Nested Optimized Adaptive Control for Linear Systems

The classical optimal control of the linear system assumes that the system is stabilizable, thereby deriving the optimal control with the outcome that the solution inherently stabilizes the system. Such optimization does not distinctly address stabilization and optimization as separate concerns, leading to a situation where, as the system expands in size and complexity, the optimal controller suffers performance decreases and becomes increasingly sensitive and fragile. In this article, nested optimized control (NOC) and nested optimized adaptive control (NOAC) are introduced to explicitly handle the stabilization, optimization/adaptation for unknown parameters separately in an effort to strike a balance between guaranteed stability and optimal control. The robustness of the classical optimal control is inherent in the design itself, and the stability margin is relatively small subject to parameter uncertainties. In our NOC, the robustness is explicitly handled by the state feedback control and its stability margin is larger than the classical one, because of the introduction of the explicit state feedback control loop, the next optimized control loop is introduced for system performance. Note that, the term optimized rather than optimal is used here as it is not the classical optimal control anymore, but a fundamental change in design methodology. To further improve the stability margin due to parameter uncertainties, adaptive control is introduced to approximate the parameters in an effort to further improve the stability margin. The effectiveness of the proposed method is demonstrated through comparative examples that highlight its advantages.