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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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