Decentralized control synthesis for bilinear systems using orthogonal functions

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1 Cent Eur J Eng DOI: 12478/s Central European Journal of Engineering Decentralized control synthesis for bilinear systems using orthogonal functions Research Article Mohamed Sadok Attia, Badii Ayadi, Naceur Benhadj Braiek Laboratoire des Systèmes Avancés - LSA - Ecole Polytechnique de Tunisie, BP 743, 278 La Marsa, Tunisia Received 2 September 213; accepted 17 December 213 Abstract: In this paper, we propose the development of a new technique of decentralized control for bilinear systems, through using orthogonal functions The use of this tool allows the conversion of differential state equations to a set of algebraic ones by projecting the system input and output variables into orthogonal functions basis and then using the operational properties of these orthogonal functions The optimum decentralized control parameters for the interconnected bilinear system can then be determined through comparison of the response behaviour with the chosen reference model Keywords: Decentralized control orthogonal functions reference model Versita sp z oo 1 Introduction Control systems are used in many area such as power generating plants, aircraft dynamics, economic models When dealing with interconnected systems, decentralized control is used to make each system perfectly regulated using only its own local state variables, and at the same time to insure the global stability of the whole system Over the past decade, different techniques of decentralized control design have been developed such as decentralized negotiation of optimal consensus [1], feedbacklinearization and feedback-feedforward decentralized control for multimachine power system [2], stabilization of decentralized control systems by means of periodic feedback attia_sadok@yahoofr BadiiAyadi@eptrnutn Naceurbenhadj@eptrnutn [3] However, the proposed approaches in general only apply to particular classes of interconnected systems, and specific conditions have to be verified in order for the techniques to be effective In this paper we design a decentralized control technique for bilinear systems This approach uses shifted Legendre polynomials as an orthogonal functions basis There are a number of different types of orthogonal function such as Chebychev [4], Hermite polynomials [5] and Walsh functions [6], which are applied for identification, model reduction, analysis and control of linear [7, 8], and some classes of nonlinear systems Orthogonal function are also used to solve problems of fractional order [9], recently, Bhrawy and Alofi [1] proposed to solve linear fractional differential equations on a finite interval a method based on the operational Chebyshev matrix of fractional integration in the Riemann- Liouville sense The important operational properties of orthogonal func- 47 Download Date 7/2/18 1:24 AM

2 Decentralized control synthesis for bilinear systems using orthogonal functions tions, such as the product and the integration operational matrices, reduce the complex differential equations of system variables to algebraic equations which are easier to process For this reason these properties are exploited in this contribution to develop a new technique leading to the determination of a decentralized control law for bilinear systems so that each controlled subsystem of the global system has the desired performance of a chosen reference model This paper is organized as follows: a short review of orthogonal functions is persented in the next section, our main contribution is developed in the third section, in which we present the proposed approach of the decentralized control synthesis for bilinear systems using orthogonal functions A numerical simulated example is provided in the final section to illustrate the proposed method 2 Review of orthogonal functions Consider a complete set of orthogonal functions φ= φ i t, i N} defined on an interval [a, b] R The principle of orthogonality leads to the property: i, j N, b a wtφ i tφ j tdt = δ ij q i 1 where wt is the weighunction and δ ij is Kronecker s symbol An integrable function f on [a, b] can be developed as: where: ft = f i φ i t 2 i= b f i = 1/q i wtftφ i tdt 3 a For obvious practical reasons, the development is truncated to order N which is large enough to allow a good approximation Thus, one has: ft N 1 = f i φ i t = F N Φ N t 4 i= F N = [f f 1 f N 1 ] Φ N t = [φ t φ 1 t φ N 1 t] T This truncated projection of scalar or vector functions can be very useful in practice in different kinds of engineering problems related to modelling, identification, analysis, simulation, control, etc Indeed, by means of the operational properties of orthogonal functions, the differential equations describing dynamic processes can be reduced into algebraic relations allowing important simplifications in the synthesis problems 21 Operational matrix of integration For a given basis of orthogonal functions φ= φ i t, i N}, the operational matrix of integration is a constant matrix P N R N N such as: a Φ N τdτ = P N Φ N t 5 Obviously, the operational matrix of integration depends on the type of orthogonal functions used In this study we use a set of Legendre polynomials as our type of orthogonal functions 22 Legendre polynomials The Legendre polynomials are orthogonal on the interval [ 1, 1], with a weighunction wτ = 1 The set of Legendre polynomials is obtained from the formula of Olinde-Rodrigues: This gives: L n τ = 1 d n τ 2 1 n 6 2 n n! dτ n 3τ 2 1 L τ = 1, L 1 τ = τ, L 2 τ = 2 These polynomials can also be obtained from the recursive relationship [11]: 7 n + 1L n+1 τ = 2n + 1τL n τ nl n 1 τ 8 L τ = 1 et L 1 τ = τ 23 The shifted Legendre polynomials To obtain orthogonal Legendre polynomials on the interval [, ], we perform the following change of variable: τ = 2t 1 with t 9 48 Download Date 7/2/18 1:24 AM

3 M Sadok Attia, B Ayadi, N Benhadj Braiek The recursive relationship 8 becomes: n + 1φ n+1 t = 2n + 1 2t 1φ n t nφ n 1 t 1 with φ n t as the shifted Legendre polynomials for t, φ t = 1 and φ 1 t = 2t 1 The principle of orthogonality of the shifted Legendre polynomials is expressed by the following equation [12]: f φ i tφ j tdt = 2i + 1 δ ij 11 A function ft, square integrable in [, tf], may be expressed in terms of Legendre polynomials as: ft = f i φ i t 12 i= where the coefficients f i are given by [13]: f i = 2i + 1 f ftφ i tdt 13 In practice, only the first N-terms Legendre polynomials are considered Then we have: and: ft N 1 = f i φ i t = F N Φ N t 14 i= F N = [f f 1 f N 1 ] Φ N t = [φ t φ 1 t φ N 1 t] T 24 The operational matrix of integration of shifted Legendre polynomials In the case of shifted Legendre polynomials, the operational matrix of integration P N is given as follows [14]: P N = N 3 2N 3 1 2N 1 25 The product matrix The product of two orthogonal functions φ i t and φ j t can be projected into orthogonal function basis Φ N t using 14 This product [15] is expressed by the next equation: i, j, 1,, N 1}, φ i tφ j t = K T ij Φ N t 15 where Kij T is a vector of constant coefficients depending on the orthogonal functions used when considering the product of an orthogonal function φ i t and the orthogonal function basis Φ N t, the equation 15 yields the following one: φ i tφ N t = φ i t [φ t φ 1 t φ N 1 t] T = MiNΦ T N t 16 M in = K T i K T in 1 26 The operational matrices of fractional differentiation and integration There are also other orthogonal functions properties to use for solving fractional differential equations for linear and nonlinear systems, such as the generalized Laguerre operational matrix of Caputo fractional derivative [16], the Jacobi operational matrix [17] and modified generalized Laguerre operational matrix of fractional integration for solving fractional differential equations on the half line [18] These tools reduce the nonlinear fractional differential equations to a system of algebraic equations 3 Problem formulation Let us consider a global large scale system S consisting of M interconnected subsystems S i described by the following state equation: ẋ S i i t = A i x i t + A bi x i tu i t + B i u i t + W t y i t = C i x i t 17 with W t = M A ij x j t where x i t R n i u i t R m i and y i t R are respectively the state vector, the control vector and the output vector of the subsystem S i A i, A bi B i, C i and A ij are the matrices characterising the subsystem S i with respective dimensions n i n i, n i n i, n i m i, 1 n i and 49 Download Date 7/2/18 1:24 AM

4 Decentralized control synthesis for bilinear systems using orthogonal functions n i n j It is desired to determine for the global system a control law with a decentralized structure of the following form: The exploitation of the operational matrix of integration of the orthogonal basis Φ N t leads to the following algebraic equation: u i t = L i r i t K i x i t 18 where K i R m i n i and L i R mi 1, are the control parameters to be determinated i = 1,, M and r i t is the input The subsystem S i with the control law 18 can be written in the following equation: ẋ i t = A i x i t + A bi L i x i tr i t A bi x i tk i x i t if we consider: B i K i x i t + B i L i r i t + H A ij x j t 19 x i tk i x i t = I n K x [2] t with x [2] t = xt xt Using now an N degree orthogonal functions basis Φ N t, then the projections of vectors x i t become: x i t = x in Φ N t and the vector r i t can be written as follow : which gives: r i t = r T t = [r T NΦ N t] T = Φ T Ntr N x i tr i t = x in Φ N tφ T Ntr in = x in M[r in ]Φ N t M[r in ] = [M r in M N r in ] The integration of the equation 19 from null initial conditions, and the use of these approximations leads to the following equation: x in Φ N t = + A i x in Φ N τdτ A bi L i x in M[r in ]Φ N τdτ A bi I n K i x [2] in Φ Nτdτ B i K i x in Φ N τdτ x in = A i x in P N + A bi L i x in M[r in ]P N A bi I n K i x [2] in P N B i K i x in P N + B i L i r in P N + H A ij x jn P N 21 Making use of the V ec operator, which transform a matrix structure into a vector and the specific property [19]: V ecabc = C T AV ecb 22 the equation 21 yields the following one: M α ii V ecx in + α i V ecx [2] in + α ij V ecx jn = β i V ecr in where: α ii = I N n P T N A i M[r in ]P N T A bi L i +P T N B i K i α i = P T N A bi I n K i α ij = P N T A ij β i = P T N B i L i 23 we aim now to determine the K i and L i matrices i = 1,, M, such that each controlled subsystem S i has an input-output behaviour as close as possible to the reference model described by the following state equation: R i ż i t = E i z i t + F i r i t y ri t = G i z i t 24 where : z i t Rñi is the state vector of the ith reference submodel, and y ri t R its output vector E i, F i and G i are the chosen matrices characterising the reference model with respective dimensions ñ i ñ i, ñ i 1 and 1 ñ i The projection of the vectors z i t and y ri t on the orthogonal functions basis, and the use of the V ec operator give the following relations: + B i L i r in Φ N τdτ + H A ij x jn Φ N τdτ 2 z i t = z in Φ N t y ri t = y ri NΦ N t 25 5 Download Date 7/2/18 1:24 AM

5 M Sadok Attia, B Ayadi, N Benhadj Braiek V ecz in = H ii V ecr in V ecy rin = γ ii V ecr in 26 H ii = [I N ñi P T N E i] 1 P T N F i γ ii = I N G i H ii 27 Having the same behaviour of each subsystem S i i = 1,, M and its reference model R i can be expressed by the following relations: and: and: V ecx in = V ecz in 28 V ecx [2] in = V ecz[2] in 29 V ecz [2] in = z in z in M 3 M = M N M N 1N Taking into account the expressions 28 and 29, the equation 23 may be written as: M α ii V ecz in + α i V ecz [2] in + α ij V ecz jn = β i V ecr in 31 which is resolved by means of the numerical minimization of the following norm ξ: α ii V ecz in + α i V ecz [2] in ξ = + M α ij V ecz jn β i V ecr in 32 and yields the different control law gains K i and L i i = 1,, M This optimization problem can be easily carried out using specific Matlab functions 4 Illustrative example To illustrate the proposed technique of decentralized control design using orthogonal functions, we consider the interconnected system composed of two second order systems characterized by a state equation of the form 17 with the following matrix parameters: A 1 = A b1 = 1 2 A 12 = B 1 = B 2 = 2 C 1 = 1 5 A 2 = A b2 = C 2 = A 21 = We aimed to determine a decentralized control law for each of the two subsystems such that they have an identical input-output dynamic evolution to the second order reference model characterized by the following matrices: E = 1 1 F = G = For this objective we have applied the proposed method using Legendre polynomials with N = 1 The obtained decentralized laws are then characterized by the following gains: K 1 = 11 7 L 1 = 6 K 2 = L 2 = 33 Figure 1 and Figure 2 show the step reponses of the two interconnected systems using the obtained decentralized control laws, along with the step response of the considered reference As can be clearly seen, the controlled system outputs are very close to the desired reference model output, which illustrates the validity of the proposed technique 5 Conclusion In this paper, a new decentralized control technique is designed by using orthogonal functions as an interesting tool of dynamic system approximation The main advantage of the proposed technique is its applicability to a large class of interconnected systems without imposing particular conditions Furthermore, the decentralized controller parameters are adjusted such that each subsystem has the specific desired performance of a chosen reference model The validity of this new approach has been illustrated in a numerical example 51 Download Date 7/2/18 1:24 AM

6 Decentralized control synthesis for bilinear systems using orthogonal functions Figure 1 Step response of subsystem 1 and the reference model Figure 2 Step response of subsystem 2 and the reference model References [1] Johansson B, Speranzon A, Johansson M, Johansson KH, On decentralized negotiation of optimal consensus, Automatica, 28,44, [2] De Tuglie E, Iannone SM, Torelli F, Feedbacklinearization and feedback-feedforward decentralized control for multimachine power system, ELECTR POW SYST RES, 28,78, [3] Lavaei J, Aghdam AG, Stabilization of decentralized control systems by means of periodic feedback, Automatica, 28,44, [4] Paraskevopoulos PN, Chebychev Series Approach to System Identification, Analysis and Optimal Control, J Franklin Inst, 1983,316, [5] Paraskevopoulos PN, Kekkeris GTh, Hermite Series Approach to System Identification, Analysis and Optimal Control, Pro Meas Contr Conf, 1983,2, [6] Chen C, Hsiao H, time-domain synthesis via walsh functions, IEEE, 1975,122, [7] Attia MS, Ayadi B, Benhadj Braiek N, Decentralized control synthesis using orthogonal functions, 12th IFAC Symposium on Large Scale Systems : Theory and Applications, 21 [8] Ayadi B, Benhadj Braiek N, MIMO PID Controllers synthesis using orthogonal functions, IFAC 16th World Congress, International Federation of Automatic Control, Prague, Czech Republic, 25 [9] Doha EH, Bhrawy AH, Ezz-Eldien SS, A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, COMPUT MATH APPL, 211,62, [1] Bhrawy AH, Alofi AS, The operational matrix of fractional integration for shifted Chebyshev polynomials, APPL MATH LETT, 213,26,25-31 [11] Gradshteyn IS, Ryzhik IM, Tables of Integrals, Academic Press, 1979 [12] Hwang C, Chen MY, Analysis and Parameter Identification of Time-Delay Systems via Shifted Legendre Polynomials, Trans IBID, 1985,41, [13] Hwang C, Guo TY, Transfer Function Matrix Identification in MIMO Systems via Shifted Legendre Polynomials, Int J Control, 1984,39, [14] Chang RY, Wang ML, Trans ASME Dynam Syst Meas Control, 1982,52,15 [15] Rotella F, Dauphin-Tanguy G, Non-linear systems identification and optimal control, IntJControl, 1988,48, [16] Baleanu D, Bhrawy AH, Taha TM, Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems, preprint available at [17] Doha EH, Bhrawy AH, Ezz-Eldien SS, A new Jacobi operational matrix: An application for solving fractional differential equations, APPL MATH MODEL, 212,36, [18] Bhrawy AH, Alghamdi MM, Taha TM, A new modified generalized Laguerre operational matrix of fractional integration for solving fractional differential equations on the half line, Adv Diff Equ 212, :179 [19] Brewer J W, Kronecker Products and Matrix Calcu- 52 Download Date 7/2/18 1:24 AM

7 M Sadok Attia, B Ayadi, N Benhadj Braiek lus in Systems Theory, Trans IEEE Circ and Syst, 1978,CAS Download Date 7/2/18 1:24 AM