Page 150 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

Matrices and Higher-Order Process Models 125

The Frequency Domain Version
As we have done for all our example processes, let's move on to the
frequency domain. Since we know that a single tank exhibits amplitude
attenuation and phase lag with increasing frequency, what do you
expect with this process? The tanks are in series. The input to the second
and third tanks is the output from the first and second tanks, respec-
tively, so, the amplitude attenuation and phase increase should be accu-
mulative. This can be demonstrated easily by using s = jm in Eq. (5-4).
R 3 1 1
G(s) = -r s + 1 -r s + 1 -r s + 1
3 2 1
G("m)= R3 1 1
1
-r jm + 1 -r jm + 1 -r jm + 1
3 2 1
R 3 1 1
=1?==~~~·-r==~==~-~==~==~
~(-r3m)2 + 1 eiB3 ~(-r2m)2 + 1 ei~ ~(-rtm)2 + 1 eiB,
(5-5)
8
~e-ifl,e-i 2e-i , 1 1
8
- ~(-r3m)2 + 1 ~(-r2m)2 + 1 ~(-rtm)2 + 1
_ ~e-i<B,~+B1 )
- ~(-r3m)2 + 1~(-r2m)2 + 1~{-rtm)2 + 1

1
9; = tan- (-r;m) i = 1,2,3
Equation (5-5) shows that the amplitude attenuations for each tank
multiply and the phase lags for each tank add. Figure 5-3 supports this.
Each tank is first order and contributes 90° of phase lag and Fig. 5-3
shows that the phase lag of the three-tank process approaches 270° at
high frequencies. Figure 5-3 also shows that the magnitudes at low
frequencies are the same.
Note that at high frequencies, the slopes of the magnitude plots for
tanks 1, 2, and 3 are -20 dB/ decade, -40 dB/ decade, and -60 dB/
decade, respectively.
The Matrix (State-Space) Version
Return to the time domain and rearrange Eq. (5-2) slightly
dX 1 =-~+~F.
dt t't t't 0

dX 2 _ ~ X 1 X 2
dt- R ~-~ (5-6)
1
dX 3 _ ~ x 2 X 3
dt-~ ~-~

Y=X 3