Galil user manual Theory of Operation DMC-2X00

The next step is to combine all the system elements, with the exception of G(s), into one function, L(s). L(s) = M(s) Ka Kd Kf H(s) =3.17∗106/[s2(s+2000)]

Then the open loop transfer function, A(s), is A(s) = L(s) G(s)

Now, determine the magnitude and phase of L(s) at the frequency ωc = 500.

L(j500) = 3.17∗106/[(j500)2 (j500+2000)] This function has a magnitude of

L(j500) = 0.00625 and a phase

Arg[L(j500)] = -180° - tan-1(500/2000) = -194°

G(s) is selected so that A(s) has a crossover frequency of 500 rad/s and a phase margin of 45 degrees. This requires that

A(j500) = 1

Arg [A(j500)] = -135° However, since

A(s) = L(s) G(s)

then it follows that G(s) must have magnitude of G(j500) = A(j500)/L(j500) = 160

and a phase

arg [G(j500)] = arg [A(j500)] - arg [L(j500)] = -135° + 194° = 59° In other words, we need to select a filter function G(s) of the form

G(s) = P + sD

so that at the frequency ωc =500, the function would have a magnitude of 160 and a phase lead of 59 degrees.

These requirements may be expressed as: G(j500) = P + (j500D) = 160

and

arg [G(j500)] = tan-1[500D/P] = 59° The solution of these equations leads to:

P= 160cos 59° = 82.4 500D = 160sin 59° = 137

Therefore,

D = 0.274

and

G = 82.4 + 0.2744s

144 • Chapter 10 Theory of Operation DMC-2X00