Problem 1 Using Jury's test, determine the stability of a discrete-time system whose characteristic polynomial is given by: Q(z) = z4+ 6z 3+ 4z 2+ 2z + 1= 0 (You may benefit by watching this video) Problem 2 Find all the critical points of the nonlinear system ẋ =x- y- x2 +xy Ẏ = -x2 - y Compute the eigenvalues and eigenvectors Problem 3 The principal comparison between continuous-time systems and discrete-time systems is from the viewpoint of stability. All left-half s-plane poles including those on the j -axis map into and on the unit circle of the z-plane, respectively. The figure below shows everything you need to know about mapping s-plane poles into and onto the unit circle of the z-plane. 1. Compute z-plane poles for a corresponding set of 10 representative s-plane poles consisting of real and complex conjugate poles. Show a tabulated result of pole-pairs in the s-plane and in the z-plane. 2. Show that for poles on a constant damping factor i line, the s-plane poles are defined as s j cot i j 3. Show that the mapping of s-plane poles along a constant damping factor line into the unit circle is a logarithmic spiral. Provide tabulated and plotted results.