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
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)
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
[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
![Page 1: Research Article Controllability and Observability of ...fractional dynamical systems by using xed point theorem. In recent paper [ ], necessary and su cient conditions of ... controllability](https://reader033.documents.pub/reader033/viewer/2022060902/609eadc79e5ea943eb627713/html5/thumbnails/1.jpg)
![Page 2: Research Article Controllability and Observability of ...fractional dynamical systems by using xed point theorem. In recent paper [ ], necessary and su cient conditions of ... controllability](https://reader033.documents.pub/reader033/viewer/2022060902/609eadc79e5ea943eb627713/html5/thumbnails/2.jpg)
![Page 3: Research Article Controllability and Observability of ...fractional dynamical systems by using xed point theorem. In recent paper [ ], necessary and su cient conditions of ... controllability](https://reader033.documents.pub/reader033/viewer/2022060902/609eadc79e5ea943eb627713/html5/thumbnails/3.jpg)
![Page 4: Research Article Controllability and Observability of ...fractional dynamical systems by using xed point theorem. In recent paper [ ], necessary and su cient conditions of ... controllability](https://reader033.documents.pub/reader033/viewer/2022060902/609eadc79e5ea943eb627713/html5/thumbnails/4.jpg)
![Page 5: Research Article Controllability and Observability of ...fractional dynamical systems by using xed point theorem. In recent paper [ ], necessary and su cient conditions of ... controllability](https://reader033.documents.pub/reader033/viewer/2022060902/609eadc79e5ea943eb627713/html5/thumbnails/5.jpg)
![Page 6: Research Article Controllability and Observability of ...fractional dynamical systems by using xed point theorem. In recent paper [ ], necessary and su cient conditions of ... controllability](https://reader033.documents.pub/reader033/viewer/2022060902/609eadc79e5ea943eb627713/html5/thumbnails/6.jpg)
![Page 7: Research Article Controllability and Observability of ...fractional dynamical systems by using xed point theorem. In recent paper [ ], necessary and su cient conditions of ... controllability](https://reader033.documents.pub/reader033/viewer/2022060902/609eadc79e5ea943eb627713/html5/thumbnails/7.jpg)
![Page 8: Research Article Controllability and Observability of ...fractional dynamical systems by using xed point theorem. In recent paper [ ], necessary and su cient conditions of ... controllability](https://reader033.documents.pub/reader033/viewer/2022060902/609eadc79e5ea943eb627713/html5/thumbnails/8.jpg)
![Page 9: Research Article Controllability and Observability of ...fractional dynamical systems by using xed point theorem. In recent paper [ ], necessary and su cient conditions of ... controllability](https://reader033.documents.pub/reader033/viewer/2022060902/609eadc79e5ea943eb627713/html5/thumbnails/9.jpg)
