Decoupling (automatic)

The goal of decoupling is to transform the transfer transfer functions or the representations of multivariable states to be able to control each output independently of the others.

That is to say the system: $\ dowry X = has \ cdot X + B \ cdot U$ and $Y = C \ cdot X$ with dim X = m; dim U = p; dim there = p

from where: $Y = F (S) \ cdot U (S)$ with $F (S) = C (if-HAVe) ^ {- 1} B$ F is called matrix of square transfer.

Approaches transfer Transfer function:

In loop closed: $Y (S) =^ {- 1} F (S) C (S) Yd (S)$ with Yd the instruction

Decoupling consists with diagonaliser the transfer transfer function of loop closed.

$FTBF = ^ {- 1} F (S) C (S)$

Thus the corrector C (S) must check: $^ {- 1} F (S) C (S) = \ Omega (S)$ with $\ Omega (S)$ = diagonal matrix

One thus has: $C (S) = F (S) ^ {- 1} \ Omega (S) ^ {- 1}$

Representation of state approaches:

there is $y = \ begin {bmatrix} y_1 \ \ \ vdots \ \ y_p \ end {bmatrix} = \ begin {bmatrix} C_1^T \ \ \ vdots \ \ C_p^T \ end {bmatrix} X$

that is to say I smallest entirety such as $C_1^TA^jB \ not= 0$

then $y^ {(j+1)}(T) =C_1^TA^ {j+1} x+C_1^TA^jBu$

in the same way for the other exits: $y_i^ {(d_i+1)}(T) =C_i^TA^ {d_i+1} x+C_i^TA^ {d_i} Bu$

One obtains $\ begin {bmatrix} y_1^ {d_1+1} \ \ \ vdots \ \ y_p^ {d_p+1} \ end {bmatrix} = \ begin {bmatrix} C_1^TA^ {d_1+1} \ \ \ vdots \ \ C_p^TA^ {d_p+1} \ end {bmatrix} X + \ begin {bmatrix} C_1^TA^ {d_1} B \ \ \ vdots \ \ C_p^TA^ {d_p} B \ end {bmatrix} U = v$

Y = Fx + Lu = v

one thus finds $u = L^ {- 1}$

The system is découplale if L is invertible.

Moreover uncoupled subsystems are form

$\ frac {Y_i (S)}{V_i (S)}= \ frac 1 {s^ {d_i+1}}$ One leads to integrators.

Random links: