lect10a - Recipes for Pole Placement Task Consider an nth...

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Recipes for Pole Placement Task: Consider an n th order linear controllable single-input (i.e., u being scalar) system ˙ x = Ax + Bu . With the linear state feedback u = Kx , the closed-loop system is ˙ x = ( A + BK ) x . Given a set of n desired eigenvalues p 1 , . . . , p n for the closed-loop system matrix ( A + BK ), find a K which places the eigenvalues of ( A + BK ) at those desired values. You can use either of the two recipes given below for this task. Recipe 1: 1. Find the controllability matrix Γ n = [ B, AB, A 2 B, . . . , A n - 1 B ]. 2. Find the characteristic polynomial p ( λ ) = λ n + a n - 1 λ n - 1 + . . . + a 1 λ + a 0 of the matrix A . 3. Find the column vector q 1 = (Γ T n ) - 1 e n where e n = [0 , . . . , 0 , 1] T is the 1 × n column vector with 1 as the n th element and zero everywhere else. Define Q = q T 1 q T 1 A q T 1 A 2 . . . q T 1 A n - 1 . If we perform the coordinate transformation ˆ x = Qx , then in the new coordinate frame, the system is in controller canonical form. 4. Find the characteristic polynomial corresponding to the desired eigen- values, i.e., find ( λ + p 1 )( λ + p 2 ) . . . ( λ + p n ). Denote this characteristic polynomial by λ n + d n - 1 λ n - 1 + . . . + d 1 λ + d 0 . 5. Let ˆ k i = a i - 1 - d i - 1 , i = 1 , . . . , n . Define ˆ K = [ ˆ
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