Model Predictive Control of Spatially Distributed Systems with Spatio-Temporal Logic Specifications

Abstract

In this paper, for spatially distributed systems, we propose a new method of model predictive control with spatio-temporal logic specifications. We formulate the finite-time control problem with specifications described by SSTL_f (signal spatio-temporal logic over finite traces) formulas. In the problem formulation, the feasibility is guaranteed by representing control specifications as a penalty in the cost function. Time-varying weights in the cost function are introduced to satisfy control specifications as well as possible. The finite-time control problem can be written as a mixed integer programming (MIP) problem. According to the policy of model predictive control (MPC), the control input can be generated by solving the finite-time control problem at each discrete time. The effectiveness of the proposed method is presented through a numerical example.

1. Introduction

A spatially distributed system is a system whose state dynamically changes over time and space [1,2]. This system represents the temporal changes in the states of other related locations using measurement values collected at a specific location. Spatially distributed systems appear in various issues such as smart buildings, advanced road traffic systems, and vibration control [3,4,5,6]. Therefore, it is important to contemplate the control problem of spatiotemporal patterns in spatially distributed systems.

In recent years, temporal logic has been employed in control issues related to IoT and cyber-physical systems. Temporal logic is a logic system where the truth value of a proposition changes over time [7]. In temporal logic, temporal operators are added to conventional logical operators, allowing the description of propositions in relation to time. In order to handle continuous signals, signal temporal logic (STL) has been proposed in [8]. In STL, specifications of properties of dense time and real-valued signals can be described. Control methods using STL have been widely studied (see, e.g., [9,10,11,12,13,14,15,16]). Moreover, signal spatio-temporal logic (SSTL) has been proposed for spatially distributed systems [17]. In SSTL, spatial operators are introduced to temporal logic. In [18], SSTL has been applied to the monitoring and planning of robotic tasks. Furthermore, SSTL_f has been proposed in the case of finite traces [6].

In [6], the finite-time optimal control problem with SSTL_f formulas for spatially distributed systems is formulated and is reduced to a mixed-integer programming (MIP) problem. When the dynamics in a spatially distributed system are linear, this problem is reduced to a mixed integer linear programming (MILP) problem. However, in [6], only the finite-time optimal control problem has been considered, and model predictive control (MPC) has not been focused on. MPC is a control method that generates the control input by solving the finite-time optimal control problem at each time and has various applications (see, e.g., [19,20,21,22,23]). MPC for spatially distributed systems has been studied in, e.g., [24]. To the best of our knowledge, MPC with spatio-temporal logical specifications for spatially distributed systems has not been studied.

In this paper, we propose a new method of MPC for spatially distributed systems. A control specification is described by an SSTL_f formula. We suppose that an SSTL_f formula is given for a sufficiently long time interval. An SSTL_f formula and spatial constraints in the finite-time optimal control problem are derived from a given SSTL_f formula. In the finite-time optimal control problem studied in this paper, an SSTL_f formula and spatial constraints are considered as a penalty in the cost function. Hence, the feasibility of the finite-time optimal control problem can be guaranteed. In other words, the control input can be necessarily generated. In a similar way to the method in [6], the finite-time optimal control problem can be reduced to an MIP problem. The proposed method provides us with an efficient control method for complex systems.

This paper is organized as follows. In Section 2, a spatially distributed system is introduced. In Section 3, SSTL_f is summarized. In Section 4, the finite-time optimal control problem is formulated. In Section 5, the reduction of the finite-time optimal control problem to an MIP problem is explained. In Section 6, a procedure of MPC is proposed. In Section 7, a numerical example is explained to show the effectiveness of the proposed method. In Section 8, we conclude this paper.

Notation: Let $\mathcal{R}$ denote the set of real numbers. For the finite set A, let $\left|A\right|$ denote the number of elements in A.

2. Spatially Distributed Systems

Consider a multi-dimensional Euclidean space partitioned into a grid, represented as an undirected graph $\mathcal{G}=(\mathcal{L},\mathcal{E})$, where $\mathcal{L}$ is the set of nodes and $\mathcal{E}\subseteq \mathcal{L}\times \mathcal{L}$ is the set of edges. The state of the node l at time t is denoted by $\mathrm{x}(t,l)\in {\mathcal{R}}^n$. Let ${\mathcal{L}}^{\prime }\subseteq \mathcal{L}$ denote the set of nodes without control inputs. Let $\mathcal{B}\left(l\right):=\{i_1^l,\cdots ,i_{\left|\mathcal{B}\right(l\left)\right|}^l\}\subseteq \mathcal{L}$ denote neighborhoods of the node l. A discrete-time spatially distributed system on graph $\mathcal{G}$ is given by

$$\mathrm{x}(t+1,l)=\left\{\begin{array}{cc}{g}_l(\mathrm{x}(t,l),\mathrm{x}(t,i_1^l),\dots ,\mathrm{x}(t,i_{\left|\mathcal{B}\right(l\left)\right|}^l),\mathrm{u}(t,l))\hfill & \mathrm{if} l\in \mathcal{L}\setminus {\mathcal{L}}^{\prime }\hfill \\ g_l(\mathrm{x}(t,l),\mathrm{x}(t,i_1^l),\dots ,\mathrm{x}(t,i_{\left|\mathcal{B}\right(l\left)\right|}^l))\hfill & \mathrm{if} l\in {\mathcal{L}}^{\prime },\hfill \end{array}\right.$$

(1)

where $\mathrm{u}(t,l)\in \mathcal{U}\subseteq {\mathcal{R}}^m$ is the control input ($\mathcal{U}$ is a set representing input constraints), and $g_l$ is a real-valued continuous function. We suppose that the centralized controller calculates the control inputs by collecting the states.

3. SSTL_f

We explain the syntax of SSTL_f (see [6] for further details).

The syntax of SSTL_f is defined over a set of m atomic predicates

$$M=\left\{{\mu }_j(x_1,\dots ,x_n)\right|j\in \{1,\dots ,m\},{\mu }_j(x_1,\dots ,x_n)\equiv \left(f_j(x_1,\dots ,x_n)\right)\ge 0\left)\right\}.$$

An SSTL_f formula is recursively defined by the following syntax:

$$\phi ::=\mathrm{True}|\neg \phi |{\phi }_1\wedge {\phi }_2|{\mathcal{F}}_{[t_1,t_2]}\phi |{\mathcal{G}}_{[t_1,t_2]}\phi \left|{◊}_{[d_1,d_2]}^l\phi \right|{⊡}_{[d_1,d_2]}^l\phi ,$$

where $\phi ,{\phi }_1$, and ${\phi }_2$ are the SSTL_f fomula, $t_1,t_2$ are times such that $t_1\le t_2$, and $d_1,d_2\in {\mathcal{R}}_{\ge 0}$ with $d_1\le d_2$. Temporal operators ${\mathcal{F}}_{[t_1,t_2]}$ and ${\mathcal{G}}_{[t_1,t_2]}$ are called eventually and globally operators, respectively. Within the time interval $[t_1,t_2]$, ${\mathcal{F}}_{[t_1,t_2]}\phi$ denotes that $\phi$ is satisfied at least once, while ${\mathcal{G}}_{[t_1,t_2]}\phi$ denotes that $\phi$ is satisfied at all times. Spatial operators ${◊}_{[d_1,d_2]}^l$ and ${⊡}_{[d_1,d_2]}^l$ are called somewhere and everywhere operators, respectively. Define $L_{[d_1,d_2]}^l:=\{l^{\prime }\in \mathcal{L}|d_1\le d(l,l^{\prime })\le d_2\}$, where $d(l,l^{\prime })$ denotes the distance between l and $l^{\prime }$. The spatial operator ${◊}_{[d_1,d_2]}^l\phi$ denotes that at least one node within $L_{[d_1,d_2]}^l$ satisfies $\phi$, while ${⊡}_{[d_1,d_2]}^l\phi$ denotes that all nodes within $L_{[d_1,d_2]}^l$ satisfy $\phi$.

For a given finite spatio-temporal trace x, the satisfaction of an SSTL_f$\phi$ formula at the time t and the location l, which is denoted by $(x,t,l)\models \phi$, is defined recursively as follows:

$$\begin{array}{cc}\hfill (x,t,l)\models {\mu }_j& \iff \left(f_j(x_1,\dots ,x_n)\ge 0\right)\hfill \\ \hfill (x,t,l)\models \neg \phi & \iff (x,t,l)⊭\phi \hfill \\ \hfill (x,t,l)\models {\phi }_1\wedge {\phi }_2& \iff (x,t,l)\models {\phi }_1\wedge (x,t,l)\models {\phi }_2\hfill \\ \hfill (x,t,l)\models {\phi }_1{\mathcal{U}}_{[t_1,t_2]}{\phi }_2& \iff \exists t^{\prime }\in \{t_1,\dots ,t_2\}.(x,t^{\prime },l)\models {\phi }_2\hfill \\ \hfill & \wedge (\forall t^{\prime \prime }\in \{t_1,\dots ,t^{\prime }-1\}.(x,t^{″},l)\models {\phi }_1)\hfill \\ \hfill (x,t,l)\models {\mathcal{G}}_{[t_1,t_2]}\phi & \iff \forall t^{\prime }\in \{t_1,\dots ,t_2\}.(x,t^{\prime },l)\models \phi \hfill \\ \hfill (x,t,l)\models {\mathcal{F}}_{[t_1,t_2]}\phi & \iff \exists t^{\prime }\in \{t_1,\dots ,t_2\}.(x,t^{\prime },l)\models \phi \hfill \\ \hfill (x,t,l)\models {◊}_{[d_1,d_2]}^l\phi & \iff \exists l^{\prime }\in L.(d_1\le d(l,l^{\prime })\le d_2)\wedge (x,t,l^{\prime })\models \phi \hfill \\ \hfill (x,t,l)\models {⊡}_{[d_1,d_2]}^l\phi & \iff (x,t,l)\models \neg ({◊}_{[d_1,d_2]}^l\neg \phi ).\hfill \end{array}$$

By utilizing SSTL_f formulas, control specifications for the system (1) can be described.

4. Problem Formulation

The time interval for control is defined as $\{0,1,\cdots ,H\}$. For this interval, we assume that the SSTL_f formula $\phi$ is given as a control specification. In MPC, we solve the finite-time optimal control problem in the time interval $\{t,t+1,\cdots ,t+N\}$, where t is the current time, and N is the prediction horizon. To generate the control input, this problem is solved until $t=H-1$.

Assume that from the SSTL_f formula $\phi$, the SSTL_f formula ${\phi }_F(t,l)$ that should be satisfied in the time interval $\{t,t+1,\cdots ,t+N\}$ can be derived. Assume also that from the SSTL_f formula $\phi$, the spatial constraint ${\phi }_T(k,l)$ at time $k=t,t+1,t+N$ such that $\phi$ is easily satisfied in the time interval $\{t+N+1,t+N+2,\cdots ,t+H\}$ can be derived. Let $z_{{\phi }_F}(t,l)$ and $z_{{\phi }_T}(k,l)$ denote binary variables, where $z_{{\phi }_F}(t,l)$ ($z_{{\phi }_T}(k,l)$) is 1 if ${\phi }_F(t,l)$ (${\phi }_T(k,l)$) is true, otherwise 0 (see also Section 5.1).

Under these preparations, consider the following finite-time optimal control problem.

Problem 1.

For the system (1), suppose that the current time t, the current state $x(t,l)$, the previous control input $u(t-1,l)$ ($u(-1,l)=0$), the SSTL_f formula ${\phi }_F(t,l)$, the spatial constraint ${\phi }_T(k,l)$, and the prediction horizon N are given. Then, find a control input sequence $u(t,l),u(t+1,l),\dots ,u(t+N-1,l)\in U$ minimizing the following cost function:

$$\begin{array}{cc}\hfill J=& \sum _{k=t}^{t+N-1}{\alpha }_k\sum _{l\in L/L^{\prime }}|u(k,l)-u(k-1,l)|\hfill \\ \hfill & +\sum _{k=t}^{t+N}\left\{\sum _{l\in L}{\beta }_{k,l}(1-z_{{\phi }_F}(t,l))+\sum _{l\in L}{\gamma }_{k,l}(1-z_{{\phi }_T}(k,l))\right\},\hfill \end{array}$$

(2)

where ${\alpha }_k$, ${\beta }_{k,l}$, and ${\gamma }_{k,l}$ are time-varying weighting coefficients.

The communication cost between the controller and the actuators can be reduced by introducing the term weighted by ${\alpha }_k$ in the cost function (2).

We suppose that there is no constraint at time $t+N$ when Problem 1 is solved at some time t. Depending on a given SSTL_f formula, when Problem 1 is solved at the next time $t+1$, a constraint at time $t+N+1$ may be newly imposed. In such a case, Problem 1 at time $t+1$ may become infeasible. To overcome this technical issue, constraints related to the SSTL_f formula $\phi$ are represented by using terms weighted by ${\beta }_{k,l}$ and ${\gamma }_{k,l}$. Hence, the feasibility of Problem 1 is always guaranteed. There is a possibility that the SSTL_f formula $\phi$ is not satisfied. The SSTL_f formula $\phi$ is satisfied as well as possible by adjusting ${\beta }_{k,l}$ and ${\gamma }_{k,l}$.

5. Reduction of Problem 1 to an MIP Problem

In this section, we consider reducing Problem 1 to an MILP problem.

First, we consider modeling the graph structure in the system (1) and introducing binary variables. Next, we consider transforming an SSTL_f formula into a set of linear inequalities. See [6] for further details. The procedure of transforming a logic formula into linear inequalities has been widely studied. See, e.g., [25,26] for further details.

5.1. Introduction of Binary Variables

Consider the system (1). For each $l_i\in L$, introduce $\left|L\right|$ binary vectors $v_{l_i}\in {\{0,1\}}^{\left|L\right|}$. Here, $v_{l_i,i}=1$, meaning the i-th element of $v_{l_i}$ is 1, and for $i\ne j$, $v_{l_i,j}=0$. The matrix $D\in {\mathbb{R}}^{\left|L\right|\times \left|L\right|}$ is a matrix representing distances, where the $(i,j)$-th element $D_{i,j}$ is given by

$$D_{i,j}=\left\{\begin{array}{cc}d(l_i,l_j)\hfill & \mathrm{if}\Sigma (l_i,l_j)\ne 0,\hfill \\ M^d\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

where $M^d$ is a sufficiently large positive number such that $M^d\ge \max \left\{d(l_i,l_j)\right|l_i,l_j\in L,\Sigma (l_i,l_j)\ne 0\}$. To convert SSTL_f expressions into linear inequalities, introduce the following binary variables $z_{\phi }(t,l)$ for each $t\in \mathbb{T}$ and $l\in L$, where $\phi$ represents an SSTL_f expression. For a given spatiotemporal trace x,

$$z_{\phi }(t,l)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}(x,t,l)\models \phi ,\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

5.2. Atomic Predicates, Boolean Operators

Next, we consider converting atomic predicates and Boolean operators into linear inequalities.

Consider an atomic predicate $\phi ={\mu }_j(x_1,\dots ,x_n)$. The satisfaction of $\phi$ is represented as

$$\begin{array}{cc}\hfill f_j\left(x(t,l)\right)& \le M^{{\mu }_j}{z}_{\phi }(t,l)-ϵ,\hfill \\ \hfill -f_j\left(x(t,l)\right)& \le M^{{\mu }_j}(1-z_{\phi }(t,l))-ϵ,\hfill \end{array}$$

where $M^{{\mu }_j}$ is a sufficiently large positive number compared to the maximum value of $f_j$ for $j\in \{1,\dots ,m\}$, and $ϵ$ is a sufficiently small number.

Next, we consider converting Boolean operators into linear inequalities. Let $\phi =\neg \psi$ denote the logical operator representing negation. Then, we can obtain

$$z_{\phi }(t,l)=1-z_{\psi }(t,l).$$

Let $\phi ={\bigwedge }_{k=1}^K{\psi }_k$ denote the logical operator representing conjunction Then, we can obtain

$$\begin{array}{cc}\hfill z_{\phi }(t,l)& \le z_{{\psi }_k}(t,l),\forall k\in [1,K],\hfill \\ \hfill z_{\phi }(t,l)& \ge 1-K+\sum _{k=1}^K{z}_{{\psi }_k}(t,l).\hfill \end{array}$$

Let $\phi ={\bigvee }_{k=1}^K{\psi }_k$ denote the logical operator representing logical disjunction. Then, we can obtain

$$\begin{array}{cc}\hfill z_{\phi }(t,l)& \ge z_{{\psi }_k}(t,l),\forall k\in [1,K],\hfill \\ \hfill z_{\phi }(t,l)& \le \sum _{k=1}^K{z}_{{\psi }_k}(t,l).\hfill \end{array}$$

Thus, any Boolean operators can be converted into linear inequalities as described above.

5.3. Temporal Operators

Next, we consider transforming temporal operators into linear inequalities for finite traces over $\{0,1,\cdots ,H\}$. We remark that there are instances where temporal operators change within the time interval for control $\{0,1,\cdots ,H\}$ and the actual computation interval for prediction $\{t,t+1,\cdots ,t+N\}$.

Consider the global operator $\phi ={\mathcal{G}}_{[t_1,t_2]}\psi$ with $t_1\ge t_2$. Then, over the time interval $\{0,1,\cdots ,H\}$, $\phi$ represents the logical conjunction of $\psi$. In this case, we can obtain

$$z_{\phi }(t,l)=\left\{\begin{array}{cc}\underset{k=t+t_1}{\overset{t+t_2}{\bigwedge }}\psi (k,l)\hfill & \mathrm{if}t\in \{0,\dots ,H-t_2\},\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

For the global operator, the linear inequalities remain unchanged within the prediction interval as well.

Consider the eventual operator $\phi =$${\mathcal{F}}_{[t_1,t_2]}\psi$ with $t_1\le t_2$. Then, over the time interval $\{0,1,\cdots ,H\}$, $\phi$ represents the logical disjunction of $\psi$. In this case,

$$z_{\phi }(t,l)=\left\{\begin{array}{cc}\underset{k=t+t_1}{\overset{t+t_2}{\bigvee }}\psi (k,l)\hfill & \mathrm{if}t\in \{0,\dots ,H-t_1\},\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Also, when $t+N<t_2$ in the predicted interval$\{t,t+1,\cdots ,t+N\}$, or when $t_1<t<t_2\le t+N$ and ${\bigvee }_{k=t_1}^t\psi (k,l)=1$, there are no constraints imposed by the temporal operator. Specifically, if $t_1\le t$ and $t+N<t_2$ with ${\bigvee }_{k=t_1}^t\psi (k,l)=1$, there are no constraint conditions even beyond $t+N$ into the future. Also, we can obtain

$$z_{\phi }(t,l)=\left\{\begin{array}{cc}\underset{k=t}{\overset{t+t_2}{\bigvee }}\psi (k,l)\hfill & \mathrm{if}(\underset{k=t_1}{\overset{t}{\bigvee }}\psi (k,l)=0)\wedge (t_1<t<t_2\le t+N),\hfill \\ \underset{k=t+t_1}{\overset{t+t_2}{\bigvee }}\psi (k,l)\hfill & \mathrm{if}(t\le t_1)\wedge (t_2\le t+N),\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

5.4. Spatial Operators

Third, we consider converting spatial operators into linear inequalities. The ‘somewhere’ operator $\phi ={◊}_{[d_1,d_2]}^l\psi$ can be represented by

$$z_{\phi }(t,l)=\underset{l^{\prime }:d_1\le v_l^{\top }D{v}_l^{\prime }\le d_2}{\bigvee }{z}_{\psi }(t,l^{\prime }).$$

The ‘everywhere’ operator $\phi ={⊡}_{[d_1,d_2]}^l\psi$ can be represented by

$$z_{\phi }(t,l)=\underset{l^{\prime }:d_1\le v_l^{\top }D{v}_l^{\prime }\le d_2}{\bigwedge }{z}_{\psi }(t,l^{\prime }).$$

5.5. MIP Problem

Let $LE\left({\phi }_F\left(t\right)\right)$ denote the linear inequalities obtained by transforming the SSTL_f formula ${\phi }_F\left(t\right)$ using the above method. Problem 1 can be equivalently reduced to the following MIP problem.

Problem 2.

Suppose that the current time t, the current state $x(t,l)$, and the previous input $u(t-1,l)$ are given. Then, find a control input sequence $u(t,l),u(t+1,l),\dots ,u(t+N-1,l)\in U$ minimizing the cost function (2) subject to conditions of $LE\left({\phi }_F\left(t\right)\right)$ and $z_{{\phi }_F\left(t\right)}=1$.

If the function $g_l$ in the system (1) is linear with respect to the state and the control input, then Problem 2 is an MILP problem.

6. Model Predictive Control

In this section, we propose a procedure of MPC using Problem 1. MPC is a control method where the control input is generated by solving the finite-time optimal control problem. We can obtain a time sequence of the control input by solving the finite-time optimal control problem. In the conventional MPC, the first one in the obtained input sequence is applied to the plant.

When a control specification is given by an SSTL_f formula, a control specification is changed at each time, because the time interval considered in the finite-time optimal control problem is shifted. Hence, a procedure of MPC under an SSTL_f formula is different from that of the conventional MPC. Based on this fact, we propose a procedure of MPC using Problem 1.

Procedure for model predictive control:

• Step 1: Set $t=0$, $\mathrm{x}(0,l)={\mathrm{x}}_{0,l}$, and an SSTL_f formula.
• Step 2: Derive an SSTL_f formula and spatial constraints at time t from a given SSTL_f formula.
• Step 3: Solve Problem 2.
• Step 4: Apply the control input $\mathrm{u}(t,l)$ obtained by solving Problem 1.
• Step 5: Measure $\mathrm{x}(t+1,l)$.
• Step 6: Update $t:=t+1$. If $t=H$, then the procedure is terminated, otherwise go to Step 2.

Since Problem 2 is always feasible, we can guarantee that the control input until $t=H-1$ is calculated.

7. Numerical Example

Consider a room in a 2-dimensional Euclidean space with seven heaters and two windows, where the temperature distribution is controlled according to specifications described by an SSTL_f formula. The room is partitioned into a $10\times 10$ grid, and is modeled by an undirected graph. Figure 1 illustrates the room setup for this simulation. By the discretization of the heat conduction equation, the dynamics of the room temperature are derived as follows:

$$\begin{array}{cc}\hfill T(t+1,l)& =(1-V·A_l)T(t,l)+Vu(t,l)+{\displaystyle \frac{W}{\left|\mathcal{B}\right(l\left)\right|}}\sum _{l^{\prime }\in \mathcal{B}\left(l\right)}(T(t,l^{\prime })-T(t,l)),l\in {\mathcal{L}}_h,\hfill \\ \hfill T(t+1,l)& =(1-V·A_l)T(t,l)+{\displaystyle \frac{W}{\left|\mathcal{B}\right(l\left)\right|}}\sum _{l^{\prime }\in \mathcal{B}\left(l\right)}(T(t,l^{\prime })-T(t,l)),l\in \mathcal{L}\setminus {\mathcal{L}}_h,\hfill \end{array}$$

where $T(t,l)$ denotes the temperature of the location l at time t, $\mathcal{L}$ is the set of partitioned areas ($\left|\mathcal{L}\right|=100$), and $A_l$, V, and W are constant. See [6] for further details. In this simulation, we set $V=1$, $W=0.4$, and

$$A_l=\left\{\begin{array}{cc}0.05\hfill & \mathrm{if} l\in {\mathcal{L}}_w,\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

We also set $H=99$, $N=15$, and $T(0,l)=15$.

The atomic propositions ${\mu }_i$ are defined as ${\mu }_1\equiv (x-18\ge 0)$ and ${\mu }_2\equiv (x-22>0)$. We give two control specifications. The first specification is that the temperature in the locations where people are present and their surroundings must be between 18 °C and 22 °C. The second specification is that in the specified location $l_{2,7}\in \mathcal{L}$ and its surroundings, there must be at least one location where the temperature is between 18 °C and 22 °C at least once during the time interval $15\le t\le 99$. These specifications are represented by the following SSTL_f formulas ($\overline{l}\in \overline{\mathcal{L}}=\{l_{4,3},l_{5,4},l_{6,5}\}$):

$$\begin{array}{cc}\hfill & (T,\overline{l})\models {\phi }_{\overline{l}}={\mathcal{G}}_{[30,59]}{⊡}_{[0,1]}^{l_{6,5}}\varphi \wedge {\mathcal{G}}_{[40,69]}{⊡}_{[0,1]}^{l_{5,4}}\varphi \wedge {\mathcal{G}}_{[50,79]}{⊡}_{[0,1]}^{l_{4,3}}\varphi ,\hfill \\ \hfill & (T,l_{2,7})\models {\phi }_{l_{2,7}}={\mathcal{F}}_{[15,99]}{◊}_{[0,1]}^{l_{2,7}}\varphi ,\hfill \\ \hfill & \varphi ={\mu }_1\wedge \neg {\mu }_2.\hfill \end{array}$$

The number of spatial constraints is four. The following four spatial constraints are imposed for Problem 1:

$${⊡}_{[0,1]}^{l_{6,5}}\varphi ,{⊡}_{[0,1]}^{l_{5,4}}\varphi ,{⊡}_{[0,1]}^{l_{4,3}}\varphi ,{◊}_{[0,1]}^{l_{2,7}}\varphi .$$

We consider $\phi ={⊡}_{[0,1]}^{l_{6,5}}\varphi$ as an example of transforming a spatial constraint into linear inequalities. This constraint can be rewritten as

$$z_{\phi }(t,l_{6,5})=z_{\varphi }(t,l_{5,5})\wedge z_{\varphi }(t,l_{6,4})\wedge z_{\varphi }(t,l_{6,5})\wedge z_{\varphi }(t,l_{6,6})\wedge z_{\varphi }(t,l_{7,5}),$$

which is equivalent to the following linear inequalities:

$$\begin{array}{cc}\hfill z_{\phi }(t,l_{6,5})& \le z_{\varphi }(t,l_{5,5}),\hfill \\ \hfill z_{\phi }(t,l_{6,5})& \le z_{\varphi }(t,l_{6,4}),\hfill \\ \hfill z_{\phi }(t,l_{6,5})& \le z_{\varphi }(t,l_{6,5}),\hfill \\ \hfill z_{\phi }(t,l_{6,5})& \le z_{\varphi }(t,l_{6,6}),\hfill \\ \hfill z_{\phi }(t,l_{6,5})& \le z_{\varphi }(t,l_{7,5}),\hfill \\ \hfill z_{\phi }(t,l_{6,5})& \ge 1-5+z_{\varphi }(t,l_{5,5})+z_{\varphi }(t,l_{6,4})+z_{\varphi }(t,l_{6,5})+z_{\varphi }(t,l_{6,6})+z_{\varphi }(t,l_{7,5}).\hfill \end{array}$$

We remark that these spatial constraints are not necessarily satisfied at each time because these constraints are considered as the penalty in the cost function. For ${\gamma }_{t,l}$ in the cost function (2), consider two cases: (i) ${\gamma }_{t,l}=0$ and (ii) ${\gamma }_{t,l}=-0.02{(t-t_{1,l})}^2+10$, where $t_{1,l_{6,5}}=30$, $t_{1,l_{5,4}}=40$, $t_{1,l_{4,3}}=50$, and $t_{1,l_{2,7}}=15$. Other coefficients ${\alpha }_k$ and ${\beta }_{t,l}=10$ are set as ${\alpha }_k=1$ and ${\beta }_{t,l}=10$, respectively. These are set through a trial and error process. One of the future efforts is to design these function/parameters in a systematic way.

In addition, when Problem 1 is solved at the initial time, no SSTL_f formula is imposed. For example, at time 10, the following SSTL_f formula is considered as a control specification (note that the prediction horizon N is given by $N=15$):

$$\begin{array}{cc}\hfill & (T,\overline{l})\models {\phi }_{\overline{l}}={\mathcal{G}}_{[10,25]}{⊡}_{[0,1]}^{l_{6,5}}\varphi ,\hfill \\ \hfill & \varphi ={\mu }_1\wedge \neg {\mu }_2.\hfill \end{array}$$

If ${◊}_{[0,1]}^{l_{2,7}}\varphi$ is not satisfied until time 83, then after time 84, we impose the constraint that this spatial constraint is satisfied at least one time.

We present the computation result. Figure 2 and Figure 3 show control inputs in two cases. From these figures, we see that the control inputs are changed to satisfy constraints depending on ${\gamma }_{t,l}$.

Next, we check whether the constraints are satisfied or not. Figure 4 and Figure 5 show whether constraints are satisfied or not in two cases. From Figure 4, we see that constraints are not satisfied at locations $l_{4,3}$ and $l_{5,4}$ in the case of ${\gamma }_{t,l}=0$. From Figure 4, we see that this technical issue is overcome by introducing a time-varying weight. Furthermore, spaces and time intervals satisfying certain conditions are explicitly characterized. Based on this characterization, we validate the proposed approach. In Case (i), the percentage that certain conditions for space/time are satisfied is $97.8\%$. In Case (ii), this percentage is $33.0\%$. Thus, it becomes easier to satisfy the SSTL_f formula by introducing a time-varying weighting coefficient.

Finally, we comment the computation time for solving the problem at each time. The mean computation time and the worst computation time were 8.7 s and 29.4 s, respectively. Here, we used the computer with CPU: Intel Core i5-12600K 3.69 GHz and Memory: 16 GB, and used the CBC (COIN-OR Brand-and-Cut) 2.10.3 [27] https://github.com/coin-or/Cbc (accessed on 1 September 2024) as an MILP solver. Reducing the computation time is one of the future efforts.

8. Conclusions

In this paper, we proposed a new method of MPC for spatially distributed systems. A control specification is given by an SSTL_f formula. To satisfy it, several conditions and time-varying weighting coefficients were introduced for the finite-time optimal control problem. The proposed method was demonstrated by a numerical example.

One of the future efforts is to develop a general procedure to derive constraint conditions and time-varying weighting coefficients in the finite-time optimal control problem. In addition, it is also important to apply the proposed method to practical and real systems. In order to implement practical and real systems, it is significant to develop a method to reduce the computation time for solving an MIP problem in spatially distributed systems.