Sensor network scheduling for identification of spatially distributed processes Dariusz Uci´ nski Institute of Control and Computation Engineering University of Zielona G´ora, POLAND e-mail:
[email protected]
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Example ctd’: Mathematical model Let y(x, t) be the temperature at spatial point x and time t. Its evolution is governed by the partial differential equation (PDE) ∂y = θ ∆y for x ∈ Ω and t ∈ [0, tf ] ∂t θ being a parameter, subject to the initial condition y = 0 in Ω at t = 0 and the boundary conditions y = 100 ∂y = −10 ∂n ∂y =0 ∂n
on the left side of Ω
(Dirichlet condition)
on the right side of Ω
(Neumann condition)
on all other boundaries
(Neumann condition)
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If thermal conductivity θ is unknown, which is rather typical, we could observe the temperature evolution at a fixed point and tune θ so that our model fits the data as best as it can.
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Example ctd’: Calibration and measurement design
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Example ctd’: Calibration and measurement design If thermal conductivity θ is unknown, which is rather typical, we could observe the temperature evolution at a fixed point and tune θ so that our model fits the data as best as it can. Problem Where to place the pyrometer so as to get the most valuable information about θ ? Such situations (PDEs and related sensor location problems) are common in engineering practice.
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Other motivating examples environment observation and forecasting systems air pollution monitoring, groundwater resources management and river monitoring, military applications: surveillance and inspection in hazardous environments,
fault detection and isolation, emerging smart materials, intelligent building monitoring, habitat monitoring, intelligent traffic systems, computer-assisted tomography, recovery of valuable minerals and hydrocarbon from underground permeable reservoirs, and many more . . . Dariusz Uci´ nski, Sensor network scheduling . . .
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Distributed parameter system—dynamic system whose state depends on both time and space; its model (a PDE) is known up to a vector of unknown parameters θ. Observations—using sensors in order to estimate θ.
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Spatiotemporal dynamics
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Subject of the talk Spatial area Ω
x1 x2
x3
Sensors
Problem: How to choose optimal sensor locations? Dariusz Uci´ nski, Sensor network scheduling . . .
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Stationary sensors Spatial area Ω
x1 x2
x3
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Strategies of taking measurements (1)
Sensors
How to determine the best sensor configuration? Dariusz Uci´ nski, Sensor network scheduling . . .
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Scanning Sensors activated at time t
(x2 , t)
Area Ω
(x3 , t)
(x1 , t)
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Strategies of taking measurements (2)
Sensors dormant at t
How to select optimal locations at given time instants? Dariusz Uci´ nski, Sensor network scheduling . . .
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Mobile sensors Spatial area Ω
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Strategies of taking measurements (3)
x1 (·)
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x2 (·)
Starting positions
Trajectories
How to design optimal trajectories? (This is the subject of the next talk by Maciej Patan.) Dariusz Uci´ nski, Sensor network scheduling . . .
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Collection of spatially distributed inexpensive, small devices equipped with sensors and interconnected by a communication network, exchanging and processing data to cooperatively monitor physical or environmental conditions (eg. temperature, sound, vibration, pressure etc.).
:À
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Enabling technology: Sensor Networks (SNs)
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sensor node gateway sensor node communication
motes
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SN nodes Motes must be low cost, low power (for long-term operation), automated (maintenance free), robust (to withstand errors and failures), and non-intrusive. Hardware: a microprocessor, data storage, sensors, AD-converters, a data transceiver (Bluetooth), controllers, and an energy source Software: TinyOS They are already manufactured (Crossbow, Intel).
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Applications: from battle field. . .
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Applications: . . . to our life
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Existing approaches to optimal sensor location Conversion to a non-linear problem of state estimation (Malebranche, 1988; Korbicz et al., 1988); Employing random fields analysis (Kazimierczyk, 1989; Sun, 1994); Formulation in the spirit of optimum experimental design (Uspenskii and Fedorov, 1975; Quereshi et al., 1980; Rafajlowicz, 1978–1995; Kammer, 1990; 1992; Sun, 1994; Point et al., 1996; Vande Wouwer et al., 1999; Uci´nski, 1999–2010; Patan, 2002-2010). Dariusz Uci´ nski, Sensor network scheduling . . .
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Existing approaches to optimal sensor location Conversion to a non-linear problem of state estimation (Malebranche, 1988; Korbicz et al., 1988); Employing random fields analysis (Kazimierczyk, 1989; Sun, 1994); Formulation in the spirit of optimum experimental design (Uspenskii and Fedorov, 1975; Quereshi et al., 1980; Rafajlowicz, 1978–1995; Kammer, 1990; 1992; Sun, 1994; Point et al., 1996; Vande Wouwer et al., 1999; Uci´nski, 1999–2011; Patan, 2001-2011). Dariusz Uci´ nski, Sensor network scheduling . . .
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System description Consider a spatial domain Ω ⊂ R2 and the PDE ∂y ∂y ∂y ∂ 2 y ∂ 2 y = F x, t, y, , , , , θ , x ∈ Ω, t ∈ T ∂t ∂x1 ∂x2 ∂x21 ∂x22 subject to the appropriate initial and boundary conditions. Notation: I x – spatial variable, t – time, T = [0, tf ]; I y = y(x, t) – state vector; I F – known function; I θ ∈ Rm – vector of unknown (constant) parameters. Dariusz Uci´ nski, Sensor network scheduling . . .
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∂ y(x, t) = ∇ · [κ(x, θ)∇y(x, t)] + v(x) · ∇y(x, t) + f (x, t) ∂t y(x, 0) = y0 (x), y(x, t) = 0
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Consider domain Ω ⊂ R2 with sufficiently smooth boundary Γ.
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Γ
Notation: y(x, t) – contaminant concentrat. at point x and time t, κ(x, θ) – diffusivity coefficient, parameterized by θ, f (x, t) – source term v(x) = [v1 (x), v2 (x)]T – wind velocity vector field T ∂y( · ) ∂y( · ) ∇y( · ) = , ∂x1 ∂x2 Dariusz Uci´ nski, Sensor network scheduling . . .
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Given N sensors and a finite set X = x1 , . . . , x` ⊂ Ω ∪ ∂Ω of points at which they can be placed, consider their fixed configuration. Spatial domain Ω ∪ ∂Ω ` potential locations
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Observations
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Given N sensors and a finite set X = x1 , . . . , x` ⊂ Ω ∪ ∂Ω of points at which they can be placed, consider their fixed configuration. Spatial domain Ω ∪ ∂Ω ` potential locations ri sensors at xi
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Observations
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Output equation For pointwise stationary sensors we have zij (t) = y(xi , t; θ) + εij (t),
t∈T
for i = 1, . . . , ` and j = 1, . . . , ri , where X` ri = N
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i=1
εij ( · ) – white Gaussian measurement noise.
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Output equation For pointwise stationary sensors we have zij (t) = y(xi , t; θ) + εij (t),
t∈T
for i = 1, . . . , ` and j = 1, . . . , ri , where X` ri = N
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i=1
εij ( · ) – white Gaussian measurement noise. Least-squares criterion The LS estimate of θ is the one which minimizes `
r
i 1 XX J (θ) = 2 i=1 j=1
Z
2 zij (t) − yˆ(xi , t; θ) dt
T
where yˆ( · , · ; θ) stands for the solution to the state equation corresponding to a given value of θ. Dariusz Uci´ nski, Sensor network scheduling . . .
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Determine r1 , . . . , r` (the numbers of sensors assigned to points x1 , . . . , x` ) so as to maximize the identification accuracy
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Optimal measurement problem
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Determine r1 , . . . , r` (the numbers of sensors assigned to points x1 , . . . , x` ) so as to maximize the identification accuracy. ←− What is it?
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Optimal measurement problem
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Optimal measurement problem Determine r1 , . . . , r` (the numbers of sensors assigned to points x1 , . . . , x` ) so as to maximize the identification accuracy. ←− What is it? Cram´er-Rao inequality n o T ˆ ˆ ˆ cov θ = E (θ − θ)(θ − θ) M −1 We have cov θˆ = M −1 provided that an estimator is efficient. But what is M ?
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Up to a constant multiplier, we have Z tf ` 1 X M= ri g(xi , t)g T (xi , t) dt N tf i=1 0 T ∂y(xj , t; θ) is the vector of sensitivity where g(x , t) = ∂θ θ=θ0 coefficients, θ0 stands for a preliminary estimate of θ. i
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Average Fisher Information Matrix (FIM)
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Up to a constant multiplier, we have Z tf ` 1 X M= ri g(xi , t)g T (xi , t) dt N tf i=1 0
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Calculation of the sensitivity vector g I Finite-difference method I Sensitivity-equation method (direct approach) I Variational method (indirect approach)
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T ∂y(xj , t; θ) is the vector of sensitivity where g(x , t) = ∂θ θ=θ0 coefficients, θ0 stands for a preliminary estimate of θ.
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Average Fisher Information Matrix (FIM)
i
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Problem of weight optimization Minimize some measure Ψ of the FIM, e.g., Ψ(M ) = ln det(M −1 ) or Ψ(M ) = trace(M −1 ) by finding a solution (called design) of the form 1 x , . . . , x` ξ= p1 , . . . , p` where pi = ri /N , i.e., the weight pi defining the proportion of sensors placed at xi , i = 1, . . . , N .
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Problem of weight optimization Minimize some measure Ψ of the FIM, e.g., Ψ(M ) = ln det(M −1 ) or Ψ(M ) = trace(M −1 ) by finding a solution (called design) of the form 1 x , . . . , x` ξ= p1 , . . ., p` where pi = ri /N , i.e., the weight pi defining the proportion of sensors placed at xi , i = 1, . . . , N .
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The problem still has a discrete nature, as the sought weights pi are integer multiples of 1/N which sum up to unity. Consequently, calculus techniques cannot be exploited in the solution.
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Problem relaxation
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Problem relaxation
The problem still has a discrete nature, as the sought weights pi are integer multiples of 1/N which sum up to unity. Consequently, calculus techniques cannot be exploited in the solution. Thus we extend the definition of the design: when N is large, the feasible pi ’s can be considered as any real numbers in the interval [0, 1] which sum up to unity, and not necessarily integer multiples of 1/N , i.e., elements of the canonical simplex.
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Ultimate formulation ?
Find ξ =
n
x1 , ..., x` p?1 , ..., p?`
o
such that
p? = arg min Ψ[M (ξ)] p∈S
where S is the canonical simplex, subject to M (ξ) =
` X
pi Υ(xi )
i=1
Rt Here Υ(xi ) = t1f 0 f g(xi , t)g T (xi , t) dt is the fixed contribution to the FIM associated with point xi . Dariusz Uci´ nski, Sensor network scheduling . . .
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Ultimate formulation ?
Find ξ =
n
x1 , ..., x` p?1 , ..., p?`
o
such that
p? = arg min Ψ[M (ξ)] p∈S
where S is the canonical simplex, subject to M (ξ) =
` X
pi Υ(xi )
i=1
Rt Here Υ(xi ) = t1f 0 f g(xi , t)g T (xi , t) dt is the fixed contribution to the FIM associated with point xi . Dariusz Uci´ nski, Sensor network scheduling . . .
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Ultimate formulation ?
Find ξ =
n
x1 , ..., x` p?1 , ..., p?`
o
such that
p? = arg min Ψ[M (ξ)] p∈S
where S is the canonical simplex, subject to M (ξ) =
` X
pi Υ(xi )
i=1
Rt Here Υ(xi ) = t1f 0 f g(xi , t)g T (xi , t) dt is the fixed contribution to the FIM associated with point xi . Dariusz Uci´ nski, Sensor network scheduling . . .
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Ultimate formulation ?
Find ξ =
n
x1 , ..., x` p?1 , ..., p?`
o
such that
p? = arg min Ψ[M (ξ)] p∈S
where S is the canonical simplex, subject to M (ξ) =
` X
pi Υ(xi )
i=1
Rt Here Υ(xi ) = t1f 0 f g(xi , t)g T (xi , t) dt is the fixed contribution to the FIM associated with point xi . Dariusz Uci´ nski, Sensor network scheduling . . .
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Defining ∂Ψ(M ) Ψ(ξ) = ∂M M =M (ξ) ◦
we introduce crucial functions h◦ i c(ξ) = − trace Ψ(ξ)M (ξ) h◦ i φ(x, ξ) = − trace Ψ(ξ)Υ(x)
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Differentiability of the design criterion
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Theorem (Existence of solution) An optimal design exists comprising no more than m(m + 1)/2 points. Moreover, the set of optimal designs is convex.
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Characterizations of solutions
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Characterizations of solutions Theorem (Existence of solution) An optimal design exists comprising no more than m(m + 1)/2 points. Moreover, the set of optimal designs is convex. Theorem (Equivalence theorem) The following characterizations of an optimal design ξ ? are equivalent: (i) (ii)
x∈X
(iii) Cambridge 2011
the design ξ ? minimizes Ψ[M (ξ)], the design ξ ? minimizes max φ(x, ξ) − c(ξ) , max φ(x, ξ ? ) = c(ξ ? ). x∈X
All the designs satisfying (i)–(iii) and their convex combinations have the same M (ξ ? ). Dariusz Uci´ nski, Sensor network scheduling . . .
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Numerical approach: A-optimality For Ψ[M (ξ)] = trace(M −1 (ξ)), we have to find p and the additional variable q ∈ Rm to minimize J(p, q) = 1T q subject to 1T p = 1, p ≥ 0 and the Linear Matrix Inequalities (LMIs) M (ξ) ej 0, j = 1, . . . , m. eTj qj where ej is j-th coordinate versor. Dariusz Uci´ nski, Sensor network scheduling . . .
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For Ψ[M (ξ)] = λmax [M −1 (ξ)] we have to find p and q ∈ R to minimize
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˜ q) = −q J(p, subject to
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Numerical approach: E-optimality
and the LMI M (ξ) qI.
p≥0
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1T p = 1,
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For Ψ[M (ξ)] = λmax [M −1 (ξ)] we have to find p and q ∈ R to minimize
subject to and the LMI M (ξ) qI.
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˜ q) = −q J(p,
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Numerical approach: E-optimality
1T p = 1,
p≥0
Note that λmax [M −1 (ξ)] is non-differentiable!
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Numerical approach: D-optimality For Ψ[M (ξ)] = ln det[M −1 (ξ)] apply 1
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2
3
(0)
Guess pi , i = 1, . . . , `. Set k = 0. If φ(xi , ξ (k) ) ≤ m, i = 1, . . . , `, then STOP. Evaluate (k+1)
pi
(k) φ(x
= pi
i
, ξ (k) ) , m
i = 1, . . . , `.
Form ξ (k+1) , increment k and go to Step 2.
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Numerical approach: D-optimality For Ψ[M (ξ)] = ln det[M −1 (ξ)] apply 1
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2
3
(0)
Guess pi , i = 1, . . . , `. Set k = 0. If φ(xi , ξ (k) ) ≤ m, i = 1, . . . , `, then STOP. Evaluate (k+1)
pi
(k) φ(x
= pi
i
, ξ (k) ) , m
i = 1, . . . , `.
Form ξ (k+1) , increment k and go to Step 2.
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Numerical approach: D-optimality For Ψ[M (ξ)] = ln det[M −1 (ξ)] apply 1
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2
3
(0)
Guess pi , i = 1, . . . , `. Set k = 0. If φ(xi , ξ (k) ) ≤ m, i = 1, . . . , `, then STOP. Evaluate (k+1)
pi
(k) φ(x
= pi
i
, ξ (k) ) , m
i = 1, . . . , `.
Form ξ (k+1) , increment k and go to Step 2.
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Consider a metallic core with a rectangular crack inside. Ω
0.4 Cross-section plane of reconstruction
x2
0
Γ1
Γ5
Γ2
−0.4
−0.8
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Γ3
1 0.8
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Simulation example: material flaw detection
Γ4
−1 −1 −0.8
−0.4
0
x1
0.4
0.8
1
The left and right boundaries Γ1 and Γ2 are subjected to constant voltages forcing the flow of a low density steady current. Dariusz Uci´ nski, Sensor network scheduling . . .
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Computer-assisted impedance tomography Electrical voltage distribution y is described by the Laplace equation −div γ(x)∇y(x) = 0, x ∈ Ω, with boundary conditions y(x) = 10 x ∈ Γ1 , y(x) = −10 x ∈ Γ2 , ∂y(x) = 0 x ∈ Γ , i = 3, 4, 5. i ∂n Conductivity coefficient γ(x) = 0.1 + 0.2 exp − θ1 (x1 − θ2 )2 − θ3 (x2 − θ4 )2 , θ0 = (30.0, 0.3, 80.0, 0.7) Dariusz Uci´ nski, Sensor network scheduling . . .
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DAE Cambridge 2011
Results
1
Cambridge 2011
DAEW01
0.8
0.23 Γ3
0.25
0
DAEW01
0.24
1
0.24 0.25
0.4
0.05 0.23
0.8
0.11 0.37
0.4
Γ1
Γ2
0
Γ5
−0.4
−0.8
Cambridge 2011
0.03
−0.4
−0.8
Γ4
−1 −1 −0.8
−0.4
0
−1 0.4
D-optimal
Dariusz Uci´ nski, Sensor network scheduling . . .
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1
−1 −0.8
−0.4
0
0.4
0.8
1
A-optimal
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It is easy to include both the support points xj and their number as additional decision variables.
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DAEW01
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Extensions
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It is easy to include both the support points xj and their number as additional decision variables. The approach is useful as It indicates spatial regions which provide the most important information about the parameters, while quantifying their influence.
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DAEW01
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Extensions
Dariusz Uci´ nski, Sensor network scheduling . . .
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DAE Cambridge 2011 DAEW01 Cambridge 2011 DAEW01 Cambridge 2011
Extensions It is easy to include both the support points xj and their number as additional decision variables. The approach is useful as It indicates spatial regions which provide the most important information about the parameters, while quantifying their influence. Instead of several sensors placed at a point, we can use a more precise measurement device at that point.
Dariusz Uci´ nski, Sensor network scheduling . . .
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DAE Cambridge 2011 DAEW01 Cambridge 2011 DAEW01 Cambridge 2011
Extensions It is easy to include both the support points xj and their number as additional decision variables. The approach is useful as It indicates spatial regions which provide the most important information about the parameters, while quantifying their influence. Instead of several sensors placed at a point, we can use a more precise measurement device at that point. Design weights can be interpreted as measurement frequencies. Dariusz Uci´ nski, Sensor network scheduling . . .
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DAE Cambridge 2011 DAEW01
With a little turning up, the results can be employed when attacking the setting with purely binary weights (ri = 0 or ri = 1), cf. (Uci´nski and Patan, 2007). This combinatorial optimization problem is dealt with the branch-and-bound method.
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Extensions
Dariusz Uci´ nski, Sensor network scheduling . . .
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With a little turning up, the results can be employed when attacking the setting with purely binary weights (ri = 0 or ri = 1), cf. (Uci´nski and Patan, 2007). This combinatorial optimization problem is dealt with the branch-and-bound method. We shall outline it for scanning observations.
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Extensions
Dariusz Uci´ nski, Sensor network scheduling . . .
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DAE Cambridge 2011 DAEW01 Cambridge 2011 DAEW01 Cambridge 2011
Scanning observations Given N 0 sensors whose spatial positions are fixed a priori, partition the observation horizon T = (0, tf ) into adjacent subintervals Tk , k = 1, . . . , K and activate a best N -element subset of sensors over each Tk . 0 Spatial domain Ω
T1 T2 T3 tf
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DAE Cambridge 2011 DAEW01 Cambridge 2011 DAEW01 Cambridge 2011
Scanning observations Given N 0 sensors whose spatial positions are fixed a priori, partition the observation horizon T = (0, tf ) into adjacent subintervals Tk , k = 1, . . . , K and activate a best N -element subset of sensors over each Tk . 0 Spatial domain Ω
T1 T2
– active – dormant
T3 tf
Dariusz Uci´ nski, Sensor network scheduling . . .
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DAE Cambridge 2011 DAEW01 Cambridge 2011 DAEW01 Cambridge 2011
Scanning observations Given N 0 sensors whose spatial positions are fixed a priori, partition the observation horizon T = (0, tf ) into adjacent subintervals Tk , k = 1, . . . , K and activate a best N -element subset of sensors over each Tk . 0 Spatial domain Ω
T1 T2
– active – dormant
T3 tf
Dariusz Uci´ nski, Sensor network scheduling . . .
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DAE Cambridge 2011 DAEW01 Cambridge 2011 DAEW01 Cambridge 2011
Scanning observations Given N 0 sensors whose spatial positions are fixed a priori, partition the observation horizon T = (0, tf ) into adjacent subintervals Tk , k = 1, . . . , K and activate a best N -element subset of sensors over each Tk . 0 Spatial domain Ω
T1 T2
– active – dormant
T3 tf
Dariusz Uci´ nski, Sensor network scheduling . . .
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DAE Cambridge 2011 DAEW01
A reason for not using all the available sensors could be the reduction of the observation system complexity and the cost of operation and maintenance. I Another interpretation: several mobile sensors. I Sensors track points providing best information about the DPS. I Rapid progress in technology makes it affordable.
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Advantages and drawbacks
Dariusz Uci´ nski, Sensor network scheduling . . .
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DAE Cambridge 2011 DAEW01 Cambridge 2011 DAEW01 Cambridge 2011
Advantages and drawbacks A reason for not using all the available sensors could be the reduction of the observation system complexity and the cost of operation and maintenance. I Another interpretation: several mobile sensors. I Sensors track points providing best information about the DPS. I Rapid progress in technology makes it affordable. But: I Lack of efficient algorithms. I Combinatorial complexity excludes direct (naive) approaches. Dariusz Uci´ nski, Sensor network scheduling . . .
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DAE Cambridge 2011 DAEW01 Cambridge 2011 DAEW01 Cambridge 2011
Observation At Stage k we have time subinterval Tk and zki (t) = vki y(xi , t; θ) + ε(xi , t) for t ∈ Tk and i = 1, . . . , I, where ε( · , · ) – white Gaussian measurement noise, Decision variables ( 1 if the i-th sensor is active over Tk vki = 0 otherwise
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Fisher information matrix We have M (v) =
K X I X
vki Mki
k=1 i=1
where Z Mki =
g(xi , t)g(xi , t)g T (xi , t) dt
Tk
and g(x, t) = ∇θ y (x, t; θ0 ) are, as usual, the sensitivity coefficients. Matrices Mki can be computered and stored before optimization. Dariusz Uci´ nski, Sensor network scheduling . . .
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(b) balanced 1
0.9
0.9
0.8
0.8
0.7
0.7
activation time
activation time
(a) unbalanced 1
0.6 0.5 0.4
0.6 0.5 0.4
0.3
0.3
0.2
0.2
0.1 0
0.1 1
2
3
4
5
6
sensor
7
0
1
2
3
4
5
6
7
sensor
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Balanced operation scheduling
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(b) balanced 1
0.9
0.9
0.8
0.8
0.7
0.7
activation time
activation time
(a) unbalanced 1
0.6 0.5 0.4
0.6 0.5 0.4
0.3
0.3
0.2
0.2
0.1 0
0.1 1
2
3
4
5
6
sensor
7
0
1
2
3
4
5
6
7
sensor
We would like have at least n sensors active at each stage on condition that each the total activation time of each sensor be at most τmax .
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Balanced operation scheduling
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activation time
(a) unbalanced
(b) balanced
1
1
0.9
0.9
0.8
0.8
0.7
0.7
activation time
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DAEW01
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Balanced operation scheduling
0.6 0.5 0.4
0.6 0.5 0.4
0.3
0.3
0.2
0.2
0.1 0
0.1 1
2
3
4
5
6
7
0
sensor
1
2
3
4
5
6
7
sensor
We would like have at least n sensors active at each stage on condition that each the total activation time of each sensor be at most τmax . Accumulated activation time τi =
K X
vki ∆tk ≤ τmax ,
∆tk = tk − tk−1
k=1 Dariusz Uci´ nski, Sensor network scheduling . . .
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DAE Cambridge 2011 DAEW01 Cambridge 2011
Problem (P) Let vk = (vk1 , . . . , vkI ). Find a sequence v = (v1 , . . . , vK ) to minimize P(v) = Ψ M (v) subject to K X vki ∆tk ≤ τmax , i = 1, . . . , N k=1 I X
vki ≥ n,
k = 1, . . . , K
vki = 0 or 1,
i = 1, . . . , N,
i=1
k = 1, . . . , K
Cambridge 2011
DAEW01
Ultimate formulation
Dariusz Uci´ nski, Sensor network scheduling . . .
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DAE Cambridge 2011 DAEW01 Cambridge 2011 DAEW01 Cambridge 2011
Ultimate formulation Problem (P) Let vk = (vk1 , . . . , vkI ). Find a sequence v = (v1 , . . . , vK ) to minimize P(v) = Ψ M (v) subject to K X vki ∆tk ≤ τmax , i = 1, . . . , N k=1 I X
vki ≥ n,
k = 1, . . . , K
vki = 0 or 1,
i = 1, . . . , N,
i=1
k = 1, . . . , K
This is a 0-1 integer programming problem.
Dariusz Uci´ nski, Sensor network scheduling . . .
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DAE Cambridge 2011 DAEW01
8 combinatorial nature of the optimization task 8 the problem dimensionality may be very high 8 difficulties in direct application of algorithms known from the area of sensor network design
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Main difficulties
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Main difficulties
8 combinatorial nature of the optimization task 8 the problem dimensionality may be very high 8 difficulties in direct application of algorithms known from the area of sensor network design Remedy A way out is to apply the well-known branch and bound technique based on an efficient ‘bounding test’.
Dariusz Uci´ nski, Sensor network scheduling . . .
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Solution via Branch&Bound – properties It consists of a systematic enumeration of all candidate solutions, where large subsets of fruitless candidates are discarded en masse, by using upper and lower estimated bounds of the quantity being optimized. Represent all feasible solutions as the leaves of a binary tree whose branches are then recursively processed as follows: Solve the ‘relaxed problem, i.e., the one with no integrality constraints. If the relaxed problem is infeasible—backtrack (there is no integer solution in this branch) If the solution is integral—terminate (‘feasible). Otherwise, split on a variable for which the assignment is non-integral, and repeat for each case. The relaxed problem is solved using simplicial decomposition (Uci´nski and Patan, 2007). Dariusz Uci´ nski, Sensor network scheduling . . .
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Branch & Bound processing V (I0 ; I1)
DAEW01
computing lower/upper bounds
V (I0 ; I1)
V (I0 ; I1)
Cambridge 2011
pruning
V (I0 ; I1)
V (I0 ; I1)
V (I0 ; I1)
V (I0 ; I1)
V (I0 ; I1)
V (I0 ; I1)
V (I0 ; I1)
V (I0 ; I1)
Cambridge 2011
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branching
Dariusz Uci´ nski, Sensor network scheduling . . .
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Computer example Consider the transport-chemistry process of an air pollutant on the observation interval T = (0, 800] [s] ∂y(x, t) + ∇ · v(x, t)y(x, t) + αy(x, t) ∂t = ∇ · κ∇y(x, t) + f1 (x) + f2 (x),
x ∈ Ω = [0, 1 km]2 ,
with boundary and initial conditions ∂y(x, t) = 0, ∂n y(x, 0) = 0,
(x, t) ∈ ∂Ω × T, x ∈ Ω.
y(x, t) – pollutant concentration, v(x, t) = 7.2 · (x1 + x2 − t · 10−3 ), (2x1 − 1)t · 10−3 + x2 − 1 – wind velocity field, κ = 50 m2 /s – diffusion coefficient, α = 0.01 s−1 – elimination rate. Dariusz Uci´ nski, Sensor network scheduling . . .
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DAE Cambridge 2011 DAEW01
f` (x) = µ` exp − 100kx − χ` k2 , ` = 1, 2 – active sources of pollutant located in (0.5 km, 0.75 km) and (0.75 km, 0.5 km), The observation interval has equipartitioned into 8 subintervals Tk = (100(k − 1), 100k], k = 1, . . . , 8, The maximum accumulated activation time per sensor was fixed as τmax = 300 s,
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Simulation setting
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DAE Cambridge 2011 DAEW01 Cambridge 2011 DAEW01 Cambridge 2011
Simulation setting
f` (x) = µ` exp − 100kx − χ` k2 , ` = 1, 2 – active sources of pollutant located in (0.5 km, 0.75 km) and (0.75 km, 0.5 km), The observation interval has equipartitioned into 8 subintervals Tk = (100(k − 1), 100k], k = 1, . . . , 8, The maximum accumulated activation time per sensor was fixed as τmax = 300 s, Objective Find a D-optimal schedule for activation of I = 460 scanning sensor nodes to identify the sources, i.e., to estimate the vector of their locations and intensities θ = (µ1 , χ11 , χ12 , µ2 , χ21 , χ22 )
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Results (1) time=0 1
0.8
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0 0
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0.8
1
n = 100 Dariusz Uci´ nski, Sensor network scheduling . . .
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Results (2) time=0 1
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0 0
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1
n = 50 Dariusz Uci´ nski, Sensor network scheduling . . .
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Finally, time-optimal paths of moving sensors Assume that the process evolves over the time horizon of [α, β], but we wish to avoid an excessive battery consumption and select its shortest subinterval [t0 , t0 + T ] to take measurements guaranteeing ‘reasonably rich’ information about θ. Therefore, for N mobile sensors we have zj (t) = y(xj (t), t; θ) + εj (t),
t ∈ [t0 , t0 + T ]
j = 1, . . . , N , where εj ( · ) is white Gaussian measurement noise.
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Dynamics of sensor movements Define the state vector s(t) = (x1 (t), x2 (t), . . . , xN (t)),
∀ t ∈ [t0 , t0 + T ]
A general model is s(t) ˙ = f (s(t), u(t)) a.e. on [t0 , t0 + T ],
s(t0 ) = s0
where s0 ∈ Rn is the initial sensor configuration, u : [t0 , t0 + T ] → Rr is a control satisfying ul ≤ u(t) ≤ uu
Dariusz Uci´ nski, Sensor network scheduling . . .
a.e. on [t0 , t0 + T ]
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Constraints on sensor movements The conditions of taking measurements only in a given admissible subdomain and keeping safe distances between the sensors are written as γ` (s(t)) ≤ 0, ∀ t ∈ [t0 , t0 + T ], ` ∈ ν¯ = 1, . . . , ν where γ` : Rn → R are C 1 functions. Then the Fisher Information Matrix is M (s) =
N Z X j=1
Dariusz Uci´ nski, Sensor network scheduling . . .
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g(xj (t), t)g T (xj (t), t) dt
t0
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In the proposed time-optimal formulation, we wish to guarantee an acceptable level of the information content of the collected observations, quantified by the D-efficiency: ED (s) =
det(M (s)) det(M (b s)
1/m
where sb stands for the D-optimal trajectories obtained for the observations over the entire feasible time interval [α, β].
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Time duration as a critical factor
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The value of det(M (ˆ s)) can be determined beforehand, and setting a reasonable positive threshold η < 1, we can introduce the constraint relation ED (s) ≥ η, which guarantees a suboptimal yet reasonable solution. This is equivalent to Ψ[M (s)] ≤ C where C = Ψ[M (ˆ s)] − m log(η).
Cambridge 2011
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Guaranteed information content
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DAE Cambridge 2011 DAEW01 Cambridge 2011 DAEW01 Cambridge 2011
Time-optimal control problem Find
min (s0 ,u,t0 ,T )∈P
T subject to h(s0 , u, t0 , T ) ≤ 0 and
J(s0 , u, t0 , T ) ≤ 0 where P = (s0 , u, t0 , T ) : s0 ∈ X N , [t0 , t0 + T ] ⊆ [α, β], u : [t0 , t0 + T ] → Rr is measurable, ul ≤ u(t) ≤ uu a.e. on [t0 , t0 + T ]
J(s0 , u, t0 , T ) = Ψ[M (s)] − C, h(s0 , u, t0 , T ) = max γ` (s(t)) . (`,t)∈¯ ν ×[t0 ,t0 +T ]
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It is convenient to cast the problem in question as a problem in canonical form which is acceptable by common numerical software for solving optimal-control problems (RIOTS 95, DIRCOL, MISER). The procedure is rather standard and reduces to augmenting the state vector by some functions of the sensitivity coefficients along the sensor trajectories.
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Reduction to canonical form
Dariusz Uci´ nski, Sensor network scheduling . . .
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Consider the heat equation ∂y(x, t) ∂ ∂y(x, t) ∂ ∂y(x, t) = κ(x) + κ(x) ∂t ∂x1 ∂x1 ∂x2 ∂x2 + 20 exp −50(x1 − t)2 , (x, t) ∈ (0, 1)3 , y(x, 0) = 0, x ∈ Ω y(x, t) = 0, (x, t) ∈ ∂Ω × T where κ(x) = θ1 + θ2 x1 + θ3 x2 , θ1 = 0.1, θ2 = −0.05, θ3 = 0.2, s(t) ˙ = u(t), s(t0 ) = s0 , |ui (t)| ≤ 0.7, i = 1, . . . , 6
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Computer example
Dariusz Uci´ nski, Sensor network scheduling . . .
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1
0.8
0.8
0.6
0.6
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x
x
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2
1
2
DAE DAEW01
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Results
0.4
0.4
0.2
0.2
0 0
0.2
0.4
0.6
0.8
x1
D-optimum solution
Dariusz Uci´ nski, Sensor network scheduling . . .
1
0 0
0.2
0.4
0.6
0.8
1
x1
Time-optimal, D-efficiency = 0.8 t0 = 0.47, T = 0.53
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Concluding remarks: Ongoing work
Extensions to Kalman filtering (joint work with Professor M. Demetriou from Worcester Polytechnic Instutite) Inverse problems for thermal processes (with Professor Laurent Autrique, Universit´e d’Angers) Development of parallel algorithms in clusters of PCs (with Tomasz Zi¸eba and Przemyslaw Baranowski, University of Zielona G´ora) Development of decentralized approaches (sensor nodes as intelligent autonomous agents) (with Maciej Patan, University of Zielona G´ora)
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More information . . .
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