Distributed-integrated model predictive control for cooperative operation with multi-vessel systems

Abstract

This paper proposes a novel Distributed-Integrated Model Predictive Control (DI-MPC) strategy for the multi-vessel cooperative path following, formation control and obstacle avoidance. Each vessel is designed with an individually distributed controller based on the MPC theory and communication graph. Subject to actuator limitations and formation constraints, the motion control and thrust allocation are integrated into a dynamic model to achieve direct control to the thrusters. A bivariate thrust efficiency matrix is embedded into the model to consider the hydrodynamic interaction effects between adjacent thrusters. The Nominal System is introduced to generate the linearized predictive model. To achieve consensus among various vessels, a real-time iterative negotiation framework is established. The Kalman Filter is utilized to estimate the low-frequency state variables from the external disturbances of environment loads and measurement noises. Numerical simulations based on the proposed distributed strategy and the centralized strategy are carried out under the scenario of cooperative operation in the Huangpu River (in Shanghai). Comparative analysis results demonstrate the high control performance of both the strategies. DI-MPC mainly contributes to the system flexibility, computational cost reduction (67.65%), energy consumption reduction (5.03%) and fault-tolerant capability. Furthermore, DI-MPC also shows strong applicability to large-scale cooperative control problems.

Appendix

Before the analysis, some assumptions and explanations are given.

1.1 Assumption 1:

The stage cost $p(\cdot )$ is positive definite. For example, the stage cost of ASV1 equals:

$$\begin{aligned} \begin{aligned}&p(\cdot )= \sum _{j=1}^{N_{p}-1}\left( Q_{e r r}\left\| \hat{Y}_{1}(k+j \mid k)-r(k+j)\right\| _{2}^{2}\right) \\&\quad +\sum _{j=1}^{N_{p}-1} \frac{Q_{o b s A v o i d}}{\left\| \hat{Y}_{1}(k+j \mid k)-Y_{o b s}(k+j)\right\| _{2}^{2}+\gamma } \\&\quad +\sum _{j=0}^{N_{c}-2}\left( Q_{i n}\left\| U_{1}(k+j \mid k)\right\| _{2}^{2}\right) \\&\quad +\sum _{j=0}^{N_{c}-2}\left( Q_{d i n}\left\| \Delta U_{1}(k+j \mid k)\right\| _{2}^{2}\right) \end{aligned} \end{aligned}$$

(41)

1.2 Assumption 2:

Terminal set ${\mathcal {X}}_f$ is invariant under the local control law:

$$\begin{aligned} \begin{aligned}&X(k+1 \mid k)=X^{e}(k+1 \mid k)+A^{J}(k \mid k)\left( X(k \mid k) \right. \\&\quad \left. -X^{e}(k \mid k)\right) +B^{J}(k \mid k)\left( U(k \mid k)-U^{e}(k \mid k)\right) \\&\quad +G^{J}(k \mid k)\left( \tau _{\text{ env }}(k \mid k)-\tau _{env}^{e}(k \mid k)\right) \\&\quad \in {\mathcal {X}}_{f}, \text{ for } \text{ all } X(k \mid k) \in {\mathcal {X}}_{f} \end{aligned} \end{aligned}$$

(42)

All state and input constraints are satisfied in ${\mathcal {X}}_f$:

$$\begin{aligned} {\mathcal {X}}_{f} \subseteq {\mathcal {X}}, U \in {\mathcal {U}}, \text{ for } \text{ all } X(k \mid k) \in {\mathcal {X}}_{f} \end{aligned}$$

(43)

1.3 Assumption 3:

The terminal cost $q(\cdot )$ is a continuous Lyapunov function in the terminal set ${\mathcal {X}}_f$ and satisfies [45]:

$$\begin{aligned} q\left( k+N_{p}+1 \mid k+1\right) -q\left( k+N_{p} \mid k\right) \le -p\left( k+N_{p} \mid k+1\right) \end{aligned}$$

(44)

For example, the terminal cost of ASV1 equals:

$$\begin{aligned} \begin{aligned}&q(\cdot )=q_{ter} Q_{err}\left\| \hat{Y}_{1}\left( k+N_{p} \mid k\right) -r\left( k+N_{p}\right) \right\| _{2}^{2}\\&\quad +\frac{q_{\text{ ter }} Q_{\text{ obsAvoid } }}{\left\| \hat{Y}_{1}\left( k+N_{p} \mid k\right) -Y_{obs}\left( k+N_{p}\right) \right\| _{2}^{2}+\gamma } \\&\quad +Q_{i n}\left\| U_{1}\left( k+N_{c}-1 \mid k\right) \right\| _{2}^{2}\\&\quad +Q_{din}\left\| \Delta U_{1}\left( k+N_{c}-1 \mid k\right) \right\| _{2}^{2} \end{aligned} \end{aligned}$$

(45)

1.4 Iteration feasibility:

1. For the system (24)-(25), the optimal input sequence at time step k is denoted as:

$$\begin{aligned} {\varvec{U}}^{*}(k)=\left[ U^{*}(k \mid k), U^{*}(k+1 \mid k), \ldots , U^{*}\left( k+N_{c}-1 \mid k\right) \right] \end{aligned}$$

(46)

The corresponding sequences of estimated states and outputs are:

$$\begin{aligned}&\widehat{{\varvec{X}}}(k)=\left[ \hat{X}(k+1 \mid k), \hat{X}(k+2 \mid k), \ldots , \hat{X}\left( k+N_{p} \mid k\right) \right] \end{aligned}$$

(47)

$$\begin{aligned}&\quad \widehat{{\varvec{Y}}}(k)=\left[ \hat{Y}(k+1 \mid k), \hat{Y}(k+2 \mid k), \ldots , \hat{Y}\left( k+N_{p} \mid k\right) \right] \end{aligned}$$

(48)

2. At next time step $k+1$, the input sequence $\left[ U^{*}(k+1 \mid k), \ldots , U^{*}\left( k+N_{c}-1 \mid k\right) , U\left( k+N_{c} \mid k+1\right) \right]$ is a feasible solution according to Assumption 2.

1.5 Definition 1:

If there exists a function $V(\cdot )$, such that

$$\begin{aligned}&\alpha _{1}(\Vert \widehat{{\varvec{X}}}(k)\Vert ) \le V(\widehat{{\varvec{X}}}(k)) \le \alpha _{2}(\Vert \widehat{{\varvec{X}}}(k)\Vert ) \end{aligned}$$

(49)

$$\begin{aligned}&\quad V(\widehat{{\varvec{X}}}(k+1))-V(\widehat{{\varvec{X}}}(k)) \le -\alpha _{3}(\Vert \widehat{{\varvec{X}}}(k)\Vert )+\sigma (\Vert g(\widehat{{\varvec{X}}}(k), {\varvec{U}}(k))\Vert ) \end{aligned}$$

(50)

where the definitions of ${\mathcal {K}}_{\infty }$-type functions $\alpha _{1}$, $\alpha _{2}$, $\alpha _{3}$ and ${\mathbf {K}}$-type function $\sigma$ can be found in [46]. Then $V(\cdot )$ is an Input to State Stable (ISS) Lyapunov function and the system (24)-(25) is ISS.

1.6 Proof:

Take the cost function (29) as an example to analyze the stability of DI-MPC controllers. Provided Assumptions 1-2 and the Iteration Feasibility are satisfied, the optimal cost at time step $k+1$ is bounded as:

$$\begin{aligned} \begin{aligned}&J^{*}(k+1) \le \sum _{j=2}^{N_{p}} \left( Q_{e r r}\left\| \hat{Y}_{1}(k+j \mid k)-r(k+j)\right\| _{2}^{2}\right) \\&\quad +\sum _{j=2}^{N_{p}} \frac{Q_{\text{ obsAvoid }}}{\left\| \hat{Y}_{1}(k+j \mid k)-Y_{\text{ obs } }(k+j)\right\| _{2}^{2}+\gamma } \\&\quad +\sum _{j=1}^{N_{c}-1}\left( Q_{i n}\left\| U_{1}^{*}(k+j \mid k)\right\| _{2}^{2}\right) \\&\quad +\sum _{j=1}^{N_{c}-1}\left( Q_{d i n}\left\| \Delta U_{1}^{*}(k+j \mid k)\right\| _{2}^{2}\right) +q\left( k+N_{p}+1 \mid k+1\right) \\&\quad =J^{*}(k)-p(k+1 \mid k)+\left( q\left( k+N_{p}+1 \mid k+1\right) \right. \\&\quad \left. -q\left( k+N_{p} \mid k\right) +p\left( k+N_{p} \mid k+1\right) \right) \end{aligned} \end{aligned}$$

(51)

where the terms of $p(\cdot )$ and $q(\cdot )$ in Equation (51) are as follows:

$$\begin{aligned}&\begin{aligned}&\quad q\left( k+N_{p}+1 \mid k+1\right) \\&\quad =q_{\text{ ter } } Q_{\text{ err } }\left\| \hat{Y}_{1}\left( k+N_{p}+1 \mid k+1\right) -r\left( k+N_{p}+1\right) \right\| _{2}^{2} \\&\quad +\frac{q_{t e r} Q_{\text{ obsAvoid } }}{\left\| \hat{Y}_{1}\left( k+N_{p}+1 \mid k+1\right) -Y_{\text{ obs } }\left( k+N_{p}+1\right) \right\| _{2}^{2}+\gamma } \\&\quad +Q_{\text{ in } }\left\| U_{1}\left( k+N_{c} \mid k+1\right) \right\| _{2}^{2}+Q_{\text{ din } }\left\| \Delta U_{1}\left( k+N_{c} \mid k+1\right) \right\| _{2}^{2} \end{aligned} \end{aligned}$$

(52)

$$\begin{aligned}&\quad \begin{aligned}&\quad p\left( k+N_{p} \mid k+1\right) \\&\quad = Q_{\text{ err } }\left\| \hat{Y}_{1}\left( k+N_{p} \mid k+1\right) -r\left( k+N_{p}\right) \right\| _{2}^{2} \\&\quad +\frac{Q_{o b s A v o i d}}{\left\| \hat{Y}_{1}\left( k+N_{p} \mid k+1\right) -Y_{o b s}\left( k+N_{p}\right) \right\| _{2}^{2}+\gamma }\\&\quad +Q_{i n}\left\| U_{1}\left( k+N_{c}-1 \mid k+1\right) \right\| _{2}^{2}+Q_{{\text {din}}}\left\| \Delta U_{1}\left( k+N_{c}-1 \mid k+1\right) \right\| _{2}^{2} \end{aligned} \end{aligned}$$

(53)

According to the second instruction of iteration feasibility, Equation (53) can be rewritten as:

$$\begin{aligned}&\begin{aligned}&p\left( k+N_{p} \mid k+1\right) =Q_{e r r}\left\| \hat{Y}_{1}\left( k+N_{p} \mid k\right) -r\left( k+N_{p}\right) \right\| _{2}^{2}\\&\quad +\frac{Q_{\text{ obsAvoid } }}{\left\| {\widehat{Y}}_{1}\left( k+N_{p} \mid k\right) -Y_{o b s}\left( k+N_{p}\right) \right\| _{2}^{2}+\gamma } \\&\quad +Q_{i n}\left\| U_{1}^{*}\left( k+N_{c}-1 \mid k\right) \right\| _{2}^{2}+Q_{d i n}\left\| \Delta U_{1}^{*}\left( k+N_{c}-1 \mid k\right) \right\| _{2}^{2} \end{aligned} \end{aligned}$$

(54)

$$\begin{aligned}&\quad \begin{aligned}&\quad p(k+1 \mid k)=Q_{e r r}\left\| \hat{Y}_{1}(k+1 \mid k)-r(k+1)\right\| _{2}^{2}+\frac{Q_{\text{ obsAvoid } }}{\left\| \hat{Y}_{1}(k+1 \mid k)-Y_{o b s}(k+1)\right\| _{2}^{2}+\gamma } \\&\quad +Q_{i n}\left\| U_{1}^{*}(k \mid k)\right\| _{2}^{2}+Q_{\text{ din } }\left\| \Delta U_{1}^{*}(k \mid k)\right\| _{2}^{2} \end{aligned} \end{aligned}$$

(55)

$$\begin{aligned}&\quad \begin{aligned}&\quad q\left( k+N_{p} \mid k\right) = q_{\text{ ter } } Q_{e r r}\left\| \hat{Y}_{1}\left( k+N_{p} \mid k\right) -r\left( k+N_{p}\right) \right\| _{2}^{2}+\frac{q_{\text{ ter } } Q_{\text{ obsAvoid } }}{\left\| \hat{Y}_{1}\left( k+N_{p} \mid k\right) -Y_{\text{ obs } }\left( k+N_{p}\right) \right\| _{2}^{2}+\gamma } \\&\quad +Q_{\text{ in } }\left\| U_{1}^{*}\left( k+N_{c}-1 \mid k\right) \right\| _{2}^{2}+Q_{d i n}\left\| \Delta U_{1}^{*}\left( k+N_{c}-1 \mid k\right) \right\| _{2}^{2} \end{aligned} \end{aligned}$$

(56)

Given the terminal constraint (Assumption 3), then we can get:

$$\begin{aligned} J^{*}(k+1)-J^{*}(k)\le -p(k+1 \mid k) \end{aligned}$$

(57)

$\alpha _{1}(\cdot )$ and $\alpha _{2}(\cdot )$ can be found that satisfy $\alpha _{1}(\cdot )\le J(k) \le \alpha _{2}(\cdot )$ since J(k) is a quadratic cost function. Then according to Definition 1, cost function (29) is an ISS function and the system is ISS under certain assumptions. Cost function (30) can be proved in the same way.