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Extra info for Adaptive Control
High-dimensional Case: In case of d > 2, φ Tθ ≤ c represents a hyperplane which splits the whole space into two half-hyperplanes. Unlike in case of d = 2, the vertices in this case generally cannot be arranged in a certain natural order (such as clock-wise order). In this case, we can use an algorithm AddLinearConstraintND which is listed in Algorithm 2. The idea of this algorithm is to classify the vertices of V first according to their relationship with the hyperplane determined by hyperplane φ Tθ ≤ c .
In this way, we can avoid increasing the vertex number by generating new vertices. The closeness between P´ and existing vertex Pj can be measured by checking the corresponding weight w. Besides the ideas of tight or loose IC estimator, to reduce the complexity of IC estimator, we can also use other flexible approaches. For example, to avoid growth in the vertex number of , we can approximate Vk by using a simple outline rectangle (see Fig. 6) every Vk as certain steps. For a polygon Vk with vertices P1, P2, · · · , Ps, we can easily obtain its outline rectangle by algorithm FindPolygonBounds listed in Algorithm 3.
E. f ( x1 ) − f ( x 2 ) ≤ L | x1 − x 2 | for some constant L > 0. • Function f is monotone (increasing or decreasing) with respect to its arguments. • Function f is convex (or concave). • Function f is even (or odd). 3 As to the unknown noise term knowledge: εk , here are some possible examples of a priori • Sequence ε k = 0. This case means that no noise/disturbance exists. • Sequence εk is is bounded in a known range, that is to say, ε ≤ εk ≤ ε for any k. One special case ε = −ε . g, | ε k |≤ 1 k for any k .