Research Article Optimal Control of Switching Topology ...downloads.hindawi.com/journals/mpe/2014/268541.pdf · Research Article Optimal Control of Switching Topology Networks EduardoMojica-Nava,

Research ArticleOptimal Control of Switching Topology Networks

Eduardo Mojica-Nava1 Jimmy Salgado2 Duvan Tellez2 and Alvaro Lopez3

1 Department of Electrical and Electronics Engineering National University of Colombia Carrera 30 No 45-03 Bogota Colombia2Departamento de Ingenierıa Electrica y Electronica Universidad de Los Andes Carrera 1 No 18A-12 Bogota Colombia3 Programa de Ingenierıa Electronica y Telecomunicaciones Universidad Catolica de Colombia Avenida CaracasNo 46-72 Bogota Colombia

Correspondence should be addressed to Eduardo Mojica-Nava eamojicaucatolicaeduco

Received 24 March 2014 Revised 20 June 2014 Accepted 21 June 2014 Published 7 July 2014

Academic Editor Kh S Mekheimer

Copyright copy 2014 Eduardo Mojica-Nava et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We present an extension of a previously proposed approach based on the method of moments for solving the optimal controlproblem for a switching system considering now a continuous external input This method is based on the transformation of anonlinear nonconvex optimal control problem into an equivalent optimal control problemwith linear and convex structure whichallows us to obtain an equivalent convex formulation more appropriate to be solved by high-performance numerical computingFinally the design of optimal logic-based controllers for networked systems with a dynamic topology is presented as an applicationof this work

1 Introduction

Thedesign of optimal logic-based controllers while satisfyingphysical and operational constraints plays a fundamentalrole in modern technological systems such as networkeddynamic systems where the information exchange is mod-eled via a communication graph In this work the com-munication graph has a dynamic topology which can beused to model asynchronous consensus problems [1 2]Networked systems with a dynamic topology are known asswitching networks [1 3] The switching network is modeledas a switching control system In general switched controlsystems are characterized by a set of several continuous statedynamics with a logic-based controller which determinessimultaneously a sequence of switching times a sequence ofmodes and a continuous external input In the last yearsseveral researchers have considered the optimal control ofswitched systems (see [4] and references therein) However itis widely perceived that the best numerical methods availablefor switched optimal control problems involve mixed integerprogramming (MIP) [5 6] Even though great progress hasbeen made in recent years in improving these methods theMIP is an NP-hard problem so scalability is problematic

One solution for this problem is to use the traditional non-linear programming techniques such as sequential quadraticprogramming (SQP) which reduces dramatically the compu-tational complexity over existing approaches [7] Howeverthe development of computational efficient tools to theseproblems is still an important research area

Recently an alternative approach to solve effectively theoptimal control problem for an autonomous (without anexternal input) nonlinear switched system based on prob-ability measures has been presented in [4] Following thisprevious work we present in this paper an extension toinclude an external continuous inputwhich presents new the-oretical results This extension is done by considering the setof probabilitymeasures associatedwith the set of both controlvariables that is switching signal and continuous input Themain contribution of this paper is twofold First the naturalextension including a continuous external input into thecontrol problem opens several possibilities for more real andmore complex applications Secondly the dynamic networkconsensus problem is treated by the proposed approach Theproposed approach is based on the fundamental concepts ofthe theory of moments introduced for global optimization

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 268541 9 pageshttpdxdoiorg1011552014268541

2 Mathematical Problems in Engineering

with polynomials in [8 9] and later extended to nonlinear 0-1 programs using an explicit equivalent positive semidefiniteprogram in [10] We also use some results recently intro-duced for optimal control problems with the control variableexpressed as polynomials [11ndash13] The moment approachfor global polynomial optimization based on semidefiniteprogramming (SDP) is consistent as it simplifies andor hasbetter convergence propertieswhen solving convex problemsThis approach works properly when the control variables(ie the switching signal and the continuous external input)can be expressed as polynomials Essentially this methodtransforms a nonlinear nonconvex optimal control problem(ie the switched system) into an equivalent optimal controlproblem with linear and convex structure which allows us toobtain an equivalent convex formulation more appropriateto be solved by high-performance numerical computing Inother words we transform a given controllable switchedsystem into a controllable continuous system with a linearand convex structure in the control variables

This paper is organized as follows In Section 2 we presentthe problem statement and the basic concepts The maincontribution based on the method of moments is developedin Section 3 Some applications of the proposed method fornetworks with switching topology are presented in Section 4and finally in Section 5 some conclusions are drawn

2 Problem Statement

21 Switching Optimal Control Problem A switched optimalcontrol problem can be stated in a general form as followsGiven the switched system in (2) and a Bolza cost functional119869 the switched optimal control problem is given by

min120590(119905)119906(119905)

119869 (1199050 119905119891 119909 (119905) 120590 (119905) 119906 (119905)) = int

119905119891

1199050

119871120590(119909 (119905) 119906 (119905)) 119889119905

(1)

subject to the state equation

(119905) = 119891120590(119909 (119905) 119906 (119905)) (2)

where 119909(119905) is the state 119891119894 R119899 times R119898 997891rarr R119899 is the 119894th

vector field 119909(1199050) = 119909

0are fixed initial values 119906(119905) isin R119898

is the continuous control function and 120590 [1199050 119905119891] 997891rarr Q isin

0 1 2 119902 is a piecewise constant function of time with1199050and 119905119891as the initial and final times respectively Every

mode of operation corresponds to a specific subsystem (119905) =

119891119894(119909(119905) 119906(119905)) for some 119894 isin Q and the switching signal 120590

determines which subsystem is followed at each point of timeinto the interval [119905

0 119905119891] The control input 120590 is a measurable

function In addition we consider a non-Zeno behaviorthat is we exclude an infinite switching accumulation pointin time Finally we assume that the state does not havejump discontinuities Moreover for the interval [119905

0 119905119891] the

control functions must be chosen so that the initial and finalconditions are satisfied and the running switched costs 119871

120590(119905)

R+ timesR119899 997891rarr R are continuously differentiable for each 120590 isin Q

Additionally we assume that each function 119891119894and 119871

119894can

be expressed as polynomials in the control variable 119906 Ingeneral we have the functionals as

119871119894(119909 119906) =

119873

sum

119895=0

120573119894119895(119909) 119906119895 (3)

and the state equations as

119891119894(119909 119906) =

119873

sum

119895=0

120585119894119895(119909) 119906119895

(4)

where119873 is the maximum degree of the polynomials in termsof 119906 setting in zero the coefficients of monomials of degreeless than 119873 and 120573

119894119895(119909) and 120585

119894119895(119909) are the coefficients in

119909 related to the respective polynomial term in 119906119895 Generalconditions for the subsystems functions should be satisfied

Assumption 1 The nonlinear switched system satisfiesgrowth Lipschitz continuity and coercivity qualificationsconcerning the mappings 119891

119894 R119899 times R119898 997891rarr R119899 and

119871119894 R119899R119899 timesR119898 997891rarr R to ensure existence of solutions of (2)

The switching optimal control problem can have theusual variations of fixed or free initial or terminal state freeterminal time and so forth

Definition 2 A control for the switched system in (2) is atriplet consisting of

(a) a finite sequence of modes 120590

(b) the optimal value for the continuous external control119906

(c) a finite sequence of switching times such that 1199050lt 1199051lt

sdot sdot sdot lt 119905119902= 119905119891

22 A Continuous Polynomial Representation A continuousnonswitched control system can be obtained from (2) as it hasbeen shown in [14]The polynomial expression in the controlvariable able to mimic the behavior of the switched systemis developed using a variable V which works as a controlvariable

A polynomial expression in the new control variable V(119905)can be obtained through Lagrange polynomial interpolationquotients defined as in [15]

119897119896(V) =

119902

prod

119894=0119894 = 119896

(V minus 119894)(119896 minus 119894)

(5)

and a constraint polynomial set defined by Ω = V isin R |

119892(V) = prod119902119896=0(V minus 119896) = 0

Consider a switched systemof the formgiven in (2)Thereexists a unique continuous state system with polynomial

Mathematical Problems in Engineering 3

dependence in the control variable V F(119909 V 119906) of degree 119902in V with V isin Ω as follows (see [14] for details)

= F (119909 V 119906) =119902

sum

119894=0

119873

sum

119895=0

120585119894119895(119909) 119906119895

119897119896(V)

=

119902

sum

119894=0

119902

sum

119896=0

119873

sum

119895=0

120585119894119895(119909) 120572119894119896V119896119906119895

(6)

where120572119894119896are the coefficient that results from the factorization

of Lagrange polynomial interpolation quotients Similarlywe define a polynomial equivalent representation for therunning cost 119871

120590(119905) Consider a switched running cost of the

form given in (1) There exists a unique polynomial runningcost equationL(119909 V 119906) of degree 119902 in V with V isin Ω which isan extension of the previous work [4 14] as follows

L (119909 V 119906) =119902

sum

119894=0

119873

sum

119895=0

120573119894119895(119909) 119906119895

119897119896(V)

=

119902

sum

119894=0

119902

sum

119896=0

119873

sum

119895=0

120573119894119895(119909) 120572119894119896V119896119906119895

(7)

We now can state an equivalent optimal control problembased on the equivalent polynomial representation presentedbefore as

minL (119909 V 119906) st Equation (6) (8)

Traditional optimization solvers perform poorly on polyno-mial constraints that are nonconvex with a disjoint feasibleset as the necessary constraint qualification is violated Thismakes this problem intractable directly by traditional nonlin-ear optimization solvers We propose then a convexificationof the equivalent polynomial representation using the specialstructure of the control variables V and 119906 which improves theoptimization process

3 The Method of Moments andSDP-Relaxations

In this section we present the first main result of this workThe inclusion of a external input in the optimal switchedsystem using the method of moments Before the method isintroduced some basic definitions of probability measuresare necessary

Let Δ = ΩtimesR119898 be the set of admissible controls V(119905) and119906(119905) The set of probability measures associated with the set Δis

Λ = 120583 = 120583119905119905isin[1199050119905119891]

supp (120583119905)

sub Δ = Ω timesR119898

ae 119905 isin [1199050 119905119891]

(9)

where 120583 is a probability measure supported (supp(sdot)) in ΔWe obtain a problem reformulation defined on the set Λ ofprobability measures as follows

min120583isinΛ

119869 (119909 V 119906) = int119905119891

1199050

intΔ

L (119909 V 119906) 119889120583 (V 119906) 119889119905 (10)

subject to

(119905) = intΔ

F (119909 V 119906) 119889120583 (V 119906) 119909 (1199050) = 1199090 120583 isin Λ (11)

This reformulation is an infinite dimensional linear pro-gram which is not tractable as it stands Considering thespecial polynomial structure of the problem in terms of thecontrol variables the theory of moments can be used toobtain a semidefinite program or linear matrix inequalityrelaxation with finitelymany constraints and variables calledthe method of moments

31 The Moments Approach When an optimal control prob-lem can be stated in terms of polynomial expressions in thecontrol variables we can use the method of moments Bymeans of moments variables an equivalent convex formula-tion can be obtained which is more appropriate to be solvedby numerical computing The method of moments takes aproper formulation in probability measures of a nonconvexoptimization problem ([9 16] and references therein) andthus when the problem can be stated in terms of polynomialexpressions in the control variable we can transform themeasures into algebraic moments to obtain a new convexproblem defined in a new set of variables that represent themoments of every measure [8 9 13]

We define the space of moments as

Γ = 119898 = 119898119896119895 119898119896119895= intΔ

V119896119906119895119889120583 (V 119906) 120583 isin Λ (Δ) (12)

where Λ(Δ) is the convex set of all probability measuressupported in Δ In addition a sequence 119898 = 119898

119896119895 has a

representingmeasure120583 supported inΩ only if thesemomentsare restricted to be entries on positive semidefinite momentsand localizing matrices [8 10]

In this work we are dealing with polynomials in twovariables 119906 and V Let a basis be defined for the vec-tor space of real-valued polynomials of degree at most 119889as 1 V 119906 V2 V119906 1199062 V2119889 1199062119889 Given a moment vector119898 = [1119898

10 11989801 11989820 11989811 11989820 119898

21198890 119898

02119889]⊤ associ-

ated with the polynomial basisThemomentmatrix119872119889(119898) is

the blockmatrix with rows and columns labeled lexicograph-ically with the polynomial basis it follows that the row 119906119896V119897

with column 119906119894V119895 entry of 119872119889(119898) is 119898

119894+119897119895+119896 For example

we consider the case with 119889 = 1 which corresponds to theapplication presented in Section 4 The polynomial basis is1 V 119906 V2 V119906 1199062 the corresponding vector of moments is119898 =

[111989810 11989801 11989820 11989811 11989802]⊤ The respective moment matrix

is

1198721(119898) = [

[

1 1198981011989801

119898101198982011989811

119898011198981111989802

]

]

(13)

The localizingmatrix is defined based on the correspond-ing moment matrix whose positivity is directly related tothe existence of a representing measure with support in Ωas follows Consider the set Ω defined by the polynomial120573(V) = 120573

0+ 1205731V + sdot sdot sdot + 120573

119889V120578 It can be represented in


moment variables as 120573(119898) = 1205730+ 120573111989810+ sdot sdot sdot + 120573

1205781198981205780 or

in compact form as 120573(119898) = sum120578

120574=01205731205741198981205740 Suppose that the

entries of the corresponding moment matrix are 1198981205880 with

120588 isin [0 1 2119889] Thus every entry of the localizing matrixis defined as 119897

1205880= sum119889

120574=01205731205740119898120574+1205880

Note that the localizingmatrix has the same dimension of the moment matrix thatis if 119889 = 1 and the polynomial 120573 = V + 2V2 then we have themoment matrix as above and the localizing matrix is

1198721(120573119898) = [

[

11989810+ 21198982011989820+ 21198983011989811+ 211989821

11989820+ 21198983011989830+ 21198984011989821+ 211989831

11989811+ 21198982111989821+ 21198983111989812+ 211989822

]

]

(14)

More details on the method of moments can be found in [1017]

Since 119869 is a polynomial in V of degree 119902 and in 119906 of degree119873 the criterion intL 119889120583 is linear in the moment variablesHence we replace 120583 with the finite sequence119898 = 119898

119895119896 of all

its moments We can then express the linear combination ofthe functional 119869 and the space of moments Γ as the followingproblem in moments variables

min119898119896119895isinΓ

119869 = int

119905119891

1199050

sum

119894

sum

119896

sum

119895

120573119894119895(119909) 120572119894119896119898119896119895 (15)

subject to

(119905) = sum

119894

sum

119896

sum

119895

120585119894119895(119909) 120572119894119896119898119896119895

119909 isin R119899

119898 isin Γ 119909 (0) = 1199090

(16)

where 120572119894119896are the coefficients resulting from the factorization

of (15) as aboveWe now have a problem inmoment variableswhich can be solved by efficient computational tools as shownbelow

32 Semidefinite Programs We can use the functional andthe state equation with moment structure to rewrite therelaxed formulation as a semidefinite program (SDP) Firstwe need to redefine the control set Δ to be coherent with thedefinitions of localizingmatrix and representation resultsWetreat the polynomial 119892(V) as two opposite inequalities that is1198921(V) = 119892(V) ge 0 and 119892

2(V) = minus119892(V) ge 0 and we redefine the

compact set to be Δ = 119892119894(V) ge 0 119894 = 1 2 Also we define

a prefixed order of relaxation which is directly related to thenumber of subsystems

Let 119908 be the degree of the polynomial 119892(V) which isequivalent to the degree of the polynomials 119892

1and 119892

2

Considering its parity we have that if119908 is even (odd) then 119903 =1199082 (119903 = (119908+1)2) In this case 119903 corresponds to the prefixedorder of relaxation We use a direct transcription method toobtain an SDP to be solved through a nonlinear programming(NLP) algorithm [18] Using a discretizationmethod the firststep is to split the time interval [119905

0 119905119891] intoN subintervals as

1199050lt 1199051lt 1199052lt sdot sdot sdot lt 119905N = 119905

119891 with a time step ℎ predefined

by the user The integral term in the functional is implicitlyrepresented as an additional state variable transforming theoriginal problem inBolza form into a problem inMayer form

which is a standard transformation [18]Therefore we obtaina set of discrete equations in moment variables Thus theoptimal control problem can be formulated as an SDP

Consider a fixed 119905 in the time interval [1199050 119905119891] and

let Assumption 1 hold We can state the following SDP ofrelaxation order 119903 (SDP

119903)

SDP119903 for every 119897 = 1 2 N a semidefinite program

SDP119903can be described by

119869lowast

119903= min119898(119905119897)

ℎ

2

Nminus1

sum

119897=0

L (119909 (119905119897) 119898 (119905

119897))

= min119898(119905119897)

ℎ

2

Nminus1

sum

119897=0

sum

119894

sum

119896

sum

119895

120573119894119895(119909 (119905119897)) 120572119894119896119898119896119895(119905119897)

st

119909 (119905119897+1) = 119909 (119905

119897) + ℎsum

119894

sum

119896

sum

119895

120585119894119895(119909 (119905119897)) 120572119894119896119898119896119895(119905119897)

119909 (1199050) = 1199090

119872119903(119898 (119905119897)) ⪰ 0 119872

0(1198921119898(119905119897)) ⪰ 0

1198720(1198922119898(119905119897)) ⪰ 0

(17)

In order to solve a traditional NLP we use the particularcharacteristics of the moment and localizing matrices formWe know that the moment and localizing matrices are sym-metric positive definite which implies that every principalsubdeterminant is positive [12] So we can use the set ofsubdeterminants of each matrix as algebraic constraints

33 ProblemAnalysis and Optimal Solutions Once a solutionhas been obtained in a subinterval [119905

119895minus1 119905119895] we obtain a vector

of moments 119898lowast(119905119895) = [119898

lowast

1(119905119895) 119898lowast

2(119905119895) 119898

lowast

119903(119905119895)] We need

to verify if we have attained an optimal solution Based on arank condition of the moment matrix [17] we can test if wehave obtained a global optimum at a relaxation order 119903 Alsobased on the same rank condition we can check whether theoptimal solution is unique or if it is a convex combination ofseveral minimizers The next result is based on an importantresult presented in [17] and used in [10] for optimization of0-1 problems

Lemma 3 Suppose that the SDP119903is solved with a moment

vector solution119898lowast(119905119897) for the 119903th relaxation If the flat extension

condition holds that is

]119903= rank119872

119903(119898 (119905119897)) = rank119872

119903minus1(119898 (119905119897)) (18)

then the global optimum has been reached and the problem has]119903global minimizers

It should be noted that for the particular case of min-imum order of relaxation the rank condition yields ]

119903=

rank119872119903(119898(119905119897)) = rank119872

0(119898(119905119897)) = 1 because119872

0= 1

Using the previous result we can state some relationsbetween solutions that can be used to obtain the switching


signal and the continuous control input for every 119905119897 First we

state the following result valid for the unique solution case

Theorem 4 If for a fixed 119905119897isin [1199050 119905119891] problem (17) is

solved and the rank condition (18) is verified with ]119903=

rank119872119903(119898lowast

(119905119897)) = 1 then the vector of moments 119898lowast(119905

119897) has

attained a unique optimal global solution and therefore theoptimal switching signal of problem (1) is obtained as

120590lowast

(119905119897) = 119898

lowast

10(119905119897) (19)

and the optimal continuous control input is obtained as

119906lowast

(119905119897) = 119898

lowast

01(119905119897) (20)

where 119898lowast10(119905119897) and 119898lowast

01(119905119897) are the first and second component

of the vector of moments119898lowast(119905119897) respectively

Proof Assume that the SDPr-program has been solved for afixed 119905

119897 Assume also that119898lowast(119905

119897) is the obtained solution and

the rank condition (18) has been verified It is known (see[10]) the equivalence between the optimization problem ina probability measure space and in the moment space that is

min120583isinΛ(Δ)

intΔ

119869 (119909 V 119906) 119889120583 (V 119906)

= min119898isinΓ

int

119905119891

1199050

sum

119894

sum

119896

sum

119895

120573119894119895(119909) 120572119894119896119898119896119895

(21)

where119898lowast(119905119897) is the vector ofmoments From this equivalence

it follows that the optimal solution in the probability measurespace corresponds with the optimal solution in the momentspace and due to the rank condition (18) this solutionis unique Therefore this solution is the solution of theequivalence polynomial problem (8) that is

119898lowast

(119905119897) = (Vlowast (119905

119897) 119906lowast

(119905119897) (Vlowast (119905

119897))2

Vlowast (119905119897) 119906lowast

(119905119897)

(119906lowast

(119905119897))2

(Vlowast (119905119895))2119889

(119906lowast

(119905119895))2119889

)

(22)

which implies that 119898lowast10= Vlowast(119905

119897) and 119898lowast

01= 119906lowast

(119905119897) On the

other hand based on the polynomial equivalence we knowthat the solutions of the polynomial problem (7) are alsosolutions of the switching system Hence we can state that120590lowast

(119905119897) = Vlowast(119905

119897) which in turn implies that 120590lowast(119905

119897) = Vlowast(119905

119897) =

119898lowast

10(119905119897) It also follows that 119906lowast(119905

119897) = 119898

lowast

01(119905119897) which is the

external input for the switching system

This result states a correspondence between theminimiz-ers of the SDP

119903and the solutions of the original switched

problem and it can be used to obtain a switching signal anda continuous control input directly from the solution of theSDP119903 However it is not always the case Sometimeswe obtain

a nonoptimal solution that arises when the rank condition isnot satisfied that is ]

119903gt 1 But we still can use information

from the solution to obtain a switching suboptimal solutionand a suboptimal continuous control input

x1x2

x4x3

x0

120590(t)

Figure 1 Network with switching topology

In [19] a sum-up rounding strategy is presented to obtaina suboptimal switched solution from a relaxed solution in thecase of mixed-integer optimal control We use a similar ideabut extended to the case when the relaxed solution is anyinteger instead of the binary case

Consider the first moment 11989810(sdot) [119905

0 119905119891] 997891rarr [0 119902]

which is a relaxed solution of theNLP problem for 119905119897when the

rank condition is not satisfiedWe can state a correspondencebetween the relaxed solution and a suboptimal switchingsolution which is close to the relaxed solution in average andis given by

120590 (119905119897) =

lceil11989810(119905119897)rceil if int

119905119897

1199050

11989810(120591) 119889120591 minus 120575119905

119897minus1

sum

119896=0

120590 (119905119896) ge 05120575119905

lfloor11989810(119905119897)rfloor otherwise

(23)

where lceilsdotrceil and lfloorsdotrfloor are the ceiling and floor functions respec-tively For the continuous control input we use the extractionalgorithm presented in [20]

4 Networks with Switching Topology

This section provides an application of the theoretical resultspresented in this work for optimal control of switchingsystems As it has been shown in [2] the possibility ofdeploying a network of small simple and cheap units oragents to execute tasks cooperatively leads to consider thecommunication network between agents as a fundamentalpart of the complete system design and control In generala given set of agents can communicate with each otherThe information exchange is modeled via a communicationgraph Each node of the graph represents an agent an edgerepresents the possibility for an agent to receive informationfrom another one For the sake of clarity a dynamic networkis shown in Figure 1 where a communication graph of fivenodes is only strongly connected through the switching signal120590(119905) which operates as the switching control variable In thecase considered in this work the communication graph has adynamic topology this could be used tomodel asynchronousconsensus or the fact that two agents are not always able tocommunicate because of possible energy limitations


12

3

u

(a)

12

3

u

(b)

Figure 2 Network with switching topology Example 1

Networked systems with a dynamic topology are knownas switching networks [1] The switching network is modeledusing a dynamic graph G

120590(119905)parameterized by a switching

signal 120590(119905) R 997891rarr Q = 0 1 119902 the set of verticesV andthe set of edges E We deal with an algorithm that is basedon the average consensus concept An average consensusalgorithm is a distributed strategy to compute the average ofthe number of each agent The average consensus protocol ismathematically expressed as follows Assume that each node120596 has a state 119909

120596(119905) which is initialized to a number 119909

1205960 that

is 119909120596(0) = 119909

1205960 Then each node updates 119909

120596(119905) according to

the iteration

119909120596(119905 + 1) = 119886

120596120596119909120596(119905) + sum

120592isinN120596

119886120596120592119909120592(119905) (24)

where N120596= 120592 isin V 120596 (120592 120596) isin E is the set of

neighbors of the agent 120596 It is also assumed that 119909120596(119905 + 1) is

a convex combination of all the states available to the agent 120596(ie 119886

120596120592ge 0 and 119886

120596120596+ sum120592isinN120596

119886120596120592= 1) The set of topologies

of the network isG119904= G1G2 G

119902

We can rewrite the previous iteration in a matrix form as

119909 (119905 + 1) = 119860120590119909 (119905) (25)

where the entries of the matrix 119860 isin R119899times119899 are 119886120596120592

in position120596 120592 consistent with the graph G Result for stability analysisof networks with switching topology is mainly based onassumptions of necessity for strong connectivity of all graphsin all time instances [1] However weaker form of networkconnectivity is crucial in analysis of asynchronous consensuswhich is the case treated in this work An interesting result forperiodically connected topologies can be used to guaranteethat the algorithm proposed can reach a consensus with aswitching sequence Consider the discrete-time consensusalgorithm in (25) a switching network with the set oftopologiesG

119904is periodically connected with a period 119879G gt 1

if the unions of all graphs over a sequence of intervals areconnected graphs

Theorem 5 (see [1]) Consider the system in (25) with G120590isin

G119904for 120590 isin Q Assume the switching network is periodically

connected Then an alignment is asymptotically reached

Considering that we are dealing with a discrete-timelinear switched system we can use a similar result fordiscrete-time linear system It is well known that the systemis stable when its poles are located in the open unit ball of the

complex plane For stabilizability of switched linear systemswe have algebraic criteria as follows [21]

Theorem 6 Suppose that the switched linear system (25) isstabilizable Then there is a 119896 isin Q such that |prod119899eig

119897=1120582119897(119860119896)| le 1

where 120582119897(119860) 1 le 119897 le 119899eig are the eigenvalues of matrix 119860

Using Theorem 6 we can state that the switched system(25) is stabilizable which implies that a switching signal existsthat leads the states of the switched system to a stable point

On the other hand the solution of (25) can be expressedas

119909 (119905119891) = (

119905119891

prod

119905=0

119860120590(119905))1199090= A1199051199090

(26)

with A119905

= 119860120590119905119891sdot sdot sdot 11986012059011990521198601205901199051 The convergence of the

algorithm depends on whether the infinite product of non-negative matrices 119860

120590has a limit The consensus value is a

quantity in the convex hull of all initial values

41 Simulation Examples

411 Example 1 In this work we consider a switchingtopology with an external input 119906 that can be used to changethe value of convergence of the average consensus In orderto illustrate the optimal problem we consider two commu-nications graphs which are shown in Figure 2 Figure 2(a)represents a network with an unconnected node (node 3)which implies that this system cannot reach consensusThe system in Figure 2(b) is the same network with node3 connected so that the system can reach consensus Thecontrol objectives are to lead the dynamics of node 3 to aparticular reference through the external input 119906 spendingminimum energy consumption which is applied to node 1as it is shown in Figure 2

The optimal control problem is described as follows Adiscrete linear switched system

119909 (119905 + 1) = 119860120590119909 (119905) + 119861

120590119906 (119905) (27)

consisting of two subsystems associated with the graphs inFigure 2 with matrices 119860

0and 119860

1 An external input 119906

connected directly to node 1 Consider

1198600=

[[[[[[

[

1

2

1

20

1

2

1

20

0 0 1

]]]]]]

]

1198601=

[[[[[[[

[

1

2

1

20

1

3

1

3

1

3

01

2

1

2

]]]]]]]

]

1198610= 1198611= [1 0 0]

⊤

(28)

The functional is the same for the two subsystems and itconsiders both optimization objectives that is a reference for1199091and 119909

3 and minimum energy consumption Notice the

polynomial form of 119906 in the functional

min120590119906

119869 = int

119905119891

1199050

((1199091minus 119909ref1)

2

+ (1199093minus 119909ref3)

2

+ 1199062

) 119889119905 (29)


0 05 1 15 2 250

5

10

15

x1(t)

andx2(t)

States x1(t) and x2(t)

t

0 05 1 15 2 250

5

10

15

x3(t)

State x3(t)

t

0 05 1 15 2 250

005

01

015

02

025

t

u(t)

External control input

0 05 1 15 2 250

05

1

15

2

t

Switching signal

120590(t)

Figure 3 Networked control system dynamic response

subject to (27) with 119909 isin R3 119909(1199050) = (1 12 01)

⊤ 120590 isin Q =

0 1 119909ref1 = 119909ref3 = 10 119906 isin R and 119905 isin [0 25]In Figure 3 the trajectories the switching signal and the

external control input of the switching system obtained foran order of relaxation 119903 = 1 are shown The simulationsshow that due to the unconnected node the system has toswitch between the two network topologies to accomplish thecontrol objective It is also shown that the external input isnecessary to change the average consensus value to a value of10 which is part of the control objective Once the system hasreached the average value the external control input is zeroIt is noted that the system response reaches a stable value andmeets the control objectives The computational efficiency ofthe proposed algorithm is based on the semidefinite structureof the relaxed problem obtained

412 Example 2 Consensus under Communication Limita-tions In this example we consider a leader-following systemconsisting of a leader and two agents The communicationnetwork is shown in Figure 4 where three switching topolo-giesG

1G2 andG

3can be observed None of the graphs has

a spanning tree which implies that the system has to switchin order to converge to a consensus (see Theorem 5) The

optimal control objectives are to minimize the disagreementas a quadratic function while the energy consumption isminimized as described in Example 1 but with matrices

1198600=

[[[[[[

[

1 0 0

01

2

1

2

01

2

1

2

]]]]]]

]

1198601=

[[[[[[

[

1

2

1

20

1

2

1

20

0 0 1

]]]]]]

]

1198602=

[[[[[[

[

1

201

2

0 1 0

1

201

2

]]]]]]

]

(30)

It is observed in the simulation that the controlled systemswitches between the three subsystems (see switching signalin Figure 5) to spread the information through all nodes andthen to reach consensus (see Figure 5)


12

3

u

(a)

12

3

u

(b)

12

3

u

(c)

Figure 4 Communication network topology in Example 2

0 005 01 015 02 025 03 035 04 045 050

10

20

xi(t)

States xi(t)

t

0 005 01 015 02 025 03 035 04 045 050

4

8

t

u(t)


0 005 01 015 02 025 03 035 04 045 050

12

t

120590(t)

Switching signal

Figure 5 Leader-following consensus Example 2

5 Conclusions

In this paper we have extended our previous work on thenew method for solving the optimal control problem ofswitched systems based on a polynomial approach includingthe external input We follow the same line of thoughttransforming the original problem into a polynomial systemwhich is able to mimic the switching behavior with acontinuous polynomial representation Then we transformthe polynomial problem into a relaxed convex problem usingthe method of moments From a theoretical point of view we

have provided sufficient conditions for the existence of theminimizer by using particular features of the relaxed convexformulation We have introduced the moment approach asa computational useful tool to solve this kind of problemswhich has been illustrated bymeans of interesting networkedcontrol systems that is a network with switching topologymodeled as switching systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[2] E Lovisari and S Zampieri ldquoPerformance metrics in theaverage consensus problem a tutorialrdquo Annual Reviews inControl vol 36 no 1 pp 26ndash41 2012

[3] NW Bauer P J HMaas andW PMHeemels ldquoStability anal-ysis of networked control systems a sum of squares approachrdquoAutomatica vol 48 no 8 pp 1514ndash1524 2012

[4] E Mojica-Nava N Quijano and Rakoto-Ravalontsalama ldquoApolynomial approach for optimal control of switched nonlinearsystemsrdquo International Journal of Robust and Nonlinear Control2013

[5] A Bemporad and M Morari ldquoControl of systems integratinglogic dynamics and constraintsrdquo Automatica vol 35 no 3 pp407ndash427 1999

[6] A Bemporad MMorari V Dua and E N Pistikopoulos ldquoTheexplicit solution of model predictive control via multiparamet-ric quadratic programmingrdquo in Proceedinhgs of the AmericanControl Conference vol 2 pp 872ndash876 Chicago Ill USA June2000

[7] SWei K UthaichanaM Zefran R A DeCarlo and S BengealdquoApplications of numerical optimal control to nonlinear hybridsystemsrdquo Nonlinear Analysis Hybrid Systems vol 1 no 2 pp264ndash279 2007

[8] J B Lasserre ldquoGlobal optimization with polynomials and theproblem of momentsrdquo SIAM Journal on Optimization vol 11no 3 pp 796ndash817 200001

[9] R Meziat ldquoThe method of moments in global optimizationrdquoJournal of Mathematical Sciences vol 116 no 3 pp 3303ndash33242003

[10] J B Lasserre ldquoAn explicit equivalent positive semidefiniteprogram for nonlinear 0-1 programsrdquo SIAM Journal on Opti-mization vol 12 no 3 pp 756ndash769 2002

[11] J B Lasserre ldquoA semidefinite programming approach to thegeneralized problem of momentsrdquoMathematical Programmingvol 112 no 1 pp 65ndash92 2008

[12] R Meziat D Patino and P Pedregal ldquoAn alternative approachfor non-linear optimal control problems based on themethod ofmomentsrdquo Computational Optimization and Applications vol38 no 1 pp 147ndash171 2007

[13] P Pedregal and J Tiago ldquoExistence results for optimal controlproblems with some special nonlinear dependence on state andcontrolrdquo SIAM Journal on Control andOptimization vol 48 no2 pp 415ndash437 2009


[14] E Mojica-Nava N Quijano N Rakoto-Ravalontsalama andA Gauthier ldquoA polynomial approach for stability analysis ofswitched systemsrdquo Systems amp Control Letters vol 59 no 2 pp98ndash104 2010

[15] R Burden and J D Faires Numerical Analysis PWS BostonMass USA 1985

[16] E Mojica Nava R Meziat N Quijano A Gauthier and NRakoto-Ravalontsalama ldquoOptimal control of switched systemsa polynomial approachrdquo in Proceedings of the 17th WorldCongress International Federation of Automatic Control (IFACrsquo08) pp 7808ndash7813 July 2008

[17] R E Curto and L A Fialkow ldquoThe truncated complex K-moment problemrdquo Transactions of the American MathematicalSociety vol 352 no 6 pp 2825ndash2855 2000

[18] D P Bertsekas Nonlinear Programming Athena ScientificBelmont Mass USA 2nd edition 1999

[19] S Sager ldquoReformulations and algorithms for the optimizationof switching decisions in nonlinear optimal controlrdquo Journal ofProcess Control vol 19 no 8 pp 1238ndash1247 2009

[20] ldquoDetecting global optimality and extracting solutions in Glop-tiPolyrdquo in Positive Polynomials in Control D Henrion andJ Lasserre Eds vol 312 of Lecture Notes in Control andInformation Sciences pp 293ndash310 Springer Berlin Germany2005

[21] Z Sun and S S Ge ldquoAnalysis and synthesis of switched linearcontrol systemsrdquo Automatica vol 41 no 2 pp 181ndash195 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


with polynomials in [8 9] and later extended to nonlinear 0-1 programs using an explicit equivalent positive semidefiniteprogram in [10] We also use some results recently intro-duced for optimal control problems with the control variableexpressed as polynomials [11ndash13] The moment approachfor global polynomial optimization based on semidefiniteprogramming (SDP) is consistent as it simplifies andor hasbetter convergence propertieswhen solving convex problemsThis approach works properly when the control variables(ie the switching signal and the continuous external input)can be expressed as polynomials Essentially this methodtransforms a nonlinear nonconvex optimal control problem(ie the switched system) into an equivalent optimal controlproblem with linear and convex structure which allows us toobtain an equivalent convex formulation more appropriateto be solved by high-performance numerical computing Inother words we transform a given controllable switchedsystem into a controllable continuous system with a linearand convex structure in the control variables

This paper is organized as follows In Section 2 we presentthe problem statement and the basic concepts The maincontribution based on the method of moments is developedin Section 3 Some applications of the proposed method fornetworks with switching topology are presented in Section 4and finally in Section 5 some conclusions are drawn

2 Problem Statement

21 Switching Optimal Control Problem A switched optimalcontrol problem can be stated in a general form as followsGiven the switched system in (2) and a Bolza cost functional119869 the switched optimal control problem is given by

min120590(119905)119906(119905)

119869 (1199050 119905119891 119909 (119905) 120590 (119905) 119906 (119905)) = int

119905119891

1199050

119871120590(119909 (119905) 119906 (119905)) 119889119905

(1)

subject to the state equation

(119905) = 119891120590(119909 (119905) 119906 (119905)) (2)

where 119909(119905) is the state 119891119894 R119899 times R119898 997891rarr R119899 is the 119894th

vector field 119909(1199050) = 119909

0are fixed initial values 119906(119905) isin R119898

is the continuous control function and 120590 [1199050 119905119891] 997891rarr Q isin

0 1 2 119902 is a piecewise constant function of time with1199050and 119905119891as the initial and final times respectively Every

mode of operation corresponds to a specific subsystem (119905) =

119891119894(119909(119905) 119906(119905)) for some 119894 isin Q and the switching signal 120590

determines which subsystem is followed at each point of timeinto the interval [119905

0 119905119891] The control input 120590 is a measurable

function In addition we consider a non-Zeno behaviorthat is we exclude an infinite switching accumulation pointin time Finally we assume that the state does not havejump discontinuities Moreover for the interval [119905

0 119905119891] the

control functions must be chosen so that the initial and finalconditions are satisfied and the running switched costs 119871

120590(119905)

R+ timesR119899 997891rarr R are continuously differentiable for each 120590 isin Q

Additionally we assume that each function 119891119894and 119871

119894can

be expressed as polynomials in the control variable 119906 Ingeneral we have the functionals as

119871119894(119909 119906) =

119873

sum

119895=0

120573119894119895(119909) 119906119895 (3)

and the state equations as

119891119894(119909 119906) =

119873

sum

119895=0

120585119894119895(119909) 119906119895

(4)

where119873 is the maximum degree of the polynomials in termsof 119906 setting in zero the coefficients of monomials of degreeless than 119873 and 120573

119894119895(119909) and 120585

119894119895(119909) are the coefficients in

119909 related to the respective polynomial term in 119906119895 Generalconditions for the subsystems functions should be satisfied

Assumption 1 The nonlinear switched system satisfiesgrowth Lipschitz continuity and coercivity qualificationsconcerning the mappings 119891

119894 R119899 times R119898 997891rarr R119899 and

119871119894 R119899R119899 timesR119898 997891rarr R to ensure existence of solutions of (2)

The switching optimal control problem can have theusual variations of fixed or free initial or terminal state freeterminal time and so forth

Definition 2 A control for the switched system in (2) is atriplet consisting of

(a) a finite sequence of modes 120590

(b) the optimal value for the continuous external control119906

(c) a finite sequence of switching times such that 1199050lt 1199051lt

sdot sdot sdot lt 119905119902= 119905119891

22 A Continuous Polynomial Representation A continuousnonswitched control system can be obtained from (2) as it hasbeen shown in [14]The polynomial expression in the controlvariable able to mimic the behavior of the switched systemis developed using a variable V which works as a controlvariable

A polynomial expression in the new control variable V(119905)can be obtained through Lagrange polynomial interpolationquotients defined as in [15]

119897119896(V) =

119902

prod

119894=0119894 = 119896

(V minus 119894)(119896 minus 119894)

(5)

and a constraint polynomial set defined by Ω = V isin R |

119892(V) = prod119902119896=0(V minus 119896) = 0

Consider a switched systemof the formgiven in (2)Thereexists a unique continuous state system with polynomial



= F (119909 V 119906) =119902

sum

119894=0

119873

sum

119895=0

120585119894119895(119909) 119906119895

119897119896(V)

=

119902

sum

119894=0

119902

sum

119896=0

119873

sum

119895=0

120585119894119895(119909) 120572119894119896V119896119906119895

(6)





L (119909 V 119906) =119902

sum

119894=0

119873

sum

119895=0

120573119894119895(119909) 119906119895

119897119896(V)

=

119902

sum

119894=0

119902

sum

119896=0

119873

sum

119895=0

120573119894119895(119909) 120572119894119896V119896119906119895

(7)







Λ = 120583 = 120583119905119905isin[1199050119905119891]

supp (120583119905)


ae 119905 isin [1199050 119905119891]

(9)


min120583isinΛ

119869 (119909 V 119906) = int119905119891

1199050

intΔ

L (119909 V 119906) 119889120583 (V 119906) 119889119905 (10)

subject to

(119905) = intΔ

F (119909 V 119906) 119889120583 (V 119906) 119909 (1199050) = 1199090 120583 isin Λ (11)




Γ = 119898 = 119898119896119895 119898119896119895= intΔ

V119896119906119895119889120583 (V 119906) 120583 isin Λ (Δ) (12)


119896119895 has a



10 11989801 11989820 11989811 11989820 119898

21198890 119898

02119889]⊤ associ-




119894+119897119895+119896 For example



is

1198721(119898) = [

[

1 1198981011989801

119898101198982011989811

119898011198981111989802

]

]

(13)


0+ 1205731V + sdot sdot sdot + 120573




1205781198981205780 or


120574=01205731205741198981205740 Suppose that the



1205880= sum119889

120574=01205731205740119898120574+1205880


1198721(120573119898) = [

[

11989810+ 21198982011989820+ 21198983011989811+ 211989821

11989820+ 21198983011989830+ 21198984011989821+ 211989831

11989811+ 21198982111989821+ 21198983111989812+ 211989822

]

]

(14)



119895119896 of all


min119898119896119895isinΓ

119869 = int

119905119891

1199050

sum

119894

sum

119896

sum

119895

120573119894119895(119909) 120572119894119896119898119896119895 (15)

subject to

(119905) = sum

119894

sum

119896

sum

119895

120585119894119895(119909) 120572119894119896119898119896119895

119909 isin R119899

119898 isin Γ 119909 (0) = 1199090

(16)








1and 119892

2









119903)



119869lowast

119903= min119898(119905119897)

ℎ

2

Nminus1

sum

119897=0

L (119909 (119905119897) 119898 (119905

119897))

= min119898(119905119897)

ℎ

2

Nminus1

sum

119897=0

sum

119894

sum

119896

sum

119895

120573119894119895(119909 (119905119897)) 120572119894119896119898119896119895(119905119897)

st

119909 (119905119897+1) = 119909 (119905

119897) + ℎsum

119894

sum

119896

sum

119895

120585119894119895(119909 (119905119897)) 120572119894119896119898119896119895(119905119897)

119909 (1199050) = 1199090

119872119903(119898 (119905119897)) ⪰ 0 119872

0(1198921119898(119905119897)) ⪰ 0

1198720(1198922119898(119905119897)) ⪰ 0

(17)





lowast

1(119905119895) 119898lowast

2(119905119895) 119898

lowast

119903(119905119895)] We need





]119903= rank119872

119903(119898 (119905119897)) = rank119872

119903minus1(119898 (119905119897)) (18)



119903=

rank119872119903(119898(119905119897)) = rank119872

0(119898(119905119897)) = 1 because119872

0= 1







rank119872119903(119898lowast


119897) has


120590lowast

(119905119897) = 119898

lowast

10(119905119897) (19)


119906lowast

(119905119897) = 119898

lowast

01(119905119897) (20)








min120583isinΛ(Δ)

intΔ

119869 (119909 V 119906) 119889120583 (V 119906)

= min119898isinΓ

int

119905119891

1199050

sum

119894

sum

119896

sum

119895

120573119894119895(119909) 120572119894119896119898119896119895

(21)



119898lowast

(119905119897) = (Vlowast (119905

119897) 119906lowast

(119905119897) (Vlowast (119905

119897))2

Vlowast (119905119897) 119906lowast

(119905119897)

(119906lowast

(119905119897))2

(Vlowast (119905119895))2119889

(119906lowast

(119905119895))2119889

)

(22)


119897) and 119898lowast

01= 119906lowast

(119905119897) On the


(119905119897) = Vlowast(119905


119897) = Vlowast(119905

119897) =

119898lowast


119897) = 119898

lowast

01(119905119897) which is the








x1x2

x4x3

x0

120590(t)




0 119905119891] 997891rarr [0 119902]



120590 (119905119897) =


119905119897

1199050

11989810(120591) 119889120591 minus 120575119905

119897minus1

sum

119896=0

120590 (119905119896) ge 05120575119905


(23)





12

3

u

(a)

12

3

u

(b)






1205960 that

is 119909120596(0) = 119909



the iteration

119909120596(119905 + 1) = 119886

120596120596119909120596(119905) + sum

120592isinN120596

119886120596120592119909120592(119905) (24)




120596120592ge 0 and 119886

120596120596+ sum120592isinN120596



119902


119909 (119905 + 1) = 119860120590119909 (119905) (25)











119897=1120582119897(119860119896)| le 1




119909 (119905119891) = (

119905119891

prod

119905=0

119860120590(119905))1199090= A1199051199090

(26)

with A119905








119909 (119905 + 1) = 119860120590119909 (119905) + 119861

120590119906 (119905) (27)


0and 119860



1198600=

[[[[[[

[

1

2

1

20

1

2

1

20

0 0 1

]]]]]]

]

1198601=

[[[[[[[

[

1

2

1

20

1

3

1

3

1

3

01

2

1

2

]]]]]]]

]

1198610= 1198611= [1 0 0]

⊤

(28)




min120590119906

119869 = int

119905119891

1199050

((1199091minus 119909ref1)

2

+ (1199093minus 119909ref3)

2

+ 1199062

) 119889119905 (29)


0 05 1 15 2 250

5

10

15

x1(t)

andx2(t)


t

0 05 1 15 2 250

5

10

15

x3(t)

State x3(t)

t

0 05 1 15 2 250

005

01

015

02

025

t

u(t)


0 05 1 15 2 250

05

1

15

2

t

Switching signal

120590(t)



⊤ 120590 isin Q =




1G2 andG




1198600=

[[[[[[

[

1 0 0

01

2

1

2

01

2

1

2

]]]]]]

]

1198601=

[[[[[[

[

1

2

1

20

1

2

1

20

0 0 1

]]]]]]

]

1198602=

[[[[[[

[

1

201

2

0 1 0

1

201

2

]]]]]]

]

(30)



12

3

u

(a)

12

3

u

(b)

12

3

u

(c)


0 005 01 015 02 025 03 035 04 045 050

10

20

xi(t)

States xi(t)

t

0 005 01 015 02 025 03 035 04 045 050

4

8

t

u(t)


0 005 01 015 02 025 03 035 04 045 050

12

t

120590(t)

Switching signal


5 Conclusions





References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











= F (119909 V 119906) =119902

sum

119894=0

119873

sum

119895=0

120585119894119895(119909) 119906119895

119897119896(V)

=

119902

sum

119894=0

119902

sum

119896=0

119873

sum

119895=0

120585119894119895(119909) 120572119894119896V119896119906119895

(6)





L (119909 V 119906) =119902

sum

119894=0

119873

sum

119895=0

120573119894119895(119909) 119906119895

119897119896(V)

=

119902

sum

119894=0

119902

sum

119896=0

119873

sum

119895=0

120573119894119895(119909) 120572119894119896V119896119906119895

(7)







Λ = 120583 = 120583119905119905isin[1199050119905119891]

supp (120583119905)


ae 119905 isin [1199050 119905119891]

(9)


min120583isinΛ

119869 (119909 V 119906) = int119905119891

1199050

intΔ

L (119909 V 119906) 119889120583 (V 119906) 119889119905 (10)

subject to

(119905) = intΔ

F (119909 V 119906) 119889120583 (V 119906) 119909 (1199050) = 1199090 120583 isin Λ (11)




Γ = 119898 = 119898119896119895 119898119896119895= intΔ

V119896119906119895119889120583 (V 119906) 120583 isin Λ (Δ) (12)


119896119895 has a



10 11989801 11989820 11989811 11989820 119898

21198890 119898

02119889]⊤ associ-




119894+119897119895+119896 For example



is

1198721(119898) = [

[

1 1198981011989801

119898101198982011989811

119898011198981111989802

]

]

(13)


0+ 1205731V + sdot sdot sdot + 120573




1205781198981205780 or


120574=01205731205741198981205740 Suppose that the



1205880= sum119889

120574=01205731205740119898120574+1205880


1198721(120573119898) = [

[

11989810+ 21198982011989820+ 21198983011989811+ 211989821

11989820+ 21198983011989830+ 21198984011989821+ 211989831

11989811+ 21198982111989821+ 21198983111989812+ 211989822

]

]

(14)



119895119896 of all


min119898119896119895isinΓ

119869 = int

119905119891

1199050

sum

119894

sum

119896

sum

119895

120573119894119895(119909) 120572119894119896119898119896119895 (15)

subject to

(119905) = sum

119894

sum

119896

sum

119895

120585119894119895(119909) 120572119894119896119898119896119895

119909 isin R119899

119898 isin Γ 119909 (0) = 1199090

(16)








1and 119892

2









119903)



119869lowast

119903= min119898(119905119897)

ℎ

2

Nminus1

sum

119897=0

L (119909 (119905119897) 119898 (119905

119897))

= min119898(119905119897)

ℎ

2

Nminus1

sum

119897=0

sum

119894

sum

119896

sum

119895

120573119894119895(119909 (119905119897)) 120572119894119896119898119896119895(119905119897)

st

119909 (119905119897+1) = 119909 (119905

119897) + ℎsum

119894

sum

119896

sum

119895

120585119894119895(119909 (119905119897)) 120572119894119896119898119896119895(119905119897)

119909 (1199050) = 1199090

119872119903(119898 (119905119897)) ⪰ 0 119872

0(1198921119898(119905119897)) ⪰ 0

1198720(1198922119898(119905119897)) ⪰ 0

(17)





lowast

1(119905119895) 119898lowast

2(119905119895) 119898

lowast

119903(119905119895)] We need





]119903= rank119872

119903(119898 (119905119897)) = rank119872

119903minus1(119898 (119905119897)) (18)



119903=

rank119872119903(119898(119905119897)) = rank119872

0(119898(119905119897)) = 1 because119872

0= 1







rank119872119903(119898lowast


119897) has


120590lowast

(119905119897) = 119898

lowast

10(119905119897) (19)


119906lowast

(119905119897) = 119898

lowast

01(119905119897) (20)








min120583isinΛ(Δ)

intΔ

119869 (119909 V 119906) 119889120583 (V 119906)

= min119898isinΓ

int

119905119891

1199050

sum

119894

sum

119896

sum

119895

120573119894119895(119909) 120572119894119896119898119896119895

(21)



119898lowast

(119905119897) = (Vlowast (119905

119897) 119906lowast

(119905119897) (Vlowast (119905

119897))2

Vlowast (119905119897) 119906lowast

(119905119897)

(119906lowast

(119905119897))2

(Vlowast (119905119895))2119889

(119906lowast

(119905119895))2119889

)

(22)


119897) and 119898lowast

01= 119906lowast

(119905119897) On the


(119905119897) = Vlowast(119905


119897) = Vlowast(119905

119897) =

119898lowast


119897) = 119898

lowast

01(119905119897) which is the








x1x2

x4x3

x0

120590(t)




0 119905119891] 997891rarr [0 119902]



120590 (119905119897) =


119905119897

1199050

11989810(120591) 119889120591 minus 120575119905

119897minus1

sum

119896=0

120590 (119905119896) ge 05120575119905


(23)





12

3

u

(a)

12

3

u

(b)






1205960 that

is 119909120596(0) = 119909



the iteration

119909120596(119905 + 1) = 119886

120596120596119909120596(119905) + sum

120592isinN120596

119886120596120592119909120592(119905) (24)




120596120592ge 0 and 119886

120596120596+ sum120592isinN120596



119902


119909 (119905 + 1) = 119860120590119909 (119905) (25)











119897=1120582119897(119860119896)| le 1




119909 (119905119891) = (

119905119891

prod

119905=0

119860120590(119905))1199090= A1199051199090

(26)

with A119905








119909 (119905 + 1) = 119860120590119909 (119905) + 119861

120590119906 (119905) (27)


0and 119860



1198600=

[[[[[[

[

1

2

1

20

1

2

1

20

0 0 1

]]]]]]

]

1198601=

[[[[[[[

[

1

2

1

20

1

3

1

3

1

3

01

2

1

2

]]]]]]]

]

1198610= 1198611= [1 0 0]

⊤

(28)




min120590119906

119869 = int

119905119891

1199050

((1199091minus 119909ref1)

2

+ (1199093minus 119909ref3)

2

+ 1199062

) 119889119905 (29)


0 05 1 15 2 250

5

10

15

x1(t)

andx2(t)


t

0 05 1 15 2 250

5

10

15

x3(t)

State x3(t)

t

0 05 1 15 2 250

005

01

015

02

025

t

u(t)


0 05 1 15 2 250

05

1

15

2

t

Switching signal

120590(t)



⊤ 120590 isin Q =




1G2 andG




1198600=

[[[[[[

[

1 0 0

01

2

1

2

01

2

1

2

]]]]]]

]

1198601=

[[[[[[

[

1

2

1

20

1

2

1

20

0 0 1

]]]]]]

]

1198602=

[[[[[[

[

1

201

2

0 1 0

1

201

2

]]]]]]

]

(30)



12

3

u

(a)

12

3

u

(b)

12

3

u

(c)


0 005 01 015 02 025 03 035 04 045 050

10

20

xi(t)

States xi(t)

t

0 005 01 015 02 025 03 035 04 045 050

4

8

t

u(t)


0 005 01 015 02 025 03 035 04 045 050

12

t

120590(t)

Switching signal


5 Conclusions





References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205781198981205780 or


120574=01205731205741198981205740 Suppose that the



1205880= sum119889

120574=01205731205740119898120574+1205880


1198721(120573119898) = [

[

11989810+ 21198982011989820+ 21198983011989811+ 211989821

11989820+ 21198983011989830+ 21198984011989821+ 211989831

11989811+ 21198982111989821+ 21198983111989812+ 211989822

]

]

(14)



119895119896 of all


min119898119896119895isinΓ

119869 = int

119905119891

1199050

sum

119894

sum

119896

sum

119895

120573119894119895(119909) 120572119894119896119898119896119895 (15)

subject to

(119905) = sum

119894

sum

119896

sum

119895

120585119894119895(119909) 120572119894119896119898119896119895

119909 isin R119899

119898 isin Γ 119909 (0) = 1199090

(16)








1and 119892

2









119903)



119869lowast

119903= min119898(119905119897)

ℎ

2

Nminus1

sum

119897=0

L (119909 (119905119897) 119898 (119905

119897))

= min119898(119905119897)

ℎ

2

Nminus1

sum

119897=0

sum

119894

sum

119896

sum

119895

120573119894119895(119909 (119905119897)) 120572119894119896119898119896119895(119905119897)

st

119909 (119905119897+1) = 119909 (119905

119897) + ℎsum

119894

sum

119896

sum

119895

120585119894119895(119909 (119905119897)) 120572119894119896119898119896119895(119905119897)

119909 (1199050) = 1199090

119872119903(119898 (119905119897)) ⪰ 0 119872

0(1198921119898(119905119897)) ⪰ 0

1198720(1198922119898(119905119897)) ⪰ 0

(17)





lowast

1(119905119895) 119898lowast

2(119905119895) 119898

lowast

119903(119905119895)] We need





]119903= rank119872

119903(119898 (119905119897)) = rank119872

119903minus1(119898 (119905119897)) (18)



119903=

rank119872119903(119898(119905119897)) = rank119872

0(119898(119905119897)) = 1 because119872

0= 1







rank119872119903(119898lowast


119897) has


120590lowast

(119905119897) = 119898

lowast

10(119905119897) (19)


119906lowast

(119905119897) = 119898

lowast

01(119905119897) (20)








min120583isinΛ(Δ)

intΔ

119869 (119909 V 119906) 119889120583 (V 119906)

= min119898isinΓ

int

119905119891

1199050

sum

119894

sum

119896

sum

119895

120573119894119895(119909) 120572119894119896119898119896119895

(21)



119898lowast

(119905119897) = (Vlowast (119905

119897) 119906lowast

(119905119897) (Vlowast (119905

119897))2

Vlowast (119905119897) 119906lowast

(119905119897)

(119906lowast

(119905119897))2

(Vlowast (119905119895))2119889

(119906lowast

(119905119895))2119889

)

(22)


119897) and 119898lowast

01= 119906lowast

(119905119897) On the


(119905119897) = Vlowast(119905


119897) = Vlowast(119905

119897) =

119898lowast


119897) = 119898

lowast

01(119905119897) which is the








x1x2

x4x3

x0

120590(t)




0 119905119891] 997891rarr [0 119902]



120590 (119905119897) =


119905119897

1199050

11989810(120591) 119889120591 minus 120575119905

119897minus1

sum

119896=0

120590 (119905119896) ge 05120575119905


(23)





12

3

u

(a)

12

3

u

(b)






1205960 that

is 119909120596(0) = 119909



the iteration

119909120596(119905 + 1) = 119886

120596120596119909120596(119905) + sum

120592isinN120596

119886120596120592119909120592(119905) (24)




120596120592ge 0 and 119886

120596120596+ sum120592isinN120596



119902


119909 (119905 + 1) = 119860120590119909 (119905) (25)











119897=1120582119897(119860119896)| le 1




119909 (119905119891) = (

119905119891

prod

119905=0

119860120590(119905))1199090= A1199051199090

(26)

with A119905








119909 (119905 + 1) = 119860120590119909 (119905) + 119861

120590119906 (119905) (27)


0and 119860



1198600=

[[[[[[

[

1

2

1

20

1

2

1

20

0 0 1

]]]]]]

]

1198601=

[[[[[[[

[

1

2

1

20

1

3

1

3

1

3

01

2

1

2

]]]]]]]

]

1198610= 1198611= [1 0 0]

⊤

(28)




min120590119906

119869 = int

119905119891

1199050

((1199091minus 119909ref1)

2

+ (1199093minus 119909ref3)

2

+ 1199062

) 119889119905 (29)


0 05 1 15 2 250

5

10

15

x1(t)

andx2(t)


t

0 05 1 15 2 250

5

10

15

x3(t)

State x3(t)

t

0 05 1 15 2 250

005

01

015

02

025

t

u(t)


0 05 1 15 2 250

05

1

15

2

t

Switching signal

120590(t)



⊤ 120590 isin Q =




1G2 andG




1198600=

[[[[[[

[

1 0 0

01

2

1

2

01

2

1

2

]]]]]]

]

1198601=

[[[[[[

[

1

2

1

20

1

2

1

20

0 0 1

]]]]]]

]

1198602=

[[[[[[

[

1

201

2

0 1 0

1

201

2

]]]]]]

]

(30)



12

3

u

(a)

12

3

u

(b)

12

3

u

(c)


0 005 01 015 02 025 03 035 04 045 050

10

20

xi(t)

States xi(t)

t

0 005 01 015 02 025 03 035 04 045 050

4

8

t

u(t)


0 005 01 015 02 025 03 035 04 045 050

12

t

120590(t)

Switching signal


5 Conclusions





References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














rank119872119903(119898lowast


119897) has


120590lowast

(119905119897) = 119898

lowast

10(119905119897) (19)


119906lowast

(119905119897) = 119898

lowast

01(119905119897) (20)








min120583isinΛ(Δ)

intΔ

119869 (119909 V 119906) 119889120583 (V 119906)

= min119898isinΓ

int

119905119891

1199050

sum

119894

sum

119896

sum

119895

120573119894119895(119909) 120572119894119896119898119896119895

(21)



119898lowast

(119905119897) = (Vlowast (119905

119897) 119906lowast

(119905119897) (Vlowast (119905

119897))2

Vlowast (119905119897) 119906lowast

(119905119897)

(119906lowast

(119905119897))2

(Vlowast (119905119895))2119889

(119906lowast

(119905119895))2119889

)

(22)


119897) and 119898lowast

01= 119906lowast

(119905119897) On the


(119905119897) = Vlowast(119905


119897) = Vlowast(119905

119897) =

119898lowast


119897) = 119898

lowast

01(119905119897) which is the








x1x2

x4x3

x0

120590(t)




0 119905119891] 997891rarr [0 119902]



120590 (119905119897) =


119905119897

1199050

11989810(120591) 119889120591 minus 120575119905

119897minus1

sum

119896=0

120590 (119905119896) ge 05120575119905


(23)





12

3

u

(a)

12

3

u

(b)






1205960 that

is 119909120596(0) = 119909



the iteration

119909120596(119905 + 1) = 119886

120596120596119909120596(119905) + sum

120592isinN120596

119886120596120592119909120592(119905) (24)




120596120592ge 0 and 119886

120596120596+ sum120592isinN120596



119902


119909 (119905 + 1) = 119860120590119909 (119905) (25)











119897=1120582119897(119860119896)| le 1




119909 (119905119891) = (

119905119891

prod

119905=0

119860120590(119905))1199090= A1199051199090

(26)

with A119905








119909 (119905 + 1) = 119860120590119909 (119905) + 119861

120590119906 (119905) (27)


0and 119860



1198600=

[[[[[[

[

1

2

1

20

1

2

1

20

0 0 1

]]]]]]

]

1198601=

[[[[[[[

[

1

2

1

20

1

3

1

3

1

3

01

2

1

2

]]]]]]]

]

1198610= 1198611= [1 0 0]

⊤

(28)




min120590119906

119869 = int

119905119891

1199050

((1199091minus 119909ref1)

2

+ (1199093minus 119909ref3)

2

+ 1199062

) 119889119905 (29)


0 05 1 15 2 250

5

10

15

x1(t)

andx2(t)


t

0 05 1 15 2 250

5

10

15

x3(t)

State x3(t)

t

0 05 1 15 2 250

005

01

015

02

025

t

u(t)


0 05 1 15 2 250

05

1

15

2

t

Switching signal

120590(t)



⊤ 120590 isin Q =




1G2 andG




1198600=

[[[[[[

[

1 0 0

01

2

1

2

01

2

1

2

]]]]]]

]

1198601=

[[[[[[

[

1

2

1

20

1

2

1

20

0 0 1

]]]]]]

]

1198602=

[[[[[[

[

1

201

2

0 1 0

1

201

2

]]]]]]

]

(30)



12

3

u

(a)

12

3

u

(b)

12

3

u

(c)


0 005 01 015 02 025 03 035 04 045 050

10

20

xi(t)

States xi(t)

t

0 005 01 015 02 025 03 035 04 045 050

4

8

t

u(t)


0 005 01 015 02 025 03 035 04 045 050

12

t

120590(t)

Switching signal


5 Conclusions





References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










12

3

u

(a)

12

3

u

(b)






1205960 that

is 119909120596(0) = 119909



the iteration

119909120596(119905 + 1) = 119886

120596120596119909120596(119905) + sum

120592isinN120596

119886120596120592119909120592(119905) (24)




120596120592ge 0 and 119886

120596120596+ sum120592isinN120596



119902


119909 (119905 + 1) = 119860120590119909 (119905) (25)











119897=1120582119897(119860119896)| le 1




119909 (119905119891) = (

119905119891

prod

119905=0

119860120590(119905))1199090= A1199051199090

(26)

with A119905








119909 (119905 + 1) = 119860120590119909 (119905) + 119861

120590119906 (119905) (27)


0and 119860



1198600=

[[[[[[

[

1

2

1

20

1

2

1

20

0 0 1

]]]]]]

]

1198601=

[[[[[[[

[

1

2

1

20

1

3

1

3

1

3

01

2

1

2

]]]]]]]

]

1198610= 1198611= [1 0 0]

⊤

(28)




min120590119906

119869 = int

119905119891

1199050

((1199091minus 119909ref1)

2

+ (1199093minus 119909ref3)

2

+ 1199062

) 119889119905 (29)


0 05 1 15 2 250

5

10

15

x1(t)

andx2(t)


t

0 05 1 15 2 250

5

10

15

x3(t)

State x3(t)

t

0 05 1 15 2 250

005

01

015

02

025

t

u(t)


0 05 1 15 2 250

05

1

15

2

t

Switching signal

120590(t)



⊤ 120590 isin Q =




1G2 andG




1198600=

[[[[[[

[

1 0 0

01

2

1

2

01

2

1

2

]]]]]]

]

1198601=

[[[[[[

[

1

2

1

20

1

2

1

20

0 0 1

]]]]]]

]

1198602=

[[[[[[

[

1

201

2

0 1 0

1

201

2

]]]]]]

]

(30)



12

3

u

(a)

12

3

u

(b)

12

3

u

(c)


0 005 01 015 02 025 03 035 04 045 050

10

20

xi(t)

States xi(t)

t

0 005 01 015 02 025 03 035 04 045 050

4

8

t

u(t)


0 005 01 015 02 025 03 035 04 045 050

12

t

120590(t)

Switching signal


5 Conclusions





References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 2 250

5

10

15

x1(t)

andx2(t)


t

0 05 1 15 2 250

5

10

15

x3(t)

State x3(t)

t

0 05 1 15 2 250

005

01

015

02

025

t

u(t)


0 05 1 15 2 250

05

1

15

2

t

Switching signal

120590(t)



⊤ 120590 isin Q =




1G2 andG




1198600=

[[[[[[

[

1 0 0

01

2

1

2

01

2

1

2

]]]]]]

]

1198601=

[[[[[[

[

1

2

1

20

1

2

1

20

0 0 1

]]]]]]

]

1198602=

[[[[[[

[

1

201

2

0 1 0

1

201

2

]]]]]]

]

(30)



12

3

u

(a)

12

3

u

(b)

12

3

u

(c)


0 005 01 015 02 025 03 035 04 045 050

10

20

xi(t)

States xi(t)

t

0 005 01 015 02 025 03 035 04 045 050

4

8

t

u(t)


0 005 01 015 02 025 03 035 04 045 050

12

t

120590(t)

Switching signal


5 Conclusions





References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










12

3

u

(a)

12

3

u

(b)

12

3

u

(c)


0 005 01 015 02 025 03 035 04 045 050

10

20

xi(t)

States xi(t)

t

0 005 01 015 02 025 03 035 04 045 050

4

8

t

u(t)


0 005 01 015 02 025 03 035 04 045 050

12

t

120590(t)

Switching signal


5 Conclusions





References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Optimal Control of Switching Topology ...downloads.hindawi.com/journals/mpe/2014/268541.pdf · Research Article Optimal Control of Switching Topology Networks EduardoMojica-Nava,