Consider the following discrete-time system x[k + 1] = Ax[k] + Bu[k], where --(2) 4-2) -(1)....

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Consider the following discrete-time system x[k + 1] = Ax[k] + Bu[k], where --(2) 4-2) -(1). C- (10). = A = B = (a) A discrete system is in equilibrium when x[k+1] = a[k] = xe for all k. Let u = r be a constant input and compute the resulting equilibrium point for the system re (b) Show that if lai < 1 for all i, all initial conditions give solutions that converge to the equilibrium point (using the same constant input as part a). To simplify, you may assume that a11 # a22, which makes A diagonalizable. Hint 1: It is often useful to pull constant terms out of summations as soon as possible. y[k] = Ca[k], Hint 2: Recall the following result from linear algebra: when A is diagonalizable, A = TAT- = T (411 *7 0 022 7-1 where T is given by @12 ( = 022 T = 11 a12 011022 l=0 122) - (TU T2). Hint 3: Recall the following Taylor Series expansion: when [g] <1 Consider the following discrete-time system x[k + 1] = Ax[k] + Bu[k], where --(2) 4-2) -(1). C- (10). = A = B = (a) A discrete system is in equilibrium when x[k+1] = a[k] = xe for all k. Let u = r be a constant input and compute the resulting equilibrium point for the system re (b) Show that if lai < 1 for all i, all initial conditions give solutions that converge to the equilibrium point (using the same constant input as part a). To simplify, you may assume that a11 # a22, which makes A diagonalizable. Hint 1: It is often useful to pull constant terms out of summations as soon as possible. y[k] = Ca[k], Hint 2: Recall the following result from linear algebra: when A is diagonalizable, A = TAT- = T (411 *7 0 022 7-1 where T is given by @12 ( = 022 T = 11 a12 011022 l=0 122) - (TU T2). Hint 3: Recall the following Taylor Series expansion: when [g] <1