Next, we determine the phase of A(s) at the crossover frequency.
A(j200) = 390,000 (j200+51)/[(j200)
2
. (j200 + 2000)]
α = Arg[A(j200)] = tan
-1
(200/51)-180° -tan
-1
(200/2000)
α = 76° - 180° - 6° = -110°
Finally, the phase margin, PM, equals
PM = 180° + α= 70°
As long as PM is positive, the system is stable. However, for a well damped system, PM should be between 30°
and 45°. The phase margin of 70° given above indicated over-damped response.
Next, we discuss the design of control systems.
System Design and Compensation
The closed-loop control system can be stabilized by a digital filter, which is preprogrammed in the DMC-41x3
controller. The filter parameters can be selected by the user for the best compensation. The following discussion
presents an analytical design method.
The Analytical Method
The analytical design method is aimed at closing the loop at a crossover frequency, ω
c
, with a phase margin PM.
The system parameters are assumed known. The design procedure is best illustrated by a design example.
Consider a system with the following parameters:
K
t
= 0.2 Nm/A Torque constant
J = 2 * 10
-4
kg.m
2
System moment of inertia
R = 2 Ω Motor resistance
K
a
= 2 Amp/Volt Current amplifier gain
N = 1000 Counts/rev Encoder line density
The DAC of theDMC-41x3 outputs ±10V for a 16-bit command of ±32768 counts.
The design objective is to select the filter parameters in order to close a position loop with a crossover frequency
of ω
c
= 500 rad/s and a phase margin of 45 degrees.
The first step is to develop a mathematical model of the system, as discussed in the previous system.
Motor
M(s) = P/I = Kt/Js
2
= 1000/s
2
Amp
K
a
= 2 [Amp/V]
DAC
K
d
= 10/32768 = .0003
Encoder
K
f
= 4N/2π = 636
ZOH
H(s) = 2000/(s+2000)
Compensation Filter
G(s) = P + sD
Chapter 10 Theory of Operation ▫ 161 DMC-41x3 User Manual
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