Stability properties of the Hadamard product of interval polynomials and SPR functions (Q5926823)

The author of this paper extends a result about the Hadamard product of Hurwitz polynomials to the case of Hurwitz interval polynomials. Thus, if NEWLINE\[NEWLINE\widetilde A(s)= \sum^n_{k=0} [\underline a_k,\overline a_k]s^k\quad\text{and} \quad \widetilde B(s)= \sum^n_{k=0} [\underline b_k,\overline b_k] s^kNEWLINE\]NEWLINE are Hurwitz interval polynomials with \(\underline a_k\geq 0\) and \(\underline b_k> 0\) for \(k= 0,1,\dots, n\), then the product NEWLINE\[NEWLINE\widetilde C(s)=\widetilde A(s)\circ\widetilde B(s)\equiv \sum^n_{k=0} [\underline a_k\underline b_k, \overline a_k\overline b_k] s^kNEWLINE\]NEWLINE is a Hurwitz interval polynomial and NEWLINE\[NEWLINE\widetilde E(s)= \sum^n_{k=0}{n\choose k} [\underline a_k\underline b_k, \overline a_k\overline b_k] s^kNEWLINE\]NEWLINE is also a Hurwitz interval polynomial.NEWLINENEWLINENEWLINELater, one introduces the definition of a SPRO function. A rational function \(p(s)\) of zero relative degree is a SPRO function ifNEWLINENEWLINENEWLINE(i) \(p(s)\) is analytic in \(\text{Re}[s]\geq 0\),NEWLINENEWLINENEWLINE(ii) \(\text{Re}[p(j\omega)]> 0\) for all \(\omega\in \mathbb{R}\).NEWLINENEWLINENEWLINEIf NEWLINE\[NEWLINEp(s)= {N_p(s)\over D_p(s)}= {a_n s^n+\cdots+ a_0\over b_n s^n+\cdots+ b_0}\quad\text{and} \quad q(s)= {N_q(s)\over D_q(s)}= {\alpha_n s^n+\cdots+ \alpha_0\over \beta_n s^n+\cdots+ \beta_0}NEWLINE\]NEWLINE are SPRO functions of the same order and if \(D_p(s)\circ D_q(s)+ j\alpha N_p(s)\circ N_q(s)\) is Hurwitz for all \(\alpha\in\mathbb{R}\), then \(p(s)\circ q(s)= N_p(s)\circ N_q(s)/ D_p(s)\circ D_q(s)\) is a SPRO function. In similar conditions, if NEWLINE\[NEWLINE\widetilde p(s)= {\widetilde N_p(s)\over\widetilde D_p(s)}= {[\underline a_k,\overline a_k] s^n+\cdots+ [\underline a_0,\overline a_0]\over [\underline b_k, \overline b_k] s^n+\cdots+ [\underline b_0,\overline b_0]}\text{ and } \widetilde q(s)= {\widetilde N_q(s)\over\widetilde D_q(s)}= {[\underline\alpha_k, \overline\alpha_k] s^n+\cdots+ [\underline\alpha_0, \overline\alpha_0]\over [\underline\beta_k, \overline\beta_k] s^n+\cdots+ [\underline\beta_0, \overline\beta_0]}NEWLINE\]NEWLINE are SPRO families of the same order, then NEWLINE\[NEWLINE\widetilde p(s)\circ\widetilde q(s)=\widetilde N_p(s)\circ \widetilde N_q(s)/\widetilde D_p(s)\circ\widetilde D_q(s)NEWLINE\]NEWLINE is also a SPRO family.NEWLINENEWLINENEWLINEFinally, the author presents an application to robust stabilization using these results.