CAREER: Multi-Variable Optimality-Guided Robust Adaptive Control Design - A Game Theoretic Approach

当前控制理论中最重要的研究方向之一是研究不确定动态系统的鲁棒控制器和滤波器设计。 对于具有显著不确定性的系统，获得控制器是一个挑战，这将导致实现良好的跟踪和干扰衰减。 一个现实的尝试，以解决这个挑战，要求自适应控制设计，将学习从系统输出，以确定底层系统，并提高控制精度的学习进展。 当今自适应控制的一个突出主题是鲁棒自适应控制，它涉及对模型不确定性具有鲁棒性且对外部干扰不敏感的控制器的设计。 鲁棒自适应控制系统的设计和分析主要有三种方法：基于确定性等价的自适应控制设计、基于非线性设计的自适应控制设计和最坏情况下的自适应控制设计。 第三种方法是最近提出的。 这种方法的主要动机是需要在一个统一的框架中同时解决自适应控制系统的瞬态性能和系统鲁棒性。 该方法通过识别H（最优控制）目标与鲁棒自适应控制目标之间的对应关系，以及H（最优控制）与零和动态博弈之间的密切关系，将自适应控制问题转化为具有不完美状态观测的非线性系统的干扰抑制问题. 针对这一普遍问题，提出了一种求解成本收益函数的对策论方法。 在这个建议中，我们建议将上述第三种方法扩展到更一般的设置，我们称之为“最优引导的鲁棒自适应控制设计”。"它的动机是建立非线性H（最优控制，零和动态博弈，风险敏感最优控制，风险中性最优控制和随机动态博弈问题之间的关系，以及它们各自的解决方案之间的相似性。 这些关系表明，两个最一般的问题，是风险敏感的最优控制问题，和随机动态博弈问题。 如果我们能够发展出风险敏感最优控制问题解的存在性、唯一性和特征的一般理论，那么我们就可以重构具有大偏差极限的零和动态博弈问题的解，进而构成非线性H（最优控制问题）的解。 此外，我们可以通过将极限作为风险敏感性参数趋于零来重建风险中性问题的解。 另一方面，随机动态博弈解的存在性、唯一性和特征的一般理论将提供非线性H（最优控制问题的设计解，以及作为特殊情况的风险中性问题的设计解。 这些问题的重要性和相关性可以通过非常流行的有源噪声消除应用得到最好的说明，其中底层系统可以是确定性的，随机的或混合的。 这些研究工作将在不完美状态测量的随机动态博弈理论方面取得根本性的进展。 同时，他们将产生新的鲁棒自适应控制设计，能够处理确定性（最坏情况）和随机干扰。 该研究对实际应用的直接影响在于非常流行的主动噪声消除领域，通过提供具有优异瞬态性能的鲁棒自适应控制律，以及对任何数量的未知频率的不可测正弦外源干扰输入的渐近跟踪。 在教育方面，研究开发和课程创新将有助于研究生课程，为控制系统中重要的工程问题提供简单而全面的解决方案。

ECS-9702702 Pan One of the most important current research directions in control theory is the study of robust controller and filter designs for uncertain dynamical systems. It is a challenge to obtain controllers for systems with significant uncertainties, that would lead to achievement of good tracking and disturbance attenuation. A realistic attempt to resolve this challenge calls for an adaptive control design that would learn from the system output to identify the underlying system and improve the control accuracy as learning progresses. A prominent topic in adaptive control today is robust adaptive control, which deals with the design of controllers that are robust to model uncertainties, and insensitive to exogenous disturbances. There are three general approaches to address the design and analysis of robust adaptive control systems: certainty-equivalence based adaptive control design, nonlinear design based adaptive control design, and worst-case adaptive control design. The third approach was introduced very recently. The main motivation for this approach has been the need to address the transient performance and system robustness of adaptive control systems simultaneously in a unified framework. Identifying the correspondence between the objectives of H( optimal control with those of robust adaptive control, and the close relationship between H(optimal control and zero-sum dynamic games, this approach poses the adaptive control problem as a disturbance attenuation problem for nonlinear systems with imperfect state measurements. A game-theoretic solution methodology of cost-to-come function has been developed to cope with this general problem. In this proposal, we propose to broaden the third approach identified above to a more general setting, which we call "optimality-guided robust adaptive control design." It is motivated by the established relationships between nonlinear H( optimal control, zero-sum dynamic game, risk-sensitive optimal control, risk-neutral optimal co ntrol, and stochastic dynamic game problems, and the similarity between their respective solutions. These relationships suggest that the two most general problems, are the risk-sensitive optimal control problem, and the stochastic dynamic game problem. If we can develop a general theory of existence, uniqueness, and characterization of a solution to the risk-sensitive optimal control problem, then we can reconstruct the solution to the zero-sum dynamic game problem with the large deviation limit, which in turn constitutes the solution to the nonlinear H( optimal control problem. Also, we can reconstruct the solution to the risk-neutral problem by taking the limit as the risk-sensitivity parameter goes to zero. On the other hand, a general theory of existence, uniqueness, and characterization of a solution to the stochastic dynamic game will offer a design solution to the nonlinear H( optimal control problem, as well as a design solution to the risk-neutral problem, as special cases. The significance and relevance of these problems can be best illustrated by the immensely popular active noise cancellation applications, where the underlying system may be deterministic, stochastic, or mixed. These research effort will yield fundamental advancement on the theoretical front for stochastic dynamic games with imperfect state measurement. At the same time, they will yield novel robust adaptive control designs that are capable of handling both deterministic (worst-case) and stochastic disturbances. The direct impact of this research on the practical applications lies in the immensely popular field of active noise cancellation, by offering robust adaptive control laws that have superb transient performance, and asymptotical tracking of any number of unmeasured sinusoidal exogenous disturbance inputs of unknown frequency. On the education side, the research development and course innovation will contribute to the graduate curriculum, in terms of providing simple but comprehensive solutions to si gnificant engineering problems ill control systems.