A Comparative Study on Algorithms of Robust and Stochastic MPC for Uncertain Systems
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摘要: 近几十年来，不确定系统模型预测控制的理论和应用得到了飞速发展.本文简要地回顾了不确定系统中鲁棒模型预测控制和随机模型预测控制的发展历史，总结了它们的相关应用，并较为细致地分析了线性不确定系统模型预测控制的各种主要算法.通过总结各种算法的通用模型、运作方式、问题规模，以及它们保证递归可行性、稳定性的方法，分析了部分算法可行域间的关系，揭示了各种算法的主要特点、适用场合和未来可发展方向，并通过仿真实例直观地分析了各种算法的性能和可靠性.Abstract: In recent years, the development of model predictive control (MPC) of uncertain systems has been remarkable. This paper briefly reviews the development of robust MPC and stochastic MPC, summarizes their applications, and expounds and discusses the main algorithms of linear uncertain systems in these two fields. By summarizing their general models, the ways they work, computational complexities, and the ideas they use to ensure recursive feasibility and stability, we reveal the relationship of feasible sets among some of them and unravel the main features of these algorithms and their application situations. Finally, we demonstrate the performance of all the algorithms through certain simulation cases and give some indication on the future development of these two fields.
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Key words:
- Uncertain system /
- robust MPC /
- stochastic MPC /
- stability /
- feasibility
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表 1 算法主要参数
Table 1 Main parameters of algorithms
说明 问题规模 变量数目:指最终OCP决策变量的数量, 包括所有的松弛变量.
约束数目:是指最终OCP的约束数目.
平均CPU时间:指每一次求解最终OCP所花费的平均时间.本质属性 可行域范围:假设OCP $^M$ 的所有决策变量为 $\Theta^M(x_k)$ , 其中 $x_k$ 是算法 $M$ 下的初始状态, OCP $^M$ 的可行域可定义为: ${F}^{M}=\{x| \exists \Theta^M(x)$ 使得OCP $^M$ 有可行解 
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表 2 Min-max算法的问题规模
Table 2 Problem scale of algorithms
算法 变量数目 约束数目 OL2M $Nm+Nr+1 $ $1LMI+Nn_{hx}+Nn_{hu}+n_S $ OL2M* $Nm+Nr+1 $ $ (q^{ N}-1)(1L+Nn_{hx}+n_S)+Nn_{hu} $ FF2M $Nm+Nr+1 $ $1LMI+N n_{hx}+Nn_{hu}+(2Nr+1)n_S $ FF2M* $Nm+Nr+1 $ $ (q^{N}-1)(N n_{hx}+Nn_{hu}+(2Nr+1)n_S+1) $ DF2M $N(N-1)/2+N m+N^2(n_{hx}+n_{hu})r+N r+1$ $1LMI+N n_{hx}+2N^2(n_{hx}+n_{hu})r+N n_{hu}+(2Nr+1)n_S $ DF2M* $N(N-1)/2+N m+N^2(n_{hx}+n_{hu})r+N r+1$ $ (q^{ N}-1)(N n_{hx}+2N^2(n_{hx}+n_{hu})r+N n_{hu}+(2Nr+1)n_S+1) $ DME2M* $ (q^{ N}-1)m/(q-1)+1$ $q^{ N}(1+ Nn_{hx} +N n_{hu} +n_S)$
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表 3 PT RMPC的Tube和参数化
Table 3 Tubes and parameterization of PT RMPC
输入Tube 状态Tube 参数化 $\left.\begin{array}{*{20}ll} \mathfrak{U}_k=\{{U}_k, {U}_{k+1},\cdots,{U}_{k+N-1}\}\\ \text{其中}\\ {U}_{k+j}=\bigoplus_{i=0}^{j} {U}_{k+j}^{i}, \forall j\in \textbf{N}_{[0,N-1]}\\ {U}_{k+j}^{i=0}=\{u_{k+j}^{i=0} \in \textbf{R}^m\}, \forall j\in \textbf{N}_{[0,N-1]}\\ {U}_{k+j}^{i}=co\{u_{k+j}^{i,l} \in \textbf{R}^m,\forall l \in \textbf{N}_{[1,q]},\\ \forall i\in \textbf{N}_{[1,j]},\forall j\in \textbf{N}_{[1,N-1]}\} \end{array}\right.$ $\left.\begin{array}{*{20}ll} \mathfrak{X}_k=\{{X}_k, {X}_{k+1},\cdots,{X}_{k+N-1}\}\\ \text{其中}\\ {X}_{k+j}=\bigoplus_{i=0}^{j} {X}_{k+j}^{i}, \forall j\in \textbf{N}_{[0,N]}\\ {X}_{k+j}^{i=0}=\{x_{k+j}^{i=0} \in \textbf{R}^n, \forall j\in \textbf{N}_{[0,N]}\}\\ {X}_{k+j}^{i}=co\{x_{k+j}^{i,l} \in \textbf{R}^n,l \in \textbf{N}_{[1,q]},\\ \forall i\in \textbf{N}_{[1,j-1]}, \forall j\in \textbf{N}_{[2,N]}\}\\ {X}_{k+j}^{i=j}=G{W}=co\{x_{k+j}^{i=j,l}=G\widetilde{w}_l,\\ \forall l\in \textbf{N}_{[1,q]}, \forall j\in \textbf{N}_{[1,N]}\}\end{array}\right.$ $\left.\begin{array}{*{20}ll} x_{k+j}=\sum_{i=0}^jx_{k+j}^i\\ \text{其中,}\forall i\in \textbf{N}_{[1,j]}, \forall j\in \textbf{N}_{[1,N]}\\ x_{k+j}^i=\sum_{l=1}^{q}\lambda^l_{k+j}x_{k+j}^{i,l}\\ \text{ 且 } \sum_{l=1}^{q}\lambda^l_{k+j}=1\\ u_{k+j}=\sum_{i=0}^ju_{k+j}^i\\ \text{其中 }~\forall i\in \textbf{N}_{[1,j]}, \forall j\in \textbf{N}_{[1,N-1]}\\ u_{k+j}^i=\sum_{l=1}^{q}\lambda^l_{k+j}u_{k+j}^{i,l},\\ \lambda^l_{k+j} \text{ 同 }~ x_{k+j}^i \end{array}\right.$
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表 4 基于Tube的RMPC算法的问题规模
Table 4 Problem scale tube-based RMPC
算法 变量数目 约束数目 RT $Nm+n$ $Nn_{hx}+Nn_{hu}+n_f+n_{sr}$ HT $Nm+ n +1$ $N(n_{hx}+n_{hu}+1)+n_f+n_{sr} $ PT $ (N-1)^2/2(mq+n_{hx}+n_{hu} )+ $ $N(N+1)/2+3N+Nn_s+N(N-1)(3(1+q)q/2+2)/2+ $ $ N(n+n_{h}+1)+N^2$ $n_{hx}+n_{hu}+N(1+q)qn_s/2+n_s$
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表 5 RMPC算法对比
Table 5 Comparison of RMPC algorithms
算法 变量数目 约束数目 平均CPU时间(s) MPC $5 $ $30 $ 0.2884 OL2M $16 $ $1LMI+35$ 0.3884 OL2M* $16 $ $25585 $ 49.4144 FF2M $16 $ $1LMI+135$ 1.1977 FF2M* $16 $ $139128 $ 267.5556 DF2M $326$ $1LMI+735 $ 1.9504 DF2M* $326$ $752928 $ 1444.8 DME2M* $ 342$ 36864 88.9460 RT $ 7$ $81$ 0.3247 HT 8 $86$ 0.3329 PT 140 636 0.7974
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表 7 RMPC和SMPC应用
Table 7 Applications of RMPC and SMPC algorithms
随机模型预测控制 鲁棒模型预测控制 无人汽车 驾驶转向控制[75] (DE)车辆导航[76] (ST)变道辅助[77] (SG)
巡航控制[78] (SG)车道保持与避障[79] (ST) [80] (DE)
自动驾驶控制[81] (SG)驾驶员行为建模[82] (DE)轨迹跟踪[83] (TB)传动动力系统控制[84] (MM)
车道保持与避障[85] (TB)智能家居 房屋气候控制[86] (SG)房屋遮阳镜片控制[87] (DE)
房屋能量控制[88] (SG)暖通空调系统建模与能量控制[89] (SG)温度控制[90] (MM)变风量空调系统控制[91] (MM) 电子电路 网络直流电机控制系统[92] (DE) 永磁同步电机驱动器[93] (TB)电子电路[94] (MM)热力电路[95] (MM) 飞行器 能源管理系统[96] (DE) 起落架系统振动抑制[97] (MM)
无人直升机轨迹跟踪控制[98] (MM)无人机飞行控制[99] (MM)
高超声速飞行器飞行控制[100] (MM)机器人 机器人导航与避障[101] (DE) 轨迹跟踪[102] (TB) 医疗 常压等离子射流控制[74] (ST)急救车辆调度[103] (SG) 药房库存管理[104](DE) 静脉麻醉控制[105] (MM) 过程工业 过程控制(四罐过程)[106] (DE) 浮选过程控制[107] (MM)热交换网控制[108] (MM)
连续搅拌反应釜[109] (MM)燃煤电站锅炉燃烧系统[110] (MM)
蒸馏塔控制[111] (MM)热轧带钢自动厚度控制[112] (MM)电网 微电网能量管理系统[113] (DE)微电网操控[114] (SG)
能源储存与生产的优化调度[115] (SG)电力调度[116] (SG)能源局域网优化调度[117] (SG)可再生能源微电网控制[118] (MM) 风力发电 风力涡轮机机械疲劳抑制[119] (DE)电池储能系统控制[120] (DE)风力发电系统控制[121] (DE) 风力涡轮机阻尼控制[122] (MM)风力涡轮机控制[123] (MM) 水资源 饮用水网络[124] (DE)水资源管理与利用[125] (DE) 城市交通 铁路货运车规模与分配控制[126](DE)能源管理[127] (SG) 城市道路交通网控制[128] (MM)地铁列车调度[129] (MM) 金融 动态套期保值和期权定价[130] (SG)欧式期权动态对冲[131] (SG) 其他 多层供应链管理[132] (SG)智能热网中的能量平衡[133](SG)云副本放置技术[134] (DE) 内燃机的热量管理[135] (MM)基于图像的视觉伺服控制[136] (MM)
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