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Youla-Kučera Parametrization

Let us consider a system
y=P(u+d),\begin{align} \bm{y}& = \bm{P}(\bm{u} + \bm{d}), \end{align}
where y\bm{y}, u\bm{u}, and d\bm{d} stand for an output, input, and disturbance, and P\bm{P} denotes a system that can be factorized using right and left coprime factors.
P≡ND−1=D~−1N~.\begin{align} \bm{P}&\equiv \bm{N}\bm{D}^{-1}=\tilde{\bm{D}}^{-1}\tilde{\bm{N}}. \end{align}
Assuming that the system is controllable and observable, the Bézout identity holds:
[YX−N~D~][D−X~NY~]=[I00I][D−X~NY~][YX−N~D~]=[I00I],\begin{align} \begin{bmatrix} \bm{Y} & \bm{X} \\ -\tilde{\bm{N}} & \tilde{\bm{D}} \end{bmatrix}\begin{bmatrix} \bm{D} & -\tilde{\bm{X}} \\ \bm{N} & \tilde{\bm{Y}} \end{bmatrix}&=\begin{bmatrix} \bm{I} & \bm{0} \\ \bm{0} & \bm{I} \end{bmatrix} \\ \begin{bmatrix} \bm{D} & -\tilde{\bm{X}} \\ \bm{N} & \tilde{\bm{Y}} \end{bmatrix}\begin{bmatrix} \bm{Y} & \bm{X} \\ -\tilde{\bm{N}} & \tilde{\bm{D}} \end{bmatrix}&=\begin{bmatrix} \bm{I} & \bm{0} \\ \bm{0} & \bm{I} \end{bmatrix}, \end{align}
where X\bm{X}, Y\bm{Y}, and Q\bm{Q} are free parameters that satisfy
{det(Y−QN~)≠0det(Y~−QN)≠0.\begin{align} \left\{\begin{matrix}{\rm det}(\bm{Y}-\bm{Q}\tilde{\bm{N}})\neq\bm{0} \\ {\rm det}(\tilde{\bm{Y}}-\bm{Q}\bm{N})\neq\bm{0}\end{matrix}\right.. \end{align}
Here, all stabilizing feedback controllers for this system is described as
C=(Y−QN~)−1(X+QD~)=(X~+DQ)(Y~−NQ)−1.\begin{align} \bm{C}&=(\bm{Y}-\bm{Q}\tilde{\bm{N}})^{-1}(\bm{X}+\bm{Q}\tilde{\bm{D}})=(\tilde{\bm{X}}+\bm{D}\bm{Q})(\tilde{\bm{Y}}-\bm{N}\bm{Q})^{-1}. \end{align}

Simplified 2-DoF Control

Let us consider a 2-DoF controller that generate an input as
u=DKr+C(NKr−y).\begin{align} \bm{u}&=\bm{D}\bm{K}\bm{r}+\bm{C}(\bm{N}\bm{K}\bm{r}-\bm{y}). \end{align}
It can be transformed into
u=DKr+(Y−QN~)−1(X+QD~)(NKr−y)⇔(Y−QN~)u=(Y−QN~)DKr+(X+QD~)(NKr−y)⇔Yu=QN~u+(Y−QN~)DKr+(X+QD~)(NKr−y)=(YD+XN)Kr−Xy+Q(N~u−D~y)+Q(D~N−N~D)Kr=Kr−Xy+QN~(u−P−1y)⇔u=Y−1(Kr−Xy)+Y−1QN~(u−P−1y).\begin{align} \bm{u}&=\bm{D}\bm{K}\bm{r}+(\bm{Y}-\bm{Q}\tilde{\bm{N}})^{-1}(\bm{X}+\bm{Q}\tilde{\bm{D}})(\bm{N}\bm{K}\bm{r}-\bm{y}) \\ \Leftrightarrow (\bm{Y}-\bm{Q}\tilde{\bm{N}})\bm{u}&=(\bm{Y}-\bm{Q}\tilde{\bm{N}})\bm{D}\bm{K}\bm{r}+(\bm{X}+\bm{Q}\tilde{\bm{D}})(\bm{N}\bm{K}\bm{r}-\bm{y}) \\ \Leftrightarrow \bm{Y}\bm{u}&=\bm{Q}\tilde{\bm{N}}\bm{u}+(\bm{Y}-\bm{Q}\tilde{\bm{N}})\bm{D}\bm{K}\bm{r}+(\bm{X}+\bm{Q}\tilde{\bm{D}})(\bm{N}\bm{K}\bm{r}-\bm{y}) \\ &=(\bm{Y}\bm{D}+\bm{X}\bm{N})\bm{K}\bm{r}-\bm{X}\bm{y} +\bm{Q}(\tilde{\bm{N}}\bm{u}-\tilde{\bm{D}}\bm{y}) +\bm{Q}(\tilde{\bm{D}}\bm{N}-\tilde{\bm{N}}\bm{D})\bm{K}\bm{r} \\ &=\bm{K}\bm{r}-\bm{X}\bm{y} +\bm{Q}\tilde{\bm{N}}(\bm{u}-\bm{P}^{-1}\bm{y}) \\ \Leftrightarrow \bm{u}&=\bm{Y}^{-1}(\bm{K}\bm{r}-\bm{X}\bm{y})+\bm{Y}^{-1}\bm{Q}\tilde{\bm{N}}(\bm{u}-\bm{P}^{-1}\bm{y}). \end{align}
It shows that an input contains a tracking controller and disturbance-suppression controller. The tracking controller works as
y=PY−1(Kr−Xy)=PY−1{Kr−(I−YD)N−1y}=PY−1{Kr−N−1y+YP−1y}=PY−1{Kr−N−1y}+y∴y=NKr.\begin{align} \bm{y}&=\bm{P}\bm{Y}^{-1}(\bm{K}\bm{r}-\bm{X}\bm{y}) \\ &=\bm{P}\bm{Y}^{-1}\left\{\bm{K}\bm{r}-(\bm{I}-\bm{Y}\bm{D})\bm{N}^{-1}\bm{y}\right\} \\ &=\bm{P}\bm{Y}^{-1}\left\{\bm{K}\bm{r}-\bm{N}^{-1}\bm{y}+\bm{Y}\bm{P}^{-1}\bm{y}\right\} \\ &=\bm{P}\bm{Y}^{-1}\left\{\bm{K}\bm{r}-\bm{N}^{-1}\bm{y}\right\}+\bm{y} \\ \therefore \bm{y}&=\bm{N}\bm{K}\bm{r}. \end{align}
The disturbance-suppression controller estimates a disturbance and suppresses it.
d^=Y−1QN~(u−P−1y)=−Y−1QN~dy=P(Y−1QN~(u−P−1y)+d)=P(−Y−1QN~d+d)=P(I−Y−1QN~)d.\begin{align} \hat{\bm{d}} &= \bm{Y}^{-1}\bm{Q}\tilde{\bm{N}}(\bm{u}-\bm{P}^{-1}\bm{y})\\ &= -\bm{Y}^{-1}\bm{Q}\tilde{\bm{N}}\bm{d}\\ \bm{y}&=\bm{P}(\bm{Y}^{-1}\bm{Q}\tilde{\bm{N}}(\bm{u}-\bm{P}^{-1}\bm{y}) + \bm{d})\\ &=\bm{P}(-\bm{Y}^{-1}\bm{Q}\tilde{\bm{N}}\bm{d} + \bm{d})\\ &=\bm{P}(\bm{I} - \bm{Y}^{-1}\bm{Q}\tilde{\bm{N}}) \bm{d}. \end{align}
The combination of these controllers generates an output as
y=NKr+P(I−Y−1QN~)d.\begin{align} \bm{y}&= \bm{N}\bm{K}\bm{r} + \bm{P}(\bm{I} - \bm{Y}^{-1}\bm{Q}\tilde{\bm{N}}) \bm{d}. \end{align}

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