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For the given example, the crossover frequency was computed numerically resulting in 200 rad/s.
Next, we determine the phase of A(s) at the crossover frequency.
Finally, the phase margin, PM, equals
As long as PM is positive, the system is stable. However, for a well damped system, PM should be
between 30 degrees and 45 degrees. The phase margin of 70 degrees given above indicated
overdamped 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
CDS-3310 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
J = 2.10 -4
R = 2
K a = 2
N = 1000
The DAC of the CDS-3310 outputs +/-10V for a 14-bit command of +/-8192 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
Amp
DAC
74 • Chapter 10 Theory of Operation
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°
PM = 180° + α = 70°
Nm/A
kg.m 2
Ω
Amp/Volt
Counts/rev
M(s) = P/I = K t /Js 2 = 1000/s 2
K a = 2
[Amp/V]
K d = 10/32768 = .0003
Torque constant
System moment of inertia
Motor resistance
Current amplifier gain
Encoder line density
CDS-3310
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