POSITIVE PARTIAL REALIZATION PROBLEM FOR LINEAR DISCRETE TIME SYSTEMS

Transcription

1 Int J Appl Math Comput Sci 7 Vol 7 No 65 7 DOI: 478/v POSITIVE PARTIAL REALIZATION PROBLEM FOR LINEAR DISCRETE TIME SYSTEMS TADEUSZ KACZOREK Faculty of Electrical Engineering Białystok Technical University Wiejska 45D 5 35 Białystok kaczorek@iseppwedupl A partial realization problem for positive linear discrete-time systems is addressed Sufficient conditions for the existence of its solution are established A procedure for the computation of a positive partial realization for a given finite sequence of the values of the impulse response is proposed The procedure is illustrated by four numerical examples Keywords: positive systems partial realization existence computation linear systems discrete-time systems Introduction In positive systems the inputs state variables outputs take only non-negative values Examples of positive systems are industrial processes involving chemical reactors heat exchangers distillation columns storage systems compartmental systems or water atmospheric pollution models A variety of models having positive linear system behaviour can be found in engineering management science economics social sciences biology medicine etc Positive linear systems are defined on cones not on linear spaces Therefore the theory of positive systems is more complicated less advanced Overviews of the state of the art in positive systems theory are given in the monographs (Farina Rinaldi ; Kaczorek ) The realization problem for positive linear systems was considered in many papers books (Benvenuti Farina 4; Farina Rinaldi ; Kaczorek 99; Kaczorek ; Kaczorek 6a; Kaczorek 6b; Kaczorek 4; Kaczorek 5; Kaczorek 6c; Kaczorek 6d; Kaczorek 6e; Kaczorek Busłowicz 4) The problem of constructing linear state variable models from given impulse responses (or Markov parameters) was considered in (Ho Kalman 966; Kaczorek 99) The partial realization problem consists in constructing state variables models from given finite sequences of impulse responses In this paper the partial realization problem will be addressed for positive linear discrete-time systems Sufficient conditions for the existence of its solution will be established a procedure for the computation of its positive realization for a given finite sequence of the values of the impulse response will be proposed To the best of the author s knowledge the positive partial realization problem for discrete-time linear system has not been considered yet Preliminaries Problem Formulation Let R n m be the set of n m real matrices R n : R n Thesetofn m real matrices with non-negative entries will be denoted by R n m + the set of nonnegative integers by Z + Consider the linear discrete-time system x i+ Ax i + Bu i i Z + y i Cx i + Du i () where 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 The system () is called (internally) positive if only if x i R n + y i R p + i Z + for every x R n + any input sequence u i R m + i Z + The system () is (internally) positive if only if (Kaczorek ; Fa-

2 66 T Kaczorek rina Rinaldi ): A R n n + B Rn m + C R p n + D Rp m + () The transfer matrix of () is given by T (z)ciz A B+D CI Ad db+d (3) where d z is the delay operator Let G(d) G i d i (4) be the impulse response matrix of the system () From the well-known equality { D for i G i CA i (5) B for i () it follows that for the positive system () we have that G i R p m + for i Z + The partial positive realization problem can be formulated as follows: Given a finite sequence of nonnegative matrices Ĝ i R p m + for i N (6) Subproblem Given a finite sequence ĝ i for i N () find n N the transfer function (8) such that ĝ i g i for i N () where g i is defined by (9) Subproblem Given the transfer function (8) find its positive realization () 3 Solution of Subproblem Taking into account that T (d) using (8) we can write g i d i () a(d)g(d) b(d) (3) find the matrices () of the positive system () such that Ĝ i G i for i N (7) a(d) ĝ(d)+ḡ(d) b(d) (4) with G i defined by (4) In this paper sufficient conditions for the existence of a solution to the above problem will be established a procedure for the computation of the matrices () for a given finite sequence G i R p m + N will be proposed 3 Problem Solution First the idea of the proposed method will be outlined for a single-input single-output (SISO m p )systemin this case the transfer function of () can be written down in the form T (d) b(d) a(d) its impulse response is g(d) b i d i n (8) a i d i g i d i g i R + i Z + (9) The partial positive realization problem can be decomposed into the following two subproblems: where the polynomial is known the sum ĝ(d) ḡ(d) N ĝ i d i (5) in+ ḡ i d i (6) is unknown Equation (4) can be rewritten in the form where a(d)ĝ(d) b(d)+ g(d) (7) g(d) a(d)ḡ(d) in+ g i d i (8) The subproblem has thus been reduced to the following one: Given the polynomial (5) find the polynomials a(d) b(d) of (8) such that (7) holds for some g(d)

3 Positive partial realization problem for linear discrete-time systems 67 Using (8) (5) we obtain a(d)ĝ(d) where ( j a i d i)( N ĝ j d j) j N i+n ĝ j d j a i ĝ k i d k N k + ( ĝ k N+n kn+ N q k d k + k ki q k ĝ k ( n a i ĝ k i ) d k ik N N+n kn+ a i ĝ k i ) d k ( n ik N a i ĝ k i )d k (9) ĝ k i a i () Let q k for k n +N Then from () we have Ĝ nn a ĝ nn () where ĝ n ĝ n ĝ ĝ n ĝ Ĝ nn ĝ N ĝ N ĝ N n a a n We assume that ĝ nn ĝ n+ ĝ N () rank ĜnN rankĝnn ĝ nn (3) Eqn () has a non-negative solution ie a R n +In (Kaczorek 6e) a method was proposed for the computation of a positive solution to () Note that (9) has the form (7) if only if q k for k n +N or equivalently the coefficients a n of the polynomial a(d) are a solution to () Knowing the coefficients a i i n we can compute the coefficients b j j nof the polynomial b(d) using the formula b k ĝ k ĝ k i a i for k n (4) It follows from the comparison of the right-h sides of (7) (9) for q k k n +N In summary we have the following result: Theorem Let n N Given a finite sequence () there exists a transfer function of the form T (z) b z n + + b n z + b n z n z n a n z a n D + b z n + + b n z + b n z n z n a n z a n (5) with non-negative coefficients if Eqn () has a nonnegative solution ie a R n + k bk ĝ k ĝ k i a i k n (6) 3 Solution of Subproblem It is well known (Kaczorek ; Benvenuti Farina 4; Kaczorek 4; Kaczorek 6a; Kaczorek 6c) that if the coefficients a n b b b n are nonnegative then there always exists a positive realization () of the transfer function (5) For example we may choose the positive realization as follows: A B a n a n a n C bn bn b D b (7a) or a n bn a n bn Ā a n B b C D b (7b) From (5) we have D b b b + b b b + b b n b n + a n b (8) using (4) we obtain (6) Therefore if the coefficients a n b b b n are non-negative then so are the coefficients b b b n the realizations (7) are then positive From (8) it follows that b k a k do not imply that b k kn

4 68 T Kaczorek If () has a non-negative solution a R n +thena positive realization () for a given finite sequence (9) can be computed using the following procedure: Procedure (SISO systems) Step Given a sequence () find a non-negative solution a R n + to () Step Using (4) compute the coefficients b b b n find the transfer function (5) Step 3 Using one of the well-known methods (Benvenuti Farina 4; Farina Rinaldi ; Kaczorek ; Kaczorek 6a; Kaczorek 6b; Kaczorek 4; Kaczorek 5; Kaczorek 6c; Kaczorek 6d; Kaczorek 6e; Kaczorek Busłowicz 4) compute a desired positive realization () of the transfer function (5) eg the positive realization (7) Remark If n is not known apriori then it is recommended to start the procedure with its small value to increase it in the next step Find a positive realization () for the se- Example quence ĝ ĝ 3 ĝ 8 ĝ 3 9 ĝ 4 46 (9) Using the procedure for N 4 n we obtain the following: Step In this case Eqn () for (9) has the form its solution is Step Using (4) we compute the coefficients b ĝ b ĝ ĝ b ĝ ĝ ĝ The desired transfer function has the form T (z) z + z + 3z + z + z z (3) z 3z + z z 3z +8z +9z 3 +46z 4 + Step 3 Using (7a) for (3) we obtain the desired positive realization A B C b b 3 D (3) Now let us assume that only the first four numbers of the sequence (9) are given ie N 3 n Inthis case Eqn () takes the form (3) One of the coefficients can be chosen arbitrarily If we choose then we obtain the same transfer function (3) its positive realization (3) If we choose then /3 b b b 4/3 The corresponding transfer function has the form T (z) 3z +6z +4 3z +5 3z + 3z z z 3 its positive realization is given by In this case 3z +5 z z 3 A 3 B (33) C 5 3 D (34) 3z +8z +9z 3 +48z 4 + Note that g 4 48is different from ĝ 4 46in the previous case Theorem Let a finite sequence ĝ i > for i n (35) be given Then a positive realization of the form (7) exists if ĝ n > ĝn+ ĝ n+ ĝ n ĝ n+ > ĝn+ ĝ n+3 ĝ n ĝ n > > > ĝ(n ) ĝ n > > ĝ(n ) ĝ n ĝ > > ĝn ĝ (n ) > ĝn ĝ n > ĝn > ĝn ĝ n (36) k bk ĝ k ĝ k i a i k n (37) with a i i n constituting the solution to () for N n

5 Positive partial realization problem for linear discrete-time systems 69 Proof Given a sequence (35) Eqn () takes the form ĝ n ĝ n ĝ ĝ ĝ n ĝ 3 ĝ ĝ (n ) ĝ n 3 ĝ n ĝ n a n ĝ n ĝ (n ) ĝ n a n ĝ n+ ĝ n ĝ n Therefore ĝ n ĝ n ĝ ĝ ĝ n ĝ 3 ĝ ĝ n+ ĝ n+ ĝ n+ ĝ n+ ĝ (n ) ĝ n 3 ĝ n ĝ n ĝ n ĝ n ĝ n ĝ n ĝ n ĝ (n ) ĝ n ĝ n ĝ n ĝ n ĝ n (38) a n a n If the conditions (36) are met then the coefficient matrix of (38) has a cyclic structure by Theorem A (see the Appendix) Eqn (38) has a positive solution a i > for i n In this case using (37) we may find the coefficients b k for k n the positive realization (7) Example (Continuation of Example ) For the sequence (9) we have that N n 4 (38) takes the form (39) It is easy to see that the coefficient matrix of (39) satisfies the conditions (36) since 8 9 > > 3 9 The transfer function has the form (3) the desired positive realization is given by (3) Theorem 3 A positive asymptotically stable realization (7) of the finite sequence (35) exists if ĝ i+ < ĝ i for i n (4) the conditions (36) (37) satisfied Proof If the conditions (36) are met then by Theorem A Eqn (38) has a positive solution a i > for i n We shall show that if (4) holds then a i < (4) Let S i j g n i+j g n+j be the sum of the entries of the i-th column of the coefficient matrix in (38) i n Adding the n equations of (38) we obtain S i a i n (4) Note that if (4) holds then each entry g n i+j / g n+j of the coefficient matrix is greater than Write S mins i Then from (4) we have i S a i n a i n S < since S >n From the well-known final value theorem we obtain g lim g i lim (z ) T (z) i z lim (z ) C I n z A B+D z b z n ++b z+b lim (z ) z z n z n a n z a since (4) holds Therefore the positive realization (7) is asymptotically stable Example 3 We wish to find a positive realization () for the sequence g g g 9 g 3 6 g 4 5 (43)

6 7 T Kaczorek This sequence satisfies the conditions (4) (36) (37) for n by Theorem 3 there exists a positive asymptotically stable realization Using Procedure we obtain the following: Step We have the system of equations g g 9 g 3 g g 3 g 4 g g 4 its solution is 4/ 9/ Step Using (37) we compute the coefficients b ĝ b ĝ ĝ 49 The desired transfer function has the form z + 49 z +49 T (z) z 4 z 9 z 4z 9 (44) Step 3 Using (7a) for (44) we obtain the asymptotically stable realization A 9 4 B 49 C b b D 4 Extension for Multi Input Multi Output Systems The method presented in Section 3 can be easily extended for linear discrete-time systems with m inputs p outputs (MIMO systems) In this case the modified procedure has the following form: Procedure (MIMO systems) Step Given a sequence (6) find a non-negative solution to () with Ĝ nn a I m Ĝ nn I m a n I m T Ĝ n Ĝ n Ĝ Ĝ n+ Ĝ n Ĝ Ĝ N Ĝ N Ĝ N n Ĝ n+ Ĝ n+ (45) Ĝ N Step Using B k Ĝk Ĝ k i a i for k n (46) compute the matrices B B B n the transfer matrix T (z) z n z n a n z a n B z n +B z n ++B n z+b n (47) Step 3 Using one of the well-known methods (Benvenuti Farina 4; Farina Rinaldi ; Kaczorek ; Kaczorek 6a; Kaczorek 6b; Kaczorek 4; Kaczorek 5; Kaczorek 6c; Kaczorek 6d; Kaczorek 6e; Kaczorek Busłowicz 4) compute the desired positive realization () of the transfer matrix (47) Example 4 We wish to find a positive realization () for the sequence Ĝ Ĝ 3 7 Ĝ Ĝ Ĝ 5 7 (48) We have m p Using Procedure for N 4 n we obtain the following: Step In this case Eqn () with (45) has the form its solution is Step Using (46) we compute the matrices B Ĝ ˆB Ĝ Ĝ ˆB Ĝ Ĝ Ĝ The desired transfer matrix (47) has the form T (z) z z z z z + z + z + z + z z +4 (49) (5)

7 Positive partial realization problem for linear discrete-time systems 7 Step 3 Using (7a) we obtain the desired positive realization () of (5) in the form A B C Concluding Remarks D (5) The partial realization problem for positive discrete-time linear systems has been formulated solved It was shown that it has a solution if Eqn () has a non-negative solution (6) holds (Theorem ) A procedure for SISO MIMO systems was proposed for the computation of positive partial realizations for a given finite sequence () (or (6)) The procedure was illustrated with four numerical examples It was shown that if the conditions (4) (36) (38) are satisfied then there exists a positive asymptotically stable realization of the form (7) The deliberations can be extended for linear continuous-time systems for D linear systems (Kaczorek ) Acknowledgment This work was supported by the Ministry of Science Higher Education in Pol under Grant No 3 TA 6 7 References Benvenuti L Farina L (4): A tutorial on the positive realization problem IEEE Trans Automat Contr Vol 49 No 5 pp Farina L Rinaldi S (): Positive Linear Systems: Theory Applications Wiley New York Ho BL Kalman RE (966): Effective construction of linear state-variable models from input/output functions Regelungstechnik Bd pp Kaczorek T (99): Linear Control Systems Vol Taunton: Research Studies Press Kaczorek T (): Positive D D Systems London: Springer Kaczorek T (4): Realization problem for positive discretetime systems with delay System Sci Vol 3 No 4 pp 7 3 Kaczorek T (5): Positive minimal realizations for singular discrete-time systems with delays in state control Bull Pol Acad Sci Techn Sci Vol 53 No 3 pp Kaczorek T (6a): Realization problem for positive multivariable discrete-time linear systems with delays in the state vector inputs Int J Appl Math Comput Sci Vol 6 No pp Kaczorek T (6b) A realization problem for positive continuous-time linear systems with reduced number of delay Int J Appl Math Comput Sci Vol 6 No 3 pp Kaczorek T (6c): Computation of realizations of discretetime cone-systems Bull Polish Acad Sci Techn Sci Vol 54 No 3 pp Kaczorek T (6d): A realization problem for positive continuous-time systems with reduced numbers of delays Int J Appl Math Comput Sci Vol 6 No 3 pp 7 Kaczorek T (6e): Minimal positive realization for discretetime systems with state time-delays Int J Comput Math Electr Eng COMPEL Vol 5 No 4 pp 8 86 Kaczorek T Busłowicz M (4): Minimal realization problem for positive multivariable linear systems with delay Int J Appl Math Comput Sci Vol 4 No pp 8 87 Merzyakov JI (963): On the existence of positive solutions of a system of linear equations Uspekhi Matematicheskikh Nauk Vol 8 No 3 pp (in Russian) Appendix Definition A A positive matrix n A a ij > a n a nn i n; j n (A) is called a matrix with a (strictly) cyclic structure if only if a ii >a i+i > >a ni >i > >a i i (A) for i n For example the matrix has a cyclic structure the matrix does not have a cyclic structure since <a 3 3 Theorem A (Merzyakov 963) The system of linear algebraic equations a i x + a i x + + a in x i n (A3) with a ij > ijnhas a positive solution x i > nif only if its coefficient matrix (A) has a cyclic structure Received: March 7

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