SYST 320: Dynamic Systems II

Homework Assignment 6

Due Mon., Oct. 17, 2022

1. Give the free response and forced response for the following dynamic system: 14 45 8x x x+ + = with (0) 1x = and (0) 1x = .

2. For the following dynamic system 3

11 18 t

x x x e−

+ + = , (0) 3, (0) 2x x= =

1. Is the system stable?
2. Find the transient response and the steady-state response.
3. For the model 16 ( )x x f t+ =
4. If x(0) = 0 and f(t) is a unit step function, what is the steady state response?
5. How long does it take before 98% of the difference between x(0) and x ss is eliminated?
6. Repeat part (b) with x(0) = 5
7. Repeat part (b) with x(0) = 5 and f(t) = 20Us(t)
8. For each of the following questions (a) – (d), choose one of the systems (i) – (iii) as your answer (you may have the same answer more than once).

(i) 10 34 1x x x+ + = (ii) 8 12 1x x x+ + = (iii) 8 41 1x x x+ + =

1. Which system takes the longest to reach steady state?
2. Which system has total response of the form: 4 4

1 2( ) cos(5 ) sin(5 )t t

x t A C e t C e t− −

= + + ?

1. Which system has the fastest oscillation (of those systems that oscillate)?
2. Which system has a time constant of 1/5?
3. Match each of the following dynamic systems with the respective graphs (in each case x(0) = 2 and

𝑥𝑥̇(0) = 0). Also, which system has the smallest time constant (#1, #2, #3, or #4)?

1. 4 104 0x x x+ + =
2. 9 14 0x x x+ + =
3. 9 20 0x x x+ + =
4. 100 0x x+ =

-3

-2

-1

0

1

2

3

0 0.5 1 1.5 2

#1

#2

#3

#4

 

 

6. Consider a dynamic systems model of a spring-mass-damper system: 3 12 ( )x cx x f t+ + = . The

designer can choose the value of the damping coefficient c among the choices c = 1, c = 12, or c = 18.

1. Which value of c corresponds to the highest time constant (1, 12 or 18)?
2. Which value of c corresponds to the lowest time constant (1, 12, or 18)?
3. Which graph (A, B, C, or D) corresponds to the free response with c = 1? (assuming initial

conditions (0) 1x = and (0) 0x = ).

7. For each of the following systems, find the roots, state if the system is stable or unstable, identify the

time constant (if it exists), and sketch the response. (“Sketch” means producing a graph that looks

something like the one below.)

1. 6 14 0x x x+ + =
2. 7 12 0x x x+ + =
3. 3 40 0x x x− − =
4. A system with transfer function 1

( ) 1

T s s

= −

Turn in all problem statements with your homework. Box all answers