Transfer Function

Transfer Function

1. How hybrid electric cars can be operated more efficiently by autonomous driving system?
2. Derive transfer function of suspension system of a car and convert it state space format.
3. Are the transfer functions of RC and CR (output across R) circuits same?

Introduction

Transfer Function

Transfer function will allow separation of the input, system, and output into three separate and distinct parts, unlike the differential equation. Transfer function will also allow us to algebraically combine mathematical representations of subsystems to yield a total system representation.

The transfer function of a linear, time-invariant, differential equation system is defined as the ratio of the Laplace transform of the output (response function) to the Laplace transform of the input (driving function) under the assumption that all initial conditions are zero.

The linear time-invariant system defined by the following differential equation:

where y is the output of the system and x is the input.

The transfer function of this system is the ratio of the Laplace transformed output to the Laplace transformed input when all initial conditions are zero,

By using the concept of transfer function, it is possible to represent system dynamics by algebraic equations in s. If the highest power of s in the denominator of the transfer function is equal to n, the system is called an nth-order system.

Importance of Transfer Function

1. The transfer function of a system is a mathematical model in that it is an operational method of expressing the differential equation that relates the output variable to the input variable.
2. The transfer function is a property of a system itself, independent of the magnitude and nature of the input or driving function.
3. The transfer function includes the units necessary to relate the input to the output; however, it does not provide any information concerning the physical structure of the system. (The transfer functions of many physically different systems can be identical.)
4. If the transfer function of a system is known, the output or response can be studied for various forms of inputs with a view toward understanding the nature of the system.
5. If the transfer function of a system is unknown, it may be established experimentally by introducing known inputs and studying the output of the system. Once established, a transfer function gives a full description of the dynamic characteristics of the system, as distinct from its physical description.

Question & Answers

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

In Hybrid Electric vehicle the word “hybrid” means a combination between two different things. The majority of hybrid electric vehicles are using a combination of gasoline engines and electric motors.

The important components related to hybrid cars are as follow:

1. A hybrid car consists of engine and mechanical systems.
2. Traction Battery Pack
3. Digital system & sensors
4. Power controller
5. Electrical Machine

If we talk about hybrid electric cars operation by combination with autonomous driving system then there is positive situation in which car will operate in full autonomous electric vehicle.

Generally, an autonomous electric vehicle has GPS (global positioning system) which uses signals to combine reading of tachometers, altimeters and gyroscopes to provide more accurate position.

Also, autonomous electric vehicle has Lidar system (lighting detection and ranging) sensor which is used to analyses and identify the lane marking and the edges of roads. Basically, Lidar sensors bounce pulse of light off the surroundings.

In autonomous electric vehicle has video camera which detects traffic light, read road signs, keep track of the position of other vehicles and look out for pedestrians and obstacles on road.

Radar sensors in autonomous car monitors the position of other vehicles nearby. Adaptive cruise control system uses this type of sensors.

Ultrasonic sensors may be used to measures the position of objects very close to the vehicle, such as curbs and other vehicle when parking.

The information from all of the sensors is analysed by a central computer that manipulate the steering, accelerator and brakes.

Also, Full autonomous electric vehicle has less chance of crash, zero emission and less traffic problem.  

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

Car suspension model transfer function

The dynamics of the suspension of a vehicle can be analysed by running simulations through a mathematical model. The simplest model of a vehicle’s suspension is called quarter-car suspension model.

The model consists of two mass bodies, quarter vehicle and wheel, and lumped parameters for stiffness and damping, for both suspension and tyre.

where:

m1 [kg] – the mass of a quarter of the vehicle body

m2 [kg] – the mass of the wheel and suspension

k1 [N/m] – spring constant (stiffness) of the suspension system

c1 [N·s/m] – damping constant of the suspension system

k2 [N/m] – spring constant (stiffness) of the wheel and tire

c2 [N·s/m] – damping constant of the wheel and tire

z1 [m] – displacement of the vehicle body (output)

z2 [m] – displacement of the wheel (output)

u [m] – road profile change (input)

Considering all initial conditions being zero and applying Lapace’s transform to equations (1) gives:

From the system of equations (1) we can determine two transfer functions, depending on which displacement (z1 or z2) we consider as the output of the system.

Replacing (4) in (3) and after doing the necessary simplification gives the first transfer function:

where displacement of the quarter vehicle body mass z1(t) is considered the output and the road irregularities u(t) the input.

Let’s consider the displacement of the wheel mass z2 as the output of the system. From equation (2) we extract Z1(s) as:

Replacing (5) in (3) and after doing the necessary simplification gives the first transfer function:

 

Derivation for transfer function and state space format of mechanical system mass-spring-friction.

Time domain equation

f(t) = M d²y (t) / dt² + B dy (t) / dt + Ky(t)…………………………...(i)

Transfer in Laplace

F(s) = M S² Y(s) + B S Y(s) + K Y(s)

Y(s) / F(s) = 1 / M S² + B S + K……………………………………. (ii)

State space format

By Assuming,

x₁ (t) = y(t)

x₂ (t) = dx₁ (t) / dt = x₁ = dy(t) / dt

f(t) = M d²y (t) / dt² + B dy (t) / dt + Ky(t) ………………………………………... From eq (i)

Now for state space format the equation (i) may be written as

Dx₂ (t) / dt = -k/M x₁ (t) – B/M x₂ (t) + f(t)/M       (we are adding assumed value in eq(i))

            x = Ax + Bu

Lets assume

            U(t) = f (t)/ M 

    

y = x₁ (t)

            y = C x + Du (State Space Format)

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

The transfer function of RC and CR circuits are not same because the output of RC circuit is across capacitor whereas the output for the CR circuit is across Resistor.

A systematic diagram of RC and CR circuit are given below

RC Circuits

 

CR Circuits

Where,

• R = Resistor
• C = Capacitor
• Vo = Voltage Out
• Vi = Voltage In

A basic Transfer Function derivation of RC circuit is as follow:

Transfer Function = Vo (s) / V_i (s) = 1 / 1 + RCs

Now applying Kirchhoff's voltage law,

V_i = V_R + V_C

V_i = i (t) R + V_O                                    

By applying Laplace transform

So, the current flowing the capacitor is i (t) = C dV_C / dt

V_i (s) = RCs V_O (s) ₊ V_O (s)

V_O (s) (1+RCs) = V_i (s)

Transfer function for RC circuit is

Vo (s) / V_i (s) = 1 / 1 + RCs

 

Now for CR circuit the transfer function will be as by deriving the equation and we get

Transfer Function = Vo (s) / V_i (s) = RCs / RCs + 1

By applying Kirchhoff's voltage law

V_i (t) = R_i (t) + 1/C i (t) dt

            V_O (t) = R_i (t)                            

Now by applying transform

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

            V_O (s) = RI (s)

            I (s) = V_O (s) / R

Therefore

            V_i (s) = V_O (s) + 1/Cs × V_O (s) / R

            V_i (s) = RCs + 1 / RCs × V_O (s)

Transfer function for CR circuit is

            Vo (s) / V_i (s) = RCs / RCs + 1

Reference :

1. Modern control Engineering/Katsuhiko ogata
2. http://ctms.engin.umich.edu/CTMS/index.php?example=Suspension&section=SystemModeling#2
3. https://x-engineer.org/graduate-engineering/signals-systems/control-systems/quarter-car-suspension-model-transfer-function/