Research ArticleControllability and Observability of Fractional Linear Systemswith Two Different Orders
Dengguo Xu Yanmei Li and Weifeng Zhou
Department of Mathematics Chuxiong Normal University Chuxiong Yunnan 675000 China
Correspondence should be addressed to Dengguo Xu dengguoxu163com
Received 17 September 2013 Accepted 5 December 2013 Published 20 January 2014
Academic Editors N Kallur and R K Naji
Copyright copy 2014 Dengguo Xu 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 is concerned with the controllability and observability for a class of fractional linear systems with two different ordersThe sufficient and necessary conditions for state controllability and state observability of such systems are established The resultsobtained extend some existing results of controllability and observability for fractional dynamical systems
1 Introduction
In the last three decades interest in fractional calculus hasexperienced rapid growth and at present we can find manypapers devoted to its theoretical and application aspects seethe work of [1] and the references therein Fractional ordermodels of real systems are often more adequate than theusually used integer order models in electrochemistry [2]advection dispersion models [3] anomalous diffusion [4]viscoelasticmaterials [5] fractal networks [6ndash8] robotics [9]and so forth Further during recent years a renewed interesthas been devoted to fractional order systems in the area ofautomatic control the reader can refer to monograph [10]Oustaloup [11] initiated the first framework for nonintegerorder systems in the automatic control area Fractional ordercontrol is the use of fractional calculus in the aforementionedtopics the system being modeled in a classical way or as afractional one From a certain point of view the applicationsof fractional calculus have experienced an evolution analo-gous to that of control following two parallel paths dependingon the starting point the time domain or the frequencydomain [12ndash14]
Controllability and observability are two of the mostfundamental concepts in modern control theory They haveclose connections to pole assignment structural decompo-sition quadratic optimal control observer design and soforth [15 16] In the past ten years many results have beenobtained on controllability and observability of fractional
order systems Chen et al [17] proposed robust controllabilityfor interval fractional order linear time invariant systemswhereas Adams and Hartley [18] studied finite time con-trollability for fractional systems The controllability condi-tions for fractional control systems with control delay wereobtained in [19] Shamardan and Moubarak [20] extendedsome basic results on the controllability and observability oflinear discrete-time fractional order systems and developedsome new concepts inherent to fractional order systems withanalytical methods for checking their properties Balachan-dran et al [21] obtained controllability criteria for fractionallinear systems and then this result is extended to nonlinearfractional dynamical systems by using fixed point theoremIn recent paper [22] necessary and sufficient conditions ofcontrollability and observability for fractional linear timeinvariant system are included
However to the best of our knowledge there has beenno result about the controllability and observability of frac-tional linear systems with different orders In this paper weinvestigated state controllability and state observability offractional linear systems with two different orders We derivethe sufficient and necessary conditions on controllabilityand observability for the fractional linear systems with twodifferent orders
The paper is organized as follows Section 2 formulatesthe problem and presents the preliminary results The mainresults about controllability and observability for the frac-tional linear systems with two different orders are given in
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 618162 8 pageshttpdxdoiorg1011552014618162
2 The Scientific World Journal
Sections 3 and 4 respectively Finally some conclusions aredrawn in Section 5
2 Preliminaries
Consider the following fractional linear systems with twodifferent orders
[1198881198631205721199051199091(119905)
1198881198631205731199051199092(119905)] = [11986011 119860
1211986021
11986022
] [1199091 (119905)1199092(119905)] + [1198611119861
2
] 119906 (119905) (1)
where 119888119863120572119905 119888119863120573119905are the Caputo derivative 0 lt 120572 lt 1 0 lt
120573 lt 1 1199091isin 1198771198991 119909
2isin 1198771198992 are the state vectors 119860
119894119895isin 119877119899119894times119899119895
119861119894isin 119877119899119894times119898 119894 119895 = 1 2 are the known constant matrices 119899
1+
1198992= 119899 119906 isin 119877119898 is the input vectorWhen 119860
12= 11986021
= 0 the system (1) reduces to thefollowing form
[1198881198631205721199051199091(119905)
1198881198631205731199051199092(119905)] = [11986011 0
0 11986022
] [1199091 (119905)1199092(119905)] + [1198611119861
2
] 119906 (119905) (2)
We first give some definitions about fractional calculusfor more details see [10 23 24]
Definition 1 Riemann-Liouvillersquos fractional integral of order120572 (120572 gt 0) for a function ℎ (0infin) rarr 119877 is defined as
0119863minus120572119905
ℎ (119905) = 1Γ (120572) int119905
0
(119905 minus 119904)120572minus1ℎ (119904) 119889119904 (3)
where Γ(120572) = intinfin0
119905120572minus1119890minus119905119889119905 is Gamma function
Definition 2 Riemann-Liouvillersquos fractional derivative oforder 120572 (0 lt 120572 lt 1) for a function ℎ (0infin) rarr 119877 is definedas
0119863120572119905ℎ (119905) = 1
Γ (1 minus 120572)119889119889119905 int1199050
(119905 minus 119904)minus120572ℎ (119904) 119889119904 (4)
Definition 3 TheCaputo fractional derivative of order 120572 (0 lt120572 lt 1) for a function ℎ (0infin) rarr 119877 is defined as
119888
0119863120572119905ℎ (119905) = 1
Γ (1 minus 120572) int1199050
(119905 minus 119904)minus120572ℎ1015840 (119904) 119889119904 (5)
Throughout the paper only the Caputo definition isused since the Laplace transform allows using initial valuesof classical integer order derivatives with clear physicalinterpretations
According to [25] the solution of the system (1) can beobtained Therefore the following lemma holds
Lemma 4 The solution of system (1) with initial conditions1199091(0) = 119909
10and 119909
2(0) = 119909
20is given by
119909 (119905) = Φ0(119905) 1199090+ int1199050
Φ (119905 minus 120591) 119861119906 (120591) 119889120591 (6)
where
119909 (119905) = [1199091 (119905)1199092(119905)] 119909
0= [1199091011990920
] 119861 = [1198611011986101
]
11986110
= [11986110 ] 11986101
= [ 01198612
] Φ (119905 minus 120591) = [Φ
1(119905 minus 120591) Φ
2(119905 minus 120591)]
Φ0(119905) =
infin
sum119896=0
infin
sum119897=0
119879119896119897
119905119896120572+119897120573Γ (119896120572 + 119897120573 + 1)
Φ1(119905) =
infin
sum119896=0
infin
sum119897=0
119879119896119897
119905(119896+1)120572+119897120573minus1Γ ((119896 + 1) 120572 + 119897120573)
Φ2(119905) =
infin
sum119896=0
infin
sum119897=0
119879119896119897
119905119896120572+(119897+1)120573minus1Γ (119896120572 + (119897 + 1) 120573)
(7)
119879119896119897
=
119868119899 119896 = 119897 = 0
[11986011 11986012
0 0 ] 119896 = 1 119897 = 0[ 0 011986021
11986022
] 119896 = 0 119897 = 111987910119879119896minus1119897
+ 11987901119879119896119897minus1
119896 + 119897 gt 1
(8)
From (8) the following lemma holds
Lemma 5 The implication
sum119896+119897=119898
119879119896119897
= 119860119898 119898 isin 119885+ (9)
Holds where
119860 = [11986011 11986012119860
2111986022
] (10)
Proof When 119898 = 1 it follows from (8) that
sum119896+119897=1
119879119896119897
= 11987901
+ 11987910
= 119860 (11)
which implies that (9) holds when 119898 = 1 Now suppose that(9) is true when 119898 = 119901 119901 isin 119885+ namely
sum119896+119897=119901
119879119896119897
= 119860119901 (12)
When 119898 = 119901 + 1 we getsum119896+119897=119901+1
119879119896119897
= sum119896+119897=119901+1
(11987910119879119896minus1119897
+ 11987901119879119896119897minus1
)
= sum119896+119897=119901+1
11987910119879119896minus1119897
+ sum119896+119897=119901+1
11987901119879119896119897minus1
= sum119896+119897=119901
11987910119879119896119897
+ sum119896+119897=119901
11987901119879119896119897
= 11987910119860119901 + 119879
01119860119901
= 119860119901+1
(13)
The Scientific World Journal 3
which means that (9) holds when 119898 = 119901 + 1 Reasoning bymathematical induction we can immediately conclude that(9) is true for any119898 isin 119885+This therefore completes the proof
3 Controllability
In this section the sufficient and necessary conditions ofcontrollability for the fractional linear system (1) with twodifferent orders are discussed based on previous definitionsand results Similar to the concepts of controllability for gen-eral fractional linear systems the definition of controllabilityfor fractional linear systems with different orders is given asfollows
Definition 6 The system (1) is called state controllable on[0 119879] if given any state 119909
0 1199091199051
isin 119877119899 there exists a controlinput signal 119906(119905) [0 119879] rarr 119877119898 such that the correspondingsolution of system (1) satisfies 119909(0) = 119909
0and 119909(119905
1) = 1199091199051
1199051isin
[0 119879]Theorem 7 The system (1) is controllable on [0 119905
1] if and only
if the controllability Gramian matrix
119882119888(0 1199051) = int11990510
Φ(1199051minus 120591) 119861119861119879Φ119879 (119905
1minus 120591) 119889120591 (14)
is nonsingular
Proof Suppose that the matrix 119882119888(0 1199051) is nonsingular
Accordingly119882119888(0 1199051) is invertibleThen given an initial state
119909(0) = 1199090
= 0 choose119906 (119905) = 119861119879Φ119879 (119905
1minus 119905)119882minus1
119888(0 1199051) [1199091199051
minus Φ0(1199051) 1199090] (15)
it follows from the solution of system (1) that
119909 (1199051) = Φ
0(1199051) 1199090+ int11990510
Φ(1199051minus 120591) 119861119906 (120591) 119889120591
= Φ0(1199051) 1199090
+ int11990510
Φ(1199051minus 120591) 119861119861119879Φ119879 (119905
1minus 120591)119882minus1
119888(0 1199051)
times [1199091199051
minus Φ0(1199051) 1199090] 119889120591
= Φ0(1199051) 1199090+ int11990510
Φ(1199051minus 120591) 119861119861119879Φ119879 (119905
1minus 120591) 119889120591
times 119882minus1119888
(0 1199051) [1199091199051
minus Φ0(1199051) 1199090]
= Φ0(1199051) 1199090+ 119882119888(0 1199051)119882minus1119888
(0 1199051) [1199091199051
minus Φ0(1199051) 1199090]
= Φ0(1199051) 1199090+ [1199091199051
minus Φ0(1199051) 1199090] = 1199091199051
(16)
Thus the system (1) is controllable on [0 1199051]
We show the converse by contradiction Suppose that thesystem (1) is controllable on [0 119905
1] but the matrix119882
119888(0 1199051) is
singularThen there exists an 119899times1 nonzero vector V such that0 = V119879119882
119888(0 1199051) V = int1199051
0
V119879Φ(1199051minus 120591) 119861119861119879Φ119879 (119905
1minus 120591) V 119889120591
= int11990510
10038171003817100381710038171003817V119879Φ(1199051minus 120591) 11986110038171003817100381710038171003817
2119889120591(17)
which implies
V119879Φ(1199051minus 120591) 119861 equiv 0 (18)
for all 120591 isin [0 1199051] If (1) is controllable there exists an input
that transfers the initial 119909(0) = 1199090to 119909(119905
1) = 0 We choose
1199090= minusΦminus10
(1199051)V then there exists an input such that
119909 (1199051) = minusΦ
0(1199051)Φminus10
(1199051) V + int1199051
0
Φ(1199051minus 120591) 119861119906 (120591) 119889120591 = 0
(19)
that is
V = int11990510
Φ(1199051minus 120591) 119861119906 (120591) 119889120591 (20)
Its premultiplication by V119879 yields
V119879V = int11990510
V119879Φ(1199051minus 120591) 119861119906 (120591) 119889120591 = 0 (21)
which contradicts V = 0 So the matrix 119882119888(0 1199051) is nonsingu-
lar The proof is thus completed
In the following we consider the special case of systems(1) with 119860
12= 11986021
= 0 The systems (1) are reduced to
[[
1198881198631205721199051199091(119905)
1198881198631205731199051199092(119905)]]
= [11986011 00 119860
22
] [1199091 (119905)1199092(119905)] + [1198611119861
2
] 119906 (119905) (22)
which can be rewritten as the following two subsystems1198881198631205721199051199091(119905) = 119860
111199091(119905) + 119861
1119906 (119905) (23a)
1198881198631205731199051199092(119905) = 119860
221199092(119905) + 119861
2119906 (119905) (23b)
Thus the following corollary is true
Corollary 8 The fractional linear system (22) is controllableon [0 119905
1] if and only if the controllability Gramian matrix
119882119888(0 1199051) = int11990510
[[[[[[
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896111198611
infin
sum119897=0
(1199051minus 120591)(119897+1)120573minus1
Γ ((119897 + 1) 120573) 119860119897221198612
]]]]]]
times[[[[[[
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896111198611
infin
sum119897=0
(1199051minus 120591)(119897+1)120573minus1
Γ ((119897 + 1) 120573) 119860119897221198612
]]]]]]
119879
119889120591
(24)
is nonsingular
4 The Scientific World Journal
Proof When 11986012
= 11986021
= 0 system (1) is reduced to thesystem (22) It follows from simple computation that
119879119896119897
=
119868119899 119896 = 119897 = 0
11987911989701 119896 = 0 119897 = 1 2
11987911989610 119897 = 0 119896 = 1 2
0119899 others
(25)
where
11987901
= [0 00 11986022
] 11987910
= [11986011 00 0]
Φ0(119905) = 119879
00+ 11987901
119905120573Γ (120573 + 1) + 119879
02
1199052120573Γ (2120573 + 1) + sdot sdot sdot
+ 11987910
119905120572Γ (120572 + 1) + 119879
20
1199052120572Γ (2120572 + 1) + sdot sdot sdot
= [1198681198991 00 1198681198992
] + 119905120573Γ (120573 + 1) [0 0
0 11986022
]
+ 1199052120573Γ (2120573 + 1)[
0 00 11986022
]2
+ sdot sdot sdot
+ 119905120572Γ (120572 + 1) [11986011 0
0 0]
+ 1199052120572Γ (2120572 + 1)[
11986011
00 0]
2
+ sdot sdot sdot
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
Φ1(119905) = 119879
00
119905120572minus1Γ (120572) + 119879
01
119905120572+120573minus1Γ (120572 + 120573) + 119879
02
119905120572+2120573minus1120574 (120572 + 2120573) + sdot sdot sdot
+ 11987910
1199052120572minus1Γ (2120572) + 119879
20
1199053120572minus1Γ (3120572) + sdot sdot sdot
= 119905120572minus1Γ (120572) [1198681198991 0
0 1198681198992
] + 119905120572+120573minus1Γ (120572 + 120573) [0 0
0 11986022
]
+ 119905120572+2120573minus1Γ (120572 + 2120573)[
0 00 11986022
]2
+ sdot sdot sdot
+ 1199052120572minus1Γ (2120572) [11986011 0
0 0]
+ 1199053120572minus1Γ (3120572)[
11986011
00 0]
2
+ sdot sdot sdot
= [[[[[
infin
sum119896=0
119905(119896+1)120572minus1Γ ((119896 + 1) 120572)119860
119896
110
0infin
sum119897=0
119905120572+119897120573minus1Γ (120572 + 119897120573)119860
119897
22
]]]]]
Φ2(119905) = 119879
00
119905120573minus1Γ (120573) + 119879
01
1199052120573minus1Γ (2120573) + 119879
02
1199053120573minus1Γ (3120573) + sdot sdot sdot
+ 11987910
119905120572+120573minus1Γ (120572 + 120573) + 119879
20
1199052120572+120573minus1Γ (2120572 + 120573) + sdot sdot sdot
= 119905120573minus1Γ (120573) [1198681198991 0
0 1198681198992
] + 1199052120573minus1Γ (2120573) [0 0
0 11986022
]
+ 1199053120573minus1Γ (3120573)[
0 00 11986022
]2
+ sdot sdot sdot
+ 119905120572+120573minus1Γ (120572 + 120573) [11986011 0
0 0] + 1199052120572+120573minus1Γ (2120572 + 120573)[
11986011
00 0]
2
+ sdot sdot sdot
= [[[[[
infin
sum119896=0
119905119896120572+120573minus1Γ (119896120572 + 120573)119860
119896
110
0infin
sum119897=0
119905(119897+1)120573minus1Γ ((119897 + 1) 120573)119860
119897
22
]]]]]
(26)
Therefore the controllability Gramian matrix in theTheorem 7 is reduced to
119882119888(0 1199051) = int11990510
[[[[[[
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896111198611
infin
sum119897=0
(1199051minus 120591)(119897+1)120573minus1
Γ ((119897 + 1) 120573) 119860119897221198612
]]]]]]
times[[[[[[
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896111198611
infin
sum119897=0
(1199051minus 120591)(119897+1)120573minus1
Γ ((119897 + 1) 120573) 119860119897221198612
]]]]]]
119879
119889120591
(27)
This completes the proof
Obviously the following proposition is true
Proposition9 The fractional linear system (22) is controllableif and only if subsystems (23a) and (23b) are all controllable
In the following we consider another special case ofsystem (1) When 120572 = 120573 in the system (1) it is reduced to
119888119863120572119905119909 (119905) = 119860119909 (119905) + 119861119906 (119905) (28)
The Scientific World Journal 5
where
119909 (119905) = [1199091 (119905)1199092(119905)] 119860 = [11986011 119860
1211986021
11986022
] 119861 = [11986111198612
] (29)
Corollary 10 The fractional linear system (28) is controllableon [0 119905
1] if and only if the controllability Gramian matrix
119882119888= int11990510
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896119861119861119879
timesinfin
sum119870=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) (119860119879)119896119889120591(30)
is nonsingular
Proof According to the result of Lemma 5 when 120572 = 120573 wecan obtain
Φ0(119905) =infin
sum119896=0
(1198791198960
119905119896120572Γ (119896120572 + 1) +119879
1198961
119905(119896+1)120572Γ ((119896 + 1) 120572 + 1) + sdot sdot sdot )
= 11987900
+ 11987901
119905120572Γ (120572 + 1) + 119879
02
1199052120572Γ (2120572 + 1)
+ 11987903
1199053120572Γ (3120572 + 1) + sdot sdot sdot + 119879
10
119905120572Γ (120572 + 1)
+ 11987911
1199052120572Γ (2120572 + 1) + 119879
12
1199053120572Γ (3120572 + 1) + sdot sdot sdot
+ 11987920
1199052120572Γ (2120572 + 1) + 119879
21
1199053120572Γ (3120572 + 1)
+ 11987922
1199054120572Γ (4120572 + 1) + sdot sdot sdot + 119879
30
1199053120572Γ (3120572 + 1)
+ 11987931
1199054120572Γ (4120572 + 1) + 119879
32
1199055120572Γ (5120572 + 1) + sdot sdot sdot
sdot sdot sdot= 11987900
+ 119905120572Γ (120572 + 1) (119879
01+ 11987910)
+ 1199052120572Γ (2120572 + 1) (119879
02+ 11987911
+ 11987920)
+ 1199053120572Γ (3120572 + 1) (119879
03+ 11987912
+ 11987921
+ 11987930) + sdot sdot sdot
= [1198681198991 00 1198681198992
] + 119905120572Γ (120572 + 1)119860 + 1199052120572
Γ (2120572 + 1)1198602
+ 1199053120572Γ (3120572 + 1)119860
3 + sdot sdot sdot =infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896(31)
For the same reason as before we get
Φ1(119905) = Φ
2(119905) =
infin
sum119896=0
infin
sum119897=0
119879119896119897
119905(119896+119897+1)120572minus1Γ ((119896 + 119897 + 1) 120572)
=infin
sum119896=0
119905(119896+1)120572minus1Γ ((119896 + 1) 120572)119860
119896(32)
Therefore by simple computation the controllabilityGramian matrix of the system (28) can be obtained as (30)The proof is thus completed
Remark 11 Corollary 10 is equivalent to the result ofTheorem 22 in [21] Therefore Theorem 7 of this paperextends the existing results to a more general case
4 Observability
In this section we treat another fundamental property ofthe fractional linear system with different orders namelyobservability with respect to a linear output Throughoutthe rest of this paper we consider the system (1) with thefollowing output equation
[1199101 (119905)1199102(119905)] = [11986211 119862
1211986221
11986222
] [1199091 (119905)1199092(119905)] (33)
where 1199101
isin 1198771199011 1199102
isin 1198771199012 are the output vectors 1199091
isin1198771198991 119909
2isin 1198771198992 are the state vectors in the system (1) 119862
119894119895isin
119877119901119894times119899119895 119894 119895 = 1 2 are the known constant matrices 1199011+1199012=
119901When 119862
12= 11986221
= 0 the output (33) is reduced to thefollowing simple form
[1199101 (119905)1199102(119905)] = [11986211 0
0 11986222
] [1199091 (119905)1199092(119905)] (34)
which is equivalent to two suboutput equations as follows
1199101= 119862111199091 (35a)
1199102= 119862221199092 (35b)
Definition 12 The system (1) with the output (33) are calledstate observable on [0 119879] if any initial state 119909(0) = 119909
0isin 119877119899
can be uniquely determined by the corresponding systeminput 119906(119905) and system output 119910(119905) for 119905 isin [0 119879]
Define observability Gramian matrix 119882119900(0 119905) as
119882119900(0 119905) = int119905
0
Φ1198790(120591) 119862119879119862Φ
0(120591) 119889120591 (36)
where
119862 = [11986211 11986212119862
2111986222
] (37)
Theorem 13 The system (1) with the output (33) is observableon [0 119905
1] if and only if the observability Gramian matrix
119882119900(0 1199051) = int11990510
Φ1198790(120591) 119862119879119862Φ
0(120591) 119889120591 (38)
is invertible
6 The Scientific World Journal
Proof It follows from Lemma 4 that the output of system (1)has the following expression
119910 (119905) = 119862119909 (119905)= 119862Φ
0(119905) 1199090+ 119862int1199050
Φ (119905 minus 120591) 119861119906 (120591) 119889120591 (39)
It is easy to see from Definition 12 that the observability ofsystem (1) is equivalent to the observability of 119910(119905) given by
119910 (119905) = 119862Φ0(119905) 1199090 (40)
as 119906(119905) = 0Multiplying both sides of (40) by Φ119879
0(119905)119862119879 and integrat-
ing with respect to 119905 from 0 to 1199051 we have
int11990510
Φ1198790(119905) 119862119879119910 (119905) 119889119905 = 119882
119900(0 1199051) 1199090 (41)
Obviously the left-hand side of (41) depends on 119910(119905) and theright-hand side in (41) does not depend on 119910(119905) 119905 isin [0 119905
1]
Thus (41) is a linear algebraic equation of 1199090 If 119882
119900(0 1199051)
is invertible then the initial state 119909(0) = 1199090is uniquely
determined by the corresponding system output 119910(119905) for 119905 isin[0 1199051] Namely the system (1) is observable on [0 119905
1]
Next we show that if 119882119900(0 1199051) is singular for all 119905
1 then
system (1) with the output (33) is not observable Suppose119882119900(0 1199051) is singular then there exists an 119899 times 1 nonzero
constant vector V such that
V119879119882119900(0 1199051) V = int1199051
0
V119879Φ1198790(120591) 119862119879119862Φ
0(120591) V 119889120591
= int11990510
1003817100381710038171003817119862Φ0(120591) V10038171003817100381710038172119889120591 = 0
(42)
which implies that
119862Φ0(120591) V equiv 0 (43)
for all 120591 isin [0 1199051] If we choose 119909(0) = 119909
0= V then the output
(33) is given by
119910 (119905) = 119862Φ0(119905) 1199090= 119862Φ
0(119905) V equiv 0 (44)
Thus the initial state 1199090cannot be uniquely determined by
119910(119905) Therefore the system (1) with the output (33) is notobservable This completes the proof
Remark 14 When 120572 = 120573 in the system (1) Φ0(119905) is
already obtained in the proof of Corollary 10 Therefore theobservability Gramian matrix in Theorem 13 is
119882119900(0 1199051) = int11990510
Φ1198790(120591) 119862119879119862Φ
0(120591) 119889120591
= int11990510
infin
sum119896=0
120591119896120572Γ (119896120572 + 1)(119860
119879)119896119862119879119862infin
sum119896=0
120591119896120572Γ (119896120572 + 1)119860
119896119889120591(45)
which is the observability Gramian matrix in paper [22]by denoting 119864
120572(119860119905120572) = suminfin
119896=0(120591119896120572Γ(119896120572 + 1))119860119896 Therefore
Theorem 13 is actually a generalization of the existing observ-ability results for the fractional linear system
When 11986012
= 11986021
= 0 and 11986212
= 11986221
= 0 the system (1)with the output (33) is reduced to the following state equationand output equation
[[
1198881198631205721199051199091(119905)
1198881198631205731199051199092(119905)]]
= [11986011 00 119860
22
] [1199091 (119905)1199092(119905)] + [1198611119861
2
] 119906 (119905) (46a)
[1199101 (119905)1199102(119905)] = [11986211 0
0 11986222
] [1199091 (119905)1199092(119905)] (46b)
which can be rewritten as the following two subsystems withinput and output
1198881198631205721199051199091(119905) = 119860
111199091(119905) + 119861
1119906 (119905) (47a)
1199101(119905) = 119862
111199091(119905) (47b)
1198881198631205731199051199091(119905) = 119860
221199092(119905) + 119861
2119906 (119905) (48a)
1199102(119905) = 119862
221199092(119905) (48b)
It follows from conditions 11986012
= 11986021
= 0 and 11986212
= 11986221
= 0that
Φ1198790(119905) 119862119879119862Φ
0(119905)
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
119879
[11986211 00 11986222
]119879
[11986211 00 11986222
][[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
The Scientific World Journal 7
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)(119860
119896
11)119879 times 119862119879
1111986211
timesinfin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)(119860
119897
22)119879 times 119862119879
2211986222
timesinfin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
= [119864120572 (119860119879
11119905120572) 11986211987911
times 11986211119864120572(11986011119905120572) 0
0 119864120573(11986011987922119905120573) 11986211987922
times 11986222119864120573(11986022119905120573)]
(49)
where 119864119904(119860119905119904) = suminfin
119896=0(119905119896119904Γ(119896119904 + 1)) 119860119896 is Mittag-Leffler function
Therefore the following corollary holds
Corollary 15 Denote 119864120572(11986011 119905) = 119864
120572(11986011987911119905120572)11986211987911
times11986211119864120572(11986011119905120572) and 119864
120573(11986022 119905) = 119864
120573(11986011987922119905120573)11986211987922
times11986222119864120573(11986022119905120573) Then the system (46a) with the output (46b) is
observable on [0 1199051] if and only if the observability Gramian
matrix
int11990510
[119864120572 (11986011 119905) 00 119864
120573(11986022 119905)] 119889119905 (50)
is nonsingular
The following proposition is also true
Proposition 16 The fractional linear system (46a) with theoutput (46b) is observable if and only if the fractional linearsubsystems (47a) with the output (47b) and (48a) with theoutput (48b) are all observable
5 Conclusions
In this paper the controllability and observability problemsfor fractional linear systems with two different orders havebeen studiedThe sufficient and necessary conditions for statecontrollability and state observability of such systems areestablishedThe results obtained will be useful in the analysisand synthesis of fractional dynamical systems Extendingthe results of this paper toward fractional linear systemsconsisting of 119899 subsystems with different orders is a futurework
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author would like to thank the editors and reviewers fortheir helpful suggestions The present work was supportedby Natural Science Foundation of Yunnan province of China(no 2012FB175) andKey Projects of ScientificResearch Fundsof educational bureau of Yunnan province of China (no2010Z002)
References
[1] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[2] M Ichise Y Nagayanagi and T Kojima ldquoAn analog simulationof non-integer order transfer functions for analysis of electrodeppocessesrdquo Journal of Electroanalytical Chemistry vol 33 no 2pp 253ndash265 1971
[3] D A Benson S W Wheatcraft and M M MeerschaertldquoApplication of a fractional advection-dispersion equationrdquoWater Resources Research vol 36 no 6 pp 1403ndash1412 2000
[4] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000
[5] M Renardy W J Hrusa and J A Nohel MathematicalProblems in Viscoelasticity vol 35 of Pitman Monographs andSurveys in Pure and Applied Mathematics Longman Scientificamp Technical Harlow UK 1987
[6] M Al-Akaidi Fractal Speech Processing Cambridge UniversityPress 2004
[7] P Arena R Caponetto L Fortuna and D Porto NonlinearNoninteger Order Circuits and Systems An Introduction vol 38of World Scientific Series on Nonlinear Science Series A WorldScientific Publishing 2000
[8] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003
[9] D Valerio and J Sa da Costa ldquoNon-integer order controlof a flexible robotrdquo in Proceedings of the IFAC Workshopon Fractional Differentiation and Its Applications (FDA rsquo04)Bordeaux France 2004
[10] C A Monje Y Chen B M Vinagre D Xue and V Feliu-Batlle Fractional-Order Systems and Controls Fundamentalsand Applications Springer 2010
[11] A Oustaloup La Commande CRONE Commande RobustedrsquoOrdre Non Entier Hermes Paris France 1991
[12] D Matignon and B Dandrea-Novel ldquoSome results on con-trollability and observability of finite-dimensional fractionaldifferential systemsrdquo in Computational Engineering in SystemsApplications vol 2 pp 952ndash956 1996
[13] J A TMachado ldquoAnalysis and design of fractional-order digitalcontrol systemsrdquo Systems Analysis Modelling Simulation vol 27no 2-3 pp 107ndash122 1997
[14] I Podlubny ldquoFractional-order systems and 119875119868120582119863120583-controllersrdquoIEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[15] W J Rugh Linear SystemTheory Prentice Hall 1996
8 The Scientific World Journal
[16] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[17] Y Chen H-S Ahn and D Xue ldquoRobust controllability ofinterval fractional order linear time invariant systemsrdquo SignalProcessing vol 86 no 10 pp 2794ndash2802 2006
[18] J L Adams and T T Hartley ldquoFinite-time controllabilityof fractional-order systemsrdquo Journal of Computational andNonlinear Dynamics vol 3 no 2 Article ID 021402 2008
[19] J Wei ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012
[20] A B Shamardan and M R A Moubarak ldquoControllabilityand observability for fractional control systemsrdquo Journal ofFractional Calculus vol 15 pp 25ndash34 1999
[21] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012
[22] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012
[23] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press 1998
[24] H M Srivastava and J J Trujillo Theory and Applications ofFractional Differential Equations Elsevier Science 2006
[25] T Kaczorek ldquoPositive linear systems with different fractionalordersrdquo Bulletin of the Polish Academy of Sciences vol 58 no3 pp 453ndash458 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 The Scientific World Journal
Sections 3 and 4 respectively Finally some conclusions aredrawn in Section 5
2 Preliminaries
Consider the following fractional linear systems with twodifferent orders
[1198881198631205721199051199091(119905)
1198881198631205731199051199092(119905)] = [11986011 119860
1211986021
11986022
] [1199091 (119905)1199092(119905)] + [1198611119861
2
] 119906 (119905) (1)
where 119888119863120572119905 119888119863120573119905are the Caputo derivative 0 lt 120572 lt 1 0 lt
120573 lt 1 1199091isin 1198771198991 119909
2isin 1198771198992 are the state vectors 119860
119894119895isin 119877119899119894times119899119895
119861119894isin 119877119899119894times119898 119894 119895 = 1 2 are the known constant matrices 119899
1+
1198992= 119899 119906 isin 119877119898 is the input vectorWhen 119860
12= 11986021
= 0 the system (1) reduces to thefollowing form
[1198881198631205721199051199091(119905)
1198881198631205731199051199092(119905)] = [11986011 0
0 11986022
] [1199091 (119905)1199092(119905)] + [1198611119861
2
] 119906 (119905) (2)
We first give some definitions about fractional calculusfor more details see [10 23 24]
Definition 1 Riemann-Liouvillersquos fractional integral of order120572 (120572 gt 0) for a function ℎ (0infin) rarr 119877 is defined as
0119863minus120572119905
ℎ (119905) = 1Γ (120572) int119905
0
(119905 minus 119904)120572minus1ℎ (119904) 119889119904 (3)
where Γ(120572) = intinfin0
119905120572minus1119890minus119905119889119905 is Gamma function
Definition 2 Riemann-Liouvillersquos fractional derivative oforder 120572 (0 lt 120572 lt 1) for a function ℎ (0infin) rarr 119877 is definedas
0119863120572119905ℎ (119905) = 1
Γ (1 minus 120572)119889119889119905 int1199050
(119905 minus 119904)minus120572ℎ (119904) 119889119904 (4)
Definition 3 TheCaputo fractional derivative of order 120572 (0 lt120572 lt 1) for a function ℎ (0infin) rarr 119877 is defined as
119888
0119863120572119905ℎ (119905) = 1
Γ (1 minus 120572) int1199050
(119905 minus 119904)minus120572ℎ1015840 (119904) 119889119904 (5)
Throughout the paper only the Caputo definition isused since the Laplace transform allows using initial valuesof classical integer order derivatives with clear physicalinterpretations
According to [25] the solution of the system (1) can beobtained Therefore the following lemma holds
Lemma 4 The solution of system (1) with initial conditions1199091(0) = 119909
10and 119909
2(0) = 119909
20is given by
119909 (119905) = Φ0(119905) 1199090+ int1199050
Φ (119905 minus 120591) 119861119906 (120591) 119889120591 (6)
where
119909 (119905) = [1199091 (119905)1199092(119905)] 119909
0= [1199091011990920
] 119861 = [1198611011986101
]
11986110
= [11986110 ] 11986101
= [ 01198612
] Φ (119905 minus 120591) = [Φ
1(119905 minus 120591) Φ
2(119905 minus 120591)]
Φ0(119905) =
infin
sum119896=0
infin
sum119897=0
119879119896119897
119905119896120572+119897120573Γ (119896120572 + 119897120573 + 1)
Φ1(119905) =
infin
sum119896=0
infin
sum119897=0
119879119896119897
119905(119896+1)120572+119897120573minus1Γ ((119896 + 1) 120572 + 119897120573)
Φ2(119905) =
infin
sum119896=0
infin
sum119897=0
119879119896119897
119905119896120572+(119897+1)120573minus1Γ (119896120572 + (119897 + 1) 120573)
(7)
119879119896119897
=
119868119899 119896 = 119897 = 0
[11986011 11986012
0 0 ] 119896 = 1 119897 = 0[ 0 011986021
11986022
] 119896 = 0 119897 = 111987910119879119896minus1119897
+ 11987901119879119896119897minus1
119896 + 119897 gt 1
(8)
From (8) the following lemma holds
Lemma 5 The implication
sum119896+119897=119898
119879119896119897
= 119860119898 119898 isin 119885+ (9)
Holds where
119860 = [11986011 11986012119860
2111986022
] (10)
Proof When 119898 = 1 it follows from (8) that
sum119896+119897=1
119879119896119897
= 11987901
+ 11987910
= 119860 (11)
which implies that (9) holds when 119898 = 1 Now suppose that(9) is true when 119898 = 119901 119901 isin 119885+ namely
sum119896+119897=119901
119879119896119897
= 119860119901 (12)
When 119898 = 119901 + 1 we getsum119896+119897=119901+1
119879119896119897
= sum119896+119897=119901+1
(11987910119879119896minus1119897
+ 11987901119879119896119897minus1
)
= sum119896+119897=119901+1
11987910119879119896minus1119897
+ sum119896+119897=119901+1
11987901119879119896119897minus1
= sum119896+119897=119901
11987910119879119896119897
+ sum119896+119897=119901
11987901119879119896119897
= 11987910119860119901 + 119879
01119860119901
= 119860119901+1
(13)
The Scientific World Journal 3
which means that (9) holds when 119898 = 119901 + 1 Reasoning bymathematical induction we can immediately conclude that(9) is true for any119898 isin 119885+This therefore completes the proof
3 Controllability
In this section the sufficient and necessary conditions ofcontrollability for the fractional linear system (1) with twodifferent orders are discussed based on previous definitionsand results Similar to the concepts of controllability for gen-eral fractional linear systems the definition of controllabilityfor fractional linear systems with different orders is given asfollows
Definition 6 The system (1) is called state controllable on[0 119879] if given any state 119909
0 1199091199051
isin 119877119899 there exists a controlinput signal 119906(119905) [0 119879] rarr 119877119898 such that the correspondingsolution of system (1) satisfies 119909(0) = 119909
0and 119909(119905
1) = 1199091199051
1199051isin
[0 119879]Theorem 7 The system (1) is controllable on [0 119905
1] if and only
if the controllability Gramian matrix
119882119888(0 1199051) = int11990510
Φ(1199051minus 120591) 119861119861119879Φ119879 (119905
1minus 120591) 119889120591 (14)
is nonsingular
Proof Suppose that the matrix 119882119888(0 1199051) is nonsingular
Accordingly119882119888(0 1199051) is invertibleThen given an initial state
119909(0) = 1199090
= 0 choose119906 (119905) = 119861119879Φ119879 (119905
1minus 119905)119882minus1
119888(0 1199051) [1199091199051
minus Φ0(1199051) 1199090] (15)
it follows from the solution of system (1) that
119909 (1199051) = Φ
0(1199051) 1199090+ int11990510
Φ(1199051minus 120591) 119861119906 (120591) 119889120591
= Φ0(1199051) 1199090
+ int11990510
Φ(1199051minus 120591) 119861119861119879Φ119879 (119905
1minus 120591)119882minus1
119888(0 1199051)
times [1199091199051
minus Φ0(1199051) 1199090] 119889120591
= Φ0(1199051) 1199090+ int11990510
Φ(1199051minus 120591) 119861119861119879Φ119879 (119905
1minus 120591) 119889120591
times 119882minus1119888
(0 1199051) [1199091199051
minus Φ0(1199051) 1199090]
= Φ0(1199051) 1199090+ 119882119888(0 1199051)119882minus1119888
(0 1199051) [1199091199051
minus Φ0(1199051) 1199090]
= Φ0(1199051) 1199090+ [1199091199051
minus Φ0(1199051) 1199090] = 1199091199051
(16)
Thus the system (1) is controllable on [0 1199051]
We show the converse by contradiction Suppose that thesystem (1) is controllable on [0 119905
1] but the matrix119882
119888(0 1199051) is
singularThen there exists an 119899times1 nonzero vector V such that0 = V119879119882
119888(0 1199051) V = int1199051
0
V119879Φ(1199051minus 120591) 119861119861119879Φ119879 (119905
1minus 120591) V 119889120591
= int11990510
10038171003817100381710038171003817V119879Φ(1199051minus 120591) 11986110038171003817100381710038171003817
2119889120591(17)
which implies
V119879Φ(1199051minus 120591) 119861 equiv 0 (18)
for all 120591 isin [0 1199051] If (1) is controllable there exists an input
that transfers the initial 119909(0) = 1199090to 119909(119905
1) = 0 We choose
1199090= minusΦminus10
(1199051)V then there exists an input such that
119909 (1199051) = minusΦ
0(1199051)Φminus10
(1199051) V + int1199051
0
Φ(1199051minus 120591) 119861119906 (120591) 119889120591 = 0
(19)
that is
V = int11990510
Φ(1199051minus 120591) 119861119906 (120591) 119889120591 (20)
Its premultiplication by V119879 yields
V119879V = int11990510
V119879Φ(1199051minus 120591) 119861119906 (120591) 119889120591 = 0 (21)
which contradicts V = 0 So the matrix 119882119888(0 1199051) is nonsingu-
lar The proof is thus completed
In the following we consider the special case of systems(1) with 119860
12= 11986021
= 0 The systems (1) are reduced to
[[
1198881198631205721199051199091(119905)
1198881198631205731199051199092(119905)]]
= [11986011 00 119860
22
] [1199091 (119905)1199092(119905)] + [1198611119861
2
] 119906 (119905) (22)
which can be rewritten as the following two subsystems1198881198631205721199051199091(119905) = 119860
111199091(119905) + 119861
1119906 (119905) (23a)
1198881198631205731199051199092(119905) = 119860
221199092(119905) + 119861
2119906 (119905) (23b)
Thus the following corollary is true
Corollary 8 The fractional linear system (22) is controllableon [0 119905
1] if and only if the controllability Gramian matrix
119882119888(0 1199051) = int11990510
[[[[[[
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896111198611
infin
sum119897=0
(1199051minus 120591)(119897+1)120573minus1
Γ ((119897 + 1) 120573) 119860119897221198612
]]]]]]
times[[[[[[
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896111198611
infin
sum119897=0
(1199051minus 120591)(119897+1)120573minus1
Γ ((119897 + 1) 120573) 119860119897221198612
]]]]]]
119879
119889120591
(24)
is nonsingular
4 The Scientific World Journal
Proof When 11986012
= 11986021
= 0 system (1) is reduced to thesystem (22) It follows from simple computation that
119879119896119897
=
119868119899 119896 = 119897 = 0
11987911989701 119896 = 0 119897 = 1 2
11987911989610 119897 = 0 119896 = 1 2
0119899 others
(25)
where
11987901
= [0 00 11986022
] 11987910
= [11986011 00 0]
Φ0(119905) = 119879
00+ 11987901
119905120573Γ (120573 + 1) + 119879
02
1199052120573Γ (2120573 + 1) + sdot sdot sdot
+ 11987910
119905120572Γ (120572 + 1) + 119879
20
1199052120572Γ (2120572 + 1) + sdot sdot sdot
= [1198681198991 00 1198681198992
] + 119905120573Γ (120573 + 1) [0 0
0 11986022
]
+ 1199052120573Γ (2120573 + 1)[
0 00 11986022
]2
+ sdot sdot sdot
+ 119905120572Γ (120572 + 1) [11986011 0
0 0]
+ 1199052120572Γ (2120572 + 1)[
11986011
00 0]
2
+ sdot sdot sdot
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
Φ1(119905) = 119879
00
119905120572minus1Γ (120572) + 119879
01
119905120572+120573minus1Γ (120572 + 120573) + 119879
02
119905120572+2120573minus1120574 (120572 + 2120573) + sdot sdot sdot
+ 11987910
1199052120572minus1Γ (2120572) + 119879
20
1199053120572minus1Γ (3120572) + sdot sdot sdot
= 119905120572minus1Γ (120572) [1198681198991 0
0 1198681198992
] + 119905120572+120573minus1Γ (120572 + 120573) [0 0
0 11986022
]
+ 119905120572+2120573minus1Γ (120572 + 2120573)[
0 00 11986022
]2
+ sdot sdot sdot
+ 1199052120572minus1Γ (2120572) [11986011 0
0 0]
+ 1199053120572minus1Γ (3120572)[
11986011
00 0]
2
+ sdot sdot sdot
= [[[[[
infin
sum119896=0
119905(119896+1)120572minus1Γ ((119896 + 1) 120572)119860
119896
110
0infin
sum119897=0
119905120572+119897120573minus1Γ (120572 + 119897120573)119860
119897
22
]]]]]
Φ2(119905) = 119879
00
119905120573minus1Γ (120573) + 119879
01
1199052120573minus1Γ (2120573) + 119879
02
1199053120573minus1Γ (3120573) + sdot sdot sdot
+ 11987910
119905120572+120573minus1Γ (120572 + 120573) + 119879
20
1199052120572+120573minus1Γ (2120572 + 120573) + sdot sdot sdot
= 119905120573minus1Γ (120573) [1198681198991 0
0 1198681198992
] + 1199052120573minus1Γ (2120573) [0 0
0 11986022
]
+ 1199053120573minus1Γ (3120573)[
0 00 11986022
]2
+ sdot sdot sdot
+ 119905120572+120573minus1Γ (120572 + 120573) [11986011 0
0 0] + 1199052120572+120573minus1Γ (2120572 + 120573)[
11986011
00 0]
2
+ sdot sdot sdot
= [[[[[
infin
sum119896=0
119905119896120572+120573minus1Γ (119896120572 + 120573)119860
119896
110
0infin
sum119897=0
119905(119897+1)120573minus1Γ ((119897 + 1) 120573)119860
119897
22
]]]]]
(26)
Therefore the controllability Gramian matrix in theTheorem 7 is reduced to
119882119888(0 1199051) = int11990510
[[[[[[
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896111198611
infin
sum119897=0
(1199051minus 120591)(119897+1)120573minus1
Γ ((119897 + 1) 120573) 119860119897221198612
]]]]]]
times[[[[[[
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896111198611
infin
sum119897=0
(1199051minus 120591)(119897+1)120573minus1
Γ ((119897 + 1) 120573) 119860119897221198612
]]]]]]
119879
119889120591
(27)
This completes the proof
Obviously the following proposition is true
Proposition9 The fractional linear system (22) is controllableif and only if subsystems (23a) and (23b) are all controllable
In the following we consider another special case ofsystem (1) When 120572 = 120573 in the system (1) it is reduced to
119888119863120572119905119909 (119905) = 119860119909 (119905) + 119861119906 (119905) (28)
The Scientific World Journal 5
where
119909 (119905) = [1199091 (119905)1199092(119905)] 119860 = [11986011 119860
1211986021
11986022
] 119861 = [11986111198612
] (29)
Corollary 10 The fractional linear system (28) is controllableon [0 119905
1] if and only if the controllability Gramian matrix
119882119888= int11990510
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896119861119861119879
timesinfin
sum119870=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) (119860119879)119896119889120591(30)
is nonsingular
Proof According to the result of Lemma 5 when 120572 = 120573 wecan obtain
Φ0(119905) =infin
sum119896=0
(1198791198960
119905119896120572Γ (119896120572 + 1) +119879
1198961
119905(119896+1)120572Γ ((119896 + 1) 120572 + 1) + sdot sdot sdot )
= 11987900
+ 11987901
119905120572Γ (120572 + 1) + 119879
02
1199052120572Γ (2120572 + 1)
+ 11987903
1199053120572Γ (3120572 + 1) + sdot sdot sdot + 119879
10
119905120572Γ (120572 + 1)
+ 11987911
1199052120572Γ (2120572 + 1) + 119879
12
1199053120572Γ (3120572 + 1) + sdot sdot sdot
+ 11987920
1199052120572Γ (2120572 + 1) + 119879
21
1199053120572Γ (3120572 + 1)
+ 11987922
1199054120572Γ (4120572 + 1) + sdot sdot sdot + 119879
30
1199053120572Γ (3120572 + 1)
+ 11987931
1199054120572Γ (4120572 + 1) + 119879
32
1199055120572Γ (5120572 + 1) + sdot sdot sdot
sdot sdot sdot= 11987900
+ 119905120572Γ (120572 + 1) (119879
01+ 11987910)
+ 1199052120572Γ (2120572 + 1) (119879
02+ 11987911
+ 11987920)
+ 1199053120572Γ (3120572 + 1) (119879
03+ 11987912
+ 11987921
+ 11987930) + sdot sdot sdot
= [1198681198991 00 1198681198992
] + 119905120572Γ (120572 + 1)119860 + 1199052120572
Γ (2120572 + 1)1198602
+ 1199053120572Γ (3120572 + 1)119860
3 + sdot sdot sdot =infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896(31)
For the same reason as before we get
Φ1(119905) = Φ
2(119905) =
infin
sum119896=0
infin
sum119897=0
119879119896119897
119905(119896+119897+1)120572minus1Γ ((119896 + 119897 + 1) 120572)
=infin
sum119896=0
119905(119896+1)120572minus1Γ ((119896 + 1) 120572)119860
119896(32)
Therefore by simple computation the controllabilityGramian matrix of the system (28) can be obtained as (30)The proof is thus completed
Remark 11 Corollary 10 is equivalent to the result ofTheorem 22 in [21] Therefore Theorem 7 of this paperextends the existing results to a more general case
4 Observability
In this section we treat another fundamental property ofthe fractional linear system with different orders namelyobservability with respect to a linear output Throughoutthe rest of this paper we consider the system (1) with thefollowing output equation
[1199101 (119905)1199102(119905)] = [11986211 119862
1211986221
11986222
] [1199091 (119905)1199092(119905)] (33)
where 1199101
isin 1198771199011 1199102
isin 1198771199012 are the output vectors 1199091
isin1198771198991 119909
2isin 1198771198992 are the state vectors in the system (1) 119862
119894119895isin
119877119901119894times119899119895 119894 119895 = 1 2 are the known constant matrices 1199011+1199012=
119901When 119862
12= 11986221
= 0 the output (33) is reduced to thefollowing simple form
[1199101 (119905)1199102(119905)] = [11986211 0
0 11986222
] [1199091 (119905)1199092(119905)] (34)
which is equivalent to two suboutput equations as follows
1199101= 119862111199091 (35a)
1199102= 119862221199092 (35b)
Definition 12 The system (1) with the output (33) are calledstate observable on [0 119879] if any initial state 119909(0) = 119909
0isin 119877119899
can be uniquely determined by the corresponding systeminput 119906(119905) and system output 119910(119905) for 119905 isin [0 119879]
Define observability Gramian matrix 119882119900(0 119905) as
119882119900(0 119905) = int119905
0
Φ1198790(120591) 119862119879119862Φ
0(120591) 119889120591 (36)
where
119862 = [11986211 11986212119862
2111986222
] (37)
Theorem 13 The system (1) with the output (33) is observableon [0 119905
1] if and only if the observability Gramian matrix
119882119900(0 1199051) = int11990510
Φ1198790(120591) 119862119879119862Φ
0(120591) 119889120591 (38)
is invertible
6 The Scientific World Journal
Proof It follows from Lemma 4 that the output of system (1)has the following expression
119910 (119905) = 119862119909 (119905)= 119862Φ
0(119905) 1199090+ 119862int1199050
Φ (119905 minus 120591) 119861119906 (120591) 119889120591 (39)
It is easy to see from Definition 12 that the observability ofsystem (1) is equivalent to the observability of 119910(119905) given by
119910 (119905) = 119862Φ0(119905) 1199090 (40)
as 119906(119905) = 0Multiplying both sides of (40) by Φ119879
0(119905)119862119879 and integrat-
ing with respect to 119905 from 0 to 1199051 we have
int11990510
Φ1198790(119905) 119862119879119910 (119905) 119889119905 = 119882
119900(0 1199051) 1199090 (41)
Obviously the left-hand side of (41) depends on 119910(119905) and theright-hand side in (41) does not depend on 119910(119905) 119905 isin [0 119905
1]
Thus (41) is a linear algebraic equation of 1199090 If 119882
119900(0 1199051)
is invertible then the initial state 119909(0) = 1199090is uniquely
determined by the corresponding system output 119910(119905) for 119905 isin[0 1199051] Namely the system (1) is observable on [0 119905
1]
Next we show that if 119882119900(0 1199051) is singular for all 119905
1 then
system (1) with the output (33) is not observable Suppose119882119900(0 1199051) is singular then there exists an 119899 times 1 nonzero
constant vector V such that
V119879119882119900(0 1199051) V = int1199051
0
V119879Φ1198790(120591) 119862119879119862Φ
0(120591) V 119889120591
= int11990510
1003817100381710038171003817119862Φ0(120591) V10038171003817100381710038172119889120591 = 0
(42)
which implies that
119862Φ0(120591) V equiv 0 (43)
for all 120591 isin [0 1199051] If we choose 119909(0) = 119909
0= V then the output
(33) is given by
119910 (119905) = 119862Φ0(119905) 1199090= 119862Φ
0(119905) V equiv 0 (44)
Thus the initial state 1199090cannot be uniquely determined by
119910(119905) Therefore the system (1) with the output (33) is notobservable This completes the proof
Remark 14 When 120572 = 120573 in the system (1) Φ0(119905) is
already obtained in the proof of Corollary 10 Therefore theobservability Gramian matrix in Theorem 13 is
119882119900(0 1199051) = int11990510
Φ1198790(120591) 119862119879119862Φ
0(120591) 119889120591
= int11990510
infin
sum119896=0
120591119896120572Γ (119896120572 + 1)(119860
119879)119896119862119879119862infin
sum119896=0
120591119896120572Γ (119896120572 + 1)119860
119896119889120591(45)
which is the observability Gramian matrix in paper [22]by denoting 119864
120572(119860119905120572) = suminfin
119896=0(120591119896120572Γ(119896120572 + 1))119860119896 Therefore
Theorem 13 is actually a generalization of the existing observ-ability results for the fractional linear system
When 11986012
= 11986021
= 0 and 11986212
= 11986221
= 0 the system (1)with the output (33) is reduced to the following state equationand output equation
[[
1198881198631205721199051199091(119905)
1198881198631205731199051199092(119905)]]
= [11986011 00 119860
22
] [1199091 (119905)1199092(119905)] + [1198611119861
2
] 119906 (119905) (46a)
[1199101 (119905)1199102(119905)] = [11986211 0
0 11986222
] [1199091 (119905)1199092(119905)] (46b)
which can be rewritten as the following two subsystems withinput and output
1198881198631205721199051199091(119905) = 119860
111199091(119905) + 119861
1119906 (119905) (47a)
1199101(119905) = 119862
111199091(119905) (47b)
1198881198631205731199051199091(119905) = 119860
221199092(119905) + 119861
2119906 (119905) (48a)
1199102(119905) = 119862
221199092(119905) (48b)
It follows from conditions 11986012
= 11986021
= 0 and 11986212
= 11986221
= 0that
Φ1198790(119905) 119862119879119862Φ
0(119905)
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
119879
[11986211 00 11986222
]119879
[11986211 00 11986222
][[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
The Scientific World Journal 7
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)(119860
119896
11)119879 times 119862119879
1111986211
timesinfin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)(119860
119897
22)119879 times 119862119879
2211986222
timesinfin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
= [119864120572 (119860119879
11119905120572) 11986211987911
times 11986211119864120572(11986011119905120572) 0
0 119864120573(11986011987922119905120573) 11986211987922
times 11986222119864120573(11986022119905120573)]
(49)
where 119864119904(119860119905119904) = suminfin
119896=0(119905119896119904Γ(119896119904 + 1)) 119860119896 is Mittag-Leffler function
Therefore the following corollary holds
Corollary 15 Denote 119864120572(11986011 119905) = 119864
120572(11986011987911119905120572)11986211987911
times11986211119864120572(11986011119905120572) and 119864
120573(11986022 119905) = 119864
120573(11986011987922119905120573)11986211987922
times11986222119864120573(11986022119905120573) Then the system (46a) with the output (46b) is
observable on [0 1199051] if and only if the observability Gramian
matrix
int11990510
[119864120572 (11986011 119905) 00 119864
120573(11986022 119905)] 119889119905 (50)
is nonsingular
The following proposition is also true
Proposition 16 The fractional linear system (46a) with theoutput (46b) is observable if and only if the fractional linearsubsystems (47a) with the output (47b) and (48a) with theoutput (48b) are all observable
5 Conclusions
In this paper the controllability and observability problemsfor fractional linear systems with two different orders havebeen studiedThe sufficient and necessary conditions for statecontrollability and state observability of such systems areestablishedThe results obtained will be useful in the analysisand synthesis of fractional dynamical systems Extendingthe results of this paper toward fractional linear systemsconsisting of 119899 subsystems with different orders is a futurework
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author would like to thank the editors and reviewers fortheir helpful suggestions The present work was supportedby Natural Science Foundation of Yunnan province of China(no 2012FB175) andKey Projects of ScientificResearch Fundsof educational bureau of Yunnan province of China (no2010Z002)
References
[1] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[2] M Ichise Y Nagayanagi and T Kojima ldquoAn analog simulationof non-integer order transfer functions for analysis of electrodeppocessesrdquo Journal of Electroanalytical Chemistry vol 33 no 2pp 253ndash265 1971
[3] D A Benson S W Wheatcraft and M M MeerschaertldquoApplication of a fractional advection-dispersion equationrdquoWater Resources Research vol 36 no 6 pp 1403ndash1412 2000
[4] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000
[5] M Renardy W J Hrusa and J A Nohel MathematicalProblems in Viscoelasticity vol 35 of Pitman Monographs andSurveys in Pure and Applied Mathematics Longman Scientificamp Technical Harlow UK 1987
[6] M Al-Akaidi Fractal Speech Processing Cambridge UniversityPress 2004
[7] P Arena R Caponetto L Fortuna and D Porto NonlinearNoninteger Order Circuits and Systems An Introduction vol 38of World Scientific Series on Nonlinear Science Series A WorldScientific Publishing 2000
[8] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003
[9] D Valerio and J Sa da Costa ldquoNon-integer order controlof a flexible robotrdquo in Proceedings of the IFAC Workshopon Fractional Differentiation and Its Applications (FDA rsquo04)Bordeaux France 2004
[10] C A Monje Y Chen B M Vinagre D Xue and V Feliu-Batlle Fractional-Order Systems and Controls Fundamentalsand Applications Springer 2010
[11] A Oustaloup La Commande CRONE Commande RobustedrsquoOrdre Non Entier Hermes Paris France 1991
[12] D Matignon and B Dandrea-Novel ldquoSome results on con-trollability and observability of finite-dimensional fractionaldifferential systemsrdquo in Computational Engineering in SystemsApplications vol 2 pp 952ndash956 1996
[13] J A TMachado ldquoAnalysis and design of fractional-order digitalcontrol systemsrdquo Systems Analysis Modelling Simulation vol 27no 2-3 pp 107ndash122 1997
[14] I Podlubny ldquoFractional-order systems and 119875119868120582119863120583-controllersrdquoIEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[15] W J Rugh Linear SystemTheory Prentice Hall 1996
8 The Scientific World Journal
[16] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[17] Y Chen H-S Ahn and D Xue ldquoRobust controllability ofinterval fractional order linear time invariant systemsrdquo SignalProcessing vol 86 no 10 pp 2794ndash2802 2006
[18] J L Adams and T T Hartley ldquoFinite-time controllabilityof fractional-order systemsrdquo Journal of Computational andNonlinear Dynamics vol 3 no 2 Article ID 021402 2008
[19] J Wei ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012
[20] A B Shamardan and M R A Moubarak ldquoControllabilityand observability for fractional control systemsrdquo Journal ofFractional Calculus vol 15 pp 25ndash34 1999
[21] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012
[22] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012
[23] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press 1998
[24] H M Srivastava and J J Trujillo Theory and Applications ofFractional Differential Equations Elsevier Science 2006
[25] T Kaczorek ldquoPositive linear systems with different fractionalordersrdquo Bulletin of the Polish Academy of Sciences vol 58 no3 pp 453ndash458 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
The Scientific World Journal 3
which means that (9) holds when 119898 = 119901 + 1 Reasoning bymathematical induction we can immediately conclude that(9) is true for any119898 isin 119885+This therefore completes the proof
3 Controllability
In this section the sufficient and necessary conditions ofcontrollability for the fractional linear system (1) with twodifferent orders are discussed based on previous definitionsand results Similar to the concepts of controllability for gen-eral fractional linear systems the definition of controllabilityfor fractional linear systems with different orders is given asfollows
Definition 6 The system (1) is called state controllable on[0 119879] if given any state 119909
0 1199091199051
isin 119877119899 there exists a controlinput signal 119906(119905) [0 119879] rarr 119877119898 such that the correspondingsolution of system (1) satisfies 119909(0) = 119909
0and 119909(119905
1) = 1199091199051
1199051isin
[0 119879]Theorem 7 The system (1) is controllable on [0 119905
1] if and only
if the controllability Gramian matrix
119882119888(0 1199051) = int11990510
Φ(1199051minus 120591) 119861119861119879Φ119879 (119905
1minus 120591) 119889120591 (14)
is nonsingular
Proof Suppose that the matrix 119882119888(0 1199051) is nonsingular
Accordingly119882119888(0 1199051) is invertibleThen given an initial state
119909(0) = 1199090
= 0 choose119906 (119905) = 119861119879Φ119879 (119905
1minus 119905)119882minus1
119888(0 1199051) [1199091199051
minus Φ0(1199051) 1199090] (15)
it follows from the solution of system (1) that
119909 (1199051) = Φ
0(1199051) 1199090+ int11990510
Φ(1199051minus 120591) 119861119906 (120591) 119889120591
= Φ0(1199051) 1199090
+ int11990510
Φ(1199051minus 120591) 119861119861119879Φ119879 (119905
1minus 120591)119882minus1
119888(0 1199051)
times [1199091199051
minus Φ0(1199051) 1199090] 119889120591
= Φ0(1199051) 1199090+ int11990510
Φ(1199051minus 120591) 119861119861119879Φ119879 (119905
1minus 120591) 119889120591
times 119882minus1119888
(0 1199051) [1199091199051
minus Φ0(1199051) 1199090]
= Φ0(1199051) 1199090+ 119882119888(0 1199051)119882minus1119888
(0 1199051) [1199091199051
minus Φ0(1199051) 1199090]
= Φ0(1199051) 1199090+ [1199091199051
minus Φ0(1199051) 1199090] = 1199091199051
(16)
Thus the system (1) is controllable on [0 1199051]
We show the converse by contradiction Suppose that thesystem (1) is controllable on [0 119905
1] but the matrix119882
119888(0 1199051) is
singularThen there exists an 119899times1 nonzero vector V such that0 = V119879119882
119888(0 1199051) V = int1199051
0
V119879Φ(1199051minus 120591) 119861119861119879Φ119879 (119905
1minus 120591) V 119889120591
= int11990510
10038171003817100381710038171003817V119879Φ(1199051minus 120591) 11986110038171003817100381710038171003817
2119889120591(17)
which implies
V119879Φ(1199051minus 120591) 119861 equiv 0 (18)
for all 120591 isin [0 1199051] If (1) is controllable there exists an input
that transfers the initial 119909(0) = 1199090to 119909(119905
1) = 0 We choose
1199090= minusΦminus10
(1199051)V then there exists an input such that
119909 (1199051) = minusΦ
0(1199051)Φminus10
(1199051) V + int1199051
0
Φ(1199051minus 120591) 119861119906 (120591) 119889120591 = 0
(19)
that is
V = int11990510
Φ(1199051minus 120591) 119861119906 (120591) 119889120591 (20)
Its premultiplication by V119879 yields
V119879V = int11990510
V119879Φ(1199051minus 120591) 119861119906 (120591) 119889120591 = 0 (21)
which contradicts V = 0 So the matrix 119882119888(0 1199051) is nonsingu-
lar The proof is thus completed
In the following we consider the special case of systems(1) with 119860
12= 11986021
= 0 The systems (1) are reduced to
[[
1198881198631205721199051199091(119905)
1198881198631205731199051199092(119905)]]
= [11986011 00 119860
22
] [1199091 (119905)1199092(119905)] + [1198611119861
2
] 119906 (119905) (22)
which can be rewritten as the following two subsystems1198881198631205721199051199091(119905) = 119860
111199091(119905) + 119861
1119906 (119905) (23a)
1198881198631205731199051199092(119905) = 119860
221199092(119905) + 119861
2119906 (119905) (23b)
Thus the following corollary is true
Corollary 8 The fractional linear system (22) is controllableon [0 119905
1] if and only if the controllability Gramian matrix
119882119888(0 1199051) = int11990510
[[[[[[
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896111198611
infin
sum119897=0
(1199051minus 120591)(119897+1)120573minus1
Γ ((119897 + 1) 120573) 119860119897221198612
]]]]]]
times[[[[[[
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896111198611
infin
sum119897=0
(1199051minus 120591)(119897+1)120573minus1
Γ ((119897 + 1) 120573) 119860119897221198612
]]]]]]
119879
119889120591
(24)
is nonsingular
4 The Scientific World Journal
Proof When 11986012
= 11986021
= 0 system (1) is reduced to thesystem (22) It follows from simple computation that
119879119896119897
=
119868119899 119896 = 119897 = 0
11987911989701 119896 = 0 119897 = 1 2
11987911989610 119897 = 0 119896 = 1 2
0119899 others
(25)
where
11987901
= [0 00 11986022
] 11987910
= [11986011 00 0]
Φ0(119905) = 119879
00+ 11987901
119905120573Γ (120573 + 1) + 119879
02
1199052120573Γ (2120573 + 1) + sdot sdot sdot
+ 11987910
119905120572Γ (120572 + 1) + 119879
20
1199052120572Γ (2120572 + 1) + sdot sdot sdot
= [1198681198991 00 1198681198992
] + 119905120573Γ (120573 + 1) [0 0
0 11986022
]
+ 1199052120573Γ (2120573 + 1)[
0 00 11986022
]2
+ sdot sdot sdot
+ 119905120572Γ (120572 + 1) [11986011 0
0 0]
+ 1199052120572Γ (2120572 + 1)[
11986011
00 0]
2
+ sdot sdot sdot
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
Φ1(119905) = 119879
00
119905120572minus1Γ (120572) + 119879
01
119905120572+120573minus1Γ (120572 + 120573) + 119879
02
119905120572+2120573minus1120574 (120572 + 2120573) + sdot sdot sdot
+ 11987910
1199052120572minus1Γ (2120572) + 119879
20
1199053120572minus1Γ (3120572) + sdot sdot sdot
= 119905120572minus1Γ (120572) [1198681198991 0
0 1198681198992
] + 119905120572+120573minus1Γ (120572 + 120573) [0 0
0 11986022
]
+ 119905120572+2120573minus1Γ (120572 + 2120573)[
0 00 11986022
]2
+ sdot sdot sdot
+ 1199052120572minus1Γ (2120572) [11986011 0
0 0]
+ 1199053120572minus1Γ (3120572)[
11986011
00 0]
2
+ sdot sdot sdot
= [[[[[
infin
sum119896=0
119905(119896+1)120572minus1Γ ((119896 + 1) 120572)119860
119896
110
0infin
sum119897=0
119905120572+119897120573minus1Γ (120572 + 119897120573)119860
119897
22
]]]]]
Φ2(119905) = 119879
00
119905120573minus1Γ (120573) + 119879
01
1199052120573minus1Γ (2120573) + 119879
02
1199053120573minus1Γ (3120573) + sdot sdot sdot
+ 11987910
119905120572+120573minus1Γ (120572 + 120573) + 119879
20
1199052120572+120573minus1Γ (2120572 + 120573) + sdot sdot sdot
= 119905120573minus1Γ (120573) [1198681198991 0
0 1198681198992
] + 1199052120573minus1Γ (2120573) [0 0
0 11986022
]
+ 1199053120573minus1Γ (3120573)[
0 00 11986022
]2
+ sdot sdot sdot
+ 119905120572+120573minus1Γ (120572 + 120573) [11986011 0
0 0] + 1199052120572+120573minus1Γ (2120572 + 120573)[
11986011
00 0]
2
+ sdot sdot sdot
= [[[[[
infin
sum119896=0
119905119896120572+120573minus1Γ (119896120572 + 120573)119860
119896
110
0infin
sum119897=0
119905(119897+1)120573minus1Γ ((119897 + 1) 120573)119860
119897
22
]]]]]
(26)
Therefore the controllability Gramian matrix in theTheorem 7 is reduced to
119882119888(0 1199051) = int11990510
[[[[[[
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896111198611
infin
sum119897=0
(1199051minus 120591)(119897+1)120573minus1
Γ ((119897 + 1) 120573) 119860119897221198612
]]]]]]
times[[[[[[
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896111198611
infin
sum119897=0
(1199051minus 120591)(119897+1)120573minus1
Γ ((119897 + 1) 120573) 119860119897221198612
]]]]]]
119879
119889120591
(27)
This completes the proof
Obviously the following proposition is true
Proposition9 The fractional linear system (22) is controllableif and only if subsystems (23a) and (23b) are all controllable
In the following we consider another special case ofsystem (1) When 120572 = 120573 in the system (1) it is reduced to
119888119863120572119905119909 (119905) = 119860119909 (119905) + 119861119906 (119905) (28)
The Scientific World Journal 5
where
119909 (119905) = [1199091 (119905)1199092(119905)] 119860 = [11986011 119860
1211986021
11986022
] 119861 = [11986111198612
] (29)
Corollary 10 The fractional linear system (28) is controllableon [0 119905
1] if and only if the controllability Gramian matrix
119882119888= int11990510
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896119861119861119879
timesinfin
sum119870=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) (119860119879)119896119889120591(30)
is nonsingular
Proof According to the result of Lemma 5 when 120572 = 120573 wecan obtain
Φ0(119905) =infin
sum119896=0
(1198791198960
119905119896120572Γ (119896120572 + 1) +119879
1198961
119905(119896+1)120572Γ ((119896 + 1) 120572 + 1) + sdot sdot sdot )
= 11987900
+ 11987901
119905120572Γ (120572 + 1) + 119879
02
1199052120572Γ (2120572 + 1)
+ 11987903
1199053120572Γ (3120572 + 1) + sdot sdot sdot + 119879
10
119905120572Γ (120572 + 1)
+ 11987911
1199052120572Γ (2120572 + 1) + 119879
12
1199053120572Γ (3120572 + 1) + sdot sdot sdot
+ 11987920
1199052120572Γ (2120572 + 1) + 119879
21
1199053120572Γ (3120572 + 1)
+ 11987922
1199054120572Γ (4120572 + 1) + sdot sdot sdot + 119879
30
1199053120572Γ (3120572 + 1)
+ 11987931
1199054120572Γ (4120572 + 1) + 119879
32
1199055120572Γ (5120572 + 1) + sdot sdot sdot
sdot sdot sdot= 11987900
+ 119905120572Γ (120572 + 1) (119879
01+ 11987910)
+ 1199052120572Γ (2120572 + 1) (119879
02+ 11987911
+ 11987920)
+ 1199053120572Γ (3120572 + 1) (119879
03+ 11987912
+ 11987921
+ 11987930) + sdot sdot sdot
= [1198681198991 00 1198681198992
] + 119905120572Γ (120572 + 1)119860 + 1199052120572
Γ (2120572 + 1)1198602
+ 1199053120572Γ (3120572 + 1)119860
3 + sdot sdot sdot =infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896(31)
For the same reason as before we get
Φ1(119905) = Φ
2(119905) =
infin
sum119896=0
infin
sum119897=0
119879119896119897
119905(119896+119897+1)120572minus1Γ ((119896 + 119897 + 1) 120572)
=infin
sum119896=0
119905(119896+1)120572minus1Γ ((119896 + 1) 120572)119860
119896(32)
Therefore by simple computation the controllabilityGramian matrix of the system (28) can be obtained as (30)The proof is thus completed
Remark 11 Corollary 10 is equivalent to the result ofTheorem 22 in [21] Therefore Theorem 7 of this paperextends the existing results to a more general case
4 Observability
In this section we treat another fundamental property ofthe fractional linear system with different orders namelyobservability with respect to a linear output Throughoutthe rest of this paper we consider the system (1) with thefollowing output equation
[1199101 (119905)1199102(119905)] = [11986211 119862
1211986221
11986222
] [1199091 (119905)1199092(119905)] (33)
where 1199101
isin 1198771199011 1199102
isin 1198771199012 are the output vectors 1199091
isin1198771198991 119909
2isin 1198771198992 are the state vectors in the system (1) 119862
119894119895isin
119877119901119894times119899119895 119894 119895 = 1 2 are the known constant matrices 1199011+1199012=
119901When 119862
12= 11986221
= 0 the output (33) is reduced to thefollowing simple form
[1199101 (119905)1199102(119905)] = [11986211 0
0 11986222
] [1199091 (119905)1199092(119905)] (34)
which is equivalent to two suboutput equations as follows
1199101= 119862111199091 (35a)
1199102= 119862221199092 (35b)
Definition 12 The system (1) with the output (33) are calledstate observable on [0 119879] if any initial state 119909(0) = 119909
0isin 119877119899
can be uniquely determined by the corresponding systeminput 119906(119905) and system output 119910(119905) for 119905 isin [0 119879]
Define observability Gramian matrix 119882119900(0 119905) as
119882119900(0 119905) = int119905
0
Φ1198790(120591) 119862119879119862Φ
0(120591) 119889120591 (36)
where
119862 = [11986211 11986212119862
2111986222
] (37)
Theorem 13 The system (1) with the output (33) is observableon [0 119905
1] if and only if the observability Gramian matrix
119882119900(0 1199051) = int11990510
Φ1198790(120591) 119862119879119862Φ
0(120591) 119889120591 (38)
is invertible
6 The Scientific World Journal
Proof It follows from Lemma 4 that the output of system (1)has the following expression
119910 (119905) = 119862119909 (119905)= 119862Φ
0(119905) 1199090+ 119862int1199050
Φ (119905 minus 120591) 119861119906 (120591) 119889120591 (39)
It is easy to see from Definition 12 that the observability ofsystem (1) is equivalent to the observability of 119910(119905) given by
119910 (119905) = 119862Φ0(119905) 1199090 (40)
as 119906(119905) = 0Multiplying both sides of (40) by Φ119879
0(119905)119862119879 and integrat-
ing with respect to 119905 from 0 to 1199051 we have
int11990510
Φ1198790(119905) 119862119879119910 (119905) 119889119905 = 119882
119900(0 1199051) 1199090 (41)
Obviously the left-hand side of (41) depends on 119910(119905) and theright-hand side in (41) does not depend on 119910(119905) 119905 isin [0 119905
1]
Thus (41) is a linear algebraic equation of 1199090 If 119882
119900(0 1199051)
is invertible then the initial state 119909(0) = 1199090is uniquely
determined by the corresponding system output 119910(119905) for 119905 isin[0 1199051] Namely the system (1) is observable on [0 119905
1]
Next we show that if 119882119900(0 1199051) is singular for all 119905
1 then
system (1) with the output (33) is not observable Suppose119882119900(0 1199051) is singular then there exists an 119899 times 1 nonzero
constant vector V such that
V119879119882119900(0 1199051) V = int1199051
0
V119879Φ1198790(120591) 119862119879119862Φ
0(120591) V 119889120591
= int11990510
1003817100381710038171003817119862Φ0(120591) V10038171003817100381710038172119889120591 = 0
(42)
which implies that
119862Φ0(120591) V equiv 0 (43)
for all 120591 isin [0 1199051] If we choose 119909(0) = 119909
0= V then the output
(33) is given by
119910 (119905) = 119862Φ0(119905) 1199090= 119862Φ
0(119905) V equiv 0 (44)
Thus the initial state 1199090cannot be uniquely determined by
119910(119905) Therefore the system (1) with the output (33) is notobservable This completes the proof
Remark 14 When 120572 = 120573 in the system (1) Φ0(119905) is
already obtained in the proof of Corollary 10 Therefore theobservability Gramian matrix in Theorem 13 is
119882119900(0 1199051) = int11990510
Φ1198790(120591) 119862119879119862Φ
0(120591) 119889120591
= int11990510
infin
sum119896=0
120591119896120572Γ (119896120572 + 1)(119860
119879)119896119862119879119862infin
sum119896=0
120591119896120572Γ (119896120572 + 1)119860
119896119889120591(45)
which is the observability Gramian matrix in paper [22]by denoting 119864
120572(119860119905120572) = suminfin
119896=0(120591119896120572Γ(119896120572 + 1))119860119896 Therefore
Theorem 13 is actually a generalization of the existing observ-ability results for the fractional linear system
When 11986012
= 11986021
= 0 and 11986212
= 11986221
= 0 the system (1)with the output (33) is reduced to the following state equationand output equation
[[
1198881198631205721199051199091(119905)
1198881198631205731199051199092(119905)]]
= [11986011 00 119860
22
] [1199091 (119905)1199092(119905)] + [1198611119861
2
] 119906 (119905) (46a)
[1199101 (119905)1199102(119905)] = [11986211 0
0 11986222
] [1199091 (119905)1199092(119905)] (46b)
which can be rewritten as the following two subsystems withinput and output
1198881198631205721199051199091(119905) = 119860
111199091(119905) + 119861
1119906 (119905) (47a)
1199101(119905) = 119862
111199091(119905) (47b)
1198881198631205731199051199091(119905) = 119860
221199092(119905) + 119861
2119906 (119905) (48a)
1199102(119905) = 119862
221199092(119905) (48b)
It follows from conditions 11986012
= 11986021
= 0 and 11986212
= 11986221
= 0that
Φ1198790(119905) 119862119879119862Φ
0(119905)
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
119879
[11986211 00 11986222
]119879
[11986211 00 11986222
][[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
The Scientific World Journal 7
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)(119860
119896
11)119879 times 119862119879
1111986211
timesinfin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)(119860
119897
22)119879 times 119862119879
2211986222
timesinfin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
= [119864120572 (119860119879
11119905120572) 11986211987911
times 11986211119864120572(11986011119905120572) 0
0 119864120573(11986011987922119905120573) 11986211987922
times 11986222119864120573(11986022119905120573)]
(49)
where 119864119904(119860119905119904) = suminfin
119896=0(119905119896119904Γ(119896119904 + 1)) 119860119896 is Mittag-Leffler function
Therefore the following corollary holds
Corollary 15 Denote 119864120572(11986011 119905) = 119864
120572(11986011987911119905120572)11986211987911
times11986211119864120572(11986011119905120572) and 119864
120573(11986022 119905) = 119864
120573(11986011987922119905120573)11986211987922
times11986222119864120573(11986022119905120573) Then the system (46a) with the output (46b) is
observable on [0 1199051] if and only if the observability Gramian
matrix
int11990510
[119864120572 (11986011 119905) 00 119864
120573(11986022 119905)] 119889119905 (50)
is nonsingular
The following proposition is also true
Proposition 16 The fractional linear system (46a) with theoutput (46b) is observable if and only if the fractional linearsubsystems (47a) with the output (47b) and (48a) with theoutput (48b) are all observable
5 Conclusions
In this paper the controllability and observability problemsfor fractional linear systems with two different orders havebeen studiedThe sufficient and necessary conditions for statecontrollability and state observability of such systems areestablishedThe results obtained will be useful in the analysisand synthesis of fractional dynamical systems Extendingthe results of this paper toward fractional linear systemsconsisting of 119899 subsystems with different orders is a futurework
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author would like to thank the editors and reviewers fortheir helpful suggestions The present work was supportedby Natural Science Foundation of Yunnan province of China(no 2012FB175) andKey Projects of ScientificResearch Fundsof educational bureau of Yunnan province of China (no2010Z002)
References
[1] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[2] M Ichise Y Nagayanagi and T Kojima ldquoAn analog simulationof non-integer order transfer functions for analysis of electrodeppocessesrdquo Journal of Electroanalytical Chemistry vol 33 no 2pp 253ndash265 1971
[3] D A Benson S W Wheatcraft and M M MeerschaertldquoApplication of a fractional advection-dispersion equationrdquoWater Resources Research vol 36 no 6 pp 1403ndash1412 2000
[4] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000
[5] M Renardy W J Hrusa and J A Nohel MathematicalProblems in Viscoelasticity vol 35 of Pitman Monographs andSurveys in Pure and Applied Mathematics Longman Scientificamp Technical Harlow UK 1987
[6] M Al-Akaidi Fractal Speech Processing Cambridge UniversityPress 2004
[7] P Arena R Caponetto L Fortuna and D Porto NonlinearNoninteger Order Circuits and Systems An Introduction vol 38of World Scientific Series on Nonlinear Science Series A WorldScientific Publishing 2000
[8] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003
[9] D Valerio and J Sa da Costa ldquoNon-integer order controlof a flexible robotrdquo in Proceedings of the IFAC Workshopon Fractional Differentiation and Its Applications (FDA rsquo04)Bordeaux France 2004
[10] C A Monje Y Chen B M Vinagre D Xue and V Feliu-Batlle Fractional-Order Systems and Controls Fundamentalsand Applications Springer 2010
[11] A Oustaloup La Commande CRONE Commande RobustedrsquoOrdre Non Entier Hermes Paris France 1991
[12] D Matignon and B Dandrea-Novel ldquoSome results on con-trollability and observability of finite-dimensional fractionaldifferential systemsrdquo in Computational Engineering in SystemsApplications vol 2 pp 952ndash956 1996
[13] J A TMachado ldquoAnalysis and design of fractional-order digitalcontrol systemsrdquo Systems Analysis Modelling Simulation vol 27no 2-3 pp 107ndash122 1997
[14] I Podlubny ldquoFractional-order systems and 119875119868120582119863120583-controllersrdquoIEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[15] W J Rugh Linear SystemTheory Prentice Hall 1996
8 The Scientific World Journal
[16] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[17] Y Chen H-S Ahn and D Xue ldquoRobust controllability ofinterval fractional order linear time invariant systemsrdquo SignalProcessing vol 86 no 10 pp 2794ndash2802 2006
[18] J L Adams and T T Hartley ldquoFinite-time controllabilityof fractional-order systemsrdquo Journal of Computational andNonlinear Dynamics vol 3 no 2 Article ID 021402 2008
[19] J Wei ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012
[20] A B Shamardan and M R A Moubarak ldquoControllabilityand observability for fractional control systemsrdquo Journal ofFractional Calculus vol 15 pp 25ndash34 1999
[21] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012
[22] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012
[23] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press 1998
[24] H M Srivastava and J J Trujillo Theory and Applications ofFractional Differential Equations Elsevier Science 2006
[25] T Kaczorek ldquoPositive linear systems with different fractionalordersrdquo Bulletin of the Polish Academy of Sciences vol 58 no3 pp 453ndash458 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 The Scientific World Journal
Proof When 11986012
= 11986021
= 0 system (1) is reduced to thesystem (22) It follows from simple computation that
119879119896119897
=
119868119899 119896 = 119897 = 0
11987911989701 119896 = 0 119897 = 1 2
11987911989610 119897 = 0 119896 = 1 2
0119899 others
(25)
where
11987901
= [0 00 11986022
] 11987910
= [11986011 00 0]
Φ0(119905) = 119879
00+ 11987901
119905120573Γ (120573 + 1) + 119879
02
1199052120573Γ (2120573 + 1) + sdot sdot sdot
+ 11987910
119905120572Γ (120572 + 1) + 119879
20
1199052120572Γ (2120572 + 1) + sdot sdot sdot
= [1198681198991 00 1198681198992
] + 119905120573Γ (120573 + 1) [0 0
0 11986022
]
+ 1199052120573Γ (2120573 + 1)[
0 00 11986022
]2
+ sdot sdot sdot
+ 119905120572Γ (120572 + 1) [11986011 0
0 0]
+ 1199052120572Γ (2120572 + 1)[
11986011
00 0]
2
+ sdot sdot sdot
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
Φ1(119905) = 119879
00
119905120572minus1Γ (120572) + 119879
01
119905120572+120573minus1Γ (120572 + 120573) + 119879
02
119905120572+2120573minus1120574 (120572 + 2120573) + sdot sdot sdot
+ 11987910
1199052120572minus1Γ (2120572) + 119879
20
1199053120572minus1Γ (3120572) + sdot sdot sdot
= 119905120572minus1Γ (120572) [1198681198991 0
0 1198681198992
] + 119905120572+120573minus1Γ (120572 + 120573) [0 0
0 11986022
]
+ 119905120572+2120573minus1Γ (120572 + 2120573)[
0 00 11986022
]2
+ sdot sdot sdot
+ 1199052120572minus1Γ (2120572) [11986011 0
0 0]
+ 1199053120572minus1Γ (3120572)[
11986011
00 0]
2
+ sdot sdot sdot
= [[[[[
infin
sum119896=0
119905(119896+1)120572minus1Γ ((119896 + 1) 120572)119860
119896
110
0infin
sum119897=0
119905120572+119897120573minus1Γ (120572 + 119897120573)119860
119897
22
]]]]]
Φ2(119905) = 119879
00
119905120573minus1Γ (120573) + 119879
01
1199052120573minus1Γ (2120573) + 119879
02
1199053120573minus1Γ (3120573) + sdot sdot sdot
+ 11987910
119905120572+120573minus1Γ (120572 + 120573) + 119879
20
1199052120572+120573minus1Γ (2120572 + 120573) + sdot sdot sdot
= 119905120573minus1Γ (120573) [1198681198991 0
0 1198681198992
] + 1199052120573minus1Γ (2120573) [0 0
0 11986022
]
+ 1199053120573minus1Γ (3120573)[
0 00 11986022
]2
+ sdot sdot sdot
+ 119905120572+120573minus1Γ (120572 + 120573) [11986011 0
0 0] + 1199052120572+120573minus1Γ (2120572 + 120573)[
11986011
00 0]
2
+ sdot sdot sdot
= [[[[[
infin
sum119896=0
119905119896120572+120573minus1Γ (119896120572 + 120573)119860
119896
110
0infin
sum119897=0
119905(119897+1)120573minus1Γ ((119897 + 1) 120573)119860
119897
22
]]]]]
(26)
Therefore the controllability Gramian matrix in theTheorem 7 is reduced to
119882119888(0 1199051) = int11990510
[[[[[[
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896111198611
infin
sum119897=0
(1199051minus 120591)(119897+1)120573minus1
Γ ((119897 + 1) 120573) 119860119897221198612
]]]]]]
times[[[[[[
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896111198611
infin
sum119897=0
(1199051minus 120591)(119897+1)120573minus1
Γ ((119897 + 1) 120573) 119860119897221198612
]]]]]]
119879
119889120591
(27)
This completes the proof
Obviously the following proposition is true
Proposition9 The fractional linear system (22) is controllableif and only if subsystems (23a) and (23b) are all controllable
In the following we consider another special case ofsystem (1) When 120572 = 120573 in the system (1) it is reduced to
119888119863120572119905119909 (119905) = 119860119909 (119905) + 119861119906 (119905) (28)
The Scientific World Journal 5
where
119909 (119905) = [1199091 (119905)1199092(119905)] 119860 = [11986011 119860
1211986021
11986022
] 119861 = [11986111198612
] (29)
Corollary 10 The fractional linear system (28) is controllableon [0 119905
1] if and only if the controllability Gramian matrix
119882119888= int11990510
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896119861119861119879
timesinfin
sum119870=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) (119860119879)119896119889120591(30)
is nonsingular
Proof According to the result of Lemma 5 when 120572 = 120573 wecan obtain
Φ0(119905) =infin
sum119896=0
(1198791198960
119905119896120572Γ (119896120572 + 1) +119879
1198961
119905(119896+1)120572Γ ((119896 + 1) 120572 + 1) + sdot sdot sdot )
= 11987900
+ 11987901
119905120572Γ (120572 + 1) + 119879
02
1199052120572Γ (2120572 + 1)
+ 11987903
1199053120572Γ (3120572 + 1) + sdot sdot sdot + 119879
10
119905120572Γ (120572 + 1)
+ 11987911
1199052120572Γ (2120572 + 1) + 119879
12
1199053120572Γ (3120572 + 1) + sdot sdot sdot
+ 11987920
1199052120572Γ (2120572 + 1) + 119879
21
1199053120572Γ (3120572 + 1)
+ 11987922
1199054120572Γ (4120572 + 1) + sdot sdot sdot + 119879
30
1199053120572Γ (3120572 + 1)
+ 11987931
1199054120572Γ (4120572 + 1) + 119879
32
1199055120572Γ (5120572 + 1) + sdot sdot sdot
sdot sdot sdot= 11987900
+ 119905120572Γ (120572 + 1) (119879
01+ 11987910)
+ 1199052120572Γ (2120572 + 1) (119879
02+ 11987911
+ 11987920)
+ 1199053120572Γ (3120572 + 1) (119879
03+ 11987912
+ 11987921
+ 11987930) + sdot sdot sdot
= [1198681198991 00 1198681198992
] + 119905120572Γ (120572 + 1)119860 + 1199052120572
Γ (2120572 + 1)1198602
+ 1199053120572Γ (3120572 + 1)119860
3 + sdot sdot sdot =infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896(31)
For the same reason as before we get
Φ1(119905) = Φ
2(119905) =
infin
sum119896=0
infin
sum119897=0
119879119896119897
119905(119896+119897+1)120572minus1Γ ((119896 + 119897 + 1) 120572)
=infin
sum119896=0
119905(119896+1)120572minus1Γ ((119896 + 1) 120572)119860
119896(32)
Therefore by simple computation the controllabilityGramian matrix of the system (28) can be obtained as (30)The proof is thus completed
Remark 11 Corollary 10 is equivalent to the result ofTheorem 22 in [21] Therefore Theorem 7 of this paperextends the existing results to a more general case
4 Observability
In this section we treat another fundamental property ofthe fractional linear system with different orders namelyobservability with respect to a linear output Throughoutthe rest of this paper we consider the system (1) with thefollowing output equation
[1199101 (119905)1199102(119905)] = [11986211 119862
1211986221
11986222
] [1199091 (119905)1199092(119905)] (33)
where 1199101
isin 1198771199011 1199102
isin 1198771199012 are the output vectors 1199091
isin1198771198991 119909
2isin 1198771198992 are the state vectors in the system (1) 119862
119894119895isin
119877119901119894times119899119895 119894 119895 = 1 2 are the known constant matrices 1199011+1199012=
119901When 119862
12= 11986221
= 0 the output (33) is reduced to thefollowing simple form
[1199101 (119905)1199102(119905)] = [11986211 0
0 11986222
] [1199091 (119905)1199092(119905)] (34)
which is equivalent to two suboutput equations as follows
1199101= 119862111199091 (35a)
1199102= 119862221199092 (35b)
Definition 12 The system (1) with the output (33) are calledstate observable on [0 119879] if any initial state 119909(0) = 119909
0isin 119877119899
can be uniquely determined by the corresponding systeminput 119906(119905) and system output 119910(119905) for 119905 isin [0 119879]
Define observability Gramian matrix 119882119900(0 119905) as
119882119900(0 119905) = int119905
0
Φ1198790(120591) 119862119879119862Φ
0(120591) 119889120591 (36)
where
119862 = [11986211 11986212119862
2111986222
] (37)
Theorem 13 The system (1) with the output (33) is observableon [0 119905
1] if and only if the observability Gramian matrix
119882119900(0 1199051) = int11990510
Φ1198790(120591) 119862119879119862Φ
0(120591) 119889120591 (38)
is invertible
6 The Scientific World Journal
Proof It follows from Lemma 4 that the output of system (1)has the following expression
119910 (119905) = 119862119909 (119905)= 119862Φ
0(119905) 1199090+ 119862int1199050
Φ (119905 minus 120591) 119861119906 (120591) 119889120591 (39)
It is easy to see from Definition 12 that the observability ofsystem (1) is equivalent to the observability of 119910(119905) given by
119910 (119905) = 119862Φ0(119905) 1199090 (40)
as 119906(119905) = 0Multiplying both sides of (40) by Φ119879
0(119905)119862119879 and integrat-
ing with respect to 119905 from 0 to 1199051 we have
int11990510
Φ1198790(119905) 119862119879119910 (119905) 119889119905 = 119882
119900(0 1199051) 1199090 (41)
Obviously the left-hand side of (41) depends on 119910(119905) and theright-hand side in (41) does not depend on 119910(119905) 119905 isin [0 119905
1]
Thus (41) is a linear algebraic equation of 1199090 If 119882
119900(0 1199051)
is invertible then the initial state 119909(0) = 1199090is uniquely
determined by the corresponding system output 119910(119905) for 119905 isin[0 1199051] Namely the system (1) is observable on [0 119905
1]
Next we show that if 119882119900(0 1199051) is singular for all 119905
1 then
system (1) with the output (33) is not observable Suppose119882119900(0 1199051) is singular then there exists an 119899 times 1 nonzero
constant vector V such that
V119879119882119900(0 1199051) V = int1199051
0
V119879Φ1198790(120591) 119862119879119862Φ
0(120591) V 119889120591
= int11990510
1003817100381710038171003817119862Φ0(120591) V10038171003817100381710038172119889120591 = 0
(42)
which implies that
119862Φ0(120591) V equiv 0 (43)
for all 120591 isin [0 1199051] If we choose 119909(0) = 119909
0= V then the output
(33) is given by
119910 (119905) = 119862Φ0(119905) 1199090= 119862Φ
0(119905) V equiv 0 (44)
Thus the initial state 1199090cannot be uniquely determined by
119910(119905) Therefore the system (1) with the output (33) is notobservable This completes the proof
Remark 14 When 120572 = 120573 in the system (1) Φ0(119905) is
already obtained in the proof of Corollary 10 Therefore theobservability Gramian matrix in Theorem 13 is
119882119900(0 1199051) = int11990510
Φ1198790(120591) 119862119879119862Φ
0(120591) 119889120591
= int11990510
infin
sum119896=0
120591119896120572Γ (119896120572 + 1)(119860
119879)119896119862119879119862infin
sum119896=0
120591119896120572Γ (119896120572 + 1)119860
119896119889120591(45)
which is the observability Gramian matrix in paper [22]by denoting 119864
120572(119860119905120572) = suminfin
119896=0(120591119896120572Γ(119896120572 + 1))119860119896 Therefore
Theorem 13 is actually a generalization of the existing observ-ability results for the fractional linear system
When 11986012
= 11986021
= 0 and 11986212
= 11986221
= 0 the system (1)with the output (33) is reduced to the following state equationand output equation
[[
1198881198631205721199051199091(119905)
1198881198631205731199051199092(119905)]]
= [11986011 00 119860
22
] [1199091 (119905)1199092(119905)] + [1198611119861
2
] 119906 (119905) (46a)
[1199101 (119905)1199102(119905)] = [11986211 0
0 11986222
] [1199091 (119905)1199092(119905)] (46b)
which can be rewritten as the following two subsystems withinput and output
1198881198631205721199051199091(119905) = 119860
111199091(119905) + 119861
1119906 (119905) (47a)
1199101(119905) = 119862
111199091(119905) (47b)
1198881198631205731199051199091(119905) = 119860
221199092(119905) + 119861
2119906 (119905) (48a)
1199102(119905) = 119862
221199092(119905) (48b)
It follows from conditions 11986012
= 11986021
= 0 and 11986212
= 11986221
= 0that
Φ1198790(119905) 119862119879119862Φ
0(119905)
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
119879
[11986211 00 11986222
]119879
[11986211 00 11986222
][[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
The Scientific World Journal 7
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)(119860
119896
11)119879 times 119862119879
1111986211
timesinfin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)(119860
119897
22)119879 times 119862119879
2211986222
timesinfin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
= [119864120572 (119860119879
11119905120572) 11986211987911
times 11986211119864120572(11986011119905120572) 0
0 119864120573(11986011987922119905120573) 11986211987922
times 11986222119864120573(11986022119905120573)]
(49)
where 119864119904(119860119905119904) = suminfin
119896=0(119905119896119904Γ(119896119904 + 1)) 119860119896 is Mittag-Leffler function
Therefore the following corollary holds
Corollary 15 Denote 119864120572(11986011 119905) = 119864
120572(11986011987911119905120572)11986211987911
times11986211119864120572(11986011119905120572) and 119864
120573(11986022 119905) = 119864
120573(11986011987922119905120573)11986211987922
times11986222119864120573(11986022119905120573) Then the system (46a) with the output (46b) is
observable on [0 1199051] if and only if the observability Gramian
matrix
int11990510
[119864120572 (11986011 119905) 00 119864
120573(11986022 119905)] 119889119905 (50)
is nonsingular
The following proposition is also true
Proposition 16 The fractional linear system (46a) with theoutput (46b) is observable if and only if the fractional linearsubsystems (47a) with the output (47b) and (48a) with theoutput (48b) are all observable
5 Conclusions
In this paper the controllability and observability problemsfor fractional linear systems with two different orders havebeen studiedThe sufficient and necessary conditions for statecontrollability and state observability of such systems areestablishedThe results obtained will be useful in the analysisand synthesis of fractional dynamical systems Extendingthe results of this paper toward fractional linear systemsconsisting of 119899 subsystems with different orders is a futurework
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author would like to thank the editors and reviewers fortheir helpful suggestions The present work was supportedby Natural Science Foundation of Yunnan province of China(no 2012FB175) andKey Projects of ScientificResearch Fundsof educational bureau of Yunnan province of China (no2010Z002)
References
[1] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[2] M Ichise Y Nagayanagi and T Kojima ldquoAn analog simulationof non-integer order transfer functions for analysis of electrodeppocessesrdquo Journal of Electroanalytical Chemistry vol 33 no 2pp 253ndash265 1971
[3] D A Benson S W Wheatcraft and M M MeerschaertldquoApplication of a fractional advection-dispersion equationrdquoWater Resources Research vol 36 no 6 pp 1403ndash1412 2000
[4] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000
[5] M Renardy W J Hrusa and J A Nohel MathematicalProblems in Viscoelasticity vol 35 of Pitman Monographs andSurveys in Pure and Applied Mathematics Longman Scientificamp Technical Harlow UK 1987
[6] M Al-Akaidi Fractal Speech Processing Cambridge UniversityPress 2004
[7] P Arena R Caponetto L Fortuna and D Porto NonlinearNoninteger Order Circuits and Systems An Introduction vol 38of World Scientific Series on Nonlinear Science Series A WorldScientific Publishing 2000
[8] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003
[9] D Valerio and J Sa da Costa ldquoNon-integer order controlof a flexible robotrdquo in Proceedings of the IFAC Workshopon Fractional Differentiation and Its Applications (FDA rsquo04)Bordeaux France 2004
[10] C A Monje Y Chen B M Vinagre D Xue and V Feliu-Batlle Fractional-Order Systems and Controls Fundamentalsand Applications Springer 2010
[11] A Oustaloup La Commande CRONE Commande RobustedrsquoOrdre Non Entier Hermes Paris France 1991
[12] D Matignon and B Dandrea-Novel ldquoSome results on con-trollability and observability of finite-dimensional fractionaldifferential systemsrdquo in Computational Engineering in SystemsApplications vol 2 pp 952ndash956 1996
[13] J A TMachado ldquoAnalysis and design of fractional-order digitalcontrol systemsrdquo Systems Analysis Modelling Simulation vol 27no 2-3 pp 107ndash122 1997
[14] I Podlubny ldquoFractional-order systems and 119875119868120582119863120583-controllersrdquoIEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[15] W J Rugh Linear SystemTheory Prentice Hall 1996
8 The Scientific World Journal
[16] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[17] Y Chen H-S Ahn and D Xue ldquoRobust controllability ofinterval fractional order linear time invariant systemsrdquo SignalProcessing vol 86 no 10 pp 2794ndash2802 2006
[18] J L Adams and T T Hartley ldquoFinite-time controllabilityof fractional-order systemsrdquo Journal of Computational andNonlinear Dynamics vol 3 no 2 Article ID 021402 2008
[19] J Wei ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012
[20] A B Shamardan and M R A Moubarak ldquoControllabilityand observability for fractional control systemsrdquo Journal ofFractional Calculus vol 15 pp 25ndash34 1999
[21] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012
[22] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012
[23] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press 1998
[24] H M Srivastava and J J Trujillo Theory and Applications ofFractional Differential Equations Elsevier Science 2006
[25] T Kaczorek ldquoPositive linear systems with different fractionalordersrdquo Bulletin of the Polish Academy of Sciences vol 58 no3 pp 453ndash458 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
The Scientific World Journal 5
where
119909 (119905) = [1199091 (119905)1199092(119905)] 119860 = [11986011 119860
1211986021
11986022
] 119861 = [11986111198612
] (29)
Corollary 10 The fractional linear system (28) is controllableon [0 119905
1] if and only if the controllability Gramian matrix
119882119888= int11990510
infin
sum119896=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) 119860119896119861119861119879
timesinfin
sum119870=0
(1199051minus 120591)(119896+1)120572minus1
Γ ((119896 + 1) 120572) (119860119879)119896119889120591(30)
is nonsingular
Proof According to the result of Lemma 5 when 120572 = 120573 wecan obtain
Φ0(119905) =infin
sum119896=0
(1198791198960
119905119896120572Γ (119896120572 + 1) +119879
1198961
119905(119896+1)120572Γ ((119896 + 1) 120572 + 1) + sdot sdot sdot )
= 11987900
+ 11987901
119905120572Γ (120572 + 1) + 119879
02
1199052120572Γ (2120572 + 1)
+ 11987903
1199053120572Γ (3120572 + 1) + sdot sdot sdot + 119879
10
119905120572Γ (120572 + 1)
+ 11987911
1199052120572Γ (2120572 + 1) + 119879
12
1199053120572Γ (3120572 + 1) + sdot sdot sdot
+ 11987920
1199052120572Γ (2120572 + 1) + 119879
21
1199053120572Γ (3120572 + 1)
+ 11987922
1199054120572Γ (4120572 + 1) + sdot sdot sdot + 119879
30
1199053120572Γ (3120572 + 1)
+ 11987931
1199054120572Γ (4120572 + 1) + 119879
32
1199055120572Γ (5120572 + 1) + sdot sdot sdot
sdot sdot sdot= 11987900
+ 119905120572Γ (120572 + 1) (119879
01+ 11987910)
+ 1199052120572Γ (2120572 + 1) (119879
02+ 11987911
+ 11987920)
+ 1199053120572Γ (3120572 + 1) (119879
03+ 11987912
+ 11987921
+ 11987930) + sdot sdot sdot
= [1198681198991 00 1198681198992
] + 119905120572Γ (120572 + 1)119860 + 1199052120572
Γ (2120572 + 1)1198602
+ 1199053120572Γ (3120572 + 1)119860
3 + sdot sdot sdot =infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896(31)
For the same reason as before we get
Φ1(119905) = Φ
2(119905) =
infin
sum119896=0
infin
sum119897=0
119879119896119897
119905(119896+119897+1)120572minus1Γ ((119896 + 119897 + 1) 120572)
=infin
sum119896=0
119905(119896+1)120572minus1Γ ((119896 + 1) 120572)119860
119896(32)
Therefore by simple computation the controllabilityGramian matrix of the system (28) can be obtained as (30)The proof is thus completed
Remark 11 Corollary 10 is equivalent to the result ofTheorem 22 in [21] Therefore Theorem 7 of this paperextends the existing results to a more general case
4 Observability
In this section we treat another fundamental property ofthe fractional linear system with different orders namelyobservability with respect to a linear output Throughoutthe rest of this paper we consider the system (1) with thefollowing output equation
[1199101 (119905)1199102(119905)] = [11986211 119862
1211986221
11986222
] [1199091 (119905)1199092(119905)] (33)
where 1199101
isin 1198771199011 1199102
isin 1198771199012 are the output vectors 1199091
isin1198771198991 119909
2isin 1198771198992 are the state vectors in the system (1) 119862
119894119895isin
119877119901119894times119899119895 119894 119895 = 1 2 are the known constant matrices 1199011+1199012=
119901When 119862
12= 11986221
= 0 the output (33) is reduced to thefollowing simple form
[1199101 (119905)1199102(119905)] = [11986211 0
0 11986222
] [1199091 (119905)1199092(119905)] (34)
which is equivalent to two suboutput equations as follows
1199101= 119862111199091 (35a)
1199102= 119862221199092 (35b)
Definition 12 The system (1) with the output (33) are calledstate observable on [0 119879] if any initial state 119909(0) = 119909
0isin 119877119899
can be uniquely determined by the corresponding systeminput 119906(119905) and system output 119910(119905) for 119905 isin [0 119879]
Define observability Gramian matrix 119882119900(0 119905) as
119882119900(0 119905) = int119905
0
Φ1198790(120591) 119862119879119862Φ
0(120591) 119889120591 (36)
where
119862 = [11986211 11986212119862
2111986222
] (37)
Theorem 13 The system (1) with the output (33) is observableon [0 119905
1] if and only if the observability Gramian matrix
119882119900(0 1199051) = int11990510
Φ1198790(120591) 119862119879119862Φ
0(120591) 119889120591 (38)
is invertible
6 The Scientific World Journal
Proof It follows from Lemma 4 that the output of system (1)has the following expression
119910 (119905) = 119862119909 (119905)= 119862Φ
0(119905) 1199090+ 119862int1199050
Φ (119905 minus 120591) 119861119906 (120591) 119889120591 (39)
It is easy to see from Definition 12 that the observability ofsystem (1) is equivalent to the observability of 119910(119905) given by
119910 (119905) = 119862Φ0(119905) 1199090 (40)
as 119906(119905) = 0Multiplying both sides of (40) by Φ119879
0(119905)119862119879 and integrat-
ing with respect to 119905 from 0 to 1199051 we have
int11990510
Φ1198790(119905) 119862119879119910 (119905) 119889119905 = 119882
119900(0 1199051) 1199090 (41)
Obviously the left-hand side of (41) depends on 119910(119905) and theright-hand side in (41) does not depend on 119910(119905) 119905 isin [0 119905
1]
Thus (41) is a linear algebraic equation of 1199090 If 119882
119900(0 1199051)
is invertible then the initial state 119909(0) = 1199090is uniquely
determined by the corresponding system output 119910(119905) for 119905 isin[0 1199051] Namely the system (1) is observable on [0 119905
1]
Next we show that if 119882119900(0 1199051) is singular for all 119905
1 then
system (1) with the output (33) is not observable Suppose119882119900(0 1199051) is singular then there exists an 119899 times 1 nonzero
constant vector V such that
V119879119882119900(0 1199051) V = int1199051
0
V119879Φ1198790(120591) 119862119879119862Φ
0(120591) V 119889120591
= int11990510
1003817100381710038171003817119862Φ0(120591) V10038171003817100381710038172119889120591 = 0
(42)
which implies that
119862Φ0(120591) V equiv 0 (43)
for all 120591 isin [0 1199051] If we choose 119909(0) = 119909
0= V then the output
(33) is given by
119910 (119905) = 119862Φ0(119905) 1199090= 119862Φ
0(119905) V equiv 0 (44)
Thus the initial state 1199090cannot be uniquely determined by
119910(119905) Therefore the system (1) with the output (33) is notobservable This completes the proof
Remark 14 When 120572 = 120573 in the system (1) Φ0(119905) is
already obtained in the proof of Corollary 10 Therefore theobservability Gramian matrix in Theorem 13 is
119882119900(0 1199051) = int11990510
Φ1198790(120591) 119862119879119862Φ
0(120591) 119889120591
= int11990510
infin
sum119896=0
120591119896120572Γ (119896120572 + 1)(119860
119879)119896119862119879119862infin
sum119896=0
120591119896120572Γ (119896120572 + 1)119860
119896119889120591(45)
which is the observability Gramian matrix in paper [22]by denoting 119864
120572(119860119905120572) = suminfin
119896=0(120591119896120572Γ(119896120572 + 1))119860119896 Therefore
Theorem 13 is actually a generalization of the existing observ-ability results for the fractional linear system
When 11986012
= 11986021
= 0 and 11986212
= 11986221
= 0 the system (1)with the output (33) is reduced to the following state equationand output equation
[[
1198881198631205721199051199091(119905)
1198881198631205731199051199092(119905)]]
= [11986011 00 119860
22
] [1199091 (119905)1199092(119905)] + [1198611119861
2
] 119906 (119905) (46a)
[1199101 (119905)1199102(119905)] = [11986211 0
0 11986222
] [1199091 (119905)1199092(119905)] (46b)
which can be rewritten as the following two subsystems withinput and output
1198881198631205721199051199091(119905) = 119860
111199091(119905) + 119861
1119906 (119905) (47a)
1199101(119905) = 119862
111199091(119905) (47b)
1198881198631205731199051199091(119905) = 119860
221199092(119905) + 119861
2119906 (119905) (48a)
1199102(119905) = 119862
221199092(119905) (48b)
It follows from conditions 11986012
= 11986021
= 0 and 11986212
= 11986221
= 0that
Φ1198790(119905) 119862119879119862Φ
0(119905)
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
119879
[11986211 00 11986222
]119879
[11986211 00 11986222
][[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
The Scientific World Journal 7
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)(119860
119896
11)119879 times 119862119879
1111986211
timesinfin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)(119860
119897
22)119879 times 119862119879
2211986222
timesinfin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
= [119864120572 (119860119879
11119905120572) 11986211987911
times 11986211119864120572(11986011119905120572) 0
0 119864120573(11986011987922119905120573) 11986211987922
times 11986222119864120573(11986022119905120573)]
(49)
where 119864119904(119860119905119904) = suminfin
119896=0(119905119896119904Γ(119896119904 + 1)) 119860119896 is Mittag-Leffler function
Therefore the following corollary holds
Corollary 15 Denote 119864120572(11986011 119905) = 119864
120572(11986011987911119905120572)11986211987911
times11986211119864120572(11986011119905120572) and 119864
120573(11986022 119905) = 119864
120573(11986011987922119905120573)11986211987922
times11986222119864120573(11986022119905120573) Then the system (46a) with the output (46b) is
observable on [0 1199051] if and only if the observability Gramian
matrix
int11990510
[119864120572 (11986011 119905) 00 119864
120573(11986022 119905)] 119889119905 (50)
is nonsingular
The following proposition is also true
Proposition 16 The fractional linear system (46a) with theoutput (46b) is observable if and only if the fractional linearsubsystems (47a) with the output (47b) and (48a) with theoutput (48b) are all observable
5 Conclusions
In this paper the controllability and observability problemsfor fractional linear systems with two different orders havebeen studiedThe sufficient and necessary conditions for statecontrollability and state observability of such systems areestablishedThe results obtained will be useful in the analysisand synthesis of fractional dynamical systems Extendingthe results of this paper toward fractional linear systemsconsisting of 119899 subsystems with different orders is a futurework
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author would like to thank the editors and reviewers fortheir helpful suggestions The present work was supportedby Natural Science Foundation of Yunnan province of China(no 2012FB175) andKey Projects of ScientificResearch Fundsof educational bureau of Yunnan province of China (no2010Z002)
References
[1] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[2] M Ichise Y Nagayanagi and T Kojima ldquoAn analog simulationof non-integer order transfer functions for analysis of electrodeppocessesrdquo Journal of Electroanalytical Chemistry vol 33 no 2pp 253ndash265 1971
[3] D A Benson S W Wheatcraft and M M MeerschaertldquoApplication of a fractional advection-dispersion equationrdquoWater Resources Research vol 36 no 6 pp 1403ndash1412 2000
[4] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000
[5] M Renardy W J Hrusa and J A Nohel MathematicalProblems in Viscoelasticity vol 35 of Pitman Monographs andSurveys in Pure and Applied Mathematics Longman Scientificamp Technical Harlow UK 1987
[6] M Al-Akaidi Fractal Speech Processing Cambridge UniversityPress 2004
[7] P Arena R Caponetto L Fortuna and D Porto NonlinearNoninteger Order Circuits and Systems An Introduction vol 38of World Scientific Series on Nonlinear Science Series A WorldScientific Publishing 2000
[8] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003
[9] D Valerio and J Sa da Costa ldquoNon-integer order controlof a flexible robotrdquo in Proceedings of the IFAC Workshopon Fractional Differentiation and Its Applications (FDA rsquo04)Bordeaux France 2004
[10] C A Monje Y Chen B M Vinagre D Xue and V Feliu-Batlle Fractional-Order Systems and Controls Fundamentalsand Applications Springer 2010
[11] A Oustaloup La Commande CRONE Commande RobustedrsquoOrdre Non Entier Hermes Paris France 1991
[12] D Matignon and B Dandrea-Novel ldquoSome results on con-trollability and observability of finite-dimensional fractionaldifferential systemsrdquo in Computational Engineering in SystemsApplications vol 2 pp 952ndash956 1996
[13] J A TMachado ldquoAnalysis and design of fractional-order digitalcontrol systemsrdquo Systems Analysis Modelling Simulation vol 27no 2-3 pp 107ndash122 1997
[14] I Podlubny ldquoFractional-order systems and 119875119868120582119863120583-controllersrdquoIEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[15] W J Rugh Linear SystemTheory Prentice Hall 1996
8 The Scientific World Journal
[16] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[17] Y Chen H-S Ahn and D Xue ldquoRobust controllability ofinterval fractional order linear time invariant systemsrdquo SignalProcessing vol 86 no 10 pp 2794ndash2802 2006
[18] J L Adams and T T Hartley ldquoFinite-time controllabilityof fractional-order systemsrdquo Journal of Computational andNonlinear Dynamics vol 3 no 2 Article ID 021402 2008
[19] J Wei ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012
[20] A B Shamardan and M R A Moubarak ldquoControllabilityand observability for fractional control systemsrdquo Journal ofFractional Calculus vol 15 pp 25ndash34 1999
[21] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012
[22] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012
[23] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press 1998
[24] H M Srivastava and J J Trujillo Theory and Applications ofFractional Differential Equations Elsevier Science 2006
[25] T Kaczorek ldquoPositive linear systems with different fractionalordersrdquo Bulletin of the Polish Academy of Sciences vol 58 no3 pp 453ndash458 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 The Scientific World Journal
Proof It follows from Lemma 4 that the output of system (1)has the following expression
119910 (119905) = 119862119909 (119905)= 119862Φ
0(119905) 1199090+ 119862int1199050
Φ (119905 minus 120591) 119861119906 (120591) 119889120591 (39)
It is easy to see from Definition 12 that the observability ofsystem (1) is equivalent to the observability of 119910(119905) given by
119910 (119905) = 119862Φ0(119905) 1199090 (40)
as 119906(119905) = 0Multiplying both sides of (40) by Φ119879
0(119905)119862119879 and integrat-
ing with respect to 119905 from 0 to 1199051 we have
int11990510
Φ1198790(119905) 119862119879119910 (119905) 119889119905 = 119882
119900(0 1199051) 1199090 (41)
Obviously the left-hand side of (41) depends on 119910(119905) and theright-hand side in (41) does not depend on 119910(119905) 119905 isin [0 119905
1]
Thus (41) is a linear algebraic equation of 1199090 If 119882
119900(0 1199051)
is invertible then the initial state 119909(0) = 1199090is uniquely
determined by the corresponding system output 119910(119905) for 119905 isin[0 1199051] Namely the system (1) is observable on [0 119905
1]
Next we show that if 119882119900(0 1199051) is singular for all 119905
1 then
system (1) with the output (33) is not observable Suppose119882119900(0 1199051) is singular then there exists an 119899 times 1 nonzero
constant vector V such that
V119879119882119900(0 1199051) V = int1199051
0
V119879Φ1198790(120591) 119862119879119862Φ
0(120591) V 119889120591
= int11990510
1003817100381710038171003817119862Φ0(120591) V10038171003817100381710038172119889120591 = 0
(42)
which implies that
119862Φ0(120591) V equiv 0 (43)
for all 120591 isin [0 1199051] If we choose 119909(0) = 119909
0= V then the output
(33) is given by
119910 (119905) = 119862Φ0(119905) 1199090= 119862Φ
0(119905) V equiv 0 (44)
Thus the initial state 1199090cannot be uniquely determined by
119910(119905) Therefore the system (1) with the output (33) is notobservable This completes the proof
Remark 14 When 120572 = 120573 in the system (1) Φ0(119905) is
already obtained in the proof of Corollary 10 Therefore theobservability Gramian matrix in Theorem 13 is
119882119900(0 1199051) = int11990510
Φ1198790(120591) 119862119879119862Φ
0(120591) 119889120591
= int11990510
infin
sum119896=0
120591119896120572Γ (119896120572 + 1)(119860
119879)119896119862119879119862infin
sum119896=0
120591119896120572Γ (119896120572 + 1)119860
119896119889120591(45)
which is the observability Gramian matrix in paper [22]by denoting 119864
120572(119860119905120572) = suminfin
119896=0(120591119896120572Γ(119896120572 + 1))119860119896 Therefore
Theorem 13 is actually a generalization of the existing observ-ability results for the fractional linear system
When 11986012
= 11986021
= 0 and 11986212
= 11986221
= 0 the system (1)with the output (33) is reduced to the following state equationand output equation
[[
1198881198631205721199051199091(119905)
1198881198631205731199051199092(119905)]]
= [11986011 00 119860
22
] [1199091 (119905)1199092(119905)] + [1198611119861
2
] 119906 (119905) (46a)
[1199101 (119905)1199102(119905)] = [11986211 0
0 11986222
] [1199091 (119905)1199092(119905)] (46b)
which can be rewritten as the following two subsystems withinput and output
1198881198631205721199051199091(119905) = 119860
111199091(119905) + 119861
1119906 (119905) (47a)
1199101(119905) = 119862
111199091(119905) (47b)
1198881198631205731199051199091(119905) = 119860
221199092(119905) + 119861
2119906 (119905) (48a)
1199102(119905) = 119862
221199092(119905) (48b)
It follows from conditions 11986012
= 11986021
= 0 and 11986212
= 11986221
= 0that
Φ1198790(119905) 119862119879119862Φ
0(119905)
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
119879
[11986211 00 11986222
]119879
[11986211 00 11986222
][[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
The Scientific World Journal 7
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)(119860
119896
11)119879 times 119862119879
1111986211
timesinfin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)(119860
119897
22)119879 times 119862119879
2211986222
timesinfin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
= [119864120572 (119860119879
11119905120572) 11986211987911
times 11986211119864120572(11986011119905120572) 0
0 119864120573(11986011987922119905120573) 11986211987922
times 11986222119864120573(11986022119905120573)]
(49)
where 119864119904(119860119905119904) = suminfin
119896=0(119905119896119904Γ(119896119904 + 1)) 119860119896 is Mittag-Leffler function
Therefore the following corollary holds
Corollary 15 Denote 119864120572(11986011 119905) = 119864
120572(11986011987911119905120572)11986211987911
times11986211119864120572(11986011119905120572) and 119864
120573(11986022 119905) = 119864
120573(11986011987922119905120573)11986211987922
times11986222119864120573(11986022119905120573) Then the system (46a) with the output (46b) is
observable on [0 1199051] if and only if the observability Gramian
matrix
int11990510
[119864120572 (11986011 119905) 00 119864
120573(11986022 119905)] 119889119905 (50)
is nonsingular
The following proposition is also true
Proposition 16 The fractional linear system (46a) with theoutput (46b) is observable if and only if the fractional linearsubsystems (47a) with the output (47b) and (48a) with theoutput (48b) are all observable
5 Conclusions
In this paper the controllability and observability problemsfor fractional linear systems with two different orders havebeen studiedThe sufficient and necessary conditions for statecontrollability and state observability of such systems areestablishedThe results obtained will be useful in the analysisand synthesis of fractional dynamical systems Extendingthe results of this paper toward fractional linear systemsconsisting of 119899 subsystems with different orders is a futurework
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author would like to thank the editors and reviewers fortheir helpful suggestions The present work was supportedby Natural Science Foundation of Yunnan province of China(no 2012FB175) andKey Projects of ScientificResearch Fundsof educational bureau of Yunnan province of China (no2010Z002)
References
[1] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[2] M Ichise Y Nagayanagi and T Kojima ldquoAn analog simulationof non-integer order transfer functions for analysis of electrodeppocessesrdquo Journal of Electroanalytical Chemistry vol 33 no 2pp 253ndash265 1971
[3] D A Benson S W Wheatcraft and M M MeerschaertldquoApplication of a fractional advection-dispersion equationrdquoWater Resources Research vol 36 no 6 pp 1403ndash1412 2000
[4] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000
[5] M Renardy W J Hrusa and J A Nohel MathematicalProblems in Viscoelasticity vol 35 of Pitman Monographs andSurveys in Pure and Applied Mathematics Longman Scientificamp Technical Harlow UK 1987
[6] M Al-Akaidi Fractal Speech Processing Cambridge UniversityPress 2004
[7] P Arena R Caponetto L Fortuna and D Porto NonlinearNoninteger Order Circuits and Systems An Introduction vol 38of World Scientific Series on Nonlinear Science Series A WorldScientific Publishing 2000
[8] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003
[9] D Valerio and J Sa da Costa ldquoNon-integer order controlof a flexible robotrdquo in Proceedings of the IFAC Workshopon Fractional Differentiation and Its Applications (FDA rsquo04)Bordeaux France 2004
[10] C A Monje Y Chen B M Vinagre D Xue and V Feliu-Batlle Fractional-Order Systems and Controls Fundamentalsand Applications Springer 2010
[11] A Oustaloup La Commande CRONE Commande RobustedrsquoOrdre Non Entier Hermes Paris France 1991
[12] D Matignon and B Dandrea-Novel ldquoSome results on con-trollability and observability of finite-dimensional fractionaldifferential systemsrdquo in Computational Engineering in SystemsApplications vol 2 pp 952ndash956 1996
[13] J A TMachado ldquoAnalysis and design of fractional-order digitalcontrol systemsrdquo Systems Analysis Modelling Simulation vol 27no 2-3 pp 107ndash122 1997
[14] I Podlubny ldquoFractional-order systems and 119875119868120582119863120583-controllersrdquoIEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[15] W J Rugh Linear SystemTheory Prentice Hall 1996
8 The Scientific World Journal
[16] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[17] Y Chen H-S Ahn and D Xue ldquoRobust controllability ofinterval fractional order linear time invariant systemsrdquo SignalProcessing vol 86 no 10 pp 2794ndash2802 2006
[18] J L Adams and T T Hartley ldquoFinite-time controllabilityof fractional-order systemsrdquo Journal of Computational andNonlinear Dynamics vol 3 no 2 Article ID 021402 2008
[19] J Wei ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012
[20] A B Shamardan and M R A Moubarak ldquoControllabilityand observability for fractional control systemsrdquo Journal ofFractional Calculus vol 15 pp 25ndash34 1999
[21] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012
[22] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012
[23] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press 1998
[24] H M Srivastava and J J Trujillo Theory and Applications ofFractional Differential Equations Elsevier Science 2006
[25] T Kaczorek ldquoPositive linear systems with different fractionalordersrdquo Bulletin of the Polish Academy of Sciences vol 58 no3 pp 453ndash458 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
The Scientific World Journal 7
= [[[[[
infin
sum119896=0
119905119896120572Γ (119896120572 + 1)(119860
119896
11)119879 times 119862119879
1111986211
timesinfin
sum119896=0
119905119896120572Γ (119896120572 + 1)119860
119896
110
0infin
sum119897=0
119905119897120573Γ (119897120573 + 1)(119860
119897
22)119879 times 119862119879
2211986222
timesinfin
sum119897=0
119905119897120573Γ (119897120573 + 1)119860
119897
22
]]]]]
= [119864120572 (119860119879
11119905120572) 11986211987911
times 11986211119864120572(11986011119905120572) 0
0 119864120573(11986011987922119905120573) 11986211987922
times 11986222119864120573(11986022119905120573)]
(49)
where 119864119904(119860119905119904) = suminfin
119896=0(119905119896119904Γ(119896119904 + 1)) 119860119896 is Mittag-Leffler function
Therefore the following corollary holds
Corollary 15 Denote 119864120572(11986011 119905) = 119864
120572(11986011987911119905120572)11986211987911
times11986211119864120572(11986011119905120572) and 119864
120573(11986022 119905) = 119864
120573(11986011987922119905120573)11986211987922
times11986222119864120573(11986022119905120573) Then the system (46a) with the output (46b) is
observable on [0 1199051] if and only if the observability Gramian
matrix
int11990510
[119864120572 (11986011 119905) 00 119864
120573(11986022 119905)] 119889119905 (50)
is nonsingular
The following proposition is also true
Proposition 16 The fractional linear system (46a) with theoutput (46b) is observable if and only if the fractional linearsubsystems (47a) with the output (47b) and (48a) with theoutput (48b) are all observable
5 Conclusions
In this paper the controllability and observability problemsfor fractional linear systems with two different orders havebeen studiedThe sufficient and necessary conditions for statecontrollability and state observability of such systems areestablishedThe results obtained will be useful in the analysisand synthesis of fractional dynamical systems Extendingthe results of this paper toward fractional linear systemsconsisting of 119899 subsystems with different orders is a futurework
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author would like to thank the editors and reviewers fortheir helpful suggestions The present work was supportedby Natural Science Foundation of Yunnan province of China(no 2012FB175) andKey Projects of ScientificResearch Fundsof educational bureau of Yunnan province of China (no2010Z002)
References
[1] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[2] M Ichise Y Nagayanagi and T Kojima ldquoAn analog simulationof non-integer order transfer functions for analysis of electrodeppocessesrdquo Journal of Electroanalytical Chemistry vol 33 no 2pp 253ndash265 1971
[3] D A Benson S W Wheatcraft and M M MeerschaertldquoApplication of a fractional advection-dispersion equationrdquoWater Resources Research vol 36 no 6 pp 1403ndash1412 2000
[4] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000
[5] M Renardy W J Hrusa and J A Nohel MathematicalProblems in Viscoelasticity vol 35 of Pitman Monographs andSurveys in Pure and Applied Mathematics Longman Scientificamp Technical Harlow UK 1987
[6] M Al-Akaidi Fractal Speech Processing Cambridge UniversityPress 2004
[7] P Arena R Caponetto L Fortuna and D Porto NonlinearNoninteger Order Circuits and Systems An Introduction vol 38of World Scientific Series on Nonlinear Science Series A WorldScientific Publishing 2000
[8] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003
[9] D Valerio and J Sa da Costa ldquoNon-integer order controlof a flexible robotrdquo in Proceedings of the IFAC Workshopon Fractional Differentiation and Its Applications (FDA rsquo04)Bordeaux France 2004
[10] C A Monje Y Chen B M Vinagre D Xue and V Feliu-Batlle Fractional-Order Systems and Controls Fundamentalsand Applications Springer 2010
[11] A Oustaloup La Commande CRONE Commande RobustedrsquoOrdre Non Entier Hermes Paris France 1991
[12] D Matignon and B Dandrea-Novel ldquoSome results on con-trollability and observability of finite-dimensional fractionaldifferential systemsrdquo in Computational Engineering in SystemsApplications vol 2 pp 952ndash956 1996
[13] J A TMachado ldquoAnalysis and design of fractional-order digitalcontrol systemsrdquo Systems Analysis Modelling Simulation vol 27no 2-3 pp 107ndash122 1997
[14] I Podlubny ldquoFractional-order systems and 119875119868120582119863120583-controllersrdquoIEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999
[15] W J Rugh Linear SystemTheory Prentice Hall 1996
8 The Scientific World Journal
[16] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[17] Y Chen H-S Ahn and D Xue ldquoRobust controllability ofinterval fractional order linear time invariant systemsrdquo SignalProcessing vol 86 no 10 pp 2794ndash2802 2006
[18] J L Adams and T T Hartley ldquoFinite-time controllabilityof fractional-order systemsrdquo Journal of Computational andNonlinear Dynamics vol 3 no 2 Article ID 021402 2008
[19] J Wei ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012
[20] A B Shamardan and M R A Moubarak ldquoControllabilityand observability for fractional control systemsrdquo Journal ofFractional Calculus vol 15 pp 25ndash34 1999
[21] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012
[22] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012
[23] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press 1998
[24] H M Srivastava and J J Trujillo Theory and Applications ofFractional Differential Equations Elsevier Science 2006
[25] T Kaczorek ldquoPositive linear systems with different fractionalordersrdquo Bulletin of the Polish Academy of Sciences vol 58 no3 pp 453ndash458 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 The Scientific World Journal
[16] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[17] Y Chen H-S Ahn and D Xue ldquoRobust controllability ofinterval fractional order linear time invariant systemsrdquo SignalProcessing vol 86 no 10 pp 2794ndash2802 2006
[18] J L Adams and T T Hartley ldquoFinite-time controllabilityof fractional-order systemsrdquo Journal of Computational andNonlinear Dynamics vol 3 no 2 Article ID 021402 2008
[19] J Wei ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012
[20] A B Shamardan and M R A Moubarak ldquoControllabilityand observability for fractional control systemsrdquo Journal ofFractional Calculus vol 15 pp 25ndash34 1999
[21] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012
[22] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012
[23] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 Academic Press 1998
[24] H M Srivastava and J J Trujillo Theory and Applications ofFractional Differential Equations Elsevier Science 2006
[25] T Kaczorek ldquoPositive linear systems with different fractionalordersrdquo Bulletin of the Polish Academy of Sciences vol 58 no3 pp 453ndash458 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of