1. Introduction

The static output feedback pole-assignment and its exact solution problems have been studied as a basic design problem in linear system theory and control engineering for last 4 decades ^(1-3). In $m$-input, $p$-output, $n^{th}$ order strictly proper linear time-invariant systems, this problem is stated by the complete algebraic real solution problem in n number of nonlinear polynomial equations with $mp$ number of real coefficients of $n^{th}$ order characteristic polynomial. There have been many various approaches on this nonlinear equation problem in linear MIMO systems. Grassmannian parametrization method could effectively handle the nonlinear geometry, but there was still a generic pole-assignment problem, which covers only open dense set of closed-loop poles ⁽¹⁾. For the evaluation of the singularities on the open dense set, a Grassmann invariant was proposed by Kacarnias and Giannakopoulos ⁽⁴⁾ and a central projection method was proposed by X. Wang ⁽⁵⁾.

The problems of pole placement by static output feedback (SOF) were w\ell-known and very critical in control engineering ⁽²⁾. In mathematical viewpoint, the SOF equations for pole placement is considered in the specific Grassmann space and the Plücker matrix formula $Lk=a$ is introduced for the real parametrization ⁽⁶⁾,⁽⁷⁾.

Through the real Grassmannian parametrization approach, it is shown that SOF pole placement can be exact pole assignable in the two-input, two-output, 4th order systems with strictly proper transfer functions (simply, (2,2,4) systems) ⁽⁸⁾.

In this paper, a complete characterization of SOF pole placement based on the real Grassmannian space is presented. As a case study, the following new results are explicitly derived:

1) The features on SOF pole placement in (2,2,4) systems can be completely reduced to 3 cases: exact pole assignable (EPA), EPA except a singular point (EPAES), and non pole assignable (NPA), where the distinctions entirely depend upon the columns’ conditions of Plücker sub-matrix and its the rank condition.

2) In 1), the real SOF gain of EPA and EPAES for the desired closed-loop pole can be calculated algebraically.

3) From 1) and 2), it is shown that several existing methods for SOF pole placement are unified.

This paper is organized as follows. A numerical construction of Plücker matrix form of $Lk=a$ in (2,2,4) systems is presented in section 2. The real Grassmannian parametrization algorithm and the total features of SOF pole placement are given in section 3. In section 4, it is demonstrated how the poles of EPA, EPAES, and NPA systems can be designated in a deterministic way in the real Grassmannian parametrization method, and compared with previous other methods. Conclusions are given in section 5.

2. Numerical Construction of Plücker Matrix Form

The Plücker matrix form in Grassmann space can be simply summarized as follows. Under the Grassmann space, the SOF equations for pole-assignment are theoretically decomposed into a linear vector equation and some quadratic constraints ⁽⁷⁾,

where $L\in{R}^{(n+1)\times(\sigma +1)}$ indicates the Plücker matrix with $\sigma =\left(\begin{matrix}m + p \\ m\end{matrix}\right)- 1$, $k$ indicates the extended vector of SOF matrix $K\in{R}^{m\times p}$, and $a =[1 a_{1}a_{2}\cdots a_{n}]^{t}$ is the coefficient vector of closed-loop characteristic polynomial.

A general numerical construction algorithm of $L k = a$ and its localized QRs in inhomogeneous coordinates for specified ($m,\: p,\: n$) systems was given in ⁽⁹⁾. The key idea is that the Binet-Cauchy theorem is applied to determinental matrix formula $\det[I +K G(s)]$ and then compares it with the signal flow graph analysis of the loop determinant $\triangle =\det[I +K G(s)]$ in Mason’s gain formula.

The localized inhomogenous coordinates in $k$ are obtained by

where $k_{11},\:\cdots ,\: k_{mp}$ indicates the entries of SOF matrix $K$ and

where $r =\sigma - mp$ in $m\ge p$ systems.

Let the $\sigma +1$ columns of Plücker matrix $L$ as one-to-one counterparts of $k$ be denoted by

$L =\left[l_{0}l_{11}\cdots l_{mp}l_{i1}\cdots l_{ir}\right]^{t}$

Then the columns of $L$ represent the ingredients of the real coefficients in the transfer function matrix.

$l_{0}$: coefficient vector of open-loop characteristic polynomial, $p(s)= s^{n}+ b_{1}s^{n-1}+\cdots + b_{n}$.

$l_{11},\:\cdots ,\: l_{mp}$: coefficient vector of numerator polynomial, $n_{11}(s),\:\cdots ,\: n_{mp}(s)$.

$l_{i1},\:\cdots ,\: l_{ir}$: coefficient vector of interacting factor,

Consider a (2,2,4) system given as

where $G(s)$ is a strictly proper transfer function matrix. A feedback control input $u(s)= - K y(s)$ is applied to the systems. It is easy to see that the desired 4th order closed-loop characteristic polynomial $p_{c}(s)$ is written as

Define $K$ and $G(s)$ as follows.

where $n_{i}(s)(i=1,\:\cdots ,\: 4)$ is a numerator polynomial of $G_{i}(s)$, respectively.

Then, the desired closed-loop characteristic polynomial $p_{c}(s)$ can be represented by

where $s_{1},\:\cdots ,\: s_{4}$ are closed-loop poles, $n_{i}(s)= n_{i1}s^{3}+$ $n_{i2}s^{2}+ n_{i3}s + n_{i4},\: i=1,\:\cdots ,\:4$, $n_{5}(s)(:= p(s)\det(G(s)))=$$n_{52}s^{2}+ n_{53}s + n_{54}$, and $k_{5}(:=\det(K))= k_{1}k_{4}- k_{2}k_{3}$.

From Eq.(9), the w\ell known Plücker matrix formula $L k = a$ in (2,2,4) systems is given as follows.

where

$L =\begin{bmatrix}1 & 0 &\cdots &0 &0 \\ b_{1}&n_{11}&\cdots &n_{41}&0 \\ b_{2}&n_{12}&\cdots &n_{42}&n_{52}\\ b_{3}&n_{13}&\cdots &n_{43}&n_{53}\\ b_{4}&n_{14}&\cdots &n_{44}&n_{54}\end{bmatrix},\: k=\begin{bmatrix}1\\ k_{1}\\\vdots \\ k_{4}\\ k_{5}\end{bmatrix},\: a=\begin{bmatrix}1 \\ a_{1}\\ a_{2}\\ a_{3}\\ a_{4}\end{bmatrix}$

The equation in (9) is rearranged into

and the reduced form for Plücker matrix formula (10) can be obtained by

where

For an elaborate examination, let’s notate the real coefficient column vectors in $L_{sub}$ by $L_{sub}=[l_{1},\: l_{2},\: l_{3},\: l_{4},\: l_{5}]$.

3. Characterization of SOF pole assignability

To consider the pole placement by SOF, the following definitions are introduced ⁽⁸⁾.

Definition 1. An $n^{th}$ order system with transfer function matrix $G(s)= N_{R}(s)D_{R}(s)^{-1}$ is exact pole assignable (EPA) if any $n^{th}$ order closed-loop polynomial $p_{c}(s)= s^{n}+ a_{1}s^{n-1}+\cdots + a_{n}$ can be achieved using some real SOF $K$.

Definition 2. An $n^{th}$ order system with transfer function matrix $G(s)$ is non pole assignable (NPA) by real SOF $K$ if given any $n^{th}$ order closed-loop characteristic polynomial, there does not exist a real SOF $K$ that covers all real coefficients $(a_{1},\:\cdots ,\: a_{n})$ in ${R}^{n}$.

Considering the invariantal necessary condition that the rank of $L_{sub}$ is 4 (simply, rk($L_{sub}$)=4). All possible geometries for the SOF pole-assignment of (2,2,4) systems are given by the following theorems.

Theorem 1 ⁽⁸⁾. The (2,2,4) systems are EPA, if the last column $l_{5}$ in $L_{sub}$ is zero under rk($L_{sub}$)=4.

Proof. If the last column $l_{5}$ in $L_{sub}k_{sub}=a_{sub}$ is zero, then the four SOF variables $k_{1},\: k_{2},\: k_{3},\: k_{4}$ in $k_{sub}$ are always determined by the real values in $L_{sub}k_{sub}=a_{sub}$ without constraint $k_{5}$. Thus, the real solution set of the linear vector equation $L_{sub}k_{sub}=a_{sub}$ is complete on the real vector field ${R}^{n}$. Thus the (2,2,4) systems are EPA by real SOF. □

From the geometric classification, it is observed that the polynomial SOF equations for closed-loop poles $L_{sub}k_{sub}=a_{sub}$, constrained by $k_{5}-k_{1}k_{4}+ k_{2}k_{3}=0$, are reduced into one of the following two equations: 1 variable 1st or 2nd order equation ⁽²⁾. The SOF pole assignabilities of exact pole assignable except a singular point (EPAES) in the following Theorem 2 can be proved in very similar to Theorem 1.

Theorem 2. The (2,2,4) systems are EPAES, if one of 4 columns, $\{l_{1},\:l_{2},\:l_{3},\:l_{4}\}$ in $L_{sub}$ is zero under rk($L_{sub}$)=4.

Proof. If one of the first 4 columns of matrix $L_{sub}$ is zero under rk($L_{sub}$)=4, then the four real SOF variables except $k_{i}$ variable are always determined from $L_{sub}k_{sub}=a_{sub}$ in (12) and (13). Substituting 4 values into the 5 variable quadratic equation (QE) reduces $k_{5}-k_{1}k_{4}+ k_{2}k_{3}=0$ to a 1 variable 1st order linear equation. Thus, the real solution of the SOF equtions, $L_{sub}k_{sub}=a_{sub}$ and QE, is complete on the real field ${R}$ except a singular point where the gain multiplied to $k_{i}$ in the QE is zero. □

The following theorems consider the other two cases.

1) Some two nonzero columns of $L_{sub}$ are linearly dependent.

2) Every two nonzero columns of $L_{sub}$ are linearly independent.

Theorem 3. The (2,2,4) systems are NPA if two columns, $\left\{\ell_{1},\: \ell_{4}\right\}$ or $\left\{\ell_{2},\:\ell_{3}\right\}$ in $L_{sub}$, are linearly dependent under rk($L_{sub}$)=4.

Proof. If two columns of $L_{sub}$, $\left\{\ell_{1},\: \ell_{4}\right\}$ or $\left\{\ell_{2},\:\ell_{3}\right\}$ are linearly dependent, then four SOF variables where one combined variable $x$ is represented by $x=k_{1}+\gamma k_{4}$or$x=k_{2}+\delta k_{3}$ for real constants $\gamma$ and$\delta$, are always determined by real values. Substituting the four real values into $k_{5}- k_{1}k_{4}+ k_{2}k_{3}= 0$ reduces the QE to a 1 variable 2nd order equation. Thus, due to the algebraic nature of the 1 variable 2nd order equation constructed for some real vector $a_{sub}$, these (2,2,4) systems are NPA within some real-disconnected interval for the SOF gain variables in ${R}$. □

Theorem 4. The (2,2,4) systems are EPAES by real SOF if two columns, $\ell_{5}$ and one column of {$\ell_{1},\: \ell_{2},\: \ell_{3},\: \ell_{4}$} in $L_{sub}$, are linearly dependent under rk($L_{sub}$)=4.

Proof. If two columns in $L_{sub}$, $\{\ell_{i},\: \ell_{5}\}$$(i = 1,\:\cdots ,\:4)$ are linearly dependent, then four variables with a combined variable $x= k_{5}+\lambda k_{i}$ for real constant $\lambda$ are always determined by real values. Substituting the four real values into $k_{5}- k_{1}k_{4}+ k_{2}k_{3}= 0$ reduces the QE to a 1 variable 1st order equation with a singular point. For example, let $k_{i}= k_{2}$, then from $x = k_{5}+\lambda k_{i}=$$\alpha_{1}\alpha_{4}-\alpha_{3}k_{2}+\lambda k_{2}=\alpha_{x}$, the $k_{2}$ is obtained by $k_{2}=(\alpha_{1}\alpha_{4}-\alpha_{x})/(\alpha_{3}-\lambda)$. In this case, $k_{i}$ is one of $\left\{k_{1},\: k_{2},\: k_{3},\: k_{4}\right\}$ and $\alpha_{x},\:\alpha_{1},\:\alpha_{3},\:\alpha_{4}$ indicate the real values of $x,\: k_{1},\: k_{3},\: k_{4}$, respectively, in $L_{sub}k_{sub}=a_{sub}$. In this way, the SOF $k_{2}$ has a singular point at $\alpha_{3}=\lambda$ for a special real vector $a_{sub}$. Thus, these (2,2,4) systems are exact pole assignable except a singular point at $k_{i}$. □

Theorem 5. The (2,2,4) systems are EPAES if two columns, $\left\{\ell_{1},\: \ell_{2}\right.$ $\left.(or \ell_{3})\right\}$ or $\left\{\ell_{4},\: \ell_{2}\right.$ $\left.(or \ell_{3})\right\}$ in $L_{sub}$, are linearly dependent under rk($L_{sub}$)=4.

Proof. If two columns in $L_{sub}$, $\left\{\ell_{1},\: \ell_{2}\right.$ $\left.(or \ell_{3})\right\}$ or $\left\{\ell_{4},\: \ell_{2}\right.$ $\left.(or \ell_{3})\right\}$, are linearly independent, then from $L_{sub}k_{sub}=a_{sub}$, four SOF variables where one combined variable $x$ is represented by $x = k_{i}+\rho k_{j}$ for real constant $\rho$, are always determined by real values. Substituting the four real values in $k_{5}$ reduces the QE to a 1 variable 1st order equation with a singular point. For example, let $k_{i}= k_{1}$ and $k_{j}= k_{2}$, then from $x = k_{1}+\rho k_{2}=\alpha_{x}$, $k_{5}$ is obtained by $\alpha_{5}= k_{1}\alpha_{4}-\alpha_{3}(\alpha_{x}- k_{1})/\rho$. Therefore, $k_{1}$ is given by $k_{1}=(\rho\alpha_{5}+\alpha_{3}\alpha_{x})/(\alpha_{3}+\rho\alpha_{4})$ and has a singular point at $\alpha_{3}+\rho\alpha_{4}= 0$ for a special real vector $a_{sub}$. Thus, these (2,2,4) systems are EPAES at $k_{i}$ and $k_{j}$. □

Theorem 6. The (2,2,4) systems are arbitrary NPA if every two columns of $L_{sub}$ are linearly independent under rk($L_{sub}$)=4.

Proof. If every 2 columns of $L_{sub}$ are linearly independent under rk($L_{sub}$)=4, then from the 1st 4 diagonalized matrix $L_{sub}^{'}$ in $L_{sub}^{'}k_{sub}=a_{sub}^{'}$, 2~4 variables among $k_{1},\:\cdots ,\: k_{4}$ in $k_{sub}$ can be expressed as a linear functions for the last remaining variable, $k_{5}$.

i) 4 variable linear combination case: The 1st 4 variables in $L_{sub}^{'}k_{sub}=a_{sub}^{'}$ depend upon the last 1 variable $k_{5}$.

In this case, let

$\beta_{1}\ell_{1}'+\beta_{2}\ell_{2}'+\beta_{3}\ell_{3}'+\beta_{4}\ell_{4}'=\ell_{5}'$

where $Ell_{i}^{'}$ indicates the $i^{th}$ column of $L_{sub}^{'}$, then all 4 variables $k_{1},\:\cdots ,\: k_{4}$ are linear functions on the variable $k_{5}$. Thus the QE, $k_{5}- k_{1}k_{4}+ k_{2}k_{3}= 0$ is always expressed as a 1 variable 2nd order equation of $k_{5}$ constructed through arbitrary selection of 4 variables for some real vector $a_{sub}^{'}$.

ii) 3 variable linear combination case: In the similar approach as i), 3 variables among 4 variables $k_{1},\:\cdots ,\: k_{4}$ depend upon the last 1 variable $k_{5}$. For example, let $\beta_{2}\ell_{2}'+\beta_{3}\ell_{3}'+\beta_{4}\ell_{4}'=\ell_{5}'$, then 3 variables $k_{2},\: k_{3},\: k_{4}$ have linear functions with the variable $k_{5}$. Thus the QE, $k_{5}- k_{1}k_{4}+ k_{2}k_{3}= 0$ is always expressed as a 1 variable 2nd order equation of $k_{5}$ constructed through arbitrary selection of 3 variables for some real vector $a_{sub}^{'}$.

iii) 2 variable linear combination case: In the similar approach as ii), 2 variables among 4 variables $k_{1},\:\cdots ,\: k_{4}$ depend upon the last 1 variable $k_{5}$. For example, let $\beta_{3}\ell_{3}'+\beta_{4}\ell_{4}'=\ell_{5}'$, then 2 variables $k_{3},\: k_{4}$ have linear functions with the variable $k_{5}$. Thus the QE, $k_{5}- k_{1}k_{4}+ k_{2}k_{3}= 0$ is always expressed as a 1 variable 1st or 2nd order equation of $k_{5}$ constructed through arbitrary selection of 2 variables for some real vector $a_{sub}^{'}$. □

Remark 1. The NPA case can be further specifically classified into rk($L_{sub}$)=4 and rk($L_{sub}$)<4. The NPA systems with rk($L_{sub}$)=4 can have some stabilizable feature if the local regions obtained by some real-connected intervals in left half s-plane, but the NPA systems have hardly stabilizable feature by intrinsic rank deficiency, rk($L_{sub}$)<4. These NPA systems can be pole assignable by properly chosen dynamic output feedback ⁽¹⁰⁾.

In Table 1, from Theorem 1 - Theorem 6, the EPA exists only on the case of the interacting column is zero ($l_{5}= 0$), and the EPAES exists only on two cases: One of numerator columns is zero and two columns except 2 crossed numerator positions, (like $\{l_{1},\: l_{4}\}$ or $\{l_{2},\: l_{3}\}$) are linearly dependent. Finally, the NPA with rk($L_{sub}$)=4 exists only on two cases: Every two columns are linearly independent and two columns between two crossed numerator positions are linearly dependent.

4. Numerical Examples

In order to show the efficiency of the proposed method, five examples are given for three cases.

Example 1. EPA case ⁽¹¹⁾

Consider a strictly proper system given by

The transfer function $G(s)(=C(s I-A)^{-1}B)$ is obtained by

From (10), $Lk=a$ is constructed by

In the rank test, rk($L_{sub}$)=4 and the last column of $L_{sub}$ is zero. From Theorem 1, this SOF system has EPA feature.

Example 2. EPAES case ⁽¹²⁾

Consider a strictly proper system given by

From $G(s)$ and (10), $Lk=a$ is constructed by

In the rank test, rk($L_{sub}$)=4 and one column $l_{3}$ of $L_{sub}$ is zero. From Theorem 2, this SOF system has EPAES feature.

Example 3. EPAES case ⁽⁹⁾

Consider a strictly proper system given by

From $G(s)$ and (10), $Lk=a$ is constructed by

In the rank test, rk($L_{sub}$)=4 and two columns, $l_{3}$ and $l_{5}$ in $L_{sub}$ are linearly dependent. Thus, this SOF system has EPAES feature from Theorem 4.

Example 4. NPA case ⁽¹³⁾

Consider a strictly proper system given by

From $G(s)$ and (10), $Lk=a$ is constructed by

In the rank test, rk($L_{sub}$)=4 and two columns, $l_{2}$ and $l_{3}$ in $L_{sub}$ are linearly dependent. Thus, from Theorem 3, this SOF system has NPA feature.

Example 5. NPA case ⁽¹⁴⁾

Consider a strictly proper system given by

From $G(s)$ and (10), $Lk=a$ is constructed by

In the rank test, rk($L_{sub}$)=4 and every two columns of are linearly independent. From Theorem 6, this system has NPA feature.

5. Conclusions

In this paper, a parametric study of the static output feedback pole placement problem for two-input, two-output, 4th order systems with strictly proper transfer functions is completely characterized by the real Grassmannian paramerization method. In order to classify the cases, Plücker matrix formula $L k = a$ is adopted. The existing pole placement methods can be unified using proposed real Grassmannian parametrization method.