An Improved Cooperative Team Spraying Control of a Diffusion Process With a Moving or Static Pollution Source

I. Introduction

IN reality, diffusion can be regarded as the net movement of molecules or atoms from a region of high concentration to a region of low concentration. It has lots of applications in chemistry [1], physics [2], biology [3], economics [4] and engineering [5]. Many diffusion processes can be modeled as distributed parameter systems (DPSs), see [6]–[8] for more knowledge.

Earlier works on controlling the diffusion process with a static pollution source were motivated by the applications of centroidal Voronoi tessellations (CVTs) [9]–[11] and mobile actuator-sensor networks (MAS-Net) [12]. Notable pioneering work on CVTs was provided by Du et al. [13]. As we know, the CVT algorithm is a kind of non-model method which can solve the problem of coverage control [14]–[17]. Motivated by the work [11], [18] that the CVT algorithm was used for the sensor placement problem in coverage control and the sensor location problem in feedback control of partial differential equation (PDE) systems, CVTs were extended to the work on diffusion control for the actuator deployment problem with the static sensor networks in [19]. Here the CVT algorithm is also applicable to our work in this paper. Actually, the monitoring and control of a diffusion process can be regarded as the optimal sensor/actuator placement or motion planning problem [20], [21]. More recently, there have been emerged related work on the use of the CVT algorithm for diffusion process control based on MAS-Net [6], [21]–[24]. And other work [25]–[27] on boundary control for PDEs based on the backstepping technique has also arisen.

To the best of our knowledge, the diffusion system in the previous research work is mostly discussed together with proportional (P) controllers and a static pollution source. In fact, a moving pollution source may be more practical than a static one. For instance, smart enemy target sources can be viewed as moving pollution sources in military warfare, which are expected to be eliminated under a group of mobile actuators (robots). Additionally, in some engineering applications, a static or steady state error can be induced by the P control method, which can be taken as a drawback of this method. Owing to this drawback, proportional integral (PI) controllers with the advantage of eliminating steady error and better flexibility (more accurate control) are used in practice instead of P controllers. However, very few results are provided on the control of a diffusion process with a moving or static pollution source based on PI controllers and the CVT algorithm currently.

Motivated by the fact a novel PI method has a better performance than a P control method (i.e., P controller for actuators motion control and spraying control) on controlling a diffusion process [19], [28], we use this PI control method to address the control problem for the diffusion process. Note that, the novelty of the PI method is a PI controller for actuator motion control together with a PI controller with time-delay (for more details, see the last part in Section II) for actuator spraying control. Moreover, a modified simulation platform (diffusion-mobile-actuators-sensors-2-dimension-proportional integral derivative (Diff-MAS2D-PID)) is built for the diffusion system together with PI, PD and PID controllers and moving or static pollution sources. In numerical simulations, we choose the appropriate parameter to minimize the difference of total pollution amount and total spraying amount, which makes the area of interest not overdosed. In addition, this paper addresses a diffusion control problem for an integer-order system by integer-order PI controllers, and the anomalous diffusion control problem in work [29] is realized by fractional-order controllers. From the theoretical point of view, here the convergence analysis of the mobile actuators location by the Lyapunov method in this paper could provide some insights into the control theory of diffusion processes.

The remainder of this paper is organized as follows. The problem statement is introduced in Section II briefly. In Section III, the control problem of a diffusion process is discussed. Section IV gives the convergence analysis of the actuators location. In Section V, a modified simulation platform (Diff-MAS2D-PID) is described. Then, numerical simulations are given to test the effectiveness of our proposed method. Finally, conclusions and some future work are provided in Section VI.

II. Problem Statement

The problem of actuator motion control and spraying control can be framed in a cyber-physical system [21]. In this section, we specifically illustrate the diffusion problem.

A diffusion process could occur in an area $\Psi$, which represents a convex polytope in ${\cal{R}}^2$. A group of $n$ mobile actuators can move freely to a certain position in this region. Denote the set $P = \{p_1, p_2, \ldots, p_n\}$ as the position coordinate set of mobile actuators, where $p_i$ represents the position coordinate of the $i$th actuator. $\phi(x, y, t) \colon \Psi \to {\mathbb R}_+$ can be described as the pollution concentration in the area $\Psi$. In order to simplify the presentation of the dynamic process, the pollution concentration $\phi(x, y, t)$ is governed by the following PDE

where $a > 0$ denotes a diffusion coefficient, $f_d(x, y, t)$ is a moving or static pollution source, $\tilde\phi$ represents the measured data of $\phi$ from the sensors, and $f_c(\tilde\phi, x, y, t)$ is a control input whose exact representation is determined by designers.

We suppose that every actuator at its position will receive information from static sensors, move to the high pollution concentration and then release chemical neutralization by the controller. In this paper, our control objectives are introduced as follows:

1) Control the diffusion process rationally and decrease the amount of total pollutants.

2) Minimize the polluted area which is heavily affected.

3) Improve the spraying speed and neutralize the pollution as quickly as possible without making the area of interest overdosed.

$\Psi$ can be divided into a collection of $n$ polytopes $\Theta = \{V_1, \ldots, V_n\}$. Suppose that $\forall i \in \{1, \ldots, n\}$, there exists

where $\vert \cdot \vert$ represents the Euclidean distance, and $s$ is an arbitrary point in the area $\Psi$. It is obvious that $V_i \cap V_j = \emptyset$ for $i\neq j$ and $\cup_{i = 1}^{n}\bar{V_i} = \bar{\Psi}$, where $\bar{V_i} = V_i \cup \partial{V_i}$, $\bar{\Psi} = \Psi \cup \partial{\Psi}$, $\partial{V_i}$ and $\partial{\Psi}$ represent boundaries of $V_i$ and $\Psi$ respectively.

The set of regions $({V_i})_{i = 1}^{n}$ is defined as the Voronoi tessellation which is generated by the set of points $({p_i})_{i = 1}^{n}$. When $V_i$ is adjacent to $V_j$, the point $p_i$ can be viewed as a neighbor of the point $p_j$. In particular, if the generators $({p_i})_{i = 1}^{n}$ of the Voronoi cells are also themselves centroids (centers of masses) of these cells, such a Voronoi tessellation is called a CVT.

Fig. 1(a) shows that a CVT can be constructed by 10 random points in the area $(0, 1)\times (0, 1)$ with a density function $\phi(x, y) = e^{-8(x+0.2)^2-8(y+0.2)^2}$. In fact, Voronoi cells are always generated by trajectories of mobile actuators at a certain range of time rather than some random points, which could be caused by the movement of actuators, as shown in Fig. 1(b). Suppose that actuators move a short distance and keep safe distance with each other in a certain time while Voronoi cells change along with moving trajectories of mobile actuators. Moreover, due to distance between any two points in the pollution area and moving trajectories of actuators, pollution area can be redistricted. Hence, moving trajectories of actuators are involved in their divisiory regions.

Figure  1.  A CVT constructed by 10 random points and trajectories of 4 mobile actuators.

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Approximately, the variation of actuators space can be neglected in a certain condition and converted into time-delay, which can be expressed as $\delta t$. It will be used for actuator neutralizing control in Section IV-C. In other words, the meaning of $\delta t$ is to reduce the impact of the measurement error of the actuator at one position on control of the actuator at another position.

III. Control of Diffusion Process

In this section, we introduce the objective optimization problem of the diffusion process for actuator motion control and spraying control.

A Actuator Position Control Problem

As we know, the CVT algorithm is used to solve the time-invariant environment problem whose density function is time-independent. Based on the arguments in [19, Section V], it still can keep the validation of solving our problem in time-varying environment when the evolution of diffusion process is slow, compared with the convergence rate of the Lloyd’s method. This method can be used as a deterministic algorithm to generate a CVT, for which more details can be found in [10], [19], [24]. In order to apply the CVT algorithm for actuator motion control, the time varying could be negligible, and then the pollutant concentration function could be taken as a time-independent function. Due to the absence of specific form of the PDE system (1), we choose the CVT algorithm for the actuator location control since the CVT algorithm is a non-model algorithm, i.e., it does not need to know the exact mathematical form of the model. The proposed control strategy may be uncommon and may have some limitations for the uncertain dynamic system.

Apparently, to control the diffusion process and minimize the heavily affected area, the mobile actuators should get close to the affected area with high pollution concentration and be far from the lightly affected area. However, it seems unreasonable to put all the mobile actuators close to the pollution source since the diffused pollutants far away from the pollution source also need to be eliminated timely. Given above control requirements, the cost function that needs to be minimized is introduced [11], [19]

where $\phi(s)$ denotes concentration at the point $s$ in the area $\Psi$, $k_{\rm vol}$ and $k_{\rm{acc} }$ are the upper bounds of the velocity and the accelerated velocity of mobile actuators, respectively. We will give more details on the optimization result of the cost function (3) later.

In what follows, we shall write

In addition, some smoothness properties of ${J_{1, P}}$ can be deduced, i.e., the function ${J_{1, P}}$ is at least continuously differentiable on ${\mathbb R}^n \setminus \{P\in {\mathbb R}^n| p_i = p_j \; {\rm{for \;some}}\; i, j \in\{1, \ldots, n\},$$i \neq j\}$ due to the dependence of Voronoi cells $({V_i})_{i = 1}^{n}$ on $P = (p_i, \ldots, p_n)$, for all $p_i \neq p_j$.

Similarly to the arguments in [11], given a region $V \subset {\mathbb R}^2$, the density function $\rho(s)\geq 0$ is defined on it. Let us give some basic definitions, including definitions of mass and center of mass [11], [19]

Next, let us consider the actuators location optimization problem (3) again. In order to illustrate this specific problem clearly, here, suppose the density function $\rho(s) = \phi(s)$. Obviously, for each Voronoi cell $V_i$, the mass and the center of mass are as follows:

Considering the parallel axis theorem, we obtain the partial derivative of the cost function ${J_{1, P}}$ as follows:

A necessary condition to minimize the $J_{1, P}$ is that $\{p_i, V_i\}^n_{i = 1}$ can be a CVT [6], [10] of $\Psi$. The local minimum points for the above cost function ${J_{1, P}}$ are centroids for their Voronoi cells, which can be described as below:

Note that we denote the actuators’ configuration as a centroidal Voronoi configuration, in which each actuator is located at the centroid of its Voronoi cell. The actuator motion control problem is essentially an optimization coverage control problem. Its purpose is to make the actuators reach a centroidal Voronoi configuration, i.e., each actuator’s location can converge to the centroid of its Voronoi cell.

For the purpose of computing the location of mobile actuators by CVTs, the Lloyd’s method is utilized to determine CVTs here. In many practical applications, the robot (actuator or sensor) has only limited communication capabilities. We assume that mobile actuators can communicate with other actuators and sensors in a certain adjustable range. A modified distribution algorithm [19] is introduced, which has the advantage in reducing the computation amount.

B Actuator Neutralizing Control Problem

Moreover, the pollution should be neutralized as soon as possible without making the area of interest overdosed. However, spraying chemicals amount may not be consistent with the amount of pollutant. In order to reduce the disparities of spraying amount and pollutant amount, we construct a cost function needed to be minimized, which can be described by the following constraint optimization problem

where $U_s(t) = \{u_{s_1}(t) \cdots u_{s_n}(t)\}$, $u_{s_i}(t)$ can be regarded as spraying chemical amount released by the $i$th actuator, and $k_s$ is an adjustable upper bound. Based on the above control objective, i.e., eliminating the pollution as much as possible without making the area overdosed, the value of $k_s$ needs to be equal to or less than the total pollution amount (i.e., $k_s \leq$ $\displaystyle\sum\nolimits_{i = 1}^n \int_{V_i} \phi(s) {{d}}s$). However, due to sensors’ measure capability and measure error, the total pollution amount may be hard to obtain accurately, then the fact the value of $k_s$ is completely equal to $\displaystyle\sum\nolimits_{i = 1}^n \int_{V_i} \phi(s) {{d}}s$ may be hard to realize. As is illustrated above, in order to avoid spraying overdosed, $k_s < \displaystyle\sum\nolimits_{i = 1}^n \int_{V_i} \phi(s) {{d}}s$ here.

In the above constrained optimization problem, it is aimed to find an optimal value $U^*_{s}(t) = \{u^*_{s_i}(t), i = 1, \ldots ,n\}$. This optimization result will be addressed in the below theorem. In this situation, the spraying control amount takes the optimal value so that the difference between total spraying chemicals amount and total pollutant amount will be minimized.

First, we will give more details on Karush-Kuhn-Tucker conditions (KKT conditions) [30], [31] as below, which are crucial to the below main theorem.

Definition 1 [30]: Let $u^*$ be a point which satisfies the below constraints

and let $I$ be the set of indices $i$ for which $n_i(u^*) = 0$, where $M = (m_1, m_2, \ldots, m_s)$, $N = (n_1, n_2, \ldots, n_t)$, $M, N \in C^1$, and $u^* \in \Omega \subset E^q$ ($s \leq q$) is a set constraint. Thus, if the gradient vectors $\nabla m_j(u^*)$, $\nabla n_i(u^*)$, $1 \leq j \leq s$, $i \in I$ are linearly independent, then $u^*$ is viewed as a regular point of the constraints (10).

Lemma 1 (KKT conditions) [30] : Let $u^*$ be a minimum point for the problem

and assume that $u^*$ is a regular point for the constraints (see Definition 1). Then there exist $\alpha \in E^s$ and $\beta \in E^t$ with $\beta \geq 0$ such that

Now, we are ready to state our main result for the optimal value $U^*_s(t)$ according to KKT conditions (Lemma 1), which are necessary conditions of optimality.

Theorem 1: For the above constrained optimization problem, the cost function (9) can approach to optimization, if the $i$th actuator’s spraying amount $u_{s_i}^{*}(t)$ satisfies the condition given by

where $k_s = \displaystyle\sum\nolimits_{i = 1}^n \int_{V_i} \phi(s) {{d}}s - \displaystyle{\lambda}/{2}$, $\lambda$ is a nonnegative constant.

Proof: By Lemma 1 (KKT conditions), we have

To find an optimal solution, we can try to set none or one constraint active. Then, we will discuss it in below two cases.

Case 1: In the first case, assume that the constraint is not active (i.e, $\lambda = 0$), we obtain

Using (9) for (16), one can readily get that

However, it contradicts the fact $k_s < \displaystyle\sum\nolimits_{i = 1}^n \int_{V_i} \phi(s) {{d}}s$ given in above analysis. Hence, this case will be abandoned.

Case 2: In the second case, suppose that the constraint is active (i.e, $\lambda \neq 0$), which yields

Then, substituting (9) into (18), we can obtain

This has the solution

where $k_s = \displaystyle\sum\nolimits_{i = 1}^n \int_{V_i} \phi(s) {{d}}s - \frac{\lambda}{2}$, $\lambda > 0$.■

As illustrated above, if we choose the optimal value $u_{s_i}^{*}(t)$ given by (21), the above constrained optimization problem can be addressed.

Remark 1: Given the above analysis, total spraying chemical amount is near to the total pollutant amount as the value of $\lambda$ becomes smaller. In other words, the tinier discrepancy (between total spraying amount and total pollutant amount) represents the better spraying control effect, so that we can conclude the actuators’ spraying amount takes $U^*_s(t) =$${\arg\min} {J_2}(U_s(t), \Theta)$ for optimal spraying control. The result in above theorem illustrates the existence of the optimal value for spraying control amount.

IV. PI Controller for Diffusion Control

In this section, we will introduce PI controllers for actuator motion planning control and actuator neutralizing control. Meanwhile, we will illustrate the convergence analysis of the actuators location and a new PI-CVT algorithm.

A PI Controller for Actuator Motion Control

Here, the mobile actuators are treated as virtual particles and their locations meet the following second-order dynamical equation

where $u_{p_i}$ is the control input for actuator motion path.

The right hand side of the above equation can be described by

where $f_i$ represents the force input to control the motion of the actuator. $f_i$ is given by a PI controller

where ${\bar p_i}$ represents the centroid of the current Voronoi cell.

Substituting (24) into (23), we can get

The second term of (23) on the right hand side is the viscous friction introduced in [32]. $k_v$ is the proportion of viscosity coefficient to actuator mass and ${\dot p_i}$ denotes the velocity of the $i$th actuator. This term is used to eliminate the oscillation behavior of actuators [33] when the actuator is close to the destination. The viscous term assures that the actuator will come to a standstill eventually without external force.

Remark 2: If $k_i$ is equal to $0$, the PI controller (24) for control input can be reduced to the P controller [19] for the optimization problem of the actuators location. The PI control method for the actuator location problem seems a more generalized alternative to the P control method.

B Convergence Analysis of the Actuators Location

Next, we consider the extension of the control design to nonlinear passive-dynamics system [11], [34]. In this paper, the second order system described by a motion equation ${\ddot p_i} = u_{p_i}$ has been investigated. Specifically, suppose that the system’s dynamics are passive with control input $u_{p_i}$ and output ${\dot p_i}$. And assume the input with the zero-dynamics manifold $\{\dot{p}_i = 0\}$ is $u_{p_i} = 0$. For such a system, we use the below PI controller for control input

where $k_{\rm{pro}}$, $k_{\rm{int}}$ and $k_v$ are positive constants, $M_{V_i} = \displaystyle\int_{V_i}\phi(s)\, {{d}}s$ represents the mass of current Voronoi cell, and ${\bar p_i}$ is the centroid of current Voronoi tessellation.

The convergence analysis on the actuators location (system state) of the closed-loop system (22) induced by (26), can be addressed by the following assumption.

Assumption 1: It is assumed that above positive constants, henceforth called scale positive gains, satisfy $k_{\rm {pro}} > k_{\rm {int}}>0$, and $k_v \geq 1$.

We use the Lyapunov stability theory to analyze the dynamic system (22) of the mobile actuators location. By the control input (26) whose gains satisfy Assumption 1, we can obtain the below main result.

Theorem 2: For the passive-dynamics system (22), asymptotic convergence of the actuators location to the set of centroidal Voronoi configurations (i.e., the set of centroids of Voronoi cells) can be achieved by the proposed control input (26) with the PI controller. If this set is finite, the actuators location will converge to a centroidal Voronoi configuration in which each actuator is located at its Voronoi cell’s centroid.

Proof: Consider the Lyapunov functional

Based on the analysis of the CVT algorithm in Section III-A, we have that $M_{V_i}$ and $\bar {p}_i$ are time-invariant, therefore their derivatives are equal to zero. Then, the derivative of the proposed Lyapunov function (27) is given by

Choosing the formulation $\displaystyle\frac{\partial{J_{1, P}}}{\partial{p_i}}$ described by (7), and ${\ddot p_i}$ given by (22), (26), then we obtain

Using Assumption 1, we have

Note that here $(p_i-\bar p_i)\int_0^t (p_i-\bar p_i){{d}}s \geq 0$. Combining with above inequality, we further get

The remainder is similar to the argument in [11, proof of Proposition 3.1 and proof of Proposition 5.1]. Based on the LaSalle’s principle [11], [35], the actuators location can asymptotically converge to the largest invariant set contained in $\{p_i = \bar p_i\}$ which is the set of centroidal Voronoi configurations. Consider the assumption of zero dynamics on the above passive system, we obtain that this set is the solution set of the closed-loop system (22) and is also invariant for (26). Thus, the largest invariant set corresponds to the set of centroidal Voronoi configurations, then the actuators location converges to this set. Moreover, when it is finite, the actuators will reach a specific centroidal Voronoi configuration where each actuator is positioned at its Voronoi cell’s centroid.■

Remark 3: Above convergence analysis of the mobile actuators location can be viewed as evolution of coverage control of mobile sensors network based on the continuous-time Lloyd descent algorithm. We refer to [11] for more details on coverage control of mobile sensors networks. In addition, if we consider the CVT algorithm in time-varying environment, the convergence problem of the mobile actuators location will be complicated. A deeper analysis on it will be investigated in future work with the help of the corresponding arguments in [36].

C PI Controller for Actuator Neutralizing Control

Motivated by [21, page 175], in this paper, we design the PI controller like the below mathematical expression for neutralizing pollutants in a diffusion process. The relationship of the chemicals amount of each robot released and the average pollutant concentration in the Voronoi cell belonging to that actuator can be described by

where

represents the average pollutant concentration, $0<\delta t<t$ (see Section II on $\delta t$ for more details), $i = 1, \ldots, n$, $k_{pr}$ and $k_{ir}$ are positive constants.

Let us consider $\tilde{V}_i$ as follows:

where $C_i = \{s \mid \vert s - p_i \vert < r_i\}$, $r_i$ represents the sensing range of the $i$th actuator, and $V_i$ is the Voronoi cell of the $i$th actuator.

Remark 4: If $k_{ir}$ is equal to $0$, the PI control method can be reduced to the P control method [19] for spraying, which takes an important effect in a diffusion process. In addition, for above PI controller (32), we could choose the appropriate coefficient (i.e., $k_{pr}$, $k_{ir}$) to minimize the pollution amount and the difference between total pollution amount and total spraying amount, which will be illustrated in the below numerical simulations (see Section V-B for more details).

D PI-CVT Algorithm

Based on the above analysis, the PI-CVT algorithm is proposed for controlling a diffusion process and will be presented as follows. Algorithm 1 contains two critical parts: actuator motion control and actuator spraying control.

Algorithm 1: PI-CVT algorithm

1. Set initial: actuator location $p_i \in \{ p_1, \ldots, p_n \}$, pollution source $f_d$, response time $t=0$.

2. Compute Voronoi cell $V_i,$ $i=1, \ldots, n$ belong to each actuator.

3. Gather the data from sensor within Voronoi cell $V_i$; compute its centroid $\bar p_i$ according to (6) and the average concentration of pollutant $\bar \phi_i$ based on (33) in range $r_i$.

4. Control actuator motion and spraying: compute motion control input $u_{p_i}$ according to (25), compute spray amount $u_{s_i}$ based on (32).

5. Repeat Steps 2–4 until there exists no pollution, then stop.

V. Numerical Simulation

The performance of the novel PI control method for the diffusion process can be demonstrated by comparing the performance between novel PI and P control methods in three aspects: 1) the amount of total pollution; 2) the difference between total pollution amount and total spraying amount; 3) the evolution of state $L_2$ norm.

In this section, a simulation platform Diff-MAS2D-PID for motion and spraying control of mobile actuators is introduced later. It is also used as the simulation platform to realize two different cases given below:

1) A diffusion process with a moving pollution source.

2) A diffusion process with a static pollution source.

In addition, this novel PI control strategy is based on the CVT algorithm and different from other PI control strategies used for a diffusion process. Therefore, it may be not available to implement the comparison with other PI control strategies which are not based on the CVT algorithm. However, the comparison of control effect between the novel PI control method and the P control method for a diffusion process with a static pollution source could be realized. Note that these two control strategies are both based on the CVT algorithm. Since the control performance of the novel PI control strategy comparing with the P control strategy for a moving disturbance is similar to the counterparts of a static disturbance case, here we only presented the comparison results of a moving pollution source.

A Diff-MAS2D-PID Simulation Platform

As the technological improvement, a simulation platform called diffusion-mobile-actuators-sensors-2-dimension (Diff-MAS2D) [37], [38] is being used for the control problem of a diffusion process. As the extension of Diff-MAS2D, new simulation platforms called fractional-order-diffusion-mobile-actuators-sensors-2-dimension (FO-Diff-MAS2D) and frac-tional-order-diffusion-mobile-actuator-sensor-2-dimension fr-actional-order-proportional-integral (FO-Diff-MAS2D-FOPI) have been introduced to the fractional order anomalous diffusion problem for optimal spraying control [28], [29], [39]. We refer [40] for more basic knowledge about above two simulation platforms.

Motivated by Diff-MAS2D [37], [38], here we develop a modified simulation platform Diff-MAS2D-PID for two-dimensional diffusion control by introducing PID controllers into it. The area concerned is given in uniformization by $\Psi = \{(x, y)\mid 0\leq x\leq1, 0\leq y \leq 1 \}.$ $M\times N$ sensors evenly distributed in $\Psi$ form a network over the area, which contains $n$ mobile robots releasing the neutralizing chemicals. In the following simulations, the system parameters are set as $M = N = 29$, $n = 4$.

The spatial domain of a diffusion system is discretized by the finite-difference method based on Diff-MAS2D-PID, then the time domain integration is left to MATLAB/Simulink simulation platform. Specifically, Diff-MAS2D-PID can be used to solve the diffusion control problem for actuator location and spraying control via PID controllers. Some main features of Diff-MAS2D-PID are shown below:

1) Sensors and actuators can be collocated or non-collocated.

2) Pollution source can be moving or static.

3) The dynamics equation of actuators location can be modeled as the first order or second order while the diffusion system can be open-loop or closed-loop.

4) The controllers for actuators motion planning and spraying control can be P controllers, PI controllers, PD controllers or PID controllers, which can be applied in $f_c(\tilde{\phi}, x, y, t)$ arbitrarily.

B Moving Pollution Source

For the case of the moving pollution source, the 4 actuators try to move near the pollution source and release chemistry to neutralize the diffusion process. And the diffusion process is depicted as

with the initial condition

and Neumann boundary conditions

where $f_{md}$ and $f_{mc}$ represent the moving pollution source and control input, respectively, for the moving issue, $\bar n$ is the outward direction normal to the boundary.

The moving pollution source is modeled as a point disturbance $f_{md}$ to the diffusion equation. Moving trajectory for this pollution source could be chosen as arbitrary functions. Without loss of generality, we choose the following form

And the disturbance equation can be described as follows:

The moving pollution source begins to diffuse at $t = 0$ to the area $\Psi$ with 4 mobile actuators layout on initial positions at $(0.33, 0.33)$, $(0.33, 0.66)$, $(0.66, 0.33)$, $(0.66, 0.66)$, respectively. In order to show the performance of mobile actuators on controlling the diffusion of the pollutants, control force is added to a diffusion process at $t = 1.0\;{\rm{s}}$. The time step is set to $\Delta{t} = 0.004\;{\rm{s}}$, then the mobile actuators recompute their desired positions every $0.2\;{\rm{s}}$. The total simulation time is $t = 6\;{\rm{s}}$.

For the actuator motion control, the viscous coefficient is given by $k_v = 1$. To obtain the PI controller for actuator motion control, we will illustrate the tuning steps below. First, choose the optimal value of $k_p$ in a certain range, which makes the total pollution amount lowest. Then, fix the optimal value of $k_p$, let the value of $k_i$ vary in a certain range, and obtain the optimal value of $k_i$ which also makes the total pollution amount lowest. We will show details in the following figure.

Specifically, Fig. 2(a) shows that the pollution amount gets to the lowest at the end of simulation time when $k_p$ is $9$. It is simple to find that the PI control method has a decided advantage over the P control method on the total pollution amount as the value of $k_i$ changes from $0$ to $3$, as shown in Fig. 2(b). Meanwhile, the PI controller has a serious disadvantage against the P controller as $k_i \geq 6$ in Fig. 2(c). Based on above analysis, we know that the PI control method can present better control performance than the P control method in the range of $0< k_i <3$. Without loss of generality, we let $k_i = 1.5$, then we can get the PI controller. Consider this controller, we can obtain the below actuator motion control input

Figure  2.  Evolution of total pollutant amount with different values of $k_p$ and $k_i$ of a static pollution source.

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In spraying process, the proportional coefficient is given by $k_{pr} = 10$. To obtain the PI controller for actuator spraying control, we will also illustrate the tuning procedure in below steps. First, give two certain ranges for $k_{ir}$ and $\delta t$ respectively. Let $k_{ir}$ and $\delta t$ vary in their ranges. Then, we can obtain the optimal values of $k_{ir}$ and $\delta t$, which make the total pollution amount and discrepancy between pollution amount and spraying amount lowest. More details can be found in following figures. For the optimal spraying control strategy, $k_{ir}$ changes from $0$ to $30$ with the step $10$ and $\delta t$ changes from $0$ to $2\;{\rm{s}}$ with the step $0.4\;{\rm{s}}$, which is displayed in Fig. 3.

Figure  3.  Spraying control with $k_{ir}=10$, $20$, $30$ and $\delta{t}=0.4\;{\rm{s}}$, $0.8\;{\rm{s}}$, $1.2\;{\rm{s}}$, $1.6\;{\rm{s}}$, $2.0\;{\rm{s}}$, of a moving pollution source.

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It can be concluded from Fig. 3 that the optimal spraying control can be realized if $\{k_{ir} = 10, \delta t = 2\;{\rm{s}}\}$, $\{k_{ir} = 20, \delta t = 1.2\;{\rm{s}}\}$, and $\{k_{ir} = 30, \delta t = 1.2\;{\rm{s}}\}$ respectively. It can be also seen that the optimal control can be obtained at the end of simulation time as $k_{ir} = 30$ and $\delta t = 1.2\;{\rm{s }}$ in Fig. 4.

Figure  4.  Spraying control with $k_{ir}=10$, $20$, $30$ and $\delta{t}=2.0\;{\rm{s}}$, $1.2\;{\rm{s}}$, $1.2\;{\rm{s}}$ of a moving pollution source.

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Furthermore, the control input could be formulated as

For maximizing spraying amount without overdosed, the effectiveness of the PI control method is verified by comparing with the P control method, as shown in Fig. 5. Moreover, from this figure especially Fig. 5 (b), we see the PI control method shows better performance on precision of control.

Figure  5.  (a) Evolution of total pollution amount under PI and P controllers of a moving pollution source. (b) difference of total pollution amount and total spraying amount under PI and P controllers of a moving pollution source. (c) evolution of state $L_2$ norm under PI and P controllers of a moving pollution source.

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Fig. 6 presents a comparison of mobile actuators trajectories under the PI control method and the P control method, along with the trajectory of a moving pollution source.

Figure  6.  Trajectories of actuators under PI and P control strategies of a moving pollution source.

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To check the validity of our proposed PI control strategy for neutralizing control, a comparison on spraying amount and spraying speed of each actuator under PI and P control methods has been shown in Fig. 7. It follows that the proposed PI control strategy outperforms the P control strategy in aspects of spraying amount and spraying speed.

Figure  7.  Evolution of spraying amount and spraying speed under PI and P control methods of a moving pollution source.

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Remark 5: If the type of disturbance is different from the exponential decay function in this paper, for example the other type decay function like $\frac{k}{p+t^2}$ ($k, p>0, t \geq 0$), then the control effect of the PI control method for the diffusion process is not as good as the one with the exponential decay function. In addition, types of moving trajectory have little effect on performance of the PI control method for the diffusion process. Due to space limitation, deeper analysis on control effect with uncertainties and more types of disturbances will be investigated in our future work.

VI. Conclusions

In this paper, the control problem of a diffusion process with a moving or static pollution source has been investigated. As the diffusion process evolves slowly enough comparing with the control efforts, CVT applied in time-invariant environment is also available to the time-varying environment. And the convergence problem of the actuators location has been specifically discussed. In addition, with the help of a modified simulation platform (Diff-MAS2D-PID) and the PI-CVT algorithm, we have demonstrated that the novel PI control method is more effective than the P control method for actuator motion and neutralizing control. In the future, the results with the single pollution source can be extended to a diffusion system with multiple pollution sources. Moreover, the proposed method can also be suitable for the diffusion control problem based on the weight cost function of actuator motion control and spraying control.