State Space and 2nd Order Systems

System of . S.S. 2n first order D.E.

y_1, y_2 (circled): state variables

State space representation

Fundamental operation (Mathematically) is two lower the order of D.E to half. If you are having 2n equation of 2^nd order this method produces 4n D.E. of 1^st order.

States of a system are the min number of variables to know at t=t_0 in order to describe the dynamics (response) at any later time.

State space representations

In physical variable form we just look at the energy storing elements

Phase variable form

Case 1: No derivative of the input

Case 2: Derivative of input is present

RHS → statically related (can be chosen) u : input, y : output

LHS: left hand side

RHS: right hand side

Burdan SS. formatına sokamayız

O yüzden bunu yapalım

y : output number of state variables n

ezberlemeye gerek yok, gerekirse sınavda verilir

Phase variable form

1st order system

III) Ramp Response

Accuracy is defined by `T'.

Ramp response =∫ unit response = impulse response

2nd order response

RLC in parallel

Spring mass Damper System

= zeta = damping ratio

no zeros but 2 poles are present

if <1 ⇒ complex conjugate poles (always oscillatory)

critically damped systems

(sorry) if =1 ⇒ repeated real valued poles

if >1 ⇒ seperate and real valued poles overdamped system

Homework (homogen solution)

Response due to external excitation (particular solution)

Specify the input

The response frequency for damped systems is ω_d

2nd order systems

If poles are C.C. → response is oscillatory and decaying

for ω_d=0 no oscillating frequency

at the same time

Step response calculation