Control Problems of Discrete-Time Dynamical Systems (Lecture by Yasumichi Hasegawa

By Yasumichi Hasegawa

This monograph offers with keep watch over difficulties of discrete-time dynamical structures, which come with linear and nonlinear input/output relatives. will probably be of renowned curiosity to researchers, engineers and graduate scholars who really expert in process concept. a brand new approach, which produces manipulated inputs, is gifted within the experience of country regulate and output keep an eye on. This monograph presents new effects and their extensions, that can even be extra appropriate for nonlinear dynamical structures. to give the effectiveness of the tactic, many numerical examples of keep an eye on difficulties are supplied besides.

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887 such that (ω (1), · · · , ω (4), x0 ) has the minimum value 0. Since the input ω 1o satisfies the input limit, we feed the system with it. 29]T . 47, we obtain the desired trajectory output. 2) For confirmation, we will consider the succeeding tracking output control problem of the same system. 3 Control Problems 29 Fg + ω (5) ∗ F 2 g + F 3 xo (4) at time 7 and x(9) = ω (8) ∗ g + ω (7) ∗ Fg + ω (6) ∗ F 2 g + ω (5) ∗ F 3 g + F 4 xxo (4) at time 8. 291 such that f (ω (5), · · · , ω (8), xo (4)) has the minimum value 0.

Algorithm for tracking output control Let σ = be a considered object which is a canonical n-dimensional so-called linear system and let an input limit be |ω (i)| ≤ r, 1 ≤ i ≤ n for some r ∈ R and let a desired function d(i) for i ∈ N be d(i) : N → Y . Then a desired trajectory output control problem is performed by the following algorithm: ((Rn , F), g, h) 44 4 Control Problems of So-Called Linear System 1) By temporarily adding an input sequence ω (n)| · · · |ω (1) with the length n into the system, the states with time can be considered as follows: x(1) := ω (1) ∗ g + Fxi0 , x(2) := ω (2) ∗ g + ω (1) ∗ Fg + F 2 x0 , · · · , x(n) := ω (n) ∗ g + ω (n − 1) ∗ Fg + · · · + ω (1) ∗ F n−1 g + F n x0 for any initial state x0 ∈ Rn .

For the other ω o ( j), let ω o(k+1) ( j) := r for ω (k+1)o ( j) > r or ω o(k+1) ( j) := −r for ω o ( j) < −r. By actually adding the input ω 0(k+1) , we make a new initial state x0(k+1) := ω o(k+1) (n) ∗ g + ω o(k+1)(n − 1) ∗ Fg + · · · + ω o(k+1)(1) ∗ F n−1 g + F n x0k . If a equation f (ω ok (1), · · · , ω ok (n), x0(k−1) ) ≤ f (ω o(k+1) (1), · · · , ω o(k+1) (n), x0k ) holds, then stop this algorithm. If a equation f (ω ok (1), · · · , ω ok (n), x0(k−1) ) > f (ω o(k+1) (1), · · · , ω o(k+1) (n), x0k ) holds, then carry on this algorithm until f (ω ok (1), · · · , ω ok (n), x0(k−1) ) > f (ω o(k+1) (1), · · · , ω o(k+1) (n), x0k ) holds, where x00 = x0 .

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