Tartalomjegyzék

2017b CCS gyak4. Subspaces

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Cascade tank system

System model

Generate symbolic system parameters $k_1 ,k_2 ,k_3 ,k_4 >0$ (flow rate of the corresponding tank).

pcz_generateSymStateVector(4,'k');
pcz_generateSymStateVector(4,'x');
assumeAlso(k>0)

State space matrices

A = [
    -k1 0 0 0
    k1 -k2 0 0
    0 k2 -k3 0
    0 0 k3 -k4
    ];
B = [ 0 1 0 0 ]';
C = [ 0 0 1 0 ];

Observability and controllability matrices

C4 = [ B A*B A^2*B A^3*B ], rank(C4)
O4 = [ C ; C*A ; C*A^2 ; C*A^3 ], rank(O4)

Output:

C4 =
[ 0,   0,              0,                              0]
[ 1, -k2,           k2^2,                          -k2^3]
[ 0,  k2, - k2^2 - k3*k2,       k3*(k2^2 + k3*k2) + k2^3]
[ 0,   0,          k2*k3, - k3*(k2^2 + k3*k2) - k2*k3*k4]
ans =
     3
O4 =
[                              0,                        0,     1, 0]
[                              0,                       k2,   -k3, 0]
[                          k1*k2,           - k2^2 - k3*k2,  k3^2, 0]
[ - k2*(k1^2 + k2*k1) - k1*k2*k3, k3*(k2^2 + k3*k2) + k2^3, -k3^3, 0]
ans =
     3

Subspaces

Controllable subspace

XC_ = orth(C4)

Output:

XC_ =
[ 0, 0, 0]
[ 1, 0, 0]
[ 0, 1, 0]
[ 0, 0, 1]

State variables $\left(x_2 ,x_3 ,x_4 \right)$ are controllable

Uncontrollable subspace

Xc_ = null(XC_')

Output:

Xc_ =
 1
 0
 0
 0

State variable $x_1$ is not controllable

Unobservable subspace

X_o = null(O4)

Output:

X_o =
 0
 0
 0
 1

State variable $x_4$ is not observable

Observable subspace

X_O = null(X_o')

Output:

X_O =
[ 1, 0, 0]
[ 0, 1, 0]
[ 0, 0, 1]
[ 0, 0, 0]

State variables $\left(x_1 ,x_2 ,x_3 \right)$ are observable

Controllable staircase form

S = [XC_ Xc_]
x_ = S'*x
A_ = S' * A * S, B_ = S' * B, C_ = C * S

Output:

S =
[ 0, 0, 0, 1]
[ 1, 0, 0, 0]
[ 0, 1, 0, 0]
[ 0, 0, 1, 0]
x_ =
 x2
 x3
 x4
 x1
A_ =
[ -k2,   0,   0,  k1]
[  k2, -k3,   0,   0]
[   0,  k3, -k4,   0]
[   0,   0,   0, -k1]
B_ =
 1
 0
 0
 0
C_ =
[ 0, 1, 0, 0]

Kalman decomposition (todo)

System model

A = [
   1.613378610435994  -3.995855597631523  -0.810450484741737  -0.168720889102838
   0.734243523886419  -1.441227814368974  -0.230398914393346  -0.184908002657739
  -0.645586131749136  -2.271869456330054  -1.257298097726161   0.738442163303650
   1.337529417392070  -5.440050610034118  -1.733590645293144  -0.143058326670208
   ];

B = [
    1.420709123535001
    0.656198450951251
   -0.385009953764639
    1.668593611633082
    ];

Observability and controllability matrices

C = [ 0.715755046956996  -3.906572661173081  -0.886008486028291   0.751557463505590 ];
rtol = 1e-10;
Cn = ctrb(A,B), rank(Cn,rtol)
On = obsv(A,C), rank(On,rtol)

Output:

Cn =
    1.4207   -0.2994    0.7760   -0.7495
    0.6562   -0.1224    0.3454   -0.3295
   -0.3850   -0.6918    0.4249   -0.6073
    1.6686   -1.2408    1.6422   -1.8125
ans =
     2
On =
    0.7158   -3.9066   -0.8860    0.7516
   -0.1364    0.6946    0.1311   -0.1602
   -0.0089    0.1175    0.0634    0.0143
    0.0501   -0.3555   -0.1243    0.0245
ans =
     2

Published with MATLAB® R2017b, and sanitarized by BeautifulSoup (Python script using BSoup is written by Polcz).