Stability Analysis for a Type of Multiswitching System ...

Research ArticleStability Analysis for a Type of Multiswitching System withParallel Structure

Zhang Yan 12 Liu Yongqiang 1 and Liu Yang 1

1School of Electric Power South China University of Technology Guangzhou Guangdong China2Centre of Faculty Development and Educational Technology Guangdong University of Finance and EconomicsGuangzhou Guangdong China

Correspondence should be addressed to Liu Yongqiang epyqliuscuteducn

Received 28 March 2018 Revised 28 June 2018 Accepted 25 July 2018 Published 8 August 2018

Academic Editor Asier Ibeas

Copyright copy 2018 Zhang Yan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposes a new type ofmultiswitching system and a subsystems-group as a basic switching unit that obeys the lawUnliketraditional switching systems the system selects multiple subsystems instead of one on each time interval Thus a framework ofparallel structure organizes the subsystems as a group A multiswitched system is widely used in engineering for modelling andcontrol this system reflects the actual industrial dynamical process Thus the stability of the system is studied Assuming thatthese continuous and discrete-time subsystems are Hurwitz and Schur stable the subsystems-groups matrices commute each otherbased on the subsystems matrices pairwise commutative Then the multiswitched system is exponentially stable under arbitraryswitching and there exists a common Lyapunov function for these subsystems The main result is extended to a parallel-likestructure therefore some stability results are gained under some reasonable assumption At last a numeral example is given toillustrate the structure and the stability of this system

1 Introduction

A switched system is a particular kind of hybrid dynamicalsystem that consist of a finite number of subsystems and aswitching signal that switches between these subsystems inan orderly manner The set of subsystems can be continuous-time and discrete-time the mathematical models are differ-ential equations and difference equations According to thedynamics behaviours it can be classified as continuous-timeswitched system discrete-time switched system or mixedswitched system (both them) [1] linear or nonlinear and soon [2] Actually many fields of complex industrial processesinclude typical switching systems which reflect the hybriddynamical characteristics that are widely used in many fields[3ndash8] such as in aircraft control the automotive industrycommunication systems electrical engineering chemicalprocesses and so on

Multiswitched systems are a novel type of system whichcan be composed of a set of continuous-time subsystems-groups

(119905) = 119864119888119894119909 (119905) 119888119894 = 1198881 119888119873119888 (1)

119864119888119894 = (((((

1198601198881 sdot sdot sdot 0 sdot sdot sdot 0 d0 119860119888119901 0 d0 sdot sdot sdot 0 sdot sdot sdot 119860119888119872

)))))

119888119901 = 1198881 119888119872(2)

and a set of discrete-time subsystems-groups119909 (119896 + 1) = 119864119889119895119909 (119896) 119889119895 = 1198891 119889119873119889 (3)

119864119889119895 = (((


)))

119889119902 = 1198891 119889119872

(4)

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 3834601 16 pageshttpsdoiorg10115520183834601

2 Mathematical Problems in Engineering

where x(t) x(k) are the states t is the time scalar and 120591 is pos-itive integer of the sample period Eci and Edj are nonnull con-stant matrices with appropriate dimensions which are com-bined by some constant matrices Acp and Adq respectivelySubsystems-groups (1) and (2) are composed of continuous-time subsystems (119905) = 119860119888119901119909(119905) with 119888119901 = 1198881 119888119872and discrete-time subsystems 119909(119896 + 1) = 119860119889119902119909(119896) with119889119902 = 1198891 119889119872 respectively The system switches betweenmatrices 1198641198881 1198641198882 119864119888119873119888 1198641198891 1198641198892 119864119889119873119889 belonging tothe set E 119864 ≜ 119864119888 119864119889 119864119888 ≜ 1198641198881 1198641198882 119864119888119873119888 and119864119889 ≜ 1198641198891 1198641198892 119864119889119873119889 1198601198881 1198601198882 119860119888119872 belong to theset 119860119888 ≜ 1198601198881 1198601198882 119860119888119872 1198601198891 1198601198892 119860119889119872 belong tothe set 119860119889 ≜ 1198601198891 1198601198892 119860119889119872 and 119860 ≜ 119860119889 119860119888 119860 and119864 are finite sets

Themajor difference between amultiswitched system anda traditional switched system is that the former selects anyamount of subsystems and the latter selects only one eachtime There is a structure for organizing the subsystems asa subsystems-group on each interval that includes mutualindependence between subsystemsMany different structurescan be considered such as parallel tandem and other mixedmodes Equations (2) and (4) show the parallel structure ofthe matrices However the traditional switching system hasonly one matrix without any structure

A multiswitched system is widely used in engineeringcases such as in the chilled water system in central airconditioning The dynamical pumping process is a kind oftypical multiswitch system and each different pump as asubsystemwith switching and variable frequency behavioursFor variable flow control there are several different pumpsworking together as a subsystem-group at each time periodAccording to the water pipe network the pumps are parallelin structure Thus this physical model illustrates the pro-posed system In this paper only parallel and parallel-likestructures are researched

Modelling [9] performance analysis [10] and optimalcontrol [11 12] are the key issues of switched systemsObviously stability analysis is a very important researchbranch that has attracted the attention of researchers globallyStability is the most basic property and the primary issueto be solved by the control system The stability problemsof switched systems are reduced to the following three basicissues [13] finding conditions to guarantee asymptoticalstability for any switching signal identifying some switchingsignals for asymptotical stability and constructing a switch-ing signal to make the system asymptotically stable Accord-ingly some excellent theoretical methods are proposed forsolving those problems such as common Lyapunov functions[14 15] multiple Lyapunov functions [16ndash18] dwell timeand average dwell time [19 20] and piecewise quadraticLyapunov functions [16]

It is worth noting that stability under arbitrary switchingis a fundamental in the design and analysis of switchedsystems [21] It is known that all the subsystems that are expo-nentially stable are not sufficient to guarantee stability underarbitrary switching expect for some reasonable assumptionssuch as the state matrices of subsystems commute pairwise[2 14] (eg 119860 119894119860119895 = 119860119895119860 119894 for all 119894 119895 isin 119868 119894 = 119895 119860119888119894119860119889119895 =119860119889119895119860119888119894 for all 119894 119895 isin 119868 c and d denote continuous-time and

discrete-time subsystems respectively) the state matricesof subsystems are symmetric [22] (eg 119860 119894 = 119860 119894119879 for all119894 isin 119868) the state matrices of subsystems are normal [23](eg 119860119888119894119879119860119888119894 = 119860119888119894119860119888119894119879 for all 119894 isin 119868 and 119860119889119895119879119860119889119895 =119860119889119895119860119889119895119879 for all 119895 isin 119868 c and d denote the same substancesabove) Thus some researchers present a sufficient conditionin terms of the Lie algebra If this Lie algebra is solvablethe exponential stability of the switched system for arbitraryswitching can be gained [20] A more general result isthat the semisimple subalgebra S is a compact Lie algebraby considered Levi decomposition and the exponentiallystability can be ensured [24] Previous research shows that thepairwise commutation of the vector fields is a sufficient con-dition for the stability [25] By using Hurwitz stable matrixpencils 120590120572[1198601 1198602] and 120590120572[1198601 1198602minus1] a common quadraticLyapunov function exists that can guarantee stability underarbitrary switching [26] Based on previous research themain emphasis is placed on the conditions of stability analysisand a common Lyapunov function is constructed under anassumption

In this paper a type of linear multiswitched systemwith parallel structure is put forward for the first time Thecomposition of the system is expounded and the structureand form of the subsystems-group are described From theswitching law and the physical meaning we illustrate thedifference between the system and the traditional switchingsystem A case of chilled water system pumps in centralair conditioning shows the engineering significance of thesystem Next stability under arbitrary switching for this typeof system is studied The inference of properties of statematrices of the subsystems-group is derived from the subsys-tems Then based on the conditions of Hurwitz stable andSchur stable for continuous-time subsystems and discrete-time subsystems respectively the stability of subsystems-groups can be obtained The matrices 119864119888 and 119864119889 of thesubsystems-groups commute pairwise can be ensured basedon the assumptions of the state matrices 119860119888 and 119860119889 [2 14]Thus the stability under arbitrary switching is guaranteedand a commonLyapunov function is given for all subsystems-groups and subsystems Finally the result is extended toa parallel-like structure which reduces conservativeness instability analysis

The body of this paper is organized as follows in Sec-tion 2 the system description the example of the multi-switching system and the preliminaries are presented InSection 3 the main stability results in the continuous-timesystem and the mixed system are expressed two examples arealso given to illustrate the results In Section 4 the studiesextended to parallel-like structure is presented Section 5concludes the paper

2 System Descriptionand Mathematical Preliminaries

21 System Description and Examples Throughout the fol-lowing notation is adopted R and C denote the fields ofreal and complex number respectively R119899 denotes the n-dimensional real Euclidean space R119899times119899 denotes the space of

Mathematical Problems in Engineering 3

119899 times 119899 matrices with real entries xi and xj denote the ith and

jth components of the vector x in R119899 respectively 1198861198881199011199011015840 and1198861198891199021199021015840 denote the entry in the (119901 1199011015840) and (119902 1199021015840) position of thematrices A or E in R119899times119899 respectively

There are three examples to show the multiswitchedsystem and the parallel structure

Example 1 Consider a set 119860 = 1198601 1198602 1198603 of a switchedsystem which are the constant matrices of 3 subsystemsrespectively Assume in the classical switched system thatthere are 3 subsystems The system switches between 1198601 1198602and 1198603 Assume in the multiswitched system that there are 7subsystems-groups The system switches between E

119864 = 1198641 1198642 1198643 1198644 1198645 1198646 1198647 = [[[1198601 0 00 0 00 0 0]]]

[[[0 0 00 1198602 00 0 0]]] [[[

0 0 00 0 00 0 1198603]]] [[[

1198601 0 00 1198602 00 0 0]]] [[[1198601 0 00 0 00 0 1198603

]]] [[[0 0 00 1198602 00 0 1198603

]]] [[[1198601 0 00 1198602 00 0 1198603

]]]

(5)

Here119873119888 = 1198621119888119872 + sdot sdot sdot + 119862119888119872119888119872 and 119873119889 = 1198621119889119872 + sdot sdot sdot + 119862119889119872119889119872Remark 2 A continuous-time subsystems-group composedof a subsystem or all the subsystems Obviously there aresubsystems but the value is null It is same with a discrete-time subsystems-group

Remark 3 Continuous-time subsystems cannot mix withdiscrete-time subsystems to be a subsystems-group Contin-uous-time subsystems-groups and discrete-time subsystems-groups must be distinguished

Remark 4 When Eci is a singular matrix zero rows or zerocolumns remain with the aim of uniform description InExample 1 1198641 = 1198601 but the uniform description is kept to

show 1198641 = [ 1198601 0 00 1198602 00 0 1198603

] = [ 1198601 0 00 0 00 0 0

] with 1198602 = 0 1198603 = 0Thus all Eci values look the same 119888119894 = 1198881 119888119873119888 The sameis true of 119864119889119895Remark 5 Equations (2) and (4) are standard parallel struc-tures A parallel-like structure will be considered in Section 4which has a coupling phenomenon based on a parallelstructure between some subsystems Assumptions are madefor reducing conservativeness to obtain ideal stability

Example 6 In a chilled water system of a central air con-ditioning system there are three pumps driving the chilledwater from the evaporator to the air conditioning unit (seeFigure 1) All the pumps can be switched ONOFF and

M

M

evaporator

pumps

air conditioningunit

chilled waterCycle

Chilled water system

Figure 1 Chilled water system of central air conditioning

0

001

002

003

004

005

006

q (k

gs)

100 200 300 400 500 600 700 800 9000time (s)

Figure 2 Total flow volume

the variable frequency obeying a range of 50-100 ratedfrequency For setting an energy-saving control strategy withvariable water volume technology the whole volume cannotbe less than 50 of the rated volume Assuming in a timeinterval that the cold load changes the switching and pumpwater volume are shown in Figures 2 and 3 The parametersand symbols are shown in Table 2

It is assumed that the cold load in a certain area requiresair conditioning to meet the cold demand within 5 minutesto achieve a balanced state Then in the first 5 minutes 3pumps work at the rated frequency After reaching the presetvalue the cold load is stable with small fluctuations In thisperiod pumpA and B exit only pumpC works in the form offrequency conversion maintaining the driving chilled waterand transferring cooling capacity In the last 3 minutes due


pump Apump Bpump C

100 200 300 400 500 600 700 800 9000time (s)

minus001

0

001

002

003

004

005

006

q (k

gs)

Figure 3 Water volumes of pumps

to the cold load increase a pump is not enough to meet thedemand and the system puts pump A into work with pumpC together with frequency conversion

From the three time intervals ([0 5min] [5min 12min]and [12min 15min]) we find that there are three oneand two subsystems working respectively According to theframework in this paper the three pumps denote differentcontinuous-time subsystems and the work combinations ofthe pumps denote the different continuous-time subsystems-groups Assuming the pump A B and C denote the subsys-tem11988311198832 and1198833 respectively and the combinationABCC and AC we just said denote the subsystems-group 1198841 1198842and 1198843 respectivelyExample 7 In the same system with Example 6 but thepumps are divided into one variable frequency pump andtwo switchable fixed frequency pumps Thus the former isa continuous-time subsystem and the latter are discrete-timesubsystems According to the energy-saving control strategyassuming pump C is always working with variable frequencyand pumps A and B are switched ON or OFF with fixedfrequency Thus the chilled water system is composed ofcontinuous and discrete-time subsystems The parametersand symbols are shown in Table 3

There are two simulation figures (Figures 4 and 5) thatillustrate the switching and working situations of the system

In the first 5 minutes 3 pumps work together thisseems a subsystems-group with one continuous and twodiscrete-time subsystems In the interval of from 5 minto 12 min only pump C works which is the only onecontinuous-time subsystem that is a subsystems-group In thelast threeminutes pumpAworkswith pumpC therefore the

100 200 300 400 500 600 700 800 9000time (s)

0

001

002

003

004

005

006

q (k

gs)


pump Apump Bpump C

minus001

0

001

002

003

004

005

006

q (k

gs)

100 200 300 400 500 600 700 800 9000time (s)

Figure 5 Pump water volume

subsystems-group has one continuous and one discrete-timesubsystem

22 Important Theories for Stability Analysis We analyse thestability of the multiswitched system based on the structuralproperty of E-matrices which cannot exist without the basiccomponent A-matrices All the inferences of multiswitchedsystem are derived from a classical switched system Thussome important theorems and propositions of the classicalswitched system should be quoted first


Theorem 8 (see [14]) Consider a classical continuous-timeswitched system = 119860119888119909(119905) with 119860119888 ≜ 1198601198881 1198601198882 119860119888119872where the matrices 119860119888119901 are asymptotically stable and commutepairwise Then we get the following

(1) The system is exponentially stable under any arbitraryswitching between the elements of 119860119888

(2) There exists a common Lyapunov function for all thesubsystems For any positive symmetric definite matrix1198751198880 let 1198751198881 1198751198882 119875119888119872 be the unique symmetric posi-tive definite solutions to the Lyapunov equations

119860119879119888119901119875119888119901 + 119875119888119901119860119888119901 = minus119875119888119901minus1 119888119901 = 1198881 1198882 119888119872 (6)

The function 119881(119909) = 119909119879119875119888119872119909 is a common Lyapunovfunction for each of the individual system = 119860119888119909(119905) andhence a Lyapunov function for the switching system

Remark 9 All of the proposed descriptions and notations areidentical to the given multisystem

Theorem 10 (see [1]) Consider a classical mixed switchedsystem composed of continuous-time subsystems 119909 = 119860119888119909(119905)with 119860119888 ≜ 1198601198881 1198601198882 119860119888119872 and discrete-time subsystems119909(119896+1) = 119860119889119909(119896)with119860119889 ≜ 1198601198891 1198601198892 119860119889119872 Let119860119888119901 beHurwitz stable and 119860119889119902 be Schur stable to commute pairwiseThen we get the following

(1) The system is exponentially stable for any arbitraryswitching between the elements of A

(2) There exists a common Lyapunov function for allthe subsystems For any positive definite matrix 1198751198890let 1198751198891 1198751198892 119875119889119872 be the unique positive definitesolutions to the Lyapunov equations

119860119879119889111987511988911198601198891 + 1198751198891 = minus1198751198890 (7)119860119879119889119902119875119889119902119860119889119902 + 119875119889119902 = minus119875119889119902minus1 (8)

let 1198751198881 1198751198882 119875119888119872 be the unique positive definite solutionsto the Lyapunov equations

119860119879119888119901119875119888119901 + 119875119888119901119860119888119901 = minus119875119888119901minus1 (9)

11986011987911988811198751198881 + 11987511988811198601198891 = minus119875119889119872 (10)

The function 119881(119909) = 119909119879119875119888119872119909 is a common Lyapunovfunction for each of the individual system 119909 = 119860119888119909(119905) 119909(119896 +1) = 119860119889119909(119896) and hence a Lyapunov function for the switchingsystem

Remark 11 Same as Remark 9

Clearly in Theorems 8 and 10 as a necessary conditionall the subsystems that are asymptotically stable still cannotensure a stable switched system under arbitrary switchingThen the sufficient condition is needed that is pairwisecommutative which guarantees the stability of the switchedsystem In other words if there exists a common Lyapunovfunction for all the subsystems the stability is ensured underarbitrary switching and the solutions are gained

In the following section matrices E is analysed and somelemmas and theories are derived The exponential stability ofthe system under any arbitrary switching is discussed and acommon Lyapunov function can be obtained

3 Main Results

In the framework of proposed multiswitched system andbased onTheorems 8 and 10 we have the following lemmas

Lemma 12 If matrices Ac and Ad are pairwise commutativematrices Ec and Ed are also pairwise commutative

Proof Consider 1198601198881198941198601198881198941015840 = 1198601198881198941015840119860119888119894 of continuous-timesubsystems 119894 = 1198941015840

We get

1198641198881198941198641198881198941015840 = 1198641198881198941015840119864119888119894

= (((((


)))))

(((((


)))))

(11)

Consider 119860119888119860119889 = 119860119889119860119888 of all subsystems we get

119864119888119864119889 = ((


))

((


))

= ((


))

= ((((


))))

= 119864119889119864119888(12)


1198641198891198941198641198891198941015840 = 1198641198891198941015840119864119889119894 is similar with (11)Therefore the matrices Ec and Ed are pairwise commuta-

tive This completes the proof for Lemma 12

Lemma 13 If matrices Ac and Ad are Hurwitz stable andSchur stable respectively matrices Ec and Ed are also Hurwitzstable and Schur stable respectively

Proof According to the Hurwitz stable criterion all the orderprincipal minors of matrices Emust be positive

The structure of (2) is parallel and matrices E are diag-onal Thus all the order principal minors are also diagonal119860119888119901 119888119901 = 1198881 119888119872 are Hurwitz stable and all the orderprincipal minors are positive thus it is proven that matricesE are Hurwitz stable119864119889 is Schur stable which is similarwith theHurwitz stablecriterion

Remark 14 Unnecessary zero rows or zero columns can beomitted

Based on Theorem 8 and Lemmas 12 and 13 we have thefollowing results

Proposition 15 Assume = 119860119888119909(119905) are Hurwitz stable andAcp are pairwise commutative Then we get the following

(1) The continuous-time multiswitched system is exponen-tially stable for any arbitrary switching between theelements of Ec

(2) There exists a common Lyapunov function for all thesubsystems-groups and subsystems For any positivesymmetric definite matrix 1198751198880 let 1198751198881 1198751198882 119875119888119872 bethe unique symmetric positive definite solutions to theLyapunov equations(1198601198791198881119875119894 + 1198751198941198601198881) + (1198601198791198882119875119894 + 1198751198941198601198882) + sdot sdot sdot+ (119860119879119888119872119875119894 + 119875119894119860119888119872) = minus119875119894minus1119888119901 = 1198881 1198882 119888119872 119894 = 1 2 119873119888

(13)

The function 119881(119909) = 119909119879119875119873119888119909 is a common Lyapunovfunction for each of the individual systems = 119860119888119909(119905) andis thus a Lyapunov function for the switching system

Proof Based on Lemmas 12 and 13 Eci are pairwise commu-tative and Ec are Hurwitz stable Combining Theorem 8 weget 119864119879119888119894119875119894 + 119875119894119864119888119894 = minus119875119894minus1 119894 = 1 119873119888 (14)We substitute Eci in (2) into (14) to obtain

(((((


)))))

119879

119875119894

+ 119875119894(((((


)))))

= minus119875119894minus1(15)

Equation (15) equals (14) which implies that the system isexponentially stable for arbitrary switching (15) transformsinto (13) which implies the solutions to the Lyapunovequations

Example 16 Consider a set 119860119888 = 1198601198881 1198601198882 1198601198883 1198601198884 1198601198885of a switched system which are the constant matrices of5 continuous-time subsystems The multiswitched systemswitches between subsystems-groups Ec Here we select onlythree subsystems-groups as an example

Let 119864119888 = 1198641 1198642 1198643 1198641 = [1198601198881] 1198642 = [ 1198601198882 00 1198601198883 ] and1198643 = [ 1198601198883 0 00 1198601198884 00 0 1198601198885

]Then for any positive symmetric definitematrix1198751198880 there

are 1198751198881 1198751198882 and 1198751198883 as the unique symmetric positive definitesolutions to the Lyapunov equations

11986411987911988811198751 + 11987511198641198881 = minus1198751198880 (16a)

11986411987911988821198752 + 11987521198641198882 = minus1198751198881 (16b)

11986411987911988831198753 + 11987531198641198883 = minus1198751198882 (16c)

Equations (16a) (16b) and (16c) can be rewritten as11986011987911988811198751 + 11987511198601198881 = minus1198751198880 (17a)

11986011987911988821198752 + 11987521198601198882 + 11986011987911988831198752 + 11987521198601198883 = minus1198751198881 (17b)

11986011987911988831198753 + 119875311986034 + 11986011987911988841198753 + 11987531198601198884 + 11986011987911988851198753 + 11987531198601198885= minus1198751198882 (17c)

The function 119881(119909) = 1199091198791198753119909 is a common Lyapunovfunction for each of the individual system = 119860119888119909(119905) with119888119901 = 1198881 1198882 1198883 1198884 and 1198885Remark 17 The above theorem and proof can be extended tothe case of discrete-time multiswitched system 119909(119896 + 1) =119864119889119895119909(119896) with 119895 = 1 119873119889 Assume the matrices 119909(119896 +1) = 119860119889119902119909(119896) with 119889119902 = 1198891 119889119872 are Schur stable andcommute pairwise Then the discrete-time multiswitchedsystem is exponentially stable for arbitrary switching betweenthe elements of Ed The solution (13) can be modified as

11986011987911988911198751198951198601198891 + 11986011987911988921198751198951198601198892 + sdot sdot sdot = 119860119879119889119872119875119895119860119889119872= 119875119895 minus 119875119895minus1 119889119902 = 1198891 1198892 119889119872 119895 = 1 119873119889 (18)

and the common Lyapunov function is 119881(119909) = 119909119879119875119873119889119909


Proposition 18 Assume 119909 = 119860119888119909(119905) are Hurwitz stable and119909(119896 + 1) = 119860119889119909(119896) are Schur stable 119860119888119901 and 119860119889119902 are pairwisecommutative Then we get the following

(1) Themultiswitched system is exponentially stable for anyarbitrary switching between the elements of E

(2) There exists a common Lyapunov function for all thesubsystems For any positive symmetric definite matrix1198751198890 let1198751198891 1198751198892 119875119889119873119889 be the unique positive definitesolutions to the Lyapunov equations

119860119879119889111987511988911198601198891 + sdot sdot sdot + 1198601198791198891198721198751198891119860119889119872 minus 1198751198891 = minus1198751198890119889119902 = 1198891 1198892 119889119872 (19a)

11986011987911988911198751198891198951198601198891 + sdot sdot sdot + 119860119879119889119872119875119889119895119860119889119872 minus 119875119889119895 = minus119875119889119895minus1119889119902 = 1198891 1198892 119889119872 119895 = 1 119873119889 (19b)

Let 1198751198881 1198751198882 119875119888119873 be the unique positive definite solutions tothe Lyapunov equations

(11986011987911988811198751198881 + 11987511988811198601198891) + sdot sdot sdot + (1198601198791198881198721198751198881 + 1198751198881119860119889119872)= minus119875119889119873119889119888119901 = 1198881 1198882 119888119872 119889119902 = 1198891 1198892 119889119872 (19c)

(1198601198791198881119875119888119894 + 1198751198881198941198601198881) + sdot sdot sdot + (119860119879119888119872119875119888119894 + 119875119888119894119860119888119872)= minus119875119888119894minus1 119888119901 = 1198881 1198882 119888119872 119894 = 1 2 119873119888 (19d)

The function 119881(119909) = 119909119879119875119888119873119888119909 is a common Lyapunovfunction for each of the individual system = 119860119888119909(119905) and119909(119896 + 1) = 119860119889119902119909(119896) hence a Lyapunov function is used for theswitching system

Proof Based onTheorem 10 as well as Lemmas 12 and 13 weget

119864119879119889111987511988911198641198891 minus 1198751198891 = minus1198751198890 (20a)

119864119879119889119895119875119889119895119864119889119895 minus 119875119889119895 = minus119875119889119895minus1 (20b)

11986411987911988811198751198881 + 11987511988811198641198891 = minus119875119889119873119889 (20c)

119864119879119888119894119875119888119894 + 119875119888119894119864119888119894 = minus119875119888119894minus1 (20d)

We substitute 119864119888119894 and 119864119889119895 in (2) into (19a) (19b) (19c) and(19d) to obtain

(((((


)))))

119879

sdot 1198751198891(((((


)))))

minus 1198751198891

= minus1198751198890(21a)

(((((


)))))

119879

sdot 119875119889119895(((((


)))))

minus 119875119889119895

= minus119875119889119895minus1

(21b)

(((((


)))))

119879

1198751198881

+ 1198751198881(((((


)))))

= minus119875119889119873119889

(21c)

((((((


))))))

119879

119888119901

119875119888119894


+ 119875119888119894(((((


)))))

= minus119875119888119894minus1(21d)

Equations (21a) (21b) (21c) and (21d) equal (20a) (20b)(20c) and (20d) which implies that the system is exponen-tially stable for any arbitrary switching and (21a) (21b) (21c)and (21d) transform into (19a) (19b) (19c) and (19d) whichimplies the solutions to the Lyapunov equations

Example 19 Consider sets 119860119888 = 1198601198881 1198601198882 1198601198883 and 119860119889 =1198601198891 1198601198892 of a switched system which are the constantmatrices of 3 continuous-time subsystems and 2 discrete-timesubsystems respectively The multiswitched system switchesbetween in subsystems-groups E Here we select only fivesubsystems-groups as an example

Let 119864119888 = 1198641198881 1198641198882 1198641198883 1198641198881 = [ 1198601198881 00 1198601198883 ] 1198641198882 = [ 1198601198882 00 1198601198883 ]and 1198641198883 = [ 1198601198881 0 00 1198601198882 0

0 0 1198601198883] let 119864119889 = 1198641198891 1198641198892 1198641198891 = [1198601198892] and1198641198892 = [ 1198601198891 00 1198601198892 ]

Then for any positive definite matrix 1198751198890 there are 1198751198891119875119889211987511988811198751198882 and1198751198883 as the unique symmetric positive definitesolutions to the Lyapunov equations

119864119879119889111987511988911198641198891 minus 1198751198891 = minus1198751198890 (22a)

119864119879119889211987511988921198641198892 minus 1198751198892 = minus1198751198891 (22b)

11986411987911988811198751198881 + 11987511988811198641198891 = minus1198751198892 (22c)

11986411987911988821198751198882 + 11987511988821198641198882 = minus1198751198881 (22d)

11986411987911988831198751198883 + 11987511988831198641198883 = minus1198751198882 (22e)

Equations (22a) (22b) (22c) (22d) and (22e) can berewritten as

119860119879119889211987511988911198601198892 minus 1198751198891 = minus1198751198890 (23a)

119860119879119889111987511988921198601198891 + 119860119879119889211987511988921198601198892 minus 1198751198892 = minus1198751198891 (23b)

11986011987911988811198751198881 + 11986011987911988821198751198881 + 11987511988811198601198892 = minus1198751198892 (23c)

11986011987911988821198751198882 + 11987511988821198601198882 + 11986011987911988831198751198882 + 11987511988821198601198883 = minus1198751198881 (23d)

11986011987911988811198751198883 + 11987511988831198601198881 + 11986011987911988821198751198883 + 11987511988831198601198882 + 11986011987911988831198751198883+ 11987511988831198601198883 = minus1198751198882 (23e)

The function 119881(119909) = 1199091198791198751198883119909 is a common Lyapunovfunction for each of the individual systems = 119860119888119909(119905) with119888119901 = 1198881 1198882 1198883 and 119909(119896 + 1) = 119860119889119909(119896) with 119889119902 = 1198891 1198892respectively

4 Results in Parallel-Like Structures

Lemma 20 If the structures of Ec are not standard paralleland contain coupled components based on a parallel framework(ie the subsystems are not independent of each other) then thematrices should be modified as

119864119888119894 = (((((((

1198601198881 sdot sdot sdot 119860119900(1119901) sdot sdot sdot 119860119900(1119888119872) d119860119900(1199011) 119860119888119901 119860119900(119901119888119872) d119860119900(1198881198721) sdot sdot sdot 119860119900(119888119872119901) sdot sdot sdot 119860119888119872

)))))))

(24)

119860119900(1199011199011015840) are matrices of the coupled components 119901 =1198881 119888119872 and 119901 = 1199011015840 Clearly there are another nonzeroelements except in the main diagonal line

Remark 21 Considering the coupled components Ed can bemodified as

119864119889119895 = (((((((


)))))))

(25)

119860119900(1199021199021015840) are matrices of the coupled components 119902 =1199021 119902119872 and 119902 = 1199021015840Remark 22 The structures proposed in Lemma 20 andRemark 21 are named as parallel-like Only the coupledcomponent is the distinction of structure between paralleland parallel-like in other words the parallel-like structure isminor alteration on the parallel structure

Proposition 23 Under the assumption of Lemma 20 theframework of Ec in Lemma 20 can be classified as thefollowing 3 types upper triangular lower triangular and othernonregular modes Then Proposition 15 can be modified asfollows

(a) If the framework of Ec is upper triangular Proposition 15is true but the solutions is


((((((

1198601198791198881119875119894 + 1198751198941198601198881 sdot sdot sdot 119875119894119860119900(1119901) sdot sdot sdot 119875119894119860119900(1119888119872) d119860119879119900(1119901)119875119894 119860119879119888119901119875119894 + 119875119894119860119888119901 119875119894119860119900(119901119888119872) d119860119879119900(1119888119872)119875119894 sdot sdot sdot 119860119879119900(119901119888119872)119875 sdot sdot sdot 119860119879119888119872119875119894 + 119875119894119860119888119872

))))))

= minus119875119894minus1 (26a)

where 119888119901 = 1198881 1198882 119888119872 119894 = 1 2 119873119888 The function119881(119909) = 119909119879119875119873119888119909 is a common Lyapunov function for each ofthe individual system 119909 = 119860119888119909(119905)

(b) Lower triangular is same as upper triangular Thesolution is

((((((

1198601198791198881119875119894 + 1198751198941198601198881 sdot sdot sdot 119860119879119900(1199011)119875119894 sdot sdot sdot 119860119879119900(1198881198721)119875119894 d119875119894119860119900(1119901) 119860119879119888119901119875119894 + 119875119894119860119888119901 119860119879119900(119888119872119901)119875119894 d119875119894119860119900(1119888119872) sdot sdot sdot 119875119894119860119900(119901119888119872) sdot sdot sdot 119860119879119888119872119875119894 + 119875119894119860119888119872

))))))

= minus119875119894minus1 (26b)

(c) Other nonregular modes must satisfy Hurwitz stable onlythis which have ideal stability The solution is

((((((

1198601198791198881119875119894 + 1198751198941198601198881 sdot sdot sdot 119860119879119900(1199011)119875119894 + 119875119894119860119900(1119901) sdot sdot sdot 119860119879119900(1198881198721)119875119894 + 119875119894119860119900(1119888119872) d119860119879119900(1119901)119875119894 + 119875119894119860119900(1119901) 119860119879119888119901119875119894 + 119875119894119860119888119901 119860119879119900(119888119872119901)119875119894 + 119875119894119860119900(119901119888119872) d119860119879119900(1119888119872)119875 + 119875119894119860119900(1119888119872) sdot sdot sdot 119860119879119900(119901119888119872)119875 + 119875119894119860119900(119901119888119872) sdot sdot sdot 119860119879119888119872119875119894 + 119875119894119860119888119872

))))))

= minus119875119894minus1 (26c)

Proof The condition of Ec that is pairwise commutative isdetermined as follows

1198641198881198941198641198881198941015840 = 1198641198881198941015840119864119888119894 = ((((((


))))))

((((((


))))))

(27)

The Hurwitz stability of Ec can be ensured by theframework of upper triangular in which all the orderprincipal minors of matrices are positive if Aci is

Hurwitz stable It satisfies the Hurwitz stable criterionwhich is the same as the framework of the lowertriangular


Thus the two frameworks of a continuous-time multi-switched system are exponentially stable for any arbitraryswitching between the elements of EcThere exists a commonLyapunov function for all the subsystems-groups and subsys-tems In the framework of the upper triangular we modify(14) as follows

(((((

1198601198881 sdot sdot sdot 119860119900(1119901) sdot sdot sdot 119860119900(1119888119872) d0 0 119860119900(119901119888119872) d0 sdot sdot sdot 0 sdot sdot sdot 119860119888119872

)))))

119879

119875119894

+ 119875119894(((((


)))))= minus119875119894minus1

(28)

Obviously (28) transforms into (26a) which implies thesolutions to the Lyapunov equations

It is similar with upper triangular in the framework of thelower triangular we modify (14) as

((((((

1198601198881 sdot sdot sdot 0 sdot sdot sdot 0 d119860119900(1199011) 119860119888119901 0 d119860119900(1198881198721) sdot sdot sdot 119860119900(119888119872119901) sdot sdot sdot 119860119888119872

))))))

119879

119875119894

+ 119875119894((((((


))))))= minus119875119894minus1

(29)

Of course (29) transforms into (26b) which implies thesolutions to the Lyapunov equations

However in other nonregular modes the stability cannotbe guaranteed The framework is nonregular so the EcHurwitz needs to be stable Then the system stability underarbitrary switching is ensured and a common Lyapunovfunction can be gained as (26c)

Remark 24 The above theorem and proof can be extendedto the structure (see (25)) of the discrete-time multiswitchedsystem 119909(119896 + 1) = 119864119889119895119909(119896) with 119895 = 1 119873119889 Remark 17 canbe modified as follows

(a) If the framework of Ed is an upper triangularRemark 17 is true however the solutions are

(((((


)))))

119879

sdot 119875119894(((((


)))))

= 119875119894

minus 119875119894minus1

(30a)

where 119889119902 = 1198891 1198892 119889119872 119895 = 1 2 119873119889 The function119881(119909) = 119909119879119875119873119889119909 is a common Lyapunov function for each ofthe individual systems = 119860119889119909(119905)


((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30b)

(c) Other nonregular modes must satisfy Schur stable whichhas ideal stability The solution is


((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30c)

If the above theorems and remarks extend to mixedswitched system we find it difficult to get the condition ofpairwise commutative whatever the triangular and otherframeworks in E In some special situations it satisfies theassuming condition 119864119888119864119889 = 119864119889119864119888 Then the matrices Ec andEd should better be in upper or lower triangular frameworkto ensure Hurwitz stable and Schur stable respectivelyOtherwise the stable condition becomes strictly to requirematrices E and be unconcerned with matrices A

If all the conditions are satisfied the goal of stabilityunder arbitrary switching can be gained and the function119881(119909) = 119909119879119875119888119873119888119909 is a common Lyapunov function foreach of the individual system = 119860119888119909(119905) and 119909(119896 +1) = 119860119889119902119909(119896) For any positive symmetric definite matrix1198751198890 thus 1198751198891 1198751198892 119875119889119873119889 are the unique positive definitesolutions to the Lyapunov equations

((((((


))))))

119879

sdot 1198751198891((((((


))))))minus 1198751198891 = minus1198751198890

(31a)

((((((


))))))

119879

sdot 119875119889119895((((((


))))))minus 119875119889119895 = minus119875119889119895minus1

(31b)

((((((


))))))

119879

1198751198881

+ 1198751198881((((((


))))))= minus119875119889119873119889

(31c)

((((((


))))))

119879

119888119901

119875119888119894

+ 119875119888119894((((((


))))))= minus119875119888119894minus1

(31d)


fresh air

cold air

room(temperature

Inside)

latentheat

Inside

air conditioningunit mixed air

random heat of occupantsand equipment

heat transferfrom building structure

Figure 6 Thermal balance of air system

5 Numerical Example

In this section an engineering application of central airconditioning is introduced as a numerical example whichshows the framework of multiswitched system with parallelstructure In addition the engineering environment is illus-trated and thermal parameters are given in Table 4 Simula-tion results are presented to illustrate the characteristics ofthe system and the situation of stability by different controlstrategies The mathematical model is given in [27] whichshows the thermal balance of a test room affected by factorssuch as the structure and materials of a building outdoorweather parameters indoor lighting radiating equipmentand number of occupants (see Figure 6)The cooling capacityis transferred from chilled water system to air system via airconditioning units by the measures of constant air volumeand variable water volumeconstant temperature difference

The thermal balance equation is

119862119886119898119886119889120579119889120591 = minus120574120576119862119908Δ120579119902119908 minus (1 minus 119877119903) 119902119904119886119862119886120579+ (1 minus 119877119903) 119902119904119886119862119886120579119900119906119905 + 119876119903119889 + 119876119902119903minus sum119870119895119860119895120579 + sum119870119895119860119895120579119895

(32)

where 120579 is the real-time indoor temperature On the left sideof the equation 119862119886119898119886(119889120579119889120591) means the time differential ofthe heat capacity of a room On the left side of the equationminus120574120576119862119908Δ120579119902119908 means the cooling capacity for chilled watersystem minus(1 minus 119877119903)119902119904119886119862119886120579 and (1 minus 119877119903)119902119904119886119862119886120579119900119906119905 represent thecooling capacity from return and fresh air systems respec-tively119876119903119889 denotes random heat of occupants and equipment119876119902119903 means latent heat inside minussum119870119895119860119895120579 + sum119870119895119860119895120579119895 is theheat transfer from building structure The description of thesymbols is presented in Table 4

A midsize conference room (length 10 m width 6 mand height 3 m) is simulated we use two different controlstrategies (the strategies 1 and 2mentioned in Examples 6 and7 respectively) to adjust cooling capacity and illustrate systemstability (corresponding Propositions 15 and 18 respectively)in the framework of multiswitching system with parallelstructure and use strategy 3 to reflect the unstable situation

pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)

Figure 7 Flow volume under strategy 1

The above three control strategies and the two types of pumpsare shown in Table 1 The range of variable volume is 50-100 and the time is divided into three intervals ([0 5min][5min 12min] and [12min 15min]) in the above threecontrol strategiesThe cold air is sent to the room for reducingthe indoor temperature The indoor temperature is requiredto be loweredwith respect to the initial temperature (1205790 30∘C)and regulated at (120579set 26∘C) as soon as possible In the last3 minutes the indoor cooling load increased significantlydue to that the number of indoor participants increased (seeFigure 10) The outdoor temperature is basically maintainedat 30∘C in the simulated 15 minutes

Figure 7 shows the switching dynamics of a continuous-time multiswitched system with parallel structure by the flowvolume of the three pumps In the first time interval all thethree pumps work as a subsystems-group in rated volume forreducing the temperature in the middle time interval only


Table 1 Control strategies and types of pumps

strategy pump working type control mode feedback coefficient switching state

1A variable volume feedbackswitching 00074 ONOFFB variable volume feedbackswitching NULL ONOFFC variable volume feedbackswitching 0021 ON

2A fixed volume switching NULL ONOFFB fixed volume switching NULL ONOFFC variable volume feedback 0019 ON


Table 2

pump rated volumeq0 (kgs)

range of variablevolumeq (kgs)

A 001 0005-001B 002 0015-002C 0033 00175-0033total 0063 00315-0063

Table 3



A 001 001-001B 002 002-002C 0033 00175-0033total 0063 00315-0063

the pump C works in variable flow mode under the feedbackcontrol in the last time interval both pump A and pumpC work as a subsystems-group under feedback control forcreasing cooling capacity because of the increase of indoorparticipants increased

Figure 8 shows the switching of a mixed multiswitchedsystem with parallel structure composed of one continuous-time subsystem and two discrete-time subsystems In thefirst two time intervals the dynamics of pumps is the sameas Figure 7 In the last time interval pump A works at therated volume as a discrete-time subsystem and the pump Cworks in variable flow mode under feedback control as acontinuous-time subsystem in other words the subsystems-group is composed with one continuous-time and onediscrete-time subsystems Figure 9 is similar to Figure 8but the switching dynamics is different (in the middle timeinterval all the three pumps work together in the last timeinterval only pump C works)

Figure 11 shows the changes of indoor temperature underthe three control strategies The indoor temperature dropsfrom the initial value (30∘C) to the set point (26∘C) in fiveminutes under the three strategies because of rated volumeby thewholewater system It isworth noting that in the last 10

pump Apump B

pump Ctotal

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)

100 200 300 400 500 600 700 800 9000t (second)


minutes the indoor temperature is always stable near the setpoint under the ideal strategies (the strategies 1 and 2) even ifthe indoor cooling load changes significantly but under thestrategy 3 the indoor temperature expresses lower and higherinstable rates in the last two time intervals respectively

6 Conclusion

In this paper a type of linear multiswitched system withparallel structure was proposed and the framework and aswitching unit were introduced Based on this various actualengineering applications were shown which illustrated theproperties of the system and differences with traditionalswitched systems Next the stability property for a typeof linear multiswitched system with parallel structure isstudied whether in continuous-time discrete-time or amixed situation A subsystems-group as a basic switchedunit instead of subsystem is proposed the matrices of whichare pairwise commutative based on some given conditionsof subsystems When all the subsystems are Hurwitz and


Table 4

Parameter Value Description119898119886 (kg) 23218 indoor air mass119902119908 (kgs) 0149 rated volume of water system119902119886 (kgs) 0022 rated volume of pump 119860119902119887 (kgs) 0044 rated volume of pump 119861119902119888 (kgs) 0083 rated volume of pump 119862119902119904119886 (kgs) 3003 volume of sending air1198601 (m2) 56 area of walls1198602 (m2) 28 area of windows1198603 (m2) 0 area of roof119862119886 (Jkglowastk) 1010 specific heat of air119862119908 (Jkglowastk) 4180 specific heat of water1198701 (Wm2lowastk) 0049 heat transfer coefficient of walls1198702 (Wm2lowastk) 0051 heat transfer coefficient of windows1198703 (Wm2lowastk) 005 heat transfer coefficient of roof119876119902119903 (J) 20 latent heat load119877119903 011 return air rateΔ120579 (∘C) 5 temperature difference120576 089 transfer efficiency from water system to air system120574 0095 coefficient of cooling capacity allocation120579119894119899119894 (∘C) 30 30 30 initial temperature120579119895 (∘C) 35 35 36 temperature of walls windows and roof respectively120579119904119890119905 (∘C) 26 setting temperature

pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)


Schur stable there exists a common Lyapunov function for allthe subsystems and subsystems-groups Then the switchedsystem is exponentially stable for any arbitrary switchingbetween the subsystems-groups The results are extended toa parallel-like structure to obtain more ideal consequence of

0

10

20

30

40

50

60

70

80

90

100

Q (K

J)

100 200 300 400 500 600 700 800 9000t (second)

Figure 10 Change of cooling load

stability A simulation example for refrigeration engineeringapplication of the system is introduced as last which showsthe characteristics of the framework and stability

Data Availability

The data used to support the findings of this study areincluded within the article


strategy 1strategy 2strategy 3

25

255

26

265

27

275

28

285

29

295

30

tem

pera

ture

insid

e (∘

C)

100 200 300 400 500 600 700 800 9000t (second)

Figure 11 Indoor temperature under different strategies

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (no 61104181)

References

[1] G Zhai H Lin A N Michel and K Yasuda ldquoStability analysisfor switched systems with continuous-time and discrete-timesubsystemsrdquo in Proceedings of the 2004 American ControlConference (AAC) pp 4555ndash4560 July 2004

[2] H Lin and P J Antsaklis ldquoStability and stabilizability ofswitched linear systems a survey of recent resultsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 308ndash3222009

[3] Z-E Lou and J Zhao ldquoStabilisation for a class of switchednonlinear systems and its application to aero-enginesrdquo IETControl Theory amp Applications vol 11 no 2 pp 237ndash244 2017

[4] Z Sun and S S Ge Stability Theory of Switched DynamicalSystems Springer London UK 2011

[5] D Liberzon Switching in Cystems and Control BirkhauserBoston Mass USA 2003

[6] R Shorten D Leith J Foy and R Kilduff ldquoTowards an analysisand design framework for congestion control in communica-tion networksrdquo in Proceedings of the 12th Yale Workshop onAdaptive and Learning Systems 2003

[7] R Shorten FWirth OMason KWulff and C King ldquoStabilitycriteria for switched and hybrid systemsrdquo SIAMReview vol 49no 4 pp 545ndash592 2007

[8] N H El-Farra and P D Christofides ldquoCoordinating feedbackand switching for control of spatially distributed processesrdquo

Computers amp Chemical Engineering vol 28 no 1-2 pp 111ndash1282004

[9] J Jiang K Song and Z Li ldquoSystem Modeling and SwitchingControl Strategy of Wireless Power Transfer Systemrdquo IEEEJournal of Emerging amp Selected Topics in Power Electronics vol1-1 Article ID 99 2018

[10] L Zhang S Zhuang and R D Braatz ldquoSwitched modelpredictive control of switched linear systems feasibility stabilityand robustnessrdquo Automatica vol 67 pp 8ndash21 2016

[11] X Liu S Li and K Zhang ldquoOptimal control of switching timein switched stochastic systems with multi-switching times anddifferent costsrdquo International Journal of Control vol 90 no 8pp 1604ndash1611 2017

[12] J Zhai T Niu J Ye and E Feng ldquoOptimal control of nonlinearswitched system with mixed constraints and its parallel opti-mization algorithmrdquo Nonlinear Analysis Hybrid Systems vol25 pp 21ndash40 2017

[13] D Liberzon and A S Morse ldquoBasic problems in stability anddesign of switched systemsrdquo IEEE Control Systems Magazinevol 19 no 5 pp 59ndash70 1999

[14] K S Narendra and J A Balakrishnan ldquoA common Lyapunovfunction for stable LTI systems with commuting A-matricesrdquoIEEE Transactions on Automatic Control vol 39 no 12 pp2469ndash2471 1994

[15] T Buyukkoroglu O Esen and V Dzhafarov ldquoCommon Lya-punov functions for some special classes of stable systemsrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 56 no 8 pp 1963ndash1967 2011

[16] R A Decarlo M S Branicky S Pettersson and B LennartsonldquoPerspectives and results on the stability and stabilizability ofhybrid systemsrdquo Proceedings of the IEEE vol 88 no 7 pp 1069ndash1082 2000

[17] A N Michel ldquoRecent trends in the stability analysis of hybriddynamical systemsrdquo IEEE Transactions on Circuits and SystemsI Fundamental Theory and Applications vol 46 no 1 pp 120ndash134 1999

[18] L Long and J Zhao ldquoAn integral-type multiple Lyapunovfunctions approach for switched nonlinear systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 61 no 7 pp 1979ndash1986 2016

[19] J P Hespanha ldquoChapter stabilization through hybrid controlrdquoEncyclopedia of Life Support Systems (EOLSS) 2004

[20] D Liberzon J P Hespanha and A S Morse ldquoStability ofswitched systems a Lie-algebraic conditionrdquo Systems amp ControlLetters vol 37 no 3 pp 117ndash122 1999

[21] A Sakly and M Kermani ldquoStability and stabilization studiesfor a class of switched nonlinear systems via vector normsapproachrdquo ISA Transactions 2014

[22] G Zhai and H Lin ldquoController failure time analysis for sym-metric Hinfincontrol systemsrdquo International Journal of Controlvol 77 no 6 pp 598ndash605 2004

[23] G Zhai X Xu H Lin and A Michel ldquoAnalysis and design ofswitched normal systemsrdquo Nonlinear Analysis Theory Methodsamp Applications An International Multidisciplinary Journal vol65 no 12 pp 2248ndash2259 2006

[24] A A Agrachev and D Liberzon ldquoLie-algebraic stability criteriafor switched systemsrdquo SIAM Journal on Control and Optimiza-tion vol 40 no 1 pp 253ndash269 2001

[25] J L Mancilla-Aguilar ldquoA condition for the stability of switchednonlinear systemsrdquo Institute of Electrical and Electronics Engi-neers Transactions on Automatic Control vol 45 no 11 pp2077ndash2079 2000


[26] R N Shorten and K S Narendra ldquoNecessary and sufficientconditions for the existence of a common quadratic Lyapunovfunction for M stable second order linear time-invariant sys-temsrdquo in Proceedings of the 2000 American Control Conferencepp 359ndash363 June 2000

[27] Yan Zhang Yongqiang Liu and Yang Liu ldquoAHybrid DynamicalModelling and Control Approach for Energy Saving of CentralAir Conditioningrdquo Mathematical Problems in Engineering vol2018 Article ID 6389438 12 pages 2018

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where x(t) x(k) are the states t is the time scalar and 120591 is pos-itive integer of the sample period Eci and Edj are nonnull con-stant matrices with appropriate dimensions which are com-bined by some constant matrices Acp and Adq respectivelySubsystems-groups (1) and (2) are composed of continuous-time subsystems (119905) = 119860119888119901119909(119905) with 119888119901 = 1198881 119888119872and discrete-time subsystems 119909(119896 + 1) = 119860119889119902119909(119896) with119889119902 = 1198891 119889119872 respectively The system switches betweenmatrices 1198641198881 1198641198882 119864119888119873119888 1198641198891 1198641198892 119864119889119873119889 belonging tothe set E 119864 ≜ 119864119888 119864119889 119864119888 ≜ 1198641198881 1198641198882 119864119888119873119888 and119864119889 ≜ 1198641198891 1198641198892 119864119889119873119889 1198601198881 1198601198882 119860119888119872 belong to theset 119860119888 ≜ 1198601198881 1198601198882 119860119888119872 1198601198891 1198601198892 119860119889119872 belong tothe set 119860119889 ≜ 1198601198891 1198601198892 119860119889119872 and 119860 ≜ 119860119889 119860119888 119860 and119864 are finite sets

Themajor difference between amultiswitched system anda traditional switched system is that the former selects anyamount of subsystems and the latter selects only one eachtime There is a structure for organizing the subsystems asa subsystems-group on each interval that includes mutualindependence between subsystemsMany different structurescan be considered such as parallel tandem and other mixedmodes Equations (2) and (4) show the parallel structure ofthe matrices However the traditional switching system hasonly one matrix without any structure

A multiswitched system is widely used in engineeringcases such as in the chilled water system in central airconditioning The dynamical pumping process is a kind oftypical multiswitch system and each different pump as asubsystemwith switching and variable frequency behavioursFor variable flow control there are several different pumpsworking together as a subsystem-group at each time periodAccording to the water pipe network the pumps are parallelin structure Thus this physical model illustrates the pro-posed system In this paper only parallel and parallel-likestructures are researched

Modelling [9] performance analysis [10] and optimalcontrol [11 12] are the key issues of switched systemsObviously stability analysis is a very important researchbranch that has attracted the attention of researchers globallyStability is the most basic property and the primary issueto be solved by the control system The stability problemsof switched systems are reduced to the following three basicissues [13] finding conditions to guarantee asymptoticalstability for any switching signal identifying some switchingsignals for asymptotical stability and constructing a switch-ing signal to make the system asymptotically stable Accord-ingly some excellent theoretical methods are proposed forsolving those problems such as common Lyapunov functions[14 15] multiple Lyapunov functions [16ndash18] dwell timeand average dwell time [19 20] and piecewise quadraticLyapunov functions [16]

It is worth noting that stability under arbitrary switchingis a fundamental in the design and analysis of switchedsystems [21] It is known that all the subsystems that are expo-nentially stable are not sufficient to guarantee stability underarbitrary switching expect for some reasonable assumptionssuch as the state matrices of subsystems commute pairwise[2 14] (eg 119860 119894119860119895 = 119860119895119860 119894 for all 119894 119895 isin 119868 119894 = 119895 119860119888119894119860119889119895 =119860119889119895119860119888119894 for all 119894 119895 isin 119868 c and d denote continuous-time and

discrete-time subsystems respectively) the state matricesof subsystems are symmetric [22] (eg 119860 119894 = 119860 119894119879 for all119894 isin 119868) the state matrices of subsystems are normal [23](eg 119860119888119894119879119860119888119894 = 119860119888119894119860119888119894119879 for all 119894 isin 119868 and 119860119889119895119879119860119889119895 =119860119889119895119860119889119895119879 for all 119895 isin 119868 c and d denote the same substancesabove) Thus some researchers present a sufficient conditionin terms of the Lie algebra If this Lie algebra is solvablethe exponential stability of the switched system for arbitraryswitching can be gained [20] A more general result isthat the semisimple subalgebra S is a compact Lie algebraby considered Levi decomposition and the exponentiallystability can be ensured [24] Previous research shows that thepairwise commutation of the vector fields is a sufficient con-dition for the stability [25] By using Hurwitz stable matrixpencils 120590120572[1198601 1198602] and 120590120572[1198601 1198602minus1] a common quadraticLyapunov function exists that can guarantee stability underarbitrary switching [26] Based on previous research themain emphasis is placed on the conditions of stability analysisand a common Lyapunov function is constructed under anassumption

In this paper a type of linear multiswitched systemwith parallel structure is put forward for the first time Thecomposition of the system is expounded and the structureand form of the subsystems-group are described From theswitching law and the physical meaning we illustrate thedifference between the system and the traditional switchingsystem A case of chilled water system pumps in centralair conditioning shows the engineering significance of thesystem Next stability under arbitrary switching for this typeof system is studied The inference of properties of statematrices of the subsystems-group is derived from the subsys-tems Then based on the conditions of Hurwitz stable andSchur stable for continuous-time subsystems and discrete-time subsystems respectively the stability of subsystems-groups can be obtained The matrices 119864119888 and 119864119889 of thesubsystems-groups commute pairwise can be ensured basedon the assumptions of the state matrices 119860119888 and 119860119889 [2 14]Thus the stability under arbitrary switching is guaranteedand a commonLyapunov function is given for all subsystems-groups and subsystems Finally the result is extended toa parallel-like structure which reduces conservativeness instability analysis

The body of this paper is organized as follows in Sec-tion 2 the system description the example of the multi-switching system and the preliminaries are presented InSection 3 the main stability results in the continuous-timesystem and the mixed system are expressed two examples arealso given to illustrate the results In Section 4 the studiesextended to parallel-like structure is presented Section 5concludes the paper

2 System Descriptionand Mathematical Preliminaries

21 System Description and Examples Throughout the fol-lowing notation is adopted R and C denote the fields ofreal and complex number respectively R119899 denotes the n-dimensional real Euclidean space R119899times119899 denotes the space of






119864 = 1198641 1198642 1198643 1198644 1198645 1198646 1198647 = [[[1198601 0 00 0 00 0 0]]]

[[[0 0 00 1198602 00 0 0]]] [[[

0 0 00 0 00 0 1198603]]] [[[

1198601 0 00 1198602 00 0 0]]] [[[1198601 0 00 0 00 0 1198603

]]] [[[0 0 00 1198602 00 0 1198603

]]] [[[1198601 0 00 1198602 00 0 1198603

]]]

(5)




show 1198641 = [ 1198601 0 00 1198602 00 0 1198603

] = [ 1198601 0 00 0 00 0 0



M

M

evaporator

pumps


chilled waterCycle



0

001

002

003

004

005

006

q (k

gs)

100 200 300 400 500 600 700 800 9000time (s)





pump Apump Bpump C

100 200 300 400 500 600 700 800 9000time (s)

minus001

0

001

002

003

004

005

006

q (k

gs)






100 200 300 400 500 600 700 800 9000time (s)

0

001

002

003

004

005

006

q (k

gs)


pump Apump Bpump C

minus001

0

001

002

003

004

005

006

q (k

gs)

100 200 300 400 500 600 700 800 9000time (s)








119860119879119888119901119875119888119901 + 119875119888119901119860119888119901 = minus119875119888119901minus1 119888119901 = 1198881 1198882 119888119872 (6)








119860119879119888119901119875119888119901 + 119875119888119901119860119888119901 = minus119875119888119901minus1 (9)

11986011987911988811198751198881 + 11987511988811198601198891 = minus119875119889119872 (10)





3 Main Results




We get

1198641198881198941198641198881198941015840 = 1198641198881198941015840119864119888119894

= (((((


)))))

(((((


)))))

(11)


119864119888119864119889 = ((


))

((


))

= ((


))

= ((((


))))

= 119864119889119864119888(12)












(13)



(((((


)))))

119879

119875119894

+ 119875119894(((((


)))))

= minus119875119894minus1(15)



Let 119864119888 = 1198641 1198642 1198643 1198641 = [1198601198881] 1198642 = [ 1198601198882 00 1198601198883 ] and1198643 = [ 1198601198883 0 00 1198601198884 00 0 1198601198885



11986411987911988811198751 + 11987511198641198881 = minus1198751198880 (16a)

11986411987911988821198752 + 11987521198641198882 = minus1198751198881 (16b)

11986411987911988831198753 + 11987531198641198883 = minus1198751198882 (16c)


11986011987911988821198752 + 11987521198601198882 + 11986011987911988831198752 + 11987521198601198883 = minus1198751198881 (17b)

11986011987911988831198753 + 119875311986034 + 11986011987911988841198753 + 11987531198601198884 + 11986011987911988851198753 + 11987531198601198885= minus1198751198882 (17c)















119864119879119889111987511988911198641198891 minus 1198751198891 = minus1198751198890 (20a)


11986411987911988811198751198881 + 11987511988811198641198891 = minus119875119889119873119889 (20c)

119864119879119888119894119875119888119894 + 119875119888119894119864119888119894 = minus119875119888119894minus1 (20d)


(((((


)))))

119879

sdot 1198751198891(((((


)))))

minus 1198751198891

= minus1198751198890(21a)

(((((


)))))

119879

sdot 119875119889119895(((((


)))))

minus 119875119889119895

= minus119875119889119895minus1

(21b)

(((((


)))))

119879

1198751198881

+ 1198751198881(((((


)))))

= minus119875119889119873119889

(21c)

((((((


))))))

119879

119888119901

119875119888119894


+ 119875119888119894(((((


)))))

= minus119875119888119894minus1(21d)



Let 119864119888 = 1198641198881 1198641198882 1198641198883 1198641198881 = [ 1198601198881 00 1198601198883 ] 1198641198882 = [ 1198601198882 00 1198601198883 ]and 1198641198883 = [ 1198601198881 0 00 1198601198882 0

0 0 1198601198883] let 119864119889 = 1198641198891 1198641198892 1198641198891 = [1198601198892] and1198641198892 = [ 1198601198891 00 1198601198892 ]


119864119879119889111987511988911198641198891 minus 1198751198891 = minus1198751198890 (22a)

119864119879119889211987511988921198641198892 minus 1198751198892 = minus1198751198891 (22b)

11986411987911988811198751198881 + 11987511988811198641198891 = minus1198751198892 (22c)

11986411987911988821198751198882 + 11987511988821198641198882 = minus1198751198881 (22d)

11986411987911988831198751198883 + 11987511988831198641198883 = minus1198751198882 (22e)


119860119879119889211987511988911198601198892 minus 1198751198891 = minus1198751198890 (23a)

119860119879119889111987511988921198601198891 + 119860119879119889211987511988921198601198892 minus 1198751198892 = minus1198751198891 (23b)

11986011987911988811198751198881 + 11986011987911988821198751198881 + 11987511988811198601198892 = minus1198751198892 (23c)

11986011987911988821198751198882 + 11987511988821198601198882 + 11986011987911988831198751198882 + 11987511988821198601198883 = minus1198751198881 (23d)

11986011987911988811198751198883 + 11987511988831198601198881 + 11986011987911988821198751198883 + 11987511988831198601198882 + 11986011987911988831198751198883+ 11987511988831198601198883 = minus1198751198882 (23e)




119864119888119894 = (((((((


)))))))

(24)



119864119889119895 = (((((((


)))))))

(25)





((((((


))))))

= minus119875119894minus1 (26a)



((((((


))))))

= minus119875119894minus1 (26b)


((((((


))))))

= minus119875119894minus1 (26c)


1198641198881198941198641198881198941015840 = 1198641198881198941015840119864119888119894 = ((((((


))))))

((((((


))))))

(27)





(((((


)))))

119879

119875119894

+ 119875119894(((((


)))))= minus119875119894minus1

(28)



((((((


))))))

119879

119875119894

+ 119875119894((((((


))))))= minus119875119894minus1

(29)





(((((


)))))

119879

sdot 119875119894(((((


)))))

= 119875119894


(30a)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30b)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30c)



((((((


))))))

119879

sdot 1198751198891((((((


))))))minus 1198751198891 = minus1198751198890

(31a)

((((((


))))))

119879

sdot 119875119889119895((((((



(31b)

((((((


))))))

119879

1198751198881

+ 1198751198881((((((


))))))= minus119875119889119873119889

(31c)

((((((


))))))

119879

119888119901

119875119888119894

+ 119875119888119894((((((


))))))= minus119875119888119894minus1

(31d)


fresh air

cold air

room(temperature

Inside)

latentheat

Inside





5 Numerical Example




(32)



pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)










Table 2



A 001 0005-001B 002 0015-002C 0033 00175-0033total 0063 00315-0063

Table 3



A 001 001-001B 002 002-002C 0033 00175-0033total 0063 00315-0063




pump Apump B

pump Ctotal

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)

100 200 300 400 500 600 700 800 9000t (second)



6 Conclusion



Table 4


pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)



0

10

20

30

40

50

60

70

80

90

100

Q (K

J)

100 200 300 400 500 600 700 800 9000t (second)



Data Availability




25

255

26

265

27

275

28

285

29

295

30

tem

pera

ture

insid

e (∘

C)

100 200 300 400 500 600 700 800 9000t (second)




Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018













119864 = 1198641 1198642 1198643 1198644 1198645 1198646 1198647 = [[[1198601 0 00 0 00 0 0]]]

[[[0 0 00 1198602 00 0 0]]] [[[

0 0 00 0 00 0 1198603]]] [[[

1198601 0 00 1198602 00 0 0]]] [[[1198601 0 00 0 00 0 1198603

]]] [[[0 0 00 1198602 00 0 1198603

]]] [[[1198601 0 00 1198602 00 0 1198603

]]]

(5)




show 1198641 = [ 1198601 0 00 1198602 00 0 1198603

] = [ 1198601 0 00 0 00 0 0



M

M

evaporator

pumps


chilled waterCycle



0

001

002

003

004

005

006

q (k

gs)

100 200 300 400 500 600 700 800 9000time (s)





pump Apump Bpump C

100 200 300 400 500 600 700 800 9000time (s)

minus001

0

001

002

003

004

005

006

q (k

gs)






100 200 300 400 500 600 700 800 9000time (s)

0

001

002

003

004

005

006

q (k

gs)


pump Apump Bpump C

minus001

0

001

002

003

004

005

006

q (k

gs)

100 200 300 400 500 600 700 800 9000time (s)








119860119879119888119901119875119888119901 + 119875119888119901119860119888119901 = minus119875119888119901minus1 119888119901 = 1198881 1198882 119888119872 (6)








119860119879119888119901119875119888119901 + 119875119888119901119860119888119901 = minus119875119888119901minus1 (9)

11986011987911988811198751198881 + 11987511988811198601198891 = minus119875119889119872 (10)





3 Main Results




We get

1198641198881198941198641198881198941015840 = 1198641198881198941015840119864119888119894

= (((((


)))))

(((((


)))))

(11)


119864119888119864119889 = ((


))

((


))

= ((


))

= ((((


))))

= 119864119889119864119888(12)












(13)



(((((


)))))

119879

119875119894

+ 119875119894(((((


)))))

= minus119875119894minus1(15)



Let 119864119888 = 1198641 1198642 1198643 1198641 = [1198601198881] 1198642 = [ 1198601198882 00 1198601198883 ] and1198643 = [ 1198601198883 0 00 1198601198884 00 0 1198601198885



11986411987911988811198751 + 11987511198641198881 = minus1198751198880 (16a)

11986411987911988821198752 + 11987521198641198882 = minus1198751198881 (16b)

11986411987911988831198753 + 11987531198641198883 = minus1198751198882 (16c)


11986011987911988821198752 + 11987521198601198882 + 11986011987911988831198752 + 11987521198601198883 = minus1198751198881 (17b)

11986011987911988831198753 + 119875311986034 + 11986011987911988841198753 + 11987531198601198884 + 11986011987911988851198753 + 11987531198601198885= minus1198751198882 (17c)















119864119879119889111987511988911198641198891 minus 1198751198891 = minus1198751198890 (20a)


11986411987911988811198751198881 + 11987511988811198641198891 = minus119875119889119873119889 (20c)

119864119879119888119894119875119888119894 + 119875119888119894119864119888119894 = minus119875119888119894minus1 (20d)


(((((


)))))

119879

sdot 1198751198891(((((


)))))

minus 1198751198891

= minus1198751198890(21a)

(((((


)))))

119879

sdot 119875119889119895(((((


)))))

minus 119875119889119895

= minus119875119889119895minus1

(21b)

(((((


)))))

119879

1198751198881

+ 1198751198881(((((


)))))

= minus119875119889119873119889

(21c)

((((((


))))))

119879

119888119901

119875119888119894


+ 119875119888119894(((((


)))))

= minus119875119888119894minus1(21d)



Let 119864119888 = 1198641198881 1198641198882 1198641198883 1198641198881 = [ 1198601198881 00 1198601198883 ] 1198641198882 = [ 1198601198882 00 1198601198883 ]and 1198641198883 = [ 1198601198881 0 00 1198601198882 0

0 0 1198601198883] let 119864119889 = 1198641198891 1198641198892 1198641198891 = [1198601198892] and1198641198892 = [ 1198601198891 00 1198601198892 ]


119864119879119889111987511988911198641198891 minus 1198751198891 = minus1198751198890 (22a)

119864119879119889211987511988921198641198892 minus 1198751198892 = minus1198751198891 (22b)

11986411987911988811198751198881 + 11987511988811198641198891 = minus1198751198892 (22c)

11986411987911988821198751198882 + 11987511988821198641198882 = minus1198751198881 (22d)

11986411987911988831198751198883 + 11987511988831198641198883 = minus1198751198882 (22e)


119860119879119889211987511988911198601198892 minus 1198751198891 = minus1198751198890 (23a)

119860119879119889111987511988921198601198891 + 119860119879119889211987511988921198601198892 minus 1198751198892 = minus1198751198891 (23b)

11986011987911988811198751198881 + 11986011987911988821198751198881 + 11987511988811198601198892 = minus1198751198892 (23c)

11986011987911988821198751198882 + 11987511988821198601198882 + 11986011987911988831198751198882 + 11987511988821198601198883 = minus1198751198881 (23d)

11986011987911988811198751198883 + 11987511988831198601198881 + 11986011987911988821198751198883 + 11987511988831198601198882 + 11986011987911988831198751198883+ 11987511988831198601198883 = minus1198751198882 (23e)




119864119888119894 = (((((((


)))))))

(24)



119864119889119895 = (((((((


)))))))

(25)





((((((


))))))

= minus119875119894minus1 (26a)



((((((


))))))

= minus119875119894minus1 (26b)


((((((


))))))

= minus119875119894minus1 (26c)


1198641198881198941198641198881198941015840 = 1198641198881198941015840119864119888119894 = ((((((


))))))

((((((


))))))

(27)





(((((


)))))

119879

119875119894

+ 119875119894(((((


)))))= minus119875119894minus1

(28)



((((((


))))))

119879

119875119894

+ 119875119894((((((


))))))= minus119875119894minus1

(29)





(((((


)))))

119879

sdot 119875119894(((((


)))))

= 119875119894


(30a)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30b)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30c)



((((((


))))))

119879

sdot 1198751198891((((((


))))))minus 1198751198891 = minus1198751198890

(31a)

((((((


))))))

119879

sdot 119875119889119895((((((



(31b)

((((((


))))))

119879

1198751198881

+ 1198751198881((((((


))))))= minus119875119889119873119889

(31c)

((((((


))))))

119879

119888119901

119875119888119894

+ 119875119888119894((((((


))))))= minus119875119888119894minus1

(31d)


fresh air

cold air

room(temperature

Inside)

latentheat

Inside





5 Numerical Example




(32)



pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)










Table 2



A 001 0005-001B 002 0015-002C 0033 00175-0033total 0063 00315-0063

Table 3



A 001 001-001B 002 002-002C 0033 00175-0033total 0063 00315-0063




pump Apump B

pump Ctotal

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)

100 200 300 400 500 600 700 800 9000t (second)



6 Conclusion



Table 4


pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)



0

10

20

30

40

50

60

70

80

90

100

Q (K

J)

100 200 300 400 500 600 700 800 9000t (second)



Data Availability




25

255

26

265

27

275

28

285

29

295

30

tem

pera

ture

insid

e (∘

C)

100 200 300 400 500 600 700 800 9000t (second)




Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018









pump Apump Bpump C

100 200 300 400 500 600 700 800 9000time (s)

minus001

0

001

002

003

004

005

006

q (k

gs)






100 200 300 400 500 600 700 800 9000time (s)

0

001

002

003

004

005

006

q (k

gs)


pump Apump Bpump C

minus001

0

001

002

003

004

005

006

q (k

gs)

100 200 300 400 500 600 700 800 9000time (s)








119860119879119888119901119875119888119901 + 119875119888119901119860119888119901 = minus119875119888119901minus1 119888119901 = 1198881 1198882 119888119872 (6)








119860119879119888119901119875119888119901 + 119875119888119901119860119888119901 = minus119875119888119901minus1 (9)

11986011987911988811198751198881 + 11987511988811198601198891 = minus119875119889119872 (10)





3 Main Results




We get

1198641198881198941198641198881198941015840 = 1198641198881198941015840119864119888119894

= (((((


)))))

(((((


)))))

(11)


119864119888119864119889 = ((


))

((


))

= ((


))

= ((((


))))

= 119864119889119864119888(12)












(13)



(((((


)))))

119879

119875119894

+ 119875119894(((((


)))))

= minus119875119894minus1(15)



Let 119864119888 = 1198641 1198642 1198643 1198641 = [1198601198881] 1198642 = [ 1198601198882 00 1198601198883 ] and1198643 = [ 1198601198883 0 00 1198601198884 00 0 1198601198885



11986411987911988811198751 + 11987511198641198881 = minus1198751198880 (16a)

11986411987911988821198752 + 11987521198641198882 = minus1198751198881 (16b)

11986411987911988831198753 + 11987531198641198883 = minus1198751198882 (16c)


11986011987911988821198752 + 11987521198601198882 + 11986011987911988831198752 + 11987521198601198883 = minus1198751198881 (17b)

11986011987911988831198753 + 119875311986034 + 11986011987911988841198753 + 11987531198601198884 + 11986011987911988851198753 + 11987531198601198885= minus1198751198882 (17c)















119864119879119889111987511988911198641198891 minus 1198751198891 = minus1198751198890 (20a)


11986411987911988811198751198881 + 11987511988811198641198891 = minus119875119889119873119889 (20c)

119864119879119888119894119875119888119894 + 119875119888119894119864119888119894 = minus119875119888119894minus1 (20d)


(((((


)))))

119879

sdot 1198751198891(((((


)))))

minus 1198751198891

= minus1198751198890(21a)

(((((


)))))

119879

sdot 119875119889119895(((((


)))))

minus 119875119889119895

= minus119875119889119895minus1

(21b)

(((((


)))))

119879

1198751198881

+ 1198751198881(((((


)))))

= minus119875119889119873119889

(21c)

((((((


))))))

119879

119888119901

119875119888119894


+ 119875119888119894(((((


)))))

= minus119875119888119894minus1(21d)



Let 119864119888 = 1198641198881 1198641198882 1198641198883 1198641198881 = [ 1198601198881 00 1198601198883 ] 1198641198882 = [ 1198601198882 00 1198601198883 ]and 1198641198883 = [ 1198601198881 0 00 1198601198882 0

0 0 1198601198883] let 119864119889 = 1198641198891 1198641198892 1198641198891 = [1198601198892] and1198641198892 = [ 1198601198891 00 1198601198892 ]


119864119879119889111987511988911198641198891 minus 1198751198891 = minus1198751198890 (22a)

119864119879119889211987511988921198641198892 minus 1198751198892 = minus1198751198891 (22b)

11986411987911988811198751198881 + 11987511988811198641198891 = minus1198751198892 (22c)

11986411987911988821198751198882 + 11987511988821198641198882 = minus1198751198881 (22d)

11986411987911988831198751198883 + 11987511988831198641198883 = minus1198751198882 (22e)


119860119879119889211987511988911198601198892 minus 1198751198891 = minus1198751198890 (23a)

119860119879119889111987511988921198601198891 + 119860119879119889211987511988921198601198892 minus 1198751198892 = minus1198751198891 (23b)

11986011987911988811198751198881 + 11986011987911988821198751198881 + 11987511988811198601198892 = minus1198751198892 (23c)

11986011987911988821198751198882 + 11987511988821198601198882 + 11986011987911988831198751198882 + 11987511988821198601198883 = minus1198751198881 (23d)

11986011987911988811198751198883 + 11987511988831198601198881 + 11986011987911988821198751198883 + 11987511988831198601198882 + 11986011987911988831198751198883+ 11987511988831198601198883 = minus1198751198882 (23e)




119864119888119894 = (((((((


)))))))

(24)



119864119889119895 = (((((((


)))))))

(25)





((((((


))))))

= minus119875119894minus1 (26a)



((((((


))))))

= minus119875119894minus1 (26b)


((((((


))))))

= minus119875119894minus1 (26c)


1198641198881198941198641198881198941015840 = 1198641198881198941015840119864119888119894 = ((((((


))))))

((((((


))))))

(27)





(((((


)))))

119879

119875119894

+ 119875119894(((((


)))))= minus119875119894minus1

(28)



((((((


))))))

119879

119875119894

+ 119875119894((((((


))))))= minus119875119894minus1

(29)





(((((


)))))

119879

sdot 119875119894(((((


)))))

= 119875119894


(30a)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30b)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30c)



((((((


))))))

119879

sdot 1198751198891((((((


))))))minus 1198751198891 = minus1198751198890

(31a)

((((((


))))))

119879

sdot 119875119889119895((((((



(31b)

((((((


))))))

119879

1198751198881

+ 1198751198881((((((


))))))= minus119875119889119873119889

(31c)

((((((


))))))

119879

119888119901

119875119888119894

+ 119875119888119894((((((


))))))= minus119875119888119894minus1

(31d)


fresh air

cold air

room(temperature

Inside)

latentheat

Inside





5 Numerical Example




(32)



pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)










Table 2



A 001 0005-001B 002 0015-002C 0033 00175-0033total 0063 00315-0063

Table 3



A 001 001-001B 002 002-002C 0033 00175-0033total 0063 00315-0063




pump Apump B

pump Ctotal

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)

100 200 300 400 500 600 700 800 9000t (second)



6 Conclusion



Table 4


pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)



0

10

20

30

40

50

60

70

80

90

100

Q (K

J)

100 200 300 400 500 600 700 800 9000t (second)



Data Availability




25

255

26

265

27

275

28

285

29

295

30

tem

pera

ture

insid

e (∘

C)

100 200 300 400 500 600 700 800 9000t (second)




Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018












119860119879119888119901119875119888119901 + 119875119888119901119860119888119901 = minus119875119888119901minus1 119888119901 = 1198881 1198882 119888119872 (6)








119860119879119888119901119875119888119901 + 119875119888119901119860119888119901 = minus119875119888119901minus1 (9)

11986011987911988811198751198881 + 11987511988811198601198891 = minus119875119889119872 (10)





3 Main Results




We get

1198641198881198941198641198881198941015840 = 1198641198881198941015840119864119888119894

= (((((


)))))

(((((


)))))

(11)


119864119888119864119889 = ((


))

((


))

= ((


))

= ((((


))))

= 119864119889119864119888(12)












(13)



(((((


)))))

119879

119875119894

+ 119875119894(((((


)))))

= minus119875119894minus1(15)



Let 119864119888 = 1198641 1198642 1198643 1198641 = [1198601198881] 1198642 = [ 1198601198882 00 1198601198883 ] and1198643 = [ 1198601198883 0 00 1198601198884 00 0 1198601198885



11986411987911988811198751 + 11987511198641198881 = minus1198751198880 (16a)

11986411987911988821198752 + 11987521198641198882 = minus1198751198881 (16b)

11986411987911988831198753 + 11987531198641198883 = minus1198751198882 (16c)


11986011987911988821198752 + 11987521198601198882 + 11986011987911988831198752 + 11987521198601198883 = minus1198751198881 (17b)

11986011987911988831198753 + 119875311986034 + 11986011987911988841198753 + 11987531198601198884 + 11986011987911988851198753 + 11987531198601198885= minus1198751198882 (17c)















119864119879119889111987511988911198641198891 minus 1198751198891 = minus1198751198890 (20a)


11986411987911988811198751198881 + 11987511988811198641198891 = minus119875119889119873119889 (20c)

119864119879119888119894119875119888119894 + 119875119888119894119864119888119894 = minus119875119888119894minus1 (20d)


(((((


)))))

119879

sdot 1198751198891(((((


)))))

minus 1198751198891

= minus1198751198890(21a)

(((((


)))))

119879

sdot 119875119889119895(((((


)))))

minus 119875119889119895

= minus119875119889119895minus1

(21b)

(((((


)))))

119879

1198751198881

+ 1198751198881(((((


)))))

= minus119875119889119873119889

(21c)

((((((


))))))

119879

119888119901

119875119888119894


+ 119875119888119894(((((


)))))

= minus119875119888119894minus1(21d)



Let 119864119888 = 1198641198881 1198641198882 1198641198883 1198641198881 = [ 1198601198881 00 1198601198883 ] 1198641198882 = [ 1198601198882 00 1198601198883 ]and 1198641198883 = [ 1198601198881 0 00 1198601198882 0

0 0 1198601198883] let 119864119889 = 1198641198891 1198641198892 1198641198891 = [1198601198892] and1198641198892 = [ 1198601198891 00 1198601198892 ]


119864119879119889111987511988911198641198891 minus 1198751198891 = minus1198751198890 (22a)

119864119879119889211987511988921198641198892 minus 1198751198892 = minus1198751198891 (22b)

11986411987911988811198751198881 + 11987511988811198641198891 = minus1198751198892 (22c)

11986411987911988821198751198882 + 11987511988821198641198882 = minus1198751198881 (22d)

11986411987911988831198751198883 + 11987511988831198641198883 = minus1198751198882 (22e)


119860119879119889211987511988911198601198892 minus 1198751198891 = minus1198751198890 (23a)

119860119879119889111987511988921198601198891 + 119860119879119889211987511988921198601198892 minus 1198751198892 = minus1198751198891 (23b)

11986011987911988811198751198881 + 11986011987911988821198751198881 + 11987511988811198601198892 = minus1198751198892 (23c)

11986011987911988821198751198882 + 11987511988821198601198882 + 11986011987911988831198751198882 + 11987511988821198601198883 = minus1198751198881 (23d)

11986011987911988811198751198883 + 11987511988831198601198881 + 11986011987911988821198751198883 + 11987511988831198601198882 + 11986011987911988831198751198883+ 11987511988831198601198883 = minus1198751198882 (23e)




119864119888119894 = (((((((


)))))))

(24)



119864119889119895 = (((((((


)))))))

(25)





((((((


))))))

= minus119875119894minus1 (26a)



((((((


))))))

= minus119875119894minus1 (26b)


((((((


))))))

= minus119875119894minus1 (26c)


1198641198881198941198641198881198941015840 = 1198641198881198941015840119864119888119894 = ((((((


))))))

((((((


))))))

(27)





(((((


)))))

119879

119875119894

+ 119875119894(((((


)))))= minus119875119894minus1

(28)



((((((


))))))

119879

119875119894

+ 119875119894((((((


))))))= minus119875119894minus1

(29)





(((((


)))))

119879

sdot 119875119894(((((


)))))

= 119875119894


(30a)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30b)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30c)



((((((


))))))

119879

sdot 1198751198891((((((


))))))minus 1198751198891 = minus1198751198890

(31a)

((((((


))))))

119879

sdot 119875119889119895((((((



(31b)

((((((


))))))

119879

1198751198881

+ 1198751198881((((((


))))))= minus119875119889119873119889

(31c)

((((((


))))))

119879

119888119901

119875119888119894

+ 119875119888119894((((((


))))))= minus119875119888119894minus1

(31d)


fresh air

cold air

room(temperature

Inside)

latentheat

Inside





5 Numerical Example




(32)



pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)










Table 2



A 001 0005-001B 002 0015-002C 0033 00175-0033total 0063 00315-0063

Table 3



A 001 001-001B 002 002-002C 0033 00175-0033total 0063 00315-0063




pump Apump B

pump Ctotal

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)

100 200 300 400 500 600 700 800 9000t (second)



6 Conclusion



Table 4


pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)



0

10

20

30

40

50

60

70

80

90

100

Q (K

J)

100 200 300 400 500 600 700 800 9000t (second)



Data Availability




25

255

26

265

27

275

28

285

29

295

30

tem

pera

ture

insid

e (∘

C)

100 200 300 400 500 600 700 800 9000t (second)




Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018



















(13)



(((((


)))))

119879

119875119894

+ 119875119894(((((


)))))

= minus119875119894minus1(15)



Let 119864119888 = 1198641 1198642 1198643 1198641 = [1198601198881] 1198642 = [ 1198601198882 00 1198601198883 ] and1198643 = [ 1198601198883 0 00 1198601198884 00 0 1198601198885



11986411987911988811198751 + 11987511198641198881 = minus1198751198880 (16a)

11986411987911988821198752 + 11987521198641198882 = minus1198751198881 (16b)

11986411987911988831198753 + 11987531198641198883 = minus1198751198882 (16c)


11986011987911988821198752 + 11987521198601198882 + 11986011987911988831198752 + 11987521198601198883 = minus1198751198881 (17b)

11986011987911988831198753 + 119875311986034 + 11986011987911988841198753 + 11987531198601198884 + 11986011987911988851198753 + 11987531198601198885= minus1198751198882 (17c)















119864119879119889111987511988911198641198891 minus 1198751198891 = minus1198751198890 (20a)


11986411987911988811198751198881 + 11987511988811198641198891 = minus119875119889119873119889 (20c)

119864119879119888119894119875119888119894 + 119875119888119894119864119888119894 = minus119875119888119894minus1 (20d)


(((((


)))))

119879

sdot 1198751198891(((((


)))))

minus 1198751198891

= minus1198751198890(21a)

(((((


)))))

119879

sdot 119875119889119895(((((


)))))

minus 119875119889119895

= minus119875119889119895minus1

(21b)

(((((


)))))

119879

1198751198881

+ 1198751198881(((((


)))))

= minus119875119889119873119889

(21c)

((((((


))))))

119879

119888119901

119875119888119894


+ 119875119888119894(((((


)))))

= minus119875119888119894minus1(21d)



Let 119864119888 = 1198641198881 1198641198882 1198641198883 1198641198881 = [ 1198601198881 00 1198601198883 ] 1198641198882 = [ 1198601198882 00 1198601198883 ]and 1198641198883 = [ 1198601198881 0 00 1198601198882 0

0 0 1198601198883] let 119864119889 = 1198641198891 1198641198892 1198641198891 = [1198601198892] and1198641198892 = [ 1198601198891 00 1198601198892 ]


119864119879119889111987511988911198641198891 minus 1198751198891 = minus1198751198890 (22a)

119864119879119889211987511988921198641198892 minus 1198751198892 = minus1198751198891 (22b)

11986411987911988811198751198881 + 11987511988811198641198891 = minus1198751198892 (22c)

11986411987911988821198751198882 + 11987511988821198641198882 = minus1198751198881 (22d)

11986411987911988831198751198883 + 11987511988831198641198883 = minus1198751198882 (22e)


119860119879119889211987511988911198601198892 minus 1198751198891 = minus1198751198890 (23a)

119860119879119889111987511988921198601198891 + 119860119879119889211987511988921198601198892 minus 1198751198892 = minus1198751198891 (23b)

11986011987911988811198751198881 + 11986011987911988821198751198881 + 11987511988811198601198892 = minus1198751198892 (23c)

11986011987911988821198751198882 + 11987511988821198601198882 + 11986011987911988831198751198882 + 11987511988821198601198883 = minus1198751198881 (23d)

11986011987911988811198751198883 + 11987511988831198601198881 + 11986011987911988821198751198883 + 11987511988831198601198882 + 11986011987911988831198751198883+ 11987511988831198601198883 = minus1198751198882 (23e)




119864119888119894 = (((((((


)))))))

(24)



119864119889119895 = (((((((


)))))))

(25)





((((((


))))))

= minus119875119894minus1 (26a)



((((((


))))))

= minus119875119894minus1 (26b)


((((((


))))))

= minus119875119894minus1 (26c)


1198641198881198941198641198881198941015840 = 1198641198881198941015840119864119888119894 = ((((((


))))))

((((((


))))))

(27)





(((((


)))))

119879

119875119894

+ 119875119894(((((


)))))= minus119875119894minus1

(28)



((((((


))))))

119879

119875119894

+ 119875119894((((((


))))))= minus119875119894minus1

(29)





(((((


)))))

119879

sdot 119875119894(((((


)))))

= 119875119894


(30a)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30b)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30c)



((((((


))))))

119879

sdot 1198751198891((((((


))))))minus 1198751198891 = minus1198751198890

(31a)

((((((


))))))

119879

sdot 119875119889119895((((((



(31b)

((((((


))))))

119879

1198751198881

+ 1198751198881((((((


))))))= minus119875119889119873119889

(31c)

((((((


))))))

119879

119888119901

119875119888119894

+ 119875119888119894((((((


))))))= minus119875119888119894minus1

(31d)


fresh air

cold air

room(temperature

Inside)

latentheat

Inside





5 Numerical Example




(32)



pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)










Table 2



A 001 0005-001B 002 0015-002C 0033 00175-0033total 0063 00315-0063

Table 3



A 001 001-001B 002 002-002C 0033 00175-0033total 0063 00315-0063




pump Apump B

pump Ctotal

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)

100 200 300 400 500 600 700 800 9000t (second)



6 Conclusion



Table 4


pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)



0

10

20

30

40

50

60

70

80

90

100

Q (K

J)

100 200 300 400 500 600 700 800 9000t (second)



Data Availability




25

255

26

265

27

275

28

285

29

295

30

tem

pera

ture

insid

e (∘

C)

100 200 300 400 500 600 700 800 9000t (second)




Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018



















119864119879119889111987511988911198641198891 minus 1198751198891 = minus1198751198890 (20a)


11986411987911988811198751198881 + 11987511988811198641198891 = minus119875119889119873119889 (20c)

119864119879119888119894119875119888119894 + 119875119888119894119864119888119894 = minus119875119888119894minus1 (20d)


(((((


)))))

119879

sdot 1198751198891(((((


)))))

minus 1198751198891

= minus1198751198890(21a)

(((((


)))))

119879

sdot 119875119889119895(((((


)))))

minus 119875119889119895

= minus119875119889119895minus1

(21b)

(((((


)))))

119879

1198751198881

+ 1198751198881(((((


)))))

= minus119875119889119873119889

(21c)

((((((


))))))

119879

119888119901

119875119888119894


+ 119875119888119894(((((


)))))

= minus119875119888119894minus1(21d)



Let 119864119888 = 1198641198881 1198641198882 1198641198883 1198641198881 = [ 1198601198881 00 1198601198883 ] 1198641198882 = [ 1198601198882 00 1198601198883 ]and 1198641198883 = [ 1198601198881 0 00 1198601198882 0

0 0 1198601198883] let 119864119889 = 1198641198891 1198641198892 1198641198891 = [1198601198892] and1198641198892 = [ 1198601198891 00 1198601198892 ]


119864119879119889111987511988911198641198891 minus 1198751198891 = minus1198751198890 (22a)

119864119879119889211987511988921198641198892 minus 1198751198892 = minus1198751198891 (22b)

11986411987911988811198751198881 + 11987511988811198641198891 = minus1198751198892 (22c)

11986411987911988821198751198882 + 11987511988821198641198882 = minus1198751198881 (22d)

11986411987911988831198751198883 + 11987511988831198641198883 = minus1198751198882 (22e)


119860119879119889211987511988911198601198892 minus 1198751198891 = minus1198751198890 (23a)

119860119879119889111987511988921198601198891 + 119860119879119889211987511988921198601198892 minus 1198751198892 = minus1198751198891 (23b)

11986011987911988811198751198881 + 11986011987911988821198751198881 + 11987511988811198601198892 = minus1198751198892 (23c)

11986011987911988821198751198882 + 11987511988821198601198882 + 11986011987911988831198751198882 + 11987511988821198601198883 = minus1198751198881 (23d)

11986011987911988811198751198883 + 11987511988831198601198881 + 11986011987911988821198751198883 + 11987511988831198601198882 + 11986011987911988831198751198883+ 11987511988831198601198883 = minus1198751198882 (23e)




119864119888119894 = (((((((


)))))))

(24)



119864119889119895 = (((((((


)))))))

(25)





((((((


))))))

= minus119875119894minus1 (26a)



((((((


))))))

= minus119875119894minus1 (26b)


((((((


))))))

= minus119875119894minus1 (26c)


1198641198881198941198641198881198941015840 = 1198641198881198941015840119864119888119894 = ((((((


))))))

((((((


))))))

(27)





(((((


)))))

119879

119875119894

+ 119875119894(((((


)))))= minus119875119894minus1

(28)



((((((


))))))

119879

119875119894

+ 119875119894((((((


))))))= minus119875119894minus1

(29)





(((((


)))))

119879

sdot 119875119894(((((


)))))

= 119875119894


(30a)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30b)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30c)



((((((


))))))

119879

sdot 1198751198891((((((


))))))minus 1198751198891 = minus1198751198890

(31a)

((((((


))))))

119879

sdot 119875119889119895((((((



(31b)

((((((


))))))

119879

1198751198881

+ 1198751198881((((((


))))))= minus119875119889119873119889

(31c)

((((((


))))))

119879

119888119901

119875119888119894

+ 119875119888119894((((((


))))))= minus119875119888119894minus1

(31d)


fresh air

cold air

room(temperature

Inside)

latentheat

Inside





5 Numerical Example




(32)



pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)










Table 2



A 001 0005-001B 002 0015-002C 0033 00175-0033total 0063 00315-0063

Table 3



A 001 001-001B 002 002-002C 0033 00175-0033total 0063 00315-0063




pump Apump B

pump Ctotal

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)

100 200 300 400 500 600 700 800 9000t (second)



6 Conclusion



Table 4


pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)



0

10

20

30

40

50

60

70

80

90

100

Q (K

J)

100 200 300 400 500 600 700 800 9000t (second)



Data Availability




25

255

26

265

27

275

28

285

29

295

30

tem

pera

ture

insid

e (∘

C)

100 200 300 400 500 600 700 800 9000t (second)




Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018









+ 119875119888119894(((((


)))))

= minus119875119888119894minus1(21d)



Let 119864119888 = 1198641198881 1198641198882 1198641198883 1198641198881 = [ 1198601198881 00 1198601198883 ] 1198641198882 = [ 1198601198882 00 1198601198883 ]and 1198641198883 = [ 1198601198881 0 00 1198601198882 0

0 0 1198601198883] let 119864119889 = 1198641198891 1198641198892 1198641198891 = [1198601198892] and1198641198892 = [ 1198601198891 00 1198601198892 ]


119864119879119889111987511988911198641198891 minus 1198751198891 = minus1198751198890 (22a)

119864119879119889211987511988921198641198892 minus 1198751198892 = minus1198751198891 (22b)

11986411987911988811198751198881 + 11987511988811198641198891 = minus1198751198892 (22c)

11986411987911988821198751198882 + 11987511988821198641198882 = minus1198751198881 (22d)

11986411987911988831198751198883 + 11987511988831198641198883 = minus1198751198882 (22e)


119860119879119889211987511988911198601198892 minus 1198751198891 = minus1198751198890 (23a)

119860119879119889111987511988921198601198891 + 119860119879119889211987511988921198601198892 minus 1198751198892 = minus1198751198891 (23b)

11986011987911988811198751198881 + 11986011987911988821198751198881 + 11987511988811198601198892 = minus1198751198892 (23c)

11986011987911988821198751198882 + 11987511988821198601198882 + 11986011987911988831198751198882 + 11987511988821198601198883 = minus1198751198881 (23d)

11986011987911988811198751198883 + 11987511988831198601198881 + 11986011987911988821198751198883 + 11987511988831198601198882 + 11986011987911988831198751198883+ 11987511988831198601198883 = minus1198751198882 (23e)




119864119888119894 = (((((((


)))))))

(24)



119864119889119895 = (((((((


)))))))

(25)





((((((


))))))

= minus119875119894minus1 (26a)



((((((


))))))

= minus119875119894minus1 (26b)


((((((


))))))

= minus119875119894minus1 (26c)


1198641198881198941198641198881198941015840 = 1198641198881198941015840119864119888119894 = ((((((


))))))

((((((


))))))

(27)





(((((


)))))

119879

119875119894

+ 119875119894(((((


)))))= minus119875119894minus1

(28)



((((((


))))))

119879

119875119894

+ 119875119894((((((


))))))= minus119875119894minus1

(29)





(((((


)))))

119879

sdot 119875119894(((((


)))))

= 119875119894


(30a)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30b)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30c)



((((((


))))))

119879

sdot 1198751198891((((((


))))))minus 1198751198891 = minus1198751198890

(31a)

((((((


))))))

119879

sdot 119875119889119895((((((



(31b)

((((((


))))))

119879

1198751198881

+ 1198751198881((((((


))))))= minus119875119889119873119889

(31c)

((((((


))))))

119879

119888119901

119875119888119894

+ 119875119888119894((((((


))))))= minus119875119888119894minus1

(31d)


fresh air

cold air

room(temperature

Inside)

latentheat

Inside





5 Numerical Example




(32)



pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)










Table 2



A 001 0005-001B 002 0015-002C 0033 00175-0033total 0063 00315-0063

Table 3



A 001 001-001B 002 002-002C 0033 00175-0033total 0063 00315-0063




pump Apump B

pump Ctotal

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)

100 200 300 400 500 600 700 800 9000t (second)



6 Conclusion



Table 4


pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)



0

10

20

30

40

50

60

70

80

90

100

Q (K

J)

100 200 300 400 500 600 700 800 9000t (second)



Data Availability




25

255

26

265

27

275

28

285

29

295

30

tem

pera

ture

insid

e (∘

C)

100 200 300 400 500 600 700 800 9000t (second)




Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018









((((((


))))))

= minus119875119894minus1 (26a)



((((((


))))))

= minus119875119894minus1 (26b)


((((((


))))))

= minus119875119894minus1 (26c)


1198641198881198941198641198881198941015840 = 1198641198881198941015840119864119888119894 = ((((((


))))))

((((((


))))))

(27)





(((((


)))))

119879

119875119894

+ 119875119894(((((


)))))= minus119875119894minus1

(28)



((((((


))))))

119879

119875119894

+ 119875119894((((((


))))))= minus119875119894minus1

(29)





(((((


)))))

119879

sdot 119875119894(((((


)))))

= 119875119894


(30a)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30b)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30c)



((((((


))))))

119879

sdot 1198751198891((((((


))))))minus 1198751198891 = minus1198751198890

(31a)

((((((


))))))

119879

sdot 119875119889119895((((((



(31b)

((((((


))))))

119879

1198751198881

+ 1198751198881((((((


))))))= minus119875119889119873119889

(31c)

((((((


))))))

119879

119888119901

119875119888119894

+ 119875119888119894((((((


))))))= minus119875119888119894minus1

(31d)


fresh air

cold air

room(temperature

Inside)

latentheat

Inside





5 Numerical Example




(32)



pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)










Table 2



A 001 0005-001B 002 0015-002C 0033 00175-0033total 0063 00315-0063

Table 3



A 001 001-001B 002 002-002C 0033 00175-0033total 0063 00315-0063




pump Apump B

pump Ctotal

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)

100 200 300 400 500 600 700 800 9000t (second)



6 Conclusion



Table 4


pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)



0

10

20

30

40

50

60

70

80

90

100

Q (K

J)

100 200 300 400 500 600 700 800 9000t (second)



Data Availability




25

255

26

265

27

275

28

285

29

295

30

tem

pera

ture

insid

e (∘

C)

100 200 300 400 500 600 700 800 9000t (second)




Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018










(((((


)))))

119879

119875119894

+ 119875119894(((((


)))))= minus119875119894minus1

(28)



((((((


))))))

119879

119875119894

+ 119875119894((((((


))))))= minus119875119894minus1

(29)





(((((


)))))

119879

sdot 119875119894(((((


)))))

= 119875119894


(30a)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30b)



((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30c)



((((((


))))))

119879

sdot 1198751198891((((((


))))))minus 1198751198891 = minus1198751198890

(31a)

((((((


))))))

119879

sdot 119875119889119895((((((



(31b)

((((((


))))))

119879

1198751198881

+ 1198751198881((((((


))))))= minus119875119889119873119889

(31c)

((((((


))))))

119879

119888119901

119875119888119894

+ 119875119888119894((((((


))))))= minus119875119888119894minus1

(31d)


fresh air

cold air

room(temperature

Inside)

latentheat

Inside





5 Numerical Example




(32)



pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)










Table 2



A 001 0005-001B 002 0015-002C 0033 00175-0033total 0063 00315-0063

Table 3



A 001 001-001B 002 002-002C 0033 00175-0033total 0063 00315-0063




pump Apump B

pump Ctotal

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)

100 200 300 400 500 600 700 800 9000t (second)



6 Conclusion



Table 4


pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)



0

10

20

30

40

50

60

70

80

90

100

Q (K

J)

100 200 300 400 500 600 700 800 9000t (second)



Data Availability




25

255

26

265

27

275

28

285

29

295

30

tem

pera

ture

insid

e (∘

C)

100 200 300 400 500 600 700 800 9000t (second)




Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018









((((((


))))))

119879

sdot 119875119895((((((


))))))= 119875119895 minus 119875119895minus1

(30c)



((((((


))))))

119879

sdot 1198751198891((((((


))))))minus 1198751198891 = minus1198751198890

(31a)

((((((


))))))

119879

sdot 119875119889119895((((((



(31b)

((((((


))))))

119879

1198751198881

+ 1198751198881((((((


))))))= minus119875119889119873119889

(31c)

((((((


))))))

119879

119888119901

119875119888119894

+ 119875119888119894((((((


))))))= minus119875119888119894minus1

(31d)


fresh air

cold air

room(temperature

Inside)

latentheat

Inside





5 Numerical Example




(32)



pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)










Table 2



A 001 0005-001B 002 0015-002C 0033 00175-0033total 0063 00315-0063

Table 3



A 001 001-001B 002 002-002C 0033 00175-0033total 0063 00315-0063




pump Apump B

pump Ctotal

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)

100 200 300 400 500 600 700 800 9000t (second)



6 Conclusion



Table 4


pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)



0

10

20

30

40

50

60

70

80

90

100

Q (K

J)

100 200 300 400 500 600 700 800 9000t (second)



Data Availability




25

255

26

265

27

275

28

285

29

295

30

tem

pera

ture

insid

e (∘

C)

100 200 300 400 500 600 700 800 9000t (second)




Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018









fresh air

cold air

room(temperature

Inside)

latentheat

Inside





5 Numerical Example




(32)



pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)










Table 2



A 001 0005-001B 002 0015-002C 0033 00175-0033total 0063 00315-0063

Table 3



A 001 001-001B 002 002-002C 0033 00175-0033total 0063 00315-0063




pump Apump B

pump Ctotal

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)

100 200 300 400 500 600 700 800 9000t (second)



6 Conclusion



Table 4


pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)



0

10

20

30

40

50

60

70

80

90

100

Q (K

J)

100 200 300 400 500 600 700 800 9000t (second)



Data Availability




25

255

26

265

27

275

28

285

29

295

30

tem

pera

ture

insid

e (∘

C)

100 200 300 400 500 600 700 800 9000t (second)




Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018














Table 2



A 001 0005-001B 002 0015-002C 0033 00175-0033total 0063 00315-0063

Table 3



A 001 001-001B 002 002-002C 0033 00175-0033total 0063 00315-0063




pump Apump B

pump Ctotal

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)

100 200 300 400 500 600 700 800 9000t (second)



6 Conclusion



Table 4


pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)



0

10

20

30

40

50

60

70

80

90

100

Q (K

J)

100 200 300 400 500 600 700 800 9000t (second)



Data Availability




25

255

26

265

27

275

28

285

29

295

30

tem

pera

ture

insid

e (∘

C)

100 200 300 400 500 600 700 800 9000t (second)




Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018









Table 4


pump Apump B

pump Ctotal

100 200 300 400 500 600 700 800 9000t (second)

0

002

004

006

008

01

012

014

016

volu

me (

kgs

)



0

10

20

30

40

50

60

70

80

90

100

Q (K

J)

100 200 300 400 500 600 700 800 9000t (second)



Data Availability




25

255

26

265

27

275

28

285

29

295

30

tem

pera

ture

insid

e (∘

C)

100 200 300 400 500 600 700 800 9000t (second)




Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018










25

255

26

265

27

275

28

285

29

295

30

tem

pera

ture

insid

e (∘

C)

100 200 300 400 500 600 700 800 9000t (second)




Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018


















Journal of












Journal of







Volume 2018






Volume 2018








Stability Analysis for a Type of Multiswitching System ...

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Transcript of Stability Analysis for a Type of Multiswitching System ...