A high order sliding mode observer for systems in triangular input observer form

Transcription

1 Insiuo Tecnológico y de Esudios Superiores de Occidene Reposiorio Insiucional del ITESO reiiesomx Deparameno de Maemáicas y Física DMAF - Arículos y ponencias con arbiraje - A high order sliding mode observer for sysems in riangular inpu observer form Sánchez-Torres, Juan D; Loukianov, Alexander; Giraldo, Berulfo; Boero- Casro, Hécor J D Sánchez-Torres, A G Loukianov, J A P Murillo, B Giraldo and H Boero, "A high order sliding mode observer for sysems in riangular inpu observer form," Roboics Symposium, IEEE IX Lain American and IEEE Colombian Conference on Auomaic Conrol and Indusry Applicaions LARC), Bogoa,, pp -6 Enlace direco al documeno: Ese documeno obenido del Reposiorio Insiucional del Insiuo Tecnológico y de Esudios Superiores de Occidene se pone a disposición general bajo los érminos y condiciones de la siguiene licencia: hp://quijoebiblioiesomx/licencias/cc-by-nc-5-mxpdf El documeno empieza en la siguiene página)

2 A high order sliding mode observer for sysems in riangular inpu observer form Juan Diego Sánchez-Torres and Alexander G Loukianov Deparmen of Elecrical Engineering and Compuer Compuer Science, Auomaic Conrol Laboraory CINVESTAV Unidad Guadalajara Av del Bosque 45, 455, Zapopan, Jalisco, México [dsanchez, louk]@gdlcinvesavmx Julián Albero Paiño Murillo, Berulfo Giraldo and Hécor Boero Universidad Nacional de Colombia, Sede Medellín Escuela de Ingeniería Elécrica y Mecánica Faculad de Minas Cr 8 No 65 -, Medellín, Anioquia, Colombia [japain, begiraldoos, haboero]@unaleduco Absrac This paper deals wih he design of a high order sliding mode observer for a class of nonlinear sysems ha can be described in he so called riangular inpu observer form The mahemaical ools required o make he sysem ransformaion o such form are also presened A las, we show he performance of he observers wih several simulaion examples, including he applicaion of a CSTR process model I INTRODUCTION Sliding mode approaches have been widely used for he problems of dynamic sysems conrol and observaion due o is characerisics of finie ime convergence, robusness o uncerainies and insensiiviy o exernal disurbances In addiion, he sae observers have anoher imporan properies like he possibiliy of obaining a sep by sep design and work wih a reduced observaion dynamics [, Ch 4], [] Ofen, sliding mode moion is obained by means of a disconinuous erm depending on he oupu error, ino he conrolling or observing sysem Addiionally, by using he sign of he error o drive he sliding mode observer, he observer rajecories become insensiive o many forms of noise Hence, some sliding mode observers have aracive properies similar o he Kalman filer bu wih simpler implemenaion [] Several researchers have deal wih he issue of designing sliding-mode observers for differen applicaions [4], [5], including he classical problem of non-linear sae esimaion [6] In [7], a sliding-mode observer for non-linear sysem based over he equivalen conrol mehod is proposed Some applicaions of he sliding mode echniques o conrol and robus differeniaion are presened in [8] [9] [] Noing ha he classical sliding mode echniques are a paricular case of he high order sliding mode concep and can be considered as a firs order sliding mode [] The high order sliding modes allows also ake ino accoun he sampling measuremen delays Some pracical examples of he use high order sliding modes observers can be found in [], [] All of hese imply ha high order sliding modes observers are very convenien for real implanaions In his work, our purpose is o discuss abou observer design for a sysem in a riangular inpu observer form We discuss his form, which has been reaed previously in [4], because for a such sysem i is possible o design a simple observer which does no use he inpu derivaive In fac, here are some applicaions, mainly in he elecrical moor conrol sysems, where he exac knowledge of he inpu derivaive is very quesionable [4] Besides his, a sysem in a riangular inpu observer form does no face he problem of singular inpu The anoher goal of his paper is o show ha he proposed observer can also be applied o a chemical process sysem like he CSTR In he following, in secion II some mahemaical preliminaries are given In secion III a higher order sliding mode observer for sysems in riangular inpu observer form is proposed Three examples of he proposed observer, including wo well-known sysems [4], [, Ch 4] and a CSTR, are presened in secion IV Finally, in secion V some conclusion are presened II MATHEMATICAL PRELIMINARIES In his secion, we presen he mahemaical conceps required o ake a given SISO sysem o he riangular inpu observer form, and is relaionship wih a robus differeniaor A Triangular inpu observer form Le us consider he following SISO sysem χ = f χ) + g χ,u) ) y = h χ) where χ R n is he sae, u R is he inpu, y R is he oupu and f, g, h are funcion vecors, wih g x,) = for all x R n In addiion for he sysem ), les suppose he following wo condiions Condiion The codisribuion is involuive and Ω i = span { dh,,dl i fh } i n ) Condiion For any u R, he vecor field g fulfills dl g L i fh Ω i i =,n )

3 From ) he coordinae change x x = x n h L f h) L n f h) is a diffeomorphism T χ) x which ransforms he sysem ) o ẋ ẋ ẋ n ẋ n χ x = x + ḡ x,u) x + ḡ x,u) x n + ḡ n x,u) f n x) + ḡ n x,u) wih ḡ i,u) = for u =, i =,n Where x = [x,,x n ] T In addiion, he equaion ) is equivalen o 4) 5) dḡ i span {dχ,,dχ i } i {,,n} 6) his reduces he sysem 5) o he so-called Triangular inpu observer form, as follows ẋ x + ḡ x,u) ẋ x + ḡ x,x,u) = 7) ẋ n x n + ḡ n x,,x n,u) ẋ n f n x) + ḡ n x,u) y = x wih ḡ i,u) = for u =, i =,n Remark The equaion ) follows from he observabiliy rank condiion dh dl f h rank dl n f h = n 8) his gives an easy way o check he Condiion, [5] B Arbirary-order exac robus differeniaor Real-ime differeniaion is a well-known problem, several approaches have been proposed o obain ime derivaives for a given signal Beween hese all soluions, sliding mode based mehods have demonsraed high accuracy and robusness For he calculaion of higher order exac derivaives, successive implemenaion of a firs order differeniaor wih finie ime convergence is used in [8] For he same objecive, an arbirary-order exac robus differeniaor based in a recursive scheme and which provides he bes possible asympoic accuracy in presence of inpu noises and discree sampling is proposed in [9] Le f ) C k [, ) be a funcion o be differeniae and le k k, hen he k-h order differeniaor is defined as follows: ż = v, v = λ k L k+ z f ) k k+ signz f )) + z ż = v, v = λ k L k z v k k sign z v ) + z 9) ż k = v k, v k = λ L zk v k sign z k v k ) + z k ż k = λ Lsign z k v k ) where z i is he esimaion of he rue signal f i) ) The differeniaor provides finie ime exac esimaion under ideal condiion when neiher noise nor sampling are presen The parameers λ =, λ = 5, λ =, λ =, λ 4 = 5, λ 5 = 8 are suggesed for he consrucion of differeniaors up o he 5-h order The parameer L is seleced such ha be a upper bound for f k+) See [9] and [] for furher deails on he esimaion of ime of convergence, he error bounds for he signal f ) and heir derivaives in presence of noise or discree sampling and oher proprieies and consrains of he differeniaor III OBSERVER DESIGN In his secion, based on he equaion of he arbirary-order exac robus differeniaor 9), we propose an observer for a sysem represened in he riangular inpu observer form 7) The use of high order sliding mode erms allow us o avoid he use of classic low pass filers employed o obain he equivalen conrols [4] Anoher feaure of he differeniaor 9) is he fac ha he oupu does no depend direcly on disconinuous funcions bu on an inegraor oupu So high frequency chaering, which can be very harmful for he sysems, can also be avoided Based in he equaion 9), we propose he following observer for 7): ˆx = ζ, ζ = ˆx + ḡ x,u) λ L n+ ˆx x n n+ sign ˆx x ) ˆx = ζ, ζ = ˆx + ḡ x, ˆx,u) λ L n ˆx ζ n n sign ˆx ζ ) ) ˆx n = ζ n, ζ n = f n ˆx) + ḡ n ˆx,u) λ n L ˆxn ζ n sign ˆx n ζ n ) For his form, as i is saed in [9] and [], we can obain he convergence of he observaion error o zero in finie ime IV APPLICATION CASE We highligh in his secion he uiliy and he advanages of he high order sliding modes of he previous recalls in he

4 resoluion of he observaion problem A firs, we use wo sysems previously proposed in he lieraure [], [, Ch 4] A las,we apply he same procedure o a Coninuous Sirred Tank Reacor CSTR) sysem [6] 5 5 x A Example 5 Using a sysem proposed in [], le he Bounded Inpu Bounded Sae BIBS) in finie ime presened in he riangular inpu observer form Fig Real x and observed ˆx - -) saes for sysem ) ẋ = x ẋ = x x + x u ) ẋ = x + x ) x + u y = x x The proposed observer as in ) for he sysem ) is shown in equaion ) The parameer values are L = 4, λ =, λ = 5 and λ = ˆx = ζ, ζ = ˆx λ L 4 ˆx x 4 sign ˆx x ) ˆx = ζ, ) ζ = ˆx ˆx + ˆx u λ L ˆx ζ sign ˆx ζ ) ˆx = ˆx + ˆx ) ˆx + u λ L ˆx ζ sign ˆx ζ ) Fig Real x and observed ˆx - -) saes for sysem ) Figures, and shows he simulaion resuls for he sysem ) The iniial condiions for his sysem were x) = [ 5 5] T and ˆx ) = [ ] T The sae ˆx converges o x in finie ime of 5s Then ˆx reaches o x in finie ime 75s Noe ha ˆx only reaches x a a ime of 75s, and afer ˆx converges o is sae Finally a s, ˆx converges o x 6 5 B Example Le us consider he following sysem in he riangular inpu observer form [, Ch 4]: 4 x ẋ = x x u ẋ = x x x u ) ẋ = x x x x u y = x Fig Real x and observed ˆx - -) saes for sysem ) For he sysem ), he observer of equaion ) akes he form shown in equaion 4), wih he parameer values L = 4, λ =, λ = 5 and λ =

5 5 4 ˆx = ζ, ζ = ˆx x u λ L 4 ˆx x 4 signˆx x ) ˆx = ζ, 4) ζ = ˆx x ˆx u + λ L ˆx ζ signˆx ζ ) ˆx = ˆx ˆx x ˆx u λ L ˆx ζ signˆx ζ ) x This approach has been esed by simulaion wih iniial condiions x) = [ 5 5] T and ˆx) = [ ] T In Figure 4, we see ha ˆx reaches x in finie ime around 5s In Figure 5, we see ha ˆx also reaches x in finie ime of abou 75s Bu ˆx will only reach x afer ˆx has been able o ge o he value of x In Figure 6, we see ha ˆx reaches x in finie ime of s 5 Fig 6 Real x and observed ˆx - -) saes for sysem ) C CSTR Applicaion The CSTR process is a recognized benchmark frequenly used for conroller proofs [6] and represens many processes ypically employed in indusry The model of he CSTR is described by he se of equaions 5) See [6] for a more in-deph descripion and modelling of he process and is parameers The simulaion values for he sysem were also exraced from he same reference, and hey are summarized in Table I x 5 dt d = F V T in T) H k C A e E UA RT + ρc p ρc p V T j T) dc A = F d V C in C A ) k C A e E RT 5) Fig 4 Real x and observed ˆx - -) saes for sysem ) Assuming as he sae variables he emperaure T of he reacive mass and he concenraion C A of he reacan respecively, he model in space-sae represenaion is shown in he se of equaions 6) We suppose ha T can be measured and acs as he model oupu y The goal is he esimaion of C A denoed by χ from T denoed by χ ) 5 4 χ = F V T in χ ) H ρc p k χ e E Rχ + UA ρc p V T j χ ) χ = F V C in χ ) k χ e E Rχ 6) y = h χ) = χ x In order o find an sae observer for he sysem, we have o describe he process in he riangular inpu observer form From 4), Fig 5 Real x and observed ˆx - -) saes for sysem ), x = χ 7) x = F V T in χ ) H k χ e E Rχ + UA ρc p ρc p V T j χ ) and he resulan sysem is:

6 TABLE I NOMINAL PARAMETERS OF CSTR Parameer Value Uni F 65 m min V 469 m C in 45 gmol m k 867 min E 7564 J gmol R 874 J gmol K T in 95 K H J gmol ρ kg m C p 57 J kg U J s m K) A 8,755 m T j 79 K T [K] [min] Fig 7 6 Temperaure x and is observaion ˆx - -) for CSTR 5 4 C A [gmolm ] ẋ = x ẋ = k e E Rx + F V Ex ) Rx 8) F V T in x ) + UA ) ρc p V T j x ) x UA + F ) x C inf H V ρc p V ρc p V k e E Rx The observer obained for he CSTR, wih parameer values L = 5, λ = 5 and λ = is presened in eq 9) ˆx = ζ, ζ = ˆx λ L x ˆx sign ˆx x ) ˆx = k e E Rx + F V Eˆx ) Rx 9) F V T in x ) + UA ) ρc p V T j x ) ˆx UA + F ) ˆx C inf H V ρc p V ρc p V k e E Rx λ L ˆx ζ sign ˆx ζ ) [min] Fig 8 Concenraion x and is observaion ˆx - -) for CSTR This approach has been esed by simulaion wih he he iniial condiions x) = [4 547] T and ˆx) = [9 ] T In Figure 7, we can see ˆx reaching he real emperaure value in less han 5 minues In Figure 8, we see ha ˆx also converges o he real value of he concenraion in finie ime of abou minues V CONCLUSION We show in his paper ha we can design a high order sliding mode observer for single oupu sysems which can be ransformed ino riangular observer form From his form, many observer designs work well, and in his case, advanages of he sliding mode observer were principally he design simpliciy and he finie ime convergence Moreover, we also showed he applicaion of he proposed schemes o a real process model like he CSTR This model is a well documened benchmark ha includes many dynamical processes; in his regard, he resuls of his work can be expanded owards a plehora of differen applicaion cases REFERENCES [] W Perruquei and J-P Barbo, Sliding Mode Conrol In Engineering Auomaion and Conrol Engineering), s ed CRC Press, [] S K Spurgeon, Sliding mode observers: a survey, Inern J Sys Sci, vol 9, pp , 8

7 [] S V Drakunov, An adapive quasiopimal filer wih disconinuous parameer, Auomaion and Remoe Conrol, vol 44, no 9, pp 67 75, 98 [4] S Drakunov and V Ukin, Sliding mode observers uorial, in Proc 4h IEEE Conf Decision and Conrol, vol 4, 995, pp [5] C Edwards and S Spurgeon, On he developmen of disconinuous observers, Inernaional Journal of Robus and Nonlinear Conrol, vol 59, p 9, 999 [6] J-J E Sloine, J K Hedrick, and E A Misawa, Nonlinear sae esimaion using sliding observers, in Proc 5h IEEE Conf Decision and Conrol, vol 5, 986, pp 9 [7] S V Drakunov, Sliding-mode observers based on equivalen conrol mehod, in Proc s IEEE Conf Decision and Conrol, 99, pp [8] A Levan, Robus exac differeniaion via sliding mode echnique, Auomaica, vol 4, no, pp 79 84, 998 [9], Higher-order sliding modes, differeniaion and oupu-feedback conrol, In J of Conrol, vol 76, no 9/, pp 94 94,, special issue on Sliding-Mode Conrol [] M T Angulo, L Fridman, and A Levan, Robus exac finie-ime oupu based conrol using high-order sliding modes, Inern J Sys Sci, vol, p, [] T Boukhobza and J-P Barbo, High order sliding modes observer, in Proc 7h IEEE Conf Decision and Conrol, vol, 998, pp 9 97 [] J Davila, L Fridman, A Pisano, and E Usai, Finie-ime sae observaion for non-linear uncerain sysems via higher-order sliding modes In J of Conrol, vol 8, pp , 9 [] L Fridman, Y Shessel, C Edwards, and X-G Yan, Higher-order sliding-mode observer for sae esimaion and inpu reconsrucion in nonlinear sysems, Inernaional Journal of Robus and Nonlinear Conrol, vol 8, no 4/5, pp 99 4, 8 [4] J P Barbo, T Boukhobza, and M Djemai, Sliding mode observer for riangular inpu form, in Proc 5h IEEE Decision and Conrol, vol, 996, pp [5] A Isidori, Nonlinear Conrol Sysems, rd ed Springer Communicaions and Conrol Engineering), 995 [6] H Boero and H Alvarez, Non linear sae and parameers esimaion in chemical processes: Analysis and improvemen of hree esimaion srucures applied o a CSTR, Inernaional Journal of Chemical Reacor Engineering, vol 9, no A6, pp 4, [Online] Available: hp://wwwbepresscom/ijcre/vol9/a6