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On critical stability of discrete-time adaptive nonlinear control

机译：离散时间自适应非线性控制的临界稳定性

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In this paper, we examine the global stability and instability problems for a class of discrete-time adaptive nonlinear stochastic control. The systems to be controlled may exhibit chaotic behavior and are assumed to be linear in unknown parameters but nonlinear in output dynamics, which are characterized by a nonlinear function (say, f(x)). It is found and proved that in the scalar parameter case there is a critical stability phenomenon for least squares (LS)-based adaptive control systems. To be specific, let the growth rate of f(x) be f(x)=O(||x||6) with b⩾0, then it is found that b=4 is a critical value for global stability, i.e., the closed-loop adaptive system is globally stable if b<4 and is unstable in general if b⩾4. As a consequence, we find an interesting phenomenon that the linear case does not have: for some LS-based certainty equivalence adaptive controls, even if the LS parameter estimates are strongly consistent, the closed-loop systems may still be unstable. This paper also indicates that adaptive nonlinear stochastic control that is designed based on, e.g., Taylor expansion (or Weierstrass approximation) for nonlinear models, may not be feasible in general
机译：在本文中，我们研究了一类离散时间自适应非线性随机控制的全局稳定性和不稳定性问题。要控制的系统可能表现出混沌行为，并且在未知参数下被认为是线性的，但是在输出动力学上却是非线性的，其特征是非线性函数（例如f（x））。发现并证明在标量参数情况下，基于最小二乘（LS）的自适应控制系统存在严重的稳定性现象。具体而言，令f（x）的增长率为b（0）的f（x）= O（|| x || 6），则发现b = 4是全局稳定性的临界值，即如果b ＜4，则闭环自适应系统是全局稳定的；如果b＆ges； 4，则闭环自适应系统通常是不稳定的。结果，我们发现了一个线性情况所没有的有趣现象：对于某些基于LS的确定性等价自适应控制，即使LS参数估计值非常一致，闭环系统仍可能不稳定。本文还表明基于非线性模型的泰勒展开式（或Weierstrass逼近）设计的自适应非线性随机控制通常可能不可行

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