SGER: Nonlinear Robust Control: A Stochastic Programming Approach

这项研究的目的是发展随机方法来进行鲁棒分析和控制设计，特别是对于不确定系统的非线性控制。随机鲁棒控制是控制设计方法与参数不确定性随机建模和鲁棒性表征的有效结合。该研究的目的是1)将一类非线性不确定系统的控制问题转化为一个可计算处理的随机规划问题，从而实现系统的鲁棒控制设计方法；2)为控制设计者提供设计灵活性和在性能、鲁棒性和计算/控制器复杂性方面的不同折衷。首先，该项目将研究以增加系统维度为代价将非线性模型代数重铸为低阶多项式向量场，其中将研究系统非线性复杂性和系统维度之间的平衡。其次，将研究随机稳健性代价函数在设计规范方面的不同公式。第三，将非线性不确定系统的鲁棒控制设计问题转化为随机规划问题，将平方和技术与统计工具和随机优化算法相结合。该项目将促进鲁棒控制理论的发展，特别是对非线性不确定系统的研究。与几个确定性的最坏情况下的鲁棒性问题相比，所提出的不确定性的随机表征和鲁棒控制具有计算可处理性的优点。此外，通过设计具有可证明的概率鲁棒性的控制器，所提出的研究提供了低复杂性控制器的系统设计，显著改善了标称性能并减少了控制工作量。它还允许控制设计者明确地在性能、稳健性和可靠性之间进行权衡。平方和技术主要用于多项式向量场的非线性稳定性分析；所提出的研究将该工具扩展到随机鲁棒非线性控制的设计，其中随机规划问题被描述为允许系统地参数化和搜索非线性控制器。预计这项研究将对理论和应用产生影响。通过直接评估和优化违反设计规范的概率，提出的随机鲁棒控制设计解决了实际问题，并有助于弥合理论和实践之间的差距。教育计划包括开发一门跨学科的控制课程，这将有助于在那些对控制实施主要感兴趣的人中普及控制理论。与宾夕法尼亚州立大学工程女性项目的合作将有助于招募代表人数不足的群体，并促进他们在工程领域的教育和研究。

The objective of this research is to develop stochastic approaches to robust analysis and control design, particularly for nonlinear control of uncertain systems. The stochastic robust control is an effective combination of control design methodologies with stochastic modeling of parametric uncertainties and robustness characterization. The proposed research aims to 1) formulate the control of a class of nonlinear uncertain systems into a computational tractable stochastic programming problem, which allows systematic robust control design methodologies; 2) provide design flexibility and different tradeoff for control designers in terms of performance, robustness and computational/controller complexity. First, the project will study algebraic recasting of a nonlinear model into low-order polynomial vector fields at the expense of increase in system dimensionality, where balance between system nonlinear complexity and system dimensionality will be investigated. Second, will be to investigate different formulations of stochastic robustness cost function in terms of design specifications. Third, the robust control design of nonlinear uncertain systems will be formulated into a stochastic programming problem, where the sum of squares technique will be combined with statistical tools and stochastic optimization algorithms. This project will advance the state of the art for robust control theory, in particular for nonlinear uncertain systems. The proposed stochastic characterization of uncertainties and robust control has the advantage of computational tractability, compared to several deterministic worst-case robustness problems. In addition, by designing controllers with provable probabilistic robustness, the proposed research offers systematic design of low-complexity controllers with significant improvement in nominal performance and reduction in control effort. It also allows the control designers to explicitly tradeoff between performance, robustness and reliability. The sum of squares technique has mainly been used for nonlinear stability analysis for polynomial vector fields; the proposed research extends this tool to the design of stochastic robust nonlinear control, where a stochastic programming problem is formulated to allow systematic parameterization and search of nonlinear controllers. This research is expected to impact both theory and applications. By directly evaluating and optimizing the probability of violating design specifications, the proposed stochastic robust control design addresses practical concerns and helps bridge the gap between theory and practice. The education plan includes developing a cross-disciplinary controls course that will help popularize control theory among those with primary interest in control implementation. The collaboration with Penn State Women in Engineering Program will help recruit underrepresented groups and facilitate their education and research in engineering.