University:
Eastern Kentucky University
Course:
CSC 310 | Data Structures
Academic year:

2020
Views:

322

Pages:

6
Author:

Lanrwst4tn

Chapter 6 61 C = [[i, 0, 0], [0, 0, - i], [0, - i, y]] P(C) = 3 controllable Q = [[1, 2, I], [- 1, - 2, - i], [I, 2, i]] P(theta) = l not obrewable [ matrix 0&AB matrix ]= [[0, i, i, 0, 0], [i, 0, 0, 0, 0], [0, 0, i, 2, 0]] It has full row rand thus con hollable \mathcal{Theta} = [[i, 0, i], [0, 3, - i], [0, - 2, 4]] P(0) = 3 obserwable 6,3 [AB AB A cap B]=A[B AB A ^ (h - 1) * B ] A([A B A^ 2 B*** A^ n B])=P([BAB*** A ^ (n - 1) * B ]) if and only if A is ronsmgular 6.4 \{A, B\} controllable> rank [ matrix A i, i -SI&A i,j &B ,, \\ A 2 ,&A j,i -s,I&0 matrix ]=n for every 3 (J Pheorem 6,1, if is stated for every sigenvalue of A. However, ifs is not an agenvalue, chen (A-SI) das rank n. Thus the statement holde for song 6.) [ matrix A 21 &A 12 - 5I matrix ] Has full now rock \ A_{2}, A_{2} ,\ controllable. 65 dot x_{1} = u - x_{1} y = - y_{2} + 2w dot x_{2} = - x_{2} dot x = [[- i, 0], [0, - i]] * x + [[i], [0]] * omega y = [0 - 1] * x + 2w zeta = [[1, - 1], [0, 0]] r(c) = 1 not contiollably O = [[0, - i], [0, i]] l(theta) = 1 not obsewable 5.6 For the state equation in Problem 6.1. we have 13. If te obrevabilig index is defined as the least integer such Char rho([ matrix epsilon A \\ 0. A ^ - 1 matrix ])= rho([ matrix epsilon A \\ 0A^ * matrix ]) then v=1. (Note that can controllability and observability indices are defined in the text for controllable and obuwable state equations.) For dhe state equation in Problem 6.2 we have mu_{1} = 2, mu_{2} = 1 = z and mu = 3 mu = kappa_{0}*ex\{mu_{1}, mu_{2}\} 6.7 A_{i} = 1 for all si and mu = 1 dot x = [[- i, omega], [phi, - i]] * x + [[t], [i]] * mu C = [[1, 3], [1, 3]] y =[*]x We ulect p ^ - 1 = [[1, 0], [1, 1]] Then P = [[i, 0], [- i, 1]] and PA * P ^ - 1 = [[1, 0], [- 1, 1]] * [[- 1, omega], [omega - 1, 0]] * [[1, 0], [1, 1]] = [[3, omega], [0, - s]] overline B = PB = [[1], [0]] overline c = c * P ^ - 1 =[1,1 Thus overline x = px will trous form the epention to overline overline x =[ matrix 3&4\\ 0&-5 matrix ] tilde x + [[1], [0]] * 4 y = [[2, 1]] * y and the equation can be reduced to 1 x 1 = 3x_{1} + 4 q =2 overline x 1 This reduced equation in obrowable 25-