Separation Principle for a Class of Takagi-Sugeno Descriptor Models

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1 International Journal of Control Science and Engineering 27, 7(): - DOI:.5923/j.control.277. Separation Principle for a Class of akagi-sugeno Descriptor Models Ilham Hmaiddouch,2, El Mahfoud El Bouatmani,2, Jalal Soulami,2,*, Abdellatif El Assoudi,2, El Hassane El Yaagoubi,2 Laboratory of High Energy Physics and Condensed Matter, Faculty of Science, Hassan II University of Casablanca, Casablanca, Morocco 2 ECPI, Department of Electrical Engineering, ENSEM, Hassan II University of Casablanca, Casablanca, Morocco Abstract his paper studies the design problem of an observer-based controller for a class of nonlinear descriptor systems described by akagi-sugeno (-S) fuzzy structure [, 2] the premise variables are unmeasurable. Based on the combination of the control law established in [3] and the fuzzy observer in explicit structure given in [4], a new procedure to investigate the separation principle problem for the considered class of akagi-sugeno descriptor models (SDMs) is then proposed. he guarantee of the global asymptotic stability of the closed-loop augmented system is proved by using the Lyapunov theory and the conditions of asymptotic stability are given in terms of linear matrix ineualities (LMIs). Finally, an application based on rolling disc process is presented as an illustrative example to show the performance of the proposed dynamic controller design. Keywords akagi-sugeno descriptor model, Fuzzy observer, State feedback controller, Lyapunov method, LMI techniue, Separation principle. Introduction It is well-known that in order to realize in practice a stabilisation by static state feedback for most industrial applications, the separation principle variously called dynamic output feedback controller or observer-based controller plays a key role since instead of a full-state feedback, we use signals reconstructed by a state observer from the on-line measurements of the input and output of the process. he aim of the paper consists to design an observer-based controller for a class of nonlinear descriptor systems described by -S representation. Notice that, descriptor models representation has been widely used in the dynamic modeling of many chemical and physical processes [5-8]. he numerical simulation of such dynamic models usually combines an ODE numerical method together with an optimization algorithm. In the literature, there have been several studies concerning the issue of stability and the stabilisation of -S models [9-]. Likewise, many works have been carried out to investigate the observer design of -S models [2-5]. he separation property of -S observers and controllers was discussed by [6-2]. For SDMs which are defined by extending the ordinary -S representation [3, 2], several research works * Corresponding author: jalal.soulami@gmail.com (Jalal Soulami) Published online at Copyright 27 Scientific & Academic Publishing. All Rights Reserved concerning the problem of control and observation design and applications have been developed [3, 4, 2-25]. Notice that, generally an interesting way to solve the various fuzzy observer and controller problems raised previously is to write the convergence conditions on the LMI form [26]. Based on the use of the state feedback given in [3] and the fuzzy observer proposed in [4], the main contribution of this paper consists to propose a new result of separation principle for a class of SDMs with unmeasurable premise variables. he global asymptotic stability of the closed-loop system is studied by using the Lyapunov theory and the stability conditions are given in terms of LMIs. Moreover, the proposed result is given without the use of an optimization algorithm. he outline of the paper is structured as follows. In Section 2, the considered class of SDMs with unmeasurable premise variables is presented. In Section 3, the main result about fuzzy observer-based controller design is established. he control and observer gains are found directly from LMI formulation. Finally, in Section 4, an illustrative application to show the good performance of the proposed result is given. hroughout the paper, some notations used are fair standard. For example, X > means the matrix X is symmetric and positive definite. X denotes the transpose of X. he symbol I (or ) represents the identity matrix (or zero matrix) with appropriate dimension.

2 2 Ilham Hmaiddouch et al.: Separation Principle for a Class of akagi-sugeno Descriptor Models µµ µµ and i j i j ij, i j X * X Y Y Z Y Z. At first, we recall some basic lemmas that are freuently used in the setting the proof of the main result of this paper. Lemma (Congruence): Let two matrices P and Q, if P is positive definite and if Q is a full column rank matrix, then the matrix QPQ is positive definite. Lemma 2 (Young's ineuality): For any matrices S and with appropriate dimensions, the following property holds for any invertible matrix J and scalar α > : - - S + S α S JS + α J () Lemma 3: Considering matrices X, Π< and a scalar γ >, the following holds: 2 - X Π X γ ( X +X) γ Π (2) 2. akagi-sugeno Descriptor Models he following class of SDMs is considered in this paper: input, Ex µ i( ξ)( Aix + Biu) (3) i y Cx n m x R is the state vector, u R is the control p n n y R is the measured output. Ai R, n m p n C R n n and E R such that <, are real known constant matrices. is Bi R, rang( E) r n the number of sub-models. ξ is the premise variable which can be here depend on completely or partially unmeasured state of the system and the µ i ( ξ ) ( i,..., ) are the weighting functions that ensure the transition between the contribution of each sub model: Ex Aix + Biu (4) y Cx hey verify the so-called convex sum properties: µ i( ξ) (5) µ i( ξ) i he main object of this paper is to design an observer-based controller for the class of SDMs (3). For this objective, we suppose that [5]: H ) ( EA, i ) is regular, i.e. det( se Ai ) s H 2 ) Sub-models (4) are impulse controllable and stabilisable. H 3 ) Sub-models (4) are impulse observable and detectable. Moreover, the following hypothesis which is necessary for the observer design given in [4] is assumed to be verified: H 4 ) E rang( ) n C Note that from the consideration of the hypothesis H 4 ), there exists a non-singular matrix Λ Λ2 such that: Λ3 Λ4 Λ E+Λ C I Λ +Λ 2 3E 4C n n Λ R n p, Λ2 R p m, Λ3 R, p p Λ4 R are constant matrices which can be found by E the singular value decomposition of C. In order to investigate the separation principle problem for the considered class of SDMs (3), firstly the following control law established in [3] is adopted: u( x) µ i( ξ) Fx i i (7) the gains Fi, i,..., can determined in order that the closed-loop system (3) is asymptotically stable. he following result has been studied in [3]. heorem [3]: he closed-loop system (3)-(7) is globally exponentially stable if there exist matrices Y >, U, i,..., verifying the following LMIs: i (6) Y E EY ϒ ii < i,, (8) ϒ ij +ϒ ji < i< j st.. µ i µ j ϒ + (9) ij AY i YAi BU i j U j Bi he fuzzy local feedback gains by: F, i,..., are given Fi UY i () On the other hand, the following fuzzy observer given in [4] is employed: z ( ˆ µ i ξ)( Niz+ Liy + L2iy+ Gu i ) () i xˆ z+ ( Λ 2+ KΛ4) y n n z R is the state of fuzzy observer and xˆ R is the estimate of x. ˆ ξ R denotes the estimated premise variables vector. he matrices N i, L i, L 2i, G i and K are unknown matrices of appropriate dimensions to be determined such that ˆx converges asymptotically to x. i

3 International Journal of Control Science and Engineering 27, 7(): - 3 Λ 2 and 4 Λ are such that euation (6) is satisfied. he convergence condition of the observer () can be formulated by the following heorem. heorem 2 [4]: he state error between the -S descriptor model (3) and its observer () converges asymptotically towards zero, if there exist matrices Y 2 >, Y 3 >, Q and W j, j,, such that the following LMIs hold : j + j Θ AY i 3+ YA 3 i < 2 Bi Y3 Θ i,j {,..., } (2) j Y2ΛAj + QΛ3Aj WC j (3) and Θ Λ + Λ Θ Λ + Λ he observer gains 3. Main Result ( Ai Aj) Y2 ( Ai Aj) 3Q 2 ( Bi Bj) Y2 ( Bi Bj) 3Q N i, L i, (4) L 2i, G i, K are given by: K Y2 Q Gi ( Λ + KΛ3) Bi L2i Y2 Wi Ni ( Λ + KΛ3) Ai L2iC Li Ni( Λ 2 + KΛ4) (5) As mentioned above, based on the use of the state feedback given by (7) and the fuzzy observer (), this section studies the problem of observer-based controller design for a class of SDMs (3). Indeed, the following augmented system in closed-loop is considered: Ex µ i( ξ)( Aix + Biu( xˆ )) i ( ˆ z µ i ξ)( Niz+ Liy + L2iy+ Gu i ( xˆ )) (6) i xˆ z+ ( Λ 2+ KΛ4) y u( xˆ ) µ ( ˆ i ξ) Fx iˆ i In order to establish the conditions for the asymptotic convergence of the system (6), we define the state estimation error: ε x xˆ (7) hen, by substituting (6) and (6) into (7) we obtain: ε Λ+ ( KΛ ) Ex z (8) 3 So, error dynamics will be: ε Λ+ ( KΛ ) Ex z (9) 3 It follows from (6) and (7) that the error dynamics can be written as: ε ˆ µξµ i( ) j( ξ)( Λ + KΛ3)( Γ ijx+ BF i jε) i,j ( ˆ) ( ˆ µξµ i j ξ)( Nz i + Liy + L2iy GF i jx+ GF i jε) i,j (2) Γ ij Ai BF i j (2) By substituting (8), euation (2) becomes: ε ˆ µξµ i( ) j( ξ)( Λ + KΛ3)( Γ ijx+ BF i jε) i,j + ( ˆ) ( ˆ µξµ i j ξ)( Niε Πijx GF i jε) i,j Πij N i ( Λ + Λ 3) + L i L 2i GF i j (22) + (23) Provided the matrices gains N i, L i, L 2i, G i and K satisfy: ( Λ + KΛ ) Γ Π (24) ij 3 Gi ( Λ + KΛ3) Bi (25) hen, from (6), (24) and (25), we have: Ni ( Λ + KΛ3 ) Ai L2iC+ ( Ni ( Λ 2+ KΛ4 ) Li ) C(26) ake: L i Ni( Λ 2+ KΛ 4) (27) hen: Ni ( Λ + KΛ3) Ai L2iC (28) It follows that the system (22) reduces to: ij Note that: ( ˆ ˆ i( ) i( )) j( )( Λ + KΛ3)( Γ ) ( ˆ ) ( ˆ ijx+ BF i j + i j ) Ni (29) i,j i,j ε µ ξ µ ξ µ ξ ε µ ξ µ ξ ε

4 4 Ilham Hmaiddouch et al.: Separation Principle for a Class of akagi-sugeno Descriptor Models B B B. Aik Ai Ak and ik i k hen, from (2), the euation (29) becomes: ( ˆ ) ˆ ˆ ˆ µ i( ξ ) µ i( ξ ) µ j( ξ ) Ai µ i( ξ ) µ j( ξ ) µ k ( ξ) Aik i,j i, jk, (3) ( ( ) ( ˆ) ) ( ˆ) ( ) ( ˆ) ( ˆ µ i ξ µ i ξ µ j ξ Bi µ i ξ µ j ξ µ k ξ) Bik i,j i, jk, ˆ ˆ ˆ ˆ i( ) j( ) k ( )( Λ + KΛ3)(( Aik Bik Fj) x+ Bik Fj ) + j( ) k ( ) Nk (3) i, jk, j,k ε µξµ ξµ ξ ε µ ξµ ξ ε By (5), euation (3) can be reduced to the following: ε µξµ ( ) ( ˆ) ( ˆ i j ξµ k ξ)( Φ ijkx +Ψijkε) (32) i,, jk Φ ijk ( Λ + KΛ3)( Aik Bik Fj) (33) Ψ ijk ( Λ + KΛ3) Bik Fj + N k hus, the global asymptotic stability of system (6) can be proved in an euivalent fashion for the following system: which can be rewritten as follows: Ex µ ( ) ( ˆ) ( ˆ i ξµ j ξµ k ξ)( Γ ijx + Bi Fjε) ij,,k ε ( ) ( ˆ) ( ˆ µξµ i j ξµ k ξ)( Φ ijk x +Ψijkε) i, jk, (34) ( ) ( ˆ) ( ˆ a µ i ξµ j ξµ k ξ) Ωijk xa (35) i,, jk Ex x E xa, E ε I (36) Γij BF i j Ω ijk ijk ijk Φ Ψ he following heorem constitutes the main result of the paper. heorem 3: he system (35) is globally asymptotically stable if for predefined scalars α >, α2 >, α3 >, α4 >, γ > ; there exist matrices Y >, Y 2 >, Q, U i, W i, i,,, verifying the following LMIs: 2 γ Z * * Π ijk γ I Π ijk * < i, jk,,, 2 Πijk α 4 I { } (37)

5 International Journal of Control Science and Engineering 27, 7(): - 5 with and Π * * * * 2 Π Φ * * * Π 3 ijk Π Φ2 * * 4 Π Φ3 * 5 α4i Π 2 2 Π ijk ( Π ) Z diag( X, X, X, X, I) X diag( Y, Y) αi Φ α I α 2I Φ 2 α2 I α3y Φ 3 α3 Y ϒ ij Π k + k 2 Λ Y2 +Λ3Q Π AY ik 3 ( Y2 3Q ) Λ +Λ Π B iku j 4 U j Bi Π ( Y2 3Q ) Π 5 Λ +Λ Π 2 ( BikU j ) I (38) (39) (4) with ϒ ij and k are defined in (9) and (3) respectively. he dynamic controller gains F i, N i, L i, L 2i, G i and K are given by: Fi UY i K Y2 Q Gi ( Λ + KΛ3) Bi L2i Y2 Wi Ni ( Λ + KΛ3) Ai L2iC Li Ni( Λ 2 + KΛ4) Λ, Λ 2, Λ 3, Λ 4 are such that (6) is satisfied. Proof of heorem 3: o prove the convergence to zero of the state variable system (35), let us consider the candidate Lyapunov function as follows: with a a a V ( x ) x EPx, P >, E P P E (42) (4)

6 6 Ilham Hmaiddouch et al.: Separation Principle for a Class of akagi-sugeno Descriptor Models P P P (43) 2 he time derivative of the Lyapunov function (42) along the trajectories of the system (35) is obtained as: herefore, we have the following stability conditions: which become from (36) and (43): ( ) ( ) ( ˆ) ( ˆ) ( a µ i ξµ j ξµ k ξ a Ω ijk + Ωijk ) (44) a i, jk, V x x P P x { } Ω P+ PΩ <, i, jk,,, (45) ijk ijk Γ ij P+ PΓ ij * 2Φ ijk + j i Ψ ijk 2+ 2Ψijk P F B P P P Applying Lemma to (46) by pre-multiplying and post-multiplying P and Y2 P2, thus we obtain: Y which become from (33): By considering the following: Matrix Θ can be rewritten again as: P { } <, i, jk,,, I YΓ ij +Γ ij Y * 2Φ ijk + j i Ψ ijk 2+ 2Ψijk Θ <, i, jk,,, Y Y F B Y Y (46) to both sides and defining new variables { } { } (47) ΘΣ+Σ+ Σ 2 < i, jk,,, (48) Y Γ ij +Γ ij Y Σ NkY2 YN + 2 k * (49) Σ Y 2( K 3) Aik Y Y2( K 3) Bik FjY Fj Bi Λ + Λ Λ + Λ + Σ2 Y 2( K 3) B Λ + Λ ik Fj + Fj Bik ( Λ + KΛ3) Y2) ( ( ) ) X Λ + KΛ3 Y2 Z ( AY ik ) X Λ + KΛ Y Z2 X3 Z3 ( I) X4 Λ + KΛ3 Y2 Z4 ( ( ) ) ( Bik FY j ) ( Fj Bi ) ( ( ) ) ( Bik Fj ) ΘΣ+ X Z + Z X + X Z + Z X + X Z + Z X + X Z + Z X (5) (5)

7 International Journal of Control Science and Engineering 27, 7(): - 7 Applying Lemma 2, the euation (5) becomes: Θ Σ+ΩΦ Ω +Ω Φ Ω +Ω Φ Ω +Ω Φ Ω (52) Y Aik Ω Y2( Λ + KΛ3) YF j Bik Ω 2 Y 2( Λ + KΛ3) BFY i j Ω 3 I Ω α 4I Φ 4 α4 I 4 Y2( Λ + KΛ3) Fj B ik (53) and Φ, Φ2, Φ 3 are defined in (39). Hence, using the Schur complement [26], the ineuality Θ< hold if the following matrix ineuality is satisfied: Σ * * * * Ω Φ * * * (54) M Ω2 Φ2 * * < Ω3 Φ3 * 4 Ω Φ4 Now, in order to establish the LMI conditions (37) of heorem 3, we rewrite the ineuality (54) as follows: with M * M < M 2 α 4 I Σ * * * * Ω Φ * * * M Ω2 Φ2 * * 3 3 * Ω Φ Ω α I ( 4 ) 4 2 M 2 Ω4 ( ( K ) Y ) ( Bik Fj ) Ω Λ + Λ 2 Ω and we consider the following matrix W defined by: (55) (56) (57) W diag(z, I ) (58) with Applying Lemma, M < which can be rewritten as follows: Z diag( X, X, X, X, I) (59) X diag( Y, Y ) (6) is euivalent to: W MW < (6) ZM Z * < M2Z α4 I Hence, applying Lemma 3 on ZMZ leads to: 2 γ M 2Z α 4 2 γz * M γ >. Using the Schur complement, we obtain: < I 2 γ Z * * Π ijk γ I M * < M 2Z α4 I (62) (63) (64) hen, the replacement of the various terms given in (64) by their expressions given before and the use of the changes of variables: U Q Wi i FY YK 2 YL i 2 2i (65) We establish the LMI conditions given in heorem 3. Finally, the dynamic controller gains given by (4) can be determined from (25), (27), (28) and (65). 4. Illustrative Example o illustrate the efficiency of the proposed dynamic controller, we consider a rolling disc process described by the following SDM with unmeasurable premise variable given in [25] such that the hypothesis H 4 ) is satisfied: x ( x x x x ) 2 Ex µ i( ξ) ( Aix + B u) i y Cx (66) is the state vector with x is the position of the center of the disc, x 2 is the translational velocity of the same point, x 3 is the angular velocity of the disc, 4 x is the contact force between the disc and the surface. u is the applied input force to the disc and y is the vector of output measurement.

8 8 Ilham Hmaiddouch et al.: Separation Principle for a Class of akagi-sugeno Descriptor Models b ξmax m m A, r b r2 ξmax + m J m b ξmin m m A 2 r b r2 ξmin + m J m E, B, r J C he weighting functions are defined by: ξ ξmin µ ( ξ) ξmax ξmin ξmax ξ µ 2( ξ) ξmax ξmin ξ is the premise variable having for expression: 2 kx 2 (67) k ξ [ ξmin, ξmax ] (68) m he physical parameters definition and their numerical values are given in [25]. As mentioned previously, the objective of this section is to use the SDM (66) as an illustrative example to demonstrate the ability of heorem 3 to be applied. herefore, the following augmented system in closed-loop is considered: 2 Ex µ i( ξ)( Aix + Bu( xˆ )) i 2 ( ˆ z µ i ξ)( Niz+ Liy + L2iy+ Gu i ( xˆ )) i xˆ z+ ( Λ 2+ KΛ4) y 2 u( xˆ ) µ ( ˆ i ξ) Fx iˆ i (69) F i, N i, L i, L 2i, G i, i, 2 and K are the gains of the proposed dynamic controller. Notice that to apply the theorem 3 in order to calculate such dynamic controller gains; it suffices to rewritten system (69) into its euivalent representation (35) as mentioned in Section 3. hus, in a first time, we calculate the matrices Λ, Λ 2, Λ 3 and Λ4 satisfying (6) from singular value decomposition of E C. Indeed, under hypothesis H 4), the following numerical expressions of Λ were obtained: Λ, Λ2, Λ 3, Λ.5 In a second time, we resolve the LMIs (37) in Y >, Y 2 >, Q, U i, W i. Indeed, with α α2 α3 α4. and γ, we obtain the following dynamic controller gains calculated by (4): , L2, , L , L L , 2, G G K In order to illustrate the performances of the proposed observer-based controller, numerical simulations of system (69) were performed using a Runge-Kutta method combined with the Newton-Raphson algorithm. he following initial conditions of system (69) were used: , xˆ x ( ), F ( ) F N , N2 he simulation results of the system (69) with the dynamic controller gains F i, N i, L i, L 2i, G i and K are given in Figures, 2, 3 and 4 the dashed lines denote the state variables estimated by the fuzzy observer. hese simulation results show the good performance of the proposed observer-based controller designed. Indeed, the i

9 International Journal of Control Science and Engineering 27, 7(): - 9 global asymptotic stability of the augmented system (69) in closed-loop with the proposed observer-based controller is satisfied..2 x -S x estimated x4 -S x4 estimated t(s) t(s) Figure. x & ˆx with fuzzy observer-based controller.5 x2 -S x2 estimated t(s) Figure 2. x 2 & ˆx 2 with fuzzy observer-based controller Figure 4. x 4 & ˆx 4 with fuzzy observer-based controller Moreover, these results of simulation show a good estimation of the unknown states of the process which is necessary for the real-time implementation of the control law (7). 5. Conclusions In this paper, a new result of separation principle for a class of SDMs with unmeasurable premise variables is suggested. More precisely, the proposed result concerns the real-time implementation without the use of an optimization algorithm of the static state feedback given in [3] by using the fuzzy observer proposed in [4]. he global asymptotic stability of the augmented system in closed-loop is studied by using the Lyapunov method and the stability conditions are given in terms of LMIs. he good performance of the proposed observer-based controller design is illustrated in simulation through a rolling disc process as an illustrative example..5 x3 -S x3 estimated REFERENCES []. akagi, M. Sugeno, Fuzzy identification of systems and its application to modeling and control, IEEE rans. Syst., Man and Cyber, Vol. 5, pp. 6-32, 985. [2]. aniguchi, K. anaka, H. Ohtake and H. Wang, Model construction, rule reduction, and robust compensation for generalized form of akagi-sugeno fuzzy systems, IEEE ransactions on Fuzzy Systems, vol. 9, No.4, pp , August t(s) Figure 3. x 3 & ˆx 3 with fuzzy observer-based controller [3]. aniguchi, K. anaka, and H. O. Wang, Fuzzy Descriptor Systems: Stability Analysis and Design via LMIs, Proceedings of the American Control Conference, San Diego, June 999. [4] I. Hmaiddouch, B. Bentahra, A. El Assoudi, J. Soulami and E. El Yaagoubi, An Explicit Fuzzy Observer Design for a Class

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