Formulation of the Estimation Problem on a Lattice

Conclusions, Future Directions, and Possible Extensions

7.2 Future Directions and Possible Extensions

7.2.2 Monitoring a Distributed Environment

7.2.2.2 Formulation of the Estimation Problem on a Lattice

Note that when an agent is in one of the interesting locations, we are not interested in tracking exactly its position; as a consequence we will represent as a point on the abscissa each interesting location. This way the not interesting locations λ₀ will appear as connectors between interesting locations (points on the abscissa). Thus, the agent abscissa is formally defined in the following definition.

Definition 7.2.4. (Agent abscissa) Let pi : [a,b] → E be the trajectory of agent Ai with

∆hi =0 in the environmentEwitha≥ 0 andt=a+k≤bfor somek ∈N. The quantity zi(t)=

Xt−1 τ=a

kp¯i(τ+1)− p¯i(τ)k,

with

¯ pi(t)=











pi(t) if pi(t)∈λ₀

p(λ_j) if pi(t)∈λ_j for j, 0,

is the agent Ai abscissa, in whichp(λ_j) denotes the position of the door of location j. The set of all points on the abscissa is denoted by

γ_i ={zi | ∃t ∈[a,b] witht =a+kfork ∈Nandzi = Xt

τ=a

kp¯i(τ+1)−p¯i(τ)k}.

In the case in which each location has more than one door, we can still collapse the location at one point. The coordinates along the abscissa of agentAi corresponding to the locationλ_jare denoted byλⁱ_j ={λ^i,1_j , ..., λ^i,n_j ⁱ}, each associated with a different time the agent visited that location along its trajectory. Also,λⁱ₀ = γ_i− ∪_jλⁱ_j. The set of positions along the abscissa at which there is a sensor j∈ {1, ...,m}, is denotedλⁱ_s,_j.

In these new coordinates, the system state is given by (z, α) ∈ Z × L^N, with z =

(z1, ...,zN) andZ= γ₁×...×γ_N. The new system dynamics is defined for anyias

z_i(k+1) = z_i(k)+v_i+ ∆v_i(k) (7.6) α_i(k+1) = λ_j ifzi(k+1)∈λⁱ_j (7.7) y_j(k+1) = 1 if∃iwithz_i(k+1)∈λⁱ_s,_j (7.8) yj(k+1) = 0 ifzi(k+1)<λⁱ_s,_j∀ i, (7.9)

z_i(0) ∈ [Li,0,U_i,0], (7.10)

in which we assume that vi is known, constant along the abscissa, and any variation to it is incubated in the uncertainty ∆v_i(k), with∆v_i(k) ∈ [∆v_i,m,∆v_i,M]. The uncertainty on the initial condition is transformed to an interval not including sensor locations along the abscissa. We do not comment at this point on the constraints specified by (7.4) and devote a later section to them.

This formulation is in accordance with the kind of model that we could have for each persons daily habits. The behavior will typically be described by sentences of the form “ agent ienters in the morning between 8am and 10am, then he usually goes to his office, where he has 30 minutes-1 hour meetings scheduled with agents (in the order) j,k,q.”

Notice also that, due to the discrete model of the agent motion, the abscissa has a grid of points on which the agent transitions, the abscissa being defined on the basis of the nominal dynamics. In order for the agent with uncertainty∆v_i(k) ,0 to still evolve on the abscissa, we require that ∆vi(k) is such that for any k it moves the agent at points on the grid of the abscissa. This technical point is due to our choice of a discrete model for the agent dynamics. This technical point would be abandoned if the equations of motion of the agent were represented in continuous time. We also assume for simplicity that there cannot be two locations next to each other on the abscissa.

In the next definition, we establish a partial order onZ × L^N.

Definition 7.2.5. For any pair of elementsw,x∈ Z × L^Nwithw=(z^w, α^w) andx= (z^x, α^x)

the partial order (Z × L^N,≤) is established by

w≤ xif and only ifz^w ≤z^x and

z^w≤ z^x if and only ifz^w_i ≤z_i^x for alli.

Since the partial order is established on the basis of thezcomponent only, and the discrete stateαcan be unequivocally determined once the continuous one has been found, we continue our arguments considering thezcomponent of the state only. Thus, we establish the partial order (χ,≤) with χ = Z and partial order as given in Definition 7.2.5. Any x≤w∈ Zdefine an interval sublattice inZdenoted [x,w]= [x1,w1]×...×[xN,wN]. Note that the discrete state evolution established by f_iwas used for defining the abscissasγ_i, as f_i establishes the sequence of locations that the agent visits, andhi establishes how the agent evolves in each location. In the new coordinate systemZ, these two different evolutions have been fused together into one variable evolutionzi for each agent. For anyx ≤w∈ Z, we define thedistance d(x,w) as follows

d(x,w) := 1 N

i=1

(wi−x_i), withw_i− x_i :=|[xi,w_i]− {x_i}|.

The system represented by equations (7.6-7.7-7.8-7.9) is denotedΣ = (Z,Y,F,g), in which F is specified by equation (7.6) and g by equations (7.8-7.9). If ∆v_i = 0 for any i, the dynamics (7.6) preserves the partial order (χ,≤), and F : [L,U] → [F(L),F(U)] is onto. If ∆v_i , 0, i.e., the system is nondeterministic, one can verify that F : [L,U] → [^VF(L),^WF(U)] is order preserving and onto.

Next, we show that the output set can be approximated by an interval in (χ,≤). Let again Oydenote the output set corresponding to a measurementy, that is,Oy ={z∈ Z |g(z)= y}. Let [L,U] ⊆ Zdenote an interval sublattice inZ, withL= (L1, ...,L_N),U = (U1, ...,U_M), Ui ∈γ_i, and Li ∈ γ_i, such that|[Li,Ui]∩λⁱ_s,_j| ∈ {1,0}for each i, j. This means that [Li,Ui] cannot contain more than one coordinate point onγ_icorresponding to the same location at which a sensor is positioned. Let Oy|_[L,U] denote the output set oncezhas been restricted

to belong to an interval sublattice [L,U], that is, Oy|_[L,U] = {z ∈ [L,U] | g(z) = y}. For simplifying notation, assume that only one sensor can fire at one time. Then, such set has the property shown in the following proposition.

ly,i =











λⁱ_s,_j∩[Li,U_i] ifλ^k_s,j∩[Lk,U_k]=∅ ∀k,i,

L_i ifλ^k_s,j∩[Lk,U_k],∅for some k ,i and L_i <λⁱ_s,k for k, j, L_i+v_i ifλ^k_s,j∩[Lk,U_k],∅for some k ,i and L_i ∈λⁱ_s,k for k, j,

(7.11)

and

u_y,i =











λⁱ_s,_j∩[Li,Ui] ifλ^k_s,_j∩[Lk,Uk]=∅ ∀k,i,

Ui ifλ^k_s,_j∩[Lk,Uk],∅for some k ,i and Li <λⁱ_s,k for k, j, Ui−vi ifλ^k_s,_j∩[Lk,Uk],∅for some k ,i and Li ∈λⁱ_s,k for k, j.

(7.12)

Proof. This can be easily proved by showing two facts. 1) If x ∈Oy|_[L,U], thenly ≤ xand uy ≥ x. 2) BothLy andUyare inOy|_[L,U].

Proof of 1). We proceed component-wise. If x∈Oy|_[L,U], theng(x)=yandxi ∈[Li,Ui].

Then, ifly,i = Lithere is nothing to show. Ifly,i =Li+vi, we need to show that there cannot be any x ∈Oy|_[L,U] with xi = Li. Ifly,i = Li+vi, then by (7.11), xi ∈λⁱ_s,k fork , j, that is, it is placed at a sensor location. By the definition ofgin equations (7.8-7.9) if xi ∈λⁱ_s,k for somek, it must be yk = 1, which is a contradiction. Finally, consider ly,i = λⁱ_s,_j∩[Li,Ui] and thusUy,i = Ly,i. Then, by (7.11) we have thatλ^k_s,_j∩[Lk,Uk] = ∅ ∀k , i. This, in turn implies that xi = λⁱ_s,_j ∩[Li,Ui] because agenti is the only one that can have caused that sensor firing.

Proof of 2). This is clear ifly,i = uy,i = λⁱ_s,_j ∩[Li,Ui]. Ifly,i = Li, we show that there is x ∈ Oy|_[L,U]such that xi = Li. If there is no xsuch that xi = Li withg(x) = y, it means that Li ∈ λⁱ_s,k for k , j, which is a contradiction by the definition (7.11). Ifly,i = Li+vi,

we show that there isx ∈Oy|_[L,U]such thatxi = Li+vi. If this is not the case, it means that Li +vi ∈λⁱ_s,k fork , j. However, by (7.11)Li = λⁱ_s,k fork , j. So we would have bothLi

andLi+vi corresponding to locations. This contradicts our assumptions that establish that on one abscissa there cannot be locations close to each other.

This proposition guarantees that the interval sublattice [ly,uy] is the smallest interval that containsOy|_[L,U], and thus it is the best representation of such a set in terms of interval sublattices. The reason whyOy|_[L,U] , [ly,uy] is because there are points in [ly,i,uy,i] where agent Ai cannot be. In fact the agent cannot be at coordinate points on the abscissa that correspond to sensor locations at which the senor has not fired. Define for eachithe set of points where the sensors have not fired as

z^{n f}_i :=∪_j{λⁱ_s,j : yj =0}. (7.13) We denoteZ_not :=Z−[(γ1−z^{n f}₁ )×...×(γN−z^{n f}_N )]. By definition, no state compatible with the output can be in such a set. In case∆vi = 0 for eachi, such a set changes dynamically in a deterministic way as positions that were not occupied at stepkmap to positions that cannot be occupied at stepk+1. This gives rise to a set U(k) to which Z must be constrained.

Such a constraint set is defined in the following definition.

Definition 7.2.6.For any executionσof the systemΣwith output sequenceg(σ)= {y(k)}_k∈N

and with∆v_i =0 for anyi, we define theconstraint setat stepk,U(k), to be the set defined as

U(0)= Z − Z_not(0),

U(k+1)= F(U(k))− Z_not(k+1).

Note that the notion of constraint set makes no sense if ∆vi , 0 as in such a case we cannot know from one step to another whereU(k) is mapped to.

In case ∆vi , 0, the functionF maps a point to a set. The supremum and infimum of this set for anyz∈ Zare defined as

_F(z)=z+v+ ∆vM

and

^F(z)= z+v+ ∆vm. If there is no uncertainty,^WF =^VF = F.

Given the systemΣ =(Z,Y,F,g) withz(0)∈ [L0,U0]⊂ Zand with output sequence {y(k)}_k∈N, we can implement the estimator for nondeterministic systems on a partial order presented in Chapter 5, that is,

L(k+1) = ^{^}F(L(k))gly(k+1)

U(k+1) = ^_F(U(k))fuy(k+1) (7.14) L(0) = L0, U(0)=U0,

in which ly(k) = ^VOy(k)|_[L⁰_,U⁰_] and uy(k) = ^WOy(k)|_[L⁰_,U⁰_] with L⁰ = ^VF(L(k)) and U⁰ =

WF(U(k)). We also assume that the set [L(k),U(k)] is such that|[Li(k),Ui(k)]∩λⁱ_s,_j| ∈ {1,0} for each i, j. This can be verified if L0 andU0 are not too far from each other compared to the distances between the coordinate of the sensor locations on the abscissas. If the system is deterministic, that is,^WF =^VF = F, then we have the following result, which is a straightforward consequence of the results in Chapter 3.

Proposition 7.2.6. Given the system Σ = (Z,Y,F,g)with output sequence {y(k)}_k∈N and with∆v_i =0for any i, then the update laws (7.14) are such that

(i) z(k)∈[L(k),U(k)]for any k;

(ii) |[L(k+1),U(k+1)]| ≤ |[L(k),U(k)]|;

(iii) ifΣis observable, then there is a k0 >0such that[L(k),U(k)]∩ U(k)= z(k)for any k≥ k0.

The only difference with the results in Chapter 3 is that in the present case the setUis not constant in time. It in fact depends on the measurements while in Chapter 3 it depended on the space structure, which was the case of Chapter 3. Despite this difference, the results follow in the same way.

If the system is nondeterministic, in order to guarantee that |[L(k),U(k)]| is bounded asymptotically, we need to ask for an additional structural property of the system. In fact, observability is not sufficient because the output set is not exactly an interval sublattice as the theory developed in Chapter 5 requires. This additional property, named interval observability, is defined in the following definition.

Definition 7.2.7. (Interval observability) Consider the systemΣ =(Z,Y,F,g) with output sequence{y(k)}_k∈N, and consider the update laws as in equations (7.14). The systemΣ is said to beinterval observableif for anyithere is an infinite sequence {ki,0, ...,ki,l, ...}such that

(i) U_i(ki,l)∈λⁱ_s,_jfor some j;

(ii) [Li(k),Ui(k)]∩λⁱ_s,_j , ∅ ⇒ [Lp(k),Up(k)]∩λ^p_s,j = ∅for any p , i, for anyki,l ≤ k ≤ ki,l+ ∆ki,l, withki,l+ ∆ki,l such thatL(ki,l+ ∆ki,l)∈λⁱ_s,_j.

This property basically requires that periodically the set [Li(k),Ui(k)], for anyi, will be the only one to contain the coordinate corresponding to the sensor jfor all the interval of time that [Li(k),Ui(k)] contains such a coordinate. This guarantees that the set [L(k),U(k)]

does not grow unbounded due to nondeterminism as the following proposition shows.

Proposition 7.2.7. Given the systemΣ = (Z,Y,F,g)with output sequence{y(k)}_k∈N, then the update laws (7.14) are such that

(i) z(k)∈[L(k),U(k)]for any k;

(ii) ifΣis interval observable, then there is a k0 >0such that d(L(k),U(k))≤ M with

M = 1 N

i=1

min di∆v_i,M−∆v_i,m vi+ ∆vi,m ,di

! ,

where di =maxl(U(ki,l+1)−U(ki,l)).

Proof. For eachi, letdi := maxj(U(ki,j+1)−U(ki,j)). Then, given the update laws (7.14), we have that agent A_i takes between d_i/(vi + ∆v_i,M) andd_i/(vi + ∆v_i,m) steps to cover the

distancedi, then we have that

U_i(k)−L_i(k)≤ min d_i∆vi,M−∆vi,m

vi+ ∆vi,m ,d_i

! .

Letdi,mbe the minimum distance between interesting locations on the abscissaγ_i. Prob- lem 3.2.1 is solved if the uncertainties on the sets [Li,U_i] are such that

min di∆v_i,M−∆v_i,m vi+ ∆vi,m ,di

≤di,m∀i. (7.15)

Note that there are two ways to satisfy such inequality:

(1) Act on the sensor position such thatdiis decreased. For example, one can put sensors such that between the points on the abscissa corresponding to U(ki,j) andU(ki,j+1), there is only one interesting location.

(2) The previous measure is not needed if the smallest velocity of the agentvi + ∆vi,mis high compared to the distancedi. In this case, the agents excite the sensor with high frequency and thus the uncertainty is decreased (we show this point in a simulation example).

Dalam dokumen State estimation in multi-agent decision and control systems (Halaman 110-118)