The Adjugate Method: Reassignment with System Classification

Abstract

Further elaborations on the adjugate method for eigenvalue–eigenvector assignment are discussed in this paper. Additional results concerning the admissible pair introduced recently are stated, verified, and commented on. Properties of the adjugate method concerning eigenvalue reassignment of controllable and uncontrollable systems are revealed and pursued further in this study A reformulation of how calculations should be carried out is investigated, pointing out merits, bounds, and limitations. It has been found that reassignment can involve a case where the closed-loop eigenvector companion $z_i$ is $z_i=0$ and the associated eligible closed-loop eigenvectors $w_i$ are open-loop ones. This can be considered an advantage in the sense of knowing $z_i$ and $w_i$ beforehand. This also applies to the case of uncontrollable eigenvalue reassignment, where enlarged closed-loop eigenvector subspaces are uncovered, enabling more flexible designs. Assessments of the traditional method compared to the adjugate method are commented on whenever appropriate. Numerical issues are pointed out where Leverrier’s algorithm is adapted for efficient matrix null space and matrix adjugate determination. A feature of the adjugate method is a result where $w_U=0$, indicating that the eigenvalue assigned is uncontrollable. Such a distinctive feature enables a classification procedure for systems regarding controllability and observability. The various concepts pointed out have been demonstrated and authenticated through carefully selected examples.

1. Introduction

In a recent paper [1], the traditional method of eigenstructure assignment [2,3,4,5,6] has been reexamined, reformulated, and manipulated through formula-wise computation. Overall, it contributes an alternative approach to eigenstructure assignment and provides a novel decomposition of the solution subspace pertinent to the traditional method. This methodology treated the case of newly assigned eigenvalues with their associated eigenvectors, culminating in an explicit approach that is formula-based and facilitates a form of parallelism.

A comeback is made to that paper with further elaborations, classifications, and extensions. Thus, this paper acts as an extension to the earlier one [1] entitled: “Eigenstructure Through Matrix Adjugates and Admissible Pairs”. The approach now concerns the notion of eigenvalue reassignment of controllable and uncontrollable systems. The references needed in this paper regarding the traditional method are curtailed in number, mainly those listed in [1], together with additional ones relevant to the new theory contributed in this paper. Additionally, textbooks’ treatment of eigenstructure is well explained in [7,8].

The versatility and applicability of the eigenstructure assignment method are well acknowledged in the recent literature, but a few applications to a three-axis dynamic flight motion simulator have been conducted in [9]. Since the differences between the two types of aircraft are reflected in their eigenvalues and eigenvectors, an approach based on multibody aircraft dynamics has been studied in [10]. Desired aircraft requirements on damping and stiffness were accomplished by implementing inner-loop control laws. In [11], the case of large-scale systems has been tackled; a partial eigenstructure assignment was used to derive new parametric solutions in addition to manipulating lower-order matrices. A problem with output feedback, where only partial assignment is possible, was addressed in [12], where it had been found that a trade-off between sensitivity and stability should be exercised. The viability of eigenstructure assignment methods is eminent regarding vibration cancellation and suppression. Large space structures have been reviewed in [13], where some experience-based designs proven effective.

The admissible pair $(w_i,z_i)$ is the backbone of the eigenstructure method. In [1], a modified form of the traditional eigenstructure method has been established. It explicitly determines the solution subspace, providing independent formula-based solutions that entail a form of parallelism. Such advantageous features have been acknowledged, identified, and proven in [1]. Such advantages are followed in this paper for the case of eigenvalue re-assignability, which demands vigilant handling when compared to the case of newly assigned eigenvalues as treated in [1].

The author in [1] went as far as considering the assignment of newly assigned eigenvalues, i.e., eigenvalues that are not eigenvalues of $A$. We foresee eigenvalue reassignment as a case worth further investigation, especially regarding uncontrollable systems. We further pursue such a quest and tackle the re-assignability problem for both controllability cases. Such pursuit leads to a new classification of system controllability and, consequently, system observability using the matrix adjugate. A role played by the open-loop eigenvectors is also exposed regarding eigenvalue re-assignability. The versatility and flexibility gained are worth pointing out.

Additional results are obtained regarding the case of re-assignability of uncontrollable eigenvalues and eigenvectors, yet demanding no distinctive treatment. They permit the possibility of a larger set of closed-loop eigenvectors. They are formally stated and proven in Section 5.

Classifying systems as controllable or uncontrollable relies on the values of the admissible pair being zero or otherwise. There is no need to resort to the rank test of the controllability matrix [14] or any others, such as the well-known PBH test [15,16].

We begin our study in Section 2, enumerating the roles of the adjugate and the null space methods in eigenstructure assignment. In Section 3, we settle on a problem formulation with certain definitions relevant to the solution process using further properties pertinent to the adjugate method and the admissible $(w_i,z_i)$ pair. The core of our study is a formulation of the admissible pair $(w_i, z_i)$ through independent and explicit formula-based relations regarding the case of re-assignability, together with insightful, relevant remarks. We consider the re-assignability of controllable eigenvalues through the closed-loop eigenvectors and the companion $z_i$ vector in Section 4. Section 5 addresses the same problem concerning uncontrollable systems, resulting in some novel results. In Section 6, eigenvalue controllability and uncontrollability classification are conveniently based on the admissible pair, preserving the open-loop eigenvectors when $z_i=0$ is proven in Section 6. Numerical considerations and some relevant MATLAB functions are outlined in Section 7. A numerically efficient method for determining the characteristic polynomial coefficients and the matrix adjugate is of great value and importance. This is accomplished using Leverrier’s algorithm to efficiently determine the admissible pair as in Section 8. The study considers three examples worked out in detail to demonstrate and validate several conceptions introduced throughout the manuscript in Section 9, together with the paper’s final conclusion in Section 10. Throughout the manuscript, the calculation and simulation tool is MATLAB.

To sum things up, the contribution and novelty of our study are as follow. The details are within the main body of the manuscript. All serve as advantageous features if compared to the traditional eigenstructure method and many other methods for eigenstructure assignment.

• Using the adjugate method, manipulating lower order matrices in the case of re-assignability when the system is controllable;
• Showing that $z_i=0$ implies re-assignability regardless of the controllability of the system. The associated open-loop eigenvector is shown as an eligible closed-loop eigenvector as well;
• Establishing bounds on the number of eligible closed-loop eigenvectors is possible depending on the controllability of a system or otherwise;
• With the help of the adjugate method, we contribute a classification procedure of systems based on the closed-loop eigenvectors $w_i$ being nonzero for controllability and zero for uncontrollability (Section 6). A similar classification applies to system observability;
• Adapting a numerically efficient method for the determination of the characteristic polynomial coefficients and the determination of the adjugate and null space of a matrix (Section 8);
• Paying attributes to the traditional eigenstructure method when it comes to uncontrollable systems. The adjugate method becomes inappropriate in providing a closed-loop eigenvector at a time when the traditional method may, in certain cases, provide as many as $n$ eigenvectors (Section 5);
• Presenting three fully worked-out examples to demonstrate the use and convenience of the adjugate method and the null space method (Section 9).

2. Features of the Adjugate and the Null Space Methods

Common characteristics are shared between the two methods [1]: both methods give the same number of $m$ closed-loop nonzero eigenvectors given by $w_i=adj({\lambda }_i{I}_n-A)B_{n\times m}$ whenever an assigned eigenvalue is not an eigenvalue of $A$. The $z_i$ pertinent to each eigenvalue is also nonzero. They are $z_i=\left|{\lambda }_i{I}_n-A\right| I_m\ne 0$ when using the adjugate method, and they are set to $z_i=I_m$ via software when using the MATLAB command $null(\left[\begin{array}{cc}{\lambda }_i{I}_n-A& -B\end{array}\right],‘r’)$. Otherwise, $z_i$ can be specified as $m\times 1$ nonzero vectors in manual determination, depending on the designer’s preference.

The adjugate method computes the eigenvectors as identically zero for uncontrollable systems. This can be considered an advantage in the sense of indicating the uncontrollability of the assigned eigenvalue. This information is not directly exposed via the null-space method. Additionally, it is not clearly pointed out that the null space method gives a $z_i$ value of zero when an eigenvalue is reassigned. Additionally, determination of $z_i$ is independent of $B$ as proven in [1]. Furthermore, the adjugate method manipulates lower-order matrices and uses independent explicit formula-based relationships, which permits a form of parallelism.

The adjugate method cannot be used whenever $rank({\lambda }_i I_n-A)<n-1$. In this case, it is identically the zero matrix, as proven in Section 3. The adjugate method gives a single eigenvector whenever reassigning a controllable eigenvalue is involved. It fails when calculating a closed-loop eigenvector associated with an uncontrollable eigenvalue.

The null space method relies on the null space solution of an $n\times n+m$ matrix, which gives at least $m$ solutions. It provides more eligible solutions than the adjugate method for both $w_i$ and $z_i$. It gives $m$ closed-loop eigenvectors when dealing with controllable systems as the nullity is always $(n+m)-n=m$ (one of them is an open-loop eigenvector) and can give as many as $n$ closed-loop eigenvectors when dealing with some uncontrollable systems (see Section 5). The adjugate method can give at most as many as $m$ solutions since the maximum rank of $w_i=adj({\lambda }_{ i}{I}_n-A )B_{n\times m}$ is $m$. This happens whenever ${\lambda }_i$ is a newly assigned eigenvalue.

3. Problem Settings, Definitions, and Preliminary Specifics

In our study, no claim regarding the controllability of the system or otherwise is presumed. The system and controller under consideration are given by

$$\stackrel{•}{x}=Ax+Bu ; x\in {ℝ}^n , u\in {ℝ}^m \mathrm{where} m\le n$$

(1)

$$u=K x+r$$

(2)

The closed-loop system is, therefore,

$$\stackrel{•}{x}=(A+BK)x+Br$$

(3)

As justified in [1], among the many rearrangements possible when treating an eigenstructure assignment involving ${\lambda }_i$ and $A$, the following presentation of

$$A_{{\lambda }_i}={\lambda }_i{I}_n-A$$

(4)

is found more convenient and is adopted throughout our investigation.

For newly assigned eigenvalues, the adjugate method of [1] obtains the two solutions $w_i$ and $z_i$ as an admissible pair, given as

$$w_i=adj({\lambda }_i{I}_n-A) B, z_i=\left|{\lambda }_i{I}_n-A\right| I_m; i= 1, 2, \cdots , n$$

(5)

While the traditional eigenstructure method [1] obtains $w_i$ and $z_i$ through a decomposition of the null space of

$$\left[\begin{array}{cc}{\lambda }_i{I}_n-A& -B\end{array}\right]$$

(6)

i.e., the admissible pair $(w_i,z_i)$ is given by what we shall call the null space method.

$$\left[\begin{array}{c}{w}_i\\ \cdots \\ z_i\end{array}\right]=null(\left[\begin{array}{ccc}{\lambda }_i{I}_n-A& ⋮& -B\end{array}\right]); i=1, 2, \cdots , n$$

(7)

Adopting either method, the feedback matrix needed for both approaches is given by

$$K=Z W^{ -1}=\left[\begin{array}{cccc}{z}_1& z_2& ⋮& z_n\end{array}\right]{\left[\begin{array}{cccc}{w}_1& w_2& ⋮& w_n\end{array}\right]}^{ -1}$$

(8)

In what follows, we anticipate that presenting our findings through theorems and corollaries using definitions and lemmas is most convenient.

Lemma 1.

$$If rank(\lambda I_n-A)<n-1,then adj(\lambda I_n-A )=0_{n\times n}$$

(9)

Proof. 

A fact of matrix algebra is that the rank of a matrix can be determined via the maximum order of a submatrix whose minor is nonzero [17] (p. 96). Also, among the many methods for determining the adjugate of an $n\times n$ matrix is one that gives the adjugate by all of its $n-1\times n-1$ minors. So, whenever $rank(\lambda I_n-A )<n-1$, this implies that all $n-1\times n-1$ minors are zero. Hence, $adj(\lambda I_n-A )=0_{n\times n}$. □

In this case, the adjugate method cannot be used to determine $w_i$, so one has to resort to the null space method.

Lemma 2.

Whenever

$$rank(\lambda I_n-A)=n-1,rank(adj(\lambda I_n-A))=1.$$

(10)

Proof. 

See [1]. □

Definition 1.

An augmented matrix is defined as

$$A_g=\left[\begin{array}{cc}\lambda I_n-A& -B\end{array}\right]$$

(11)

where $\lambda$ is an eigenvalue of $A$.

Lemma 3.

For a controllable eigenvalue ${\lambda }_C$ of $A$, ${\lambda }_C{I}_n-A$ is singular and $rank(\left[\begin{array}{cc}{\lambda }_C{I}_n-A& -B\end{array}\right])=n$.

Proof. 

Standard pieces of information in control theory [14,15,16]. □

Theorem 1.

For a controllable eigenvalue ${\lambda }_C$ of $A$, $adj({\lambda }_C{I}_n-A)B\ne 0_{n\times m}$.

Proof. 

For any eigenvalue of $A$, $rank({\lambda }_C{I}_n-A)\le n-1$, to restore the rank of the augmented matrix of a controllable system to $n$, some or all columns of $B$ should be independent of the columns of ${\lambda }_C{I}_n-A$. Since $adj({\lambda }_C{I}_n-A )$ left-annihilates ${\lambda }_C{I}_n-A$ due to their singularity, it cannot totally annihilate $B$ in the augmented matrix $A_g$. Hence, there is at least a column of $B$ such that $adj({\lambda }_C{I}_n-A)B\ne 0_{n\times m}$. □

In the case of a controllable system where $m>1$, it may happen that certain columns of $B$ are dependent on the columns of ${\lambda }_C{I}_n-A$. In this case, $adj({\lambda }_C{I}_n-A)B$ is still nonzero but contains some zero columns.

Corollary 1.

When reassigning a controllable eigenvalue, the adjugate method gives only a single independent closed-loop eigenvector given by a nonzero column of $w_C=adj({\lambda }_C{I}_n-A)B$.

Proof. 

As proven in [1], $rank(adj({\lambda }_C{I}_n-A))=1$. Using Theorem 1, we are left with the conclusion that

$$rank(w_C)=rank(adj({\lambda }_C{I}_n-A)B)=\min \{rank(adj({\lambda }_C{I}_n-A)), rank(B ) \}=\min (1,m)=1$$

i.e., no more than a single closed-loop eigenvector can be obtained using the adjugate method when eigenvalue reassignment is considered regardless of $m$. □

4. Re-Assignability: Nonzero Eigenvectors with Zero z

Theorem 2.

Using the adjugate method to reassign a controllable eigenvalue ${\lambda }_C$, the associated $z_C$ is always zero. Furthermore, the closed-loop eigenvector $w_C=adj({\lambda }_C{I}_n-A)B\ne 0$ is an open-loop one.

Proof. 

Referring to (7), the solution of $w_C$ and $z_C$ is the null space of

$$\left[\begin{array}{cc}{\lambda }_C{I}_n-A& -B\end{array}\right]\left[\begin{array}{c}{w}_C\\ z_C\end{array}\right]=0_{n\times m}$$

(12)

As ${\lambda }_C$ is a reassigned eigenvalue of $A$, ${\lambda }_C{I}_n-A$ and, consequently, $adj({\lambda }_C{I}_n-A)$ are singular, yielding their product an $n\times n$ zero matrix.

Involving $adj({\lambda }_C{I}_n-A)$ in the solution of (12), we premultiply (12) via $adj({\lambda }_C{I}_n-A)$, obtaining,

$$\left[\begin{array}{cc}{0}_{n\times n}& -adj({\lambda }_C{I}_n-A)B\end{array}\right]\left[\begin{array}{c}{w}_C\\ z_C\end{array}\right]=0_{n\times m}$$

(13)

The difficulty is that the solution of $w_C$ and $z_C$ is now within a larger null space than of (12). Using corollary 1, as compared with (12), the $0_{n\times n}$ matrix in (13) lowers its rank to 1, thus increasing its null space dimension to $n+m-1\ge n$ whenever $m\ge 1$ [14] (p. 50). Additionally, possible solutions for (13) allow $w_C$ to have $n$ arbitrary values.

We, therefore, resort to the joint solution of (12) and (13). It is the intersection of the two solutions given by (12) and (13). It is systematically obtained via the null space solution of the following augmented matrix [18] (p. 8), [19] (p. 328)

$$\left[\begin{array}{cc}{\lambda }_C{I}_n-A& -B\\ 0_{n\times n}& -adj({\lambda }_C{I}_n-A)B\end{array}\right]\left[\begin{array}{c}{w}_C\\ z_C\end{array}\right]=0_{2n\times m}$$

(14)

The nonzero solution, which satisfies (14) and is based on the adjugate of ${\lambda }_C{I}_n-A$, is a nonzero column of

$$\left[\begin{array}{c}{w}_C\\ z_C\end{array}\right]= \left[\begin{array}{c}adj({\lambda }_C{I}_n-A)B_{n \times m}\\ 0_{m\times m}\end{array}\right]$$

(15)

i.e., one can only pick a nonzero column of $adj({\lambda }_C{I}_n-A)B_{n \times m}$ for $w_C$ together with $z_C$, which is always $0_{m\times 1}$. Recalling [1], and since ${\lambda }_C$ is a reassigned eigenvalue, i.e., an eigenvalue of $A$, columns of $adj({\lambda }_C{I}_n-A)$ are the same open-loop eigenvector. Post-multiplication via $B$ only results in a linear combination of them.

Additionally, careful examination of (12) when ${\mathrm{z}}_{\mathrm{C}}=0_{m\times 1}$ yields

$$\left[\begin{array}{cc}{\lambda }_C{I}_n-A& -B\end{array}\right]\left[\begin{array}{c}{w}_C\\ 0_{m\times 1}\end{array}\right]=0_{n\times 1}$$

(16)

In this case, $({\lambda }_C{I}_n-A)w_C=0_{n\times m}$ it implies that $w_C$ is an open-loop eigenvector. □

Reassignment of controllable systems using the adjugate method is accompanied by $z=0$ and the compulsory use of open-loop eigenvectors, unless $m>1$, in which case, the null space method can be used with nonzero $z_i$. It is important to note that the solutions associated with nonzero $z_C$ cannot be obtained using the adjugate method.

Contrary to the adjugate method, the null space method can give solutions associated with zero and nonzero $z_C$. In other words, the remaining additional $m-1$ solutions are given by the null space solutions associated with nonzero $z_C$.

5. Re-Assignability: Uncontrollable Systems

Theorem 3.

For an uncontrollable eigenvalue ${\lambda }_U$,

$$adj({\lambda }_U{I}_n-A)\ast B=0_{n\times m}$$

(17)

Proof. 

Utilizing the Popov–Belevitch–Hautus (PBH) test [15,16,18] (p. 77), for an uncontrollable eigenvalue ${\lambda }_U$, there exists a row left eigenvector $q_U\ne 0$, such that the

$$q_U({\lambda }_U{I}_n-A)=0, \mathrm{and} q_{U}B=0,$$

(18)

Combining the two facts presented in (18), we obtain,

$$q_U\left[\begin{array}{cc}{\lambda }_U{I}_n-A& B\end{array}\right]=0_{1\times n+m}$$

(19)

Yielding,

$$rank(\left[\begin{array}{cc}{\lambda }_U{I}_n-A& B\end{array}\right] )<n$$

(20)

This implies that $range(B)$ has to lie in $range( {\lambda }_U{I}_n-A )$, otherwise $q_U\left[\begin{array}{cc}{\lambda }_U{I}_n-A& B\end{array}\right]\ne 0$. In other words, $B=({\lambda }_U{I}_n-A) Q$ for some $n\times m$ matrix $Q$. Thus, using the fact that ${\lambda }_U{I}_n-A$ is singular,

$$adj({\lambda }_U{I}_n-A)B=\stackrel{= 0_{n\times n}}{\cancel{adj({\lambda }_U{I}_n-A)({\lambda }_U{I}_n-A)}}Q=0_{n\times m}$$

(21)

□

Theorem 4.

For an uncontrollable system with an admissible pair ($w_U$, $z_U$), the solutions $w_U$ and $z_U$ cannot be obtained using the adjugate method but are given by the null space of

$$\left[\begin{array}{cc}{\lambda }_U{I}_n-A& -B\end{array}\right]\left[\begin{array}{c}{w}_U \\ z_U\end{array}\right]=0_{n\times m}$$

(22)

Proof. 

In attempting to obtain solutions based on the adjugate method, we premultiply (7) by $adj({\lambda }_U{I}_n-A)$, obtaining,

$$adj({\lambda }_U{I}_n-A)\left[\begin{array}{cc}{\lambda }_U{I}_n-A& -B\end{array}\right]\left[\begin{array}{c}{w}_U \\ z_U\end{array}\right]=0_{n\times m}$$

(23)

$$\left[\begin{array}{cc}{0}_{n\times n}& -adj({\lambda }_U{I}_n-A)B\end{array}\right]\left[\begin{array}{c}{w}_U \\ z_U\end{array}\right]=0_{n\times m}$$

(24)

The difficulty now is that the solutions of $w_U$ and $z_U$ are within a larger null space than that of (22). This is due to the pre-multiplication by $adj({\lambda }_U{I}_n-A)$, which is singular. Back to Theorem 2, the solution now is that of the combined solution of (22) and (24), i.e., the solution of

$$\left[\begin{array}{cc}{\lambda }_U{I}_n-A& -B\\ 0_{n\times n}& -adj({\lambda }_U{I}_n-A)B\end{array}\right]\left[\begin{array}{c}{w}_U \\ z_U\end{array}\right]=0_{2n\times m}$$

(25)

As proven in (21), $adj({\lambda }_U{I}_n-A)B=0$. Hence,

$$\left[\begin{array}{cc}{\lambda }_U{I}_n-A& -B\\ 0_{n\times n}& 0_{n\times m}\end{array}\right]\left[\begin{array}{c}{w}_U \\ z_U\end{array}\right]=0_{2n\times m}$$

(26)

Effectively, we are back to where we started, i.e., the solutions still have to satisfy the equations imposed by the top $n$ rows, i.e., by determining the null space of $\left[\begin{array}{cc}{\lambda }_U{I}_n-A& -B\end{array}\right]$. In other words, using the adjugate method is inappropriate in this case. □

A careful examination of the solution in this case reveals a similar case as that encountered in the controllable case, that is $z_U=0$ is a viable solution with an associated $w_U$ as an open-loop eigenvector. The remaining solutions are given by the left-behind null space solutions associated with nonzero $z_U$.

It is important to note that the solutions associated with $z_U\ne 0$ cannot be obtained using the adjugate method since $w_U$ is always $0_{n\times m}$ as given by (21).

The number of solutions can now be as high as $n$ solutions with some cases of uncontrollable systems. To verify this, consider the following theorem.

Theorem 5.

For uncontrollable systems, where $\nu$ is the nullity of the matrix $A_g=\left[\begin{array}{cc}{\lambda }_U{I}_n-A& -B\end{array}\right]$, the number $\nu$ of the eligible closed-loop eigenvectors is bounded by $m+1 \le \nu \le n$.

Proof. 

The number of solutions of (22) is given by the nullity $\nu$ of the augmented matrix $A_g=\left[\begin{array}{cc}{\lambda }_U{I}_n-A& -B\end{array}\right]$, where [14] (p. 50)

$$\begin{array}{l}\nu =\mathrm{number} \mathrm{of} \mathrm{columns} \mathrm{of} A_g-\mathrm{rank} \mathrm{of} A_g\\ =(n+m)-rank(\left[\begin{array}{cc}{\lambda }_U{I}_n-A& -B\end{array}\right])\\ =(n+m)-(n-d)=m+d\end{array}$$

(27)

where $d$ is the rank deficiency of (22). For an uncontrollable system, it is bounded by $1\le d\le n-m$, since $n$, $m$, and $d$ are all positive with $m \le n$, $\min (d)=1$, and $\max (d)=n-m$. Hence,

$$\min (\nu )=\min (m+d)=m+\min (d)=m+1$$

And

$$\max (\nu )=\max (m+d)=m+\max (d)=m+n-m=n$$

Therefore, $m+1 \le \nu \le n$. □

For controllable systems $d=0$, hence, the number of qualified closed-loop eigenvectors is always $m$.

Note that the null space method may give as many as $n$ closed-loop eigenvectors when the system has $n-m$ uncontrollable eigenvalues as the rank deficiency is now $n-m$. This offers maximum control over the selection of the closed-loop eigenvectors. For some systems, they can be totally arbitrary. In this case, there is no need to calculate generalized eigenvectors either via differentiation or otherwise [1]. This is considered a valuable advantage.

6. Classification of Systems Based on the Admissible Pair

In addition to the many tests already used to determine the controllability or uncontrollability of eigenvalues and systems, the adjugate method further introduces a new approach to do that. This is possibly guided by the fact that controllability is invariant under state feedback, so tests developed based on closed-loop information are equally valid.

With regard to $A_{{\lambda }_i}={\lambda }_i{I}_n-A$, the adjugate method can be used to an advantage when it comes to the classification of eigenvalues as controllable or uncontrollable. With such classification, the use of the matrix rank test is avoided. The Faddeeva–Leverrier algorithm outlined in Section 8 is used to facilitate an efficient method for matrix adjugate determination. It can be easily adapted to fulfill such a job.

Based on the theory that has been developed in Section 4 and Section 5, we arrive at the following classification of systems regarding the notion of controllability:

• Whenever $z_i=0$ and $w_i\ne 0$, the eigenvalue is controllable, and the eigenvalue is a reassigned one, not a newly assigned eigenvalue;
• Whenever $w_i=0$, the eigenvalue is deemed uncontrollable due to the partial input involved, irrespective of the value of $z_i$;
• The case $z_i\ne 0$ together with $w_i\ne 0$ is inapplicable. In this case, it should be understood that ${\lambda }_i$ is not an eigenvalue of $A$ but a newly assigned one where the controllability tests based on matrix ranks, such as the PBH test [15,16], are irrelevant.

The information above is best put in the form of a theorem.

Theorem 6.

A system given by the tuple $( A, B, C, D )$ is controllable if $w=adj( \lambda I_n-A^{ }) B$ is nonzero for all $n$ eigenvalues $\lambda$ of $A$. Furthermore, a zero column of $w$ indicates that the mode associated with $\lambda$ is uncontrollable from that associated input.

Proof. 

To avoid repetition, refer to the theory developed in Section 4 and Section 5. □

Remark 1.

With the help of the adjugate method, any eigenvector associated with $z_i=0$ is an open-loop one, which also qualifies as a closed-loop eigenvector. This is not given using the null-space method as direct information. Therefore, the traditional null space method does not distinguish between open-loop and closed-loop eigenvectors, while the adjugate method does.

Remark 2.

Since controllability and observability are dual, the classification mentioned above generalizes equally well to arrive at a similar test for observability.

Corollary 3.

A system given by the tuple $( A, B, C, D )$ is observable if $w_{ T} =( C \times Adj( \lambda I_n-A))$ is nonzero for all $n$ eigenvalue $\lambda$ of $A$. Furthermore, a zero row of $w_{ T}$ indicates that the mode associated with $\lambda$ is unobservable from that associated output.

Proof. 

Owing to the duality of linear systems, the observability properties of a system represented by the tuple $( A, B, C, D )$ are identical to the controllability properties of the dual system given by the tuple $( A{ }^T, C^{ T}, B^{ T}, D^{ T} )$. Thus, using the adjugate test and recalling that $adj(M{ }^T)={(adj(M))}^{ T}$ [18] (p. 22).

$$w= adj( \lambda I_n-A^{ T}{}^{ }) C^{ T}={(adj( \lambda I_n-A))}^{ T} C^{ T} ={( C \times adj( \lambda I_n-A))}^{ T}$$

For $w$ being zero or nonzero is also that of its transpose $w_T$ being zero or nonzero. Hence, observability is determined by the rows of $w_{ T}$ where

$$w_{ T} =( C \times adj( \lambda I_n-A))$$

(28)

Any nonzero row of $w_T$ confirms observability. □

7. Invariances under State Feedback in the Case of Reassignment

As it is well known, the eigenstructure method provides control over the selection of the closed-loop eigenvectors. This is the case with the right eigenvectors, whether the system is controllable or otherwise. However, this is not the case with the left eigenvectors, as left eigenvectors are invariant under state feedback [20].

The right eigenvector corresponding to any eigenvalue $\lambda$ is associated with $z_K=0$.

Assuming that $w_O$ is an open-loop eigenvector different from the closed-loop eigenvector $w_K$, and recalling that $K w_K=z_K$, then,

$$\begin{array}{l}A w_O=\lambda w_O\\ (A+B K)w_K=\lambda w_K\\ A(w_K-w_O) + B K w_K=\lambda (w_K-w_O)\\ A(w_K-w_O) + B z_K=\lambda (r_K-w_O)\end{array}$$

(29)

If $\lambda$ is reassigned, then the adjugate method has shown that $z_K=0$. In which case, $B z_K=0$, leading to either $w_K-w_O$ as a right eigenvector of $A$, or $w_K-w_O=0$. In both cases, either $w_K$ is a scalar multiple of $w_O$ or is identical to $w_O$. For both cases, it means that the closed-loop eigenvector is an open-loop eigenvector.

Such a fact provides a useful distinction between the adjugate and classical eigenstructure methods. Knowing this in advance is important to avoid the evaluation of closed-loop eigenvectors if the open-loop ones are already known.

8. Numerical Considerations and Some Relevant MATLAB Functions

The matrix adjugate is computable without ever using division. In this regard, and from a numerical analysis point of view, it is well conditioned due to the absence of division.

Following the usual determination of a nonsingular matrix inverse, one may be tempted to determine the adjugate using $adj(A)=\left|A\right|A^{ -1}$. The catch is: this only applies when $A$ is nonsingular. Additionally, matrix inverses involve $O(4n^3)$ flops when using the Gauss–Jordan method.

The cofactor method for adjugate determination is simple in concept and programming but involves $O(n!)$ flops. Unfeasible unless $n$ is very low [17] (p. 92).

To gain maximum numerical efficiency when evaluating polynomials such as $\left|{\lambda }_i{I}_n-A\right|$, Horner’s algorithm is used. A polynomial

$$p(\lambda )={\lambda }^n+a_1{\lambda }^{n-1}\lambda +a_2{\lambda }^{n-2}+\cdots +a_{n-1}\lambda +a_n$$

(30)

requires a number of multiplications and additions of order $n+n(n+1)/2$ when coded as it is. If coded using Horner’s algorithm, where the same polynomial is formulated as

P = ((((L + a1)*L + a2)*L + a3)*L + a4)*L + a5

(31)

L represents $\lambda$ and a1, …, a5 are the polynomial coefficients. It now requires an order of $2n$ multiplications and additions.

A MATLAB program implementing Horner’s algorithm at, say, ${\lambda }_i=-5$ is as simple as

$$>>\mathrm{p}=\mathrm{poly}\left(\mathrm{A}\right);\mathrm{lam}=-5;\mathrm{n}=\mathrm{length}\left(\mathrm{p}\right);\mathrm{z}=\mathrm{p}\left(1\right);\mathrm{for} \mathrm{k}=1:\mathrm{n}-1; \mathrm{z}=\mathrm{z}\ast \mathrm{lam}+\mathrm{p}\left(\mathrm{k}+1\right);\mathrm{end}; \mathrm{z}$$

To determine the null space of a generic $p\times q$ matrix $M$ of rank $r$, an augmented matrix is generated as $\left[\frac{M_{p\times q}}{I_q}\right]$. Elementary column operations (equivalent to post-multiplication by a product of elementary matrices $P$) are then performed till the following form is obtained,

$$\left[\frac{M_{p\times q}}{I_q}\right]P\sim \left[\frac{E_{p\times r}\hspace{1em}⋮\hspace{1em}{0}_{p\times q-r}}{F_{p\times r}\hspace{1em}⋮\hspace{1em}{R}_{p\times q-r}}\right]$$

(32)

where $\sim$ stands for elementary column operations and $E_{p\times r}$ and $F_{q\times r}$ are in reduced column echelon forms. The null space is taken as the columns immediately below the resulting zero matrix shown in (32), i.e., the matrix named $R$. Thus, $null(M)=R$ [21].

In our study, $M$ is usually $M=\left[\begin{array}{cc}{\lambda }_i{I}_n-A& -B\end{array}\right]$, in which case $p=n$ and $q=n+m$. Therefore, the augmented column matrix, as in (32), is of order $(2n+m)\times (n+m)$. A convincing higher-order matrix compared to the $n\times n$ matrix is needed via the adjugate determination.

The MATLAB function $null(M)$ should be used with care for checking purposes. The reason is that it uses the Gram–Schmidt process, providing an orthonormalized set of vectors for the null space, rendering its answers indirect when compared to those given using the adjugate method. Instead, MATLAB $null(M,‘r’ )$ is used, as illustrated in the examples in Section 10.

However, a numerically efficient method is to use the Faddeeva–Leverrier algorithm. It is of an iterative nature and is available as a MATLAB program on the web [22]. In addition, it is an efficient method for finding the coefficients of the characteristic polynomial. Additionally, a further advantage is that the inverse of a nonsingular matrix and the null space of the $A$ matrix (assuming it to be singular) is a giveaway at no extra computational expense. It is outlined in the following section and adapted to efficiently calculate the matrix adjugate.

9. The Leverrier’s Algorithm for Efficient Determination of the Admissible Pair

A numerically efficient method for determining the $adj({\lambda }_{i}I-A)$ is an indispensable part of our study. Such necessity is fulfilled using Leverrier’s algorithm (also known as the Faddeev–Leverrier algorithm) [23].

An important fact is that the highest power of $\lambda$ involved in evaluating $adj(\lambda I_n-A)$ for any $A$ is $n-1$. Therefore, let

$$adj(\lambda I-A)={\lambda }^{n-1}{R}_1+{\lambda }^{n-2}{R}_2+\cdots +\lambda R_{n-1}+R_n$$

(33)

Assuming

$$\left|\lambda I-A\right|={\lambda }^n+a_1{\lambda }^{n-1}+\cdots +a_{n-1}\lambda +a_n$$

(34)

$$( \lambda I_n-A ) adj(\lambda I_n-A )=\left|\lambda I_n-A\right|I_n=({\lambda }^n+a_1{\lambda }^{n-1}+\cdots +a_{n-1}\lambda +a_n)I_n$$

(35)

Based on the term-by-term multiplication of the left-hand side of (35), we equate coefficients of the $n$ powers of $\lambda$, together with using the identity

$$\frac{d}{d\lambda }(\left|\lambda I-A\right|)=tr(adj(\lambda I-A))$$

(36)

The coefficients $R_1, R_2,\cdots R_n$, $a_1,a_2,\cdots a_n$ and $A^{-1}$ are determined iteratively (see Appendix A for the derivation and extension). The coefficients are given by [23]

$$\begin{array}{l}{R}_1=I_n ; a_1=-tr(A{R}_1)=-tr(A)\\ R_2=A R_1 + a_1{I}_n ; a_2=-\frac{1}{2}tr(A R_2)\\ ⋮\\ R_n=A R_{n-1} + a_{n-1}{I}_n ; a_n=-\frac{1}{n}tr(A R_n)\\ \mathrm{and} A R_n + a_n{I}_n =0 \Rightarrow A^{-1} =-\frac{1}{a_n}{R}_n ; iff a_n\ne 0\end{array}$$

(37)

The iterative algorithm realizes an efficient method for determining $adj(\lambda I-A)$, together with the coefficients of the characteristic polynomial, which are frequently referred to in our investigation. Additional byproducts are:

• To determine the adjugate of $A$, let $\lambda =0$, in which case, $adj(-A)=R_n$. Invoking a property of matrix adjugates (where $\forall c, adj(cM)=c^{n-1}adj(M)$), [18].

By letting $c=-1$, one obtains $adj(-A)= {(-1)}^{n-1}adj(A)=R_n$, yielding,

$$adj(A)={(-1)}^{1-n}{R}_n={(-1)}^{n+1}{R}_n$$

(38)

• When a matrix $A$ is singular, it has at least a zero eigenvalue. Substituting $\lambda =0$ in (33), we obtain $adj(-A)=R_n$.

To prove this, since $A$ is singular, $A adj(A)=0= A \times {(-1)}^{n+1}{R}_n=0$

$$\Rightarrow A\times R_n =0 \Rightarrow null(A) =R_n$$

(39)

• The characteristic polynomial coefficients $a_1,a_2,\cdots a_n$, determined iteratively, can be used together with Horner’s algorithm to efficiently evaluate $z_i=\left|{\lambda }_{ i}{I}_n-A\right|$, as explained in Section 8;
• To determine $adj({\lambda }_{i}I-A)$ for a specific ${\lambda }_i$, matrices $R_1, R_2,\cdots R_n$ are substituted in (33). Furthermore, additional numerical efficiency can be gained using the nesting nature of Horner’s algorithm. In which case,

  $$adj({\lambda }_{i}I-A)=(\cdots (({\lambda }_i{R}_1+R_2) {\lambda }_i+R_3){\lambda }_i+\cdots +R_{n-1}){\lambda }_i+R_n$$

  (40)

Recently, the algorithm has been presented as a pseudo-code in [24]. It requires $O(n^{3.5})$ operations for matrix multiplications and additions.

10. Examples

Example 1.

An unstable controllable system considered in [25] is given by

$$\stackrel{•}{x}=\left[\begin{array}{ccc}-5.5& 3& 3\\ -6& 2.5& 4\\ 0& 1& -0.5\end{array}\right]x+\left[\begin{array}{c}1\\ 2\\ 5\end{array}\right]u$$

(41)

The open-loop eigenvalues are $0.5$, $-1.5$, and $-2.5$. Obviously, it is an unstable system.

It is required to assign the two eigenvalues, $-0.5$ and $-3$, and to keep (reassign) $-2.5$.

The characteristic polynomial is

$$\Delta (\lambda )={\lambda }^3+3.5{\lambda }^2+1.75\lambda -1.875$$

Following the adjugate method using (5)

$$\begin{gathered} w_1=adj(-0.5I_3-A)\times B=\left[\begin{array}{c}17\\ 10\\ 19\end{array}\right] ; z_1=\left|-0.5I_3-A\right|= \Delta (-0.5)=-2\ne 0 \\ {w}_2=adj(-3I_3-A)\times B=\left[\begin{array}{c}-21.75\\ -37.5\\ 20.25\end{array}\right] ; z_2=\left|-3I_3-A\right|= \Delta (-3)=-2.625\ne 0 \\ {w}_3=adj(-2.5I_3-A)\times B=\left[\begin{array}{c}-15\\ -30\\ 15\end{array}\right] ; z_3=\left|-2.5I_3-A\right|= \Delta (-2.5)=0 \end{gathered}$$

(42)

As the theory predicts, $z_1$ and $z_2$ are nonzero since the associated eigenvalues are newly assigned [1], but $z_3=0$ indicates a reassigned eigenvalue. This is confirmed by the fact that $-2.5$ is already an eigenvalue of $A$.

Hence,

$$\begin{array}{l}{K}_1=\left[\begin{array}{ccc}{z}_1& z_2& z_3\end{array}\right]{\left[\begin{array}{ccc}{w}_1& w_2& w_3\end{array}\right]}^{ -1}\\ K_1=\left[\begin{array}{ccc}-2& -2.625& 0\end{array}\right]{\left[\begin{array}{ccc}17& -21.75& -15\\ 10& -37.5& -30\\ 19 & 20.25& 15\end{array}\right]}^{ -1}=\left[\begin{array}{ccc}16& -13& -10\end{array}\right]/24\end{array}$$

(43)

Note that $z_3=0$ and that $w_3$ is an open-loop eigenvector as predicted by the theory in Section 4. The other two are not. In fact, their associated eigenvectors are incomparable to those of $A$, since $-0.5$ and $-3$ are not eigenvalues of $A$.

When checking our answers for $K$ using those given by the two MATLAB $acker$ or $place$ commands [26], we observed a sign reversal. This is because the two MATLAB functions are based on systems where $u$ is considered as $u=-K x$ as opposed to $u=K x$ used by the adjugate method, see (4).

One may be tempted to explore new alternative closed-loop eigenvectors using the null space method as in (7), where $w_i=null(\left[\begin{array}{cc}{\lambda }_i{I}_n-A& -B\end{array}\right],‘r’ )$. In this case, non-identical new numbers are obtained. However, they are actually a scalar multiple of those already obtained using the adjugate method, as the theory in [1] established.

As predicted by the theory in Section 4, neither the use of the adjugate method nor the null space method can give more than a single eligible eigenvector. This is because the system is controllable with $m=1$.

Example 2.

Consider the following uncontrollable system. The example will be used to verify many facts mentioned in the main body of the paper.

$$\stackrel{•}{x}=\left[\begin{array}{ccc}0& 1& 1\\ -2& -3& -2\\ 0& 0& -4\end{array}\right]x+\left[\begin{array}{c}1\\ 0\\ 2\end{array}\right]u$$

(44)

The rank of the controllability matrix is $2$, with $-2$ being the uncontrollable eigenvalue. It is required to assign $-5$ as a new eigenvalue and to retain (reassign) the $-4$ eigenvalue.

The open-loop characteristic polynomial is

$$\Delta (\lambda )=\left|\lambda I_3-A\right|={\lambda }^3+7{\lambda }^2+14\lambda +8$$

(45)

Using the adjugate method with ${\lambda }_1=-5$, gives

$$w_1=adj(-5I_3-A)B={\left[\begin{array}{ccc}-6& 18& 24\end{array}\right]}^{ T}; z_1=\left|-5 I_3-A\right|=\Delta (-5)=-12$$

(46)

For the reassigned ${\lambda }_2=-4$ and $z_2=0$. Hence,

$$w_2=adj(-4I_3-A)B={\left[\begin{array}{ccc}-6& 12& 12\end{array}\right]}^T; z_2=0$$

(47)

Since the eigenvector $w_2$ is nonzero, then $-4$ is a controllable eigenvalue as the new classification presented in Section 6 has established.

Since $-2$ is an uncontrollable eigenvalue, the adjugate method gives a zero eigenvector, confirming the classification presented in Section 6.

So, we resort to the null space method to determine $w_3$

$$w_u=\left[\begin{array}{c}{w}_{32}\hfill \\ z_{32}\hfill \end{array}\right]=null( \left[\begin{array}{cc}-2I_3-A& -B\end{array}\right],‘r’)=\left[\begin{array}{cc}-0.5\hfill & -1\hfill \\ 1\hfill & 0\hfill \\ 0\hfill & 1\hfill \\ 0\hfill & 1\hfill \end{array}\right]$$

(48)

We now have two closed-loop eigenvectors to choose from. The first one happens to be the open-loop eigenvector characterized by an associated $z=0$. The second one is a new qualifying eigenvector characterized by a nonzero $z=1$. Let us choose the first one, hence,

$$w_3={\left[\begin{array}{ccc}-0.5& 1& 0\end{array}\right]}^{ T} ; z_3=0$$

In which case,

$$K_1=\left[{\begin{array}{ccc}{z}_1& z_2& z\end{array}}_3\right]{\left[\begin{array}{ccc}{w}_1& w_2& w_3\end{array}\right]}^{ -1}=\left[\begin{array}{ccc}-12& 0& 0\end{array}\right]{\left[\begin{array}{ccc}-6& -6& -0.5\\ 18& 12& 1\\ 24& 12& 0\end{array}\right]}^{ -1}=\left[\begin{array}{ccc}-4& -2& 0\end{array}\right]$$

(49)

If the second eigenvector $w_3={\left[\begin{array}{ccc}-1& 0& 1\end{array}\right]}^{ T}$ with $z_3=1$ was chosen, then

$$K_2=\left[\begin{array}{ccc}-12& 0& 1\end{array}\right]{\left[\begin{array}{ccc}-6& -6& -1\\ 18& 12& 0\\ 24& 12& 1\end{array}\right]}^{ -1}=\left[\begin{array}{ccc}-2& 0& -1\end{array}\right]$$

(50)

$K_1$ and $K_2$ are different due to the system being uncontrollable. They are also different from the feedback matrix given by the MATLAB place command, which turns out to have the following value:

$$K_{place}=\left[\begin{array}{ccc}845/382& 317/1495& 995/1113\end{array}\right]=\left[\begin{array}{ccc}2.2120& 0.2120& 0.8940\end{array}\right]$$

(51)

Any of the three feedback matrices above results in assigning the desired eigenvalues but with a different closed-loop eigenvector associated with the uncontrollable eigenvalue in each case.

Example 3.

Consider the uncontrollable system [27].

$$\stackrel{•}{x}=\left[\begin{array}{cccc}2& 3& 2& 1\\ -2& -3& 0& 0\\ -2& -2& -4& 0\\ -2& -2& -2& -5\end{array}\right]x+\left[\begin{array}{cc} 0\hfill & 1\hfill \\ 1 \hfill & -2\hfill \\ -2\hfill & 1\hfill \\ 1\hfill & 0\hfill \end{array}\right]u$$

(52)

It is required to assign the two eigenvalues, $-5$ and $-6$. The system has two uncontrollable eigenvalues, $-1$ and $-4$.

Since $-5$ and $-6$ are newly assigned eigenvalues, the adjugate matrix is nonsingular. Hence, each eigenvalue has $m$ independent closed-loop eigenvectors to choose from or to make a linear combination out of them. This we shall perform in the alternate solution later on.

To determine the two eigenvectors associated with $-5$, we use the adjugate method, which gives two qualifying eigenvectors $w{w}_1$. For the sake of demonstration [1], we pick the first eigenvector of $w{w}_1$ with its corresponding first column of $z{z}_1$, i.e.,

$$\begin{gathered} w{w}_1=adj(-5I_4-A)B=\left[\begin{array}{cccc}0& 6& 4& 2\\ 0& -6& 4& 2\\ 0& 0& -8& 8\\ -24& -24& -24& -36\end{array}\right]B=\left[\begin{array}{cc} 0\hfill & -8\hfill \\ -12\hfill & 16\hfill \\ 24\hfill & -8\hfill \\ -12\hfill & 0\hfill \end{array}\right]\Rightarrow w_1= w{w}_1\left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{c}0\\ -12\\ 24\\ -12\end{array}\right] \\ z{z}_1=\left|-5I_4-A\right|I_2=polyval(poly(A),-5)\ast eye(2)=\left[\begin{array}{cc}24& 0\\ 0& 24\end{array}\right] \Rightarrow z_1 = z{z}_1 \left[\begin{array}{c}1\\ 0\end{array}\right] = \left[\begin{array}{c}24\\ 0\end{array}\right] \end{gathered}$$

(53)

To determine the eigenvector associated with $-6$, and for the sake of demonstration, we pick the second eigenvector of $w{w}_2$ with its corresponding second column of $z{z}_2$, i.e.,

$$\begin{gathered} w{w}_2=adj(-6I_4-A)B=\left[\begin{array}{cccc}-6& 18& 12& 6\\ -4& -28& 8& 4\\ -10& -10& -40& 10\\ -40& -40& -40& -80\end{array}\right]B=\left[\begin{array}{cc} 0\hfill & -30\hfill \\ -40\hfill & 60\hfill \\ 80\hfill & -30\hfill \\ -40\hfill & 0\hfill \end{array}\right] \Rightarrow w_2= w{w}_2\left[\begin{array}{c}0\\ 1\end{array}\right]=\left[\begin{array}{c}-30\\ 60\\ -30\\ 0\end{array}\right] \\ z{z}_2=\left|-6I_4-A\right|I_2=polyval(poly(A),-6)\ast eye(2)=\left[\begin{array}{cc}120& 0\\ 0& 120\end{array}\right] \Rightarrow z_2 = z{z}_2 \left[\begin{array}{c}0\\ 1\end{array}\right] = \left[\begin{array}{c}0\\ 120\end{array}\right] \end{gathered}$$

(54)

If we try to determine the eigenvectors associated with $-1$ and $-4$, the adjugate method fails as $-1$ is an uncontrollable eigenvalue. In this case, it gives zero eigenvectors. So, we resort to the null space method. For the sake of demonstration, we select a closed-loop eigenvector $w_3$ as the sum of the three eligible vectors, obtaining,

$$v_{wz3}=null(\left[\begin{array}{cc}-I_4-A& -B\end{array}\right],‘r’)\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{ccc}-1\hfill & 0.5 \hfill & -1\hfill \\ 1\hfill & 0\hfill & 0\hfill \\ 0\hfill & -1\hfill & 1\hfill \\ 0\hfill & 0.5\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right] \left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]= \left[\begin{array}{c}-1.5\hfill \\ 1\hfill \\ 0\hfill \\ 0.5\hfill \\ 1\hfill \\ 1\hfill \end{array}\right] = \left[\begin{array}{c}{w}_3\hfill \\ z_3\hfill \end{array}\right]$$

(55)

The same is obtained from the $-4$ eigenvalue, obtaining,

$$v_{wz4}=null(\left[\begin{array}{cc}-4I_4-A& -B\end{array}\right],‘r’)\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{ccc}0\hfill & 0 \hfill & -0.5\hfill \\ 0\hfill & -1\hfill & 1\hfill \\ -0.5\hfill & 1.5\hfill & -0.5\hfill \\ 1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right] \left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]= \left[\begin{array}{c}-0.5\hfill \\ 0\hfill \\ 0.5\hfill \\ 1\hfill \\ 1\hfill \\ 1\hfill \end{array}\right] =\left[\begin{array}{c}{w}_4\hfill \\ z_4\hfill \end{array}\right]$$

(56)

Hence,

$$\begin{gathered} K_1=\left[\begin{array}{cccc}{z}_1\hfill & z_2\hfill & z_3\hfill & z_4\hfill \end{array}\right]{\left[\begin{array}{cccc}{w}_1\hfill & w_2\hfill & w_3\hfill & w_4\hfill \end{array}\right]}^{ -1}=\left[\begin{array}{cccc}24\hfill & 0\hfill & 1\hfill & 1\hfill \\ 0\hfill & 120\hfill & 1\hfill & 1\hfill \end{array}\right]{\left[\begin{array}{cccc}0& -30& -1.5& -0.5\\ -12& 60& 1& 0\\ 24& -30& 0& 0.5\\ -12& 0& 0.5& 1\end{array}\right]}^{ -1} \\ K_1 =\left[\begin{array}{cccc}-0.2\hfill & 0.6\hfill & 1.4\hfill & 0.2\hfill \\ 3\hfill & 5\hfill & 3\hfill & 1\hfill \end{array}\right] \end{gathered}$$

(57)

An alternate $K$ is obtained if the two vectors $w_1$ and $w_2$ are shaped as in [1], as demonstrated below:

Suppose we require the eigenvector associated with $-5$ to have a ratio of $-1:2$ between its second and third components. In this case, with reference to [1], we obtain a single eigenvector out of $w{w}_1$ and its companion $z{z}_1$, see [1], ending with $w_{1s}$ and

$$\left[\begin{array}{cc}-12& 16\\ 24& -8\end{array}\right]M_{2\times 1}=\left[\begin{array}{c}-1\\ 2\end{array}\right]\Rightarrow M_{2\times 1}=\left[\begin{array}{c}1/12\\ 0\end{array}\right] \Rightarrow w_{1s}=w_1\ast M=\left[\begin{array}{c}0\hfill \\ -1\hfill \\ 2\hfill \\ -1\hfill \end{array}\right] , z_{1s}=z_1\ast M=\left[\begin{array}{c}2\\ 0\end{array}\right]$$

(58)

Similarly, an eigenvector associated with $-6$ having a ratio of $1:4$ between its first and third components is obtained as follows. Again, with reference to [1], we obtain a single eigenvector out of $w{w}_2$ and its companion $z{z}_2$, see [1], ending with $w_{2s}$ and $z_{2s}$.

$$\left[\begin{array}{cc}0& -30\\ 80& -30\end{array}\right]M_{2\times 1}=\left[\begin{array}{c}1\\ 4\end{array}\right]\Rightarrow M_{2\times 1}= \left[\begin{array}{c}3/80\\ -1/30\end{array}\right] \Rightarrow w_{2s}=w_2\ast M=\left[\begin{array}{c}1\hfill \\ -3.5\hfill \\ 4\hfill \\ -1.5\hfill \end{array}\right] , z_{2s}=z_2\ast M=\left[\begin{array}{c}4.5\\ -4\end{array}\right]$$

(59)

Hence,

$$\begin{gathered} K_2=\left[\begin{array}{cccc}{z}_{1s}\hfill & z_2{}_s\hfill & z_3\hfill & z_4\hfill \end{array}\right]{\left[\begin{array}{cccc}{w}_{1s}\hfill & w_{2s}\hfill & w_3\hfill & w_4\hfill \end{array}\right]}^{ -1}=\left[\begin{array}{cccc}2\hfill & 4.5\hfill & 1\hfill & 1\hfill \\ 0\hfill & -4\hfill & 1\hfill & 1\hfill \end{array}\right]{\left[\begin{array}{cccc}0& 1& -1.5& -0.5\\ -1& -3.5& 1& 0\\ 2& 4& 0& 0.5\\ -1& -1.5& 0.5& 1\end{array}\right]}^{ -1} \\ K_2 =\left[\begin{array}{cccc}-1.55\hfill & -1.35\hfill & 0.35\hfill & 0.05\hfill \\ 3\hfill & 5\hfill & 3\hfill & 1\hfill \end{array}\right] \end{gathered}$$

(60)

Due to the shaping of the eigenvectors used, an alternative feedback matrix is expected, but then again, resulting in the same assigned eigenvalues.

Finally, as explained in Section 6, to demonstrate the use of the adjugate method to determine the system controllability, we consider the $-2$ eigenvalue, obtaining

$$w_{UC}=adj(-2I_4-A)B=\left[\begin{array}{cccc}6& 6& 4& 2\\ -12& -12& -8& -4\\ 6& 6& 4& 2\\ 0& 0& 0& 0\end{array}\right]B=\left[\begin{array}{cc}0\hfill & -2\hfill \\ 0\hfill & 4\hfill \\ 0\hfill & -2\hfill \\ 0\hfill & 0\hfill \end{array}\right]$$

(61)

The nonzero second column indicates the controllability of the $-2$ eigenvalue owing to the second input. It is uncontrollable from the first input.

If we perform the same to the $-1$ and $-4$ eigenvalues, both columns of $w_{UC}$ will be zero, indicating they are uncontrollable.

11. Conclusions

Additional insights have been gained through the admissible pair approach to eigenstructure reassignment concerning the cases of single, multi-input, controllable, and uncontrollable systems.

Compared to the traditional null space method, the study reveals information that would not have been exposed otherwise, such as the reassignment of controllable systems being accompanied with $z_i=0$, the compulsory use of open-loop eigenvectors unless $m>1$, and that, in some cases of uncontrollable reassignment, as many as $n$ closed-loop eigenvectors can be involved in shaping the time response.

The case $z_i=0$ indicates that reassignment is entangled with an assigned eigenvector being an open-loop one. The case $w_i=0$ indicates that the associated eigenvalue ${\lambda }_i$ is an uncontrollable one. The open-loop eigenvectors are closed-loop eigenvectors whenever reassignment is involved in the case of single-input controllable systems.

For single-input controllable systems, using the adjugate method, a necessary and sufficient condition for reassignment is $z_i=0$. It is sufficient for multi-input systems. An advantage of using the adjugate method and obtaining that $w_U=0$ is an indication that the eigenvalue is uncontrollable. Such facts have been used in developing a novel classification method for testing controllability based on column-wise elements of the input matrix $B$.

Relevant, efficient computations of the characteristic polynomial coefficients and the matrix adjugates are achieved using Leverrier’s algorithm, which was also shown to offer further byproducts, such as the determination of matrix null spaces. Together with the use of Horner’s method, the computation efficiency can be boosted even further.

The adjugate method proved commendable for exposing internal structures relevant to the reassignment problem. It offers advantageous decoupled solutions for each component of the admissible pair $(w_i,z_i)$. It manipulates lower dimensional matrices. It distinguishes between the eigenvectors associated with controllable and uncontrollable eigenvalues depending on $w_i$ being zero or nonzero. The same distinction carries over to eigenvalues being newly assigned or reassigned depending on $z_i$ being nonzero or otherwise.

The null space method shows advantages over the adjugate method regarding eigenvalue reassignment. Exclusively, when it comes to determining the closed-loop eigenvectors, as well as providing more eligible closed-loop eigenvectors with their associated $z_i$. The study shows an expanded solution subspace in this case to the extent of providing as many as $n$ closed-loop eigenvectors in some cases of uncontrollable reassignment.

The three examples presented illustrate the many concepts, notions, and new findings stated within the paper. This has been obtained with the help of MATLAB R2018a, which is a worthy software when it comes to control theory studies.

Future works should consider methods that simplify reassignment issues related to partial eigenstructure assignment, controller design involving reduced order matrices, and system matrices having special forms.