煤矿安全外文翻译文献
SHa(25)
?EHa??????eHma?,
?0?SQ????SQc?eQm??SQa?,
(26)
?0?Sd?????1?(27)
With (13), (24), differentiating (3), we have
2Ha??RQQDR??QQ??dd.
,
(28) Where
?RQ??EQcKcSHa?Ka??1EQK???RQc?RQa?(29)
,
?Q???EQcKcSHa?Ka??1EQcKcRmSQ???Qc(30)
?Qa?,
?d???EQcKcSHa?Ka??1EQcKcSd(31)
,
One should mention, that the inverse of EQcKcSHa +Ka, from Eqs. (29)–(31), exists, which will be shown in Lemma 2.1. Substituting (28) into (24), it can be expressed as
2Hc?SHa?RQQDR??SHa?Q?RmSQ?Q??Sd?SHa?d?d.
(32)
With (11), (28) and (32), we have
?SHa?RQ?2?SHa?Q?RmSQ??Sd?SHa?d?H???QDR???Q???d ???Qd???RQ???2?YRQQDR?YQQ?Ydd,
(33)
煤矿安全外文翻译文献
where
?SHa?RQ??SHa?Q?RmSQ??Sd?SHa?d?, , . YRQ??Y?Y????Qd???Q?d????RQ???Substituting (33) into (13), rewrite it as
???K?I?Y?Q2R?KYQ?KYd QRQDQd=
(34) where
2AQDR?BQ?Cd,
A??K?I?YRQ?, B?KYQ, C?KYd. Lemma 2.1. The inverse of EQcKcSHa +Ka exists.
Proof. We number the branches in the following way: the links are enumerated from 1 to l=N -nc +1, the rst branch connects with the fan branch, and the tree branches are enumerated from l to N, where the fan branch is the last one. The loop and the node equations, including the fan branch, can be expressed as
?EHc?e?HmcEHaeHma?Hc?0????Ha??01???H??m?,
(35)
?EQc??e?QmcEQaeQma?Q?0??c??Qa??01???Q??m?.
(36)
It can be shown [3, p. 493], that
?EHa?e?Hma(37) or
0?T??EQc1???T?eQmc?,
?E?T ?Ha???EQc.
?eHma?煤矿安全外文翻译文献
From (19) and (37), EQ is of full rank. So EQK can be factorized by singular value decomposition [4]as
EQK12?U?V.
(38)
?(39)
??1?0???0?0?0??n?2c?0??0?0??,
?i?0 , i=1,…,nc-2.
With (38) and (39), we can write
TEQKEQ?U?VVT?TUT?U??TUT.
So we have
det(40)
Substituting (17) and (19) into (40), we get
det(41)
With (25), (19), (37) and (41), we have the following:
???eHma?T? ?Ka??det?-EK?Kadet?EQcKcEQc?Qcc?E???Ha????EKE??0QcTQc,
?EQcTKcEQc?Ka?0?,
=det
(42)
So the inverse of EQcKcSHa +Ka exists. 2.3. Minimal model of the network
?EQcTKcEQc?Ka?0?.
In the previous subsection we have established the full model of mine ventilation network, which is of order n. The states of the system are not independent, so one needs to nd the minimal representation. In this subsection, we establish a minimal model of the mine ventilation network.
Define
煤矿安全外文翻译文献
2QcD?diag(Q1Q1,...,QN?nc?1QN?nc?1),
(43)
2QaD?diag(QN?nc?2?Qc?QN?nc?2?Qc?,...,Qn?Qc?Qn?Qc?),
(44)
TR?Rc?TRa?T,
(45)
where the dependence on Qc in (44) should be understood in the sense of (21). Proposition 2.2. There exist matrices Ac, Aca, Bc and Cc of appropriate dimensions so that the minimal model of mine ventilation network system can be expressed as
??AQ2R?AQ2R?BQ?CdQcccDccaaDaccc,
(46)
22?Qc?Ra??QcQc??ddHa??RQcQcDRc??RQaQaD,
(47)
where Qc is a state , Rc , Ra and d are the control inputs, and Ha is the system output. Proof. First we should mention, that SQa =0, which follows from (19) and (26). Also, from (30),?Qa?0.
Substituting (44) into (32), we get
22?Qc?Ra??SHa?Qc?RmSQc?Qc??Sd?SHa?dd?d. (48) Hc?SHa?RQcQcDRc?SHa?RQaQaDFrom (13), we have
???KQ2R?KHQcccDccc(49)
Substituting now (48) into (49),
22???K?KS?QcccHaRQcQcDRc?KcSHa?RQaQaD?Qc?Ra
,
??+Kc?SHa?Qc?RmSQc?Qc?Kc?Sd?SHa?d?d =
(50)
22AcQcDRc?AcaQaDRa?BcQc?Ccd,
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