WSEAS TRANSACTONS on SYSTEMS and CONTROL M. Khadhraoui M. Ezzine H. Messaoud M. Darouach Design of Unknown nput Functional Observers for Delayed Singular Systems State Variable Time Delay M. Khadhraoui M. Ezzine H. Messaoud Laboratoire de recherche LARATS Ecole Nationale d ngénieurs de Monastir Université de Monastir Avenue bn El Jazzar 519 Monastir Tunisie malek.enim@gmail.com montassarezzine@yahoo.fr Hassani.Messaoud@enim.rnu.tn M. Darouach Centre de Recherche en Automatique de Nancy CRAN - CNRS UMR 739 Université de Lorraine UT de Longwy 186 rue de Lorraine 544 Cosnes et Romain France darouach@iut-longwy.uhp-nancy.fr Abstract: n this paper we propose a new time-domain design of an Unknown nput functional Observer (UFO for delayed singular systems known and unknown inputs. The order of this observer is equal to the dimension of the vector to be estimated. Constant and variable delays act on the known input vectors and a variable state delay is also taken into account. The proposed approach is based on the unbiasedness of the estimation error and Lyapunov-Krasovskii stability theorem. The observer optimal gain satisfies a sufficient condition of the observer stability dependent on the state delay. This condition is expressed in term of Linear Matrix nequalities (LMs formulation. The proposed approach is tested on a numerical example. Key Words: LM observer stability singular system unknown input variable time delay. 1 ntroduction Singular models are of great interest. Despite their complicated structure these models known also as generalized mathematical representations are the most adopted in researches for physical system description particularly when singular systems are concerned [18]. Functional observing problems is of great interest ([3 6 8 1]. n fact it s equivalent to find an observer that estimates a linear combination of the states of a considered systems using the input and output measurement. This functional state can be used on control purposes ([9 16]. n addition linear modeling delayed state interests a large class of physical processes. t highlights the input propagation delay through the dynamics of the system to reach the output which clearly affect by the system stability [17]. Therefore we review some techniques of stability detection based on Lyapunov-Krasovskii stability theorem in order to ensure the stability of the observer dynamics in spite of delays [15]. Observer design theorem for delayed systems has been investigated over the last decade and several design techniques have been proposed ([3 5 14]. n addition the observers for systems unknown input are of great interest in the fault detection and the control of systems in presence of disturbances [5 6]. However there is less literature about observer design unknown input for singular systems variable state delay [13]. n this paper and based on [7] a time domain method of a functional observer design for delayed singular systems unknown input is proposed. n fact we aim to reconstruct a functional state independently from the considered constant time delay acting on the known input vector and the unknown input vector. The observer design is based on a sufficient condition dependent on the state variable time delay and based on the Lyapunov-Krasovskii stability theorem [15]. This paper is organized as follows. Section 2 gives assumptions used through this paper and formulates the functional observer problem to be solved. Section 3 presents the contribution of the paper by giving the design procedure of a functional observer in the time domain. Using the unbiasedness condition the problem is transformed into a matrix inequalities. LM approach is then applied and the E-SSN: 2224-2856 53 Volume 1 215
WSEAS TRANSACTONS on SYSTEMS and CONTROL observer optimal gain is given as a solution of an LM condition depending on the variable state delay. The fourth section summarizes the UFO design steps. Section 5 gives a numerical example to illustrate our approach and section 6 concludes the paper. 2 Problem Formulation Let s consider the following continuous-time linear time-delay singular system described by: Eẋ(t = Ax(t A d x(t τ 1 (t q v q d B vj u(t τ 2j (t B dj u(t τ 3j Bu(t E 1 v(t (1a y(t = Cx(t (1b m(t = Lx(t (1c x(t = φ (1d x(t R n is the state vector m(t R mz is the functional of the state to be estimated y(t R p is the output vector u(t R q is the known input vector and v(t R r is the unknown input. E A A d B C B vj(1 j qv B dj (1 j qd E 1 and L are known matrices of appropriate dimensions. φ is the initial state τ 1 (t R is the state variable delay τ 2j (t is the known input variable delay and τ 3j R is the known input constant delay. Note that q v and q d are positive integers. The variable state delay satisfies the following condition: τ 1 (t τ t R (2 τ R n the sequel we suppose that : Hypothesis 1 [7] 1. rank (E = r 1 n [ E 2. rank C ] = n The main objective of this paper is to design in time domain an unknown input functional observer for delayed singular linear systems. The considered systems is affected by a bounded variable time delay acting on the state vector and by a variable and a constant delay associated both to the known input vector u(t. 3 UFO Time Domain Design Under hypothesis 1 there exists a non singular matrix ( a b S = (3 c d such that a E b C = n (4 c E d C = p n (5 The seeked functional observer for system (1 is of the form : ż(t = Nz(t N d z(t τ 1 (t q v q d H vj u(t τ 2j (t H dj u(t τ 3j Hu(t D 1 y(t D 2 y(t τ 1 (t (6a ˆm(t = z(t My(t (6b M = L(b E 2 d (7 z is the state of the observer and ˆm(t R n is the estimate of the functional m(t. Matrices N N d H vj(1 j qv H dj (1 j qd H D 1 D 2 and M will be determined in the sequel using LM approach. 3.1 UFO conditions of time-delay singular systems The estimation error e(t can be given from (1c and (6b using (4 and (5 as : M. Khadhraoui M. Ezzine H. Messaoud M. Darouach e(t = m(t ˆm(t (8a = L( n b C E 2 d Cx(t z(t (8b = GEx(t z(t (8c G = L(a E 2 c (9 Purpose : Given the singular system (1 and the functional observer (6 we aim to design the observer matrices N N d H H vj(1 j qv H dj (1 j qd D 1 D 2 and E 2 so that ˆm converges asymptotically to m so : lim e(t = (1 e(t is given by (8a. To do so we propose the following theorem : E-SSN: 2224-2856 54 Volume 1 215
Theorem 1 The functional observer (6 is an UFO for singular model (1 if and only if the following conditions are satisfied : i ė(t = Ne(t N d e(t τ 1 (t is asymptotically stable. ii GA NGE D 1 C = iii GA d N d GE D 2 C = iv GE 1 = v H = GB vi H vj = GB vj 1 j q v vii H dj = GB dj 1 j q d Proof 1 The derivative of (8c is given as follows : ė(t = GEẋ(t ż(t (11 By replacing Eẋ(t and ż(t by their expressions in (1 and (6a respectively relation (11 becomes : ė(t = Ne(t N d e(t τ 1 (t GE 1 v(t q v q d (GB vj H vj u(t τ 2j (t (GB dj H dj u(t τ 3j (GA d N d GE D 2 Cx(t τ 1 (t (GA NGE D 1 Cx(t (GB Hu(t (12 the initial condition e = m ˆm. So the unknown input functional observer (6 will estimate asymptotically the real functional of the state m(t for any initial conditions any u(t u(tτ 2j (t u(t τ 3j and independently of the unknown input v(t if and only if conditions i - vii are satisfied. 3.2 UFO time domain design By replacing G by its expression given by (9 in conditions ii - iv of theorem 1 and according to (4 and (5 we have : WSEAS TRANSACTONS on SYSTEMS and CONTROL La A = NLa E F 1 C LE 2 c A (13 La A d = N d La E F 2 C LE 2 c A d (14 La E 1 = LE 2 c E 1 (15 F 1 = D 1 NLE 2 d (16 F 2 = D 2 N d LE 2 d (17 Equations (13 - (15 can be written in the following matrix form : XΣ = Θ (18 M. Khadhraoui M. Ezzine H. Messaoud M. Darouach X = [ N N d F 1 F 2 LE 2 ] La E Σ = La E C C c A c A d c E 1 Θ = [ La A La A d ] La E 1 (19 (2 (21 Note that a general solution of (18 exists if and only if [ ] Σ rank = rank(σ (22 Θ So under condition (22 we can have : X = ΘΣ Z( ΣΣ (23 Σ is the generalized inverse of the matrix Σ and Z is an arbitrary matrix of appropriate dimensions that will be determined in the sequel using LM approach. is the identity matrix of appropriate dimension. The unknown matrix N in (19 can be given by: N = X (24 By replacing (23 in (24 we obtain : N = ΘΣ Let s consider : Z( ΣΣ A 11 = ΘΣ (25 (26 E-SSN: 2224-2856 55 Volume 1 215
and Then B 11 = ( ΣΣ (27 N = A 11 ZB 11 (28 Similarly for matrix N d we obtain : and N d = A 22 ZB 22 (29 A 22 = ΘΣ B 22 = ( ΣΣ (3 (31 At this stage and based on theorem 1 and Lyapunov- Krasovskii stability theorem one can get the gain matrix Z which parametrizes the observer matrices as proposed in theorem 2. Theorem 2 : The proposed observer (6 is an UFO for singular system (1 the delay τ 1 (t satisfying (2 if there exist P = P T > X = X T > and X 1 = X T 1 > such that : Ψ = Q τ P N d τ P N d τ Nd T P τ 2 X τ N T d P τ 2 X 1 < (32 Q = α α T τ 2 N T X N τ 2 N d T X 1 N d (33 and WSEAS TRANSACTONS on SYSTEMS and CONTROL Note that τ is given by (2. α = (N N d T P (34 Proof 2 The chosen Lyapunov functional is : V (t = V 1 (t τ 1 (tv 2 (t τ 1 (tv 3 (t (35 whith V 1 (t = e(t T P e(t (36 V 2 (t = V 3 (t = τ1 (t t tθ τ1 (t t tθ e(s T N T X Ne(sdsdθ e(s T N T d X N d e(sdsdθ (37 (38 Where P X and X 1 are symmetric positive definite matrices of dimension n. According to condition (2 V (t t R and knowing that e(s T N T X Ne(s and e(s T N T d X 1N d e(s are positive scalars we can write : M. Khadhraoui M. Ezzine H. Messaoud M. Darouach V (t V (t (39 V (t = V 1 (t τ V 21 (t τ V 31 (t (4 V 21 (t = V 31 (t = t tθ t tθ e(s T N T X Ne(sdsdθ e(s T N T d X N d e(sdsdθ (41 (42 So according to (39 if lim V (t = then lim V (t =. To prove that V (t when t we can prove that V (t < (See [1]. The derivative of the functional V (t is : V (t = V 1 (t τ V21 (t τ V31 (t (43 By applying the Leibniz s transformation on (6 we have: ė(t = (N N d e(t N d N τ 1 (t e(t θdθ τ1 (t N d N d e(t θdθ (44 2τ 1 (t E-SSN: 2224-2856 56 Volume 1 215
so : V 1 (t = e(t T [(N N d T P P (N N d ]e(t We have : so τ 1 (t e(t T P N d N V 21 (t = τ1 (t 2τ 1 (t e(t T P N d N e(t θ T (NN d T P e(tdθ τ 1 (t e(t θdθ e(t θ T (NN d T P e(tdθ τ1 (t 2τ 1 (t [e(t T N T X Ne(t e(t θdθ (45 e(t θ T N T X Ne(t θ]dθ (46 V 21 (t = τ e(t T N T X Ne(t Let s so we write : and Let s : so WSEAS TRANSACTONS on SYSTEMS and CONTROL e(t θ T N T X Ne(t θ]dθ (47 ɛ(t θ = Ne(t θ R n (48 V 21 (t = τ e(t T N T X Ne(t V 31 (t = ɛ(t θ T X ɛ(t θ]dθ [e(t T N T d X 1N d e(t (49 e(t θ T N T d X 1N d e(t θ]dθ (5 ɛ d (t θ = N d e(t θ R n (51 V 31 (t = τ e(t T N T d X 1N d e(t ɛ d (t θ T X 1 ɛ d (t θdθ (52 Uniform asymptotic stability of (4 implies that : lim V (t (53 As θ is bounded the quantities ɛ(t θ and ɛ d (t θ respectively given by (48 and (51 satisfy : and and consequently lim ɛ(t θ = lim ɛ(t (54 lim ɛ d(t θ = lim ɛ d(t (55 lim ( ɛ(t θ T X ɛ(t θdθ = τ lim ɛ(tt X ɛ(t (56 and lim ( ɛ d (t θ T X 1 ɛ d (t θdθ = τ lim ɛ d(t T X 1 ɛ d (t (57 We set the variable s changes: γ = lim ɛ(t (58 ν = lim d(t (59 The equations (56 and (57 can be written as : lim ( ɛ(t θ T X ɛ(t θdθ = τ γ T X γ (6 and lim ( ɛ d (t θ T X 1 ɛ d (t θdθ = τ ν T X 1 ν (61 lim M. Khadhraoui M. Ezzine H. Messaoud M. Darouach We suppose that ξ = lim e(t we have : V (t = [ξ T [(N N d T P P (N N d ]ξ τ 2 ξ T N T X Nξ τ 2 ξ T N d T X 1 N d ξ] [τ 1 (tγ T N T d P ξ τ 1(tξ T P N d γ] [τ 1 (tν T N T d P ξ τ 1(tξ T P N d ν] [τ 2 γ T X γ τ 2 ν T X 1 ν] (62 E-SSN: 2224-2856 57 Volume 1 215
and since γ T Nd T P ξ is a scalar term we can write τ 1 (tγ T N T d P ξ τ 1(tξ T P N d γ = 2τ 1 (tγ T N T d P ξ (63 Let us define : ρ = P N d γ (64 and K a positive definite matrix such that : ρ = Kξ (65 Since that for any vector a R n and b R n we have: so 2a T b a T K 1 a b T Kb (66 2τ 1 (tρ T ξ τ 1 (t(ρ T K 1 ρ ξ T Kξ (67 Since ρ T K 1 ρ R ξ T Kξ R and < τ(t τ then τ 1 (t(ρ T K 1 ρ ξ T Kξ τ (ρ T K 1 ρ ξ T Kξ (68 Which results in : τ 1 (t(ρ T K 1 ρ ξ T Kξ τ (ρ T ξ ξ T ρ (69 Similarly: τ 1 (tν T N T d P ξ τ(tξt P N d ν Therefore: lim WSEAS TRANSACTONS on SYSTEMS and CONTROL τ ν T N T d P ξ τ ξ T P N d ν (7 V (t [ξ T [(N N d T P P (N N d ]ξ which can be written as : lim τ 2 ξ T N T X Nξ τ 2 ξ T N d T X 1 N d ξ] [τ γ T N T d P ξ τ ξ T P N d γ] [τ ν T N T d P ξ τ ξ T P N d ν] [τ 2 γ T X γ τ 2 ν T X 1 ν] (71 V (t [ ξ T γ T ν T ] Ψ Ψ is given by (32. According to condtition (53 if then Ψ which satisfies theorem 2. lim ξ γ ν (72 V (t To avoid the quadratic form present in equation (32 of theorem 2 we propose a congruence transformation of Ψ using the Schur lemma ([2]. Once equivalence is established we can express the observer optimal gain Z as a solution of a LM. n fact the chosen Lyapunov functional given by (35 can be modified by choosing X = X 1 = P and theorem 2 is equivalent to : Theorem 3 The proposed observer (6 is an UFO for singular system (1 the delay τ 1 (t satisfying (2 if there exist P = P T > and Y such as the symmetrical matrix Π is negative : β 11 β 12 β 13 β 14 β 15 β 16 β 22 β 23 β 24 β 25 β 26 Π = β 33 β 34 β 35 β 36 β 44 β 45 β 46 (73 β 55 β 56 β 66 β 11 = β 22 = β 33 = β 56 = P (74a β 12 = β 13 = β 15 = β 16 = β 23 = β 24 = β 35 = β 36 = n n (74b β 14 = τ P A 11 τ Y B 11 (74c β 25 = β 26 = P (74d β 34 = β 45 = β 46 = τ P A 22 τ Y B 22 (74e β 44 = A T 11P B11Y T T P A 11 Y B 11 A T 22P B T 22Y T P A 22 Y B 22 (74f β 55 = β 66 = (τ 2 1P (74g The observer gain Z is given by : Z = P 1 Y (75 Proof 3 Matrix Ψ can be written as : M. Khadhraoui M. Ezzine H. Messaoud M. Darouach S = H = M = Ψ = M S T H 1 S (76 τ N n n n n n n n n τ N d n n n n P 1 n n n n n n P 1 n n n n n n P 1 α α T τ P N d τ P N d β 55 P β 66 (77a (77b (77c E-SSN: 2224-2856 58 Volume 1 215
According to the Schur lemma Ψ < and H < if and only if the : ( H S Ω = S T < (78 M We apply a congruence transformation to Ω such as : Π = T T ΩT < (79 T is a non singular matrix given by : P....... P.... T =.... P........ n....... n......... n (8 then by replacing N and N d by their expressions given respectively by (28 and (29 in Π theorem 3 holds. Once Z is calculated using (75 all observer matrices can be given by : F 1 = A 33 ZB 33 (81 A 33 = ΘΣ B 33 = ( ΣΣ The matrix F 2 is given by : WSEAS TRANSACTONS on SYSTEMS and CONTROL (82 (83 F 2 = A 44 ZB 44 (84 A 44 = ΘΣ B 44 = ( ΣΣ (85 (86 and M. Khadhraoui M. Ezzine H. Messaoud M. Darouach LE 2 = A 55 ZB 55 (87 A 55 = ΘΣ B 55 = ( ΣΣ 4 UFO design steps Summary Step 1 Verify hypothesis 1. Step 2 Get the non singular matrix S verifying (5 and (6. Step 3 Compute matrices Σ and Θ from (19 and (2. Step 4 Verify the regular condition (21. Step 5 Solve the LM (73 to obtain P and Y. Step 6 Compute the matrix Z from (75. Step 7 Compute N N d F 1 F 2 and LE 2 using equations (28 (29 and (81-(89. Step 8 Compute M using equation (7. Step 9 Compute D 1 and D 2 using (16 and (17. Step 1 Get matrices H H vj(1 j qv and H dj(1 j qd using respectively conditions v vi and vii from theorem 1. So all observer matrices are known. 5 Numerical example (88 (89 Let s consider system (1 q v = 1 q d = 1 and ( ( ( 1 2 3 1 E = A = B = 2 4 1 A d = B v = (.5 1 1 ( 1 L = ( B d = 1 ( 1 E 1 =.279 (.5 5.23 3.87 C = ( 1 1 The variable state delay is a sinusoid such : E-SSN: 2224-2856 59 Volume 1 215
WSEAS TRANSACTONS on SYSTEMS and CONTROL τ 1 (t = 1 4 (sin(2πt 1 Obviously we have τ 1 (t τ =.5 t R. The constant known input delay is evaluated at τ 31 = 1s. The variable input delay τ 21 (t has the following form: ( E We have rank C verified. Figure 1: The nput Delay = 2 so hypothesis 2 is According to equations (4 and (5 we have: ( (.2.4 a = b.2.4 = 1 c = ( 1.5 d =. The resolution of the LM (73 gives : ( 3.2351 44.7667 P = 4.7667 66.2824 M. Khadhraoui M. Ezzine H. Messaoud M. Darouach Z 1 = (.144.75 2.3217.75.476 1.568 Z 2 = ( 1.568.2958.916.181 1.59.1998.6187.1222 So the functional observer matrix values are given as follows : N = N d = ( 126.185 181.3236 84.9996 122.1363 ( 15.545 24.557 1.3118 16.3489 D 1 = D 2 = H = H v = H d = LE 2 = ( 6.1162 4.138 ( 3.3295 2.2487 (.9289.6274 (.9289.6274 ( 3.3295 2.2487 ( 2.749 1.414 Next simulations are carried out using a known input given by Figure 2. Y 1 = Y = ( Y 1 Y 2 ( 9.9282 14.6993 7.1634 14.6994 21.7644 1.5945 Y 2 = ( 1.5945.7.22.4 15.6943.5.15.3 Computing Z from equation (75 gives : Z = 1 8 ( Z 1 Z 2 Figure 2: The known input : u(t Figures 3 and 4 show the real and the estimated E-SSN: 2224-2856 51 Volume 1 215
WSEAS TRANSACTONS on SYSTEMS and CONTROL M. Khadhraoui M. Ezzine H. Messaoud M. Darouach components of the state functional and Figures 5 and 6 illustrate the evolution of the state error components. We note the observer independence of the unknown input and so the effectiveness of the proposed approach. Figure 6: The Second Compoent of the Estimation Error Figure 3: First States Component 6 Conclusion n this paper a new time-domain design is proposed of an unknown input functional observer for singular systems variable time delays acting on both state vector and known input vector. A constant time delay is also considered in the known input vector. First we ensured the unbiasedness of the estimation error. Then the observer gain that parametrizes all functional observer matrices is an optimal solution of LMs conditions dependent on the bounded state delay. So the estimation error converges for any constant time delay any known input and independently from the unknown input. The application of the proposed design procedure on a numerical example shows the effectiveness of the proposed approach. References: Figure 4: Second States Component [1] Y. Ariba Sur la stabilité des systèmes à retards variant dans le temps : Théorie et application au contrôle de congestion d un routeur. Ph.D. dissertation Université de Toulouse 3 Paul Sabatier France 29. [2] S. Boyd and L. Vandenberghe "Convex Optimization" Cambridge University Press First edition 24. [3] M. Darouach "Linear Functional Observers for Systems With Delays in State Variables : The Discrete-Time Case" EEE Transactions on Automatic Control Vol. 5 No. 2 pp. 228-233 25. Figure 5: The First Component of the Estimation Error [4] M. Darouach and M. Boutayeb "Design of observers for descriptor systems" 4 th EEE Transactions on Automatic Control no. 7 1323-1327 1995. E-SSN: 2224-2856 511 Volume 1 215
WSEAS TRANSACTONS on SYSTEMS and CONTROL M. Khadhraoui M. Ezzine H. Messaoud M. Darouach [5] M. Ezzine M. Darouach H. Souley Ali and H. Messaoud "Synthèse temporelle et fréquentielle de filtres H d ordre plein pour les systèmes singuliers" Conférence nternationale Francophone d Automatique 21. [6] M. Ezzine M. Darouach H. Souley Ali and H. Messaoud "functional H filtering for Linear systems in the frequency domain : The time and frequency domain cases" nternational Journal of Control Automation and Systems pp. 558-565 211. [7] M. Ezzine M. Darouach H. Souley Ali and H. Messaoud "Unknown nputs Functional Observers Designs For Descriptor Systems Constant Time Delay" 18 th FAC Milano-taly 211. [8] M. Ezzine M. Darouach H. Souley Ali and H. Messaoud "Time and Frequency domain design of Functional Filters" American Control Conference pp. 599-64 Baltimore MD USA 21. [9] M. Ezzine M. Darouach H. Souley Ali and H. Messaoud "A controller design based on a functional H filter for descriptor systems : The time and frequency domain cases" Automatica pp. 542-549 212. [1] A. Fattouh O. Sename and J-M. Dion "Robust observer design for time-delay systems: a Riccati equation approach" Kybernetika Vol. 35 No. 6 pp. 753-764 1999. [11] J. Hale and S. Lunel "ntroduction to functional differential equations" Applied Mathematical Sciences 99 Springer-Verlag 1991. [12] M. Khadhraoui M. Ezzine H. Messaoud and M. Darouach "Design of Full Order Observers Unknown nputs for Delayed Singular Systems Constant Time Delay" nternational Conference on Control Decision and nformation Technologies (CODT214 pp. 423-428 Metz-France 214. [13] D. Koenig D. Jacquet and S. Mammar "Delaydependent H observer of linear delay descriptor systems" Proc. American Control Conference pp.3813-3817 26. [14] D. Koenig and B. Marx "Design of Observers for descriptor systems delayed state and unknown inputs" Proc. American Control Conference pp.486-481 24. [15] N. Krasovskii "Stability of Motion" Stanford University Press 1963. [16]. Masubuchi Y. Kamitane A. Ohara and N. Suda "H Control for descriptor systems: A matrix inequalities approach" Automatica vol. 33 no. 4 pp. 669-673 1997. [17]. S. Niculescu "Delay effect on stability A robust control approach" Springer Verlag 21. [18] G. Verghese B. Levy and T. Kailath "A generalized state-space for singular systems" EEE Transactions on Automatic Control pp. 811-831 1981. [19] M. Vidyasagar " Control systems Synthesis: A factorization approach" M..T Press Cambridge 1985. E-SSN: 2224-2856 512 Volume 1 215