IEEE Electrification Magazine - December 2017 - 69

Vin (V)

Torque
Uncertainty
T - (N . m)

Vo (V)

Maximum

Minimum
-1

0

18.5
0.450

Normalized
Torque
δT - [p .u]

0.455

0.460

0.455

0.460

0.455
Time (s)

0.460

5.10
5.05
0.450
3.3

Io (A)

Robust
Stability Margin

Robust
Stability Range

Critical

20.5

2.3
0.450

1/µ = 0.42 +1

uncertainty in torque.

Figure 13. Case 1.1: The experimental results for the system with
load uncertainty. The system is at the boundary of stability, with
R = 1.63 X from t = 0.453 s to 0.460 s.

supplied from an ideal voltage source through a line
cable, was set to regulate the output voltage to a resistive
load. Sumsurooah et al. present the detailed modeling
and analysis of the system.
Seven case studies were performed using the n anal-
ysis tool, as given in Table 1. The buck converter sys-
tem has a conversion rate equal to its duty cycle, which
has nonlinear dependence on the output resistive load.
The nonlinearities in the duty cycle must be accounted
for in the modeling of the analyzed system (Sumsurooah
et al. 2015). Further, the duty cycle, which has been
obtained as an irrational term in the system model, must
be approximated by polynomial expansion to suit n
analysis. Case studies 1.1-1.2 and 2.1-2.2 employed the
first and zeroth order of approximation for the duty
cycle, respectively. Since case studies 3.1-3.3 investigated
the impact of model uncertainties on stability robust-
ness, they treated the duty cycle as an uncertain parame-
ter about its nominal value. It is also added that the case
studies 1.1, 1.2, 2.1, and 2.2 employed accurately mea-
sured nominal values for the system parameters, while
case studies 3.1-3.3 were based on available rough esti-
mates of the nominal values of the system parameters.

Case study 1.1 investigated the robust stability of the
buck converter system when it is exposed to a large vari-
ation in its resistive load, i.e., 2.5 X ! 50%. The n analysis
determined that the analyzed system becomes unstable
when the output power is increased to 16 W, based on a
robust stability margin 1 /n of 0.696, as shown in Table 1.
The robust stability margin 1 /n of unity would mean
that the system is guaranteed stable on the entire uncer-
tainty set and for the output power up to 20.8 W. The n
predicted critical output power of 16 W was verified in
experiment. In the laboratory, the electronic resistive
load was decreased in small steps from a peak value
of 2 X. At time 0.453 s, when the resistive load was
decreased to 1.63 X (i.e., 16.0 W ), the system reached
boundary stability, as shown by the sustained oscilla-
tions in input voltage Vin, output voltage Vo and output
current I o in Figure 13.
The line resistance is not known accurately at the
design stage but is dependent on the final assembly of the
system components. Case study 1.2 investigates the effect
of the line resistance on robust stability margin, when
both the load and the line resistance are uncertain. The
line resistance is assumed to vary within ! 50% of its

Figure 12. The robust stability margin of the EMA system under

TABLE 1. The buck converter: robust stability studies results.
Case Study

Uncertain Parameters

Robust Stability Margin 1/n

Critical Load Power (W)

1.1

Load (with fixed line resistance)

0.696

16.0

1.2

Load and line resistance

0.803

17.3

2.1

Load and temperature

0.505

13.9

2.2

Load (with fixed temperature)

0.745

16.6

3.1

Load (with no model uncertainty)

0.614

15.0

3.2

Load and model uncertainties (wide range)

0.210

11.6

3.3

Load and model uncertainties (narrow range)

0.288

12.2

IEEE Elec trific ation Magazine / D EC EM BE R 2 0 1 7

69



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