Recap of Session VII

Chapter II: Mathematical Modeling

• Mathematical Modeling of Mechanical systems

•Mathematical Modeling of Electrical systems

• Models of Hydraulic SystemsLiquid Level SystemFluid Power System

Recap of Session VII

Mathematical Modeling: Thermal Systems

qin = heat inflow rate

Tov = Temperature of the oven

Tamb = Ambient Temperature

T = Rise in Temperature = (Tov - Tamb)

Mathematical Modeling: Thermal Systems

Parts

Tov

Oven

Tamb

qout

qin

Example: Heat treatment oven

Mathematical Modeling: Thermal Systems-I

From Law of Conservation Energy

qin = heat inflow rate

qin = qs + qout --- (1)

qout = heat loss through the walls of the oven

qs = Rate at which heat is stored (Rate at which heat is absorbed by the parts)

Mathematical Modeling: Thermal Systems-II

Thermal Resistance: R=outq

T

R

Tqout --- (a)

Thermal Capacitance = C = Q/T

Heat stored =

dt

dTCqs --- (b)

Mathematical Modeling: Thermal Systems-IIISubstitute (a) and (b) in (1)

qin = qs + qout --- (1)

R

T

dt

dTCqin .

Model inRqTdt

dTRC

Chapter III: System ResponseChapter III: System Response

• Prediction of the performance of control systems requires

1. Obtaining the differential equations2. Solutions

System behaviour can be expressed as a function of time

Such a study: System response or system analysis in time domain

System Response in Time DomainSystem Response in Time Domain

System Response: The output obtained corresponding to a given Input.

Total response: Two parts•Transient Response (yt)•Steady state response (yss)

•Total response is the sum of steady state response and transient response

y = yt + yss

Transient Response (yt):Transient Response (yt):

•Initial state of response and has some specific characteristics which are functions of time.

•Continues until the output becomes steady.

•Usually dies out after a short interval of time.

•Tends to zero as time tends to ∞

Steady State Response (yss)Steady State Response (yss)

• Ultimate Response obtained after some interval of time

• Response obtained after all the transients die out

• It is not independent of time

• As time approaches to infinity system response attains a fixed pattern

Transient and Steady-state Response of a spring system

Transient and Steady-state Response of a spring system

Transient SS

• When the weight is added the deflection abruptly increases•System oscillates violently for some time (Transient)•Settles down to a steady value (Steady state)

Steady State ErrorSteady State Error

• Steady State Response may not agree with Input

• Difference is called steady state error

Steady state error = Input – Steady state response

Input or Response

Steady state error

Timet =0

Input

Response

t ∞

Test Input SignalsTest Input Signals

• Systems are subjected to a variety of input signals (working conditions)

• Most cases it is very difficult to predict the type of input signal

• Impossible to express the signals by means of Mathematical Models

• Common Input Signals

- Step Input

- Ramp Input

- Sinusoidal

- Parabolic

- Impulse functions, etc.,

Common Input Signals

• In system analysis one of the standard input signal is applied and the response produced is compared with input

• Performance is evaluated and Performance index is specified

• When a control system is designed based on standard input signals – generally, the performance is found satisfactory

Standard input signals

Common System Input Signalsa) Step Input

i (t)

t = 0 time

K

Common System Input Signals

a) Step Input

Input is zero until t = 0

Then takes on value K which remains constant for t > 0

Signal changes from zero level to K instantaneously

Mathematically

i (t) = K for t > 0

= 0 for t < 0

for t = 0, step function is not defined

When a system is subjected to sudden disturbance step input can be used as a test signal

Common System Input Signalsa) Step Input-I

Common System Input Signalsa) Step Input- Examples

Examples

Angular rotation of the Shaft when it starts from rest

Change in fluid flow in a hydraulic system due to sudden opening of a valve

Voltage applied on an electrical network when it is suddenly connected to a power source

b) Ramp Input

i (t)

t = 0 time

K*t

Input

Signals is linear function of time

Increases with time

Mathematically i (t) = K*t for t > 0= 0 for t < 0

Example: Constant rate heat input in thermal system

Common System Input Signalsb) Ramp Input

imei (t)

Input k Sin t

i (t) = k Sin t

c) Sinusoidal Input

Mathematically

i (t) = k Sin t

System response in frequency domain

Frequency is varied over a range

Example: Voltage, Displacement, Force etc.,

Common System Input Signalsc) Sinusoidal Input

Order of the SystemOrder of the System

The responses of systems of a particular order are Strikingly similar for a given input

Order of the system: It is the order of the highest derivative in the ordinary linear differential equation with constant coefficients, which represents the physical system mathematically.

kxkydt

dyC .

Illustration: First order systemIllustration: First order system

x (t) i/p

y (t) o/p

C

K

Cy + ky = kx.

kxkydt

dyC .

Order: Order of the highest derivative = 1 First order system

Illustration: Second order systemIllustration: Second order system

m

x (t)

y (t)

K

C

kxkydt

cdy

dt

ydm

kxkycymy

2

2

.

.. .

Order: Order of the highest derivative = 2 Second order system

Response of First Order Mechanical

Systems to Step Input

Response of First Order Mechanical Systems to Step

Input