To Study about the Autonomous Driving System and Transfer Function Equation.

AIM : To Study about the Autonomous Driving System and Transfer Function Equation.

Transfer Function :

In control engineering and control theory the transfer function of a system is a very common concept. A transfer function is determined using Laplace transform and plays a vital role in the development of the automatic control systems theory.

A system can be defined as a mathematical relationship between the input, output and the states of a system. In control theory, a system is represented a a rectangle with an input and output.

where :

u(t) – input
y(t) – output

Both input and output are variable in time. A system with only one input and output is called SISO (Single Input Single Output) system. Control theory also applies to MIMO (Multi Input Multi Output) systems, but for an easier understanding of the concept we are going to refer only to SISO systems.

The simplest representation of a system is through ODE (Ordinary Differential Equation). When dealing with ordinary differential equations, the dependent variables are function of a positive real variable t (often time). By applying Laplace’s transform we switch from a function of time to a function of a complex variable s (frequency) and the differential equation becomes an algebraic equation.

The transfer function defines the relation between the output and the input of a dynamic system, written in complex form (s variable). For a dynamic system with an input u(t) and an output y(t), the transfer function H(s) is the ratio between the complex representation (s variable) of the output Y(s) and input U(s).

By using the concept of transfer function, it is possible to represent system dynamics by algebraic equations in s.

Importance of Transfer Function :

• 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.
• The transfer function is a property of a system itself, independent of the magnitude and nature of the input or driving function.
• 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.
• 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.
• 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.

OBJECTIVES :

1. How Hybrid Electric Cars can be operated more efficiently by Autonomous Driving System ?

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

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

STUDY - 1 :

Autonomous Driving System :

• Autonomous vehicles (AVs) use technology to partially or entirely replace the human driver in navigating a vehicle from an origin to a destination while avoiding road hazards and responding to traffic conditions.
• The Society of Automotive Engineers (SAE) has developed a widely-adopted classification system with six levels based on the level of human intervention. The U.S. National Highway Traffic Safety Administration (NHTSA) uses this classification system.

Levels of Automation -

The SAE AV classification system is broken down by level of automation :

The Society of Automotive Engineers (SAE) has classified self-driving vehicles into stages which range from level 0 to 5. Let's take a brief look at each stage -

Level 0 - It doesn't feature any self-driving technique at all.

Level 1 - These cars offer at least one system that helps the driver to brake, steer, accelerate, but if there are multiple systems then they are not capable of communicating with each other.

Level 2 - These cars simultaneously control speed 7 steering, even if the driver is not driving for a short period of time. Think lane-centering technology combined with advanced cruise control, as an example.

Level 3 - These vehicles are fully autonomous but require driver’s attention. These cars are not yet available but are being tested by some tech start-ups.

Level 4 - These cars once programmed to destination won’t need driver’s input, but the controls are available for driver’s interference for some instances.

Level 5 - These cars are fully autonomous without driver’s interference at all.

Currently very few vehicles offer Level 3 technologies & no level 4 or level 5 systems have been commercially deployed.

Hybrid Electric Cars :

• A hybrid electric vehicle (HEV) has two types of energy storage units, electricity and fuel. Electricity means that a battery is used to store the energy, and that an electromotor will be used as traction motor.
• Fuel means that a tank is required, and that an Internal Combustion Engine is used to generate mechanical power, or that a fuel cell will be used to convert fuel to electrical energy. In the latter case, traction will be performed by the electromotor only. In the first case, the vehicle will have both an engine and a motor.
• Depending on the drive train structure (how motor and engine are connected), we can distinguish between parallel, series or combined HEVs.
• Depending on the share of the electromotor to the traction power, we can distinguish between mild or micro hybrid (start-stop systems), power assist hybrid, full hybrid and plug-in hybrid.
• Depending on the nature of the non-electric energy source, we can distinguish between combustion (ICE), fuel cell, hydraulic or pneumatic power, and human power.
• In the first case, the ICE is a spark ignition engines(gasoline) or compression ignition direct injection(diesel) engine. In the first two cases, the energy conversion unit may be powered by gasoline, methanol, compressed natural gas, hydrogen, or other alternative fuels.

Hybrid Electric Cars can be operated more efficiently by Autonomous Driving System :

• Applying hybrid-electric technology to our self-driving vehicles delivers several benefits, including maximum mileage to keep the vehicle on the road. Plus, hybrids help provide the significant amount of electrical power required for self-driving sensors and computing systems without having a significant impact on the mileage.
• Computer efficiency is at the top of mind for everyone involved in the race to develop self-driving cars.
• The location is detected by the Global Positioning System (GPS). It is new and promissing technology, which uses real time geogeaphical data received from several GPS satellites to calculate coordinates, speed and course to help navigate the car. GPS integrates the data received from tachometer, gyoscope, and aitimeters to provide more accurate positioning.
• For identifyning the lanes marking and the edges of roads, LIDAR(Light Dtecting and Ranging) sensors are used. These sensors use the pulses of light off the surroungings. LIDAR consists of emitter, mirrorand receiver. The emitter emitter sends a laser beam bouncesof the mirror 10-15 revolutions per min. After bouncing off objects, the laser beam reutrns through the mirror to the receiver. The data of such points is stored as a point cloud data (pcd).
• 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-brakers and other vehicle parking.
• Cameras are also mounted to detect traffic signals, keep track of other vehicles and look out for pedestrains and obstacles on the road. The infromation from all of the sensors is analysed by the central computer that maipulates the steering, accelerator and brakes.
• In the autonomous cars the switching of fuel system to battery system and vice-versa can be done by the car only with out human assitance. By that the car can be work more efficiently.

Other Benifits of Autonomous Cars are -

• Reduce Accidents

• Reduce Traffice congestion

• Reduce CO2 Emissions

• Increased lane capacity

• Lower fuel consumption

• Transportation accessibility

• Reduce travel tie and transportation cost

• More effective and affordable taxis

• More efficient parking

STUDY - 2 :

Transfer Function of the Suspension System :

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

m₁ (Kg) = The mass of a quarter of a vehicle body.

m₂ (Kg) = The mass of the wheel and suspension.

k₁ (N/m) = Spring Constant (Stiffness) of the suspension system.

k₂ (N/m) = Spring Constant (Stiffness) of the wheel & tyre.

b₁ (N-s/m) = Damping constant of the suspension system.

b₂ (N-s/m) = Damping constant of the wheel & tyre.

x₁ (m) = Displacement of the Vehicle Body. (Output)

x₂ (m) = Displacement of the wheel & tyre. (Output)

Transfer Function –

By Newton’s 2 Law of Motion –

Now applying Laplace Transform (Assuming initial condition = 0) on both side of equation –

From equation {i} –

Putting x in equation (ii) we get the transfer function –

State-Space Model –

By Newton’s 2 Law of Motion –

Suppose,

Putting these state variables in equation (iii) & (iv) –

Putting all this in the form of the state space model –

STUDY - 3 :

Derivation of the Transfer Functions of RC (output across C) and CR (output across R) circuits :

The transfer function of RC (output across C) :

' V_i(t) ' = Input voltage in time domain

' V_i(s) ' = Input voltage in s-domain

' i(t) ' = Current in the time 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 counter clockwise direction -

By applying Kirchhoff's voltage law,

' V_i(t) = R_i(t) + 1/C∫i(t)dt ------------(1) '

' V_o(t) = 1/C∫i(t)dt ---------(2) '

Now applying L.T. on both sides of equations (1) & (2) -

' therefore V_i(s) = RI(s) + (1/C_s) * (I(s)) --------(3) '

' V_o(s) = (1/C_s) * (I(s)) --------(4) '

' i.e. : I(s) = V_o(s)C_s -------(5) '

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

' therefore V_i(s) = RV_o(s)C_s + (1/C_s)V_o(s)C_s '

' V_i(s) = RV_o(s)C_s + V_o(s) '

' V_i(s) = V_o(s) * (RC_s+1) '

' V_o(s) / V_i(s) = 1/(RC_s+1) '

The transfer function of CR output accross Resistor :

Current is flowing in the counter clockwise direction -

By applying Kirchhoff's voltage law,

' V_i(t) = R_i(t) + 1/C∫i(t)dt ------------(1) '

' V_o(t) = R_i(t) ---------(2) '

Now applying L.T. on both sides of equations (1) & (2) -

' therefore V_i(s) = RI(s) + (1/C_s) * (I(s)) -----------(3) '

' V_o (s) = RI(s) ----------(4) '

' I(s) = V_o(s)/R ------(5) '

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

' therefore V_i(s) = V_o(s) + (1/C_s) * (V_o(s)/R) '

' V_i(s) = ((RC_s+1) / (RC_s)) * (V_o(s)) '

' V_o(s) / V_i(s) = (RC_s) / (RC_s+1) '

Hence, The transfer function of RC output accross capacitor and The transfer function of CR output accross Resistor are not same.

CONCLUSION :

• Discussed about the Hybrid Electric Cars can be operated more efficiently by Autonomous Driving System.
• Successfully Derive the transfer function of the suspension system of a car and convert it to state-space format.
• Compared the transfer functions of RC(output across C) and CR (output across R) circuits.

REFERENCES :

https://x-engineer.org/graduate-engineering/signals-systems/control-systems/find-transfer-function-system/

http://css.umich.edu/factsheets/autonomous-vehicles-factsheet

https://skill-lync.com/knowledgebase/derivation-of-transfer-function-of-a-suspension-system-and-converting-into-a-state-space-format