WSEAS RANSACONS on SYSEMS A Controller A Controller Design Design based based on on a Functional H1 H Filter Filterfor for Delayed Delayed Singular Systems : he ime ime Frequency Domain Domain Cases Cases M. KHADHRAOU, M. Khadhraoui, M. M. EZZNE, Ezzine, H. Messaoud MESSAOUD M. DAROUACH Darouach Laboratoire de recherche LARAS Centre de Recherche en Automatique de Nancy Ecole Nationale d ngénieurs de Monastir CRAN - CNRS UMR 739 Université de Monastir Université de Lorraine, U de Longwy Avenue bn El Jazzar, 519 Monastir 186 rue de Lorraine 544 Cosnes et Romain UNSAunisie FRANCErance malek.enim@gmail.com darouach@iut-longwy.uhp-nancy.fr montassarezzine@yahoo.fr Hassani.Messaoud@enim.rnu.tn Abstract: his paper deals the time frequency domain of a controller design based on a function reduced order filter for linear multi-variable delayed singular systems measurements are affected by bounded disturbances. he control gain is designed using H techniques. he time procedure design is based on the unbiasedness of the estimation error using Sylvester equation on Lyapunov-Krasovskii stability theory. hen a new method to avoid the time derivative of the disturbance in filtering error is proposed the problem is solved by means of Linear Matrix nequalities (LMs. Both cases the H technique is dependent or independent from the delay are dervied separetely. he frequency domain approach is derived from the time domain one by applying the factorization approach. A numerical example is given to illustrate the proposed approach. Key Words: Controller, singular systems, H functional filter, delay, disturbance. 1 ntroduction Singular models or descriptor models have been a topic of recurring interests of many researches. n fact, these generalized mathematical representations better describe physical systems than regular ones [23]. So a great deal of work based on the theory of state-space systems has been extended to the descriptor models. Furthermore, delay modeling has been extensively studied as it influences the stability robustness other performances of the systems [22]. his situation becomes obvious when dealing communication networks, economic systems chemical processes [1, 22, 2]. On the other h, a great deal of work has been devoted to the design of the filter-based controller for delayed singular systems [5, 16, 8, 9]. his controller is of major importance, mainly, when the states of the systems is partially measurable. his kind of controller is getting more more interesting especially that a great part of control designs are developed the assumption that the state components of the system are available for the feedback [11]. However, only a few part of the state can be measured. Motivated by these facts, a recurring interests of researches are focused on the development of filtering techniques to estimate a functional of state which can be used, also, on control laws based on the state feedback principle [6]. n addition, these controllers are hled by H filters able to minimize the perturbation effect on the estimation error. However, in the frequency domain, few results has been developed to the controller design based on a functional H filter for singular delayed systems [1, 11]. n this framework, a new time frequency domain design procedure of filter-based controller for singular delayed systems is proposed. he time domain approach is obtained into two steps. Firstly, we give conditions ensuring the admissibility of the H problem. Secondly, we propose a functional H filter design essential to derive the control law. he estimation problem is extended to a singular one in order to avoid the time derivative of the disturbance (ẇ(t on the estimation error dynamic. his filter, based on unbiasedness conditions, estimate a functional of state according to a H criteria E-SSN: 2224-2678 131 Volume 16, 217
WSEAS RANSACONS on SYSEMS is proposed by means of LMs conditions. hese conditions are of two kinds. One satisfies the H criteria independently from the delay the second respects the same criteria but dependently on the state delay. Furthermore based on the time domain results, a frequency domain approach is set to the design of the functional H filter based controller using MFDs. he main reason of formulating the results of the time domain in the frequency one is the advantages that it presents for the filter-based control. n fact, the compensator is driven by the input the output of the system. So, only the input-output behavior of the compensator (characterized by its transfer function influences the properties of the closed-loop system. he outlines of the paper are as follows. Section 2 gives assumptions used through this paper presents the functional filter-based controller problem that we propose to solve. Section 3, presents the first contribution of the paper by giving the time domain design of the controller. his contribution is presented in two parts. First, the state feedback gain is designed respect to the H performance criteria satisfying the problem admissibility. Second, we propose to design the filter-based controller using the unbiasedness condition, dependently independently from the state delay. he problem is transformed into a matrix inequalities to solve. A LM approach is then applied to optimize the gain implemented in the filter. he fourth section presents the second result of the paper by giving a frequency domain description of the controller using polynomial MFDs. A summary of the filter based controller is presented in the fifth part of the article. Section 6 gives numerical examples to illustrate our approaches section 7 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 d + Bu(t +B d u(t d + D 1 w(t (1a z(t = F 1 x(t (1b y(t = Cx(t + D 2 w(t (1c x(t = φ (1d x(t R n is the state vector, y(t R q is the output vector, u(t R p is the input vector, w(t R m is the bounded disturbance z(t R mz is the functional state. E, A, A d, B, C, B d, D 1, F 1 D 2 are known matrices of appropriate dimensions. φ is the value of the initial state, d R + is the considered delay. n the sequel, we suppose that : Hypothesis 1. [5] 1. rank(e = r 1 n [ E 2. rank( C ] = n Purpose: he main objective of this paper is to design in the time the frequency domain a controller based on a functional H filter for delayed singular systems. he same delay is assumed in the state the input vectors. 3 ime domain design of the filterbased controller We propose to solve the observer-based controller problem into two steps. First, we propose to design the control gain K c satisfying the admissibility of the subsystem {(1a-(1b}. Second, we aim to design a filter-based controller in order to reconstruct the control law by estimating only the state functional essential to the controller design. Under hypothesis 1, ( there exists a non singular a b matrix S S = a c d R n n, b R n q, c R q n d R q q such that : a E + b C = n (2a c E + d C = q n (2b E C are given in (1. he purpose of the paper is to design a functional filter-based controller for system (1 of the form : χ(t = F χ(t + F d χ(t d + H d u(t d +Hu(t + L 1 y(t + L 2 y(t d (3a u(t = χ(t + My(t (3b M = K c b + Kd. Matrices F, F d, H, H d, L 1, L 2, K c K are to be designed. Problem: E-SSN: 2224-2678 132 Volume 16, 217
Our main objective is to build a functional filter a control law following (3a (3b in time domain its equivalent descriptor in the frequency domain such that : a lim u K c x =, if w =. b he system state vector the filtering error are asymptotically stable satisfy the H performance. he H criteria is given by: < H ɛw = sup w ɛ 2 w 2 < γ (4 ɛ(s w(s H ɛw (s = is a transfer matrix, γ is a positive scalar ɛ = u K c x is the estimation error. 3.1 State feedback synthesis Let us consider the subsystem given by (1a (1b : Eẋ(t = Ax(t + A d x(t d + Bu(t +B d u(t d + D 1 w(t (5a z(t = F 1 x(t (5b Lemma 1. he system (5 is admissible satisfies the H performance requirement given by (4 if only if there exist a matrix X a symmetric definite positive matrix Z such that : WSEAS RANSACONS on SYSEMS V = X E = EX (6 V < (7 α A d X X F 1 Z n (8 α = AX + X A + Z + γ 2 D 1 D 1 (9 By replacing u by K c x in system (5, we have: Eẋ(t = (A + BK c x(t + (A d + B d K c x(t d +D 1 w(t (1a z(t = F 1 x(t (1b So results given by Lemma 1 can be applied on system (1 in order to design the state feedback K c according to the next proposed theorem. heorem 2. he system (1 is admissible satisfies the H criteria if only if there exist matrices : X c, Y c Z c = Z c > such that : V c = X c E = EX c (11 V c < (12 α c β x X c F 1 Z c n α c = AX c + BY c + (AX c + BY c (13 +Z c + γ 2 D 1 D 1 (14 β x = A d X c + B d Y c (15 So the state feedback gain is given by : K c = Y c X 1 c (16 Proof 1. t s obtained by applying the results of [25] on system (1 using a transformation of the main result according to the Schur lemma [3]. n fact, the system considered on [25] is a delayed singular system uncertain parameters. When setting the uncertainty components to zero, the admissibility conditions of the H problem is given by: M c = X c E = EX c (17 α c β x Xc F1 Z c n ɛ c < (18 ɛ c is a positive scalar. By realizing that M c is a bloc-diagonal matrix ɛ c is a negative matrix, then equation (13 is the condition for the H problem admissibility. 3.2 Filter-based controller synthesis 3.2.1 he unbiasedness conditions of the filterbased controller Let ɛ(t be the estimation error: Considering (2 (3b, ɛ(t is given by : ɛ(t = u(t K c x(t (19a = χ(t (K c a + Kc Ex(t +(K c b + Kd D 2 w(t (19b = χ(t Ψ 1 Ex(t + Ψ 2 w(t (19c E-SSN: 2224-2678 133 Volume 16, 217
Ψ 1 = K c a + Kc (2 Ψ 2 = (K c b + Kd D 2 (21 Given the singular system (1 the functional filter-based controller (3, we aim to design the filter matrices F, F d, H, H d, L 1, L 2 M which verify the unbiasedness estimation error conditions (if w(t = the attenuation of the disturbance effect given by (4 (if w(t he unbiasedness of the estimation error dynamics is verified according to the following proposed theorem : heorem 3. he unbiasedness of the estimation error given by (19 relative to system (1 filter (3 is verified such that: ɛ(t = F ɛ(t + F d ɛ(t d +αw(t + βw(t d ζẇ(t (22 if only if the following equations are satisfied : i L 1 C + F Ψ 1 E Ψ 1 A = ii L 2 C + F d Ψ 1 E Ψ 1 A d = iii H = Ψ 1 B iv H d = Ψ 1 B d WSEAS RANSACONS on SYSEMS α = L 1 D 2 F Ψ 2 Ψ 1 D 1 (23 β = L 2 D 2 F d Ψ 2 (24 ζ = Ψ 2 (25 Proof 2. he derivative of the estimation error is given as follows : ɛ(t = χ(t Ψ 1 Eẋ(t + Ψ 2 ẇ(t (26 By replacing in (26 Eẋ(t χ(t by their expressions given by (1 (3 respectively, we have: ɛ(t = F ɛ(t + F d ɛ(t d (Ψ 1 B Hu(t (Ψ 1 B d H d u(t d +(L 2 D 2 F d Ψ 2 w(t d +(L 1 D 2 F Ψ 2 Ψ 1 D 1 w(t +(L 2 C + F d Ψ 1 E Ψ 1 A d x(t d +(L 1 C + F Ψ 1 E Ψ 1 Ax(t +Ψ 2 ẇ(t (27 the initial condition ɛ = u K c x Purpose 1. At this stage, we propose to design the filter-based controller dependently independently from the state delay using the obtained state feedback gain (16. By replacing, in condition i of theorem 1, Ψ 1 Ψ 2 by their expressions in (2 (21 respectively, we have : : F K c a E + JC Kc A = K c a A (28 J = L 1 F Kd (29 Similarly for condition ii of theorem 1, we obtain : F d K c a E + J d C Kc A d = K c a A d (3 J d = L 2 F d Kd (31 Equations (28-(31 can be written in the following matrix form : XΣ = Θ (32, X = [ F F d K J J d ] K c a E Σ = K c a E c A c A d C C Θ = [ K c a A ] K c a A d (33 (34 (35 Note that a general solution of (32, exists if only if [ ] Σ rank = rank(σ (36 Θ n this case, the general solution for (32 is given by: X = ΘΣ + Z( ΣΣ + (37 Σ + is the generalized inverse of matrix Σ given by (34 Z is an arbitrary matrix of appropriate dimensions, that will be determined in the sequel using LM approach. he unknown matrix F in (33 can be given by : F = X (38 E-SSN: 2224-2678 134 Volume 16, 217
WSEAS RANSACONS on SYSEMS By replacing (37 in (38, we obtain : F = ΘΣ + Z( ΣΣ+ Let s consider : hen, F 11 = ΘΣ + F 22 = ( ΣΣ + Similarly for matrix F d, we obtain : We have : (39 (4 (41 F = F 11 ZF 22 (42 F d = F d11 ZF d22 (43 F d11 = ΘΣ + F d22 = ( ΣΣ + (44 (45 J = J 11 ZJ 22 (46 J 11 = ΘΣ + J 22 = ( ΣΣ + (47 (48 Similarly for matrix J d, we obtain :, : J d = J d11 ZJ d22 (49 J d11 = ΘΣ + J d22 = ( ΣΣ + (5 (51 K = K 11 + ZK 22 (52 K 11 = ΘΣ + K 22 = ( ΣΣ + By combining (21 (52, we obtain : (53 (54 Ψ 2 = Ψ 211 ZΨ 222 (55 Ψ 211 = N 11 D 2 (56 Ψ 222 = N 22 D 2 (57 N 11 = K c b + K 11 d (58 N 22 = K 22 d (59 n order to avoid the derivative component of the perturbation (ẇ(t which affect the filtering error dynamic (22, we propose to reformulate the equation (22 on the following singular state-space form: [ n ζ ] [ ɛ(t ẇ(t ] = + + [ F α m [ Fd β [ m ] [ ] ɛ(t w(t ] [ ] ɛ(t d w(t d ] w(t (6 E-SSN: 2224-2678 135 Volume 16, 217
[ ɛ(t Let us consider ξ = w(t ], so we have : ρ ξ(t = F ξ(t + F d ξ(t d 1 + Bw(t (61 ρ = F = F d = [ ] n ζ [ F α m [ ] Fd β [ B = m ] ] (62 (63 (64 (65 Since the matrix ζ in (62 depends on the unknown matrix Z (See (25 (55. We assume that the gain matrix Z satisfies the following relation : ZΨ 22 = Zζ 22 = (66 his assumption enables us to avoid an unknown (to be designed gain matrix Z in the singular matrix ρ given by (62. So there always exists a matrix Z 1 such that: Z = Z 1 ( ζ 22 ζ + 22 (67 ζ 22 = N 22 D 2 (68 ζ + 22 is the pseudo inverse of ζ 22 such that : n fact, WSEAS RANSACONS on SYSEMS then, we have : ζ 22 ζ + 22 ζ 22 = ζ 22 (69 Zζ 22 = Z 1 ( ζ 22 ζ + 22 ζ 22 (7 = Z 1 (ζ 22 ζ 22 ζ 22 + ζ 22 (71 = Z 1 (ζ 22 ζ 22 (72 = (73 3.2.2 Filter-based controller design independent from the delay he design procedure is based on Lyapunov- Krasovskii stability theory using LMs approach. he filter-based controller stability conditions are independent from the delay, so the estimated state converges asymptotically to the real one for any constant time delay the satisfaction of condition (4. At this stage, based on theorem 2 Lyapunov- Krasovskii stability theory, one can get the gain matrix Z which parametrizes the filter matrices, as proposed in the following theorem. heorem 4. he filter-based controller in the form of (3 is a H controller for system (1 if there exist matrices P 1s = P 1s, P 2s, P 3s = P 3s, Q 1 = Q 1, Q 4 = Q 4, Q 2 Y s satisfying the following linear matrix inequalities : ( Q1 Q Q s = 2 Q > (76 2 Q 4 P s ρ = ρ P s > (77 ( P1s P P s = 2s P 2s P 3s (78 P 2s = LP 1s (79 satisfying α 11 α 12 α 13 α 14 α 15 α 22 α 23 α 24 α 25 α 33 α 34 α 35 α 44 α 45 < (8 α 55 α ij1 i,j 5 are given in appendix A. then the gain Z 1 is given by: Z 1 = P 1 1 s Y s (81 We note that according to (77, we have: ζ = ζ 11 Zζ 22 = ζ 11 (74 ζ 11 = N 11 D 2 (75 L = N 11 D 2 (82 n fact when replacing P 2s in (78 by its expression in (79, we have : ( P1s P P s ρ = 1s ζ 11 (83 LP 1s LP 1s ζ 11 E-SSN: 2224-2678 136 Volume 16, 217
( ρ Ps P1s P = 1s L ζ11 P 1s ζ11 P 1sL So, according to (77 by identification we have: (84 L = ζ 11 (85 when using (75, equation (82 holds. Proof 3. Let V (ξ, t be the Lyapunov-Krasovskii (See [16] functional of the form : V (ξ, t = ξ (t P s ρ ξ(t + t t d ξ(µ Q s ξ(µdµ (86 Q s P s verify respectively (76 (77. n order to establish sufficient conditions for existence of (3 according to (4, we should verify the following inequality: H(ɛ, w = V (ξ, t + ɛ (tɛ(t Since w w >, we can write: γ 2 w (tw(t < (87 H(ɛ, w < H(ɛ, w + w (tw(t (88 Equation (88 can be written as: H(ɛ, w < V (ξ, t + ɛ (tɛ(t By considering γ 2 x = γ 2 1 since (γ 2 1w (tw(t (89 ξ ξ γ 2 w w = ɛ ɛ γ 2 xw w (9 Equation (89 can be written like: With: WSEAS RANSACONS on SYSEMS H(ɛ, w < H x (ξ, w (91 H x (ξ, w = V (ξ, t+ξ (tξ(t γ 2 xw (tw(t (92 So it s sufficient to impose that: H x (ξ, w < (93 By differentiating V (ξ, t along the solution (61, we obtain : H x (ξ, w = ξ (t[ F P s + P s F + Qs + ]ξ(t +ξ (t d F d P s ξ(t +ξ (tp s Fd ξ(t d ξ (t dq s ξ(t d +w (t BP s ξ(t + ξ (tp s Bw(t γ 2 xw (tw(t < (94 it can be written like : v α s P s Fd P s B Q s v < (95 γx 2 m α s = F Ps + P s F + Qs + (96 v = [ ξ ξ (t d w (t ] From (95, H(ξ, w < if α s P s Fd P s B Q s < (97 γx 2 m By replacing F, F d, B, Q s P s by their expressions given, respectively, by (63, (64, (65, (76 (78 in (97 according to equations (42, (43, (46, (49 (52, the matrix in (97 equals that in (8 which prove theorem 3. Once Z 1 is calculated using (81 Z is calculated using (67, all filter matrices can also be given by equations (42, (43, (46, (49 (52. 3.2.3 Filter-based controller design dependent on the delay n this paragraph, we aim to design a filter-based controller dependently on the delay. Based on the Lyapunov-Krasovskii stability theory respect to the H criteria given by (4, one can get the gain matrix Z which parametrizes the filter matrices, as proposed in theorem 4. his type of design is of great importance, especially, when dealing unknown or variable delay known bounds such that: τ1 τ 2 are scalars. < τ 1 d τ 2 (98 heorem 5. he filter-based controller in the form of (3 is a H filter for system (1 if there exist matrices P 1 = P1, P 2, P 3 = P3 Y satisfying the following linear matrix inequalities : satisfying P ρ = ρ P > (99 ( P1 P P = 2 P 2 P 3 (1 P 2 = LP 1 (11 ( Ξ Q Ω = Q < (12 U E-SSN: 2224-2678 137 Volume 16, 217
Where Ξ U are symmetric matrices of dimension 3m z + 4m, Q R (3mz+4m (3mz+4m. Ξ, U Q are given in Appendix B. we have : Z 1 = P 1 1 Y (13 We note that according to (99, we have: L = N 11 D 2 (14 Proof 4. he chosen Lyapunov functional is (See [14]: V (t = V 1 (t + d[v 2 (t + V 3 (t] (15 V 1 (t = ξ(t P ρξ(t (16 V 2 (t = V 3 (t = d t t θ d t t θ ξ(s F P F ξ(sdsdθ ξ(s F d P F d ξ(sdsdθ (17 (18 n order to establish sufficient conditions for existence of (3 according to (4, we should verify the inequality (93. he derivative of the functional V (t is : V (t = V 1 (t + d V 2 (t + d V 3 (t (19 According to equations (61 (99, we have: hen, so, WSEAS RANSACONS on SYSEMS V 2 (t = V 1 (t = ξ(t [ F P + P F ]ξ(t d +ξ(t d F d P ξ(t +ξ(t P F d ξ(t d +w (t B P ξ(t +ξ (tp Bw(t (11 [ξ(t F P F ξ(t ξ(t θ F P F ξ(t θ]dθ (111 V 2 (t = dξ(t F P F ξ(t d ξ(t θ F P F ξ(t θ]dθ (112 Let s so we write : Υ(t θ = F ξ(t θ R n (113 V 2 (t = dξ(t F P F ξ(t V 3 (t = Let s : so, d d Υ(t θ P Υ(t θ]dθ (114 [ξ(t F d P F d ξ(t ξ(t θ F d P F d ξ(t θ]dθ (115 Υ d (t θ = F d ξ(t θ R n (116 V 3 (t = dξ(t F d P F d ξ(t d Υ d (t θ P Υ d (t θdθ (117 Uniform asymptotic stability implies that : lim V (t (118 As θ is bounded, the quantities Υ(t θ Υ d (t θ, respectively, given by (113 (116 satisfy :, Consequently, lim Υ(t θ = lim Υ(t (119 lim Υ d(t θ = lim Υ d(t (12 d lim ( Υ(t θ P Υ(t θdθ, = d lim Υ(t P Υ(t (121 d lim ( Υ d (t θ P Υ d (t θdθ We set the variable s changes: = d lim Υ d(t P Υ d (t (122 γ v = lim Υ(t (123 ν = lim d(t (124 E-SSN: 2224-2678 138 Volume 16, 217
WSEAS RANSACONS on SYSEMS he equations (121 (122 can be written as : d lim ( Υ(t θ P Υ(t θdθ, 2d lim ( Υ d (t θ P Υ d (t θdθ d = dγ v P γ v (125 = dν P ν (126 We suppose that ξ = lim ξ(t, we have : lim V (t = [ξ [( F + F d P + P ( F + F d ]ξ +d 2 ξ F P F ξ +d 2 ξ F d P F d ξ] +[ d 2 γ v P γ v d 2 ν P ν] (127 herefore, according to equation (98 we have : lim V (t [ξ [( F + F d P + P ( F + F d ]ξ +τ 2 2 ξ F P F ξ +τ 2 2 ξ F d P F d ξ] +[ξ (tp Bw(t + w (t BP ξ(t] τ 1 2 γ v P γ v hen, according to (87 we have : τ 1 2 ν P ν] (128 H(ξ, t < [ ξ γv ν ] Ψ Ψ = ξ γ v ν < (129 β c P B τ 1 2 P τ 1 2 P B P γx 2 m β c = F P + P F + F d P + P F d (13 +τ 2 2 F P F + τ 2 2 F d P F d + (131 Since Ψ is a symmetric matrix then relation (129 is equivalent to : Ψ < (132 n order to avoid the quadratic form present in β c, we propose to transform the inequality given by (132 in an other form according to the Schur Lemma (See [3]. n fact, matrix Ψ can be written as : Ψ = U Q v Γ 1 Q v (133 Γ = Ξ 1 (134 According to the Schur lemma, Ψ < Γ < if only if : Q v = ( Γ Qv Ω v = < (135 U Q v τ 2 F τ 2 F d γ x (136 Now, we apply a congruence transformation to Ω v such that: Π = Ω v < (137 Where is a non singular matrix given by : P............. P........ P... =................................ (138 hen, by replacing P, F, F d, α β by their expressions given, respectively, by (1, (42, (43, (23 (24 in (137 considering equation (13, theorem 4 holds. 4 Filter-based controller design in the frequency domain n this section based on time domain results, we propose the filter-based controller design procedure that operates in the frequency domain, dependently independently form the delay, using left co-prime factorization of a transfer matrix [9, 24]. So, the filter transfer function is given by the following theorem : E-SSN: 2224-2678 139 Volume 16, 217
heorem 6. he frequency domain description of the H functional filter-based controller (3 for the linear singular delayed system (1 is given by: u(s = 1 (s y(s + 2 (se ds y(s (139 1 (s = [ N1 1 1(s N2 1 2(s] N3 1 3(s = [ c (s(h + H d e ds ] 1 [M + c (sl 1 ] (14a 2 (s = [ N1 1 1(s N2 1 2(s] N4 1 4(s = [ c (s(h + H d e ds ] 1 WSEAS RANSACONS on SYSEMS c (sl 2 (14b c (s = s mz F x (s (141 F x (s = F + F d e ds (142, using left coprime factorization [9], all matrices implemented in this design are given by : N 1 (s = (s mz F x (s + X 1 1 + mz (143 M 1 (s = (s mz F x (s + X 1 1 H (144 N 2 (s = (s mz F x (s X 2 1 X 2 + mz (145 M 2 (s = (s mz F x (s X 2 1 H d e ds (146 N 3 (s = (s mz F x (s + X 3 1 + mz (147 M 3 (s = (s mz F x (s + X 3 1 (L 1 X 3 M +M (148 N 4 (s = (s mz F x (s X 4 1 X 4 + mz (149 M 4 (s = (s mz F x (s X 4 1 L 2 e ds (15 Note that X 1, X 2, X 3 X 4 are matrices of appropriate dimensions such that, respectively, det(s mz F x (s + X 1, det(s mz F x (s X 2, det(s mz F x (s + X 3, det(s mz F x (s X 4 are Hurwitz. Proof 5. By applying the Laplace transform to (3a taking into accounts (141, we write : χ(s = (s mz F x (s 1 Hu(s +(s mz F x (s 1 H d e ds u(s +(s mz F x (s 1 L 2 e ds y(s +(s mz F x (s 1 L 1 y(s (151 By replacing in (3b χ(s by its expression in (151, we have: u(s = [ mz c (s(h + H d e ds ] 1 [M + c (sl 1 ]y(s +[ mz c (s(h + H d e ds ] 1 c (sl 2 e ds y(s (152 So, the proposed frequency domain description holds. 5 filter-based controller design steps summary 5.1 State Feedback Synthesis Step 1 Compute matrices X c, Y c Z c using (11 (12. Step 2 Compute matrix K c using (16. 5.2 ime Domain Functional Filter-Based Controller Design Step 1 Compute matrix S using (2. Step 2 Compute matrices Θ Σ using (34 (35. Step 3 Compute matrices F 11, F 22, F d11, F d22, J 11, J 22, J d11 J d22 using (4, (41, (44, (45, (47, (48, (5 (51. 5.2.1 ime Domain Design ndependent from the state delay Step 4 Compute matrix L using (82. Step 5 Compute matrices P s, Q s Y s by solving the LMs given by (76, (77 (8. Step 6 Compute matrix gain Z 1 using (81. Step 7 Compute matrix gain Z using (67. Step 8 Compute F F d using equations (42 (43. Step 9 Compute J, J d K using respectively equations (46, (49 (52. Step 1 Get matrices L 1 L 2 from (29 (31. Step 11 Get matrices H H d using, respectively, conditions iii iv from theorem 2. 5.2.2 ime Domain Design Dependent on the state delay Step 4 Compute matrix L using (14. Step 5 Compute matrices P, Q Y by solving the LMs given by (99, (12 (169. Step 6 Compute matrix gain Z 1 using (13. Step 7 Compute matrix gain Z using (67. Step 8 Compute F F d using equations (42 (43. E-SSN: 2224-2678 14 Volume 16, 217
WSEAS RANSACONS on SYSEMS Step 9 Compute J, J d K using respectively equations (46, (49 (52. Step 1 Get matrices L 1 L 2 from (29 (31. Step 11 Get matrices H H d using, respectively, conditions iii iv from theorem 2. 5.3 Functional Filter-Based Controller Design in the Frequency Domain Step 1 Calculate X 1, X 2, X 3 X 4 using MFDs such that, det(s mz F x (s + X 1, det(s mz F x (s X 2,det(s mz F x (s +X 3, det(s mz F x (s X 4 are Hurwitz. Step 2 Compute N 1 (s, M 1 (s, N 2 (s, M 2 (s, N 3 (s, M 3 (s, N 4 (s, M 4 (s, using (143-(15, F x (s is given by (142. Step 3 Deduce from step 2 the values of 1 (s, 2 (s, 3 (s 4 (s. Step 4 Compute the filter-based controller design given by (139. 6 Numerical Examples Let s consider system (1, : E = A d = ( 1 1 (.5 1 1 1 ( 1 1, A = 2 1 ( 1, B d = 1 C = ( 1 1, F = ( 1 2, D 1 = ( E We have rank C verified. (, B = 1, D 2 = 1 4, ( 1 1 = 2, so hypothesis 1 is According to conditions given by equation (2, we have: a = (.5.5.5.5 ( 1, b = c = ( 1 1, d =., Figure 1: he Disturbance w(t 6.1 he feedback gain synthesis: he resolution of the LM system given by equations (11 (12 gives : Z c = X c = (.586.917.917 1.463 (.716 1.33.64.64,, Y c = (.244.948 he resolution of equation (16 leads to : K c = (.59.2772 6.2 Filter-based controller design independent from the delay: n this paragraph we impose : γ = 7.7, d = 1s. Solving LM (8, we get : (.824 4.862 1 P s = 7 4.862 1 7 38.3992 Q s = ( 44.3913.18.18 27.89, Y s = ( 44.56 31.19 1.94 5.83 2.91, Z = ( Z 1 Z 2, Z 1 = ( 541.4 378.73, Figure 1 illustrates the used bounded disturbance such that w 2 = 419.6 units Z 2 = ( 23.61 7.83 35.41, E-SSN: 2224-2678 141 Volume 16, 217
WSEAS RANSACONS on SYSEMS So, the filter-based controller matrices values are given as follows: F = 541.33, F d = 378.823, L 1 = 7.7845 L 2 = 35.3923, H = 23.5221, Figure 4: he Estimation Error H d =.2182, K = 23.6312 Figures 2 3 show a comparison between the desired law control defined by K c x(t the output of the filter-based controller given by equation (3b. (a ransient Phase of the Estimation Error. (b Disturbance Effect on the Estimation Error(Zone :(22s >24s. Figure 5: Zoom n of the Estimation Error. hen the signal (y(t to noise (w(t ratio is evaluated by: Figure 2: he Control Laws Evolution SNR = 2Log 1 ( y 2 w 2 = 7.3843dB the norm H of the the transfer function of the error to the disturbance is evaluated by: H ɛw = 3.6546 < γ. 6.3 Filter-based controller design dependent on the delay Similarly to paragraph (6.2, we suppose that: γ = 7.7, d = 1s. (a ransient Phase of the Control Laws Evolution. (b Delay Effect on the Control Laws Evolution (t=1s. Figure 3: Zoom n of the Control Laws Evolution. Figures 4 5 represent the evolution of the estimation error. We can remark as it shown in figure (4a that the response time of the filter based controller is relatively small, so it has a rapid estimation dynamic. hen the disturbance effect on the estimation error (Figure (4b is not noticed. hen, we can consider, as an application of theorem 4 on a constant known delay, according to equation (98 that : τ 1 = τ 2 = 1s. Using equation (12, we get : (.5349 2.975 1 P = 6 2.975 1 6 41.9921 Y = (.58.41.2.8.4,, E-SSN: 2224-2678 142 Volume 16, 217
WSEAS RANSACONS on SYSEMS Z = ( 1.1.77.5.15.7, So, the filter-based controller matrices values are given as follows: F = 1.599, F d =.842, L 1 =.993, L 2 =.496, H =.428, H d =.2277, K =.71 Using the same disturbance function w(t as used in paragraph (6.2, we draw the estimation error the evolution of the law control. H ɛw = 3.8624 < γ. Comparison between the design dependent on the delay the independent from delay technique : Figures 8 9 represent a comparison between the estimation errors using the two mentioned methods. hen, we note a quicker dynamic when using the independent from state delay technique but a greater magnitude (figure 9a. t s obvious in (figure 9b that the dependent on delay method leads to a better error magnitude during the permanent phase. Figure 6 shows the disturbance effect on the estimation error figure 7 represents a comparison between the real law control the estimated one during the ransient phase, permanent phase the ransient duration when disturbance is applied. Figure 8: he Estimation Error Figure 6: he Estimation Error (a ransient Phase of the Evolution of the estimation Error. (b Permanent Phase of the evolution of the Estimation Error. Figure 9: Zoom n of the Estimation Error : Dependent ndependent techniques. 6.4 Filter-based controller design Dependent on the delay: Application on a linear singular system variable state delay n this paragraph, we switch the constant state delay to a variable one d(t such that : Figure 7: he Control Laws Evolution he signal (y(t to noise (w(t ratio is evaluated by : SNR = 2Log 1 ( y 2 w 2 = 7.7223dB the norm H of the the transfer function of the error to the disturbance is evaluated by: hen, we have : d(t =.3sin(t +.7 τ 1 =.4s. τ 2 = 1s. Using equation (12, we get : (.5312 3.1338 1 P = 6 3.1338 1 6 42.79, E-SSN: 2224-2678 143 Volume 16, 217
WSEAS RANSACONS on SYSEMS Y = (.6.42.2.7.4, Z = ( 1.13.79.4.14.7, So, the filter-based controller matrices values are given as follows: F = 1.924, F d =.8647, L 1 =.994, L 2 =.497, H =.396, H d =.2182, K =.695 When using the same disturbance function w(t as used in paragraph (6.2, we draw the estimation error as shown in Figure 11 the evolution of the function u(t K c x(t : 6.5 Filter-based controller synthesis in the frequency domain 6.5.1 Filter-based controller design independent from the delay he considered controller is the same as in paragraph (6.2. By using the left co-prime factorization, matrices of the frequency domain description of the filter-based controller for singular system (1 are given by : X 1 = X 3 =.9575, X 2 = X 4 =.9649 N 1 (s = s + 378.8e s + 54.9 s + 378.8e s + 541.9, M 1 (s = 23.52 s + 378.8e s + 541.9 hen : N 1 1 M 1 (s = 23.52 s + 378.8e s + 54.9 ; we have : N 2 (s = s + 365.5e s + 522 s + 378.8e s + 54, Figure 1: he Control Laws Evolution M 2 (s =.2e s s + 378.8e s + 54 hen : N 2 1 M 2 (s =.2e s s + 365.5e s + 522 ; Similarly to N 1 (s M 1 (s, we get : N 3 (s = N 1 (s M 3 (s =.6s + 22.3e s + 38.8 s + 378.8e s + 541.9 Figure 11: he Estimation Error he signal (y(t to noise (w(t ratio is evaluated by : SNR = 2Log 1 ( y 2 w 2 = 7.5782dB the norm H of the the transfer function of the error to the disturbance is evaluated by: H ɛw = 3.6831 < γ. hen : N 3 1 M 3 (s =.6s + 22.3e s + 38.8 s + 378.8e s + 54.9 ; And finally : M 4 (s = N 4 (s = N 2 (s 35.4e s s + 378.8e s + 54.3 E-SSN: 2224-2678 144 Volume 16, 217
hen WSEAS RANSACONS on SYSEMS N 4 1 M 4 (s = 35.4e s s + 365.5e s + 522 ; he singular values plot is given by figure 12. Similarly to N 1 (s M 1 (s, we get : hen : N 3 (s = N 1 (s M 3 (s =.86s +.64e s +.34 s +.74e s + 1.79 N 1 3 M 3 (s =.86s +.64e s +.34 s +.74e s ; +.79 And finally : N 4 (s = N 2 (s Figure 12: he singular values plot 6.5.2 Filter-based controller design dependent on the delay he considered controller is the same as in paragraph (6.3. By using the left co-prime factorization, matrices of the frequency domain description of the filter-based controller for singular system (1 are given by : hen : M 4 (s = N 4 1 M 4 (s =.38e s s +.74e s.492.38e s.97s +.72e s +.95 ; he singular values plot is given by figure 13. X 1 = X 3 =.1576, X 2 = X 4 =.976 N 1 (s = s +.74e s +.79 s +.74e s + 1.79, hen : M 1 (s = N 1 1 M 1 (s = we have : hen :.47 s + 378.8e s + 541.9.47 s +.74e s +.79 ; N 2 (s =.97s +.72e s +.95 s +.74e s.492, M 2 (s = N 2 1 M 2 (s =.22e s s +.74e s.492.22e s.97s +.72e s +.95 ; Figure 13: he Singular values plot n this section, we show the filter-based controller designs effectiveness in numerical examples. So, we highlight the effectiveness of the design techniques independently from the state delay dependently on the delay an application on a variable state delay. 7 Conclusion n this paper, we have studied the problem of controller design based on a functional H filter for singular systems delay in both state input vector. he controller is set in time frequency domains. he time domain method begin computing the feedback gain for the control law design the respect to the admissibility problem a H criteria by means of LMs. hen, a functional filter techniques are used to reconstruct this control E-SSN: 2224-2678 145 Volume 16, 217
WSEAS RANSACONS on SYSEMS law. Note that the filter synthesis verifies a LM condition dependently independently from the delay based on Lyapunov-Krasovskii theory. he frequency domain approach is based on the time domain result. So using some useful MFDs, functional H filter description is given. he proposed approaches have been applied on a numerical example they show their effectiveness. References: [1] Y. Ariba, Sur la stabilité des systèmes à retards variant dans le temps : héorie et application au contrôle de congestion d un routeur. Ph.D. dissertation, Université de oulouse 3 Paul Sabatier, France, 29. [2] E. K. Boukas, N. F. Al-Muthairi, (26. Delay-dependent stabilization of singular linear systems delays.,nternational Journal of nnovative Computing, nformation Control, 283-291. [3] S. Boyd L. Venberghe, "Convex Optimization", Cambridge University Press, First edition, 24. [4] M. Darouach, Linear Functional Filters for Systems With Delays in State Variables : he Discrete-ime Case.(25. EEE ransactions on Automatic Control. Vol. 5 No. 2, pp. 228-233. [5] M. Darouach, M. Boutayeb, (1995. Design of filters for descriptor systems, 4 th EEE ransactions on Automatic Control, no. 7, 1323-1327. [6] M. Ezzine, M. Darouach, H. Souley Ali H. Messaoud, (21a.A controller design based on function observer for singular linear systems : ime frequency domains approaches, 18th Mediterranean Conference, pp. 413-418, Marrakech, Morocco. [7] M. Ezzine, M. Darouach, H. Souley Ali H. Messaoud, (21b. ime Frequency domain design of Functional Filters, American Control Conference, pp. 599-64, Baltimore, MD, USA. [8] M. Ezzine, M. Darouach, H. Souley Ali H. Messaoud, (21c. Synthése temporelle et fréquentielle de filtres H d ordre plein pour les systèmes singuliers [Design of full order H filters for singular systems in the time the frequency domain],conférence nternationale Francophone d Automatique, Nancy, France. [9] M. Ezzine, M. Darouach, H. Souley Ali H. Messaoud, (211a. Unknown nputs Functional Filters Designs for Descriptor Systems Constant ime Delay,18 th FAC, Milano- taly. [1] M. Ezzine, M. Darouach, H. Souley Ali H. Messaoud, (211b. Full order H filtering for Linear systems in the frequency domain : he time frequency domain cases,nternational Journal of Control, Automation, Systems, pp. 558-565. [11] M. Ezzine, M. Darouach, H. Souley Ali H. Messaoud, (212. A controller design based on a functional H filter for descriptor systems : he time frequency domain cases,automatica, pp. 542-549. [12] A. Fattouh, O. Sename J-M. Dion,D. Koenig, D. Jacquet S. Mammar, Robust filter design for time-delay systems: a Riccati equation approach,kybernetika, Vol. 35, No. 6, pp. 753-764. [13] J. Hale, S. Lunel, (1991. ntroduction to functional differential equations,applied Mathematical Sciences 99, Springer-Verlag. [14] M. Khadhraoui, M. Ezzine, H. Messaoud, M. Darouach, (214a. Design of full order unknown input observers for delayed singular systems tate variable time delay, nternational Symposium on Communications, Control Signal Processing (SCCSP214, pp. 624-627, Athens, Grec. [15] M. Khadhraoui, M. Ezzine, H. Messaoud, M. Darouach, (214b. Functional Observers design unknown inputs for delayed singular systems : he discrete time case, Mediterranean Conference of Control Automation (MED214c, pp. 127-132, Palermo, taly. [16] M. Khadhraoui, M. Ezzine, H. Messaoud, M. Darouach, (214c. Design of Full Order H Filter for Delayed Singular Systems Unknown nput Bounded Disturbance, nternational Conference on Control, Decision nformation echnologies (COD214, pp. 429-434, Metz-France. [17] M. Khadhraoui, M. Ezzine, H. Messaoud, M. Darouach, (214d. Design of Full Order Ob- E-SSN: 2224-2678 146 Volume 16, 217
WSEAS RANSACONS on SYSEMS servers Unknown nputs for Delayed Singular Systems Constant ime Delay, nternational Conference on Control, Decision nformation echnologies (COD214, pp. 423-428, Metz-France. [18] D. Koenig, D. Jacquet S. Mammar, (26. Delay-dependent H filter of linear delay descriptor systems,proc. American Control Conference, pp.3813-3817, Minneapolis, MN, USA. [19] D. Koenig B. Marx, (24. Design of Observers for descriptor systems delayed state unknown inputs,proc. American Control Conference, pp.486-481, Boston, MA, USA. [2] N. Krasovskii, (1963. Stability of Motion,Stanford University Press. [21]. Masubuchi, Y. Kamitane, A. Ohara N. Suda, (1997. H Control for descriptor systems: A matrix inequalities approach,automatica, vol. 33, no. 4, pp. 669-673. α 13 = P 1s F d11 Y s F d22 (155 α 14 = P 1s F d11 K c b D 2 Y s F d22 K c b D 2 Y s J d22 D 2 + P 1s J d11 D 2 (156 α 15 = P 1s L (157 α 22 = LP 1s K c a D 1 LP 1s K 11 c D 1 + LP 1s J 11 D 2 LP 1s F 11 K c b D 2 + LY s F 22 K c b D 2 LY s J 22 D 2 + (LP 1s a D 1 LP 1s K 11 c D 1 LP 1s F 11 K c b D 2 + LY s F 22 K c b D 2 +LP 1s J 11 D 2 LY s J 22 D 2 LP 1s L P 3s P 3s + Q 4 + m (158 α 23 = LP 1s F d11 LY s F d22 (159 α 24 = LP 1s F d11 K c b D 2 LY s F d22 K c b D 2 +LY s J d22 D 2 LP 1s J d11 D 2 (16 α 25 = P 3s (161 α 33 = Q 1 (162 α 34 = Q 2 (163 α 35 = n m (164 α 44 = Q 4 (165 α 45 = m (166 α 55 = (γ 2 1 m (167 [22]. S. Niculescu, (21. Delay effect on stability, A robust control approach, Springer Verlag. = ( ζ 22 ζ + 22 (168 [23] Verghese, G., Levy, B. Kailath,. (1981. A generalized state-space for singular systems,eee ransactions on Automatic Control, pp. 811-831. [24] M. Vidyasagar, (1985. Control systems Synthesis: A factorization approach,m.., Press, Cambridge. [25] S. Xu, B. J. Lam C. Yang, (23. Robust H control for uncertain singular systems state delay,nternational Journal of Robust Nonlinear Control, vol. 13, pp. 1213-1223. Appendix A : heorem 4 equations: α 11 = P 1s F 11 Y s F 22 + (P 1s F 11 Y s F 22 +Q 1 + n (153 α 12 = P 1s K c a D 1 P 1s K 11 c D 1 P 1s F 11 K c b D 2 + Y s F 22 K c b D 2 Y s J 22 D 2 + (LP 1s F 11 LY s F 22 P 1s L + Q 2 + P 1s J 11 D 2 (154 Appendix B: heorem 5 equations: Ξ = P P P m (169 U 11 = h 1 + h 1 + mz (17 U 22 = h 2 + h 2 + m (171 U 33 = U 55 = U 34 = (1 + τ 1 2 P 1 (172 U 44 = U 66 = (1 + τ 1 2 P 3 (173 U 77 = 2γx 2 m (174 U 56 = (1 + τ 1 2 P 1 N 11 D 2 (175 U 12 = h 3 + h 4 + h 5 L (176 U 13 = U 35 = mz mz (177 U 14 = U 36 = U 37 = U 57 = mz m (178 U 15 = h 5 (179 E-SSN: 2224-2678 147 Volume 16, 217
WSEAS RANSACONS on SYSEMS U 16 = h 5 L (18 U 17 = P 1 N 11 D 2 (181 U 23 = U 45 = m mz (182 U 24 = U 46 = U 47 = U 67 = m (183 U 25 = h 4 (184 U 26 = h 4 L (185 U 27 = P 3 (186 h 1 = P 1 F 11 Y F 22 + P 1 F d11 Y F d22 (187 h 2 = LP 1 K c a D 1 LP 1 K 11 c D 1 +LY F 22 K c b D 2 LP 1 F 11 K c b D 2 LY J 22 D 2 +(LP 1 a D 1 LP 1 K 11 c D 1 LP 1 F 11 K c b D 2 LP 1 L Q 33 = Q 55 = P 1 (2 Q 34 = Q 56 = P 1 N 11 D 2 (21 Q 44 = Q 66 = P 3 (22 Q 77 = m (23 Q 13 = Q 15 = Q 35 = Q 31 = Q 53 = mz m (24 z Q 24 = Q 26 = Q 46 = Q 42 = Q 64 = Q 72 = Q 74 = Q 76 = Q 27 = Q 47 = Q 67 = m m (25 Q 14 = Q 16 = Q 32 = Q 54 = Q 17 = Q 37 = Q 57 = Q 36 = mz m (26 Q 23 = Q 25 = Q 43 = Q 45 = Q 41 = Q 63 = Q 71 = Q 65 = Q 73 = Q 75 = m mz (27 = ( ζ 22 ζ + 22 (28 +LY F 22 K c b D 2 +LP 1 J 11 D 2 LY J 22 D 2 +LP 1 F d11 K c b D 2 + LY J d22 D 2 LY F d22 K c b D 2 LP 1 J d11 D 2 +LP 1 J 11 D 2 P 3 P 3 (188 h 3 = P 1 K c a D 1 P 1 K 11 c D 1 P 1 F 11 K c b D 2 + Y F 22 K c b D 2 +Q 2 + (LP 1 F 11 LY F 22 P 1 L Y J 22 D 2 + P 1 J 11 D 2 (189 h 4 = P 1 F d11 K c b D 2 Y F d22 K c b D 2 Y J d22 D 2 + P 1 J d11 D 2 (19 h 5 = (P 1 F d11 Y F d22 (191 Q 11 = τ 2P 1 F 11 τ 2Y F 22 (192 Q 12 = τ 2( P 1 K c a D 1 P 1 K 11 c D 1 P 1 F 11 K c b D 2 + Y F 22 K c b D 2 Y J 22 D 2 P 1 L + P 1 J 11 D 2 (193 Q 21 = τ 2(LP 1 F 11 LY F 22 (194 Q 22 = τ 2( LP 1 K c a D 1 LP 1 K 11 c D 1 LP 1 F 11 K c b D 2 + LY F 22 K c b D 2 +LP 1 J 11 D 2 LY J 22 D 2 P 3 (195 Q 51 = τ 2(P 1 F d11 Y F d22 (196 Q 52 = τ 2( P 1 F d11 K c b D 2 Y F d22 K c b D 2 Y J d22 D 2 + P 1 J d11 D 2 (197 Q 61 = τ 2(LP 1 F d11 LY F d22 (198 Q 62 = τ 2(LP 1 F d11 K c b D 2 LY F d22 K c b D 2 +LY J d22 D 2 LP 1 J d11 D 2 (199 E-SSN: 2224-2678 148 Volume 16, 217