# Hierarchical Coordinated Predictive Control of Multiagent Systems for Process Industries
## 2 Preliminaries
### Problem Statement  
Consider an asymptotically stable process control system with \(N_{\text{O}}\) outputs and \(N_{\text{a}}\) controllers, and assume that the system is controllable and observable, where each output is affected by all inputs, which is denoted as  
\[\boldsymbol{x}(k+1)=\boldsymbol{A}\boldsymbol{x}(k)+\boldsymbol{B }\Delta\boldsymbol{U}(k) \tag{1a}\] \[\boldsymbol{y}(k)=\boldsymbol{C}^{\text{T}}\boldsymbol{x}(k) \tag{1b}\]  
where \(\boldsymbol{x}(k)\) is the system state, \(\Delta\boldsymbol{U}(k)\) is the control input increment and \(\boldsymbol{y}(k)\) is the system output. The triple \((\boldsymbol{A},\boldsymbol{B},\boldsymbol{C})\) is not unique, and the dimension of the state space is not necessarily minimum.The system matrix \(\boldsymbol{A}\) describes how the system state changes from the current time step to the next time step with unchanged inputs, the control matrix \(\boldsymbol{B}\) describes how changes in the control inputs affect the system states and the output matrix \(\boldsymbol{C}\) maps the system states to the outputs. To deploy the proposed HCPC strategy of MASs to process industries, the process system shown in Equation (1) is considered a process industry MAS, where each control input and related component parts are considered an agent. Thus, the corresponding local states and system outputs of agent \(v_{i}\) under a simple complete graph (i.e., where each agent can communicate directly with the others) are denoted by \[\mathbf{x}_{i}(k+1)=A_{i}\mathbf{x}_{i}(k)+\mathbf{B}_{i}\Delta\mathbf{U}(k)=\mathbf{A}_{i}\mathbf{x}_{i} (k)+\mathbf{b}_{i}\Delta u_{i}(k)+\sum_{j\
eq i}\mathbf{b}_{j}\Delta u_{j}(k) \tag{2a}\] \[\mathbf{y}_{i}(k)=C^{\mathrm{T}}\mathbf{x}_{i}(k) \tag{2b}\]  
where \(x_{i}(k)\) is the local state with respect to agent \(v_{i}\) at time step \(k\) and \(\Delta u_{i}(k)\) is the control input increment of agent \(v_{i}\) at time step \(k\). \(\mathbf{B}_{i}=\text{diag}(\mathbf{b}_{1},\cdots,\mathbf{b}_{N_{\mathrm{C}}})\) from Equation (2a). Let the step response of the \(n\)-th system output to the \(i\)-th control input at sampling moment \(t\) be denoted as \(r_{i,n,t}\) with \(t=1,\cdots,N\) and \(n=1,\cdots N_{\mathrm{O}}\), where \(N\) is the total number of samples, \(r_{i,n,t}\) tends toward a steady-state with \(t\geq N\). Let \(N_{\mathrm{C}}\) be the control horizon, \(N_{\mathrm{P}}\) be the prediction horizon and \(N_{\mathrm{C}}\leq N_{\mathrm{P}}\leq N\). If the realization condition shown in Equation (4) is satisfied, the step response-based state-space realization of the system with respect to agent \(v_{i}\), as shown in Equation (3), is obtained. \[\mathbf{x}_{i}(k+1)=\mathbf{M}_{\mathrm{S}}\mathbf{x}_{i}(k)+\mathbf{r}\Delta\mathbf{U}(k) \tag{3a}\] \[\mathbf{y}_{i}(k)=\mathbf{C}^{\mathrm{T}}\mathbf{x}_{i}(k) \tag{3b}\]  
\[\mathbf{C}^{\mathrm{T}}\mathbf{A}_{i}^{t-1}\mathbf{B}_{i}=\mathbf{C}^{\mathrm{T}}\mathbf{M}_{ \mathrm{S}}^{t-1}\mathbf{r}=\mathbf{r}_{t},t=1,\cdots,N\;\;\text{and}\;\;n=1,\cdots,N_ {\mathrm{O}} \tag{4}\]  
where \(\mathbf{M}_{\mathrm{S}}=\text{diag}(\mathbf{m}_{s},\cdots,\mathbf{m}_{\mathrm{S}})\) is the shifting matrix, \(\mathbf{C}^{\mathrm{T}}=\left[\mathbf{c}_{1}^{\mathrm{T}}\;\mathbf{c}_{2}^{\mathrm{T}}\; \cdots\;\mathbf{c}_{N_{\mathrm{O}}}^{\mathrm{T}}\right]\), \(\mathbf{c}_{n}^{\mathrm{T}}=\left[1\;\;0\;\cdots\;0\right]_{N}\), \(\mathbf{r}_{t}=\left[\sum_{n=1}^{\mathrm{NO}}r_{1,n,t}\;\sum_{n=1}^{\mathrm{NO}}r _{2,n,t}\;\cdots\;\sum_{n=1}^{\mathrm{NO}}r_{N_{\mathrm{O}},n,t}\right]\), \(\mathbf{r}\) is the matrix composed of step responses, and the forms of \(r\) and \(\mathbf{m}_{\mathrm{S}}\) are provided in Appendix A.