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Some Recent Developments in Positive and Compartmental Systems Invited Paper Some recent developments in positive and compartmental systems Tadeusz Kaczorek Warsaw University of Technology Institute of Control and Industrial Electronics 00-662 Warszawa, Koszykowa 75,Poland e-mail: [email protected] ABSTRACT Notions of the externally and internally positive standard and singular discrete-time and continuous-time linear systems are introduced. Necessary and sufficient conditions for the external and internal positivity of linear systems are given. It is shown that the reachability and controllability of the internally positive linear systems are not invariant under the state- feedbacks. By suitable choice of the state-feedbacks an unreachable internally positive linear systems can be made reachable and a controllable internally positive system can be made uncontrollable. The basic properties of continuous- time and discrete-time linear compartmental systems are derived and the relationships between the compartmental and positive systems are established. The realization problem for compartmental systems is formulated and partly solved. Keywords: discrete-time and continuous-time linear systems, ,reachabilityand controllability 1. INTRODUCTION In the last decade a dynamic development in positive and compartmental systems has been observed [2-21, 23-26]. Roughly speaking, positive systems are systems whose inputs, state variables and outputs take only nonnegative values. Examples of positive systems 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 system behaviour can be found in engineering, management science, economics, social sciences, biology and medicine, etc. The basic mathematical tools for analysis and synthesis of linear systems are linear spaces and the theory of linear operators. Positive linear systems are defined on cones and not on linear spaces. This is why the theory of positive systems is more complicated and less advanced. The theory of positive and compartmental systems has some elements in common with theories of linear and non-linear systems. Schematically the relationship between the theories of linear, non-linear, positive and compartmental systems is shown in the Fig. Fig. Positive linear systems, for example, satisfy the superposition principle. Limiting the consideration of positive linear systems only to R (the first quarter of R ')showsthat the theory of positive linear systems has some elements in common with the theory of non-linear systems. An overview of the state of art in positive systems theory is given in the monographs [8, 20]. Compartmental systems is a special subclass of the positive systems. Photonics Applications in Astronomy, Communications, Industry, and High-Energy 1 Physics Experiments II, edited by Ryszard S. Romaniuk, Proceedings of SPIE Vol. 5484 (SPIE, Bellingham, WA, 2004) · 0277-786X/04/$15 · doi: 10.1117/12.568841 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 12/04/2015 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx In this paper an overview of recent developments in positive and compartmental systems will be presented. Special attention will be devoted to the relationships between positive and compartmental systems and to the realization problem for compartmental systems. Besides known results some new results will be also presented. 2. EXTERNALLY AND INTERNALLY POSITIVE SYSTEMS 2.1. Discrete-time systems Let R be the set of n X m matrices with entries from the field of real numbers R and R := R .Theset of n x m matrices with real non-negative entries will be denoted by R' and R := R" .Theset of non-negative integers will be denoted by Z. Consider the discrete-time linear system (1 a) Ex1 =Ax2+ Bu1 , iEZ (ib) y=Cx+Du where X1E R , U1R' and y1 ERare the state, input and output vectors and E, A ER''BE CE R2,DE The system (1) is called singular if detE =0.IfdetE 0 then premultiplying (la) by E1 we obtain the standard system (2a) xi+1=A'x1 + B'u1 , iE Z (2b) yi=c,vi+D'u For the singular system (1) it is assumed that (3) det[Ez —A]0 for some zEC (the field of complex numbers) Definition 1. The standard system (2) is called externally positive if for x0 = 0 and every U1E R,iE Z we have yiE Rfor i Z. Theorem 1. [18] The standard system (2) is externally positive if and only if its impulse response matrix [CAB for i>O (4) giH [Dfori=O is non-negative g E R><m for i E Z Definition 2. The standard system (2) is called internally positive if for every x0 E R and all inputs U R, j E Z we have x E Rf and y E R for i E Z. Theorem 2. [20] The standard system (2) is internally positive if and only if (5) A E R.Xfl ,BE R><m ,CE RXfl ,DE R><m The standard internally positive system (2) is always externally positive. 2.2. Continuous-time systems. Consider the continuous-time linear system (6a) EAx+Bu 2 Proc. of SPIE Vol. 5484 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 12/04/2015 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx (6b) y=Cx+Du x= x(t) R ,u= u(t) R' ,y =y(t) R are the state, input and output vectors, and dt E ,AE R flXflBR nxrn cER ,DE R The system (6) is called singular if det E = 0 .Ifdet E 0 then premultiplying (6a) by E1 we obtain the standard system (7a) xAx+Bu (7b) y=C+D'u For the singular system (6) it is assumed that (8) forsome SEC Definition 3. The standard system (7) is called externally positive if for x0 x(O) = 0 and every u(t) E R ,t 0 wehave y(t)ER, for t0. Theorem 3. [20] The standard system (7) is externally positive if and only if its impulse response matrix ICeAtBfor t>0 (9) g(t)= [D(t) for t=0 is non-negative g (t) E Rm for t 0 ,where6 (t) is the Dirac impulse. Definition 4. The standard system (7) is called internally positive if for every x0 E R and all inputs u(t) E R, t 0 we have x(t) E R and y(t) E R for t 0. Theorem 4. [20] The standard system (7) is internally positive if and only if A is a Metzler matrix (all off-diagonal . Xm,-' entries are non-negative) and iE E pXn E pxin The standard internally positive system (7) is always externally positive. The standard internally positive system (2) and (7) will be shortly called positive. 2.3. Reachability and controllability of positive 1D systems without and with feedbacks. Definition 5. The positive system (2) is called h-step reachable if for every XfcR (and x0 = 0 ) there exists a input = — sequence u. CR,i 0,1,... , h 1such that XhXf. Definition6. The positive system (2) is called reachable if for every x1 e R (and x0 = 0)there exists hEZ and E R, i =0,1,...,h —1 such that Xh= Xf. Definition7. Thepositive system (2) iscalled controllable if for every nonzero Xf, x0ERthere exists hE Z÷ and ER, i =0,1 h—i suchthat Xh= Xf. Definition8. The positive system (2) is called controllable to zero if for every x0 ERthere exists hE Z and u E R, i =0,1,...,h —1 such that Xh =0. Theorem 5. [7,20] The positive system (2) is n-step reachable if and only if: i. rank R =ii Proc. of SPIE Vol. 5484 3 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 12/04/2015 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx 'H JqI S1STX IflUT5U0U XTJNUJ UI1STSUO3 JO U SUUJflJO3 JO 3flS T- JO 1 i/ "PuTT 1upudpuT suuinjo ipi uruniuo cjuo uo AT1TSOd i'ijU (oT) =: 'uv'u] [u1_v" ><i qi ATTSOd ffl1S1cS (j) T uq TT T ds-u '[9'c] o LU.IOflJ 9 L] [oz 'a AT1TSOd mSs (z) T Jq13JJoi1uoo JT pu ícjuo :JT 'T qi XUIIEftU uJ SlEfti i/ jJiuij iupudpui suwnjo qi uiuniuoo juo uo AT1TSOd '1cJ1U SnTPIII JO V ST > IT T1 JJSU1ll1 WOJJ T PM0TI UI Ut? 1TUTJUT ro 'U pu13dS ()d (V)d t 0X O X qmnu — sdis pui ()d 0 JT qi juiii moq °x o T P1Tflb1 ui 1JU iqmnu JO sdis 0 ic-j Sn fflfl55 liftil JOJ UI = t qi S3TJ1W Ut El JO (z) A4 qi TETU0U3 U_uOJ 0 T •••0 0 0 0 0 I •• 0 (ii) uI+ 00 ••O t 0 . • O7_ n— '0v— T II ST 'S1 01 S 1T41 JOJ i) (1 (zi) =[UT_uV''GV'EJ])IU1L1 U — inq qi UOT1TpUO (o JO miotj c T IOU PTJST1S JT 11E 1S1T UO 1) 0 JOJ = • U' t • UJ SR S13 AT1TSOd m1S1cS (z) qITM i) (1 T iou dais-u •jqii.pi.i 1PT5UO i-T1 TU155 (z) TIITM )puqpJ-1131S (ci) 1X)I+4::fl ST i.ji JqM )J UI !(t u InduT uoT1n1T1sqnJo (;j) oui (z) spj1c = (j) 'x X3V + W ! +z (ci) xu+v3v ioj (JJ) U13 (91) T_Uv...TvcOvIXF ji XTJIiTU (çj) siq mJoj 0 1 0 0 0 0...OtO 0 0 t " 0 o...too V(Lt) + V ... :4 0 0 0 ••• I T-1T0O 1...000 13— V— 13— 1J—••• 0 I Z T— t o...000 uisç (LI) uT11qo to...oo El d•3 : :.-i" a vu] 3 [WT-Uv3 oo...1o oo...o1 ujj qi suoT1Tpuo jo miop c pqsnts pur qi doo-psop misis si dais-u •JqBqo11 iopi tp uiojjoj mro siq uq pAoJd '6T] [oz m.ioqj L icr. qi AflTsod miss () qITM j) (j q iou dis-u Jqiq3iJ ujj qi dooj-psop mss (i'T) LflJM (Li) ST dis-u jqiqoii JT T1 qpj-is UT13 XTJW )j sij qi TUJoj (çj) D 4 Proc.
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