acker -Pole placement design for single-input systems

Syntax

k = acker(A,b,p)

Description

k = acker(A,b,p)

Given the single-input system

and a vector p of desired closed-loop pole locations, acker (A,b,p) uses Ackermann's formula [1] to calculate a gain vector k such that the state feedback u = −kxplaces the closed-loop poles at the locations p. In other words, the eigenvalues of A − bk match the entries of p (up to ordering). Here A is the state transmitter matrix and b is the input to state transmission vector.

>> %Example 9.2 - State feedback gain calculation 
>> A=[0 1 0; 0 0 1; -6 -11 -6]; 
>> p1=poly(A)

p1 =

    1.0000    6.0000   11.0000    6.0000

>> P=[-3+3j -3-3j -6]; 
>> p2=poly(P) %the "alpha" coefficients

p2 =

     1    12    54   108

>> p2-p1  %Ki as in Eq. (9-21)

ans =

     0     6    43   102

>> %Now re-do using Ackermann's formula 
>> B=[0; 0; 1]; 
>> acker(A,B,P)

ans =

   102    43     6

>> %Redo Ackermann's formula using its definitions 
>> M=[B A*B A^2*B];    %controllability matrix 
>> ac=polyvalm(p2,A);  %Eq.(9-23) 
>> [0 0 1]*inv(M)*ac   %Eq.(9-22)

ans =

   102    43     6

最后

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