**2. Preliminaries**

Before we proceed, we give some essential definitions that will act as prerequisites for the subsequent results.

**Definition 1** ([23])**.** *The set of block diagonal matrices X is defined as*

*X* := - *diag*(*αiIi*; *<sup>β</sup>jIj*; *Ct*) : *<sup>α</sup><sup>i</sup>* <sup>∈</sup> <sup>R</sup>, *<sup>β</sup><sup>j</sup>* <sup>∈</sup> <sup>C</sup>, *Ct* <sup>∈</sup> <sup>C</sup>*mt*×*mt* ,

*where αiIi, βjIj, and Ct denote the number of repeated real scalar blocks with different sizes for all i* = 1, ··· ,*r, the number of complex scalar blocks with different sizes for all j* = 1, ··· , *c and the number of full complex blocks with different sizes for all t* = 1, ··· , *k, respectively.*

**Definition 2** ([1])**.** *For a given n-dimensional complex valued matrix <sup>M</sup>* <sup>∈</sup> <sup>C</sup>*n*×*n*, *the structured singular value with respect to X is defined as*

$$\mu\_X(M) := \begin{cases} 0, & \text{if } \det(I - M\Delta) \neq 0, \,\forall \Delta \in X \\ \frac{1}{\min\{\|\|\Delta\|\_2 \colon \Delta \in X, \det(I - M\Delta) = 0\}}, & \text{else}. \end{cases} \tag{1}$$

*The matrix valued function* Δ *is an uncertainty that occurs in the linear feedback system.*

**Definition 3** ([23])**.** *The sets DX and GX*

$$D\_X := \left\{ \operatorname{diag} \left( P\_{1'}, \dots, P\_r; P\_{1'}, \dots, P\_c; P\_{1'}, \dots, P\_t \right) \right\}.$$

*and*

$$G\_X := \left\{ \operatorname{diag} \left( H\_{1'}, \dots, H\_{\mathbf{r}}; \mathbf{O}\_{1'}, \dots, \mathbf{O}\_{\mathbf{c}'}; \mathbf{O}\_{1'}, \dots, \mathbf{O}\_{\mathbf{t}} \right) \right\}$$

*contains positive definite matrices Pi for all i* = 1, ··· ,*r, Pj for all j* = 1, ··· , *c and Pt for all t* = 1, ··· , *k and Hermitian matrices Hi for all i* = 1, ··· ,*r, and repeated null complex scalar blocks Oj for all j* = 1, ··· , *c and number of null full complex blocks Ot for all t* = 1, ··· , *k, respectively.*

**Definition 4** ([23])**.** *For a given <sup>n</sup> dimensional complex valued square matrix <sup>M</sup>* <sup>∈</sup> <sup>C</sup>*n*×*<sup>n</sup> and <sup>β</sup>* <sup>∈</sup> <sup>R</sup>*, the matrix-value function fβ*(*D*, *<sup>G</sup>*) *is defined as*

$$f\_{\beta}(D,G) := M^HDM + i(GM - M^HG) - \beta^2 D,$$

*where matrices D*, *G belongs to DX and GX, receptively.*

**Definition 5** ([23])**.** *The upper bound of μX*(*M*) *is denoted by νX*(*M*) *and is defined as*

$$\nu\_X(M) := \inf\_{\beta > 0} \left\{ \beta : \exists \ D \in D\_X \text{ and } \ G \in G\_X \text{ s.t. } f\_{\beta}(D, G) < 0 \right\}.$$

Let *<sup>M</sup>* <sup>∈</sup> <sup>C</sup>*m*×*<sup>n</sup>* be a given matrix and (*mr*, *mc*, *mC*) represent an *<sup>m</sup>*-tuples of positive integers and let

$$\mathcal{K} = (k\_1, \dots, k\_{m\_{r'}} k\_{m\_{r+1'}}, \dots, k\_{m\_r + m\_{c'}} k\_{m\_r + m\_{c+1'}}, \dots, k\_{m\_C})\_t \tag{2}$$

where ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *ki* = *n*.

**Definition 6** ([23])**.** *The set of block diagonal matrices is defined as*

$$X\_K := \{ \Delta = \text{diag}\left(\delta\_1 I\_1, \dots, \delta\_r I\_r, \delta\_1 I\_1, \dots, \delta\_\varepsilon I\_\varepsilon, \Delta\_1, \dots, \Delta\_\ell\right) \}.$$

*In Definition 6, <sup>δ</sup><sup>i</sup>* <sup>∈</sup> <sup>R</sup> <sup>∀</sup>*<sup>i</sup>* <sup>=</sup> 1, ··· ,*r*, *<sup>δ</sup><sup>j</sup>* <sup>∈</sup> <sup>C</sup> <sup>∀</sup>*<sup>j</sup>* <sup>=</sup> 1, ··· , *<sup>c</sup>*, <sup>Δ</sup>*<sup>t</sup>* <sup>∈</sup> <sup>C</sup>*t*×*<sup>t</sup>* <sup>∀</sup>*<sup>t</sup>* <sup>=</sup> 1, ··· , *<sup>k</sup>*. *The set XK is pure real if δ<sup>j</sup>* = 0 *and pure complex if δ<sup>i</sup>* = 0*, otherwise it is with mixed real and complex block perturbation. For* <sup>Δ</sup>*<sup>t</sup>* <sup>∈</sup> <sup>C</sup>*t*×*<sup>t</sup> , the set XK turns out to be a set of full complex blocks perturbation.*

**Definition 7** ([1])**.** *For given matrix <sup>M</sup>* <sup>∈</sup> <sup>C</sup>*m*×*<sup>n</sup> and XK, the structured singular value; denoted by μXK* (*M*) *and is defined as*

$$\mu\_{X\_K}(M) := \begin{cases} 0, & \text{if } \det(I - M\Delta) \neq 0, \Delta \in X\_K\\ \frac{1}{\min\limits\_{\Delta \in X\_K} \{||\Delta||\_2 \colon \det(I - M\Delta) = 0\}}, & \text{else}, \end{cases} \tag{3}$$

*where* Δ<sup>2</sup> *denotes the largest singular value of* Δ*.*

**Definition 8** ([23])**.** *The set YK of block diagonal structure is defined as*

$$\mathcal{Y}\_K: \left\{ \Delta\_{\mathcal{V}} = \text{diag}\left(\delta\_1 I\_1, \dots, \delta\_r I\_r, \delta\_1 I\_1, \dots, \delta\_c I\_c; \delta\_1 I\_1, \dots, \delta\_c I\_c; \Delta\_{\mathcal{V}}, \Delta\_{\mathcal{V}}, \dots, \Delta\_{\mathcal{V}}\right) \right\}.$$

$$\text{In Definition 8, } \delta\_i \in \mathbb{R} \,\,\forall i = 1, \dots, r, \,\,\delta\_j \in \mathbb{C} \,\,\forall j = 1, \dots, c, \,\,\Delta\_l \in \mathbb{C}^{t \times t} \,\,\forall t = 1, \dots, k.$$

**Definition 9** ([23])**.** *The secondary set ZK*<sup>ˆ</sup> *of block diagonal structure is defined as*

$$Z\_{\hat{\mathbb{K}}} \colon \{ \Delta\_{\nu} = \operatorname{diag} \{ \delta\_1 I\_1, \dots, \delta\_r I\_r, \delta\_1 I\_1, \dots, \delta\_\varepsilon I\_\varepsilon; \Delta\_{1\prime}, \dots, \Delta\_{\ell} \} \}.$$

$$\text{In Definition 9, } \delta\_i \in \mathbb{R} \text{ } \forall i = 1, \dots, r, \text{ } \delta\_j \in \mathbb{C} \text{ } \forall j = 1, \dots, c, \text{ } \Delta\_l \in \mathbb{C}^{t \times t} \text{ } \forall t = 1, \dots, k.$$

**Definition 10** ([23])**.** *The set ZK is restricted to the unit ball and is defined as*

$$BZ\_{\mathcal{Z}} = \left\{ \Delta\_f \in Z\_{\mathcal{K}} : \left\| \Delta\_f \right\|\_2 \le 1 \right\}.$$

**Definition 11** ([23])**.** *The block structure WK*,*K*<sup>ˆ</sup> *is defined as*

$$\mathcal{W}\_{\mathbf{K},\hat{\mathbb{K}}} = \left\{ \boldsymbol{\Delta} = \operatorname{diag} (\boldsymbol{\Delta}\_{f'}, \boldsymbol{\Delta}\_{\boldsymbol{\nu}}) \right\}\_{\mathsf{f}'} \tag{4}$$

*or*

$$
\Delta = \begin{pmatrix} \Delta\_f & 0 \\ \hline 0 & \Delta\_V \end{pmatrix}. \tag{5}
$$

**Definition 12** ([23])**.** *For given matrix <sup>M</sup>* <sup>∈</sup> <sup>C</sup>*m*×*<sup>n</sup> and ZK*<sup>ˆ</sup> *, the skewed structured singular value is denoted by μZK*ˆ(*M*) *and is defined as*

$$\mu\_{Z\_{\tilde{\mathcal{K}}}}(M) := \begin{cases} 0, & \text{if } \det(I - M\Delta) \neq 0, \Delta \in \mathcal{W}\_{\mathcal{K}, \tilde{\mathcal{K}}}\\ \frac{1}{\min\_{l \in \mathcal{W}\_{\mathcal{K}, \tilde{\mathcal{K}}}} \{||\Delta\_{\nu}||\_{2} : \det(I - M\Delta) = 0\}}, & \text{else}. \end{cases} \tag{6}$$

#### **3. The Main Results**

In the section, we present some new results on the computation of structured singular values and skewed structured singular value.
