First and Second Order Systems Consider the system x+ax = u. (a) (b) (c) (a) (b) (c) Let u = 0 for all time, and consider x(t). If x(0) and x(2) = e−8, what is the value of a? = 2 Now let a be the value found in Part (a). If u(t) = 1(t) is the unit step function and x(0) = 0, derive the response of x(t). For a 6, what is the value of u, so that x(t) approaches = 2 as t tends to infinity for any initial value x(0)? Consider the following second-order system x+7x12x = U. What are the poles of the system? What is the meaning that the system be stable in terms of system response x(t)? Is the system stable or not? Design a rate-feedback PD controller - u(t) = K₂(r — x) – Kax - so that the system response to a step input has a settling time around 4 sec and an overshoot of about 5%. Show all working.
First and Second Order Systems Consider the system x+ax = u. (a) (b) (c) (a) (b) (c) Let u = 0 for all time, and consider x(t). If x(0) and x(2) = e−8, what is the value of a? = 2 Now let a be the value found in Part (a). If u(t) = 1(t) is the unit step function and x(0) = 0, derive the response of x(t). For a 6, what is the value of u, so that x(t) approaches = 2 as t tends to infinity for any initial value x(0)? Consider the following second-order system x+7x12x = U. What are the poles of the system? What is the meaning that the system be stable in terms of system response x(t)? Is the system stable or not? Design a rate-feedback PD controller - u(t) = K₂(r — x) – Kax - so that the system response to a step input has a settling time around 4 sec and an overshoot of about 5%. Show all working.
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Transcribed Image Text:First and Second Order Systems
Consider the system
x+ax = u.
(a)
(b)
(c)
(a)
(b)
(c)
Let u = 0 for all time, and consider x(t). If x(0)
and x(2) = e−8, what is the value of a?
= 2
Now let a be the value found in Part (a). If u(t) = 1(t)
is the unit step function and x(0) = 0, derive the response of x(t).
For a 6, what is the value of u, so that x(t) approaches
=
2 as t tends to infinity for any initial value x(0)?
Consider the following second-order system
x+7x12x = U.
What are the poles of the system?
What is the meaning that the system be stable in terms
of system response x(t)? Is the system stable or not?
Design a rate-feedback PD controller
-
u(t) = K₂(r — x) – Kax
-
so that the system response to a step input has a settling time
around 4 sec and an overshoot of about 5%. Show all working.
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