基于有限步控制李雅普诺夫函数的分布式模型预测控制
Distributed model predictive control via finite-step control Lyapunov functions
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中文总结 AI 辅助
本文提出一种基于有限步控制李雅普诺夫函数的分布式模型预测控制框架，针对受约束互联非线性离散时间系统，保证递归可行性与渐近收敛，在直流微电网场景中得到验证。
中文摘要 AI 辅助
与经典的逆李雅普诺夫定理不同，有限步逆结果具有构造性，且提供了不同的起点：对于足够大的有限预测步长（记为$M$），在显式意义下，任何缩放范数都可作为逆有限步李雅普诺夫函数。对于互联离散时间系统，类似的论证可导出“无保守性”的小增益条件。受此观点启发，本文针对受约束互联非线性离散时间系统，提出一种分布式模型预测控制框架。每个子系统在每个系统时间步通过以下方式求解局部优化问题：将局部阶段函数设为局部控制类有限步李雅普诺夫函数，使用时间对齐的邻居预测、优化后的状态约束收紧半径，以及有限步小增益终端不等式。由于接收的邻居预测无需等于未来滚动时域优化生成的轨迹，名义逆证书无法单独保证递归可行性或稳定性。因此，本文针对控制仿射网络，提出了移位兼容的约束裕度、局部单步终端可行性测试，以及预测与重优化失配的解析界。所得分析保证了递归可行性、约束满足，以及实用的$M$步李雅普诺夫估计，当预测与重优化失配界趋于零时，可实现渐近收敛。对于受约束线性网络，条件简化为有限维矩阵、QP（二次规划）和SOCP（二阶锥规划）测试。该框架被应用于双分布式发电单元（DGU）直流微电网中，在DC/DC功率转换器的运行约束下实现电流共享和终端母线电压安全，并在小型实验室规模原型上进行了验证。
英文摘要
As opposed to classical converse Lyapunov theorems, finite-step converse results are constructive and offer a different starting point: for sufficiently large finite step ahead, say $M$, in an explicit sense, any scaled norm can serve as a converse finite-step Lyapunov function. As for interconnected discrete-time systems, similar lines of argument lead to ``non-conservative'' small-gain conditions. Motivated by this viewpoint, this paper develops a distributed model predictive control framework for constrained interconnected nonlinear discrete-time systems. Each subsystem solves one local optimization problem at each system time step instant by setting the local stage function in form of a local control finite-step like Lyapunov function, using time-aligned neighbor predictions, optimized state-constraint tightening radii, and a finite-step small-gain terminal inequality. Since received neighbor predictions need not equal the trajectories generated by future receding-horizon optimizations, the nominal converse certificates do not alone ensure recursive feasibility or stability. We therefore develop shift-compatible constraint margins, a local one-step terminal feasibility test for networks that are affine in control, and an analytical bound for the prediction and reoptimization mismatch. The resulting analysis gives recursive feasibility, constraint satisfaction, and a practical $M$-step Lyapunov estimate, with asymptotic convergence when the prediction and reoptimization mismatch bound tends to zero. For constrained linear networks, the conditions reduce to finite-dimensional matrix, QP, and SOCP tests. The framework is specialized to current sharing and terminal bus voltage safety under DC/DC power converters' operational constraints in a two-DGU DC microgrid evaluated on a small laboratory-scale prototype.
