PARTIAL STATE OBSERVER DESIGN FOR A CLASS OF NONLINEAR DESCRIPTOR SYSTEMS

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1 INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 7, Numbe 4, Pages c 2 Institute fo Scientific Computing and Infomation PARTIAL STATE OBSERVER DESIGN FOR A CLASS OF NONLINEAR DESCRIPTOR SYSTEMS JUAN ZHOU AND BO MEN Abstact. This pape studies the patial state obseve design fo a class of nonlinea descipto systems. Necessay and sufficient conditions ae deived fo the existence of patial state obseves. Based on Lyapunov stability theoem, the stability condition fo eo systems is analyzed by using linea matix inequality appoach. Futhemoe, a design method of the state obseves is given by Cayle-Hamilton theoem and genealized invese theoy. Finally, two numeical examples ae given to illustate the effectiveness of the obseve design. Key Wods. nonlinea descipto system, patial state obseve, stability, Cayley-Hamilton theoem. Intoduction In contol systems, depending on the input and output of the oiginal system a new dynamical system can be constucted to estimate the state of the oiginal system. This new dynamical system is called an obseve. The obseve design includes full-ode state obseve design, educed ode state obseve design and patial state obseve design. To design a full-ode state obseve fo a system we need to obseve all state component. When designing a educed ode state obseve, we need to estimate the patial component of the state vaiable which can t be measued by output. And we need not to conside the pat fom measuable output. Then the dimension of the system can be educed to make the system simple, while no design cost is incease. But when educed ode state obseve is designed, the state component which measued by obseve is detemined by measuable output of the oiginal system. So when we obseve the state component by educed ode state obseve, we can t obseve the specific component we need. But we can design a patial state obseve to solve this poblem. Theefoe, to study patial state obseve has impotant theoetical meaning and pactical significance. The achievement of obseve design fo nonlinea system is studied fist in the 97s [2. In ecent yeas obseve design poblem becomes a hot issue in contol theoy and signal pocessing. The study fo designing state obseve has eceive moe and moe attention [9, 3-5, 8-9. With futhe development of nonlinea system theoy, eseaches have done much wok and got many accomplishments of obseve design. The theoy of full-ode o educed ode state obseve design of nomal system is elatively matue. In this study sufficient condition full-ode o educed ode state obseve design of invaiant descipto system is studied in [8, and the design method is given. Based γ-positive lemma, the method to design obust adaptive full ode obseve fo sensoless vecto contolled induction motos is given in [9. H patial state obseve state design fo nomal nonlinea systems This eseach was suppoted by National Natual Science Foundation of China (69744). 357

2 358 J. ZHOU AND B. MEN is consideed in [3 and the condition that the matix should satisfy and specific expession ae given. Uncetain obseve design fo uncetain system is studied in [4. By using unifom LMI method, full-ode o educed ode state obseve design fo Lipschitz descipto systems is obtained in [2. Fo nomal nonlinea system patial state obseve design is studied and the necessay and sufficient conditions of the existence of state obseve is given in [. A new ecusion state obseve design pogam is poposed fo multiple input multiple output contol system, patial lowe tiangula matix in [. Base on absolute stability theoy, both types of full-ode and educed-ode obseves ae constucted fo a class of nonlinea descipto systems in [2. At pesent thee ae many methods applied to design state obseve. [5 with the coodinates tansfomation of the state vaiable, the state obseve is designed and the method of patial asymptote obseve design fo a class of nonlinea descipto systems is obtained. In the study [2, the state obseve is design diectly by solving linea matix inequality. [6 and [7 based on mean value theoem, a eo system is tansfomed to a linea vaiable paamete system and an obseve is designed. In this pape [6, the state obseve of descipto system is designed based on singula value decomposition and genealized invese matix method. This pape studies the patial state obseve design fo a class of nonlinea descipto systems. Necessay and sufficient conditions ae deived fo the existence of patial state obseves. Based on Lyapunov stability theoem, the stability condition fo eo systems is analyzed by using linea matix inequality appoach. Futhemoe, a design method of the state obseves is given. Finally, two numeical examples ae given to illustate the effectiveness of the obseve design. 2. Main esults () (2) (3) Conside the following nonlinea descipto system Eẋ = Ax+f(t,y,z,u)+Bu y = Cx z = Lx whee x R n is the state, y R p is the output, u R m is input, z R q is the vecto to be estimated, A,B,C,L,E ae matices with appopiate dimension, anke = l n,ankc = p,ankl = q, nonlinea function f(t,x,z,u) R n. suppose the descipto ()-(3) is contollable. In this pape we will design an obseve with espect to vecto z, then we design the following obseve fo the system ()-(3) (4) (5) ẇ = Dw+Fy +Gf(t,y,ẑ,u)+GBu ẑ = w+hy whee w R q,d,f,g,h ae undetemined matices with appopiate dimension. Set eo vaiable e = ẑ z, we intoduce a new vaiable (6) ε = w GEx Theoem Fo abitay x(), ẑ(), u(t), ẑ in system (4)-(5) is the asymptotically estimate with espect of z in system ()-(3), if and only if the following condition is satisfied () by obseving the eo system with detemining eo (7) ε = Dε+G[f(t,y,ẑ,u) f(t,y,z,u)

3 PARTIAL STATE OBSERVER DESIGN FOR NONLINEAR DESCRIPTOR SYSTEMS 359 the solution appoaches to asymptotically. (2) the coefficient matix satisfied the following equation (8) (9) GE +HC L = FC GA+DGE = Poof: Sufficiency: Eo e = ẑ z = w +Hy Lx = ε+(ge +HC L)x, by ()-(6) ε = ẇ GEẋ = Dw+Fy +Gf(t,y,ẑ,u)+GBu G(Ax+f(t,y,z,u)+Bu) = D(ε+GEx)+FCx+Gf(t,y,ẑ,u)+GBu G(Ax+f(t,y,z,u)+Bu) = Dε+G(f(t,y,ẑ,u) f(t,y,z,u))+(dge +FC GA)x is obtained. When condition () and (2) ae satisfied, we get e = ε, and when t, e, that is ẑ in system (4)-(5) is the asymptotically estimate with espect of z in system ()-(3). Necessity: If condition () can not be satisfied, then even u(t) =,x() =,f(t,x,z,u) =, because when t, ε, then, when t, e, so the necessay condition () is indispensable. Howeve if condition (2) can not be satisfied, by the contollability of the system (), we can choose special u, to contol x so that when t, x(t), thus, even if ε asymptotically appoaches to, we can t make e(t) appoach to. In conclusion, the necessity is hold. Lemma Set L q n is full ow ank matix, then thee exist full column ank matix L n q, then L L = I q, and ank [ L In LL = n Poof: By linea algeba L exists, L is full ow ank matix, then thee exists invetiblematixγ n n, then LΓ = [ I q, howeve Lisfull [ column ankmatix, Iq then thee exists invetible matix Λ n n, such that Λ L =, thus [ L In LL [ [ [ = Λ Iq I n Λ Iq [ Iq Γ [ [ [ = Λ Iq Iq [ Λ Iq Γ [ = Λ Iq Λ Γ Λ 2 Γ 2 Λ 2 Λ 22 hence ank [ L In LL [ = ankλ Iq Λ Γ Λ 2 Γ 2 Λ 2 Λ 22 [ Iq Λ = ank Γ Λ 2 Γ 2 = n Λ 2 Λ 22 Lemma 2 Suppose A is m n ode matix, B is p n ode matix, then matix B satisfies the following elation BA + A = B if and only if thee exists p m ode matix C so that B = CA whee A + is the genealized invese matix of A, that is AA + A = A.

4 36 J. ZHOU AND B. MEN Fistly, conside the design of patial state obseve of the nonlinea descipto system ()-(3). Set the chaacteistic polynomial of matix E is ϕ(s) = det(si E) = (s s )(s s 2 ) (s s )s n = ( α i s i )s n whee s i,i =,2,,, is the nonzeo eigenvalue of matix E, α =, = ( ) s i. i= Theoem 2 ẑ in system (4)-(5) is the asymptotically estimate with espect of z in system ()-(3), if and only if the following condition is satisfied () i= () ank =ank () ank =ank C (I n E n E n + )A i= α ie i A C + i= α ie i L C (I n E n E n + )A i= α ie i A C + i= α ie i i= α ie i A L+ i= α ie i L sl+ i= α ie i A L+ i= α ie i C (I n E n E n + )A i= α ie i A C + i= α ie i C (I n E n E n + )A i= α ie i A C + i= α ie i, s C,Re(s) > L (2) function f(t,y,z,u) with espect to z is a Lipschitz function, Lipschitz constant is γ >, that is (2) f(t,y,z,u) f(t,y,ẑ,u) γ z ẑ, t,x,u (3) thee exist positive definite matix P = P T >, matix N and positive eal numbe µ,µ 2, such that the following linea matix inequality is hold. PG NG 2 (3) G T P µ I n < G T 2N T µ 2 I n whee N = Pν, = PD +D T P +ND 2 +D T 2 NT +γ 2 (µ +µ 2 )I q

5 PARTIAL STATE OBSERVER DESIGN FOR NONLINEAR DESCRIPTOR SYSTEMS 36 D = L α i E i A L+ΞΘ + α (I n E n E n + )A L i= C i= α ie i A L D 2 = (I 2p+n ΘΘ + ) (I n E n E n + )A L C i= α ie i A L G = L α i E i +ΞΘ + α (I n E n E n + ) i= C i= α ie i G 2 = (I 2p+n ΘΘ + ) (I n E n E n + ) C i= α ie i C(I n LL) Θ = (I n E n E n + )A(I n LL) i= α ie i A(I n LL) C + C i= α ie i Ξ = [ i= α ie i A(I n LL) L+ i= α ie i Poof: () By Cayley-Hamilton theoem, we get then next (4) G ϕ(e) = ( (G α i E i )E n = i= α i E i )E n = i= α i E i = µ(i n E n E n + ) i= whee matix E n + is the genealized invese matix of E n, µ is q n ode constant matix. Notice that, then by (8) and (4), we get (5) hence (6) G = α i GE i + µ(i n E n E n + ) i= = (L HC) α i E i + µ(i n E n E n + ) i= L HC = GE = (L HC) plug (5) in (9), we get FC + (L HC) α i E i ++ µ i= α i E i A µ(i n E n E n + )A+DL DHC = i=

6 362 J. ZHOU AND B. MEN simplified, we get (7) DL = L i= α i E i A+ [ DH F µ H C (I n E n E n + )A i= α ie i A by lemma, post-multiply both sides of (7) by the matix L and I n LL to espectively, we get (8) D = L α i E i A L+ [ DH F µ H α (I n E n E n + )A L i= i= α ie i A L (9) C(I [ n LL) DH F µ H (I n E n E n + )A(I n LL) C i= α ie i A(I n LL) = L α i E i A(I n α LL) i= by equation (6) and (9) we get (2) [ DH F µ H Θ = Ξ [ Θ bylineaalgeba,theeexiststhesolutiontoequation(2)ifandonlyifank Ξ ankθ, by lemma, matix [ L In LL is full ow ank matix, so C ank (I n E n E n + )A C n i= α ie i A C + i= α ie i L C =ank (I n E n E n + )A (E E n [ E n + E) C n i= α ie i A C + i= α L In LL ie i I n L C(I n LL) =ank (I n E n E n + )A L (I n E n E n + )A(I n LL) C n i= α ie i A L C n i= α ie i A(I n LL)E i I q C + i= α ie i =

7 PARTIAL STATE OBSERVER DESIGN FOR NONLINEAR DESCRIPTOR SYSTEMS 363 C (I n E n E n + )A ank C n i= α ie i A C + i= α ie i i= α ie i A L+ i= α ie i L C (I n E n E n + )A =ank C n i= α ie i A C + C [ i= α ie i L In i= α ie i A L+ L LL i= α ie i I n L C(I n LL) (I n E n E n + )A L (I n E n E n + )A(I n LL) =ank C n i= α ie i A L C n i= α ie i A(I n LL) L n i= α ie i A L L n i= α ie i A(I n LL) I q C + i= α ie i L+ i= α ie i [ Θ by (), we get ank = ankθ, then equation (2) has solution, and the Ξ solution is [ DH F µ H = ΞΘ + +ν(i 2p+n ΘΘ + ) whee ν is q (2p+n) ode constant matix. So D = L α i E i A L+(ΞΘ + +ν(i 2p+n ΘΘ + )) α (I n E n E n + )A L i= i= α ie i A L = D +νd 2 G = L α i E i +(ΞΘ + +ν(i 2p+n ΘΘ + )) α (I n E n E n + ) i= C i= α ie i = G +νg 2 (2) By (), the state expession in the system (7) is tansfomed to (2) ε = (D +νd 2 )ε+(g +νg 2 )[f(t,x,z,u) f(t,x,ẑ,u) let f = f(t,y,z,u) f(t,y,ẑ,u), then the following is the necessay condition of the solution to system (2) asymptotically appoaches to. Matix D = D +νd 2 is Huwitz matix, and D = D +νd 2 is Huwitz matix if and only if (D 2,D ) is testable, that is [ siq D ank = q, s C,Re(s) > D 2

8 364 J. ZHOU AND B. MEN by lemma, matix [ L In LL is full ow ank matix and lemma 2, then (22) ank =ank =ank sl+ i= α ie i A L+ i= α ie i C (I n E n E n + )A i= α ie i A C + i= α ie i sl+ i= α ie i A L+ i= α ie i C (I n E n E n + )A i= α ie i A C + i= α ie i si + i= α ie i A L i= α ie i A(I n LL) C(I n LL) (I n E n E n + )A L L+ i= α ie i (I n E n E n + )A(I n LL) i= α ie i A L i= α ie i A(I n LL) C + i= α ie i si + i= α ie i A L Ξ =ank (I n E n E n + )A L C Θ i= α ie i A L =ank I si + q ΞΘ + i= α ie i A L Ξ (I 2p+n ΘΘ + ) ΘΘ + (I n E n E n + )A L C Θ i= α ie i A L si q D D 2 =ank ΘΘ + (I n E n E n + )A L C Θ i= α ie i A L I q Θ + (I n E n E n + )A L C I n i= α ie i A L [ siq D =ank +ank(θ) D 2 (23) C ank (I n E n E n + )A i= α ie i A C + i= α ie i L [ L In LL I n

9 PARTIAL STATE OBSERVER DESIGN FOR NONLINEAR DESCRIPTOR SYSTEMS 365 C =ank (I n E n E n + )A [ i= α ie i A C + i= α L In LL ie i I n L C(I n LL) =ank (I n E n E n + )A L (I n E n E n + )A(I n LL) i= α ie i A L i= α ie i A(I n LL) I q C + i= α ie i =q +ank(θ) thus, by (22) and (23) we get (). (3) Notice that, () is a necessay condition of that the solution to the system (2) asymptoticallyappoachedto. Next considenonlinea function f, thus, take Lyapunov function by (2) we get (24) V(ε,t) = ε T (t)pε(t)+ε T (t)p ε(t) V(ε,t) = ε T (t)pε(t) =ε T (D T +DT 2 νt )Pε+ f T (G T +GT 2 νt )Pε+ε T P(D +νd 2 )ε+ε T P(G +νg 2 ) f ε T (D T +DT 2 νt )Pε+ε T P(D +νd 2 )ε+µ ft f + µ ε T PG G T Pε+µ 2 f T f + µ 2 ε T PνG 2 G T 2 νt Pε ε T (D T +DT 2 νt )Pε+ε T P(D +νd 2 )ε+ µ ε T PG G T Pε+ µ 2 ε T PνG 2 G T 2 νt Pε +γ 2 (µ +µ 2 )ε T ε =ε T (D T P +PD )ε+ε T (D T 2 NT +ND 2 )ε+ µ ε T PG G T Pε+ µ 2 ε T NG 2 G T 2 NT ε +γ 2 (µ +µ 2 )ε T ε by Schu lemma, we get that () is equivalent to that (24) is less than. So, when () is hold, ẑ in system (4)-(5) is an asymptotically estimate of z in ()-(3). Remak Now the esults of patial state obseve is less, in [6 patial state obseve fo nomal systems is studied, and the design method of the obseve is obtained. In this pape the patial state obseve design poblem is consideed. If the coefficient matix E = I n of system ()-(3) is invese matix, the system is nomal, then the theoem 2 is still hold, the conclusion is moe geneal than that in [6. Remak 2 Thee ae many conclusions about state obseve design implying E T P = P T E,A T P + P T A <, that is, the system is egula, stable, impulsive, such as [2. But in this pape it is not equied in theoem 2.

10 366 J. ZHOU AND B. MEN Remak 3 The difficult to veify theoem 2 is how to the sepaate the coefficient matix E fom the constaint condition. We use the chaacteistic polynomial of E and Cayley-Hamilton theoem, this method efficiently avoid tansfomation of coodinates, and we can design an obseve diectly. By theoem 2, we get the design method of obseve (4)-(5) D = D +P ND 2,G = G +P NG 2,H = [ΞΘ + +P N(I 2p+n ΘΘ + ) F = DH [ΞΘ + +P N(I 2p+n ΘΘ + ) I p I p Next, conside a class of special nonlinea descipto system. [ I Coollay Suppose in system ()-(3) the coefficient matix E =, then ẑ in system (4)-(5) is the asymptote estimate of z in system ()-(3), if the following ae hold () (25) ank (26) ank CA C(E I n ) C JA L = ank sl LA L(E I n ) CA C(E I n ) C JA = ank CA C(E I n ) C JA LA L(E I n ) L CA C(E I n ) C JA L, s C,Re(s) > wheej = [ I n (2) function f(t,y,z,u) with espect to z satisfied the expession (2). (3) thee exist positive definite matix P = P T >, matix M and positive eal numbe µ,µ 2, such that the following inequality holds (27) whee PG MG 2 G T P µ I n G T 2 MT µ 2 I n < M = Pλ, = PD +D T P MD 2 D2 T M T +γ 2 (µ +µ 2 )I q D = LA L ΨΩ + CA L,D 2 = (I 2p+n ΩΩ + ) CA L JA L JA L G = L ΨΩ + C,G 2 = (I 2p+n ΩΩ + ) C J J Ω = CA(I n LL) C(E I n ) C(I n LL) JA(I n LL),Ψ = [ LA(I n LL) L(E I n )

11 PARTIAL STATE OBSERVER DESIGN FOR NONLINEAR DESCRIPTOR SYSTEMS Figue : The esponse cuve fo eo system in example 3. numeical example In the system ()-(3) conside the coefficient matix C = [,L = [, then L = [ T. Take nonlinea function f(t,y,z,u) = [ x x 2.5sin 2 (z) 3.5x x 2 T then f(t,y,z,u) satisfied the condition (2) of theoem 2. Example Take coefficient matix E =,A = 2 we can easily veify that the matices taken above satisfy the condition () in theoem 2. Set µ = µ 2 =, solve LMI (3), then P =.492,N = [ then D = 2.275,F =.275,H =.,G= [ T theefoe, we get the figue of the esponse cuve fo eo system, see Figue Example 2 Take coefficient matix E =,A = 2 we can easily veify that the matices taken above satisfy the condition () in coollay 2. Set µ = µ 2 =, solve LMI (27), then P =.5766,M = [ then D =.2334,F =.789,H =.2334,G= [ T theefoe, we get the figue of the esponse cuve fo eo system, see Figue 2: Remak 4 Note that, in example the system is impulsive, in example 2 the system is impulsive fee, so fom these examples, the conclusions in this pape can be applied to the two conditions.

12 368 J. ZHOU AND B. MEN Figue 2: The esponse cuve fo eo system in example 2 4. Conclusions This pape studies the patial state obseve design fo a class of nonlinea descipto systems by Lyapunov stability theoem, expanding the existing esults. Stability condition fo eo systems is given. The method is simple and effective. Finally, two numeical examples ae given to illustate the effectiveness of the obseve design. Refeences [ H. Tinh, T. Fenando, and S. Nahawandi. Patial State Obseves fo Nonlinea Systems. IEEE, 26, 5(): [2 Guoping Lu and Daniel W. C. Ho. Full-Ode and Reduced-Ode Obseves fo Lipschitz Descipto Systems :The Unified LMI Appoach. IEEE Tans. Cicuits and Syst. II: Expess Biefs, 26, 53(7): [3 Jia xiuqin, Liu yungang. H patial state obseve design fo nonlinea systems.journal OF SHANDNG UNIVERSITY(ENGINEERING SCIENCE), 27, 37(5): [4 Li wenlin, Li juntao. Uncetainty obseves fo uncetain systems. Contol theoy and applications, 27 24(6): [5 Daouach M, Zasadzinski M, Haya M., Reduced-ode obseve design fo descipto systems with unknown inputs [J. IEEE Tans, Automal. Cont., 996, 4: [6 Zemouche A, Doutayeb M and Baa G., Obseve design fo nonlinea systems: An appoach based on the diffeential mean value theoem[c. Poceedings of the 44th IEEE Confeence on Decision and Contol and Euopean Contol Confeence, 25: [7 Pan haipeng, Ke ting, Dai wenzhan. Obseve design fo geneal nonlinea system with linea output.contol theoy and applications, 24, 2(): [8 M. Daouach and M. Boutayeb. Design of Obseves fo Descipto Systems. IEEE Tans,Automal.Cont., 995, 4(7): [9 Masau Hasegawa. Robust-Adaptive-Obseve Design Based on γ-positive Real Poblem fo Sensoless Induction-Moto Dives. IEEE TRANSACTIONS ONINDUSTRIAL ELEC- TRONICS, 26, 53(): [ Hyungbo Shim and Jin H. Seo. Recusive Nonlinea Obseve Design: Beyond the Unifom Obsevability.IEEE TRANSACTIONS ON AUTOMATIC CONTROL, 23, 48(2): [ J. Tsinias, Obseve Design fo nonlinea systems. Systems and Contol Lettes, 989, 3(2): [2 F. Thau, Obseving the states of nonlinea dynamical systems. Intenational Jounal of Contol, 973, 8: [3 Ben Bian, Jing Wang, and Zhihua Qu, Nonlinea Positive Obseve Design fo Positive Dynamical Systems. 2 Ameican Contal Confeence, 2: [4 M. M. Fahmy and J.O Reilly, Obseves fo descipto systems, Int. J.Contol, 989, 49(6):

13 PARTIAL STATE OBSERVER DESIGN FOR NONLINEAR DESCRIPTOR SYSTEMS 369 [5 Salim Ibi, Algebaic obseve design fo a class of unifomly-obsevable nonlinea systems: Application to 2-link obotic manipulato, Poceedings of the 7-th Asian Contol Confeence, 29: [6 El-Tohami E, Lovass-Nagy V, Mukundan R. On the design of obseves fo genealized state space systems using singula value decomposition. Int. J. Cont., 983, 38(3): [7 M. Daouach and M. Boutayeb, Design of obseves fo descipto systems, IEEE Tans. Autom. Contol., 995, 4(7): [8 Sheyas Sundaam,, Chistofoos N. Hadjicostis. Patial state obseves fo linea systems with unknown inputs. Automatica, 28, 44: [9 Yang chunyu,zhang qingling, Chai tianyou, Nonlinea obseves fo a class of nonlinea descipto systems. Optim Contol Appl.Meth. 22, : -6. [2 José M. Aaújo, Péicles R.Baos, Calos E. T. Dóea, Design of obseves with eo limitation in discete-time descipto systems: A case study of a hudaulic tank system. IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY. 2: -7. Institute of Systems Science, Notheasten Univesity, Shenyang, Liaoning, 4, P.R. China @qq.com School of Mathematics and Systems Science, Shenyang Nomal Univesity, Shenyang, Liaoning, 34, P.R. China menbobo2@yahoo.com.cn