Transfer Function

1. How hybrid electric cars can be operated more efficiently by the autonomous driving system?

The hybrid /electric autonomous cars are the future of automotive industries. The location is detected by the Global Positioning System (GPS). It is a new and promising technology, which uses real-time geographical data received from several GPS satellites to calculate coordinates, speed, and course to help navigate the car. GPS integrates the data received from tachometer, gyroscope, and altimeters to provide more accurate positioning. For identifying the lanes marking and the edges of roads, LIDAR (Light Detection and Ranging) sensors are used. These sensors use the pulses of light off the surroundings. Radar sensors used as the adaptive cruise control (ACC) system to monitor the position of in-front vehicles. Also, Ultrasonic sensors are used to measure the position of objects very close to the vehicle such as speed breakers and other vehicle parking. Cameras are also mounted to detect traffic lights, read road signs, keep track of the position of other vehicles and look out for pedestrians and obstacles on the road. The information from all of the sensors is analyzed by the central computer that manipulates the steering, accelerator, and brakes.

2. Derive transfer function of suspension system of a car and convert it state-space format.

`m_1`= sprung mass of 1/4^th of vehicle, `m_2`= un-sprung mass of the vehicle, `k_1`= spring constant of the suspension system, `b_(1 )`= damping constant of suspension constant, `b_2`= damping constant of wheel and tire, `k_2`= spring constant of wheel and tire, and `f (t)` = control force

1. a) Transfer Function

By Newton’s 2^nd law of Motion,

`m_1 x ̈_1+ k_1 x_1- k_1 x_2+b_1 x ̇_1-b_1 x ̇_2=0`

`m_2 x ̈_2+ k_2 x_2+b_2 x ̇_2 + k_1 (x_2-x_1 )+b_1 (x ̇_2-x ̇_1)=f(t)`

Now applying Laplace Transform (assuming initial conditions = 0) on both side of equations,

`s^2 m_1 X_1+k_1 X_1- k_1 X_2+sb_1 X_1-sb_1 X_2=0` (i)

`s^2 m_2 X_2+k_2 X_2+sb_2 X_2+k_1 (X_2-X_1 )+sb_1 (X_2-X_1 )=F(s)` (ii)

From equation (i),

`X_2 (s)= (s^2 m_1+ k_1+sb_1)/(k_1+sb_1 ) X_1 (s)`

Putting `X_2` in equation (ii) we get the transfer function

`(X_1 (s))/(F(s))=(k_1+sb_1)/(m_1 m_2 s^4+(m_1 b_1+m_1 b_2+m_2 b_1 ) s^3+(m_1 k_1+m_1 k_2+m_2 k_1+b_1 b_2 ) s^2+(k_1 b_2+k_2 b_1 )s+k_1 k_2 )`

`(X_2 (s))/(F(s))=(s^2 m_1+k_1+sb_1)/(m_1 m_2 s^4+(m_1 b_1+m_1 b_2+m_2 b_1 ) s^3+(m_1 k_1+m_1 k_2+m_2 k_1+b_1 b_2 ) s^2+(k_1 b_2+k_2 b_1 )s+k_1 k_2 )`

b) State Space Model

By Newton’s 2^nd Law of Motion

`m_1 x ̈_1+ k_1 x_1- k_1 x_2+b_1 x ̇_1-b_1 x ̇_2=0` (iii)

`m_2 x ̈_2+ k_2 x_2+b_2 x ̇_2 + k_1 (x_2-x_1 )+b_1 (x ̇_2-x ̇_1)=f(t)` (iv)

Suppose,

`X_1=x_1`,`X_2= x ̇_1`and `X ̇_1=x ̇_1=X_2`putting these in equation (iii)

`X ̇_2=x ̈_1= -k_1/m_1 X_1-b_1/m_1 X_2+k_1/m_1 X_3+b_1/m_1 X_4`

Similarly,

`X ̇_4=x ̈_2= k_1/m_2 X_1+b_1/m_2 X_2-(k_1+k_2)/m_2 X_3-(b_1+b_2)/m_2 X_4+(f(t))/m_2`

Putting these in the form of the state-space model

`dotbar X =barA barX+barB barU`and `barY=barC barX + barD bar U`

Where = Time derivative of , = State Vector, = Matrix of state variables, = Input matrix, = Vector of input function, = Output matrix, = feed forward matrix, = Output vector

`[[dotX_1],[dotX_2],[dotX_3],[dotX_4]] = [[0, 1 ,0 ,0],[-k_1/m_1,-b_1/m_1,k_1/m_1,b_1/m_1],[0,0,0,1],[k_1/m_2,b_1/m_2,-(k_1+k_2)/m_2,-(b_1+b_2)/m_2]]*[[X_1],[X_2],[X_3],[X_4]]+[[0],[0],[0],[1/m_1]]*f(t)`

`Y_1=X_1=x_1 and Y_2=X_3=x_2`

`[[Y_1],[Y_2]]=[[1,0,0,0],[0,0,1,0]]*[[X_1],[X_2],[X_3],[X_4]]+[0]*f(t)`

3. Are the transfer functions of RC (output across C) and CR (output across R) circuits the same?

a) The transfer function for the RC circuit with output measured across capacitor:

`V_i (t)`= Input voltage in time domain, `i (t)`= Current in time domain, `V_o (t)` = Input voltage in s-domain, `I (s)` = Current in s-domain, C = Capacitance of Capacitor, R = Resistance of resistor, t = Time, s = Complex Variable, `V_o(t)`= Output voltage in time domain, `V_o (s)` = Output voltage in s-domain

Current is flowing in the counterclockwise direction

By Applying Kirchhoff’s Voltage Law,

`V_i (t)=Ri(t)+ 1/C ∫i(t)dt` (1)

`V_o (t)= 1/C ∫i(t)dt` (2)

Now applying Laplace Transform on both sides of equations (1) and (2),

`∴V_i (s)=RI(s)+ 1/(Cs) I(s)` (3)

`∴V_o (s)= 1/(Cs) I(s)` (4)

i.e.`I(s)=V_o (s)Cs` (5)

Putting equation (5) in equation (3) we get,

`∴V_i (s)=RV_o (s)Cs+ 1/Cs V_o (s)Cs`

`∴V_i (s)=RV_o (s)Cs+V_o (s)`

`∴V_i (s)=V_o (s)(RCs+1)`

`(V_o (s))/(V_i (s))= 1/(RCs+1)`

b) The transfer function for the CR circuit with output measured across resistor:

`V_i (t)`= Input voltage in time domain, `i (t)`= Current in time domain, `V_o (t)` = Input voltage in s-domain, `I (s)` = Current in s-domain, C = Capacitance of Capacitor, R = Resistance of resistor, t = Time, s = Complex Variable, `V_o(t)`= Output voltage in time domain, `V_o (s)` = Output voltage in s-domain

The current is flowing in the counter clockwise direction.

By applying Kirchhoff’s Voltage Law,

`V_i (t)=Ri(t)+ 1/C ∫ i(t)dt`

`V_o (t)=Ri(t)`

Now applying Laplace Transform on both sides of equations,

`V_i (s)=RI(s)+ 1/(Cs) I(s)`

`V_o (s)= RI(s)`

`I(s)=(V_o (s))/R`

Putting equation (5) in equation (3) we get,

`∴V_i (s)=V_o (s)+ 1/(Cs) (V_o (s))/R`

`∴V_i (s)=(RCs+1)/(RCs)V_o (s)`

`(V_o (s))/(V_i (s))= (RCs)/(RCs+1)`