*3.2. Duty Ratio Regulator*

In fact, according to the traditional PI-SVPWM method, the locus of applied vectors is generally round in shape within the hexagon range illustrated in Figure 2a, where the null vectors are commonly adopted for the better steady-state performance. Therefore, apart from two active vectors determined in Section 3.1, two null vectors *V*<sup>0</sup> and *V*<sup>7</sup> (counted as *u*0) are inserted during each sampling period as well. This subsection introduces a formal mathematical methodology and targets the calculation of the duration for *u*0, *u*<sup>1</sup> and *u*2.

Indeed, the value of cost function (7) expresses the square of current error, termed as ε2. Since the sampling period is extremely short in general, this error could be mentioned as a linear variable relevant to the duration of the corresponding vector [27]. Hence, considering the case where *u*0, *u*<sup>1</sup> and *u*<sup>2</sup> are applied and their durations are *d*0, *d*<sup>1</sup> and *d*2, the value of cost function can be approximated as:

$$\mathcal{g} = \varepsilon^2 = \varepsilon (u\_0)^2 d\_0^2 + \varepsilon (u\_1)^2 d\_1^2 + \varepsilon (u\_2)^2 d\_2^2 \tag{8}$$

where ε(*uk*) <sup>2</sup> (*k*=0, 1 or 2) is equal to the corresponding cost function value generated by *uk*.

Note that *d*0, *d*<sup>1</sup> and *d*<sup>2</sup> subject to the following constraints:

$$\begin{cases} \, \, d\_0 + d\_1 + d\_2 = 1 \\ \, \, \, 0 \le d\_0 \le 1 \\ \, \, \, 0 \le d\_1 \le 1 \\ \, \, \, 0 \le d\_2 \le 1 \end{cases} \tag{9}$$

In this manner, the objective is to determine an optimal set of duty ratios with respect to a given (*u*0, *u*1, *u*2), thereby minimizing (8) accordingly. This optimization along with constraints is usually called as the conditional extremum problem, which could be effectively solved by employing the well-noted Lagrange multipliers method in conjunction with the Hessian matrix. By these means, the optimal set of duty ratios, termed as (*d*<sup>0</sup> \* , *d*<sup>1</sup> \* , *d*<sup>2</sup> \* ), can be calculated by the utilization of (10) and the Hessian matrix *G* is provided in (11). Particularly, *G* is a positive definite matrix with respect to the calculated (*d*<sup>0</sup> \* , *d*<sup>1</sup> \* , *d*<sup>2</sup> \* ); namely, this set of duty ratios corresponds to the minimum point of (8). In fact, this problem can be solved using the convex optimization theory and the same result can be deduced as well.

⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ *d*∗ 0 *d*∗ 1 *d*∗ 2 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ε(*u*1) 2 ε(*u*2) 2 ε(*u*0) 2 ε(*u*1) 2 +ε(*u*1) 2 ε(*u*2) 2 +ε(*u*2) 2 ε(*u*0) 2 ε(*u*0) 2 ε(*u*2) 2 ε(*u*0) 2 ε(*u*1) 2 +ε(*u*1) 2 ε(*u*2) 2 +ε(*u*2) 2 ε(*u*0) 2 ε(*u*0) 2 ε(*u*1) 2 ε(*u*0) 2 ε(*u*1) 2 +ε(*u*1) 2 ε(*u*2) 2 +ε(*u*2) 2 ε(*u*0) 2 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (10) *G* ! *d*∗ <sup>0</sup>, *d*<sup>∗</sup> <sup>1</sup>, *d*<sup>∗</sup> 2 " = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ∂<sup>2</sup> *g* ∂*d*<sup>2</sup> 0 ∂<sup>2</sup> *g* ∂*d*0∂*d*<sup>1</sup> ∂<sup>2</sup> *g* ∂*d*0∂*d*<sup>2</sup> ∂<sup>2</sup> *g* ∂*d*1∂*d*<sup>0</sup> ∂<sup>2</sup> *g* ∂*d*<sup>2</sup> 1 ∂<sup>2</sup> *g* ∂*d*1∂*d*<sup>2</sup> ∂<sup>2</sup> *g* ∂*d*2∂*d*<sup>0</sup> ∂<sup>2</sup> *g* ∂*d*2∂*d*<sup>1</sup> ∂<sup>2</sup> *g* ∂*d*<sup>2</sup> 2 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ - - - - - - - - - - - - *d*<sup>0</sup> = *d*<sup>∗</sup> 0 *d*<sup>1</sup> = *d*<sup>∗</sup> 1 *d*<sup>2</sup> = *d*<sup>∗</sup> 2 = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ 2ε(*u*0) <sup>2</sup> 0 0 0 2ε(*u*1) <sup>2</sup> 0 0 02ε(*u*2) 2 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ (11)
