 4 is 132 (see Table II).
1
1 0
0.28 , L22 0.11 0.69 H1 0.32
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