This is all that was given in the question.

A quartz clock uses a quartz crystal resonator to keep time. The displacement of the crystal z is effected by a thermal expansion parameter a > 0 and can be moderated using the regulator input u. The governing dynamics of the system is (t) = at(t) - ar"(t) + u(t)e(t). The output of the system is r(t) and the input is u(t). (a) Show that (To, , ) = (-1,0,0) is an equilibrium of the system. [ 7 marks (b) Find the linearised state space model around the equilibrium from part (a). (15 marks (c) Consider the system operating with u(t) = 0 and (x(0), (0)) close to (T, ). If the quartz clock is running too slowly, how could you vary a to make the clock run more quickly? Explain and support your answer. [8 marks] A quartz clock uses a quartz crystal resonator to keep time. The displacement of the crystal z is effected by a thermal expansion parameter a > 0 and can be moderated using the regulator input u. The governing dynamics of the system is (t) = at(t) - ar"(t) + u(t)e(t). The output of the system is r(t) and the input is u(t). (a) Show that (To, , ) = (-1,0,0) is an equilibrium of the system. [ 7 marks (b) Find the linearised state space model around the equilibrium from part (a). (15 marks (c) Consider the system operating with u(t) = 0 and (x(0), (0)) close to (T, ). If the quartz clock is running too slowly, how could you vary a to make the clock run more quickly? Explain and support your answer. [8 marks]