Distributed Model Predictive Control Strategy for a Class of Interconnected Discrete-Time Systems with Communication Delay

Abstract

In this paper, we present a distributed model predictive control strategy for discrete-time interconnected linear systems. The overall system is composed of a number of discrete-time interconnected linear subsystems. The MPC controllers of subsystems communicate their information to handle the interconnection effects. It is known that distributed MPC relies on communication networks to transmit and share information among subsystems. Communication delays, resulting in delayed information exchange among subsystems, may inhibit the use of any distributed MPC approach. To address this problem, a delay handling mechanism is employed. Besides, it is showed that the proposed method guarantees input-to-state stability characterization for both local subsystems and the global system under some predetermined assumptions. The simulation results are exploited to illustrate the effectiveness of the proposed method.

Appendix

One can use the following proposition to extend the results to nonlinear case:

Problem 2 (Naghavi et al. 2014)

Let $\alpha_{3i} \in K_{\infty } ,J_{{{\text{MPC}}i}} ( \cdot )$ in (5.7) and a CLF $V_{i} (.)\,$ be given for the $i$th subsystem. At time k measure the state $x_{i} (k)\,$ and minimize the cost $J_{{\rm MPC}i} (.)$ over $\bar{U}_{i} (k): = (\bar{u}_{i} (k\left| {k),\bar{u}_{i} (k + 1|k), \ldots ,\bar{u}_{i} (k + N_{c} - 1} \right|k))$ subject to the following constraints

$$\varPsi \left\{ \begin{aligned} & x_{i} \in X_{i} ,\quad u_{i} \in U_{i} \\ & V_{i} \left( {f_{i} (x_{i} ,u_{i} } \right) - V_{i} \left( {x_{i} \left( k \right)} \right) \le - \alpha_{3i} \left( {\left\| {x_{i} \left( k \right)} \right\|} \right) \\ & \bar{x}_{i} \left( {k + j + 1} \right) = \left( {f_{i} (\bar{x}_{k + j} ,\bar{u}_{k + j} } \right);\quad j = 0,1, \ldots ,N_{p} \\ & \bar{x}_{i} \left( {k|k} \right) = x_{i} \left( k \right) \\ & \bar{u}_{i} \left( {k + j} \right) = \bar{u}_{i} \left( {k + N_{c} - 1} \right);\quad j \ge N_{c} - 1 \\ & \bar{x}_{j}^{a} \left( {k + j} \right) = \left( {f_{j} (\bar{x}_{k + j} ,\bar{u}_{{k + j - \tau_{k} }} } \right);\quad j = 0,1, \ldots ,N_{p} \\ \end{aligned} \right.$$

(21)

Note that $\bar{u}_{i} \left( {k |k} \right) = u_{i} (k$).

Let $\pi_{o} (x_{i} (k)): = \left\{ {u_{i} (k)\left| {\varPsi \;{\text{holds}}} \right.} \right\}$ and $x_{i} (k + 1)\, \in \,f_{oi} (x_{i} (k),\pi_{o} (x_{i} (k))): = \,\left\{ {f_{i} (x_{i} (k),u_{i} (k))\left| {u_{i} } \right. \in \pi_{o} (x_{i} (k))} \right\}$ denote the difference inclusion corresponding to the system without considering the disturbance in a closed loop with the set of feasible solutions obtained by solving Problem 1 at each instant $k \in Z_{ + }$.

Theorem 2

(Naghavi et al. 2014) Let$\alpha_{1i} (s) = a_{i} s^{{\delta_{i} }} ,\alpha_{2i} (s) = b_{i} s^{{\delta_{i} }} ,\alpha_{3i} (s) = c_{i} s^{{\delta_{i} }}$for some$a_{i} ,b_{i} ,c_{i} ,\delta_{i} \in R_{ + }$, $b_{i} \ge c_{i}$, continuous and convex CLFs$V_{i} ( \cdot )$and costs$J_{{{\text{MPC}}i}} ( \cdot )$be given for all systems indexed by$i = 1,2, \ldots ,n$. Suppose Problem 1 is feasible for each subsystem$i = 1,2, \ldots ,n$and for all states$x_{i} \in X_{i}$. Then, each ith subsystem is ISS ($X_{i} ,\,\,W_{i}$) and the global interconnected dynamically coupled nonlinear system described by the collection of difference inclusions

$$x_{i} (k + 1)\, \in \,f_{i}^{CL} (x_{i} (k),\pi_{i} (x_{i} (k))) + \,w_{i} (k)$$

is ISS(X,W).