Use the unilateral z-transform

Use the unilateral \(z\)-transform. A unity-negative-feedback loop has open-loop pulse transfer \[ G_{\mathrm{ol}}(z)=\frac{0.200K}{z^{2}(z-0.500)}, \] with real gain \(K\) and sampling implicit in \(G_{\mathrm{ol}}\). The closed-loop characteristic polynomial, made monic, is \[ \chi(z)=z^{3}-0.500\,z^{2}+0.200K. \] Apply the Jury test in the following exact form. For a cubic \[ \chi(z)=a_{3}z^{3}+a_{2}z^{2}+a_{1}z+a_{0} \] with \(a_{3}>0\), Schur stability holds if and only if all four of the following are true: \[ \lvert a_{0}\rvert<a_{3},\qquad \chi(1)>0,\qquad \chi(-1)<0,\qquad \bigl\lvert a_{0}^{2}-a_{3}^{2}\bigr\rvert>\bigl\lvert a_{0}a_{2}-a_{1}a_{3}\bigr\rvert. \] (The last inequality is the \(n=3\) Jury-table condition.) (a) Identify \(a_{3},a_{2},a_{1},a_{0}\) in terms of \(K\). (b) Convert each of the four Jury inequalities into a condition on \(K\). Treat positive and negative \(K\) separately where the absolute values require it. (c) Intersect the four conditions and report the complete open set of real \(K\) for which the closed loop is Schur stable. (d) For \(K=2.00\) and for \(K=4.50\), evaluate all four Jury quantities and decide stability in each case. For each unstable case, identify at least one violated inequality.

** The four listed Jury inequalities are necessary *and* sufficient for a cubic with positive leading coefficient.