3-3 2th order system analysis

- 3-3 2th order system analysis 一. Math model position following system is a typical 2th order system 。 block diag.

Whereξdamped ratio. wn natural oscillation frequency （also called no damped natural oscillation frequency）

Closed-loop characteristic equ： Its characteristic roots are the poles of closed-loop characteristic equ: 1.when 0< ξ <1，a couple of conjugate complex roots with negative part called under-damped。(fig .a)

2.whenξ=1，2 equal negative roots ，called critically damped。（fig.b） 3. whenξ>1， 2 different negative roots， called over damped。（ fig. c） 4. whenξ=0， a couple of conjugate imaginary roots ， called non-damped or 0 -damped。（fig.d） following calculate respectively。

二. 2th order system unit step response 1.over-damped。 whenξ>1， 2 different negative roots， closed-loop TF

where： Over damped 2th order system can be seen to have two first order system series with different time constants. When input signal is an unit step

output inverse Laplace transformation：

Starting speed is small, then gradually increases, different from first order system.it is difficult to solve ts, generally use computer to solve it.。 response curve： From the curve，when ， when ，，so when It can approximate a first order system. Response is very slowly,little adopted。

when (critically damped)，

2.under-damped When 0< ξ <1,closed-loop characteristic equ. wn natural oscillation frequency （also called no damped natural oscillation frequency）。

when input=step,output

curve：

under-damped 2th order system unit step response attenuates to its steady-state value according to exponent rule，attenuating speed depends on -ξwn， attenuating frequency depends on wd。 angle definition

from the curve, ξ ↑，σ%↓；ξ ↓ ， σ% ↑ 。When ξ＝0，0 damped response is： constant amplitude oscillation curve,， oscillation frequency is wn wn is called no dampedoscillation frequency 。

if ξ ↑↑→ ts ↑ ； if ξ ↓↓→ ts andtp ↓,but σ% ↑ →ts ↑ ↑ 。

1.rise time tr from definition：tr ---first reach steady-state value，set n=1。

When wn is definite,the smallerξ，the smaller tr； When ξ is definite,the bigger wn, the smaller tr. 2.peak time tp ① ②

Differentiate equ.①，set it=0： replace

Tp--first reach peak value，set n=1。 so When wn is definite,the smallerξ，the smaller tp； When ξ is definite,the bigger wn, the smaller tp. 3.over-shootσ%

So over-shoot is a function ofξ，independent of wn。

Relation curve betweenσ% andξ

ξ↑，σ% ↓ ，in general,takeξin 0.4-0.8, σ% in 25%-2.5%。 4.settling time ts from definition： difficult to solve ts，but get the curve between wnts and ξ：

ts not continuous diag. Small ξ change can produce ts change violently.

whenξ=0.68（5% error band）orξ=0.76（2% error band ），ts is the shortest,the system is often designed under damped 。 un-continuous curve is because the small ξ variant can cause ts variant。 when approximately calculate，usually use embracing curve to calculate ts。

take both sides logarithms, get That is： while design system, the ξ is usually selected by request, but settling time by wn.

5.oscillation No.N definition:in ts,half of the No.response curve cross to steady state value. Td:damped oscillation period。

Exam.1:given a unit feedback system.its open-loop TF assume input r(t)=1(t) ,when KA=200,solve transient response performance。When KA changes to 1500 or KA =13.5,how does the transient response performance?

Closed-loop TF:

from this,the bigger KA, the smaller ξ, the bigger wn, the smaller tp, the bigger б%,whereas ts seldom changes。 System works with over-damped, tp, б% ,N do not exist, but ts.can be approximately calculated by a first order system with a big time constant T,that is: .

Ts is much longer than the two KA above.although response has no over-shoot,response process is slow,the curve as follows：

KA ↑ ， tp ↓， tr ↓，improve speed,but б% ↑. For improving transient performance ,the proportion-differential control or speed feedback control can be adopted ,namely introduce a correction.。

Homework: Chapter 3-6,3-7

3-3 2th order system analysis