Quadratic Lyapunov Functions and State Feedback via InteriorPoint Optimization

适当设计的反馈控制系统的最有用的品质之一是鲁棒性，即，闭环系统在（开环）设备动态变化较大的情况下继续良好工作的能力。 这个属性也是控制系统分析和综合中一些最困难问题的根源。 例如，从一组给定的模型中确定一个反馈律，保证它与任何一个被控对象模型相连接时都能很好地工作，这就是鲁棒综合问题。 最近的进展是一个特定类型的鲁棒综合问题的解析解。 这个解析解可以在几个不同的框架中导出： Hoo最优控制，线性二次博弈论控制，或线性指数二次高斯（LEQG）最优控制。 相比之下，本研究围绕一种将李雅普诺夫理论与（数值）凸优化耦合的方法（但与上述解析解方法密切相关）。 特别是，新开发的内点优化方法将应用于控制理论（分析和综合）中出现的一些凸和拟凸问题。 初步调查表明，内点方法对李雅普诺夫函数和状态反馈搜索问题的影响将是深刻的，因为它已经在线性和二次规划。 研究的目标是发展控制系统鲁棒性分析和鲁棒控制器综合方法的基础，该方法用凸优化代替上述解析解中所需的计算，但反过来解决了更大一类的问题。 此外，利用内点优化方法，计算工作量的增加超过解析解所需的计算工作量，可能是相当适度的。 ***

9222391 Boyd One of the most useful qualities of a properly designed feedback control system is robustness, i.e., the ability of the closed- loop system to continue working well despite large variations in the (open-loop) plant dynamics. This property is also the source of some of the hardest problems in control system analysis and synthesis. For example, the problem of determining a feedback law that is guaranteed to work well when connected to any plant model from a given set of models-the robust synthesis problem-is in general quite difficult. A recent advance is an analytic solution to a specific type of robust synthesis problem. This analytic solution can be derived in several different frameworks: Hoo-optimal control, linear-quadratic game-theoretic control, or linear exponential quadratic Gaussian (LEQG) optimal control. In contrast, this research centers around a method that couples Lyapunov theory with (numerical) convex optimization (but is closely connected with the analytic solution method mentioned above). In particular, newly developed interior point optimization methods will be applied to some convex and quasiconvex problems that arise in control theory (analysis and synthesis). Preliminary investigations suggest that the impact of interior point methods on Lyapunov function and state feedback search problems will be as profound as it has been on linear and quadratic programming. The goal of the research is to develop the foundations for method of control system robustness analysis and robust controller synthesis that substitutes convex optimization for the computations required in the analytic solution mentioned above, but in return solves a much larger class of problems. Moreover, with interior point optimization methods, the increase in computation effort over that required for the analytic solution, may be quite modest. ***