Application of Mathematics for Robust Stability and for Robustly Strictly Positive Real on an Uncertain Interval Plant

In this article, we present a robust control problem wherein $P=\left\{P\left(s,l,m\right)=U\left(s,l\right)/V\left(s,m\right):l\in L,m\in M\right\}$ represents a family of interval plants. For this particular problem, we introduce four Kharitonov polynomials uniquely by minimizing and maximizing the concept of multilinear functions with uncertain parameters $l$ and $m$ for the plant $P\left(j\omega ,l,m\right)=\mathrm{Re}\frac{U(j\omega ,l)}{V(j\omega ,m)}=\mathrm{ReU}(j\omega ,l)V^{*}(j\omega ,m)$ and $P\left(j\omega ,l,m\right)=\mathrm{Im}\frac{U(j\omega ,l)}{V(j\omega ,m)}=\mathrm{ImU}(j\omega ,l)V^{*}(j\omega ,m)$ where $V^{*}(j\omega ,m)$ denotes the conjugate of $V(j\omega ,m)$ and $s=j\omega$, with $\omega$ representing frequency within a specified domain. This technique yields a Kharitonov rectangle or box $P\left(j\omega ,L,M\right)$ whose every four vertices are represented by four unique Kharitonov polynomials $P\left(j\omega ,l,m\right)$, each is denoted by ${\tilde{k}}_1\left(s\right),{\tilde{k}}_2\left(s\right),{\tilde{k}}_3\left(s\right)$ and ${\tilde{k}}_4\left(s\right)$ and this rectangle characterizes both robust stability and robustly strictly positive real (SPR) for the family of interval plants ($P$). Thus, the introduction of this Kharitonov rectangle stands as a novel innovation in this article. Our contribution encompasses two key aspects. Initially, we demonstrate the stability of the four unique Kharitonov polynomials employing Hurwitz stability criteria, followed by the utilization of Kharitonov's theorem to ensure robust stability of $P$. Subsequently, to establish robustly SPR of $P$, we analyze the SPR of each interval plant $P\left(s,l,m\right)$ concerning the stability of $U\left(s,l\right)$ and $V\left(s,m\right)$, adhering to the condition $\underset{l\in L,m\in M}{\min }\mathrm{Re}U(j\omega ,l)V^{*}(j\omega ,m)>0$. Additionally, we provide a detailed illustrative example. Furthermore, we demonstrate the motion of the Kharitonov rectangle and the robust stability test through simulation.