Decentralized state estimation in connected systems

7/28/2019 Decentralized state estimation in connected systems

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Decentralized state estimation in connected systems

Alessandro Pilloni Alessandro Pisano Elio Usai

Purushothama Prathyush Menon Christopher Edwards

Dept. of Electrical and Electronic Engineering (DIEE),University of Cagliari, Cagliari, Italy

(email: {alessandro.pilloni,pisano,eusai}@diee.unica.it). College of Engineering, Mathematics and Physical Sciences, University of

Exeter, United Kingdom (email: [email protected]). Control and Instrumentation Research Group, University of Leicester,

Leicester, United Kingdom (email: [email protected]).

Abstract: This paper considers the problem of state estimation and unknown input reconstruction of aclass of connected heterogeneous LTI MIMO systems. Local high order sliding mode observers at eachnode of the network are designed for this purpose. The proposed method, under some network structural

conditions, is inherently robust, nonlinear and totally independent of the time-varying network topology.Knowledge of the number of nodes that belong to the network is not required. At the supervisory level,decentralized control signals are computed based on the state estimates in order to operate the networkingsynchronization. By mean of simulation, the effectiveness of the proposal procedure is shown.

Keywords: Observers for linear systems; Complex systems; Structural properties.

1. INTRODUCTION

Control applications in distributed and cooperative environ-ments has been a subject growing interest in the last decades be-coming one of the most important research fields in the control

and decision theory (see Siljak [1991]). Analysis and control

of complex behaviors in large networks attracted the attentionof researcher from different fields; an overview of the problemsrelated to networks of dynamic systems is given in Newmanet al. [2006], and contribution in synchronization of networksand in cooperative control can be found in Wu [2007].

Complex networks are usually characterized by several distinc-tive properties including complexity, topological structure, dy-namical evolution, time-varying coupling strengths and interac-tions between nodes. The first model of networks was proposedin Erdos and Renyi [1960] (ER-Model). In that model each pairof elements was randomly connected with the same probability.However in the real world, connectivity between each elementis neither completely regular nor completely random. Thanks

to Watts and Strogatzs (WS-Model) a more realistic represen-tation has been given (see Watts and Strogatz [1998]). Anothersignificant recent discovery in the field of complex networksis the observation that a number of large-scale and complexnetworks are scale-free, that is, their connectivity distributionshave the power-law form (see Barabasi et al. [1999]). In thiswork, a scale-free dynamical network representation, consistentwith Wang and Chen [2002], will be utilized.

The motivation for the present work is to increase the levelof autonomy for a class of scale-free networks composed ofheterogeneous systems created to perform synchronization ona common reference trajectory. Network of mobile robots,

The research leading to these results has received funding from the EuropeanCommunitys Seventh Framework Programme FP7/2007-2013 under Grant

Agreement n257462 [Research Project HYCON2-Network of excellence].

unmanned aerial vehicle, satellites and localization systems arejust some examples of cooperative systems (see Frew et al.[2005]). It is noteworthy that in autonomous application, theagents are often monitored at a supervisory level, and in mostcases a supervisory level node use to drive the systems (seeFrew et al. [2005], Edwards and Menon [2008]).

A great deal of attention has been paid to the problem of de-centralized state-space estimation in complex networks. Mo-tivated by a large amount of important practical problems,the estimation of uncertain systems has become an importantarea of research. Such problem arises in systems subject todisturbances or with inaccessible or unmeasurable inputs andin many applications such as fault detection isolation. In theliterature several approaches to deal with this class of problemhas been proposed (see Edwards and Menon [2008], Stankovicet al. [2009], Pillosu et al. [2011])), however, recently, slidingmode control (SMC) theory has been further extended in thearea of networked systems (see Yan et al. [2006]). The mainfeatures of SMC are insensitivity to external disturbances, high

accuracy and finite time convergence, which make it one of theuseful tools in robust state estimation. The main drawbacks ofclassical SMC are principally related to the so-called chatteringeffect. High-order sliding modes (HOSM) have been suggestedto attenuate this phenomena whilst maintaining the mentionedrobustness properties (see Levant [1993, 2005], Bartolini et al.[2003]). Furthermore the combination of HOSM control algo-rithms and sliding mode differentiators (Levant [1998]) pro-duces effective observers (see Davila et al. [2009]).

In this paper, network structural conditions for designing localnonlinear observers independent from any topological changingare provided. Complete finite-time state estimation and theunknown input reconstruction of each system operating over a

network are fulfilled. The estimated state variables of each nodeare then used to synchronize the whole network. The paper is


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organized as follows; in Section 2 a brief description of scale-free system is presented; Section 3 provides a reminder of theconcepts of strong observability, and an innovative approachbased on the observability indices is presented; in Section 4conditions for the design of a decentralized observer are given.The proposed framework is verified by means of simulationsin Section 5. Section 6 provides some concluding remarks andhints for further research.

2. THE SCALE-FREE DYNAMICAL NETWORK MODEL

Many real-world networks are scale-free. The main featuresof a scale-free network are growth and preferential attach-ment. These refer to networks continuously evolving by theinsertion/removal of nodes and changing interconnection. Suchtopological changes can be described using graph theory andthe notation G = {G1,...,GM}, where each Gi represents an in-teraction topology for a particular time period. For the graphG the adjacency matrix A[G](t) = [ai j(t)] is a binary matrixwhose entries depend on the current graph Gi at time t, the

entry ai j(t) = 1 if the i-th and j-th node are adjacent at time t,and zero otherwise. Let[G](t) = [i j (t)] be a diagonal matrixwhich represents the in-degree matrix for the considered graphGi at time t, ii(t) is the input-degree of the i-th vertex. TheLaplacian matrix ofG,L[G] = [Li j] is defined as the difference[G] A[G].

Suppose that the scale-free network consists ofNheterogenous,dynamical, linearly and diffusively coupled nodes. Accordingto Wang and Chen [2002], Edwards and Menon [2008], eachnodei of the graph G can be described as follow:

xi = Aixi +Biui +Difi c(t)N

j=1

Li j i j xj (1)

yi = Cixi (2)

where xi = [xi,1 . . .xi,ni ]T Rni and yi = [yi,1 . . .yi,pi ]

T Rpi

represent the ni-dimensional state vector and the output vectorof the i-th node of the network. Functionsui(xi) Rmi and fi R

qi represent respectively the control inputs and the unknowninputs of the i-th node. For the sake of simplicity it is assumedqi = 1. The unknown term is assumed to satisfy certain sectorsbounds which will be defined later. The term c(t) is the timevarying coupling strength between nodes and it is assumed tobe for simplicity identical for all links between the nodes. ThematricesAi,Bi,Ci, Di describe the dynamics of node i and areassumed to be of appropiatedimensions.i j R

ninj is a binarymatrix and represents the node-to-node coupling configuration

among the i-th and j-th node. The entries ofi j are nonzero ifa communication channel among different states of neighborsnodes exist.

Assumption 1. Each node i is assumed to be observable.

Assumption 2. For achieving network synchronization, we as-sume each pair(Ai,Bi) to be controllable.

3. STRONG OBSERVABILITY AND UNKNOWN INPUTRECONSTRUCTION

Consider a LTI system:

x =Ax+Df (3)

y =Cx (4)where x Rn, y Rp and f Rq represent the state, theoutput and the unknown input vector of. Generic well-known

strong observability conditions for LTI systems with unknowninputs f based on the study of the invariant zeros of the triple(A,D,C) are summarized in Trentelman et al. [2001].

Whereas in Fridman et al. [2007] necessary conditions forstrong observability with respect to the unknown inputf, underthe assumption that q = p, has been given.

In particular, let ci and dj be the rows ofC and the columnsofD. The output vector y = Cx is said to have vector relativedegree r = (r1, . . . , rp) with respect to the unknown input f ifthe following conditions hold:

ciAlD = 0(1q)

ciAri1D = 0(1q)

with

i = 1,2, . . . ,pl = 0,1, . . . , ri 2

(5)

det(Q) = 0 with Q =

c1A

r1 1d1 c1Ar1 1dq

.

.

.

.

.

.

cpArp1d1 cpA

rp1dq

(6)

The next lemma gives sufficient conditions for guarantee thatthe system (3)-(4) is strong observable:

Lemma 1. Let the output y of have vector relative degree

r. Then the vectors c1, . . . ,c1Ar11, . . . ,cp, . . . ,cpArp1 are lin-early independent. is strongly observable if the total relativedegree of the system r =

pk=1 rk is equal to n.

Remarks 1. Generalizing Lemma 1 to the the case of rectan-gular systems (q < p), it is obvious that c1, . . . ,c1A

r11, . . . ,cp, . . . ,cpArp1 can not be linearly independent even if the sys-tem is strongly observable. This assertion can be easily provedapplying the same procedure shown in Fridman et al. [2007]because Q is a non-square matrix.

Currently the concept of strong observability with respect to theunknown input for non-square LTI systems (q < p) is not welldefined and remains an open problem in the field of estimation

and control. In the rest of this section we revert to the conceptof observability indices in order to define a possible approachfor such class of systems.

Consider the system in (3)-(4), the observability index for thei-th output of can be defined as follows:

Definition 1. The maximum number (vi) of successive linearlyindependent derivatives of the i-th output of, it is called theobservability index and represents the number of system statewhich can be reconstructed from yi.

The set V := {v1, . . . ,vp} is called the observability indices ofthe pair (A,C). It is obvious that each entry vi can not begreater then the order of the system n. Recalling the definition

of relative degree ri of the i-th output of the system with respectto an unknown inputf in (5), we can assert that if the followinglemma holds, the system is strongly observable and unknowninput reconstruction (UIR) is practicable.

Lemma 2. The system is strongly observable if it is possi-ble to define a set of positive integers U := {1, . . . ,h} withh p, in which each element is associated with one outputscomponent, such that the following conditions are satisfied:

i vii ri

,1 +2 + +h = n

with i = 1,2, . . . ,h(7)

det {M} = 0, M=

M1...

Mi

.

.

.

Mh

, Mi =

ciciA

ciA2

..

.

ciAi1

Rin (8)


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Proof. Suppose the pair (A,C) is observable,O has n linearlyindependent rows. After choosing a set U which satisfies theconditions (7)-(8), it is obvious that the matrix M is full-rank because it is obtained combining linear independent blocklike ci,ciA, . . . ,ciA

i1 linearly independent each other. Then,applying the following bijection mapping x =Mxo the fol-lowing canonical observable representation is obtained:

xo = Aoxo +Dofyo = Coxo

(9)

where

Ao =

A11 A1h..

.. . .

.

.

.

Ah1 Ahh

, Co =

c11 c1h..

.. . .

.

.

.

ch1 chh

(10)

Aii =

0 1 0

.

.

.

.

.

.. .

....

0 0 1aii,1 aii,2 aii,i

, Ai j =

0 0

.

.

.

.

.

.

0 0

aij ,1 aij ,j

(11)

Do = MD =1..

.

h

, i =0

..

.

0

ciAi1D

Riq (12)

cii = (1 0 0) R1i , ci j = (0 0) R

1j (13)

Note that the previous mapping emphasized the strong observ-ability property of. Since the condition y 0 implies x = 0for any unknown input f, if i is chosen lower than ri theentries ciA

i1D in i are zero, then f has no effect on thelinear combinationMi obtained by the i-th component ofy.

Corollary 1. If Lemma 2 is satisfied and

rankDTD

1DTM1T

= q (14)

with T= diag (t1, . . . ,ti, . . . ,th) andti = (0 0 1)T Ri1,it is possible to reconstruct completely the unknown vectorf= [f1, . . . , fq]

T by a suitable robust observer.

Proof. Consider the system (9) and the following observer:

xo = Aoxo +Tyo = Coxo

(15)

where xo, yo and represent the estimated states, the observedoutput and the injection term. If condition (14) holds, it isobvious that the mapping x= Mxo implies that the matrixrank{Do} = rank{D} = q. Let eo = xo xo be the state ob-servation error, its dynamic takes the following form:

eo = Aoeo +TDof (16)Let F= [F1, . . . ,Fq]

T be a constant vector which constitutes anupper bound on the input f so that |fi| < Fi, then we can designan algorithm which drivesto zero the observation error dynamic(e, e) (0,0) , allowing us to reconstruct f as follows:

f=D+o T= D+M1T (17)

where the index + indicates the Moore-Penrose pseudo-inverseof the matrix. It is obvious that by construction, the solutionof system (17) gives an unique solution with respect to theunknown input f if and only if the condition (14) is satisfied.

Remarks 2. Consider the observer structure (15). Applying theinverse mapping xo = M

1x, we can obtain a completely

equivalent representation for the observer (15) which dispenseswith the need to work in a transformed domain. In the rest ofthe paper will be used this observer representation.

4. LOCAL NETWORK OBSERVER DESIGN

Consider the scale-free network ofNheterogeneous dynamical,coupled nodes in (1)-(2). Hereinafter is presented a frameworkfor designing local nonlinear observers for the complete finite-time state estimation and the UIR of each node i of thenetwork. This is an extension of the strategy for SISO systemspresented in Davila et al. [2009], for networks of MIMOsystems. Conditions to achieve this goal will be discussedbelow in details. Consider the following observer structure:

xi =Aixi +Biui +Gii (18)

yi =Cixi (19)

where xi Rni and yi R

pi represent the estimated state andthe observed output for the node i. In order to capitalize onthe advantages of the sliding mode algorithms, all the equa-tions will be understood in a Filippov sense. Filippovs solutioncoincides with the classical one for ODEs with continuousright-hand side (Filippov [1960]). Gi and the injection term

i = [i,1, . . . ,i,h] Rh

i will be designed in the sequel, in sucha way that each component of the nodes outputs has a suitablepreassigned relative degree with respect to the associated injec-tion term. Observing the structure of the scale-free dynamicalnetwork in (1)-(2), due to the presences of the coupling termrelated to the i-th node with its neighbors, Lemma 2 is notapplicable in the present form. Before presenting the conditionsunder which the state estimation and the UIR are practicable,we need to define the concept of relative degree of the k-th out-put of the i-th node yi,k with respect to the local unknown inputsignal (ri,k) and the coupling terms with the j-th neighbor (wi,k).Let ci,k be the k-th row of the output matrix Ci, in accordancewith the definition of relative degree in (5), we define for eachoutput component ofi the following indices as:

ci,kAliDi = 0(1qi)

ci,kAri,k1

i Di = 0(1qi)with

i = 1,2, . . . ,pj = 1,2, . . . ,Nl = 0,1, . . . , ri,k 2

(20)

ci,kAlii j = 0(1qi)

ci,kAwi,k1

i i j = 0(1nj)with

i = 1,2, . . . ,p

l = 0,1, . . . ,wi,k 2(21)

Then in order to adapt Lemma 2 for the class of connectedsystems in (1)-(2), we can reformulate it as follows:

Lemma 3. The scale-free network of N heterogeneous dynam-ical, coupled nodes in (1)-(2) is strongly observable, if it is

possible to find a set of integerUi := {i,1, . . . ,i,h} with h pin which each element is associated to one outputs componentand such that the following conditions are satisfied:

i,k vi,ki,k min

ri,k,wi,k

, i,1 + +i,hi = niwith k= 1,2, . . . ,hi

(22)

det {Mi} = 0,Mi =

Mi,1...

Mi,k...

Mi,h

,Mi,k =

ci,kci,kAici,kA

2i

.

.

.

ci,kAi,k1

i

(23)

Proof. The proof of this lemma can be obtained in a similarmanner to Lemma 2 by inspection of the canonical observablerepresentation obtained by the following mappingxi =Mixo,i.

In light of Lemma 3, we now analyze how to design an observer

which allows to achieve the state space estimation for thecomplex network in (1)-(2) and we provideadditional structuralconditions for the UIR. Supposing Lemma 3 is satisfied. Let


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Ti = diag(ti,1, . . . , ti,k, . . . ,ti,hi ) Rnihi be a block-diagonal

matrix with each k-th block designed as follows:

ti,k = (0 0 1)T

Ri,k1 (24)

The matrix Gi in (18) can be selected as follows

MiGi = Ti Gi =M1i Ti (25)

In light of (22)-(25), taking a generic observer output yi,kand differentiating it i,k times, we obtain the following I/Odynamic:

d

dt

yi,kyi,k

.

.

.

y

i,k1

i,k

=

ci,kAici,kA

2i

.

.

.

ci,kAi,ki

xi +

ci,kBici,kAiB1

.

.

.

ci,kAi,k1

i Bi

ui +

0

.

.

.

0

1

i,k

(26)which can be rewritten as

yi,k =Mi,kAixi +Mi,kBiui + tki,k (27)

where the matrix Mi,kAi takes the following form:

Mi,kAi =

0 1 0 0 0

0...

. . . ......

..

.

0 0 1 0 0

Ri,kni (28)

It is easy to see that the k-th observers output yi,k has rel-ative degree i,k with respect to the injection term i,k. Letei = xi xi be the state error for the agent i, we can definethe output error dynamic for the k-th output yi,k as follows:

ei,yk = yi,kyi,k = ci,kei for k= 1,2, . . . ,h (29)

i,k =i,k,1 i,k,i,k

T=

ei,yk ei,yk e(i,k1)

i,k

T(30)

i,k,1 = i,k,2i,k,2 = i,k,3

... ... ...i,k,i,k = i,k,(i,k+1) +i,k

(31)

with

i,k,(i,k+1) = ci,kAi,ki ei +i,kfi +

N

j=1

i,k(t)xj (32)

i,k = ci,kAi,k1

i Di (33)

i,k(t) = c(t)Li jci,kAi,k1

i i j (34)

where the I/O dynamic (31) has the so-called Brunovsky chain-of-integrator canonical form. Note that the presence ofi,k andi,k is strictly related to the choice of the differentiation indexi,k. The following cases list explain all the possible scenarios:

1) ifi,k = ri,k = wi,k, from (20)-(21), both termsi,k and i,kappear in (32);

2) ifi,k = ri,k< wi,k, i,k = 0, thus only the Unknown InputTerm i,k appears in (32);

3) if i,k = wi,k < ri,k, i,k = 0 thus only the DynamicalCoupling Term i,k appears in (32);

4) ifi,k is smaller than both ri,k and wi,k according to (20)-(21) these terms are both i,k and i,k equal to zero.

Following similar reasoning, the complete observation errordynamic i = [i,1, . . . ,i,hi ]

T can be obtained by putting incolumns the hi Brunovsky canonical blocks (31) calculatedat the time. Note that each sub-dynamic has relative degree

i,k.In order to estimate the whole state space of each node i,we need suitable injections signals which are able to drive tozero the error ei = [ei,1, . . . ,ei,ni ]

T. Since i = Miei with Mi

invertible, it is obvious that i = 0 implies ei = 0. Thus if weare able to drive to zero all the hi sub-dynamics expressed by(31), automatically the whole observation error ei goes to zero.Make the following boundedness assumption:

Assumption 3. There are known constants Fi,k such that thefunction i,k,(i,k+1) satisfies:

|i,k,(i,k+1)| < Fi,k k= 1,2, . . . ,hi (35)

Under this assumption we can guarantee the finite-time con-vergence to zero of i by a properly designed HOSM. Thechoice of the most suitable HOSM algorithm is strictly relatedto the relative degree i,k of each sub-block (31). Table 1shows the best choices depending on i,k. Note that the so-called quasi-continuous arbitrary-order (QCAO) SMC is ableto provide finite-time stabilization of arbitrary relative degreedynamics. For relative degree one or two systems, the bestchoices are respectively the Super-Twisting (STW) algorithmand the Generalized Sub-Optimal algorithm (GSO). In particu-lar STW gives rise to a continuous control action which pos-

sesses significant robustness properties against nonlinearitiesand disturbances, whilst the GSO algorithm stabilizes secondorder dynamics without any derivative estimation, simplifyingthe complexity of the algorithm (see Bartolini et al. [2003]).However the GSO has a discontinuous behavior, whereas theQCAO SMC has discontinuities only during the sliding motion

i,k = i,k = = i,k1i,k (see Levant [2005]).

Taking into account the case list presented earlier, we can adjustCorollary 1 in order that UIR becomes practicable.

Corollary 2. If Lemma 3 is satisfied along with

rankD+i M

1i Ti= qi (36)

where Ti = diag(ti,1, . . . , ti,hi ), ti,k = (0, . . . ,0,1) Ri,k1

MiDi =

i,1..

.

i,h

,i,k =

0

ci,kAi,k1

i Di

Ri,kni (37)

and each sub-blocki,k has its index i,k which satisfies

i,k = ri,k< wi,k (38)

then it is possible to reconstruct completely the unknown vectorfi = [fi,1, . . . , fi,q]

T by a suitable robust observer.

Proof. The proof of this corollary can be obtained similarly to

Corollary 1 by inspection of the following error dynamic:

ei = Aiei +Gii Difi + c(t)N

i=1

Li ji jxj (39)

Since the error dynamic (39) can be driven to zero in finite time,once (ei, ei) = (0,0) UIR can be achieved as follows

fi = D+M1Tii c(t)

N

i=1

Li jD+i jxj (40)

If the state xj is accessible only at the supervisory level,it would be impossible to locally reconstruct fi without theknowledge of the neighbors states. However, if both conditions

(36) and (38) hold,D+

i j =0

and the complete information offi is contained in i. UIR can be obtained as the unique solutionof the following equalities fi = (D

TD)1DTM1Tii.


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5. NUMERICAL EXAMPLE

5.1 Observer design

Before presenting results for a scale-free network, for the sakeof completeness an application of the design strategy shown in

Lemma 2 for an rectangual unstable system in form (3)-(4) ispresented. The following matrices describes the system:

A=

0 1 1 0 01 0.2 0 0 0

0 0 6 1 00 0 0 0 1

1 0 0 1 2

,D =

0 1 0

0 0 0

1 1 0

0 0 0

0 0 1

,C=

0 1 0 0 0

0 0 1 1 1

0 0 0 1 0

1 1 0 1 1

f=

f1f2

f3

=

3(2(t 3) 2(t 4))cos (2t)

cos2 (2t) 3(2(t 3) 2(t 4))

(41)

From Definition 1 the set of observability indices is V ={5,5,5,5}, while the vector relative degree is r = {2,1,2,1}.By means of Lemma 2 and Corollary 1, the observer matricesM and T can be easily derived by the following possibleset of indices U = {i}i=1,...,4 = {2,1,1,1}. Then in order

to reduce the need of sliding differentiators needed for theinjection term = [1, . . . ,4]

T, we design, in accordance withTable 1, as the STW and the GSO algorithms for the relativedegree one and two dynamics, respectively. For further detailson these algorithms refer to the next section and references inTable 1. In Fig. 1 the estimation errors (upper plot) and thecomparison between the f components and the reconstructedunknown input vector (lower plot) as in (17) after proper low-pass Butterworth filtering are shown.

5.2 Observer design for agents networks

To demonstrate the theory developed in this paper, the time-

varying networkG of six heterogeneous chaotic circuits showin Fig. 2 is considered. In light of (1)-(2), hereinafter the dy-namics of each node i of the complex network are presented:

Nodes i with i = 1,2,3 and j = 1,2, . . . ,6 are repre-sented by a Rossler dynamic as follows:

Ai =

0 1 11 0.2 0

0 0 6

,Di =

0

0

1

,i j =

1 10 00 0

(42)

Bi = I33,Ci =

0 1 0

0 0 1

, fi(xi) = (0.2 +xi,1xi,3) (43)

Nodes i with i = 4,5,6 and j = 1,2, . . . ,6 are repre-sented by an Hyperchaotic Rossler dynamic as follows:

Ai =0 1 1 01 0.25 0 1

0 0 0 0

0 0 0.5 0.05

,Di =

001

0

,i j =

0 00 01 11 1

(44)

Bi =I44,Ci =

1 0 0 0

0 1 0 0

, fi(xi) = (3 +xi,1xi,3) (45)

Note that for both circuits, the coefficients are such that Rosslerchaotic and hyper-chaotic attractors dynamic are presented. Awindow of 100 seconds of simulation is considered. The timevarying coupling strength is c(t) = sin(250t).

Full static state error feedback synchronization

Consistent with Suykens et al. [1999], the control strategy

adopted in the supervisory level for synchronization in thenetwork is a full static feedback rule as ui = Fi(z xi), wherez and xi are the master state vector and the estimated state

Table 1. Sliding mode and relative degree

Relative Degree HOSM Algorithm

1 Super-Twisting SMC (Levant [1993])

2 Generalized Sub-Optimal SMC (Bartolini et al. [2003])

2 Quasi-Continuous Arbitr. Order SMC (Levant [2005])

of the i-th node. Fi has been constructed with the intention tosynchronize each node with the following unstable limit cycle:

z1 = cos( )(a cos(t) + c sin(2t)) + sin( )(b sin(t) + c sin(2t))

z2 = 0.6 [a sin( ) cos(t) b cos( ) sin(t)] ++0.3 [c sin( ) cos(2t) c cos( ) sin(2t)]

z3 = z3(0) expt

to

exp(t) z2d, z4 = 0

(46)

where a = 2.6, b = 1.2, c = 0.2, = /18, = 1.7. Note thatthe fourth control involve only the nodes 4, 5, 6.

State Estimation and Unknown Input Reconstruction

The objective is to demonstrate the robustness of the presentedframework to time varying coupling strengths and varyingnetwork topologies at different time intervals (see Fig. 2). Notethat it is not necessary to have any a prior knowledge of thenumber of nodes, but only the knowledge of the matrix i j ofeach node part of the network or of a new potential one.

Recalling Definition 1, and the definitions of relative degreewith respect to the unknown input and the coupling terms withthe j-th neighbor, (in (20) and (21)), and Lemma 3, the onlycombination of the designing indices i,k which satisfy for eachnode conditions in (22) and (23), is the following one:

for the first three nodes i with i = 1,2,3:

(vi,1 = 3,ri,1 = 3,wi,1 = 2) i,1 = 2 (47)(vi,2 = 1,ri,2 = 1,wi,2 = ) i,2 = 1 (48)

for the remaining nodes i with i = 4,5,6:

(vi,1 = 3,ri,1 = 2,wi,1 = 2) i,1 = 2 (49)

(vi,2 = 2,ri,2 = 3,wi,2 = 2) i,2 = 2 (50)

Thus from equations (23) and (25) we can derive the designmatrices Mi and Gi for each observer (18)-(19). Then inaccordance with Table 1, and in light of (47)-(50), we canchoose the following sliding mode algorithm as injection terms:

for the first three nodes i with i = 1,2,3:

i = (i,1 i,2)

T

=

GSOi,1

STWi,2T

(51) for the last three nodes i with i = 4,5,6:

i = (i,1 i,2)T

=

GSOi,1

GSOi,2

T(52)

where GSOi,k

and STWi,k

are

GSOi,k

= Ui,ksign(i,k,2 i,kMi,k,2 ) (53)

STWi,k

= i,k1 +i,k2 with

i,k1 = i,k1 | i,k,1 |1/2 sign (i,k,1)

i,k2 = i,k2 sign(i,k,1) , i,k2 (0) = 0(54)

where Ui,k > Fi,k is the control magnitude, i,k = 0.5 the an-ticipation parameter, Mi,k,2 is the latest singular point of the

sliding surface i,k,2, while i,k1 > 2Fi,k and i,k2 > Fi,k aresuitable constant gains. For further details on these algorithmsrefer to the references in Table 1. The good performances ofthe estimation process are shown in Fig. 3. The actual and


6/6

0 2 4 6 8 101

0.5

0

0.5

1

Time [sec]

0 2 4 6 8 10

0

3

3

Time [sec]

f1

f2

f3

f1 est.

f2 est.

f3 est.

e1

e2

e3

e4

Fig. 1. Estimation errors and UIR after proper filtering.

1

2 3

4

56

1

2 3

4

56

1

2 3

4

56

Time [0 to 12 sec.] Time [12 to 24 sec.] Time [24 to 36 sec.]

1

2 3

4

56

1

2 3

4

56

1

2 3

4

56

Time [36 to 50 sec.] Time [50 to 65 sec.] Time [65 to 100 sec.]

Fig. 2. Time-Varying network topology.

Fig. 3. State reconstruction process in 1 (top) and4 (down).

estimated states of1 and4 are depicted. No synchronizationhas been applied (ui = 0 i).

From Corollary 2 and the chosen indices in (47)-(50), we canassert that due to the structure of the coupling matrix i j for thelast three nodes, UIR is practicable only in1,2,3, becausei,2 = ri,2 < wi,2 and rank{D

+i MiTi} = 1. However if the third

row ofi j in (45) was all zero, taking i,1 = i,2 = 2 the UIReither for the last three nodes would have been possible. Fig. 4(top) shows the convergence of the six nodes to the referencetrajectory, whereas in the bottom, the UIR of the signal f1obtained by means of1,2 = STW1,2 is shown.

6. CONCLUSION

A new approach for designing HOSM observers based on theconcept of observability indices for rectangular MIMO systemsaffected by multiple unknown input signals has been proposed.The framework has been extended to the problem of decentral-ized state estimation and unknown input reconstruction from aclass of connected heterogeneous systems. Conditions for com-plete finite-time state estimation and the UIR in each systemoperating over a network are fulfilled. The proposed frameworkis inherently robust and is totally independent to the networkconfiguration or to the number of nodes. Simulation resultsconfirmed the effectiveness of the presented methods.

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