By Gang Tao

Perceiving a necessity for a scientific and unified figuring out of adaptive regulate conception, electric engineer Tao offers and analyzes universal layout ways with the purpose of masking the basics and state-of-the-art of the sector. Chapters conceal platforms thought, adaptive parameter estimation, adaptive kingdom suggestions regulate, continuous-time version reference adaptive regulate, discrete-time version reference adaptive regulate, oblique adaptive regulate, multivariable adaptive keep an eye on, and adaptive keep an eye on of platforms with nonlinearities.

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**Example text**

There can be no path from a node in this second set to the first. For if such a path existed (say between i and j) then there would be a path from n to j via i. 22 General Concepts in Network Theory If there is no path between two nodes then they are certainly not adjacent. Hence aij = 0 if r < i ≤ n and 1 ≤ j ≤ r, thus A= X O Y Z where X ∈ Rr×r and Z ∈ R(n – r)×(n – r) are square. If a network is not connected then it can be divided into components each of which is connected. In an undirected network the components are disjointed.

The idea is that you show something for small networks that can be inductively extended to all the networks you are studying to prove the result. 2 Suppose that you have been asked to do the following. Prove that a connected network with n nodes is a tree if and only if it has exactly m = n – 1 edges. You can proceed in the following way to obtain the proof. 1. First, gain some insights by drawing some pictures as we have recommended before. 2. For the if (⇒ ) part of the theorem we proceed as follows.

16 we illustrate a network and two of the possible spanning trees. 16 A network and two of its spanning trees (ii) The notion of a spanning tree can be expanded to disconnected networks. If we form a spanning tree for each component and take their union, the result is a spanning forest. (iii) The number of edges in a spanning tree/forest of G is called its cut-set rank, denoted by ξ (G). If a network has k components ξ (G) = n – k. The number of edges removed from G to form the forest, m – n + k, is known as the cycle rank and is denoted by γ (G).