Difficulties in pole placement of an observer usign "place" command

I have this MIMO system, which as you can see is fully observable and fully controllable: yet if I attempt to place poles in a desired position using 'place' i obtain the error message:

A=[0  1  0 0;1690.3  0  -17.71 0.065111;0 0 -234.42  0; 0    0    0  -1.0882e+05];
B=[0 0; 0 0;1031.5 0;0  2.9056e+05];
C=[1 0 0 0]

C = 1×4

1 0 0 0

D=[0]

D = 0

rank(ctrb(A,B),10e-4)

ans = 4

rank(obsv(A,C))

ans = 4

L=place(A',C',[-1 -2 -3 -4])'

The only way to make it works is to place one of the poles very far,but I receive e warning, for example:

L1=place(A',C',[-1 -2 -3 -4e5])'

Warning: Achieved pole locations differ by more than 10 percent from specified locations.

L = 4×1

1.0e+16 * 0.0000 -0.0000 0.0000 5.5048

Answer 1

[Sam Chak]

Hi @Salvatore De Luca

In general, the Observer is typically designed to respond at least twice as fast as the full-state closed-loop system. However, the initial eigenvalue selection of [-1 -2 -3 -4] for the observer appears to be too slow.

%% MISO system
A   = [0      1    0     0;
       1690.3 0  -17.71  0.065111;
       0      0 -234.42  0; 
       0      0    0    -1.0882e+05];
B   = [0      0; 
       0      0;
       1031.5 0;
       0      2.9056e+05];
C   = [1 0 0 0];
D   =  0*C*B;
sys = ss(A, B, C, D)

sys =
 
  A = 
               x1          x2          x3          x4
   x1           0           1           0           0
   x2        1690           0      -17.71     0.06511
   x3           0           0      -234.4           0
   x4           0           0           0  -1.088e+05
 
  B = 
              u1         u2
   x1          0          0
   x2          0          0
   x3       1032          0
   x4          0  2.906e+05
 
  C = 
       x1  x2  x3  x4
   y1   1   0   0   0
 
  D = 
       u1  u2
   y1   0   0
 
Continuous-time state-space model.

%% Check stability
isstable(sys)

ans = logical
   0

%% Check Controllability and Observability
rkC = rank(ctrb(A, B))

rkC = 2

rkO = rank(obsv(A, C))

rkO = 4

%% Controller gain matrix
[K, S, Cp] = lqr(A, B, eye(4), eye(2))

K = 2×4

-203.5183 -5.0501 0.8796 -0.0000 0.7470 0.0186 -0.0003 0.6933

S = 4×4

513.1477 12.2520 -0.1973 0.0000 12.2520 0.3045 -0.0049 0.0000 -0.1973 -0.0049 0.0009 -0.0000 0.0000 0.0000 -0.0000 0.0000

Cp = 4×1

1.0e+05 * -0.0003 -0.0005 -0.0106 -3.1027

%% Observer gain matrix
Op  = 2*Cp  % Observer should respond at least 2 times faster than the closed-loop system

Op = 4×1

1.0e+05 * -0.0007 -0.0010 -0.0212 -6.2054

L   = place(A', C', Op)'

L = 4×1

1.0e+16 * 0.0000 -0.0000 -0.0000 9.1314

%% Observed-state feedback control system
cls = ss([A-B*K B*K; zeros(size(A,1)) A-L*C], eye(2*size(A,1)), eye(2*size(A,1)), eye(2*size(A,1)));

%% Check stability
isstable(cls)

ans = logical
   1