Behaviours Flashcards
whats difference between steady state gain and steady state output?
s-s gain: –> K=lim s–>0 G(s)
G(0) –> asymptotic(s-s) ratio of the sytem o/p to i/p. G(s)=Y(s)/U(s). Assumes i/p is constant. It assumes that the o/p is convergent and thus dont need to check poles- we assume u(t) is such that limits exists and therefore works. therefore system gain always meaningfully defined. Could be infinite if G(s) has poles on origin aka p=0.
ss- output: –> asymptotic(basically at s-s) o/p of system. in practice thats what we are interested in. A=i/p magnitude.
We use FVT:
lim t–> infinity y(t) = lim s–>0 sY(s).
note: only applied if signal convergent, so MUST check first if poles are stable.
laplace tranforms are defined for a…
signal
what is a TF?
it is the system, G(s).
it gives the relationship btw the i/p and o/p of the system
U(s)—> G(s) —-> Y(s)
Y(s)=G(s)U(s)
what do the o/p Y(s) depend on
Y(s) contains poles from both i/p, u and the system,G and therfore the dynamics of the o/p depends on G(s) and U(s).
G(s) is the natural dynamics of the system(free response)
and
U(s) is the behaviour that enters the sytem(forced response)
what do we mean when we talk about instability?
When the signal Y(s) include RHP poles and therfore the signal Y(s) is divergent and goes to infinity.
As Y=GU…
… it can be unstable if i/p, U OR the inherent sytem dynamics, G OR both are divergent.
where does the instability originate from in an open and closed loop system?
but think about it. in open loop system you choose the input, why would put a divergent signal into sytem. Thats MAD! hence instability originates in the TF.
In a closed sytem, u is a target/ set point. instabilty willl typically arise from Gc(s) [closed loop TF] having RHP poles.
loop i/p is again assumed to be convergent so instability originates from TF.
In Summary, so even though Y depends on G and U, most of time instability originates from G(s). so if Y has RHP poles its cuz they originated from the RHP poles in G.
thats why convergence of Y(s) can be assessed solely by analysing G(s).
what is steady state gain?
what is steady-state output?
when U(s)=A/s
Only if G(s) has not integrators[when G(0)=infinity
Y(s) = G(s)U(s) = G(s)A/s
We use FVT:
lim t–> infinity y(t) = lim s–>0 sY(s).
sG(s)A/s =
G(0)A (as s in num and den cancel out)
The only converging signal with a non-zero asymptotic value is what?
when it is a contains a constant, aka a step function as L[u] will therefore give us A/s.
Therefore using FVT lim s–>0 (sF(s))
asymptotic value is A if all others converge to 0. The s at num and den cancel out that’s why so not timesing by 0/s anymore
if a transform does not contain a single ‘s’ in the denominator, what will Final Value be?
zero or infinite(except pure sinusoids)
FVT only applies if signal=converging.
what’s condition of FVT
FVT only applies if signal=converging.
if applied to Laplace transform of diverging signal, answer is meaningless
FVT w/ Matlab
y=impulse(tf([1 6],[1 4 3]),1000);
y(end)
Analysing behaviours, what are the things to look at. (4) SSSO
- Stability
- Settling time/ Speed of Response
- Shape of Response–> Smooth/Oscillitary
- Offset –> Do we reach the Target? How far off?
If closed loop, we need the closed-loop gain/offset Gc(s)
How to calculate offset?
1/ 1 + G(0)M(0)
How to calculate s-s offset with a simple feedback loop?
Since e = 1/ 1 + GM
lim t–> ∞ e(t) = lim s–>0 ( s / 1 + G(0)M(0) ) * R(s)
note: only meaning full in step input R(s)=A/s
Therefore the s at num and den in e (error) cancel out
When considering offset and s-s, what must you ask yourself? Its only meaningful when…
Offset and s-s are only meaningful for STEP signal. Therefore we assume R(s)=A/s (as r=A)
Must check for stability first as well. ‘ iff closed-loop is stable’ If unstable, no offset as its diverging–> ∞
What is the offset of a simple closed loop system?
e= ( 1 / 1 + GM ) * r
lim t–> ∞ e(t) = lim s–>0 ( s / 1 + GM) * R(s)
With a stable closed loop system. G has an integrator. M=2 and r(s)= 3/s. What is the s-s offset?
lim t–> ∞ e(t) = lim s–>0 ( s / 1 + GM) * R(s)
G(0)–>∞
lim s–>0 3/∞ = 0 offset iff stable
