Algoritmo Escada e Modos do Sistema

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Problem 7.30PP
Staircase Algorithm (Var) Dooren et al., 1978):Any realization (A.B.C) can be transformed by an 
orthogonal similarity transformation to (A, B, C), where A is an upper Hessenberg matrix 
(having one nonzero diagonal above the main diagonal) given by
' * ai • O ' ■ 0 ■
a = t’'at = * * '. 0 , B = T’'B =
* ♦ Oj,_l 0
. «l .
where g1 # 0, and
C = CT = [ c , C l - c . 1, T U T r
Orthogonai transformations correspond to a rotation of the vectors (represented by the matrix 
columns) being transformed with no change in length.
fat Prove that if ai = 0 and ai+1...... an-^ # 0 for some /. then the controllable and uncontrollable
Orthogonal transformations correspond to a rotation of the vectors (represented by the matrix
columns) being transformed with no change in length.
(a) Prove that if ai = 0 and ai+1...an-1 ^ 0 for some /, then the controllable and uncontrollable
modes of the system can be identified after this transformation has been done.
(b) How would you use this technique to identify the observable and unobservable modes of (A, 
B,C)?
(c) What advantage does this approach for determining the controllable and uncontrollable 
modes have over transforming the system to any other form?
(d) How can we use this approach to determine a basis for the controllable and uncontrollable 
subspaces, as in Problem?
This algorithm can also be used to design a numerically stable algorithm for pole placement [see 
Minimis and Paige (1982)]. The name of the algorithm comes from the multi-input version in 
which the ai are the blocks that make A resemble a staircase. Refer to ctrbf, obsvf commands in 
Matlab.
Problem
Consider the system y+3y+2y=u+u.
(a) Find the state matrices Ac, Be, and Cc in control canonical form that correspond to the given 
differential equation.
(b) Sketch the eigenvectors of Ac in the (x1, x2) plane, and draw vectors that correspond to the 
completely observable (xO) and the completely unobservable (xO) state-variables.
(c) Express xO and xCin terms of the observability matrix O.
(d) Give the state matrices In observer canonical form and repeat parts (b) and (c) in terms of 
controllability instead of observability.
Step-by-step solution
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