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ELEC 3200 - System Modeling, Analysis and Control

This cheat sheet summarizes key concepts in System Modeling, Analysis and Control.

Transfer Function

For an LTI system, we can take the Laplace transform with zero initial conditions

sX(s)=AX(s)+bU(s)Y(s)=cX(s)+dU(s)

Using linear algebra, we can express X(s) as:

X(s)=(sIA)1bU(s)

The transfer function of an LTI system is defined as the ratio of the Laplace transform of the output to that of the input when the initial conditions are zero:

G(s)=Y(s)U(s)=c(sIA)1b+d

Different state space representations can be derived for a particular system due to the infinite number of linear transformations, but the transfer function remains unique.

Routh Stability Criterion

Given the characteristic polynomial:

c(s)=1s5+2s4+3s3+4s2+5s1+6s0
s5135
s4246
s312|1324|=12(1×42×3)=31×42=112|1526|=12(1×62×5)=51×62=2
s242×21=06
s12
s06

Kharitonov Theorem

Let

P={a(s)=a0sn+a1sn1++an1s+an | ai[ai,ai]}

All members of P are stable if and only if the following four polynomials are stable

a1(s)=a0sn+a1sn1+a2sn2+a3sn3+a4sn4+a2(s)=a0sn+a1sn1+a2sn2+a3sn3+a4sn4+a3(s)=a0sn+a1sn1+a2sn2+a3sn3+a4sn4+a4(s)=a0sn+a1sn1+a2sn2+a3sn3+a4sn4+

Prototype 2nd-Order System

The transfer function for a second-order system is given by:

H(s)=kωn2s2+2ζωns+ωn2=k(σ2+ωd2)(s+σ)2+ωd2

Time Constants

The following approximations hold:

{tr1.8ωntp=πωn1ζ2=πωdts3ζωn=3σ

Percent Overshoot

The percent overshoot (PO) is defined as:

PO=(y(tp)y()1)×100%=exp(ζπ1ζ2)

Final Value Theorem

The final value theorem states that:

FVT=H(0)=k

Bode's Sensitivity

In the nominal situation, we have the motor with DC gain = A, and the overall transfer function, either open- or closed-loop, has some other DC gain (call it T).

Perturbations

Let:

A^=A+δAT^=T+δT

Then, the sensitivity can be approximated as:

δTdTdAδA

Sensitivity Function

The sensitivity S is given by:

S=δTTδAA=δTAδATdTdAAT

Root Locus

The standard form of the root locus is:

1+KL(s)=0

Transformation to Standard Form

Change to standard form:

a(s)+Kb(s)=0

This can be expressed as:

1+Kb(s)a(s)=0

Root Locus Rules

Rule An branches
Rule Bstarts at s = x, x, ...
Rule Cends at s = x, x, ...
Rule DReal locus: (-xx,-xx) U (-xx,-xx)
Rule En - m =xx, l = 0,1,...,xx-1
Asymptotes = xxx°, xxx°
Rule Fa(s)+Kb(s)=0
Routh Table => K∈(xx,xx)
j·w?, w=?