Analysis and System Design of Multi-convex Hull Stabilization Domain for Double-layered Model Predictive Control System
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摘要: 针对双层模型预测控制（Model predictive control，MPC）中出现的由于系统状态在动态控制（Dynamic control，DC）过程中超出约束集，导致下层优化不可行的问题，本文在综合控制方法的基础上提出一种新的动态控制策略，引入多包镇定域（Multi-convex hull stabilization domain，MHSD）的概念.通过离线计算多包镇定域，并根据系统每一时刻的实测状态值，在线决定（Dynamic control）层的镇定域以及相应的控制时域，结合变约束思想，保证动态控制过程递归可行，从而有效控制在大范围内变化的系统状态.另外，本文通过设计非线性反馈控制器，扩大了终端不变集和多包镇定域的范围，提高了DC层对稳态目标值的跟踪效果.本文的控制算法可以使得DC层在目标跟踪过程中保证递归可行性，并最大程度地实现无静差跟踪.仿真算例验证了本文算法对稳定系统和不稳定系统都有效.Abstract: In order to solve the feasibility problem in double-layered model predictive control (MPC) caused by some states in dynamic control (DC) that violate the constraints, this paper proposes a new control strategy based on the overall control solution and introduces a new definition multi-convex hull stabilization domain (MHSD). This strategy designs the MHSD off line and then chooses a proper stabilization range online according to the real-time system state. The control horizon of the DC layer can be calculated at the same time. What is more, this paper enlarges the invariant sets and the stabilization domain through designing a nonlinear feedback controller so that the system states varying in a wide range can be controlled and the tracking effect is significantly improved. By using the above algorithm, the control process is recursively feasible and the optimal targets can be tracked precisely. The effectiveness of this method is verified in both stable and unstable systems through two examples.1) 本文责任编委 谢永芳
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图 3 控制时刻$k$从31到60对应的纸机系统控制过程
Fig. 3 Control process of the paper system with the control moment $k$ from 31 to 60
图 4 控制时刻$k$从61到90对应的纸机系统控制过程
Fig. 4 Control process of the paper system with the control moment $k$ from 61 to 90
图 5 控制时刻$k$从31到60对应的双积分系统控制过程
Fig. 5 Control process of the double-integrator system with the control moment $k$ from 31 to 60
表 1 本文符号及其含义
Table 1 The meanings of the notations in this paper
符号 含义 $x^*$ $x$的最优值 ${\bf R}^n$ $n$维欧氏空间 $k$ 离散采样间隔 $x$ 系统状态, $x \in {\bf R}^{n_x}$ $u$ 系统输入, $u \in {\bf R}^{n_u}$ $x_s(u_s)$ 稳态状态(输入) $x_t(u_t)$ 期望稳态状态(输入) $\bar{x}(\bar{u})$ 状态(输入)上界 $I_n $ $n$维单位矩阵 $Q_s, R_s$ 适维权重矩阵 $N_i$ 第$i$个镇定域所对应的控制时域 ${\| x\|}_{Q_s}^2$ $x^{\rm T}$$Q_s$$x$ $x(k+i|k)$ $k$时刻对未来状态的预测值 $u(k+i|k) $ $k$时刻对未来输入的预测值 
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表 2 纸机系统的稳态目标计算结果
Table 2 The results of the SSTC in the AS DPS system
$k$ $u_{s, 1}$ $u_{s, 2}$ $x_{s, 1}$ $x_{s, 2}$ $1\sim90$ -0.39 -0.41 -0.32 -0.33
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表 3 双积分器系统的稳态目标计算结果
Table 3 The results of SSTC in the double-integrator system
k $u_t$ $x_t$ $u_s$ $x_s$ $1\sim30$ (0, 0) (2, -2) (0, 0) (2, 0.5) $31\sim60$ (0, 0) (0, -2) (-0.1, 0.2) (-0.38, 0.2998) $61\sim90$ (0, 0) (0, 0) (0, 0) (0, 0)
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