Coordinated Sliding Mode Control of Fuel Cell Systems Based on Partial State Feedback
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摘要: 质子交换膜燃料电池因其高效清洁的特性, 成为替代传统内燃机的理想选择. 在质子交换膜燃料电池系统中, 空气供给子系统的氧气过量比与阴极压力是影响其性能和寿命的关键变量. 然而, 这些变量在实际应用中通常难以直接测量, 且系统模型存在参数不确定性. 为应对上述挑战, 提出一种部分状态反馈预设时间协同控制策略. 该策略的核心在于, 首先创新性地设计了仅依赖于可测状态与目标设定值的“引导变量”, 并借助输入——状态稳定性理论, 将原控制问题转化为“引导变量”的镇定问题. 随后, 选取“引导变量”及其导数的线性组合来构建滑模面, 并提出一种基于障碍李雅普诺夫函数的自适应滑模控制律, 确保滑动变量在预设时间内收敛至指定小邻域内, 从而间接实现对氧气过量比和阴极压力的精确控制, 同时抑制测量噪声的干扰. 该方法规避了对关键状态的直接测量需求, 且不依赖于精确的系统模型参数. 仿真与硬件在环实验结果共同验证了所提策略具有优异的动态响应性能和对参数不确定性的鲁棒性.Abstract: Proton exchange membrane fuel cell (PEMFC) has emerged as an ideal alternative to traditional internal combustion engines due to their high efficiency and clean operation. Within PEMFC systems, the oxygen excess ratio and cathode pressure in the air supply subsystem are critical variables that directly affect performance and durability. However, these variables are often not directly measurable in practical applications, and the system model is subject to parameter uncertainties. To address these challenges, this paper proposes a partial state feedback coordinated control strategy with a a prescribed time. The core of this strategy is the innovative design of “guidance variables”, which depend solely on measurable states and target setpoints. By leveraging input-to-state stability theory, the original control problem is transformed into a stabilization problem for these guidance variables. Subsequently, a sliding surface is constructed as a linear combination of the guidance variables and their derivatives. Building upon this, an adaptive sliding mode control law based on a barrier Lyapunov function is proposed. This ensures that the sliding variable converges to a specified small neighborhood within a prescribed time, thereby achieving indirect and precise control of the oxygen excess ratio and cathode pressure while the measurement noise vanishes. This method circumvents the need for direct measurement of critical states and does not rely on an exact system parameter model. Both simulation and hardware-in-the-loop experimental results validate that the proposed strategy exhibits excellent dynamic response and strong robustness against parameter uncertainties.
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图 1 质子交换膜燃料电池发电原理示意图
Fig. 1 Schematic diagram of power generation principle in proton exchange membrane fuel cell
图 2 质子交换膜燃料电池空气供给子系统原理示意图
Fig. 2 Schematic diagram of the air supply subsystem in a proton exchange membrane fuel cell
图 3 系统控制目标与参考值的对比((a)氧气过量比; (b)阴极压力)
Fig. 3 Comparison of control targets with reference values in the system ((a) Oxygen excess ratio; (b) Cathode pressure)
图 4 所设计“引导变量”的变化((a)$\sigma_1 $; (b)$\sigma_2 $)
Fig. 4 Variation of the designed guiding variables ((a) $ \sigma_1$; (b) $ \sigma_2$)
图 5 滑动变量与其预设值的对比((a)$s_1$; (b)$s_2$; (c)$s_1$的误差; (d)$s_2$的误差)
Fig. 5 Comparison of sliding variables with their prescribed values ((a) $s_1$; (b) $s_2$; (c) Error of $s_1$; (d) Error of $s_2$)
图 6 系统状态变化((a)空压机转速; (b)空压机出口流量; (c)供气歧管压力; (d)阴极压力; (e)排气歧管压力; (f)背压阀开度)
Fig. 6 Variation of system states ((a) Compressor speed; (b) Compressor outlet flow rate; (c) Supply manifold pressure; (d) Cathode pressure; (e) Exhaust manifold pressure; (f) Backpressure valve opening)
图 7 控制器输出((a)空压机转速指令; (b)背压阀开度指令)
Fig. 7 Controller outputs ((a) Compressor speed command; (b) Backpressure valve opening command)
图 8 硬件在环实验平台示意图
Fig. 8 Schematic diagram of the hardware-in-the-loop experimental platform
图 9 实验中系统控制目标与参考值的对比((a)氧气过量比; (b)阴极压力)
Fig. 9 Comparison of control targets with reference values in the system in experiment ((a) Oxygen excess ratio; (b) Cathode pressure)
图 10 实验中所设计“引导变量”的变化((a)$\sigma_1$; (b)$\sigma_2$)
Fig. 10 Variation of the designed guiding variables in experiment ((a) $\sigma_1$; (b) $\sigma_2$)
图 11 实验中滑动变量与其预设值的对比((a)$s_1$; (b)$s_2$; (c)$s_1$的误差; (d)$s_2$的误差)
Fig. 11 Comparison of sliding variables with their desired values in experiment ((a) $s_1$; (b) $s_2$; (c) Error of $s_1$; (d) Error of $s_2$)
图 12 实验中系统状态变化((a)空压机转速; (b)空压机出口流量; (c)供气歧管压力; (d)阴极压力; (e)排气歧管压力; (f)背压阀开度)
Fig. 12 Variation of system states in experiment ((a) Compressor speed; (b) Compressor outlet flow rate; (c) Supply manifold pressure; (d) Cathode pressure; (e) Exhaust manifold pressure; (f) Backpressure valve opening)
图 13 实验中控制器输出((a)空压机转速指令; (b)背压阀开度指令)
Fig. 13 Controller outputs in experiment ((a) Compressor speed command; (b) Backpressure valve opening command)
表 1 系统状态方程中的参数$ a_i $和$ c_i $
Table 1 Parameters $ a_i $ and $ c_i $ in the system state equations
参数 表达式 确定性 $ a_1 $ $ p_{atm} $ $ \checkmark $ $ a_2 $ $ \frac{\gamma-1}{\gamma} $ $ \checkmark $ $ a_3 $ $ k_{ca,\; in} $ $ \checkmark $ $ a_4 $ $ k_{ca,\; out} $ $ \checkmark $ $ a_5 $ $ \frac{R_{O_2} n_{cell} M_{O_2}}{4R_a F} $ $ \checkmark $ $ a_6 $ $ \frac{n_{cell} M_{O_2}}{4F \omega_{O_2}} $ $ \checkmark $ $ c_1 $ $ \frac{1}{J_{cp}} \left(\frac{k_t \eta_{cm} k_{cm,\; p}}{R_{cm}} + f\right) $ $ \times $ $ c_2 $ $ \frac{C_p T_{atm}}{J_{cp} \eta_{cp}} $ $ \times $ $ c_3 $ $ \frac{k_t \eta_{cm} k_{cm,\; p}}{J_{cp} R_{cm}} $ $ \times $ $ c_4 $ $ \frac{k_t \eta_{cm} k_{cm,\; i}}{J_{cp} R_{cm}} $ $ \times $ $ c_5 $ $ \frac{A_{cp}}{L_{cp}} $ $ \times $ $ c_6 $ $ \frac{R_a T_{atm}}{V_{sm}} $ $ \times $ $ c_7 $ $ \frac{1}{\eta_{cp}} $ $ \times $ $ c_8 $ $ \frac{R_a T_{st}}{V_{ca}} $ $ \times $ $ c_9 $ $ \frac{R_a T_{st}}{V_{rm}} $ $ \times $ $ c_{10} $ $ \frac{C_{D} A_{T}}{\sqrt{R_a T_{st}}} \gamma^{\frac{1}{2}} \left(\frac{2}{\gamma +1}\right)^{\frac{\gamma +1}{2\gamma - 1}} $ $ \times $ $ c_{11} $ $ \frac{1}{\tau_{rm}} $ $ \times $ 
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表 2 模拟参数表
Table 2 Simulation parameter table
参数 含义 数值 确定性 物理系统参数 $ J_{{\rm{cp}}} $ 压缩机转动惯量 $ 4.8 \times 10^{-5} \,\; {\rm{kg m}}^2 $ $ \checkmark $ $ \eta_{{\rm{cm}}} $ 电机效率 $ 0.85 $ $ \times $ $ R_{{\rm{cm}}} $ 电机内阻 $ 0.7 \,\; \Omega $ $ \times $ $ k_t $ 电机转矩常数 $ 0.0161 \,\; {\rm{N m/A}} $ $ \times $ $ k_{cm,\; p} $ 电机比例常数 $ 0.0005 \,\; {\rm{V/rpm}} $ $ \times $ $ k_{cm,\; i} $ 电机积分常数 $ 0.0005 \,\; {\rm{V/rpm}} $ $ \times $ $ C_p $ 空气定压比热容 $ 1004 \,\; {\rm{J/(kg K)}} $ $ \checkmark $ $ T_{{\rm{atm}}} $ 大气温度 $ 298.15 \,\; {\rm{K}} $ $ \checkmark $ $ \eta_{{\rm{cp}}} $ 压缩机效率 $ 0.7 $ $ \times $ $ p_{{\rm{atm}}} $ 大气压力 $ 101.315 \,\; {\rm{kPa}} $ $ \checkmark $ $ \gamma $ 空气绝热指数 $ 1.4 $ $ \checkmark $ $ f $ 压缩机摩擦系数 $ 3 \times 10^{-5} \,\; {\rm{N m/rpm}} $ $ \times $ $ A_{{\rm{cp}}} $ 压缩机流通面积 $ 0.03 \,\; {\rm{m}}^2 $ $ \times $ $ L_{{\rm{cp}}} $ 压缩机腔体长度 $ 0.5 \,\; {\rm{m}} $ $ \times $ $ R_a $ 空气气体常数 $ 287 \,\; {\rm{J/(kg K)}} $ $ \checkmark $ $ V_{{\rm{sm}}} $ 进气歧管容积 $ 0.0043 \,\; {\rm{m}}^3 $ $ \times $ $ k_{{\rm{ca,\; in}}} $ 阴极入口流阻系数 $ 1.4 \times 10^{-6} \,\; {\rm{kg/(Pa s)}} $ $ \checkmark $ $ T_{{\rm{st}}} $ 电堆温度 $ 349 \,\; {\rm{K}} $ $ \checkmark $ $ V_{{\rm{ca}}} $ 电堆阴极容积 $ 0.005 \,\; {\rm{m}}^3 $ $ \times $ $ R_{{\rm{O}}_2} $ 氧气气体常数 $ 259 \,\; {\rm{J/(kg K)}} $ $ \checkmark $ $ n_{{\rm{cell}}} $ 电堆单体数量 $ 280 $ $ \checkmark $ $ M_{{\rm{O}}_2} $ 氧气摩尔质量 $ 0.032 \,\; {\rm{kg/mol}} $ $ \checkmark $ $ F $ 法拉第常数 $ 96485 \,\; {\rm{C/mol}} $ $ \checkmark $ $ \omega_{{\rm{O}}_2} $ 空气中氧气质量分数 $ 0.233 $ $ \checkmark $ $ k_{{\rm{ca,\; out}}} $ 阴极出口流阻系数 $ 2.1 \times 10^{-6} \,\; {\rm{kg/(Pa s)}} $ $ \checkmark $ $ V_{{\rm{rm}}} $ 排气歧管容积 $ 0.001 \,\; {\rm{m}}^3 $ $ \times $ $ A_T $ 背压阀最大流通面积 $ 0.0025 \,\; {\rm{m}}^2 $ $ \times $ $ C_D $ 背压阀流量系数 $ 0.025 $ $ \times $ $ \tau_{{\rm{rm}}} $ 背压阀响应时间常数 $ 0.2 \,\; {\rm{s}} $ $ \times $ 运行条件与设定值 $ \checkmark $ $ I_{{\rm{st}}} $ 电堆负载电流 200A $ \checkmark $ $ y_{1,\; {\rm{ref}}} $ 氧气过量比设定值 3 $ \checkmark $ $ y_{2,\; {\rm{ref}}} $ 阴极压力设定值 $ 150 \,\; {\rm{kPa}} $ $ \checkmark $
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