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On critical stability of discrete-time adaptive nonlinear control. (English) Zbl 0898.93020
The author considers a stochastic system with output nonlinearities in the form $y_{t+1} =\theta^T f_t(y_t, y_{t-1}, \dots, y_{t-p+1}) +u_t+ w_{t+1},\;t>0, \tag{1}$ where $$y_t$$, $$u_t$$ and $$w_t$$ are the system output, input and noise sequences, respectively, $$f_t(\cdot): \mathbb{R}^{p+q} \to\mathbb{R}^d$$ is a known nonlinear function and $$\theta \in\mathbb{R}^d$$ is an unknown parameter. The author shows that if the function $$f_t(x)$$ satisfies the condition $\| f_t(x) \| \leq k_1+ k_2 \| x\|^b, \quad \forall t\in \mathbb{R}^+,\;\forall x\in \mathbb{R}^{p+q}, \tag{2}$ where $$k_1\geq 0$$ and $$k_2\geq 0$$ are constants, then $$b=4$$ is a critical value for global stability.
This paper also indicates that adaptive nonlinear stochastic control that is designed based on, e.g., Taylor expansion (or Weierstrass approximation) for a nonlinear model may not be feasible in general.

##### MSC:
 93C40 Adaptive control/observation systems 93E35 Stochastic learning and adaptive control 93C55 Discrete-time control/observation systems 93D21 Adaptive or robust stabilization 93C10 Nonlinear systems in control theory
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