i.e., Given a linear harmonic oscillator ÿ+y=u, y(0) = yo, y(0) = vo, y(t) = 0, y(t) = 0, x Ax+ Bu = 0 [191 -1 0 x + [i]- U₂ a. Determine et, b. Determine We(t) = f eAo BBT ATo do; Yo x(0) = [ 10 ], x(tr) = - [8]. c. Determine a minimum energy controller by u = - =-BT₂AT (te-t) W-¹(te) [etx(0) - x(t₁)]; d. Let tf = π/4, yo = 1 and vo = 5, plot x1(t), x2(t) and u(t) for 0 ≤ t ≤ Tπ/4 and compute the energy of u, π/4 f** u²(t)dt;
i.e., Given a linear harmonic oscillator ÿ+y=u, y(0) = yo, y(0) = vo, y(t) = 0, y(t) = 0, x Ax+ Bu = 0 [191 -1 0 x + [i]- U₂ a. Determine et, b. Determine We(t) = f eAo BBT ATo do; Yo x(0) = [ 10 ], x(tr) = - [8]. c. Determine a minimum energy controller by u = - =-BT₂AT (te-t) W-¹(te) [etx(0) - x(t₁)]; d. Let tf = π/4, yo = 1 and vo = 5, plot x1(t), x2(t) and u(t) for 0 ≤ t ≤ Tπ/4 and compute the energy of u, π/4 f** u²(t)dt;
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![i.e.,
Given a linear harmonic oscillator
ÿ+y=u, y(0) = yo, ý (0) = vo, y(tf) = 0, ý(t£) = 0,
x = Ax+ Bu =
=
1
[99]++8)
x
0
=
a. Determine e¹¹t;
b. Determine We(tf) = ſtf eªº BBT eA¹º do;
c. Determine a minimum energy controller by u =
π/4, yo 1 and vo
d. Let tf
compute the energy of u,
e. Repeat part d for tf = π/2.
U₂
-
x(0) =
> = [ 2 ] ₁ (t) = [8] ·
200
—BT¸AT
BT₁A¹ (₁-¹) W¯¯¯¹(tx) [eAttx(0) − x(te)];
(tf-t
5, plot x₁(t), x2(t) and u(t) for 0 ≤ t ≤ π/4 and
π/4
[*/ u² (1) dt;](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab5c0065-215e-4cda-b148-780e56759579%2Fee10f0f1-3ba4-43dd-8ead-927865a0e03e%2Fpqs3bg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:i.e.,
Given a linear harmonic oscillator
ÿ+y=u, y(0) = yo, ý (0) = vo, y(tf) = 0, ý(t£) = 0,
x = Ax+ Bu =
=
1
[99]++8)
x
0
=
a. Determine e¹¹t;
b. Determine We(tf) = ſtf eªº BBT eA¹º do;
c. Determine a minimum energy controller by u =
π/4, yo 1 and vo
d. Let tf
compute the energy of u,
e. Repeat part d for tf = π/2.
U₂
-
x(0) =
> = [ 2 ] ₁ (t) = [8] ·
200
—BT¸AT
BT₁A¹ (₁-¹) W¯¯¯¹(tx) [eAttx(0) − x(te)];
(tf-t
5, plot x₁(t), x2(t) and u(t) for 0 ≤ t ≤ π/4 and
π/4
[*/ u² (1) dt;
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