This is an exercise around linearization of a scalar system. the scalar nonlinear differential equation we have is r(t)= sin(x(t)) + u(t). (a) the first thing we want to do is find equilibria (dc operating points) that this system can support. suppose we want to investigate potential expansion points (x",u") with u. = 0. sketch sin(x*) for 4π x< 4π and intersect it with the horizontal line at 0 this will show us the equilibria points. where sin(x0. (b) show that all r(r) = satisfy (7) together with u" = 0. let us zoom in on two choices: π and representative points.) = o. (looking at the sketch we made, these seem like (c) linearize the systern (7) around the equilibrium (x0,u") = (0,0). what is the resulting linearized scalar differential equation for (t)-x-x(t)-0, involving udr) = u(r) _ u" = u(t) 0? (d) for the linearized approximate system model that you found in the previous part, what happens if we try to discretize time to intervals of duration δ? assume now we use a piecewise constant control input over duration δ, that δ is small relative to the ranges of controls applied, and that we sample the state x every δ (that is, at every 1 na where n is an integer) as well write out the resulting scalar discrete-time control system model. this model is an approximation of what will happen if we actually applied a piecewise constant control input to the original nonlinear differential equation. (e) is the (approximate) discrete-time system you found in the previous part stable or unstable? (f) now linearize the system (7) around the equilibrium (x, u") ( π.0), what is the resulting approximate scalar differential equation ) 0? (g) for the linearized system model that you found in the previous part, what happens if we try to discretize time to intervals of duration δ? assume now we use a piecewise constant control input over duration that δ is small relative to the ranges of controls applied, and that we sample the state x every δ (that is, at every , = na, where n is an integer) as well, write out the resulting scalar discrete-time control system. this model is an approximation of what will happen if we actually applied a piecewise constant control input to the original nonlinear differential equation. (h) is the (approximate) discrete-time system you found in the previous part stable or unstable? (i) suppose for the two discrete-time systems above, we chose to apply a feedback law u(r) =-k(x(1) x"). for what range of k values, would the resulting linearized discrete-time systems be stable? your answer will depend on δ.