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Consider the control process x' = - y + u(t), y' = x. Find to what point (in R^(2)) the point \begin{pmatrix}1\\ 0\end{pmatrix} gets steered to by a control function u\left(t\right)\:=\:\left\{1\:if\:\:0\:\le \:t\:\le \:1,\:0\:if\:\:1\:\le \:t\:\le \:2\right\}\:in\:time\:T\:=2. Use this Theorem : Consider x' = Ax + Bu, u(t) in Λ ⊆ R^(m). Suppose 1. System is null controllable, 2. 0 in int Λ, Re λ ≤ 0 for all eigenvalue λ of A. Then the where control system with negotiable constraints is global null controllable. Lemma : Consider the control system x' = Ax + Bu - 1, y' = P^(-1) APy + PBy - 2, P is a non singular nxn - matrix.

Consider the control process x' = - y + u(t), y' = x. Find to what point (in R^(2)) the point \begin{pmatrix}1\\ 0\end{pmatrix} gets steered to by a control function u\left(t\right)\:=\:\left\{1\:if\:\:0\:\le \:t\:\le \:1,\:0\:if\:\:1\:\le \:t\:\le \:2\right\}\:in\:time\:T\:=2. Use this Theorem : Consider x' = Ax + Bu, u(t) in Λ ⊆ R^(m). Suppose 1. System is null controllable, 2. 0 in int Λ, Re λ ≤ 0 for all eigenvalue λ of A. Then the where control system with negotiable constraints is global null controllable. Lemma : Consider the control system x' = Ax + Bu - 1, y' = P^(-1) APy + PBy - 2, P is a non singular nxn - matrix.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the control process x' = - y + u(t), y' = x. Find to what point (in R^(2)) the point \begin{pmatrix}1\\ 0\end{pmatrix} gets steered to by a control function u\left(t\right)\:=\:\left\{1\:if\:\:0\:\le \:t\:\le \:1,\:0\:if\:\:1\:\le \:t\:\le \:2\right\}\:in\:time\:T\:=2. Use this Theorem : Consider x' = Ax + Bu, u(t) in Λ ⊆ R^(m). Suppose 1. System is null controllable, 2. 0 in int Λ, Re λ ≤ 0 for all eigenvalue λ of A. Then the where control system with negotiable constraints is global null controllable. Lemma : Consider the control system x' = Ax + Bu - 1, y' = P^(-1) APy + PBy - 2, P is a non singular nxn - matrix.

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