SENSOR DEPLOYMENT FOR PIPELINE LEAKAGE DETECTION VIA OPTIMAL BOUNDARY CONTROL STRATEGIES

. We consider a multi-agent control problem using PDE techniques for a novel sensing problem arising in the leakage detection and localization of oﬀshore pipelines. A continuous protocol is proposed using parabolic PDEs and then a boundary control law is designed using the maximum principle. Both analytical and numerical solutions of the optimality conditions are studied.


Introduction.
1.1. Background and motivations. Pipeline transport is the transportation of mass flow (such as gas or crude oil) from one place to another through a pipeline network. Generally, pipelines are probably the most economical way of transporting any chemically stable substance over land, such as large quantities of oil, refined oil products or natural gas. Pipelines are also an important choice to transport oil from the platform to the tanker ships or directly to the land refinery factories in the offshore oil industry. Based on 2008 statistics, more than 6000 kilometers of pipelines have been constructed in China and more than 2000 kilometers of them are offshore pipelines [9].
The safety and security of pipeline networks is tightly controlled by government regulations and policies [29]. For example, it is a mandatory rule for pipeline operators in the State of Washington (US) to be able to detect and locate leaks of 8 graphs (e.g., [6,21,24,27,31]). When the agent population is large, the performance of the multi-agent system degrades, and thus continuation approaches can be introduced to derive unified protocols whose representations are usually governed by partial differential equations (PDEs) (e.g., [18,19,32]). In [13], the framework of partial difference equations (PDEs) over graphs is proposed to analyze the behavior of multi-agent systems equipped with decentralized control schemes. The resulting PDEs enjoy properties that are similar to those of well-known PDEs like the heat equation, which coincides with the graph Laplacian control approach proposed in [28]. Wave-like PDE models have been introduced in [4,16,17] to study the scaling laws of stability margin and robustness to external disturbances for large-scale vehicular formations. To achieve formation control of agents, linear diffusion-advection-reaction equations were investigated in [14] with dynamic boundary feedback control using backstepping design techniques [20]. To deal with the parameter uncertainties for MAS modeled by PDEs, adaptive control methods have been considered in [18,19]. A systematic flatness-based motion planning using formal power series and suitable summability methods is considered in [25] for the finite-time deployment of multi-agent systems into planar formation profiles along predefined spatial-temporal paths governed by the Burgers equation. For sensor scheduling problems in terms of given physical dynamic models (both continuous and discrete cases), optimal discrete-valued control problems (e.g., [38], [40]) have been formulated to make optimal sensor deployments (e.g., [8], [11], [12], [37]).

Contribution.
In this paper, we use the same idea in [14] to use the linear diffusion-advection-reaction PDE to model the protocol of the sensors. Instead of using the backstepping technique to design a boundary controller to stabilize the desired profile governed by the equilibrium of the PDE system, the optimal control of evolutionary PDE systems is considered in this work. For the optimal control of the linear diffusion-advection-reaction PDEs with boundary actuation, the calculus of variations method is used to derive the optimality condition, which consists of two coupled PDE boundary value problems. An iterative algorithm is then proposed to obtain the numerical solution. To the best of our knowledge, this is the first paper on sensor deployment arising in leakage detection of pipelines. For the optimal control of PDE systems, much work has been done (e.g., [1,7,10,39,26,34,22]).
1.3. Paper organization. We organize this paper as follows. In Section 2, we develop the mathematical model of the sensor deployment problem using the first order mass point kinetic assumption. Then, the continuation approach is used to obtain diffusion and diffusion-reaction protocols. In Section 3, the statement of the optimal control problem is given and the calculus of variations method is then used to derive the optimality condition. In Section 4, the analytical solutions are given for a particular special case using the method of separation of variables. For the general case, a numerical solution framework is needed. An iteration scheme is discussed in Section 5 that gives the numerical solution of the optimality condition, which consists of two coupled boundary value problems. We conclude the paper in Section 6 by stating remarks and future research work.

2.
Modeling collective dynamics of agents. Given an agent i, i ∈ {0, 1, 2, . . . , n}, the mass point dynamics can be described as dxi(t) dt = u i (t), where x i (t) denotes the position of agent i at time t and u i (t) is the control input for agent i at time t. When the agent population is large, i.e., n is large, then we consider the following dynamical model using the continuation approach [26], where x(θ, t) represents the position of agent θ at time t, and u(θ, t) is the control input for agent θ at time t. We consider the agent's identification (ID) number θ as the spatial variable of a PDE model for the collective dynamics. Note that θ is only an auxiliary map (θ : I → [0, 1], where I = {0, 1, 2, · · · , n}) to label the sensors and does not represent the spatial coordinate of the pipeline. Usually, the following control protocol is used to achieve consensus, where N i denotes the set of agents/neighbors that communicate with agent i. By introducing a vector X(t) = [x 0 (t), x 1 (t), . . . , x n (t)] T , we can rewrite the multiagent system into the following matrix representation where the matrix M is given by Formally, equation (3) can be demonstrated to coincide with the heat equation [33] ∂x(θ, t) ∂t where each agent employs the diffusion-like feedback protocol, i.e., u(θ, t) = ∂ 2 x(θ,t) ∂θ 2 . The equilibrium ( ∂ 2 x(θ,t) ∂θ 2 = 0) of this type of strategy only generates linear formations (the hollow circles and dot line shown in Fig. 2). In this work, we generalize SENSOR DEPLOYMENT FOR PIPELINE LEAKAGE DETECTION 203 the diffusion-based feedback protocol to a more complex case where the equations in the second line represents the boundary agents' dynamics at θ = 0 and θ = 1, respectively. f and g represent the velocities of the end agents.
We discretize the PDE model spatially to obtain implementable control laws for the agents, e.g., the three-point central difference scheme. By using the continuous representation, PDE control techniques can be introduced to handle large agent systems. We do not need to make the number of sensors extremely large to cover the whole pipeline, but only to ensure the sensor group has a comparably dense formation. The advantage of using mobile sensors is to realize spatial coverage. In practice, less than ten sensors are sufficient to implement the PDE-based protocol but comparably dense formation is needed to guarantee the accuracy.
In order to make a standard Dirichlet boundary condition, we integrate the boundary condition of (6) over [σ, t) (0 ≤ σ < t), then the boundary condition becomes x( where the initial time constant is usually set to be σ = 0. Thus, we consider a standard Dirichlet boundary control problem To simplify the problem, we suppose that the right end position is fixed atḠ, i.e., G(t) =Ḡ. The equilibria for (9) are much more general than linear in θ. These equilibria are governed by whereF is a designated point of the left end, for example,F = 0. Although the linear reaction-advection-diffusion feedback protocol can generate very complex deployments (the solid circles and curve shown in Fig. 2), it does not guarantee stability for an arbitrary array of (α, β, γ) without the boundary control actuation. For example, when α = 1, β = 0 and γ is sufficiently large, the solution of the uncontrolled system is unstable [33]. Now we assume the desired target is denoted byx(θ), then we can introduce a new variablex(θ, t) := x(θ, t) −x(θ) to denote the

flow direction
Sensor group-1 Sensor group-2 difference, which satisfies whereF (t) is the Dirichlet boundary control function. The stabilization problem using boundary actuation can be handled very well using the backstepping technique developed in [20]. In contrast, in this paper, we deploy an optimal control strategy.
3. The PDE control problem. We first give the following definitions for the L 2 inner product and linear operators that will be used later in this paper for the PDE control problem: where x 1 (θ) and x 2 (θ) are square integrable functions in a Hilbert space and S : subject to where Q : L 2 (0, 1) → L 2 (0, 1), is symmetric and satisfies x, Qx ≥ 0 for all x ∈ L 2 (0, 1). The control weight factor R is a strictly positive number. S is symmetric operator, and x, Sx ≥ 0 for all x ∈ L 2 (0, 1). Note that the motivation for the cost function is to makex(θ, t) small, so that the system moves to on equilibrium, or steady-state.

Numerical experiments.
To obtain the optimal control law and the solution of the system state profile, we need to solve the state and the costate equations simultaneously. These two equations are coupled through the boundary control. For general system coefficients, an analytical solution is not available and thus a numerical method must be used to obtain an approximate solution. The algorithm, which is based on the iterations between the state and costate equations, can be summarized as follows: 1. Give the initial guess ofF * [0] (t) in the admissible set; i.e., set j = 0 to start the iteration; 2. Solve the state equation to obtainx [j] (θ, t) with the controlF * [j] (t): 3. Reverse the time scale from t to T − τ and solve the costate equation to obtaiñ λ [j] (θ, τ ) and λ [j] (θ, t): 5. If the following conditions are satisfied: (t) and stop. 6. Let j = j + 1. If j > N max where N max is a given integer, then stop. Otherwise, goto Step 2.

Remark 2.
The numerical solution is based on the iteration procedures in the above algorithm which can be considered as a Rosen-type method and widely used to solve optimal control of diffusive models. For further details on the convergence of the Roson-type numerical algorithms, one may read the recent volumes [3,15] on computational optimal control of systems governed by PDEs.
Remark 3. The result in Theorem 3.1 can be extended to the case of convex constrained control sets where the Pontryagin Maximum Principle of the optimal control problem can be established without making too many changes to Theorem 3.1. More details on general optimal control of PDEs can be found in [34].  In addition, the numerical algorithm can be modified to apply the Rosen-type projected gradient algorithm to approximate the optimal control over a certain convex set [1,2].
In the numerical illustration, we let α = 0.1, β = 0, γ = 0.4, T = 3 and the initial profile isx(θ, 0) = sin(πθ), respectively. The weighting functions are set to be Q(θ, η) = θ 2 +η 2 and S(θ, η) = θ 2 +η 2 . To start the numerical iterations, we first set the control function to be zero, then the behavior of the freely driven system is shown in Fig. 3a while the corresponding costate behavior is shown in Fig. 3b, where the uncontrolled trajectoryx(θ, t) is used for the term in the costate equation (48). The Matlab function pdepe is used to generate the numerical evolutions of both the state trajectoryx(θ, t) (forward-in-time) and the costate trajectory λ(θ, t) (backward-in-time), where t ∈ [0, T ]. Now we can continue this iteration process by updating the control input (which is set to be zero) following the optimal control law (16). Then, one can immediately update the state and costate equations using the updated control input sequence. Some of the iterations for the control input sequences are shown in Fig. 4 where convergence can be observed after 13 iterations. We compare the final profilex(θ, T ) in Fig. 5 associated with the control sequences of each iteration. One can observe that the control input obtained using the optimality condition can improve the stability and convergence. The evolutions of both the state and the costate dynamics are shown in Fig. 6a and Fig. 6b, respectively.
One can observe that the compatibility condition of the Dirichlet control and the initial profile is not satisfied at the left boundary end. In practice, this is impossible and a modification function m(t) = 1−exp(−t/τ ) (by changing the control function locally at θ = 0,F (t) ←F (t)m(t)) can be introduced to satisfy the compatibility condition, i.e.,F (0) =x(0, 0), where τ is a positive constant. 6. Conclusions and future work. We proposed in this paper a methodology to solve the sensor deployment problem along the offshore pipelines to detect and localize the leakage points. By introducing the PDE protocol technique, we can handle the MAS when the population of autonomous sensors becomes quite large. To increase the flexibility of sensor distributions, the PDE protocol is more complex than the heat equation and the linear reaction-advection-diffusion PDE is used instead with a boundary control designed using the optimal control of PDEs.   Notice from Fig. 3a, that the simulation case is stable without boundary stabilizing actuation, but the dissipation is comparably slow. As the next step, the closed-loop control synthesis will be explored to give a feedback control formulation, where a nonlinear Riccati PDE will be solved to obtain the feedback gain kernel.
In addition, this paper only considers a simplified case by converting the unique dynamic boundary conditions to normal Dirichlet boundary conditions (7)- (8). A new theoretical work should be done toward the new boundary condition for the future work, including the solution, and feedback control synthesis of the PDE with the boundary conditions with time derivatives.