Answer Happy

Consider the discrete-time system given by the state equation X[K + 1] = |x[K]+[K] x[k 1+ u[k] y[k] = 0 1 x[k]. (a) [2 marks] What is the characteristic polynomial of this system? (b) [3 marks] Characterise the internal and BIBO stability of this system: (i) Is the system asymptotically stable? (ii) Is the system Lyapunov stable? (iii) Is the system BIBO stable? (c) [5 marks] Design a state feedback control law of the form u[k] = r[k] - Kx[k] such that the closed-loop system response is dead-beat. What is the closed-loop transfer function obtained from r[k] to y[k]? Can the closed-loop system output asymptotically track constant input references? (d) [6 marks] Now suppose only the output y[k] is available for feedback. Design an observer for all the states of the system and compute the observer matrix L such that the observer characteristic polynomial is det(z/2x2-[A-LC]) = z²z+-- 4