acker

Ackermann's formula (pole placement gain selection for single-input systems).

Syntax

K = acker(A, B, P)

Inputs

A
The state matrix (n x n), where n is the number of states.
B
The input matrix (n x 1).
P
The desired closed-loop pole location vector.

Outputs

K
Feedback gain matrix K.

Example

Control design of a magnetic levitation system with Ackermann's formula using pole placement:
Figure 1. 

A=[0 1;20.6 0];
B=[0;1];
C=[1 0];
D=[0];                   
t = 0:0.01:2;                 %Time Vector
u = zeros(size(t));           %Input vector
x0 = [0.01 0];                %Initial Condition
%-----Closed loop Poles location
p1 = -1.80000 + 2.40000i;
p2 = -1.80000 - 2.40000i;
K = acker(A,B,[p1 p2])
sys_cl = ss(A-B*K,B,C,0);  
lsim(sys_cl,u,t,x0);       %Simulate time response of dynamic system to arbitrary inputs
xlabel('Time (sec)');
ylabel('Ball Position (m)');
grid;
K = [Matrix] 1 x 2
29.60000  3.60000
Figure 2.

Comments

K = acker(A, B, P) computes the feedback gain matrix K such that the single input systemx = Ax + Bu with a feedback law of u = -K * x has the closed loop poles specified in vector P, with P = eig(A - B * K).