# Hierarchical Coordinated Predictive Control of Multiagent Systems for Process Industries
## 3 Methodology
### Game-Based Priority Coordinating
#### 3.2.2 Economic Performance-Based Gaming for Priority  
A mechanism of gaming that is governed by the economic performance of the first item \(\Delta u_{p_{i}^{\text{PC}}}^{\text{PC}}(k|k)\) of independent control scheme is designed to allow agents to determine their priorities in real time. Based on the game criterion that governed by the economic performance, \(v_{i}\)'s priority is reduced by \(1\) level (reflected in the value of the priority \(+1\)) if \(v_{j}\)'s independent control scheme reduces the cost more; otherwise \(v_{i}\)'s priority remains unchanged.  
A cost function that combines the economic performance and the game behavior of the agents is designed as a criterion for each game, which is represented as \[J_{\text{EO},ij}(k)=s_{ij}(k)(P_{j}(k)-P_{i}(k))+s_{ji}(k)(P_{i}(k)-P_{ j}(k))+\lambda_{ij}(s_{ij}(k)(j-i)+s_{ji}(k)(i-j))\] (22a) \[\lambda_{ij}=\left\{\begin{array}{l}1,P_{i}(k)=P_{j}(k)\\ 0,P_{i}(k)\
eq P_{j}(k)\end{array}\right.\] (22b) \[P_{i}(k)=\alpha_{i}\Delta u_{\text{P}j}^{\text{PC}}(k|k)-\beta \Delta y_{\text{ss},j}(k)=(\alpha_{i}-\beta\mathbf{r}_{i,N})\Delta u_{\text{P}j}^{ \text{PC}}(k|k)=\gamma_{i}\Delta u_{\text{P}j}^{\text{PC}}(k|k)\] (22c) \[\text{where }P_{i}(k)\text{ is a normalized cost for evaluating the economic performance of the independent control scheme of agent }v_{i}\), \(\alpha_{i}\) is the cost coefficient of agent \(v_{i}\), \(\beta=\begin{bmatrix}\beta_{1}&\beta_{2}&\cdots&\beta_{N_{0}}\end{bmatrix}\) is a vector consisting of the payoff coefficient of each system output, \(\gamma_{i}\) is the vector of the comprehensive cost coefficients, \(\mathbf{r}_{i,N}=\begin{bmatrix}r_{i,1,N}&r_{i,2,N}&\cdots&r_{i,N_{0},N}\end{bmatrix}^ {\mathsf{T}}\), \(\Delta y_{\text{ss},j}(k)=\begin{bmatrix}\Delta y_{\text{ss},i,1}(k)&\Delta y_{ \text{ss},i,2}(k)&\cdots&\Delta y_{\text{ss},i,N_{0}}(k)\end{bmatrix}^{ \mathsf{T}}\) is the vector of steady-state increments and \(\Delta y_{\text{ss},i,n}(k)=r_{i,n,N}\Delta u_{\text{P}j}^{\text{PC}}(k|k)\) is the steady-state increment of the \(n\)-th system output corresponding to \(\Delta u_{\text{P}j}^{\text{PC}}(k|k)\). \(s_{ij}(k)\) is the behavior chosen by agent \(v_{i}\) when playing the game with agent \(v_{j}\); \(s_{ij}(k)\in\{0,1\}:=\{\text{\emph{priority remains unchanged, priority is reduced by }1\text{\emph{level}}}\}\).  
The smaller the normalized cost \(P_{i}(k)\) in the cost function (22) is, the better the economic performance. Therefore, to optimize economic performance, agent \(v_{i}\) and its adjacent agent \(v_{j}\) obtain their respective optimal game behaviors by solving the optimization problem shown in Equation (23).  
\[s_{ij}^{*}(k)=\operatorname*{argmin}_{s_{ij}(k)}J_{\text{EO},ij}( k,s_{ij}(k),s_{ji}(k)) \tag{23a}\] \[s_{ji}^{*}(k)=\operatorname*{argmin}_{s_{ji}(k)}J_{\text{EO},ji}( k,s_{ji}(k),s_{ij}(k))\] (23b) \[\text{s.t.}\left\{\begin{array}{l}s_{ij}(k)\in\{0,1\}\\ s_{ji}(k)\in\{0,1\}\end{array}\right.\]  
where \(s_{ij}^{*}(k)\) and \(s_{ji}^{*}(k)\) are the optimal game behaviors of agent \(v_{i}\) and its adjacent agent \(v_{j}\) when playing the game, respectively. On the basis of cost function (22) and optimization problem (23), the priority obtained by agent \(v_{i}\) after playing games with all its neighbors is represented as  
\[\psi_{i}(k)=\sum_{j\
eq i,v_{j}\in\mathbf{N}_{i}}s_{ij}^{*}(k) \tag{24}\]  
Accordingly, the set of adjacent agents with higher priority than agent \(v_{i}\) is denoted as  
\[\mathbf{\mathcal{N}}_{i}^{\mathsf{h}}=\left\{\begin{array}{l}\{v_{j}|(v_{j},v_{ i})\in\mathbf{\mathcal{E}},\psi_{j}(k)<\psi_{i}(k)\},\psi_{j}(k)\
eq\psi_{i}(k)\\ \{v_{j}|(v_{j},v_{i})\in\mathbf{\mathcal{E}},j(k)<i(k)\},\psi_{j}(k)=\psi_{i}(k) \end{array}\right. \tag{25}\]