DESCRIPTOR FRACTIONAL LINEAR SYSTEMS WITH REGULAR PENCILS

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1 Int. J. Appl. Math. Comput. Sci., 3, Vol. 3, No., DOI:.478/amcs-3-3 DESCRIPTOR FRACTIONAL LINEAR SYSTEMS WITH REGULAR PENCILS TADEUSZ KACZOREK Faculty of Electrical Engineering Białystok Technical University, ul. Wiejska 45D, 5-35 Białystok, Pol Methods for finding solutions of the state equations of descriptor fractional discrete-time continuous-time linear systems with regular pencils are proposed. The derivation of the solution formulas is based on the application of the Z transform, the Laplace transform the convolution theorems. Procedures for computation of the transition matrices are proposed. The efficiency of the proposed methods is demonstrated on simple numerical examples. Keywords: descriptor system, fractional, system, regular pencil.. Introduction Descriptor (singular linear systems with regular pencils have been considered in many papers books (Dodig Stosic, 9; Wang,, Dai, 989; Fahmy O Reill, 989; Kaczorek, 4; 99; 7a; 7b; Kucera Zagalak, 988, Luenberger, 978; Van Dooren, 979. The eigenvalue invariant assignment by state output feedbacks was investigated by Dodig Stosic (9, Wang (, Dai (989, Fahmy O Reill (989 as well as Kaczorek (4; 99, while the realization problem for singular positive continuous-time systems with delays was discussed by Kaczorek (7b. The computation of Kronecker s canonical form of a singular pencil was analyzed by Van Dooren (979. A delay dependent criterion for a class of descriptor systems with delays varying in intervals was proposed by Wang (. Fractional positive continuous-time linear systems were addressed by Kaczorek (8, along with positive linear systems with different fractional order (Kaczorek, 7a. A new concept of practical stability of positive fractional D systems was proposed by Kaczorek (b, who also presented an analysis of fractional linear electrical circuits (Kaczorek, a some selected problems in the theory of fractional linear systems (Kaczorek, b. A new class of descriptor fractional linear systems electrical circuits was introduced, their solution of state equations was derived a method for decomposition of the descriptor fractional linear systems with regular pencils into dynamic static parts was proposed by Kaczorek (a, who also considered positive fractional continuous-time linear systems with singular pencils (Kaczorek, b. Fractional-order iterative learning control for fractional-order systems was addressed by Yan et al. (c. In this paper, methods of finding solutions of the state equations of descriptor fractional discrete-time continuous-time linear systems with regular pencils will be proposed. The paper is organized as follows. In Section the solution to the state equation of the descriptor system is derived using the method based on the Z transform the convolution theorem. A method for computation of the transition matrix is proposed illustrated on a simple numerical example in Section 3. In Section 4 the proposed method is extended to continuous-time linear systems. Concluding remarks are given in Section 5. The following notation will be used: R is the set of real numbers, R n m is the set of real n m matrices R n R n, Z is the set of nonnegative integers, I n is the n n identity matrix.. Discrete-time fractional linear systems Consider the descriptor fractional discrete-time linear system EΔ α x i Ax i Bu i, i Z {,,,...}, <α<, (

2 3 T. Kaczorek α is the fractional order, x i R n is the state vector u i R m is the input vector E,A R n n, B R n m. It is assumed that det E, but the pencil (E, A is regular, i.e., detez A for somez C. ( Without lost of generality we may assume that E E R n n, E R r r rank E ranke r<n. (3 Admissible boundary conditions for ( are given by x. The fractional difference of the order α, is defined by i Δ α x i c k x i k, (4a ( α k { c k ( k ( α k α(α...(α k k!, k,,... (4b for for k, k,,... (4c Substitution of (4a into ( yields i Ex i Fx i c k x i k Bu i, i Z, (5 k F A Ec A Eα. Applying to (5 the Z-transform taking into account that (Kaczorek, 99 Zx i p p z p X(zz p x j z j, p,,..., (6 j we obtain X(z Ex F {Ex z H(zBU(z}, X(z Zx i x i z i, U(z Zu i i u i z i, i i H(z Zh i, h i Ec k x i k. k (7a (7b Let Ex F j μ ψ j z (j, (8 μ is the positive integer defined by the pair (E,A (Kaczorek, 99; Van Dooren, 979. Comparison of the coefficients at the same powers of z of the equality Ex F j μ j μ ψ j z (j ψ j z (j Ex F I n (9a yields Eψ μ ψ μ E (9b Eψ k Eψ k ψ k E ψ k E { In for k, for k μ, μ,...,,,,... (9c From (9b (9c we have the matrix equation ψμ V G, (a ψ N G G R G G (Nμn (Nμn,... F... G RNn (μn,... E G R (μn (μn, G E R Nn Nn,

3 Descriptor fractional linear systems with regular pencils ψ μ ψ μ ψ μ.. R(μn n, ψ N V ψ ψ ψ. ψ N. I n RNn n, R(μn n. Equation (a has the solution ψμ ψ N (b for given G V if only if { } V rank G, rankg. ( It is easy to show that the condition ( is satisfied if the condition ( is met. Substituting (8 into (7a we obtain X(z ψ j z (j Ex z H(zBU(z. j μ ( Applying the inverse transform Z the convolution theorem to (, we obtain iμ k x i ψ i Ex ψ i k c j x k j iμ j ψ i k Bu k. (3 To find the solution to Eqn. (, first we compute the transition matrices ψ j for j μ, μ,...,,,... next, using (3, we obtain the desired solution. 3. Computation of transition matrices To compute the transition matrices ψ k for k μ, μ,...,n,..., the following procedure is recommended. Procedure. Step. Find a solution ψ μ of the equation G ψ μ V, (4 G, ψ μ V are defined by (b. Note that, if the matrix E has the form (3, then the first r rows of the matrix ψ μ are zero its n r last rows are arbitrary. 3 Step. Choose n r arbitrary rows of the matrix ψ so that { E rank the equation E } In Fψ, E rank ψ ψ In Fψ (5 has a solution with arbitrary last n r rows of the matrix ψ. Step 3. Knowing ψ μ, choose the last n r rows of the matrix ψ so that { } E Fψ E rank, rank (6 the equation E ψ F ψ ψ (7 has a solution with arbitrary last n r rows of the matrix ψ. Repeating the last step for ψ ψ3,,... ψ 3 ψ 4 we may compute the desired matrices ψ k for k μ, μ,... The details of the procedure will be shown on the following example. Example. Find the solution to Eqn. ( for α.5 with the matrices E, A, B (8 the initial condition x u i, i Z. In this case the pencil ( of (8 is regular since detez A z z, (9 μ α A α.5. (

4 3 T. Kaczorek Using Procedure, we obtain the following. Step. In this case Eqn. (4 has the form ψ α ψ, ( its solution with the arbitrary second row ψ ψ of ψ is given by ψ ψ.5 ψ ψ. ( Step. We choose the row ψ ψ of ψ so that (5 holds, i.e., rank α the equation α has the solution ψ rank ψ ψ ψ α.5 α ψ ψ, (3 (4 (5 with the second arbitrary row ψ ψ of ψ. Step 3. We choose ψ ψ so that the equation α α ψ ψ α α ψ ψ (6 has the solution ψ ψ α.5α α ψ ψ (7 with arbitrary ψ ψ. Continuing the procedure, we obtain ψ, ψ.5 k ( k α k.5α k (8 for k,,... Using (3, (8 (, we obtain the desired solution of the form x i ψ i i i ψ i k k ψ i k j c j are defined by (4b. c j x k j u k, (9 4. Continuous-time fractional linear systems Consider the descriptor fractional continuous-time linear system described by the state equations ED α x(t Ax(tBu(t, n <α n {,,...} y(t Cx(tDu(t, (3a (3b D α is the Caputo differentiation operator, x(t R n,u(t R m, y(t R p are the state, input output vectors E,A R n n,b R n m,c R p n,d R p m. It is assumed that det E the pencil (E,A is regular, i.e., deteλ A for some λ C. (3 Admissible initial conditions for (3a are given by x (k ( x k, for k,,...,n. (3 Applying the Laplace transform (L to Eqn. (3a, we obtain (Kaczorek, b Es α AX(s BU(s s α k x k,, (33 X(s Lx(t U(s Lu(t. Ifthe condition (3 is satisfied, then from (33 we obtain X(s Es α A (BU(s s α k x k,. (34

5 Descriptor fractional linear systems with regular pencils In a particular case when det E, from (34 we have X(s (E A i E s (iα (35 since i ( BU(s s α k x k, Es α A Es α (I n (Es α A (E A i E s (iα. i Using the inverse Laplace transform (L the convolution theorem, we obtain x(t L X(s t (E i (t τ(iα A E Bu(τdτ Γ(i α i t iαk Γ(iα k (E A i x k,, (36 Γ(α is the gamma function (Kaczorek, b. Therefore, the following theorem for det E has been proved. Theorem. The solution of Eqn. (3a for det E the initial conditions (3 is given by (36. If E I n, then (36 takes the form (Kaczorek, b x(t n Φ l (tx (l ( l Φ l (t t A k t (kαl Γ(kα l, Φ(t τbu(τdτ, (37 (38a A k t (kα Φ(t Γ(k α. (38b If det E butthepencil is regular (the condition (3 is met, then Es α A T i satisfy the equality T i s (iα, (39 i μ ET i AT i T i E T i A { In for i, for i, (4 33 T i for i< μ, T μ E ET μ. The equality (4 follows from the comparison of the coefficients at the same powers of s in the equality Es α A T i s (iα i μ i μ Substitution of (39 into (34 yields X(s i μ i T i s (iα Es α A I n. T i s (iα BU(s s iα k x k, T i s (iα BU(s s iα k x k, i μ T i s (iα BU(s s iα k x k,. (4 Applying the inverse Laplace transform the convolution theorem to (4, we obtain x(t t (t τ(iα T i B Bu(τdτ Γ(i α i T i μ i i t iαk Γ(iα k Ex k, T i Bu(t (i α δ (iα Ex k, (4 or t x(t (T A i (t τ(iα T B u(τdτ i Γ(i α (T A i t iαk T Γ(iα k x k, μ T i Bu(t (i α δ (iα Ex k,, (43 since by (4 T i (T A i T for i,,... δ (k is the k-th derivative of the delta impulse function δ. Therefore, the following theorem has been proved. Theorem. If the condition (3 is satisfied, then the solution of Eqn. (3a with the admissible initial conditions (3 is given by (4 or (43.

6 34 T. Kaczorek In a particular case of <α, from (43 we have x(t t (T A i (t τ(iα T B u(τdτ i Γ(i α t iα Γ(iα Ex μ T i Bu(t (i α δ (iα Ex. i (44 To compute the matrices T i for i μ, μ,...,the procedure given in Section 3 is recommended. Example. Consider Eqn. (3a for α.5with the matrices E, A, B (45 the zero initial condition x. The pencil is regular since deteλ A λ λ, (λ sα (46 Eλ A T λ.5λ.5.5 T T λ,, T.5 (47a. (47b Using (4, (43 (47b we obtain t (t τ.5 x(t T B u(τdτ T Bu(t Γ(.5 t (t τ.5 u(τdτ u(t..5 Γ(.5 (48 5. Concluding remarks New methods of finding solutions of the state equations of descriptor fractional discrete-time continuous-time linear systems with regular pencils have been proposed. Derivation of the solution formulas has been based on the application of the Z-transform, the Laplace transform the convolution theorems. A procedure for computation of the transition matrices has been proposed its application has been demonstrated on simple numerical examples. An open problem is the extension of the method for D descriptor fractional discrete continuous-discrete linear systems. Acknowledgment This work was supported by the National Science Centre in Pol under the grant no. N N References Dodig, M. Stosic, M. (9. Singular systems state feedbacks problems, Linear Algebra Its Applications 43(8: Dai, L. (989. Singular Control Systems, Lectures Notes in Control Information Sciences, Vol. 8, Springer-Verlag, Berlin. Fahmy, M.H O Reill, J. (989. Matrix pencil of closed-loop descriptor systems: Infinite-eigenvalues assignment, International Journal of Control 49(4: Gantmacher, F.R. (96. The Theory of Matrices, Chelsea Publishing Co., New York, NY. Kaczorek, T. (a. Descriptor fractional linear systems with regular pencils, Asian Journal of Control 5(4: 4. Kaczorek, T. (b. Positive fractional continuous-time linear systems with singular pencils, Bulletin of the Polish Academy of Sciences: Technical Sciences 6(: 9. Kaczorek, T. (a. Positive linear systems consisting of n subsystems with different fractional orders, IEEE Transactions on Circuits Systems 58(7: 3. Kaczorek, T. (b. Selected Problems of Fractional System Theory, Springer-Verlag, Berlin. Kaczorek, T. (a. Positive linear systems with different fractional orders, Bulletin of the Polish Academy of Sciences: Technical Sciences 58(3: Kaczorek, T. (b. Practical stability asymptotic stability of positive fractional D linear systems, Asian Journal of Control (: 7. Kaczorek, T. (8. Fractional positive continuous-time linear systems their reachability, International Journal of Applied Mathematics Computer Science 8(: 3 8, DOI:.478/v Kaczorek, T. (7a. Polynomial Rational Matrices. Applications in Dynamical Systems Theory, Springer-Verlag, London. Kaczorek, T. (7b. Realization problem for singular positive continuous-time systems with delays, Control Cybernetics 36(: Kaczorek, T. (4. Infinite eigenvalue assignment by an output feedbacks for singular systems, International Journal of Applied Mathematics Computer Science 4(: 9 3.

7 Descriptor fractional linear systems with regular pencils Kaczorek, T. (99. Linear Control Systems, Vol., Research Studies Press J. Wiley, New York, NY. Kucera, V. Zagalak, P. (988. Fundamental theorem of state feedback for singular systems, Automatica 4(5: Luenberger, D.G. (978. Time-invariant descriptor systems, Automatica 4: Podlubny, I. (999. Fractional Differential Equations, Academic Press, New York, NY. Wang, C. (. New delay-dependent stability criteria for descriptor systems with interval time delay, Asian Journal of Control 4(: Van Dooren, P. (979. The computation of Kronecker s canonical form of a singular pencil, Linear Algebra Its Applications 7: 3 4. Yan L., YangQuan C., Hyo-Sung A., (c. Fractional-order iterative learning control for fractional-order systems, Asian Journal of Control 3(: Tadeusz Kaczorek received the M.Sc., Ph.D. D.Sc. degrees in electrical engineering from the Warsaw University of Technology in 956, , respectively. In the years he was the dean of the Electrical Engineering Faculty, in the period of he was a deputy rector of the Warsaw University of Technology. In 97 he became a professor in 974 a full professor at the same university. Since 3 he has been a professor at Białystok Technical University. In 986 he was elected a corresponding member in 996 a full member of the Polish Academy of Sciences. In the years he was the director of the Research Centre of the Polish Academy of Sciences in Rome. In 4 he was elected an honorary member of the Hungarian Academy of Sciences. He has been granted honorary doctorates by nine universities. His research interests cover systems theory, especially singular multidimensional systems, positive multidimensional systems, singular positive D D systems, as well as positive fractional D D systems. He initiated research in the field of singular D, positive D positive fractional linear systems. He has published 4 books (six in English over scientific papers. He has also supervised 69 Ph.D. theses. He is the editor-in-chief of the Bulletin of the Polish Academy of Sciences: Technical Sciences a member of editorial boards of ten international journals. Received: 8 May Revised: September