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Xie and Guo obtained the following results: if L ≥ 3 + 2 f 2 , then there exists a feedback control law such that for any F(L), the corresponding closed-loop control system is globally stable; and if 3 L< + 2 2 , then for any feedback control law and any y 0 ∈ R1 , there always exists Adaptive Estimation and Control for Systems with Parametric and Nonparametric Uncertainties some 23 f ∈ F (L) such that the corresponding closed-loop system is unstable. 6), the “magic” number 3 + 2 characterizes the capability and limits of the whole 2 feedback mechanism.

E. θ ≤ θ ≤ θ , where θ and θ are constant vector and the relationship “≤” means element-wise “less or equal”. e. ||θ − θ0 || ≤ rθ, where rθ ≥ 0 is a known constant and θ0 is the center of set Θ. • The unknown parameter lies in a known countable or finite set of values, that is to say, θ { θ1, θ2, θ 3, · · · }. 2 As to the unknown function f(·), here are some possible examples of a priori knowledge: • f(x) = 0 for all x. This case means that there is no unmodeled dynamics. Adaptive Estimation and Control for Systems with Parametric and Nonparametric Uncertainties • Function f is bounded by other known functions, that is to say, 25 f ( x) ≤ f ( x) ≤ f ( x) for any x.

Now we discuss how to implement the algorithm AddLinearConstraint. 2D Case: In case of d = 2, φ Tθ ≤ c represents a straight line which splits the plane into two half-planes (see Fig. 3). In this case, we can use an efficient algorithm AddLinearConstraint2D which is listed in Algorithm 1. Its basic idea is to simply test each vertex of V to see whether to keep original vertex or generate new vertex. The time Adaptive Estimation and Control for Systems with Parametric and Nonparametric Uncertainties 33 complexity of Algorithm 1 is O(s), where s is the number of vertices of domain V.