Classical discrete-time adaptive controllers provide asymptotic stabilization and tracking; neither exponential stabilization nor a bounded noise gain is typically proven. In our recent work, it is shown, in both the pole placement stability setting and the first-order one-step-ahead tracking setting, that if the original, ideal, projection algorithm is used (subject to the common assumption that the plant parameters lie in a convex, compact set and that the parameter estimates are restricted to that set) as part of the adaptive controller, then a linear-like convolution bound on the closed-loop behaviour can be proven; this immediately confers exponential stability and a bounded noise gain, and it can be leveraged to provide tolerance to unmodelled dynamics and plant parameter variation. In this paper, we solve the much harder problem of adaptive tracking; under classical assumptions on the set of unmodelled parameters, including the requirement that the plant be minimum phase, we are able to prove not only the linear-like properties discussed above, but also very desirable bounds on the tracking performance. We achieve this by using a modified version of the ideal projection algorithm, termed as vigilant estimator: it is equally alert when the plant state is large or small and is turned off when it is clear that the disturbance is overwhelming the estimation process.

经典的离散时间自适应控制器提供渐近稳定和跟踪；通常没有证明指数稳定或有界噪声增益。在我们最近的工作中，无论是在极点放置稳定性设置还是一阶单步跟踪设置中，都表明如果使用原始的理想投影算法（通常假设工厂参数作为自适应控制器的一部分，它位于凸的紧集上，并且参数估计值仅限于该集合），然后可以证明闭环行为上的线性卷积边界；这立即赋予了指数稳定性和有限的噪声增益，并且可以利用它来为未建模的动力学和植物参数变化提供容差。在本文中，我们解决了更难的自适应跟踪问题；在未建模参数集的经典假设下，包括要求植物必须为最小相位，我们不仅能够证明上面讨论的线性性质，而且还可以证明跟踪性能非常理想。我们通过使用理想投影算法的修改版本（称为警惕的估计器：当工厂状态较大或较小时，它同样会发出警报；当明显的干扰使估计过程不堪重负时，它会关闭。