AI 中文总结
针对含参数不确定性与对抗干扰的离散时间正线性系统，提出极小极大自适应控制框架，通过重构为零和动态博弈求解，得到保正稳定策略，具鲁棒性，数值实验验证了其有效性。
AI 中文摘要
我们针对含参数不确定性和对抗干扰的离散时间正线性系统，提出了一种极小极大自适应控制框架。系统动力学的不确定性被假设位于有限个可能的被控对象集合中。我们将该问题表述为控制器与对抗者之间的动态博弈，控制器最小化代价，对抗者则选择干扰和被控对象动力学以最大化代价。对原博弈的等价重构将问题转化为标准的极小极大两人零和动态博弈，从而可通过极小极大动态规划求解。我们给出了贝尔曼不等式的显式解，得到了稳定、保正性的策略，且无需初始稳定控制器。所得控制器具有有界的ℓ₁增益（从干扰到误差）的鲁棒性保证；当不确定参数被充分估计后，其表现类似正线性系统的标准ℋ∞型最优控制器。数值实验验证了该自适应控制器的有效性，支撑了上述理论结果。
英文摘要
We present a minimax adaptive control framework for discrete-time positive linear systems with parametric uncertainty and adversarial disturbances. The uncertainty in the system dynamics is assumed to lie in a finite set of possible plants. We formulate the problem as a dynamic game between the controller, which minimizes the cost, and an adversary, which selects both the disturbances and the plant dynamics to maximize the cost. An equivalent reformulation of the original game transforms the problem into a standard minimax two-player zero-sum dynamic game. This enables the problem to be addressed via minimax dynamic programming. We provide an explicit solution to the Bellman inequality, yielding stabilizing, positivity preserving policies without requiring an initially stabilizing controller. The resulting controller enjoys robustness guarantees in the form of bounded $\ell_1$-gain from disturbances to errors. Once the uncertain parameters have been sufficiently estimated, the controller behaves like a standard $\mathcal H_\infty$-type optimal controller for positive linear systems. The theoretical findings are supported by numerical experiments illustrating the resulting adaptive controller in action.
