# Linear system theory by Wilson J. Rugh

By Wilson J. Rugh

*Linear method conception, moment Edition*, outlines the elemental idea of linear platforms in a unified, obtainable, and cautious demeanour, with parallel, self sufficient therapy of continuous-time and discrete-time linear structures.

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5) becomes the standard extended Kalman ﬁlter [2, 8]. Usually Pi and Qi are initialized as diagonal matrices, with entries Pi (0) and Qi (0), respectively. It is important to remark that Hi (k), Ki (k), and Pi (k) for the EKF are bounded; for a detailed explanation of this fact see [6]. 7) can be expressed as ei (k + 1) = wi (k)zi (x(k), u(k)) + zi . 4) is wi (k + 1) = wi (k) − ηi Ki (k)e(k). 10) Now, we establish the ﬁrst main result of this chapter in the following theorem. 1. 7) is semiglobally uniformly ultimately bounded (SGUUB); moreover, the RHONN weights remain bounded.

10) Now, we establish the ﬁrst main result of this chapter in the following theorem. 1. 7) is semiglobally uniformly ultimately bounded (SGUUB); moreover, the RHONN weights remain bounded. Proof. 1. 11), T ∆Vi (k) = [wi (k) − ηi Ki (k)ei (k)] [wi (k) − ηi Ki (k)ei (k)] + [wi (k)zi (x(k), u(k)) + − wi (k)wi (k) − e2i (k). 12) can be expressed as ∆Vi (k) = wiT (k)wi (k) − wiT (k)wi (k) + η 2 e2i (k)K T Ki (k) + 2 zi wi (k)zi (x(k), u(k)) + ziT (x(k), u(k))wiT (k)wi (k)zi (x(k), u(k)) + 2zi − 2ηi ei (k)wiT (k)Ki (k) − e2i (k), ∆Vi (k) ≤ |ei (k)|2 ηKi 2 − |ei (k)|2 − |2ηi ||ei (k)| wi (k) Ki (k) + | 2 zi | + |2 zi | wi (k) zi (x(k), u(k)) + wi (k) 2 zi (x(k), u(k)) 2 .

Vi (k) = wi (k)Pi (k)wi (k) + xi (k)Pi (k)xi (k), ∆Vi (k) = V (k + 1) − V (k), = wi (k + 1)Pi (k + 1)wi (k + 1) + xi (k + 1)Pi (k + 1)xi (k + 1) − wi (k)Pi (k)wi (k) − xi (k)Pi (k)xi (k). 14) can be expressed as ∆Vi (k) = wiT (k)Pi (k)wi (k) − wiT (k)[Bi (k)]wi (k) + η 2 xT (k)C T K T [Ai (k)]Ki (k)C x(k) + f T (k)Pi (k)f (k) − f T (k)[Bi (k)]f (k) + xT (k)C T giT [Ai (k)]gi C x(k) − wiT (k)Pi (k)wi (k) − xT i (k)Pi (k)xi (k), ∆Vi (k) ≤ x(k) 2 ηKi C 2 Ai (k) − x(k) 2 gi C 2 Ai (k) − x(k) 2 Pi (k) − wi (k) 2 Bi (k) + | zi |2 Ai (k) + 2 wi (k) + wi (k) 2 zi (x(k), u(k)) | zi (x(k), u(k)) 2 zi | Ai (k) Ai (k) , with Bi (k) = Di (k) − Qi , ∆Vi (k) ≤ − x(k) 2 Ei (k) − wi (k) 2 Fi (k) + | 2 zi | Ai (k) + 2Gi (k), with Ei (k) = Pi (k) − ηKi C 2 Ai (k) − gi C 2 Ai (k) , Fi (k) = Bi (k) − zi (x(k), u(k)) 2 Ai (k) , Gi (k) = wi∗ − wi max zi (x(k), u(k)) | zi | Ai (k) .