System Theory

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By Gang Tao

Perceiving a necessity for a scientific and unified figuring out of adaptive keep watch over thought, electric engineer Tao offers and analyzes universal layout techniques with the purpose of overlaying the basics and cutting-edge of the sphere. Chapters hide platforms concept, adaptive parameter estimation, adaptive kingdom suggestions keep an eye on, continuous-time version reference adaptive keep watch over, discrete-time version reference adaptive keep watch over, oblique adaptive keep watch over, multivariable adaptive keep an eye on, and adaptive keep an eye on of platforms with nonlinearities.

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Example text

15 that (a) is equivalent to mean-square stability. The fact that R and C may be interchanged is proven as follows. Obviously, the existence of Gj ∈ B(Rn ) n in (b) is sufficient for Gj ∈ B(Cn ). The necessity is due to A√ i ∈ B(R ) for all R I i ∈ S. In this case, whenever (b) is true, we have G = G + −1G for some R I I I n GR = (GR 1 , . . , GN ) and G = (G1 , . . , GN ) in H . 4 Mean-Square Stability for the Homogeneous Case 51 transpose we have (bearing in mind that the entries of Ai are real) √ √ Li GR + −1Li GI < 0, Li GR − −1Li GI < 0, i ∈ S, so that, summing both expressions, we obtain the real version of (b).

B) For some Gj > 0 in B(Cn ), j ∈ S, we have Li (G) < 0, i ∈ S. 4 Mean-Square Stability for the Homogeneous Case 49 (c) For any Si > 0 in B(Cn ), i ∈ S, there is a unique G = (G1 , . . , GN ), Gi > 0 in B(Cn ), i ∈ S, such that L(G) + S = 0. 45) Moreover, Gi = ϕˆ −1 −A−1 ϕ(S) ˆ , i i ∈ S. Furthermore, the above results also hold if we replace L by T and A by A∗ , or C by R. Proof Clearly (c) implies (b). Suppose now that (b) holds. 46) where 2 ϕˆ Hn+ = y ∈ CN n ; y = ϕ(Q), ˆ Q ∈ Hn+ . 8 we have that −1 ϕˆj−1 y(t) ˙ = Tj ϕˆ 1−1 y(t) , .

1 in [302]). For Di ∈ B(Rn ), i ∈ S, diag(Di ) is an N n square matrix where the matrices Di are put together corner-to-corner diagonally, with all other entries being zero. Define now the operators ϕ, ϕˆ i , and ϕˆ in the following way: for V = n m m (V1 , . . , VN ) ∈ Hn,m C with Vi = [vi1 . . vin ] ∈ B(C , C ), vij ∈ C , ⎡ ⎤ vi1 ⎢ ⎥ ϕ(Vi ) := ⎣ ... 6 The Space of Sequences of N Matrices 25 ⎤ ϕ(V1 ) ⎥ ⎢ ϕ(V) ˆ := ⎣ ... ⎦ ∈ CN mn . 34) ϕ(VN ) Furthermore, for ⎤ v1 ⎢ ⎥ v = ⎣ ... ⎦ ∈ CN mn , ⎡ vN vi ∈ Cmn , we define Vi ∈ B(Rn , Rm ) such that Vi := ϕ −1 (vi ) and V = (V1 , .

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