Question EEET 3046 Control Systems

Q Problem 1. Referring to the RLC circuit in Fig. 1, let the source voltage be v, the voltage over the inductor be vL, and the voltage over the capacitor be vC. Show that v = RCv˙C + vL + vC, (1)

vL L +

1 R v˙L = Cv¨C. (2)

(2 marks) V R C

R L

VL VC + -

+ -

Fig. 1. A circuit with some resistors, an

inductor and a capacitor. 𝑚𝑚 𝑚𝑚 𝑏𝑏 𝑘𝑘 𝑘𝑘 𝑓𝑓

𝑥𝑥1 𝑥𝑥2 𝑚𝑚 𝑘𝑘 𝑏𝑏 𝑏𝑏

𝑓𝑓 𝑥𝑥

Fig. 2. A mass-spring-damper system, with spring constant k, damping

coefficient b, mass m, and external force f. Problem 2. Referring to the mass-spring-damper system in Fig. 2, use Newton's second law to derive the differential equations that relate the external force f to x1 and x2. The friction between the masses and

the floor is negligible. Hint: Get ideas from the mass-spring-damper example in the lecture. (2 marks) Problem 3. Referring to the mass-spring-damper system in Fig. 3, derive the differential equations that relate the external force f to x1, x2 and x3. The friction between the mass and the floor is negligible.

Hint: Place a zero mass at x2(t). (3 marks) 𝑚𝑚 𝑚𝑚 𝑏𝑏 𝑘𝑘 𝑘𝑘

𝑓𝑓 𝑥𝑥1 𝑥𝑥2 𝑚𝑚 𝑘𝑘 𝑏𝑏1 𝑏𝑏2

𝑓𝑓

𝑥𝑥1 𝑥𝑥2 𝑥𝑥3 Fig. 3. A mass-spring-damper system, with spring constant k, damping coefficient b, mass m, and external force f. Page 2 of 8

EEET 3046 Control Systems / EEET 5148 Control Systems M Part III: System modelling using Laplace transform [18 marks] Problem 1. If L{tf(t)} = − dF(s)

ds , and L{e

at} = 1 s−a

, then L{teat} =? (1 mark) Problem 2. If the Laplace transform of f(t) = (t + a)(t + b) is F(s) = 3s 2+4s+2

s 3

, what are the values of a and b (assuming a > b)? (2 marks)

Problem 3. If the inverse Laplace transform of F(s) = 2 s 2+as+b

is f(t) = e −t

sin(2t), what are the values of a and b? (2 marks) Problem 4. L{R t

0 f(τ )g(t − τ ) dτ} =? (1 mark) Problem 5. Apply Laplace transform to Eq. (1) and Eq. (2) and determine the transfer function VC (s)

V (s) ,

where VC(s) is the Laplace transform of vC, and V (s) is the Laplace transform of v. (3 marks) Problem 6. Apply Laplace transform to the differential equations derived for Part II Problem 2, and show

the transfer functions X1(s) F(s) and X2(s)

F(s) are

X1(s) F(s)

= ms2 + bs + k (ms2 + bs + k)(ms2 + 2k) − k

2 ,

X2(s) F(s) =

k (ms2 + bs + k)(ms2 + 2k) − k 2 , (3)

where F(s), X1(s) and X2(s) are the Laplace transforms of f, x1 and x2 respectively. (4 marks) Problem 7. Apply Laplace transform to the differential equations derived for Part II Problem 3, and determine the transfer function X2(s) F(s)

, where F(s) and X2(s) are the Laplace transforms of f and x2

respectively. Hints: • Laplace transformation gives you three linear equations. Solve this three linear equations for X1(s), X2(s) and X3(s). Due to the algebraic complexity, you are encouraged to use MATLAB's Symbolic

Math Toolbox or Mathematica. If you are using MATLAB, then the code below will help: syms s k b1 b2 m F ; A = [ . . . ] ; % f i l l t h i s i n by y o u r s e l f b = [ . . . ] ; % f i l l t h i s i n by y o u r s e l f

s o l s = l i n s o l v e (A, b ) • So that you can check your result, X2(s) F(s)

is a second-order system that depends only on m, b1 and b2. (5 marks) Page 3 of 8 EEET 3046 Control Systems / EEET 5148 Control Systems M

Part IV: System modelling using state space equations [10 marks] Problem 1. Taking v as the input, y = vC as the output, write down the state space equations corresponding to Eq. (4). (3 marks) LCv¨C + RCv˙C + vC = v. (4) Problem 2. Suppose a system is governed by the following differential equation: d 4

x dt 4

= 3 d 3

x dt 3

+ 2 d 2

x dt 2 + dx

dt

− x + u. (5) Taking u as the input, and y = dx

dt as the output, write down the state space equations. Furthermore, determine the corresponding transfer function.

Hint: Owing to the algebraic complexity of inverting a 4-by-4 matrix, you are encouraged to use MATLAB's Symbolic Math Toolbox or Mathematica. Later in the course however, you will learn to write down the transfer function just by inspecting the state space matrices. (5 marks)

Problem 3. Based on the definition of the matrix exponential, show that Ae

A = e AA, where A is a square matrix.