 Circuits and Systems, 2010, 1, 41-48 doi:10.4236/cs.2010.12007 Published Online October 2010 (http://www.SciRP.org/journal/cs) Copyright © 2010 SciRes. CS Decoupling Zeros of Positive Discrete-Time Linear Systems* Tadeusz Kaczorek Faculty of Electrical Engineering, Bialystok University of Technology, Bialystok, Poland E-mail: kaczorek@isep.pw.edu.pl Received July 13, 2010; revised August 16, 2010; accepted August 20, 2010 Abstract The notions of decoupling zeros of positive discrete-time linear systems are introduced. The relationships between the decoupling zeros of standard and positive discrete-time linear systems are analyzed. It is shown that: 1) if the positive system has decoupling zeros then the corresponding standard system has also decoup-ling zeros, 2) the positive system may not have decoupling zeros when the corresponding standard system has decoupling zeros, 3) the positive and standard systems have the same decoupling zeros if the rank of reachability (observability) matrix is equal to the number of linearly independent monomial columns (rows) and some additional assumptions are satisfied. Keywords: Input-Decoupling Zeros, Output-Decoupling Zeros, Input-Output Decoupling Zeros, Positive, Discrete-Time, Linear, System 1. Introduction In positive systems inputs, state variables and outputs take only non-negative values. Examples of positive sys-tems are industrial processes involving chemical reactors, heat exchangers and distillation columns, storage systems, compartmental systems, water and atmospheric pollution models. A variety of models having positive linear be-havior can be found in engineering, management science, economics, social sciences, biology and medicine, etc. An overview of state of the art in positive linear theory is given in the monographs [1,2]. The notions of controllability and observability and the decomposition of linear systems have been introduced by Kalman [3,4]. Those notions are the basic concepts of the modern control theory [5-9]. They have been also ex-tended to positive linear systems [1,2]. The reachability and controllability to zero of standard and positive fractional discrete-time linear systems have been investigated in . The decomposition of positive discrete-time linear systems has been addressed in . The notion of decoupling zeros of standard linear systems have been introduced by Rosebrock [8,12]. In this paper the notions of decoupling zeros will be extended for positive discrete-time linear systems. The paper is organized as follows. In Section 2 the ba-sic definitions and theorems concerning reachability and observability of positive discrete-time linear systems are recalled. The decomposition of the pair (A,B) and (A,C) of positive linear system is addressed in Section 3. The main result of the paper is given in Section 4 where the definitions of the decoupling-zeros are proposed and the relationships between decoupling zeros of standard and positive discrete-time linear systems are discussed. Con-cluding remarks are given in Section 5. 2. Preliminaries The set of nm real matrices will be denoted by nm and 1:.nn The set of mn real matrices with nonnegative entries will be denoted by mn and 1:.nn The set of nonnegative integers will be de-noted by Z and the nn identity matrix by .nI Consider the linear discrete-time systems 1,iiiiiixAxBui ZyCxDu  (2.1) where ,nix miu, piy are the state, input and output vectors and ,nnA nmB , pnC , pmD . Definition 2.1. The system (2.1) is called (internally) positive if and only if ,nix and ,piy Zi for every 0,nx and any input sequence ,miu iZ. Theorem 2.1. [1,2] The system (2.1) is (internally) positive if and only if *This work was supported by Ministry of Science and Higher Educa-tion in Poland under work No NN514 1939 33. T. KACZOREK Copyright © 2010 SciRes. CS 42 ,,,nn nmpnpmAB CD    . (2.2) Definition 2.2. The positive system (2.1) is called reachable in q steps if there exists an input sequence ,miu 0,1,..., 1iq which steers the state of the system from zero (00x) to any given final state ,nfx i.e., 0fxx. Let ei, i = 1,…,n be the ith column of the identity ma-trix In. A column aei for a > 0 is called the monomial column. Theorem 2.2. [1,2] The positive system (2.1) is reach-able in q steps if and only if the reachability matrix 1[...]qnqmqRBAB AB (2.3) contains n linearly independent monomial columns. Theorem 2.3. [1,2] The positive system (2.1) is reach-able in q steps only if the matrix ][ AB (2.4) contains n linearly independent monomial columns. Definition 2.3. The positive systems (2.1) is called observable in q steps if it is possible to find unique initial state 0nx of the system knowing its input sequence ,miu 0,1,..., 1iq and its corresponding output sequence ,piy 0,1,..., 1iq. Theorem 2.4. [1,2] The positive systems (2.1) is ob-servable in q steps if and only if the observability matrix 1qp nqqCCAOCA (2.5) contains n linearly independent monomial rows. Theorem 2.5. [1,2] The positive system (2.1) is ob-servable in q steps only if the matrix CA (2.6) contains n linearly independent monomial rows. 3. Decomposition of Positive Pair (A,B) and (A,C) of Positive Linear Systems Let the reachability matrix 1[...]nnmnnRBAB AB (3.1) of the positive system (2.1) has n1 < n linearly inde-pendent monomial columns and let the columns 12,,..., kiiiBBB (km) (3.2) of the matrix nmB be linearly independent mono-mial columns. We choose from the sequence 11 1221 1,..., ,,...,,...,,...,kk knniii iiiABABABABABAB (3.3) monomial columns which are linearly independent from (3.2) and previously chosen monomial columns. From those monomial columns we build the monomial matrix 1112 221112[............ ][... ]kkiidi ididnnnPPPPP PPPPP P (3.4a) where 111 111222 222111,..., ,,...,,..., kkk kdii ididdii idi idiPB PABPB PABPAB  (3.4b) and di (1,..., )ik are some natural numbers. Theorem 3.1. Let the positive system (2.1) be unrea- chable, the reachability matrix (3.1) have n1 < n linearly independent monomial columns and the assumption 110 for 1,...,;1,...,TkjPAPknnjn (3.5) be satisfied. Then the pair (,)AB of the system can be reduced by the use of the matrix (3.4) to the form 112 212 111112 1212 2112 1,,00,,(),nnnnnn nmAA BAPAPBPBAAA nnnAB      (3.6) where the pair 11(,)AB is reachable and the pair 22(, 0)AB is unreachable. Proof is given in . Theorem 3.2. The transfer matrix 1() []nTzCIz AB D (3.7) of the positive system (2.1) is equal to the transfer matrix 1111 11() []nTzCIz ABD (3.8) of its reachable part 111(,,)ABC , where 12[],CPC C 11pnC . Proof is given in . By duality principle  we can obtained similar (dual) result for the pair (A,C) of the positive system (2.1). Let the observability matrix 1qn nnnCCAOCA (3.9) has 1nn linearly independent monomial rows. In a similar way as for the pair (A,B) by the choice of n1 linearly independent monomial row for the pair (A,C) we may find the monomial matrix nnQ of the form  12 1112 21[............ ]llTT TTT TTTjj nnjdj djdQQQ QQQ QQ (3.10a) T. KACZOREK Copyright © 2010 SciRes. CS 43where 111 111222 222111,..., ,,...,,..., lllldjj jjdddjj jjjd jdQC QCAQC QCAQCA  (3.10b) and (1,..., )jdj l are some natural numbers. Theorem 3.3. Let the positive system (2.1) be unob-servable, the matrix (3.9) has n1 < n linearly independent monomial rows and the assumption 0TkjQAQ  for 11,...,kn; 11,...,jn n (3.11) be satisfied. Then the pair (A,C) of the system can be reduced by the use of the matrix (3.10) to the form 112 221 1111121 212 2121 1ˆ0ˆˆˆ,ˆˆˆˆ,,()ˆˆ,nnnnnn pnAAQAQCCQCAAAA nnnAC    (3.12) where the pair 11ˆˆ(, )AC is observable and the pair 22ˆˆ(, 0)AC is unobservable. Proof is given in . Theorem 3.4. The transfer matrix (3.7) of the positive system (2.1) is equal to the transfer matrix 1111 11ˆˆˆ() []nTzCIz ABD (3.13) where 121122ˆˆˆ,,ˆnmn mBQB BBB. (3.14) Proof is given in . Remark 3.1. From Theorem 3.1 and 3.3 it follows that the conditions for decomposition of the pair (A,B) and (A,C) of the positive system (2.1) are much stronger than of the pairs of the standard system. 4. Decoupling Zeros of the Positive Systems It is well-known [5-8] that the input-decupling zeros of standard linear systems are the eigenvalues of the matrix A2 of the unreachable (uncontrollable) part of the system. Similarly, the output-decoupling zeros of standard linear systems are the eigenvalues of the matrix of the un-reachable and unobservable parts of the system. In a similar way we will defined the decoupling zeros of the positive linear discrete-time systems. Definition 4.1. Let 222nnA be the matrix of un-reachable part of the system (2.1). The zeros 212, ,...,ii inzz z of the characteristic polynomial 222212110det[ ]...nnnnIzAza zaza  (4.1) of the matrix 2A are called the input-decoupling zero of the positive system (2.1). The list of the input-decoupling zeros will be denoted by 212{, ,...,}iii inZzz z. Example 4.1. Consider the positive system (2.1) with the matrices 102 1020, 0003 0AB (4.2) Note that the pair (4.2) has already the form (3.6) with 1122112102020,000310, (1,2)00AAAA ABBBnn   (4.3) In this case the characteristic polynomial of the matrix 22003A has the form 22220det[]03(2)(3)56zIz Azzz zz , (4.4) the input-decoupling zeros are equal to 122, 3iizz and {2, 3}iZ. Definition 4.2. Let 22ˆˆ2ˆnnA be the matrix of un-observable part of the system (2.1). The zeros 212, ,...,oo onzz z of the characteristic polynomial 2222ˆˆ1ˆˆ2110ˆˆˆˆdet[ ]...nnnnIzAza zaza  (4.5) of the matrix 2ˆA are called the output-decoupling zero of the positive system (2.1). The list of the output-decoupling zeros will be denoted by 2ˆ12{ ,,...,}oooonZzz z. Example 4.2. Consider the positive system (2.1) with the matrices (4.2) and [0 1 0],CD. (4.6) The observability matrix 32010020040COCACA (4.7) has only one monomial row 1Q. In this case the monomial matrix (3.10) has the form 123010100001QQQQ (4.8) T. KACZOREK Copyright © 2010 SciRes. CS 44 and the assumption (3.11) is satisfied since 12 310210[]02000 00301TTQAQ Q (4.9) Using (3.12) and (4.8) we obtain 111221 21101010 2010ˆ100020100001003001200 ˆ0012,( 1,2)ˆˆ003ˆˆAQAQAnnAACCQC   Characteristic polynomial of the matrix 212ˆ03A has the form 22212ˆdet[]( 1)(3)4303zIz Azzzzz  the output-decoupling zeros are equal to 121, 3oozz and {1, 3}oZ. Definition 4.3. Zeros (1)(2)( )00 0,,..., kii izzz which are si-multaneously the input-decoupling zeros and the out-put-decoupling zeros of the positive system (2.1) are called the input-output decoupling zeros of the positive system, i.e., ijiZz )(0 and ()0jiozZ for j = 1,…,k; 22ˆmin( ,)knn (4.10) The list of input-output decoupling zeros will be de-noted by (1)(2)()000 0{ ,,...,}kiii iZzz z. Example 4.3. Consider the positive system (2.1) with the matrices (4.2) and (4.6). The system has the input- decoupling 122, 3iizz and {2, 3}iZ (Example 4.1) and the output-decoupling zero 121, 3oozz and {1, 3}oZ (Example 4.2). Therefore, by Definition 4.3 the positive system has one input-output decoupling zero 3ioz, {3}ioZ. This zero is the eigenvalue of the matrix 12 A of the unreachable and unobservable part of the system. Note that the transfer function of the system is zero, i.e., 11() []10210100200 00 30nTsCIzABDzzz (4.11) since it represents the reachable and observable part of the system. Example 4.4. Consider the positive system (2.1) with the matrices 102 1021,0 ,,003 0ABCD. (4.12) Note that the matrices B, C, D are the same as in Ex-ample 4.1 and 4.2 and the matrix A differs by only one entry 123a. The pair (A,B) has already the form (3.6) since 1122112102021,000310, (1,2)00AAAA ABBBn n  . (4.13) The observability matrix 32010021045COCACA (4.14) has only one monomial row 1[0 1 0]Q and 010100001Q (4.15) is the same as in Example 4.2. The positive pair (A,C) can not be decomposed because the assumption (3.11) is not satisfied, i.e., 12310210[ ]0210000301[0 1]TTQAQQ (4.16) Now let us consider the standard system (2.1) with (4.12). In this case the matrix (4.14) has two linearly independent rows and 010021100Q (4.17) Using (3.12) and (4.17) we obtain 1121 20101020 01ˆ0210211 00100003 210010 ˆ0650 ˆˆ421AQAQAAA       (4.18a) T. KACZOREK Copyright © 2010 SciRes. CS 45and 11ˆˆCCQC , 21()nnn (4.18b) The matrix 2ˆA of the unobservable part of the standard system has one eigenvalue which is equal to the output-decoupling zero 11oz. Note that the standard system has two input-decoupling zeros 12iz, 32iz and has no input-output decoupling zeros. The transfer function of the positive and standard sys-tem is equal to zero. Consider the positive pair 012 1010... 00001... 01,0000... 1... 0nn nnABaaa a(4.19) with 010aa. The reachability matrix of the pair (4.19) 1010...0100...0[...]000...0...000...0nnnnRBAB AB  (4.20) has rank equal to two and two linearly independent mo-nomial columns. In this case the monomial matrix (3.4) has the form 1010...0100...0[...]001...0000...1nnnPP P  (4.21) and the assumption (3.5) is satisfied since 2[0 ... 0]TAP  and 20TkPAP for k = 3,…,n. Using (3.6) and (4.21) we obtain 1012 11122010 ...00 1 0...0001 ...01 00... 00 0 1...0000 ...1...000...1010...0100...0,001...0 0000 ...1nAPAPaaa aAAA 222( 2)112(2)(2)2234100 10...0,,10 00...0010... 0001... 0000... 1...nnnnAAAaaa a     (4.22a) and 1 211010...00100...011,001...00 00000...10BBPB B   (4.22b) Theorem 4.1. If the rank of the reachability matrix (4.20) is equal to the number of linearly independent monomial columns then the input-decoupling zeros of the standard and positive system with (4.19) are the same and they are the eigenvalues of the matrix 2A. The state vector xi of the system is independent of the input-de- coupling zeros for any input ui and zero initial conditions (x0 = 0). Proof. By Definition 4.1 the input-decoupling zeros are the eigenvalues of the matrix 2A and they are the same for standard and positive system since the similar-ity transformation matrix P has in both cases the same form (4.21). If the initial conditions are zero then the zet transformation of xi is given by 11111112 1211111()()[]()()001[] () ()000XzPXzP IzA BUzIz AABPUsIz AzIz ABPUzUz (4.23) where U(z) is the zet transform of ui. Dual result we obtain for the positive pair 0121100...010...001...0 ,00...1[0 1 0...0]nnnnaaAaaC  (4.24) T. KACZOREK Copyright © 2010 SciRes. CS 46 with 010aa. The observability matrix of the pair (4.24) 1010...0100...0000...0...000...0nnnnCCAOCA   (4.25) has rank equal to two and two linearly independent mo-nomial rows. In this case the monomial matrix (3.10) has the form 1010...0100...0001...0000...1nnnQPQ  (4.26) and the assumption (3.11) is satisfied since 2[0 ... 0]QA and 20kQAQ  for k = 3,…,n. Using (3.12) and (4.25) we obtain 10121121 21ˆ00...00 1 0...010...01 00... 001...00 0 1...000...1000...1010...0100...0 ˆ0,001...0 ˆˆ000 ...101ˆ00nAQAQaaaaAAAA 22( 2)22123(2)(2)2411000ˆ,,0000...010...0ˆ01...000...1nnnnAaaAaa  (4.27a) and 11211010...0100...0ˆ[010...0] 001... 0000...1ˆˆ[1 0 ...0],CCQCC  (4.27b) Theorem 4.2. If the rank of the observability matrix (4.25) is equal to the number of linearly independent monomial rows then the output-decoupling zeros of the standard and positive system with (4.24) are the same and they are the eigenvalues of the matrix 2ˆA. The out-put yi of the system is independent of the output-decoup- ling zeros for any input 'iiuBu and zero initial condi-tions (x0 = 0). Proof is similar (dual) to the proof of Theorem 4.1. Example 4.5. For the positive pair 000100,( 0),01AaCa (4.28) the matrix (4.26) has the form 010100001Q. (4.29) Using (3.12) and (4.29) we obtain 1121 21212010 ˆ0ˆ000 ,ˆˆ1001ˆˆˆ,,[]00AAQAQAAaAAAa  (4.30a) and 111ˆˆ[1 00],CCQCC (4.30b) The pair 11ˆˆ(, )AC is observable since 111ˆ10ˆˆ 01CCA and the positive system has one ouput- decoupling zero 01za. The zet transform of the output for 0ox and )()(' zBUzU is given by 111()[] '()ˆˆ[]'() '()nnTsCIzAU zCIzAUzz Uz (4.31) and it is independent of the output-decoupling zero. The presented results can be extended to multi-input multi-output discrete-time linear systems as follows. Theorem 4.3. Let the reachability matrix (3.1) of the positive system (2.1) have rank equal to its 1nn line- T. KACZOREK Copyright © 2010 SciRes. CS 47arly independent monomial columns and the assumption (3.5) be satisfied. Then the input-decoupling zeros of the standard and positive system are the same and they are the eigenvalues of the matrix 2A. The state vector xi of the system is independent of the input-decoupling zeros for any input vector ui and zero initial conditions. Proof. If the reachability matrix (3.1) of the system (2.1) has rank equal to its n1 linearly independent mono-mial columns and the assumption (3.5) is satisfied then the similarity transformation matrix P has the same form for standard and positive system. In this case the matrix 2A is the same for standard and positive system. There-fore, the input-decoupling zero for the standard and posi-tive system is the same. The second part of the Theorem can be proved in a similar way as of Theorem 4.1. Theorem 4.4. Let the observability matrix (3.9) of the positive system (2.1) has rank equal to its 1nn line-arly independent monomial rows and the assumption (3.11) be satisfied. Then the output-decoupling zeros of the standard and positive system are the same and they are the eigenvalues of the matrix 2ˆA. The output vector yi of the system is independent of the output-decoupling zeros for any input vector 'iiuBu and zero initial conditions. Remark 4.1. Note that if the positive pair (A,B) can be decomposed then the corresponding standard pair (A,B) can also be decomposed. Therefore, if the positive sys-tem (2.1) has input-decoupling zeros then the standard system (2.1) has also input-decoupling zeros. Similar (dual) remark we have for the pair (A,C) and the output-decoupling zeros. The following example shows that the positive system (2.1) may not have input-decoupling zeros but the stan-dard system has input-decoupling zeros. Example 4.6. The reachability matrix for the positive pair 121 1010,0131 1AB (4.32) has the form 2124[] 000124BABAB. (4.33) It has no monomial columns and it can not be decom-posed (as the positive pair) but it can be decomposed as a standard pair since the rank of the reachability matrix (4.33) is equal to one. The similarity transformation ma-trix has the form 100010101P (4.34) and we obtain  0101],12[],2[,0010010122212121211AAAAAAAPPA (4.35a) ]1[,0001111 BBBPB (4.35b) The matrix 2A has the eigenvalues 11iz, 20iz. Therefore, the positive system with (4.32) has not in-put-decoupling zeros but it has input-decoupling zeros (11iz, 20iz) as a standard system. 5. Concluding Remarks The notions of the input-decoupling zero, output-decoup- ling zero and input-output decoupling zero for positive discrete-time linear systems have been introduced. The necessary and sufficient conditions for the reachability (observability) of positive linear systems are much stronger than the conditions for standard linear systems (Theorem 2.2 and 2.4). The conditions for decomposition of positive system are also much stronger than for the standard systems. Therefore, the conditions for the exis-tence of decoupling zeros of positive systems are more restrictive. It has been shown that: 1) if the positive sys-tem has decoupling zeros then the corresponding stan-dard system has also decoupling zeros, 2) the positive system may not have decoupling zeros when the corre-sponding standard system has decoupling zeros (Exam-ple 4.6), 3) the positive and standard system have the same decoupling zeros if the rank of reachability (ob-servability) matrix is equal to the number of linearly in-dependent monomial columns (rows) and the assumption (3.5) ((3.11)) is satisfied (Theorem 4.3 and 4.4). The considerations have been illustrated by numerical examples. 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