Question

System Dynamics

1. Given the differential equation x + 3x = f,where f(t) is the input function and x(t) is the response (output), find the transfer function X(s)/F(s)

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### Question 36905

System Dynamics

10.4 Figure Pl0.4 shows a closed-loop control system.
a. Compute the controller gain Kp so that the undamped natural frequency of the closed-loop system is w, = 4 rad/s.
b. Compute the controller gain Kp so that the damping ratio of the closed-loop system is = 0.7.
Compute the steady-state output for a step reference input r(t)=4u(t) and controller gain kr=2

### Question 36904

System Dynamics

Figure P10.3 shows a general closed-loop control system. The plant transfer function is
a. Determine whether the closed-loop system is stable for control gain K, 2.
b. Compute the controller gain Kp so that step response shows 25% overshoot.
c. Estimate the settling time for a step reference input if the control gain is K, 0.5.
G_{p}(s)=\frac{1}{s^{2}+6 s+8}

### Question 36903

System Dynamics

IFigure P10.1 shows a general feedback control system with forward-path transfer functions Ge(s) (controller) and Gp(s) (plant) and feedback transfer functions H(s). Given the following transfer functions,determine the closed-loop transfer function T(s) = Y(s)/R(s).

### Question 35645

System Dynamics

Problem 5. (5 points) Consider the dynamic system that has negative real poles only.Determine the transfer function from the asymptote of the Bode magnitude plot shown below:

### Question 35644

System Dynamics

Problem 4. The closed-loop system is shown below. WVe want to draw its root-locus and design the positive constant K to achieve closed-loop stability.
Find the departure angles at pi and p2, and the arrival angle at 21.
\text { Departure angle: } \phi_{d e p}=\sum_{j=1}^{m} \psi_{j}-\sum_{i \neq d e p}^{n} \phi_{i}-180(2 k+1)
\text { Arrival angle: } \psi_{a r r}=\sum_{i=1} \dot{\phi}_{i}-\sum_{j \neq a r r}^{m} \psi_{j}+180(2 k+1)
(2) (2.5 points) Find the range of K for closed-loop stability using the Routh stability criterion.

### Question 35643

System Dynamics

(2) (2.5 points) Determine K, and K2 such that wn = 4 rad/sec, and t, = 1 sec. Note:uhere u and t are the natural freguency and damning ratio respectivel:
whcre K1 and K2 arc the positive constants.
Derive the closed-loop sensitivity function: S(s) = E(s)/R(e).
Consider the closed-loop control system shown below:
t_{s}=\frac{4}{\omega_{n}} \text {, where } \omega_{n} \text { and } \zeta \text { are the natural frequency and damping ratio, respectively. }

### Question 35642

System Dynamics

Problem 2. (1) (2.5 points) DErive the Equations of motion of a quarter-car model shown below. (2) (2.5 points) Obtain the state-space model. (3) (2.5 points) Obtain the transfer function: G(s) = Y(s)/R(s).

### Question 35641

System Dynamics

1. (2.5 points) Solve the following ODE using the Laplace Transform approach.
\ddot{y}(t)+7 \dot{y}(t)+10 y(t)=4
y(0)=\dot{y}(t)=0
\text { Note: } \mathcal{L}(1)=\frac{1}{s} \text { and } \mathcal{L}\left(e^{-a t}\right)=\frac{1}{s+a} \text {. }

### Question 34038

System Dynamics

a. Obtain the 1/O equation for this system where y is the output and u is the input.
a. Obtain the 1/O equation for this system where y is the output and u is the input.
\dot{x}=\left[\begin{array}{cc} 0 & 1 \\ -20 & -4 \end{array}\right] x+\left[\begin{array}{c} 0 \\ 0.2 \end{array}\right] u \quad y=\left[\begin{array}{ll} 1 & 0 \end{array}\right] x
b. Obtain the transfer function for this system.

### Question 34037

System Dynamics

Given the following system equation
4 z+20 z+84 z=0.12 u
b. Derive the system transfer function G(s) = Z(s)/U(s)
a. Obtain a complete SSR with input u and output y 2.
Derive the transfer function Y(s)/U(s) where the output is y = ż.C.