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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