A Coalitional Distributed Model Predictive Control Perspective for a CyberPhysical MultiAgent Application
Abstract
:1. Introduction
2. Problem Formulation
2.1. Robust Positive Invariant Set Computation
 the default working framework is noncooperative DMPC, which implies that each agent ${A}_{i}$, $\forall i\in \mathcal{N}$, from the multiagent application communicates with its neighbour, in order to compute the local solution;
 each subsystem model ${S}_{i}$, $\forall i\in \mathcal{N}$, is subject to input uncertainties received from the subsystem to whom it is connected (in our case its predecessor);
 to provide a simplified algorithm with minimal communication load in the network, only the selfimposed upper bound for the local input trajectory is broadcast in the network (i.e., the optimization variable ${\widehat{u}}^{\mathrm{max}}$ introduced in (6));
 a table with different predefined robust positive invariant sets ${\mathsf{\Omega}}_{i}$ is computed using the constraints limits from (5), in which each element is a particular combination of the variable bounds (see Algorithm 1);
 at each sampling period, after the uncertainty upper bound is received from the neighbour, each agent ${A}_{i}$ uses this information to compute the uncertainty polytope. Next, from the predefined terminal sets table, a set ${\mathsf{\Omega}}_{i}$ is searched for, which includes the received uncertainty polytope (i.e., which will ensure a local feasible solution in the terminal state framework).
Algorithm 1 
For$\alpha ={u}^{max}:ste{p}_{\alpha}:0.1$ 
For $\beta =0.1:ste{p}_{\beta}:{w}^{max}$ 
1. Compute the inequality constraints: 
$$\begin{array}{c}\hfill {A}_{u}u\le \alpha {b}_{u};\phantom{\rule{1.em}{0ex}}{A}_{w}w\le \beta {b}_{w};\phantom{\rule{1.em}{0ex}}{A}_{x}x\le {b}_{x}\end{array}$$

2. Compute the robust positiveinvariant set: 
$$\begin{array}{c}\hfill \mathsf{\Omega}(A,B,K,{A}_{u},{b}_{u},{A}_{w},{b}_{w},{A}_{x},{b}_{x})\end{array}$$

3. Save the information $\alpha $, $\beta $, $\mathsf{\Omega}$ 
end 
end 
2.2. Coalitional Distributed Model Predictive Control (CDmpc) Methodology
2.2.1. Coalition Dynamics
2.2.2. Coalition Problem Definition
2.2.3. CDmpc Algorithm
 the default uncertainty value used in Step 1 is selected to ensure that optimization problems from Step 3 are feasible, thus ensuring that the proposed methodology is recursively stable (i.e., the terminal set for the coalition is obtained by aggregating the terminal sets of the involved individual agents).
 if the condition from Step 6 is satisfied, then at that sampling period, the working framework is noncooperative DMPC; otherwise the framework changes to coalitional DMPC (since at least one coalition is activated).
 the priority value, which is used as a condition term to initialize a coalition, is defined by each agent as a random subunitary number. In this manner, there is no use of a hierarchical control level to assign these priorities.
 in the extreme, all the agents can be combined in a coalition ($\mathcal{C}=\mathcal{N}$), which corresponds to a centralized MPC working framework.
 one or more coalitions can be active simultaneously and are dissolved at the end of each sampling period.
3. Illustrative Example
 Four heterogeneous discretetime subsystems ${S}_{i}$, $\forall i\in \{1,\dots ,4\}$, coupled in a chain architecture were defined using (1), with the following numerical matrices:$$\begin{array}{c}\hfill {S}_{1}:{A}_{p}^{1,1}=\left(\right)open="["\; close="]">\begin{array}{cc}0.7913& 0.2020\\ 0.1010& 0.8417\end{array}\phantom{\rule{1.em}{0ex}}{B}_{p}^{1,1}=\left(\right)open="["\; close="]">\begin{array}{c}0.0271\\ 0.2291\end{array}\end{array}\hfill {B}_{p}^{1,0}=\left(\right)open="["\; close="]">\begin{array}{c}0\\ 0\end{array}\phantom{\rule{1.em}{0ex}}{C}_{p}^{1}=\left[\begin{array}{cc}0& 1\end{array}\right]\phantom{\rule{1.em}{0ex}}$$$$\begin{array}{c}\hfill {S}_{2}:{A}_{p}^{2,2}=\left(\right)open="["\; close="]">\begin{array}{cc}0.7936& 0.1996\\ 0.1198& 0.8236\end{array}\phantom{\rule{1.em}{0ex}}{B}_{p}^{2,2}=\left(\right)open="["\; close="]">\begin{array}{c}0.0269\\ 0.2265\end{array}\end{array}\hfill {B}_{p}^{2,1}=\left(\right)open="["\; close="]">\begin{array}{c}0.0004\\ 0.0034\end{array}\phantom{\rule{1.em}{0ex}}{C}_{p}^{2}=\left[\begin{array}{cc}0& 1\end{array}\right]\phantom{\rule{1.em}{0ex}}$$$$\begin{array}{c}\hfill {S}_{3}:{A}_{p}^{3,3}=\left(\right)open="["\; close="]">\begin{array}{cc}0.7888& 0.2043\\ 0.0817& 0.8604\end{array}\phantom{\rule{1.em}{0ex}}{B}_{p}^{3,3}=\left(\right)open="["\; close="]">\begin{array}{c}0.0273\\ 0.2316\end{array}\end{array}\hfill {B}_{p}^{3,2}=\left(\right)open="["\; close="]">\begin{array}{c}0.0004\\ 0.0034\end{array}\phantom{\rule{1.em}{0ex}}{C}_{p}^{3}=\left[\begin{array}{cc}0& 1\end{array}\right]\phantom{\rule{1.em}{0ex}}$$$$\begin{array}{c}\hfill {S}_{4}:{A}_{p}^{4,4}=\left(\right)open="["\; close="]">\begin{array}{cc}0.7912& 0.1994\\ 0.0997& 0.8211\end{array}\phantom{\rule{1.em}{0ex}}{B}_{p}^{4,4}=\left(\right)open="["\; close="]">\begin{array}{c}0.0269\\ 0.2263\end{array}\end{array}\hfill {B}_{p}^{4,3}=\left(\right)open="["\; close="]">\begin{array}{c}0.0004\\ 0.0034\end{array}\phantom{\rule{1.em}{0ex}}{C}_{p}^{4}=\left[\begin{array}{cc}0& 1\end{array}\right]\phantom{\rule{1.em}{0ex}}$$
 The limit constraints for the inputs, disturbances and outputs are the following:$$\begin{array}{c}\left(\right)open="\{"\; close>\begin{array}{c}{u}_{i}^{\mathrm{min}}=5\hfill \\ {u}_{i}^{\mathrm{max}}=5\hfill \end{array}\hfill \end{array}\left(\right)open="\{"\; close>\begin{array}{c}{w}_{i}^{\mathrm{min}}=8\hfill \\ {w}_{i}^{\mathrm{max}}=8\hfill \end{array}\begin{array}{}\forall i\in \{1,\dots ,4\}\end{array}$$
 For all subsystems ${S}_{i}$, $\forall i\in \{1,\dots ,4\}$, the following optimization parameters are used: the prediction horizon ${N}_{p}=5$, the input weights ${R}_{{u}_{i}}=0.1$ and ${R}_{{w}_{i}}=0.01$.
 The feedback laws were computed using classical statefeedback control based on the Ackermann’s formula [31], applied for the extended model (4), obtaining:$$\begin{array}{c}{K}_{1}=[0.5494\phantom{\rule{1.em}{0ex}}2.6061\phantom{\rule{1.em}{0ex}}0.7488],\hfill \\ {K}_{2}=[0.3473\phantom{\rule{1.em}{0ex}}2.5199\phantom{\rule{1.em}{0ex}}0.7047],\hfill \\ {K}_{3}=[0.6249\phantom{\rule{1.em}{0ex}}2.6565\phantom{\rule{1.em}{0ex}}0.7401],\hfill \\ {K}_{4}=[0.4355\phantom{\rule{1.em}{0ex}}2.5107\phantom{\rule{1.em}{0ex}}0.7051].\hfill \end{array}$$
 The reference tracking scenario was constructed for 12 time samples, using a sampling period ${T}_{s}=0.25s$, with the following imposed references:$$\begin{array}{c}{r}_{1}=\left[0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.5\phantom{\rule{1.em}{0ex}}0.5\phantom{\rule{1.em}{0ex}}0.5\phantom{\rule{1.em}{0ex}}0.5\phantom{\rule{1.em}{0ex}}0.5\phantom{\rule{1.em}{0ex}}0.5\right],\hfill \\ {r}_{2}=\left[0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\right],\hfill \\ {r}_{3}=\left[0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.5\phantom{\rule{1.em}{0ex}}0.5\phantom{\rule{1.em}{0ex}}0.5\phantom{\rule{1.em}{0ex}}0.5\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\right],\hfill \\ {r}_{4}=\left[0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.2\right].\hfill \end{array}$$
 1.
 default case—no coalitions between ${A}_{1}$, ${A}_{2}$, ${A}_{3}$, ${A}_{4}$;
 2.
 coalition ${\mathcal{C}}_{12}$ between ${A}_{1}$ and ${A}_{2}$, while ${A}_{3}$, ${A}_{4}$ remain outside the coalition but interconnected;
 3.
 coalition ${\mathcal{C}}_{123}$ between ${A}_{1}$, ${A}_{2}$, and ${A}_{3}$, while ${A}_{4}$ remains outside the coalition but interconnected;
 4.
 twosimultaneous active coalitions ${C}_{12}$ and ${\mathcal{C}}_{34}$ between ${A}_{1}$ and ${A}_{2}$ and ${A}_{3}$ and ${A}_{4}$, respectively, which are interconnected;
 5.
 coalition ${\mathcal{C}}_{23}$ between ${A}_{2}$ and ${A}_{3}$, while ${A}_{1}$, ${A}_{4}$ remain outside the coalition but interconnected;
 6.
 coalition ${\mathcal{C}}_{234}$ between ${A}_{2}$, ${A}_{3}$ and ${A}_{4}$, while ${A}_{1}$ remains outside the coalition but interconnected;
 7.
 coalition ${\mathcal{C}}_{34}$ between ${A}_{3}$ and ${A}_{4}$, while ${A}_{1}$, ${A}_{2}$ remain outside the coalition but interconnected;
 8.
 extreme case: coalition ${\mathcal{C}}_{1234}$ between all agents ${A}_{1}$, ${A}_{2}$, ${A}_{3}$, ${A}_{4}$.
Algorithm 2 
Initialization: For each agent ${A}_{i}$, $\forall i\in \mathcal{N}$, compute a table ${\mathcal{T}}_{i}$, with potential terminal sets ${\mathsf{\Omega}}_{i}$. 
At each sampling time k, each agent ${A}_{i}$, $\forall i\in \mathcal{N}$, receives the local state value and performs the following steps: 
1. Computes the uncertainty polytope using default limit values for the constraints: 
$$\begin{array}{ccc}\hfill {\mathcal{W}}_{i}& =& {B}_{i,i1}{\mathcal{U}}_{i1}^{0}\hfill \\ \hfill {U}_{i1}& =& [{u}_{i1}^{\mathrm{max},0};{u}_{i1}^{\mathrm{max},0}].\hfill \end{array}$$

2. Searches in the predefined table ${\mathcal{T}}_{i}$ for a terminal set ${\mathsf{\Omega}}_{i}^{0}$ that includes the default uncertainty ${\mathcal{W}}_{i}\subseteq {\mathsf{\Omega}}_{i}^{0}$. 
3. Solves the local optimization problem (6) and obtains the optimal values ${U}_{i}^{*,0}$, ${\widehat{u}}_{i}^{\mathrm{max},0}$ using the default values ${\mathsf{\Omega}}_{i}={\mathsf{\Omega}}_{i}^{0}$ for the terminal set and the uncertainty constraint limit (${w}_{i}^{\mathrm{max}}={u}_{i1}^{\mathrm{max},0}$). 
4. Broadcasts to its successor the local optimal value ${\widehat{u}}_{i}^{\mathrm{max},0}$ and receives the corresponding value ${\widehat{u}}_{i1}^{\mathrm{max},0}$ from its predecessor. 
5. Repeats Steps 1–3 using the uncertainty constraint value received in Step 4. 
6. Checks the feasibility of the local optimization problem: 
If the optimization problem from Step 5 is feasible: 
then: Coalitions between agents are not necessary.Each local agent ${A}_{i}$ sends to its subsystem ${S}_{i}$, the first value from the optimal trajectory ${U}_{i}^{*}$; 
else: Coalitions between agents are necessary. In this case, in order to be included in a coalition, each agent ${A}_{i}$, $\forall i\in \mathcal{N}$, performs the following steps: 
a. Receives, from its predecessor, a coalitional report containing the following information: the feasibility status (for the local optimization problem solved at Step 5) and priority value relating to all the predecessor agents from the chain architecture. 
b. Sends to its successor, the updated coalitional report (i.e., all the relevant information received, together with its own local feasibility and priority data). 
c. Initializes a coalition only if its local priority is the highest from the report. Within a coalition between two agents, the following steps are performed: 
i. the coalition model is defined as (8); 
ii. the optimization problem (10) subject to (9) is solved. 
iii. the relevant information is broadcast to the coalition’s neighbour. 
iv. a feasibility check for all the optimization problems is done. 
If the all the optimization problems are feasible: 
then: The existing coalition was successful and can be dissolved after every subsystem ${S}_{i}$ receives the first value from the optimal trajectory ${U}_{i}^{*}$; 
else: The existing coalition was not successful. Another agent must be included in the existing coalition (if the coalition’s status is infeasible), or another coalition can be activated (if more agents outside the existing coalition have infeasible problems). At this stage, Step (c) is repeated as necessary. 
7. End algorithm. 
4. Conclusions
5. Materials and Methods
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
MPC  Model Predictive Control 
DMPC  Distributed Model Predictive Control 
CMPC  Coalitional Model Predictive Control 
CDMPC  Coalitional Distributed Model Predictive Control 
CPMAS  CyberPhysical MultiAgent System 
CPsS  CyberPhysical subsystem 
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Maxim, A.; Caruntu, C.F. A Coalitional Distributed Model Predictive Control Perspective for a CyberPhysical MultiAgent Application. Sensors 2021, 21, 4041. https://doi.org/10.3390/s21124041
Maxim A, Caruntu CF. A Coalitional Distributed Model Predictive Control Perspective for a CyberPhysical MultiAgent Application. Sensors. 2021; 21(12):4041. https://doi.org/10.3390/s21124041
Chicago/Turabian StyleMaxim, Anca, and ConstantinFlorin Caruntu. 2021. "A Coalitional Distributed Model Predictive Control Perspective for a CyberPhysical MultiAgent Application" Sensors 21, no. 12: 4041. https://doi.org/10.3390/s21124041