IEEE Electrification Magazine - December 2017 - 67

Robust stability
domains can be
viewed as subsets
of the much wider
stability domains in
the multidimensional
parametric space.

Robust
Stability Margin
(1/µ)

can be computed as 11.53 W (S. Sum-
surooah 2017).
Based on the technique of linear
fractional transformation, the nor-
malized uncertain parameters d Pin
may be extracted from the state
space system model and grouped in a
diagonal matrix D in feedback form
(S. Sumsurooah 2017). In this case
study, the uncertainty matrix D has
the size of 3 # 3, as the uncertain
parameter Pin appears three times in
the state space system model. n anal-
ysis identifies the smallest distur-
bance matrix that can destabilize the system. The
normalized uncertain parameter d Pin in the critical uncer-
tainty matrix D is equal to + 0.331.
For the single parametric uncertainty, the n tool has iden-
tified the largest line segment of coordinate size 1/n = 0.331
centered about the nominal point, within which the sys-
tem is guaranteed robustly stable, as illustrated in Figure 8.
The given line segment corresponds to the robust uncer-
tainty sets of 69.3 W, 11.5 W@ for the input power. Provided
that the system under investigation operates with an input
power that lies within the aforementioned robust stability
margin, the system is ensured to be robustly stable.
The second case considers that the system under study
is exposed to an additional parametric uncertainty, i.e., to
the input capacitance uncertainty C in, which is allowed to
vary within 95 nF ! 10%. For this case study, the resulting
n is equal to 4.03, and the robust stability margin is
| d Pin | = | d Cin | = 1/n = 0.248.
In parametric space, for a system subject to two para-
metric uncertainties, the n tool identifies the largest
square of coordinate size 1/n, which is equal to 0.248 for
this case study, centered about the nominal point within
which the system can be guaranteed to be robustly stable,
as illustrated in Figure 9. The square corresponds to the
robust uncertainty sets [9.5 W, 11.3 W] and [92.6 nF,

Stable
-1

-0.331

A
0

Unstable
+0.331 +1

97.4 nF ] for the input power and
capacitance, respectively. It implies
that if the analyzed system operates
within the given boundary, the sys-
tem is guaranteed robustly stable.
A last case is presented in this
section whereby in addition to
uncertainties in the input power and
the input filter capacitance, the input
filter inductance L in is also allowed
to vary within ! 10% of its nominal
value of 511.8 mH . For this three un-
certain parameters system, the peak
value of n is 4.974, and the corre-
sponding robust stability margin 1/n is 0.201 (Figure 10).
When considering a system subject to three parametric
uncertainties, n analysis identifies the largest cube with-
in which system robust stability is guaranteed, which
in this  case study is of coordinate size | d Pin | = | d Cin | = | d Lin |
| d Lin | = 1/n = 0.201 about the nominal point. It can be
shown that the hypercube corresponds to the input
power, capacitance and inductance lying within the
robust uncertainty sets of [9.7 W, 11.1 W], [93.1 nF, 97.0 nF ]
and [501.5 mH, 522.1 mH], respectively.
By extrapolating on the ideas presented in this section,
for a system subject to N parametric uncertainties, n anal-
ysis provides the largest hypercube of dimension Ncen-
tered about the nominal point and of coordinate size 1/n,
within which system robust stability can be guaranteed
(Sumsurooah 2017, Ferreres 1999).
This section has shown, through application and illus-
trations, the robust stability domains as subsets of the
wider stability domains in the multidimensional paramet-
ric space. The concept presented in this section has many
practical implications. It offers the design engineer a pa-
rametric space within which to maneuver and choose
optimum parameters while ensuring stability robustness.
The n-based robust stability domains can be extended
to more complex studies. The stability robustness of

δ Pin

1

Normalized
Power
δPin - [p.u]

Robustly
Stable

Unstable

0.5
0.248
0
-0.248
-0.5
-1
-1

A
Robustly
Stable

Stable
-0.5 -0.248 0 0.248 0.5
δ Cin

1

Figure 8. The single uncertain parameter system is robustly stable

Figure 9. The two uncertain parameters system is robustly stable

within the largest line segment of coordinate size (1/n = 0.331),
which is centered about the nominal point (d Pin = 0) and connects the
stability margin point A. p.u.: per unit.

within the largest square of coordinate size (1/n = 0.248), which is
centered about the nominal point (0, 0) and connects the system
stability line at the point A.

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

67

n



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