A new approach to the realization problem for fractional discrete-time linear systems
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1 BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vol 64 No 26 DOI: 55/bpasts-26-2 A new approach to the realization problem for fractional discrete-time linear systems T KACZOREK Faculty of Electrical Engineering Białystok University of Technology 45D Wiejska St 5-35 Bialystok Pol Abstract A new approach to the realization problem for fractional discrete-time linear systems is proposed A procedure for computation of fractional realizations of given transfer matrices is presented illustrated by numerical examples Key words: fractional linear discrete-time system computation procedure realization transfer matrix Introduction Determination of the state space equations for given transfer matrices is a classical problem called the realization problem which has been addressed in many papers books 8 An overview of the positive realization problem is given in The realization problem for positive continuous-time discrete-time linear system has been considered in for linear systems with delays in The realization problem for fractional linear systems has been analyzed in for positive 2D hybrid linear systems in for fractional systems with delays in A new modified state variable diagram method for determination of positive realizations with reduced number of delays for given proper transfer matrices has been proposed in 35 In this paper a new approach to the realization problem for fractional discrete-time linear systems will be proposed The paper is organized as follows Some preliminaries problem formulation are given in Sec 2 In Sec 3 the solution to the realization problem for fractional discrete-time linear systems is presented illustrated by numerical examples Concluding remarks are given in Sec 4 The following notation will be used: R the set of real numbers R n m the set of n m real matrices R n m (w) - the set of n m rational matrices in w with real coefficients Z + the set of nonnegative integers I n the n n identity matrix 2 Preliminaries problem formulation Consider the fractional discrete-time linear system α x i = Ax i + Bu i i Z + = { } (a) y i = Cx i + Du i (b) α x i = i c j x i j j= c j = ( ) j ( { = ( ) j α(α )(α j+) j! α j ) for for j = j = 2 (c) x i R n u i R m y i R p are the state input output vectors A R n n B R n m C R p n D R p m Using the Z-transformation to (a) (b) for zero initial conditions we obtain 6 Z α x i = wx(z) = AX(z) + BU(z) i Z + = { } Y (z) = CX(z) + DU(z) Z α x i = ( z ) α X(z) = w(z)x(z) = wx(z) w = w(z) = ( z ) α = c i z i X(z) = Zx i = i= x i z i i= U(z) = Zu i Y (z) = Zy i From (2) we have the transfer matrix (2a) (2b) (2c) T(w) = CI n w A B + D (3) The transfer matrix T(z) is called proper if only if lim T(w) = D Rp m (4) it is called strictly proper if only if D = From (3) we have since lim I nw A = lim T(w) = D (5) kaczorek@iseppwedupl 9 Download Date /2/8 9:5 AM
2 T Kaczorek Definition The matrices A B C D are called a fractional realization of a given transfer matrix T(w) if they satisfy the equality (3) A fractional realization A B C D is called minimal if the dimension of the matrix A is minimal among all realizations of T(w) The fractional realization problem can be stated as follows Given a proper transfer matrix T(w) R p m (w) find a fractional realization A B C Dof the matrix T(w) 3 Problem solution 3 Single-input single-output systems First the essence of the proposed method is presented for single-input singleoutput (SISO) fractional discrete-time linear systems with the transfer function T(w) = b nw n + b n w n + + b w + b w n + a n w n + + a w + a (6) Using (4) for (6) we obtain T sp (w) = T(w) D = D = lim T(w) = b n (7) b n w n + + b w + b w n + a n w n + + a w + a (8a) b k = b k a k b n k = n (8b) Therefore the realization problem has been reduced to finding matrices A R n n B R n m C R p n for given strictly proper transfer function (8a) Multiplying the numerator the denominator of (8a) by w n we obtain T sp (w) = Y U = b n w + + b w n + b w n + a n w + + a w n + a w n (9) Y U are the Z-transforms of y i u i respectively Define E = U + a n w + + a w n () + a w n From (9) () we have E = U (a n w + + a w n + a w n )E Y = (b n w + + b w n + b w n )E (a) (b) From () follows the block diagram shown in Fig Assuming as the state variables x i x 2i x ni the outputs of the delay elements we may write the equations α x i = x 2i α x 2i = x 3i α x n i = x ni α x ni = a x i a x 2i a n x ni + u i y i = b x i + b x 2i + + b n x ni The Eq (2) can be written in the form α x i = Ax i + Bu i y i = Cx i x i = x i x 2i x ni T i Z + A = a a a 2 a n B = C = b b b n (2a) (2b) (3a) (3b) (4) Fig State diagram for transfer function (9) Bull Pol Ac: Tech 64() 26 Download Date /2/8 9:5 AM
3 A new approach to the realization problem for fractional discrete-time linear systems Remark If we choose the state variables so that x k = x n k+ for k = n then the realization of (8) has the form a n a n 2 a a A = B = C = b n b n 2 b (5) Remark 2 Note that the transposition (denoted by T ) of the transfer function does change it ie T sp (w) T = T sp (w) = CI n w A B T = B T I n w A T C T the matrices a a A 2 = A T = a 2 a n (6) B 2 = C T = b b b n C 2 = B T = a n a n 2 A 3 = A T = a a B 3 = C T = b n b n 2 b C 3 = B T = (7) are also the realizations of the transfer function (8) Example Find the fractional realization of the transfer function T(w) = 2w2 + w + w 2 + 3w + 4 (8) Using (7) we obtain D = lim T(w) = 2 (9) T sp (w) = T(w) D = In this case we have E = 5 w 2 + 3w + 4 = 5w + 2w 2 + 3w + 4w 2 (2) U + 3w + 4w 2 (2) E = U (3w + 4w 2 )E Y = (5w + 2w 2 )E (22a) (22b) The block diagram corresponding to (22) is shown in Fig 2 Fig 2 State diagram for transfer function (2) For the choice of the state variables shown in Fig 2 we obtain the equations α x i = x 2i α x 2i = 4x i 3x 2i + u y i = 2x i + 5x 2i the realization A = B = 4 3 (23a) (23b) C = 2 5 (24) 32 Multi-input multi-output systems Consider a strictly proper transfer matrix T sp (w) R p m (w) Let D i (w) = w di (a idi w di + + a i w + a i ) i = m (25) be the least common denominator of all entries of the i-th column of T sp (w) Using (25) we may write T sp (w) in the form N (w) N m (w) D (w) D m (w) T sp (w) = = N(w)D (w) N p (w) D (w) N pm (w) D m (w) N (w) N m (w) N(w) = N p (w) N pm (w) D(w) = diag D (w) D m (w) (26a) (26b) Bull Pol Ac: Tech 64() 26 Download Date /2/8 9:5 AM
4 T Kaczorek From (25) it follows that Note that if then ¾ C = D(w) = diag w di w dm A m W (27a) A m = blockdiag a a m a i = a i a idi W = blockdiag W W m W i = w w di (27b) (27c) N ij (w) = c dj ij w dj + + c ij w + c ij (28a) N(w) = CW c c c d c m c m c dm m c p c p c d p c pm c pm c dm pm We shall show that the matrices A = blockdiag A A m A i = a i a i a i2 a idi i = m B = blockdiag b b m b i = T R di i = m (28c) are the desired realization of (26) Using (27) (28) it is easy to verify that w b i D i (w) = I n w A i w di (28b) (28c) (29) (3) BD(w) = I n w AW (3) Premultiplication of (3) by CI n w A postmultiplication by D (w) yields CI n w A B = CWD (w) = N(w)D (w) = T sp (w) (32) Therefore we have the following procedure for finding a fractional realization of a given proper transfer matrix T(w) Procedure Step Using (4) find the matrix D the strictly proper transfer matrix T sp (w) Step 2 Find the least common denominators D (w) D m (w) write T sp (w) in the form (26) Step 3 Knowing D(w) find the indices d d m the matrices W A m Step 4 Knowing N(w) find the matrix C defined by (28c) Step 5 Using (29) find the matrices A B Remark 3 Similar results can be obtained for the least common denominator of all entries of the j-th row of T sp (w) Example 2 Find the fractional realization of the transfer matrix 2w + w + 3 w w + (33) T(w) = 3w + 8 2w + 5 Using Procedure (33) we obtain the following: Step Using (4) (33) we obtain = lim D = lim T(w) w + 3 w + 2w + w 3w + 8 T sp (w) = T(w) D = 2w + 5 w = w + (34) (35) Step 2 From (35) we have D (w) = w() D 2 (w) = (w + )() D(w) = N(w) = T sp (w) = N(w)D (w) 2() 2w w + w() (w + )() Step 3 From (36b) we have d = d 2 = 2 w W = A 2 m = 2 3 w (36a) (36b) (37) since w() w 2 D(w) = = (w + )() w 2 2 w 2 3 w (38) 2 Bull Pol Ac: Tech 64() 26 Download Date /2/8 9:5 AM
5 A new approach to the realization problem for fractional discrete-time linear systems Step 4 Using (36b) we obtain 2w + 4 N(w) = 2w w w = 2 = CW w C = Step 5 Using (29) (37) we obtain 2 A = B = 2 3 (39a) (39b) (4) The desired fractional realization of (33) is given by (4) (39b) (34) 4 Concluding remarks A new approach to finding fractional realizations of given transfer matrices of discrete-time linear systems has been proposed It has been shown that for any given proper transfer matrix there exist always many fractional realizations A procedure for computation a fractional realization of a given transfer matrix has been proposed The effectiveness of the procedure has been demonstrated on numerical examples The classical Gilbert method 29 can also be applied to compute the fractional realizations of the given transfer matrices of discrete-time linear systems The presented method can be easily extended to positive fractional linear discrete-time systems without with delays Acknowledgements This work was supported by the grant No S/WE//25 REFERENCES L Benvenuti L Farina A tutorial on the positive realization problem IEEE Trans on Automatic Control 49 (5) (24) 2 L Farina S Rinaldi Positive Linear Systems Theory Applications J Wiley New York 2 3 T Kaczorek Existence determination of the set of Metzler matrices for given stable polynomials Int J Appl Math Comput Sci 22 (2) (22) 4 T Kaczorek Linear Control Systems vol Research Studies Press J Wiley New York T Kaczorek Polynomial Rational Matrices Springer London 29 6 T Kaczorek Ł Sajewski The Realization Problem for Positive Fractional Systems Springer London 24 7 T Kaczorek Selected Problems in Fractional Systems Theory Springer London 2 8 U Shaker M Dixon Generalized minimal realization of transfer-function matrices Int J Contr 25 (5) (977) 9 T Kaczorek Positive D 2D Systems Springer London 22 T Kaczorek A realization problem for positive continuoustime linear systems with reduced numbers of delays Int J Appl Math Comput Sci 6 (3) (26) T Kaczorek Computation of positive stable realizations for discrete-time linear systems Computational Problems of Electrical Engineering 2 () 4 48 (22) 2 T Kaczorek Computation of positive stable realizations for linear continuous-time systems Bull Pol Ac: Tech 59 (3) (2) 3 T Kaczorek Computation of realizations of discrete-time cone systems Bull Pol Ac: Tech 54 (3) (26) 4 T Kaczorek Positive asymptotically stable realizations for descriptor discrete-time linear systems Bull Pol Ac: Tech 6 () (23) 5 T Kaczorek Positive minimal realizations for singular discrete-time systems with delays in state delays in control Bull Pol Ac: Tech 53 (3) (25) 6 T Kaczorek Positive realizations for descriptor continuoustime linear systems Measurement Automation Monitoring 56 (9) (22) 7 T Kaczorek Positive realizations for descriptor discrete-time linear systems Acta Mechanica et Automatica 6 (2) 58 6 (22) 8 T Kaczorek Positive stable realizations of continuous-time linear systems Proc Conf Int Inf Eng Syst CD- ROM (22) 9 T Kaczorek Positive stable realizations of discrete-time linear systems Bull Pol Ac: Tech 6 (3) (22) 2 T Kaczorek Positive stable realizations with system Metzler matrices Archives of Control Sciences 2 (2) (2) 2 T Kaczorek Realization problem for positive discretetime systems with delays System Science 3 (4) 7 3 (24) 22 T Kaczorek Realization problem for positive multivariable discrete-time linear systems with delays in the state vector inputs Int J Appl Math Comput Sci 6 (2) 6 (26) 23 T Kaczorek Determination of positive realizations with reduced numbers of delays or without delays for discrete-time linear systems Archives of Control Sciences 22 (4) (22) 24 T Kaczorek Positive realizations with reduced numbers of delays for 2-D continuous-discrete linear systems Bull Pol Ac: Tech 6 (4) (22) 25 T Kaczorek Positive stable realizations for fractional descriptor continuous-time linear systems Archives of Control Sciences 22 (3) (22) 26 T Kaczorek Positive stable realizations of fractional continuous-time linear systems Int J Appl Math Comput Sci 2 (4) (2) 27 T Kaczorek Realization problem for descriptor positive fractional continuous-time linear systems Theory Applications of Non-integer Order Systems eds W Mitkowski pp 3 3 Springer London 23 Bull Pol Ac: Tech 64() 26 3 Download Date /2/8 9:5 AM
6 T Kaczorek 28 T Kaczorek Realization problem for fractional continuoustime systems Archives of Control Sciences 8 () (28) 29 Ł Sajewski Positive stable minimal realization of fractional discrete-time linear systems Advances in the Theory Applications of Non-integer Order Systems eds W Mitkowski pp Springer London 23 3 Ł Sajewski Positive stable realization of fractional discretetime linear systems Asian J Control 6 (3) DOI: 2/asjc75 (24) 3 T Kaczorek Positive realizations of hybrid linear systems described by the general model using state variable diagram method J Automation Mobile Robotics Intelligent Systems 4 3 (2) 32 T Kaczorek Realization problem for positive 2D hybrid systems COMPEL 27 (3) (28) 33 Ł Sajewski Positive realization of fractional continuous-time linear systems with delays Measurement Automation Monitoring 58 (5) (22) 34 Ł Sajewski Positive realization of fractional discrete-time linear systems with delays Measurements Automatics Robotics 2 CD-ROM (22) 35 T Kaczorek A modified state variables diagram method for determination of positive realizations of linear continuoustime systems with delays Int J Appl Math Comput Sci 22 (4) (22) 4 Bull Pol Ac: Tech 64() 26 Download Date /2/8 9:5 AM
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