Decoupling Zeros of Positive Discrete-Time Linear Systems*

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1 Circuits ad Systems,,, 4-48 doi:.436/cs..7 Published Olie October ( Decouplig Zeros of Positive Discrete-Time Liear Systems* bstract Tadeusz Kaczorek Faculty of Electrical Egieerig, Bialystok Uiversity of Techology, Bialystok, Polad Received July 3, ; revised ugust 6, ; accepted ugust, The otios of decouplig zeros of positive discrete-time liear systems are itroduced. The relatioships betwee the decouplig zeros of stadard ad positive discrete-time liear systems are aalyzed. It is show that: ) if the positive system has decouplig zeros the the correspodig stadard system has also decouplig zeros, ) the positive system may ot have decouplig zeros whe the correspodig stadard system has decouplig zeros, 3) the positive ad stadard systems have the same decouplig zeros if the rak of reachability (observability) matrix is equal to the umber of liearly idepedet moomial colums (rows) ad some additioal assumptios are satisfied. Keywords: Iput-Decouplig Zeros, Output-Decouplig Zeros, Iput-Output Decouplig Zeros, Positive, Discrete-Time, Liear, System. Itroductio I positive systems iputs, state variables ad outputs take oly o-egative values. Examples of positive systems are idustrial processes ivolvig chemical reactors, heat exchagers ad distillatio colums, storage systems, compartmetal systems, water ad atmospheric pollutio models. variety of models havig positive liear behavior ca be foud i egieerig, maagemet sciece, ecoomics, social scieces, biology ad medicie, etc. overview of state of the art i positive liear theory is give i the moographs [,]. The otios of cotrollability ad observability ad the decompositio of liear systems have bee itroduced by Kalma [3,4]. Those otios are the basic cocepts of the moder cotrol theory [5-9]. They have bee also exteded to positive liear systems [,]. The reachability ad cotrollability to zero of stadard ad positive fractioal discrete-time liear systems have bee ivestigated i []. The decompositio of positive discrete-time liear systems has bee addressed i []. The otio of decouplig zeros of stadard liear systems have bee itroduced by Rosebrock [8,]. I this paper the otios of decouplig zeros will be exteded for positive discrete-time liear systems. The paper is orgaized as follows. I Sectio the basic defiitios ad theorems cocerig reachability ad *This work was supported by Miistry of Sciece ad Higher Educatio i Polad uder work No NN observability of positive discrete-time liear systems are recalled. The decompositio of the pair (,B) ad (,C) of positive liear system is addressed i Sectio 3. The mai result of the paper is give i Sectio 4 where the defiitios of the decouplig-zeros are proposed ad the relatioships betwee decouplig zeros of stadard ad positive discrete-time liear systems are discussed. Cocludig remarks are give i Sectio 5.. Prelimiaries m The set of m real matrices will be deoted by ad :. The set of m real matrices with oegative etries will be deoted by m ad :. The set of oegative itegers will be deoted by Z ad the idetity matrix by I. Cosider the liear discrete-time systems xi xi Bui, i Z (.) yi Cxi Dui m p where xi, ui, yi are the state, iput m p ad output vectors ad, B, C, pm D. Defiitio.. The system (.) is called (iterally) p positive if ad oly if xi, ad yi, i Z m for every x, ad ay iput sequece ui, i Z. Theorem.. [,] The system (.) is (iterally) positive if ad oly if Copyright SciRes.

2 4 T. KCZOREK m p pm, B, C, D. (.) Defiitio.. The positive system (.) is called reachable i q steps if there exists a iput sequece m ui, i,,..., q which steers the state of the system from zero ( x ) to ay give fial state x f, i.e., x f x. Let e i, i =,, be the ith colum of the idetity matrix I. colum ae i for a > is called the moomial colum. Theorem.. [,] The positive system (.) is reachable i q steps if ad oly if the reachability matrix q qm Rq [ B B... B] (.3) cotais liearly idepedet moomial colums. Theorem.3. [,] The positive system (.) is reachable i q steps oly if the matrix [ B ] (.4) cotais liearly idepedet moomial colums. Defiitio.3. The positive systems (.) is called observable i q steps if it is possible to fid uique iitial state x of the system kowig its iput sequece m ui, i,,..., q ad its correspodig output p sequece yi, i,,..., q. Theorem.4. [,] The positive systems (.) is observable i q steps if ad oly if the observability matrix C C qp Oq (.5) q C cotais liearly idepedet moomial rows. Theorem.5. [,] The positive system (.) is observable i q steps oly if the matrix C (.6) cotais liearly idepedet moomial rows. 3. Decompositio of Positive Pair (,B) ad (,C) of Positive Liear Systems Let the reachability matrix m R [ B B... B] (3.) of the positive system (.) has < liearly idepedet moomial colums ad let the colums Bi, B,..., i B i k (k m) (3.) m of the matrix B be liearly idepedet moomial colums. We choose from the sequece Bi,..., B,,...,,...,,..., i B k i B i B k i B i (3.3) k moomial colums which are liearly idepedet from (3.) ad previously chose moomial colums. From those moomial colums we build the moomial matrix P [ Pi... P id P i P id P ikd P k... P] [ P P... P ] (3.4a) where d Pi B i,..., P, id B i d dk P B,..., P B,..., P B (3.4b) i i id i ikdk ik ad d i ( i,..., k) are some atural umbers. Theorem 3.. Let the positive system (.) be ureachable, the reachability matrix (3.) have < liearly idepedet moomial colums ad the assumptio T P P for k,..., ;,..., (3.5) k be satisfied. The the pair (, B ) of the system ca be reduced by the use of the matrix (3.4) to the form B P P, B P B,,, ( ) (3.6) m, B where the pair (, B ) is reachable ad the pair (, B ) is ureachable. Proof is give i []. Theorem 3.. The trasfer matrix T( z) C[ I z] B D (3.7) of the positive system (.) is equal to the trasfer matrix T ( z) C [ I z ] B D (3.8) of its reachable part (, B, C ), where CP [ C C], C. p Proof is give i []. By duality priciple [] we ca obtaied similar (dual) result for the pair (,C) of the positive system (.). Let the observability matrix C C q O (3.9) C has liearly idepedet moomial rows. I a similar way as for the pair (,B) by the choice of liearly idepedet moomial row for the pair (,C) we may fid the moomial matrix Q of the form [] Q [ Q... Q Q... Q... Q Q... Q ] (3.a) T T T T T T T T d d d l l Copyright SciRes.

3 T. KCZOREK 43 where Q C Q C d,...,, d d d d l l l l Q C,..., Q C,..., Q C d (3.b) ad d (,..., l) are some atural umbers. Theorem 3.3. Let the positive system (.) be uobservable, the matrix (3.9) has < liearly idepedet moomial rows ad the assumptio T QQ k for k,..., ;,..., (3.) be satisfied. The the pair (,C) of the system ca be reduced by the use of the matrix (3.) to the form QQ, C CQ [ C ],, ( ) (3.) p, C where the pair (, C ) is observable ad the pair (, C ) is uobservable. Proof is give i []. Theorem 3.4. The trasfer matrix (3.7) of the positive system (.) is equal to the trasfer matrix T( z) C [ I z ] B D (3.3) where B m m QB, B, B. (3.4) B Proof is give i []. Remark 3.. From Theorem 3. ad 3.3 it follows that the coditios for decompositio of the pair (,B) ad (,C) of the positive system (.) are much stroger tha of the pairs of the stadard system. 4. Decouplig Zeros of the Positive Systems It is well-kow [5-8] that the iput-decuplig zeros of stadard liear systems are the eigevalues of the matrix of the ureachable (ucotrollable) part of the system. Similarly, the output-decouplig zeros of stadard liear systems are the eigevalues of the matrix of the ureachable ad uobservable parts of the system. I a similar way we will defied the decouplig zeros of the positive liear discrete-time systems. Defiitio 4.. Let be the matrix of ureachable part of the system (.). The zeros z, z,..., z of the characteristic polyomial i i i det[ I z ] z a z... az a (4.) of the matrix are called the iput-decouplig zero of the positive system (.). The list of the iput-decouplig zeros will be deoted by Zi { zi, zi,..., zi }. Example 4.. Cosider the positive system (.) with the matrices, B (4.) 3 Note that the pair (4.) has already the form (3.6) with, 3 (4.3) B B B, (, ) I this case the characteristic polyomial of the matrix 3 has the form z det[ Iz ] z 3, (4.4) ( z)( z3) z 5z6 the iput-decouplig zeros are equal to zi, zi 3 ad Zi {, 3}. Defiitio 4.. Let be the matrix of uobservable part of the system (.). The zeros z, z,..., z of the characteristic polyomial o o o det[ I z ] z a z... a z a (4.5) of the matrix  are called the output-decouplig zero of the positive system (.). The list of the output-decouplig zeros will be deoted by Zo { zo, zo,..., zo }. Example 4.. Cosider the positive system (.) with the matrices (4.) ad C [ ], D []. (4.6) The observability matrix C O3 C (4.7) C 4 has oly oe moomial row Q [ ]. I this case the moomial matrix (3.) has the form Q Q Q (4.8) Q 3 Copyright SciRes.

4 44 T. KCZOREK ad the assumptio (3.) is satisfied sice T T QQ [ Q3 ] [ ] [ ] 3 (4.9) Usig (3.) ad (4.8) we obtai QQ 3, (, ) 3 [ ] [ ] C CQ C Characteristic polyomial of the matrix 3 has the form z det[ Iz ] ( z)( z3) z 4z3 z 3 the output-decouplig zeros are equal to zo, zo 3 ad Zo {, 3}. () () ( k ) Defiitio 4.3. Zeros zi, zi,..., z i which are simultaeously the iput-decouplig zeros ad the output-decouplig zeros of the positive system (.) are called the iput-output decouplig zeros of the positive system, i.e., z ( ) i Zi ad ( z ) i Zo for =,,k; k mi(, ) (4.) The list of iput-output decouplig zeros will be deoted by Zi { zi, zi,..., zi }. Example 4.3. Cosider the positive system (.) with () () ( k ) the matrices (4.) ad (4.6). The system has the iputdecouplig zi, zi 3 ad Zi {,3} (Example 4.) ad the output-decouplig zero zo, zo 3 ad Zo {, 3} (Example 4.). Therefore, by Defiitio 4.3 the positive system has oe iput-output decouplig zero zio 3, Zio {3}. This zero is the eigevalue of the matrix [3] of the ureachable ad uobservable part of the system. Note that the trasfer fuctio of the system is zero, i.e., T() s C[ I z ] BD z z [] [] z 3 (4.) sice it represets the reachable ad observable part of the system. Example 4.4. Cosider the positive system (.) with the matrices, B, C [ ], D []. 3 (4.) Note that the matrices B, C, D are the same as i Example 4. ad 4. ad the matrix differs by oly oe etry a 3. The pair (,B) has already the form (3.6) sice, 3. (4.3) B B B, (, ) The observability matrix C O3 C (4.4) C 4 5 has oly oe moomial row Q [ ] ad Q (4.5) is the same as i Example 4.. The positive pair (,C) ca ot be decomposed because the assumptio (3.) is ot satisfied, i.e., T T QQ [ Q3 ] [ ] (4.6) 3 [ ] [ ] Now let us cosider the stadard system (.) with (4.). I this case the matrix (4.4) has two liearly idepedet rows ad Q (4.7) Usig (3.) ad (4.7) we obtai QQ 3 (4.8a) Copyright SciRes.

5 T. KCZOREK 45 ad C CQ [ C ] [ ], ( ) (4.8b) The matrix [] of the uobservable part of the stadard system has oe eigevalue which is equal to the output-decouplig zero zo. Note that the stadard system has two iput-decouplig zeros zi, z i 3 ad has o iput-output decouplig zeros. The trasfer fuctio of the positive ad stadard system is equal to zero. Cosider the positive pair......, B (4.9)... a a a... a with a a. The reachability matrix of the pair (4.9) R [ B B... B] (4.) has rak equal to two ad two liearly idepedet moomial colums. I this case the moomial matrix (3.4) has the form P [ P... P ]... (4.)... ad the assumptio (3.5) is satisfied sice P [... ] T T ad Pk P for k = 3,,. Usig (3.6) ad (4.) we obtai P P a a a... a , ( ),, ( ) ( )... a a3 a4... a (4.a) ad B B P B..., B... (4.b) Theorem 4.. If the rak of the reachability matrix (4.) is equal to the umber of liearly idepedet moomial colums the the iput-decouplig zeros of the stadard ad positive system with (4.9) are the same ad they are the eigevalues of the matrix. The state vector x i of the system is idepedet of the iput-decouplig zeros for ay iput u i ad zero iitial coditios (x = ). Proof. By Defiitio 4. the iput-decouplig zeros are the eigevalues of the matrix ad they are the same for stadard ad positive system sice the similarity trasformatio matrix P has i both cases the same form (4.). If the iitial coditios are zero the the zet trasformatio of x i is give by X( z) P X( z) P [ Iz] BU( z) Iz B Iz P U() s z [ Iz ] B P U( z) U( z) where U(z) is the zet trasform of u i. Dual result we obtai for the positive pair... a... a... a,... a C [... ] (4.3) (4.4) Copyright SciRes.

6 46 T. KCZOREK with a a. The observability matrix of the pair (4.4)... C... C O C... (4.5) has rak equal to two ad two liearly idepedet moomial rows. I this case the moomial matrix (3.) has the form... Q... P... Q... ad the assumptio (3.) is satisfied sice Q [... ] ad QQ k for k = 3,,. Usig (3.) ad (4.5) we obtai ad QQ a a a ,... ( ),,... a... a 3 ( ) ( )... a 4... a a (4.6) (4.7a) C CQ [... ] [... ] [ C ], C [ ] (4.7b) Theorem 4.. If the rak of the observability matrix (4.5) is equal to the umber of liearly idepedet moomial rows the the output-decouplig zeros of the stadard ad positive system with (4.4) are the same ad they are the eigevalues of the matrix Â. The output y i of the system is idepedet of the output-decouplig zeros for ay iput ui ' Bui ad zero iitial coditios (x = ). Proof is similar (dual) to the proof of Theorem 4.. Example 4.5. For the positive pair, ( a ), C [ ] (4.8) a the matrix (4.6) has the form Q. (4.9) Usig (3.) ad (4.9) we obtai QQ, a (4.3a), [ ], [ a] ad C CQ [ ] [ C ], C [ ] (4.3b) The pair (, C ) is observable sice C C ad the positive system has oe ouput- decouplig zero z a. The zet trasform of the output for x o ad U '( z) BU ( z) is give by T() s C[ I z] U '() z (4.3) CIz [ ] U'( z) z U'( z) ad it is idepedet of the output-decouplig zero. The preseted results ca be exteded to multi-iput multi-output discrete-time liear systems as follows. Theorem 4.3. Let the reachability matrix (3.) of the positive system (.) have rak equal to its lie- Copyright SciRes.

7 T. KCZOREK 47 arly idepedet moomial colums ad the assumptio (3.5) be satisfied. The the iput-decouplig zeros of the stadard ad positive system are the same ad they are the eigevalues of the matrix. The state vector x i of the system is idepedet of the iput-decouplig zeros for ay iput vector u i ad zero iitial coditios. Proof. If the reachability matrix (3.) of the system (.) has rak equal to its liearly idepedet moomial colums ad the assumptio (3.5) is satisfied the the similarity trasformatio matrix P has the same form for stadard ad positive system. I this case the matrix is the same for stadard ad positive system. Therefore, the iput-decouplig zero for the stadard ad positive system is the same. The secod part of the Theorem ca be proved i a similar way as of Theorem 4.. Theorem 4.4. Let the observability matrix (3.9) of the positive system (.) has rak equal to its liearly idepedet moomial rows ad the assumptio (3.) be satisfied. The the output-decouplig zeros of the stadard ad positive system are the same ad they are the eigevalues of the matrix Â. The output vector y i of the system is idepedet of the output-decouplig zeros for ay iput vector ui ' Bui ad zero iitial coditios. Remark 4.. Note that if the positive pair (,B) ca be decomposed the the correspodig stadard pair (,B) ca also be decomposed. Therefore, if the positive system (.) has iput-decouplig zeros the the stadard system (.) has also iput-decouplig zeros. Similar (dual) remark we have for the pair (,C) ad the output-decouplig zeros. The followig example shows that the positive system (.) may ot have iput-decouplig zeros but the stadard system has iput-decouplig zeros. Example 4.6. The reachability matrix for the positive pair, B (4.3) 3 has the form [ B B B] 4. (4.33) 4 It has o moomial colums ad it ca ot be decomposed (as the positive pair) but it ca be decomposed as a stadard pair sice the rak of the reachability matrix (4.33) is equal to oe. The similarity trasformatio matrix has the form P (4.34) ad we obtai P P, (4.35a) [], [ ], B B P B, B [] (4.35b) The matrix has the eigevalues zi, zi. Therefore, the positive system with (4.3) has ot iput-decouplig zeros but it has iput-decouplig zeros ( zi, zi ) as a stadard system. 5. Cocludig Remarks The otios of the iput-decouplig zero, output-decouplig zero ad iput-output decouplig zero for positive discrete-time liear systems have bee itroduced. The ecessary ad sufficiet coditios for the reachability (observability) of positive liear systems are much stroger tha the coditios for stadard liear systems (Theorem. ad.4). The coditios for decompositio of positive system are also much stroger tha for the stadard systems. Therefore, the coditios for the existece of decouplig zeros of positive systems are more restrictive. It has bee show that: ) if the positive system has decouplig zeros the the correspodig stadard system has also decouplig zeros, ) the positive system may ot have decouplig zeros whe the correspodig stadard system has decouplig zeros (Example 4.6), 3) the positive ad stadard system have the same decouplig zeros if the rak of reachability (observability) matrix is equal to the umber of liearly idepedet moomial colums (rows) ad the assumptio (3.5) ((3.)) is satisfied (Theorem 4.3 ad 4.4). The cosideratios have bee illustrated by umerical examples. Ope problems are extesio of these cosideratios to positive cotiuous-time liear systems ad to positive D liear systems. 6. Refereces [] L. Faria ad S. Rialdi, Positive Liear Systems, Theory ad pplicatios, Wiley, New York,. [] T. Kaczorek, Positive D ad D Systems, Spriger Verlag, Lodo,. [3] R. E. Kalma, Mathematical Descriptios of Liear Systems, SIM Joural o Cotrol, Vol., No., 963, pp [4] R. E. Kalma, O the Geeral Theory of Cotrol Sys- Copyright SciRes.

8 48 T. KCZOREK tems, Proceedigs of the First Iteratioal Cogress o utomatic Cotrol, Butterworth, Lodo, 96, pp [5] P. J. tsaklis ad. N. Michel, Liear Systems, Birkhauser, Bosto, 6. [6] T. Kaczorek, Liear Cotrol Systems, Vol., Wiley, New York, 993. [7] T. Kailath, Liear Systems, Pretice-Hall, Eglewood Cliffs, New York, 98. [8] H. H. Rosebrock, State-Space ad Multivariable Theory, Wiley, New York, 97. [9] W.. Wolovich, Liear Multivariable Systems, Spriger-Verlag, New York, 974. [] T. Kaczorek, Reachability ad Cotrollability to Zero Tests for Stadard ad Positive Fractioal Discrete-Time Systems, Joural Europée des Systèmes utomatisés, Vol. 4, No. 6-8, 8, pp [] T. Kaczorek, Decompositio of the Pairs (,B) ad (,C) of the Positive Discrete-Time Liear Systems, Proceedigs of TRNSCOMP, Zakopae, 6-9 December. [] H. H. Rosebrock, Commets o Poles ad Zeros of Liear Multivariable Systems: Survey of the lgebraic Geometric ad Complex Variable Theory, Iteratioal Joural o Cotrol, Vol. 6, No., 977, pp Copyright SciRes.