positive linear systems with delays

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1、8PositiveLinearSystemswithDelays8.1PositiveDiscrete-timeandContinuous-timeSystems8.1.1Discrete-timeSystemsnìmLetbethesetofnìmmatriceswithentriesfromthefieldofrealnumbersandnnì1=.Thesetofnìmmatriceswithrealnonnegativeentrieswillbedenotedbynìmnnì1+and+=+Theset

2、ofnonnegativeintegerswillbedenotedby+Considerthediscrete-timelinearsystemwithdelaysdescribedbytheequationshqxxuik+--1=+ƒƒABikjij,i≠D+,(8.1.1a)kj==00yxu=+CD,(8.1.1b)iiinmpwherehandqarepositiveintegers,xi≠,ui≠,yi≠arethestate,inputandnìnnìmoutputvectors,respect

3、ively,andAk≠(k=0,1,…,h),Bj≠(j=0,1,…,q),pìnpìmC≠,D≠.Theinitialconditionsfor(8.1.1a)aregivenbynmx≠=<<,(ih0,1,...,),ujq≠=(1,2,...,).(8.1.2)--ijTheorem8.1.1.Thesolutionto(8.1.1a)isgivenby422PolynomialandRationalMatrices--11hj++11qj++xik=+ĭĭ(ix)01ƒƒ()ik-A--jxj+ƒƒ

4、ĭ()ik-Bk--1jujjhk=-=11jqk=-=(8.1.3)i-1q+-ƒƒĭ(1ik--ju),Bkjjk==00where-1½°°⊬·h--1kĭ()iZz=-®¾∆¸IAnkƒzz(8.1.4)°°¯¿«¹k=0-1isthestate-transitionmatrixandZdenotestheinversez-transform.Thestate-transitionmatrixF(i)satisfiestheequationĭ(1ii+=)Aĭ()+Aĭ(1i-++)...Aĭ()i-h

5、,(8.1.5)01hwiththeinitialconditionsĭ(0)==

6、¬k=0--11hj++1ik-qºq+-+-ƒƒĭ(2)ikBkj--1ujƒƒĭi(2ikj--1)BBkju»+ƒjiju-jqk=-=10jk==0¼j=0-1hj++1=++ĭĭ(1ix)01ƒƒ(ik-+1)Akj--xjjhk=-=1-1qj++1iq+-ƒƒĭ(1ik+)Bkj--1uj+-ƒƒĭ()ikjux-=Bkji+1.jqk==10j==k0Then(8.1.3)satisfies(8.1.1a).

7、nternally)positiveifxi≠+andpnmyi≠+(i≠+)foreveryx-i≠+,u-j≠+,i=0,1,…,h,j=1,2,…,qandallinputsmui≠+,i≠+.Theorem8.1.2.Thesystem(8.1.1)isinternallypositiveifandonlyifnnìììnmpnpmìABC≠=<<<,(kh0,1,...,),≠=,(jq0,1,...,),≠≠,D<.(8.1.7)kj++++PositiveLinearSystemswithDela

8、ys423Proof.Defining»ºxi»ºui…»…»xu…»i-1…»i-1=≠=≠…»nm!!…»xu!!@<<,@,(8.1.8)ii++…»…»xu…»ih-+1…»iq-+1…»x…»u¬¼ih-¬¼iq-»ºAA01?AAhh-1…»I00?0…»nA=…»@@B@@,(8.1.9a)…»…»00?00…»00?I0¬¼n»ºBB01?BBqq-1…»00?00…»