Stable SISO plant

Let ${\displaystyle P(s)}$ be a transfer function of a stable single-input single-output system (SISO) system. Further, let ${\displaystyle \Omega }$ be a set of stable and proper functions of ${\displaystyle s}$. Then, the set of all proper stabilizing controllers for the plant ${\displaystyle P(s)}$ can be defined as

${\displaystyle \left\{{\frac {Q(s)}{1-P(s)Q(s)}},Q(s)\in \Omega \right\}}$,

where ${\displaystyle Q(s)}$ is an arbitrary proper and stable function of s. It can be said, that ${\displaystyle Q(s)}$ parametrizes all stabilizing controllers for the plant ${\displaystyle P(s)}$.

General SISO plant

Consider a general plant with a transfer function ${\displaystyle P(s)}$. Further, the transfer function can be factorized as

${\displaystyle P(s)={\frac {N(s)}{M(s)}}}$, where ${\displaystyle M(s)}$, ${\displaystyle N(s)}$ are stable and proper functions of s.

Now, solve the Bézout's identity of the form

${\displaystyle \mathbf {N(s)Y(s)} +\mathbf {M(s)X(s)} =\mathbf {1} }$,

where the variables to be found ${\displaystyle (X(s),Y(s))}$ must be also proper and stable.

After proper and stable ${\displaystyle X,Y}$ are found, we can define one stabilizing controller that is of the form ${\displaystyle C(s)={\frac {Y(s)}{X(s)}}}$. After we have one stabilizing controller at hand, we can define all stabilizing controllers using a parameter ${\displaystyle Q(s)}$ that is proper and stable. The set of all stabilizing controllers is defined as

${\displaystyle \left\{{\frac {Y(s)+M(s)Q(s)}{X(s)-N(s)Q(s)}},Q(s)\in \Omega \right\}}$.

General MIMO plant

In a multiple-input multiple-output (MIMO) system, consider a transfer matrix ${\displaystyle \mathbf {P(s)} }$. It can be factorized using right coprime factors ${\displaystyle \mathbf {P(s)=N(s)D^{-1}(s)} }$ or left factors ${\displaystyle \mathbf {P(s)={\tilde {D}}^{-1}(s){\tilde {N}}(s)} }$. The factors must be proper, stable and doubly coprime, which ensures that the system ${\displaystyle \mathbf {P(s)} }$ is controllable and observable. This can be written by Bézout identity of the form:

${\displaystyle \left[{\begin{matrix}\mathbf {X} &\mathbf {Y} \\-\mathbf {\tilde {N}} &{\mathbf {\tilde {D}} }\\\end{matrix}}\right]\left[{\begin{matrix}\mathbf {D} &-\mathbf {\tilde {Y}} \\\mathbf {N} &{\mathbf {\tilde {X}} }\\\end{matrix}}\right]=\left[{\begin{matrix}\mathbf {I} &0\\0&\mathbf {I} \\\end{matrix}}\right]}$.

After finding ${\displaystyle \mathbf {X,Y,{\tilde {X}},{\tilde {Y}}} }$ that are stable and proper, we can define the set of all stabilizing controllers ${\displaystyle \mathbf {K(s)} }$ using left or right factor, provided having negative feedback.

${\displaystyle {\begin{aligned}&\mathbf {K(s)} ={{\left(\mathbf {X} -\mathbf {\Delta {\tilde {N}}} \right)}^{-1}}\left(\mathbf {Y} +\mathbf {\Delta {\tilde {D}}} \right)\\&=\left(\mathbf {\tilde {Y}} +\mathbf {D\Delta } \right){{\left(\mathbf {\tilde {X}} -\mathbf {N\Delta } \right)}^{-1}}\end{aligned}}}$

where ${\displaystyle \Delta }$ is an arbitrary stable and proper parameter.

Let ${\displaystyle P(s)}$ be the transfer function of the plant and let ${\displaystyle K_{0}(s)}$ be a stabilizing controller. Let their right coprime factorizations be:

${\displaystyle \mathbf {P(s)} =\mathbf {N} \mathbf {M} ^{-1}}$

${\displaystyle \mathbf {K_{0}(s)} =\mathbf {U} \mathbf {V} ^{-1}}$

then all stabilizing controllers can be written as

${\displaystyle \mathbf {K(s)} =(\mathbf {U} +\mathbf {M} \mathbf {Q} )(\mathbf {V} +\mathbf {N} \mathbf {Q} )^{-1}}$

where ${\displaystyle Q}$ is stable and proper.^[2]