Appendix 1
Here, it will be shown that (48) can be formulated as (50) for each possible value of \({ L}_{{ c}}{ L}_{ c} \in \left\{ {1,2,3} \right\} \).
$$\begin{aligned} 0= & {} -( {{ y}_{\mathrm{des}} -\gamma })\left( {\sum \limits _{{ j}=1}^{{ L}_{ c} } {\overline{ f}^{ j}} +\sum \limits _{{ j}={ L}_{ c} +1}^{ N} {\underline{{ f}^{ j}}} } \right) \nonumber \\&+\left( {\sum \limits _{{ j}=1}^{{ L}_{ c} } {\overline{{ f}^{ j}} \underline{{ c}}_{ j} +\sum \limits _{{ j}={ L}_{ c} +1}^{ N} {\underline{{ f}^{ j}}\underline{{ c}}_{ j}} } }\right) \end{aligned}$$
(48)
where the total firing set is:
$$\begin{aligned}&{\underline{f}^{1}}=[ {\underline{\mu }{\tilde{{ A}}}_{1}^{1} \underline{\mu }\tilde{{ B}}_1^{1} } ] \quad \overline{f }^1=[ {{\overline{ \mu }} {\tilde{{ A}}}_{1}^{1} {\overline{ \mu }} \tilde{{ B}}_{1}^{1} } ]\nonumber \\&{{\underline{f}^{2}}=[ {\underline{\mu }{\tilde{ A}}_{1}^{1} \underline{\mu }\tilde{{ B}}_{1}^{2} } ]} \quad \overline{f }^2=[ {{\overline{ \mu }} {\tilde{ A}}_{1}^{1} {\overline{ \mu }} \tilde{{ B}}_{1}^{2} } ]\nonumber \\&{{\underline{f}^{3}}=[ {\underline{{ \mu }}{{\tilde{ A}}_{1}^{2}} \underline{\mu }\tilde{{ B}}_{1}^{1} } ]} \quad \overline{f }^3=[ {{\overline{ \mu }} {\tilde{ A}}_{1}^{2} {\overline{ \mu }} \tilde{{ B}}_{1}^{1} } ] \nonumber \\&{{\underline{f}^{4}}=[ {\underline{{ \mu }}{\tilde{ A}}_{1}^{2} \underline{{ \mu }}\tilde{{ B}}_{1}^{2} } ]} \quad \overline{f }^4=[ {{\overline{ \mu }} {\tilde{ A}}_{1}^{2} {\overline{ \mu }} \tilde{{ B}}_{1}^{2} } ] \end{aligned}$$
(49)
$$\begin{aligned} \psi _{1}^{{ L}} (.){\overline{\mu }} \tilde{{ B}}_{1}^{1} +\psi _{2}^{{ L}} (.){\overline{\mu }} \tilde{{ B}}_{1}^{2} +\psi _{3}^{{ L}} (.)\underline{{ \mu }} \tilde{{ B}}_{1}^{2} +\psi _{4}^{{ L}} (.)\underline{{ \mu }}\tilde{{ B}}_{1}^{1} =0\nonumber \\ \end{aligned}$$
(50)
Here \(\psi _{{ i}}^{{ L}} (.),{{ i}}=1,...,4\) is a function of \(( {{{ y}}_{\mathrm{des}} ,\gamma ,\underline{{ c}}_{{ j}} })\) and a possible combination of the following parameters \({\overline{\mu }} {\tilde{{ A}}}_{1}^{1} ,{\overline{\mu }} \tilde{{ A}}_{1}^{2} ,\underline{{ \mu }}{\tilde{{ A}}}_{1}^{2}\, \text{ and }\, \underline{{ \mu }}{\tilde{{ A}}}_{1}^{1} \).
(i) For \({ L}_{{ c}} =1\), (48) can be formulated as:
$$\begin{aligned} 0= & {} -( {{{ y}}_{\mathrm{des}} -\gamma })\left( {\sum \limits _{{ j}=1}^{{{ L}}_{{ c}} =1} {\overline{{ f}}^{{ j}}} +\sum \limits _{{{ j}}={{ L}}_{{ c}} +1}^{{{ N}}=4} {\underline{{{ f}}^{{ j}}}}}\right) \nonumber \\&+\left( {\sum \limits _{{{ j}}=1}^{{{ L}}_{{ c}} =1} {\overline{{ f}}^{{ j}}} \underline{{ c}}_{{ j}} +\sum \limits _{{{ j}}={{ L}}_{{ c}} +1}^{{{ N}}=4} \underline{{{ f}}^{{ j}}}{\underline{{ c}}_{{ j}} } }\right) \end{aligned}$$
(51)
$$\begin{aligned} 0= & {} -( {{{ y}}_{\mathrm{des}} -\gamma })( {\overline{{ f}^1} +{\underline{{ f}}^{2}}+{\underline{{ f}}^{3}}+\underline{{{ f}}^{4}}})\nonumber \\&+( {\underline{{ c}}_{1} \overline{{{ f}}^1} +\underline{{ c}}_{2} {\underline{{ f}}^{2}} +\underline{{ c}}_{3} {\underline{{ f}}^{3}}+ \underline{{ c}}_{4} \underline{{{ f}}^{4}}}) \end{aligned}$$
(52)
$$\begin{aligned} 0= & {} \overline{{{ f}}^1} ( {1}+\underline{{ c}}_1 ( {-{{ y}}_{\mathrm{des}} +\gamma }))+{\underline{{ f}}^2}({1} +\underline{{ c}}_2 ( {-{{ y}}_{\mathrm{des}} +\gamma }))+ \cdots \nonumber \\&\cdots +{\underline{{ f}}^3}( {1} +\underline{{ c}}_3 ( {-{{ y}}_{\mathrm{des}} +\gamma }))+\underline{{{ f}}^4}( {1} +\underline{{ c}}_4 ( {-{{ y}}_{\mathrm{des}} +\gamma }))\nonumber \\ \end{aligned}$$
(53)
Replacing total interval firing set (49) into (54), we can obtain:
$$\begin{aligned} 0= & {} ( {{\overline{ u}} {\tilde{ A}}_{1}^{1} {\overline{ u}} \tilde{{ B}}_{1}^{1} })( {1}+\underline{{ c}}_1 ( {-{{ y}}_{\mathrm{des}} +\gamma }))\nonumber \\&+( {\underline{\mu }{\tilde{{ A}}}_{1}^{1} \underline{\mu }\tilde{{ B}}_{1}^{2} })( {1} +\underline{{ c}}_2 ( {-{{ y}}_{\mathrm{des}} +\gamma }))+\cdots \nonumber \\&\cdots +( {\underline{\mu }{\tilde{{ A}}}_{1}^{2} \underline{\mu }\tilde{{ B}}_{1}^{1} })( {1} +\underline{{ c}}_3 ( {-{{ y}}_{\mathrm{des}} +\gamma }))\nonumber \\&+( {\underline{\mu }{\tilde{ A}}_{1}^{2} \underline{\mu }\tilde{{ B}}_{1}^{2} })({1} +\underline{{ c}}_4 ( {-{{ y}}_{\mathrm{des}} +\gamma })) \end{aligned}$$
(54)
Reformulating (55)
$$\begin{aligned} 0= & {} ( {{\overline{\mu }} {\tilde{{ A}}}_1^1 +\underline{{ c}}_1 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^1 -\underline{{ c}}_1 {\overline{\mu }} {\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}} })\overline{\mu }\tilde{{ B}}_1^1 +( 0){\overline{\mu }} \tilde{{ B}}_1^2 +\cdots \nonumber \\&\cdots +( \underline{\mu }{\tilde{{ A}}}_{1}^{1} +\underline{\mu }{\tilde{{ A}}}_{1}^{2} +\underline{{ c}}_{2} \gamma \underline{\mu }{\tilde{{ A}}}_{1}^{1} +\underline{{ c}}_{4} \gamma \underline{\mu }{\tilde{{ A}}}_{1}^{2}\nonumber \\&-\underline{{ c}}_{2} \underline{\mu }\tilde{{ A}}_{1}^{1} {{ y}}_{\mathrm{des}} -\underline{{ c}}_{4} \underline{\mu }{\tilde{{ A}}}_{1}^{2} {{ y}}_{\mathrm{des}} )\underline{\mu }\tilde{{ B}}_{1}^{2} \nonumber \\&\cdots +( {\underline{\mu }{\tilde{{ A}}}_{1}^{2} +\underline{{ c}}_{3} \gamma \underline{\mu }{\tilde{{ A}}}_{1}^{2} -\underline{{ c}}_{3} \underline{{ \mu }}{\tilde{{ A}}}_{1}^{2} {{ y}}_{\mathrm{des}} })\underline{{ \mu }}\tilde{{ B}}_{1}^{1} \end{aligned}$$
(55)
Then the following expression is obtained:
$$\begin{aligned} \psi _1^{{ L}} (.){\overline{\mu }} \tilde{{ B}}_1^1 +\psi _2^{ {L}} (.){\overline{\mu }} \tilde{{ B}}_1^2 +\psi _3^{{ L}} (.)\underline{\mu }\tilde{{ B}}_1^2 +\psi _4^{{ L}} (.)\underline{\mu }\tilde{{ B}}_1^1 =0\nonumber \\ \end{aligned}$$
(56)
where
$$\begin{aligned} \psi _1^{{ L}} (.)&=( {{\overline{\mu }} {\tilde{{ A}}}_1^1 +\underline{{ c}}_1 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^1 -\underline{{ c}}_1 {\overline{\mu }} {\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}} }) \nonumber \\ \psi _2^{{ L}} (.)&=0 \nonumber \\ \psi _3^{{ L}} (.)&=( \underline{\mu }{\tilde{{ A}}}_1^1 +\underline{\mu }{\tilde{{ A}}}_1^2 +\underline{{ c}}_2 \gamma \underline{\mu }{\tilde{{ A}}}_1^1 +\underline{{ c}}_4 \gamma \underline{\mu }{\tilde{{ A}}}_1^2\nonumber \\&\quad -\underline{{ c}}_2 \underline{\mu }{\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}} -\underline{{ c}}_4 \underline{\mu }{\tilde{{ A}}}_1^2 {{ y}}_{\mathrm{des}}) \nonumber \\ \psi _4^{{ L}} (.)&=( {\underline{{ \mu }}{\tilde{{ A}}}_1^2 +\underline{{ c}}_3 \gamma \underline{{ \mu }}{\tilde{{ A}}}_1^2 -\underline{{ c}}_3 \underline{{ \mu }}{\tilde{{ A}}}_1^2 {{ y}}_{{ \mathrm{des}}} }) \end{aligned}$$
(57)
(ii) For \({ L}_{{ c}}=2\), (48) can be formulated as:
$$\begin{aligned} 0= & {} -\left( {{{ y}}_{\mathrm{des}} -\gamma }\right) \left( {\sum \limits _{{{ j}}=1}^{{{ L}}_{{ c}} =2} {\overline{{ f}}^{ j}} +\sum \limits _{{{ j}}={{ L}}_{{ c}} +1}^{{{ N}}=4} {\underline{{{ f}}^{{ j}}}} } \right) \nonumber \\&+\left( {\sum \limits _{{{ j}}=1}^{{{ L}}_{{ c}} =2} {\overline{{ f}}^{{ j}}} \underline{{ c}}_{{ j}} +\sum \limits _{{{ j}}={{ L}}_{{ c}} +1}^{{{ N}}=4} {\underline{{{ f}}^{{ j}}}\underline{{ c}}_{{ j}} } } \right) \end{aligned}$$
(58)
$$\begin{aligned} 0= & {} -( {{{ y}}_{\mathrm{des}} -\gamma })({\overline{{ f}}^{1}}+{\overline{{ f}}^{2}}+{\overline{{ f}}^{3}}+{\overline{{ f}}^{4}})\nonumber \\&+(\underline{{ c}}_{1}{\overline{{ f}}^{1}}+\underline{{ c}}_{2}{\overline{{ f}}^{2}}+\underline{{ c}}_{3}\underline{{ f}}_{3}+ \underline{{ c}}_{4}\underline{{ f}}_{4}) \end{aligned}$$
(59)
$$\begin{aligned} 0&=\overline{{{ f}}^1} ( {1}+\underline{{ c}}_1 ( {-{{ y}}_\mathrm{\mathrm{des}} +\gamma }))+\overline{{{ f}}^2} ( {1} +\underline{{ c}}_2 ( {-{{ y}}_{\mathrm{des}} +\gamma }))+\cdots \nonumber \\&\quad \cdots +{\underline{{ f}}^3}( {1} +\underline{{ c}}_3 ( {-{{ y}}_{\mathrm{des}} +\gamma }))+\underline{{{ f}}^4}( {1} +\underline{{ c}}_4 ( {-{{ S}}_{\mathrm{des}} +\gamma }))\nonumber \\ \end{aligned}$$
(60)
Replacing total interval firing set (49) into (61)
$$\begin{aligned} 0= & {} \left( \overline{\mu }\tilde{A}^1_1+\underline{c}_1\gamma \overline{\mu }\tilde{A}^1_1-\underline{c}_1 \overline{\mu }\tilde{A}^1_1y_\mathrm{des}\right) \overline{\mu }\tilde{B}^1_1\nonumber \\&+ \left( \overline{\mu }\tilde{A}^1_1+\underline{c}_2\gamma \overline{\mu }\tilde{A}^1_1-\underline{c}_2 \overline{\mu }\tilde{A}^1_1y_\mathrm{des}\right) \overline{\mu }\tilde{B}^2_1\ldots \nonumber \\&\ldots +\left( \underline{\mu }\tilde{A}^2_1+\underline{c}_4\gamma \underline{\mu }\tilde{A}^2_1 -\underline{c}_4 \underline{\mu }\tilde{A}^2_1y_\mathrm{des}\right) \overline{\mu }\tilde{B}^2_1\nonumber \\&+ \left( \underline{\mu }\tilde{A}^2_1+\underline{c}_3\gamma \underline{\mu }\tilde{A}^2_1-\underline{c}_3 \underline{\mu }\tilde{A}^2_1y_\mathrm{des}\right) \overline{\mu }\tilde{B}^1_1 \end{aligned}$$
(61)
Reformulating (62)
$$\begin{aligned} 0= & {} ( {{\overline{\mu }} {\tilde{{ A}}}_1^1 +\underline{{ c}}_1 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^1 -\underline{{ c}}_1 {\overline{\mu }} {\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}} })\overline{\mu }\tilde{{ B}}_1^1\nonumber \\&+( {{\overline{\mu }} {\tilde{{ A}}}_1^1 +\underline{{ c}}_2 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^1 -\underline{{ c}}_2 {\overline{\mu }} {\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}} }){\overline{\mu }} \tilde{{ B}}_1^2 +\cdots \nonumber \\&\cdots +( {\underline{\mu }{\tilde{{ A}}}_1^2 +\underline{{ c}}_4 \gamma \underline{\mu }{\tilde{{ A}}}_1^2 -\underline{{ c}}_4 \underline{\mu }{\tilde{{ A}}}_1^2 {{ y}}_{\mathrm{des}} })\underline{\mu }\tilde{ B}_1^2\nonumber \\&+( {\underline{\mu }\tilde{{ A}}_1^2 +\underline{{ c}}_3 \gamma \underline{\mu }\tilde{{ A}}_1^2 -\underline{{ c}}_3 \underline{\mu }\tilde{{ A}}_1^2 {{ y}}_{\mathrm{des}} })\underline{\mu }\tilde{{ B}}_1^1 \end{aligned}$$
(62)
Then the following expression is obtained:
$$\begin{aligned} \psi _1^{{ L}} (.){\overline{\mu }} \tilde{{ B}}_1^1 +\psi _2^{{ L}} (.){\overline{\mu }} \tilde{{ B}}_1^2 +\psi _3^{{ L}} (.)\underline{{ \mu }}\tilde{{ B}}_1^2 +\psi _4^{{ L}} (.)\underline{{ \mu }}\tilde{{ B}}_1^1 =0\nonumber \\ \end{aligned}$$
(63)
where
$$\begin{aligned}&\psi _1^{{ L}} (.)=( {{\overline{\mu }} {\tilde{{ A}}}_1^1 +\underline{{ c}}_1 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^1 -\underline{{ c}}_1 {\overline{\mu }} {\tilde{{ A}}}_2^1 {{ y}}_{\mathrm{des}} }) \nonumber \\&\psi _2^{{ L}} (.)=( {{\overline{\mu }} {\tilde{{ A}}}_1^1 +\underline{{ c}}_2 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^1 -\underline{{ c}}_2 {\overline{\mu }} {\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}} }) \nonumber \\&\psi _3^{{ L}} (.)=( {\underline{\mu }{\tilde{{ A}}}_1^2 +\underline{{ c}}_4 \gamma \underline{\mu }{\tilde{{ A}}}_1^2 -\underline{{ c}}_4 \underline{\mu }{\tilde{{ A}}}} _1^2 {{ y}}_{\mathrm{des}} ) \nonumber \\&\psi _4^{{ L}} (.)=( {\underline{{ \mu }}{\tilde{{ A}}}_1^2 +\underline{{ c}}_3 \gamma \underline{{ \mu }}{\tilde{{ A}}}_1^2 -\underline{{ c}}_3 \underline{{ \mu }}{\tilde{{ A}}}_1^2 {{ y}}_{\mathrm{des}} }) \end{aligned}$$
(64)
(iii) For \({ L}_{{ c}}=3\), (48) can be formulated as:
$$\begin{aligned}&0=-\left( {{{ y}}_{\mathrm{des}} -\gamma }\right) \left( {\sum \limits _{{{ j}}=1}^{{{ L}}_{{ c}} =3} {\overline{{ f}}^{{ j}}} +\sum \limits _{{{ j}}={{ L}}^{*} +1}^{{{ N}}=4} {\underline{{{ f}}^{{ j}}}} } \right) \nonumber \\&\quad \qquad +\left( {\sum \limits _{{{ j}}=1}^{{{ L}}_{{ c}} =3} {\overline{{ f}}^{{ j}}} \underline{{ c}}_{{ j}} +\sum \limits _{{{ j}}={{ L}}^{*} +1}^{{{ N}}=4} {\underline{{{ f}}^{{ j}}}\underline{{ c}}_{{ j}} } }\right) \end{aligned}$$
(65)
$$\begin{aligned}&0=-\left( {{{ y}}_{\mathrm{des}} -\gamma }\right) ( {\overline{{{ f}}^{1}} +\overline{{{ f}}^2} +\overline{{{ f}}^{3}} +\underline{{{ f}}^{4}}})\nonumber \\&\quad \,\,\quad +( {\underline{{ c}}_1 \overline{{{ f}}^{1}} +\underline{{ c}}_{2} \overline{{{ f}}^{2}} +\underline{{ c}}_{3} \overline{{{ f}}^{3}} +\underline{{ c}}_4 \underline{{{ f}}^{4}}}) \end{aligned}$$
(66)
$$\begin{aligned}&0=\overline{{{ f}}^1} ( {1}+\underline{{ c}}_1 ( {-{{ y}}_{\mathrm{des}} +\gamma }))+\overline{{{ f}}^2} ( {1} +\underline{{ c}}_2 ( {-{{ y}}_{\mathrm{des}} +\gamma }))+ \cdots \nonumber \\&\quad \cdots +\overline{{{ f}}^3} ( {1} +\underline{{ c}}_3 ( {-{{ y}}_{\mathrm{des}} +\gamma }))+\underline{{{ f}}^4}( {1} +\underline{{ c}}_4 ( {-{{ y}}_{\mathrm{des}} +\gamma }))\nonumber \\ \end{aligned}$$
(67)
Replacing total interval firing set (49) into (68)
$$\begin{aligned}&0=( {{\overline{\mu }} {\tilde{{ A}}}_{1}^{1} {\overline{\mu }} \tilde{{ B}}_{1}^{1} })\left( {1}+\underline{{ c}}_1 ( {-{{ y}}_{\mathrm{des}} +\gamma })\right) +( {{\overline{\mu }} {\tilde{{ A}}}_{1}^{1} {\overline{\mu }} \tilde{{ B}}_{1}^{2} })\nonumber \\&\quad \quad \times \left( {1} +\underline{{ c}}_2 ( {-{{ y}}_{\mathrm{des}} +\gamma })\right) + \cdots \nonumber \\&\quad \quad \cdots +( {{\overline{\mu }} {\tilde{{ A}}}_{1}^{2} {\overline{\mu }} \tilde{{ B}}_{1}^{1} })\left( {1} +\underline{{ c}}_3 ( {-{{ y}}_{\mathrm{des}} +\gamma })\right) +( {{\overline{\mu }} {\tilde{{ A}}}_{1}^{2} {\overline{\mu }} \tilde{{ B}}_{1}^{2} })\nonumber \\&\quad \quad \times \left( {1} +\underline{{ c}}_4 ( {-{{ y}}_{\mathrm{des}} +\gamma })\right) \end{aligned}$$
(68)
Reformulating (69)
$$\begin{aligned} 0&=({\overline{\mu }} {\tilde{{ A}}}_1^1 +{\overline{\mu }} \tilde{{ A}}_1^2 +\underline{{ c}}_1 \gamma {\overline{\mu }} \tilde{{ A}}_1^1 -\underline{{ c}}_1 {\overline{\mu }} {\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}}\nonumber \\&\quad \quad +\underline{{ c}}_3 {\overline{\mu }} \tilde{{ A}}_1^2 -\underline{{ c}}_3 {\overline{\mu }} \tilde{{ A}}_1^2 {{ y}}_{\mathrm{des}}){\overline{\mu }} \tilde{{ B}}_1^1 \nonumber \\&\quad \quad +( {{\overline{\mu }} {\tilde{{ A}}}_1^1 +\underline{{ c}}_2 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^1 -\underline{{ c}}_2 {\overline{\mu }} {\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}} })\overline{\mu }\tilde{{ B}}_1^2\nonumber \\&\quad \quad +( {\underline{{ \mu }}{\tilde{{ A}}}_1^2 +\underline{{ c}}_4 \gamma \underline{{ \mu }}{\tilde{{ A}}}_1^2 -\underline{{ c}}_4 \underline{{ \mu }}{\tilde{{ A}}}_1^2 {{ y}}_{\mathrm{des}} })\underline{\mu }\tilde{{ B}}_1^2 +\left( 0\right) \underline{{ \mu }}\tilde{{ B}}_1^1 \end{aligned}$$
(69)
Then the following expression is obtained:
$$\begin{aligned} \psi _1^{{ L}} (.){\overline{\mu }} \tilde{{ B}}_1^1 +\psi _2^{{ L}} (.){\overline{\mu }} \tilde{{ B}}_1^2 +\psi _3^{{ L}} (.)\underline{{ \mu }}\tilde{{ B}}_1^2 +\psi _4^{{ L}} (.)\underline{{ \mu }}\tilde{{ B}}_1^1 =0\nonumber \\ \end{aligned}$$
(70)
where
$$\begin{aligned}&\psi _1^{{ L}} (.)={\overline{\mu }} {\tilde{{ A}}}_1^1 +\overline{\mu }{\tilde{{ A}}}_1^2 +\underline{{ c}}_1 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^1 -\underline{{ c}}_1 {\overline{\mu }} \tilde{{ A}}_1^1 {{ y}}_{\mathrm{des}}\nonumber \\&\qquad \quad \ \quad +\,\underline{{ c}}_3 \overline{\mu }{\tilde{{ A}}}_1^2 -\underline{{ c}}_3 {\overline{\mu }} \tilde{{ A}}_1^2 {{ y}}_{\mathrm{des}} \nonumber \\&\psi _2^{{ L}} (.)=( {{\overline{\mu }} {\tilde{{ A}}}_1^1 +\underline{{ c}}_2 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^1 -\underline{{ c}}_2 {\overline{\mu }} {\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}} }) \nonumber \\&\psi _3^{{ L}} (.)=( {\underline{{ \mu }}{\tilde{{ A}}}_1^2 +\underline{{ c}}_4 \gamma \underline{{ \mu }}{\tilde{{ A}}}_1^2 -\underline{{ c}}_4 \underline{{ \mu }}{\tilde{{ A}}}_1^2 {{ y}}_{\mathrm{des}} }) \nonumber \\&\psi _4^{{ L}} (.)=( 0) \end{aligned}$$
(71)
As it has been shown above, (48) can be represented in form given in (50) for each possible value of \({ L}_{{ c}} ( { L}_{ c} \in \left\{ {1,2,3} \right\} )\).
Appendix 2
In this part, it will be shown that (72) can be formulated as (74) for each possible value of \({{ R}}_{{ c}} \in \left\{ {1,2,3} \right\} \).
$$\begin{aligned} 0= & {} -( {{{ y}}_{\mathrm{des}} +\gamma })\left( {\sum \limits _{{{ j}}=1}^{{{ R}}_{{ c}} } {\underline{{ f}}^{{ j}}+\sum \limits _{{{ j}}={{ R}}_{{ c}} +1}^{{ N}} {\overline{{ f}}^{{ j}}} } }\right) \nonumber \\&+\left( {\sum \limits _{{{ j}}=1}^{{{ R}}_{{ c}}} {\underline{{{ f}}}^{{ j}}\overline{{ c}}_{{ j}} +\sum \limits _{{{ j}}={{ R}}_{{ c}} +1}^{{ N}} {\overline{{ f}}^{{ j}}\overline{{ c}}_{{ j}}}}}\right) \end{aligned}$$
(72)
where the total firing set is:
$$\begin{aligned}&{\underline{{{ f}}}^{{1}}=[ {\underline{{ \mu }}{\tilde{{ A}}}_{1}^{1} \underline{{ \mu }}\tilde{{ B}}_1^{1} } ]} \quad {\overline{{{ f}}}^1=[ {{\overline{\mu }} {\tilde{{ A}}}_{1}^{1} {\overline{\mu }} \tilde{{ B}}_{1}^{1} } ]} \nonumber \\&{\underline{{{ f}}}^{{2}}=[ {\underline{{ \mu }}{\tilde{{ A}}}_{1}^{1} \underline{{ \mu }}\tilde{{ B}}_{1}^{2} } ]} \quad {\overline{{ f}}^2=[ {{\overline{\mu }} {\tilde{{ A}}}_{1}^{1} {\overline{\mu }} \tilde{{ B}}_{1}^{2} } ]} \nonumber \\&{\underline{{{ f}}}^{{3}}=[ {\underline{\mu }{\tilde{{ A}}}_{1}^{2} \underline{{ \mu }}\tilde{{ B}}_{1}^{1} } ]} \quad {\overline{{ f}}^3=[ {{\overline{\mu }} {\tilde{{ A}}}_{1}^{2} {\overline{\mu }} \tilde{{ B}}_{1}^{1} } ]} \nonumber \\&{\underline{{{ f}}}^{{4}}=[ {\underline{{ \mu }}{\tilde{{ A}}}_{1}^{2} \underline{{ \mu }}\tilde{{ B}}_{1}^{2} } ]} \quad {\overline{{ f}}^4=[ {{\overline{\mu }} {\tilde{{ A}}}_{1}^{2} {\overline{\mu }} \tilde{{ B}}_{1}^{2} } ]} \end{aligned}$$
(73)
$$\begin{aligned} \psi _1^{{ R}} (.)\underline{{ \mu }}\tilde{{ B}}_1^1 +\psi _2^{{ R}} (.){\overline{\mu }} \tilde{{ B}}_1^2 +\psi _3^{{ R}} (.)\underline{{ \mu }}\tilde{{ B}}_1^2 +\psi _4^{{ R}} (.)\underline{{ \mu }}\tilde{{ B}}_1^1 =0\nonumber \\ \end{aligned}$$
(74)
where \(\psi _{{ i}}^{{ R}} (.),{{ i}}=1,...,4\) is a function of \(( {{{ y}}_{\mathrm{des}} ,\gamma ,\overline{{ c}} _{{ j}} })\) and a possible combination of the following parameters \(( {{\overline{ u}} {\tilde{{ A}}}_1^1 ,{\overline{ u}} \tilde{{ A}}_1^2 ,\underline{{ u}}{\tilde{{ A}}}_1^2 ,\underline{\mu }{\tilde{{ A}}}_1^1 })\).
(i) For \({{ R}}_{{ c}} =1\), (72) can be formulated as:
$$\begin{aligned} 0= & {} -( {{{ y}}_{\mathrm{des}} +\gamma })\left( {\sum \limits _{{{ j}}=1}^{{{ R}}_{{ c}} =1} {\underline{{ f}}^{{ j}}+\sum \limits _{{{ j}}={{ R}}_{{ c}} +1}^{{ N}} {\overline{{ f}}^{{ j}}} } }\right) \nonumber \\&+\left( {\sum \limits _{{{ j}}=1}^{{{ R}}_{{ c}} =1} {\underline{{ f}}^{{ j}}\overline{{ c}}_{{ j}} +\sum \limits _{{{ j}}={{ R}}_{{ c}} +1}^{{ N}} {\overline{{ f}}^{{ j}}\overline{{ c}}_{{ j}} } } }\right) \end{aligned}$$
(75)
$$\begin{aligned} 0= & {} -( {{{ y}}_{\mathrm{des}} +\gamma })( {\underline{{{ f}}}^1}+\overline{{{ f}}^2} +\overline{{{ f}}^3} +\overline{{{ f}}^4} )\nonumber \\&+( {\overline{{ c}}_1 \underline{{{ f}}^1}+\overline{{ c}}_2 \overline{{{ f}}^2} + \overline{{ c}}_3 \overline{{{ f}}^3} +\overline{{ c}}_4 \overline{{{ f}}^4} }) \end{aligned}$$
(76)
$$\begin{aligned} 0= & {} \underline{{{ f}}^1}\left( {1}+\overline{{ c}}_1 ( {-{{ y}}_{\mathrm{des}} -\gamma })\right) +\overline{{{ f}}^2} \left( {1} +\overline{{ c}}_2 ( {-{{ y}}_{\mathrm{des}} -\gamma })\right) + \cdots \nonumber \\&\cdots + \overline{{{ f}}^3} \left( {1} +\overline{{ c}}_3 ( {-{{ y}}_{\mathrm{des}} -\gamma })\right) +\overline{{{ f}}^4} \left( {1} +\overline{{ c}}_4 ( {-{{ y}}_{\mathrm{des}} -\gamma })\right) \nonumber \\ \end{aligned}$$
(77)
Replacing total interval firing set (73) into (78)
$$\begin{aligned} 0= & {} ( {\underline{{ \mu }}{\tilde{{ A}}}_{1}^{1} \underline{\mu }\tilde{{ B}}_1^{1} })( {1}+\overline{{ c}}_1 ( {-{{ y}}_{\mathrm{des}} -\gamma }))\nonumber \\&+( {{\overline{\mu }} {\tilde{{ A}}}_{1}^{1} {\overline{\mu }} \tilde{{ B}}_{1}^{2} })( {1} +\overline{{ c}}_2 ( {-{{ y}}_{\mathrm{des}} -\gamma }))+ \cdots \nonumber \\&\cdots +( {{\overline{\mu }} {\tilde{ A}}_{1}^{2} {\overline{\mu }} \tilde{{ B}}_{1}^{1} })( {1} +\overline{{ c}}_3 ( {-{{ y}}_{\mathrm{des}} -\gamma }))\nonumber \\&+( {{\overline{\mu }} {\tilde{{ A}}}_{1}^{2} {\overline{\mu }} \tilde{{ B}}_{1}^{2} })( {1} +\overline{{ c}}_4 ( {-{{ y}}_{\mathrm{des}} -\gamma })) \end{aligned}$$
(78)
Reformulating (78)
$$\begin{aligned} 0= & {} ( {{\overline{\mu }} {\tilde{{ A}}}_1^2 -\overline{{ c}}_3 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^2 -\overline{{ c}}_3 {\overline{\mu }} {\tilde{{ A}}}_1^2 {{ y}}_{\mathrm{des}} }){\overline{\mu }} \tilde{{ B}}_1^1 \nonumber \\&+\cdots +({\overline{\mu }} {\tilde{{ A}}}_1^1 +{\overline{\mu }} \tilde{{ A}}_1^2 +\overline{{ c}}_2 \gamma {\overline{\mu }} \tilde{{ A}}_1^1 +\overline{{ c}}_4 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^2 -\overline{{ c}}_2 {\overline{\mu }} {\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}}\nonumber \\&-\overline{{ c}}_4 {\overline{\mu }} {\tilde{{ A}}}_1^2 {{ y}}_{\mathrm{des}}){\overline{\mu }} \tilde{{ B}}_1^2 +\cdots \nonumber \\&\cdots +( 0)\underline{{ \mu }}\tilde{{ B}}_1^2 +( {\underline{\mu }{\tilde{{ A}}}_1^1 -\overline{{ c}}_1 \gamma \underline{\mu }{\tilde{{ A}}}_1^1 -\overline{{ c}}_1 \underline{{ \mu }}\tilde{{ A}}_1^1 {{ y}}_{\mathrm{des}} })\underline{\mu }\tilde{{ B}}_1^1\nonumber \\ \end{aligned}$$
(79)
Then the following expression is obtained:
$$\begin{aligned} \psi _1^{{ R}} (.)\underline{{ \mu }}\tilde{{ B}}_1^1 +\psi _2^{{ R}} (.){\overline{\mu }} \tilde{{ B}}_1^2 +\psi _3^{{ R}} (.)\underline{{ \mu }}\tilde{{ B}}_1^2 +\psi _4^{{ R}} (.)\underline{{ \mu }}\tilde{{ B}}_1^1 =0\nonumber \\ \end{aligned}$$
(80)
where
$$\begin{aligned}&\psi _1^{{ R}} (.)=( {{\overline{\mu }} {\tilde{{ A}}}_1^2 -\overline{{ c}}_3 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^2 -\overline{{ c}}_3 {\overline{\mu }} {\tilde{{ A}}}_1^2 {{ y}}_{\mathrm{des}} }) \nonumber \\&\psi _2^{{ R}} (.)=({\overline{\mu }} {\tilde{{ A}}}_1^1 +{\overline{\mu }} {\tilde{{ A}}}_1^2 +\overline{{ c}}_2 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^1 +\overline{{ c}}_4 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^2\nonumber \\&\qquad \quad \qquad -\overline{{ c}}_2 {\overline{\mu }} {\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}} -\overline{{ c}}_4 {\overline{\mu }} {\tilde{{ A}}}_1^2 {{ y}}_{\mathrm{des}}) \nonumber \\&\psi _3^{{ R}} (.)=( 0) \nonumber \\&\psi _4^{{ R}} (.)=( {\underline{{ \mu }}{\tilde{{ A}}}_1^1 -\overline{{ c}}_1 \gamma \underline{{ \mu }}{\tilde{{ A}}}_1^1 -\overline{{ c}}_1 \underline{{ \mu }}{\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}} }) \end{aligned}$$
(81)
(ii) For \({ R}_{{ c}}\)=2, (72) can be formulated as:
$$\begin{aligned} 0= & {} -( {{{ y}}_{\mathrm{des}} +\gamma })\left( {\sum \limits _{{{ j}}=1}^{{{ R}}_{{ c}} =2} {\underline{{ f}}^{{ j}}+\sum \limits _{{{ j}}={{ R}}_{{ c}} +1}^{{ N}} {\overline{{ f}}^{{ j}}}}}\right) \nonumber \\&+\left( {\sum \limits _{{{ j}}=1}^{{{ R}}_{{ c}} =2} {\underline{{ f}}^{{ j}}\overline{{ c}}_{{ j}} +\sum \limits _{{{ j}}={{ R}}_{{ c}} +1}^{{ N}} {\overline{{ f}}^{{ j}}\overline{{ c}}_{{ j}}}}}\right) \end{aligned}$$
(82)
$$\begin{aligned} 0= & {} -( {{{ y}}_{\mathrm{des}} +\gamma })( {\underline{{{ f}}}^1}+{\underline{{ f}}^2}+\overline{{{ f}}^3} +\overline{{{ f}}^4})\nonumber \\&+( {\overline{{ c}}_1 \underline{{{ f}}^1}+\overline{{ c}}_2 {\underline{{ f}}^2} + \overline{{ c}}_3 \overline{{{ f}}^3} +\overline{{ c}}_4 \overline{{{ f}}^4} }) \end{aligned}$$
(83)
$$\begin{aligned}&0=\underline{{{ f}}^1}\left( {1}+\overline{{ c}}_1 ( {-{{ y}}_{\mathrm{des}} -\gamma })\right) +{\underline{{ f}}^2}\left( {1} +\overline{{ c}}_2 ( {-{{ y}}_{\mathrm{des}} -\gamma })\right) + \cdots \nonumber \\&\quad \cdots + \overline{{{ f}}^3} \left( {1} +\overline{{ c}}_3 ( {-{{ y}}_{\mathrm{des}} -\gamma })\right) +\overline{{{ f}}^4} \left( {1} +\overline{{ c}}_4 ({-{{ y}}_{\mathrm{des}} -\gamma })\right) \nonumber \\ \end{aligned}$$
(84)
Replacing total interval firing set (73) into (84)
$$\begin{aligned} 0= & {} ( {\underline{{ \mu }}{\tilde{ A}}_{1}^{1} \underline{\mu }\tilde{{ B}}})_1^{1} ( {1}+\overline{{ c}}_1 ( {-{{ y}}_{\mathrm{des}} -\gamma }))\nonumber \\&+\,( {\underline{{ \mu }}{\tilde{{ A}}}_{1}^{1} \underline{{ \mu }}\tilde{{ B}}_{1}^{2} })( {1} +\overline{{ c}}_2 ( {-{{ y}}_{\mathrm{des}} -\gamma }))+ \cdots \nonumber \\&\cdots + ( {{\overline{\mu }} {\tilde{{ A}}}_{1}^{2} {\overline{\mu }} \tilde{{ B}}_{1}^{1} })( {1} +\overline{{ c}}_3 ( {-{{ y}}_{\mathrm{des}} -\gamma }))\nonumber \\&+\,{\overline{\mu }} {\tilde{{ A}}}_{1}^{2} \overline{\mu }\tilde{{ B}}_{1}^{2} ( {1} +\overline{{ c}}_4 ( {-{{ y}}_{\mathrm{des}} -\gamma })) \end{aligned}$$
(85)
Reformulating (85)
$$\begin{aligned} 0= & {} ( {{\overline{\mu }} {\tilde{{ A}}}_1^2 -\overline{{ c}}_3 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^2 -\overline{{ c}}_3 {\overline{\mu }} {\tilde{{ A}}}_1^2 {{ y}}_{\mathrm{des}} }){\overline{\mu }} \tilde{{ B}}_1^1\nonumber \\&+( {{\overline{\mu }} {\tilde{{ A}}}_1^2 -\overline{{ c}}_4 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^2 -\overline{{ c}}_4 {\overline{\mu }} {\tilde{{ A}}}_1^2 {{ y}}_{\mathrm{des}} }){\overline{\mu }} \tilde{{ B}}_1^2 +\cdots \nonumber \\&\cdots +( {\underline{{ \mu }}{\tilde{{ A}}}_1^1 -\overline{{ c}}_{2} \gamma \underline{{ \mu }}{\tilde{{ A}}}_1^1 -\overline{{ c}}_{2} \underline{{ \mu }}{\tilde{{ A}}}_{1}^{1} {{ y}}_{\mathrm{des}} })\underline{{ \mu }}\tilde{{ B}}_{1}^{2}\nonumber \\&+( {\underline{{ \mu }}\tilde{{ A}}_{1}^{1} -\overline{{ c}}_1 \gamma \underline{{ \mu }}\tilde{{ A}}_1^1 -\overline{{ c}}_{1} \underline{{ \mu }}{\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}} })\underline{{ \mu }}\tilde{{ B}}_{1}^{1} \end{aligned}$$
(86)
Then the following expression is obtained:
$$\begin{aligned} \psi _1^{{ R}} (.)\underline{{ \mu }}\tilde{{ B}}_1^1 +\psi _2^{{ R}} (.){\overline{\mu }} \tilde{{ B}}_1^2 +\psi _3^{{ R}} (.)\underline{{ \mu }}\tilde{{ B}}_1^2 +\psi _4^{{ R}} (.)\underline{{ \mu }}\tilde{{ B}}_1^1 =0\nonumber \\ \end{aligned}$$
(87)
where
$$\begin{aligned}&\psi _1^{{ R}} (.)=( {{\overline{\mu }} {\tilde{{ A}}}}_1^2 -\overline{{ c}}_3 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^2 -\overline{{ c}}_3 {\overline{\mu }} {\tilde{{ A}}}_1^2 {{ y}}_{\mathrm{des}} ) \nonumber \\&\psi _2^{{ R}} (.)=( {{\overline{\mu }} {\tilde{{ A}}}}_1^2 -\overline{{ c}}_4 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^2 -\overline{{ c}}_4 {\overline{\mu }} {\tilde{{ A}}}_1^2 {{ y}}_{\mathrm{des}} ) \nonumber \\&\psi _3^{{ R}} (.)=( {\underline{\mu }{\tilde{{ A}}}}_1^1 -\overline{{ c}}_2 \gamma \underline{\mu }{\tilde{{ A}}}_1^1 -\overline{{ c}}_2 \underline{\mu }{\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}} ) \nonumber \\&\psi _4^{{ R}} (.)=( {\underline{\mu }{\tilde{{ A}}}}_1^1 -\overline{{ c}}_1 \gamma \underline{\mu }{\tilde{{ A}}}_1^1 -\overline{{ c}}_1 \underline{\mu }{\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}} ) \end{aligned}$$
(88)
(iii) For \({{ R}}_{{ c}}=3\), (72) can be formulated as:
$$\begin{aligned} 0= & {} -( {{{ y}}_{\mathrm{des}} +\gamma })\left( {\sum \limits _{{{ j}}=1}^{{{ R}}_{{ c}} =3} {\underline{{ f}}^{{ j}}+\sum \limits _{{{ j}}={{ R}}_{{ c}} +1}^{{ N}} {\overline{{ f}}^{{ j}}} } }\right) \nonumber \\&\quad +\,\left( {\sum \limits _{{{ j}}={1}}^{{{ R}}_{{ c}} =3} {\underline{{ f}}^{{ j}}\overline{{ c}}_{{ j}} +\sum \limits _{{{ j}}={{ R}}_{{ c}} +1}^{{ N}} {\overline{{ f}}^{{ j}}\overline{{ c}}_{{ j}}}}}\right) \end{aligned}$$
(89)
$$\begin{aligned} 0= & {} -( {{{ y}}_{\mathrm{des}} +\gamma })( \underline{{{ f}}^{1}}+\underline{{{ f}}^{2}}+\underline{{{ f}}^{3}}+\overline{{{ f}}^{4}} )\nonumber \\&\quad +( {\overline{{ c}}_1 \underline{{{ f}}^1}+\overline{{ c}}_{2} {\underline{{ f}}^{2}} + \overline{{ c}}_{3} {\underline{{{ f}}}^{3}}+\overline{{ c}}_{4} \overline{{{ f}}^{4}} }) \end{aligned}$$
(90)
$$\begin{aligned}&0=\underline{{{ f}}^1}( {1}+\overline{{ c}}_1 ( {-{{ y}}_{\mathrm{des}} -\gamma }))+{\underline{{ f}}^2}( {1} +\overline{{ c}}_2 ( {-{{ y}}_{\mathrm{des}} -\gamma }))+ \cdots \nonumber \\&\quad \cdots + {\underline{{ f}}^{3}}( {1} +\overline{{ c}}_3 ( {-{{ y}}_{\mathrm{des}} -\gamma }))+\overline{{{ f}}^{4}} ( {1} +\overline{{ c}}_{4} ( {-{{ y}}_{\mathrm{des}} -\gamma }))\nonumber \\ \end{aligned}$$
(91)
Replacing total interval firing set (73) into (91)
$$\begin{aligned} 0= & {} ( {\underline{\mu }{\tilde{ A}}_{1}^{1} \underline{\mu }\tilde{{ B}}_1^{1} })\left( {1}+\overline{{ c}}_1 ( {-{ y}_{\mathrm{des}} -\gamma })\right) +( {\underline{{ \mu }}{\tilde{{ A}}}_{1}^{1} \underline{{ \mu }}\tilde{{ B}}_{1}^{2} })\nonumber \\&\left( {1} +\overline{{ c}}_2 ( {-{{ y}}_{\mathrm{des}} -\gamma })\right) + \cdots \nonumber \\&\cdots + ( {\underline{\mu }{\tilde{{ A}}}_{1}^{2} \underline{\mu }\tilde{{ B}}_{1}^{1} })\left( {1} +\overline{{ c}}_3 ( {-{ y}_{\mathrm{des}} -\gamma })\right) +( {{\overline{\mu }} {\tilde{{ A}}}_{1}^{2} {\overline{\mu }} \tilde{{ B}}_{1}^{2} })\nonumber \\&\quad \left( {1} +\overline{{ c}}_4 ( {-{{ y}}_{\mathrm{des}} -\gamma })\right) \end{aligned}$$
(92)
Reformulating (92)
$$\begin{aligned} 0= & {} ( 0){\overline{\mu }} \tilde{{ B}}_{1}^{1} +( {{\overline{\mu }} \tilde{{ A}}_{1}^{2} -\overline{{ c}}_{4} \gamma {\overline{\mu }} \tilde{{ A}}_{1}^{2} -\overline{{ c}}_{4} {\overline{\mu }} {\tilde{{ A}}}_{1}^{2} {{ y}}_{\mathrm{des}} }){\overline{\mu }} \tilde{{ B}}_{1}^{2} +\cdots \nonumber \\&\quad \cdots +( {\underline{{ \mu }}{\tilde{{ A}}}_1^1 -\overline{{ c}}_2 \gamma \underline{{ \mu }}{\tilde{{ A}}}_1^1 -\overline{{ c}}_2 \underline{{ \mu }}{\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}} })\underline{{ \mu }}\tilde{{ B}}_1^2 +\cdots \nonumber \\&\quad \cdots +(\underline{{ \mu }}{\tilde{{ A}}}_1^1 +\underline{\mu }{\tilde{{ A}}}_1^2 -\overline{{ c}}_1 \gamma \underline{\mu }{\tilde{{ A}}}_1^1 -\overline{{ c}}_3 \gamma \underline{\mu }{\tilde{{ A}}}_1^2 -\overline{{ c}}_1 \underline{{ \mu }}\tilde{{ A}}_1^1 {{ y}}_{\mathrm{des}}\nonumber \\&\quad \qquad -\overline{{ c}}_3 \underline{\mu }{\tilde{{ A}}}_1^3 {{ y}}_{\mathrm{des}})\underline{\mu }\tilde{{ B}}_1^1 \end{aligned}$$
(93)
Then the following expression is obtained:
$$\begin{aligned} \psi _{1}^{{ R}} (.)\underline{{ \mu }}\tilde{{ B}}_{1}^{1} +\psi _{2}^{{ R}} (.){\overline{\mu }} \tilde{{ B}}_{1}^{2} +\psi _{3}^{{ R}} (.)\underline{{ \mu }}\tilde{{ B}}_{1}^{2} +\psi _4^{{ R}} (.)\underline{{ \mu }}\tilde{{ B}}_{1}^{1} =0\nonumber \\ \end{aligned}$$
(94)
where
$$\begin{aligned} \psi _1^{{ R}} (.)&=( 0) \nonumber \\ \psi _2^{{ R}} (.)&=( {{\overline{\mu }} {\tilde{{ A}}}_1^2 -\overline{{ c}}_4 \gamma {\overline{\mu }} {\tilde{{ A}}}_1^2 -\overline{{ c}}_{4} {\overline{\mu }} {\tilde{{ A}}}_{1}^{2} {{ y}}_{\mathrm{des}} }) \nonumber \\ \psi _{3}^{{ R}} (.)&=( {\underline{\mu }{\tilde{{ A}}}_{1}^{1} -\overline{{ c}}_2 \gamma \underline{\mu }{\tilde{{ A}}}_{1}^{1} -\overline{{ c}}_{2} \underline{{ \mu }}{\tilde{{ A}}}_{1}^{1} {{ y}}_{\mathrm{des}} }) \nonumber \\ \psi _{4}^{{ R}} (.)&=(\underline{{ \mu }}{\tilde{{ A}}}_{1}^{1} +\underline{{ \mu }}{\tilde{{ A}}}_1^2 -\overline{{ c}}_{1} \gamma \underline{{ \mu }}{\tilde{{ A}}}_1^1 -\overline{{ c}}_{3} \gamma \underline{{ \mu }}{\tilde{{ A}}}_1^2\nonumber \\&\qquad -\overline{{ c}}_{1} \underline{{ \mu }}{\tilde{{ A}}}_1^1 {{ y}}_{\mathrm{des}} -\overline{{ c}}_3 \underline{{ \mu }}{\tilde{{ A}}}_1^3 {{ y}}_{\mathrm{des}}) \end{aligned}$$
(95)
As it has been shown above, (72) can be represented in form given in (74) for each possible value of \({ R}_{{ c}} ( { R}_{ c} \in \left\{ {1,2,3} \right\} )\).
