Hybridized Ackermann’s Methods

Further elaborations on the modified Ackermann’s method (MAM) for eigenvalue assignment are considered in this paper. Additional results concerning the incomplete eigenvalue assignment (IEA) are stated, verified, and commented. The advantages of IEA are pursued even further in this study beyond that mentioned in [1]. The study proposes two newly appended approaches based on MAM; named spectral and truncated methods. They are grounded on IEA, which fundamentally exemplify a hybridized approach to eigenvalue assignment. Necessary and sufficient conditions for stability of the truncated hybridized method are established, proved, and validated by examples. All results obtained apply equally-well to identical eigenvalue assignment, complex eigenvalue assignment, as well as to uncontrollable systems. Besides, they lead to simplified state feedback matrix determination. Three numerical examples are fully worked out to substantiate the nature of the IEA and the two hybridized methods. Simulation and visualization using MATLAB demonstrate the flexibility of the proposed methods.


I. INTRODUCTION
In a recent paper [1], the classical method of Ackermann [2][3][4][5] has been revisited, extended, and generalized; contributing an alternative proof and newly compact expositions. A modified Ackermann's method (MAM) proposed in [1], enabled incomplete eigenvalue assignment(IEA) and a generalization to uncontrollable systems. We make a comeback in this paper with further elaborations, extensions, and systemization to MAM and IEA. Doing so, we are led to new additional forms of hybridized nature. This paper acts as a continuation or as part II of an earlier one [1] entitled: Ackermann's Method : Revisited, Extended, and Generalized to Uncontrollable Systems. Thus, the references needed in this paper are limited in number; mainly those listed in [1], together with additional relevant ones. It's worth pointing out that IEA is different from partial eigenvalue assignment as treated in [6][7][8].
In essence, IEA enjoys the liberty of assigning an incomplete set of eigenvalues say, q where 1 qn  without having to assign the remaining nq  eigenvalues.
In other words, we are left with nq  eigenvalues that are enforced. Their values are shown to be subject to certain limitations. They depend on the structure of the system and on the specific choice of an n n q  matrix N chosen to generate a square nonsingular modified controllability matrix (MCM).
The controllability matrix C is the backbone of the classical method. In [1], a modified form of C was proposed to enable IEA and to generalize the method to uncontrollable systems. The modified method determines a state feedback matrix which assigns say q eigenvalues explicitly and implicitly enforces the assignment of the remaining n-q eigenvalues. Limitations regarding those enforced eigenvalues were recognized and identified in [1]. Such limitations are followed in this paper. They are shown to be not as stringent, and can be relaxed in certain cases. This is discussed in Section. IV and Section V.
We envisage incomplete assignment as a form of hybridization worth additional investigation. Two hybridized methods, termed spectral and truncated are the subject of our study shedding light on their nature, structure, and advantages. The entire n eigenvalues can be assigned through what we call the q-phase and through what we call the N-phase.
A particular matrix g N AN arose when deriving the sum of the remaining nq  enforced eigenvalues encountered when applying IEA. The author in [1], went as far as showing that the trace of g N AN equals the sum of the enforced eigenvalues. We pursue such assertion further and show that g N AN encompasses the remaining nq  enforced eigenvalues as well.
In conjunction with the use of IEA as a mechanism of extending the method to uncontrollable systems, it can be justified on its own right in obtaining other methods beyond the assignment of a single eigenvalue as has been done in [1]. It enables the introduction and derivation of two new methods named: the spectral and the truncated methods. They are systematic and of hybridized nature. They are formally stated and proved in Section III and Section IV.
The spectral hybridized method relies on a particular choice of N based on certain candidate vectors obtained when solving a particular set of linear equations. An advantage of the spectral hybridized method is the ability to affect closed loop system structure beyond merely shifting the eigenvalues.
The truncated method is based on polynomials of order nq  . It yields a systematic and simplified feedback controller of lower dimensionality. In fact, the enforced eigenvalues turn out to be the roots of sub-polynomials; those trruncated from the open loop characteristic polynomial. A stable open loop system is shown to always result in a stabilized system. An unstable open loop system can sometimes result in a stabilized closed loop system conditionally. A necessary condition for stability when employing any order of truncated polynomial is a negative trace of A. Moreover, this condition is also sufficient when employing a first order sub-polynomial. These facts are proved in Section V. The sub-polynomials are dictated by the coefficients in the last column of a transformed matrix 1 C AC  which happen to be the coefficients of the open loop characteristic polynomial as proved in Section IV. A particular similarity transformation is therefore derived to facilitate our analysis.
The case of repeated and complex eigenvalue assignment pose no problem within the classical Ackermann's method. The same applies with the hybridized methods. The case of uncontrollable eigenvalue assignment enabled by IEA requires no special treatment, and is shown to even simplify computations in terms of reduced complexity and reduced dimensionality. However, knowledge of the actual values of the uncontrollable eigenvalues as required by other methods [9][10] is not required by ours. Furthermore, as an added advantage, a simplified feedback matrix results. The number of terms and the highest power of A are now reduced adding to the numerical attractiveness of the proposed methods.  Demonstrating the spectral method ability to provide more control over the entire spectrum beyond ordinary eigenvalue assignment .
 Knowledge of the number and values of the uncontrollable eigenvalues is optional. Assignment can be carried out regardless. This is not the case with most methods of eigenvalue assignment.

II. PRELIMINARY SETTINGS FOR THE HYBRIDIZED METHODS
The system considered is a linear time invariant system.
Since matrix B has full rank 1 it is routinely replaced by b . The state feedback controller used is where K is a 1 n  row matrix yielding the closed loop system.
The method has been revisited in [1], where the author derived an alternate compact form of (4) given by Where (.) g stands for a specialized left inverse of a matrix.
A modification to the controllability matrix in [1] led to the introduction of what has been termed IEA, which explicitly assign qn  eigenvalues and implicitly assign nq  eigenvalues . The introduction of IEA facilitated additional enrichment to (5) culminating in a generalization of the method to uncontrollable systems.
The controllability matrix and later the modified controllability matrix(MCM) play central roles in our study.
The controllability matrix C is well known; given by The following partitioning of the inverse of C is most convenient.
Each row matrix with the superscript g is a unique generalized inverse of a column of the C matrix, [11][12][13].
The generalized inverses are unique in our case since they satisfy the additional conditions given in (8 ) and (9). To cater for IEA and assignment of uncontrollable systems, the controllability matrix was modified in [1] and justifiably given the name MCM. In our study, MCM is denoted by a matrix E given by Where N is n n q  matrix ensuring the nonsingularity of E together with other terms and conditions on its selection, which will be uncovered later. Consequently, the inverse of E assumes the following form shown in (11) together with the adjoining conditions listed below: Conditions (12) and (13) will be referred to quite frequently. As proved in [1], such modification to C As shown in [1], Where the 1 in the first row matrix is positioned at the qth column. The remaining nq  eigenvalues are implicitly assigned and are dependent on N . See example 1, where 1 q  and N is an arbitrary vector. Methods for selecting appropriate N is the main theme of this paper as stated in Section III and Section IV.

III. THE SPECTRAL METHOD
The classical Ackermann's method offers no control over the system spectrum other than changing the eigenvalues. Association between eigenvalues and eigenvectors exist in the sense that knowledge of one leads to knowledge of the second. Therefore, one may think of the columns of N as a kind of eigenvectors resulting in assignment of the associated eigenvalue. In which case, we have a blending of two approaches to eigenvalue assignment; one explicit through the q-phase and another implicit through the Nphase. A type of hybridization we shall call the spectral method. The spectral method is justifiable whenever more control on the structural properties of the closed loop system beyond eigenvalue assignment are desired, and whenever control over the entire spectrum beyond stabilizability is required as demonstrated in example 3. It has the advantage of knowing the closed loop eigenvectors prior to the calculation of K. When used with uncontrollable systems, the freedom in choosing the eigenvectors is even broaden resulting in diversified choices of K. By and large, the spectral method provides additional freedom utilized in shaping the transient and the steady state response. See example 3. To develop the spectral method we proceed as follows.
Referring to (3), using the following state transformation  (17), The essence of the spectral method is to assign the remaining nq  eigenvalues through a specific choice of g K N A N , which will be shown later to equal g N AN .
As proved in [1], the q-phase ensures assignment of q eigenvalues using (15). What remains now is to select an appropriate N matrix to ensure assignment of the rest nq  eigenvalues.
Note that due to controllability of the system, Or, for repeated eigenvalue assignment.  (18) is thus determined as.
Collectively, and based on (23) and (24), we now have . In a compact form, Using the decompositions as in (10) (12) gives

IV. THE TRUNCATED METHOD
In principle, other blends involving the q-phase and Nphase are possible. The difficulty is in selecting an N matrix that results in at least a stabilized To develop the proposed method, we need to expose the system structure through a suitable similarity transformation T followed by relevant stability study.
The similarity transformation T is worked out as follows.
Equivalently, using To get the most simple form for (36) , let T is the controllability matrix as in (6). In which case, And,   11 0 0 1 ( ) Hence, It can be shown that ( see Appendix).    Therefore, the stability of the closed loop system is dictated by (49), which depends on the roots of (50), which is a polynomial extracted out of the original open loop system characteristic polynomial. Conditions governing closed loop stability are now investigated.

A linear time-invariant system is asymptotically stable if and only if matrix A is Hurwitz. A more popular approach is that of Routh's which is based on the coefficients of the characteristic polynomial. A system has all poles in the open left half plane if and only if all first-column elements of the Routh's array have the same sign.
Alongside Routh's method, [20], [21] give simple ratio checking inequalities that determine stability of a system. A necessary condition for stability is that all the coefficients of the characteristic polynomial have the same sign.
Given a system characteristic equation Assume all i a have the same sign. This condition is necessary. Otherwise, the system is unstable since a change in sign in the characteristic polynomial renders the system unstable as established in control theory.
So, to establish stability, the following inequalities regarding the ratios of the coefficients of () n  should be satisfied [20], [21]. Based on the inequalities given in (52), we now claim the following assertion: If an open loop system is known to be stable and hence, satisfying (52), then any subsystem having a truncated characteristic equation of order p where pn  and, will be stable.
The coefficients i a are those of () n as in (52).
The proof is straightforward. It follows from (52). Since the system is stable then (52) is satisfied.
Since fewer terms as dictated by (51) are involved, satisfaction of the inequalities in (52) is preserved. Hence, the subsystem as given by its characteristic polynomial as in (50) will be stable.
However, the roots of the subsystem given by (50) are generally different from those of the original open loop system given by (51).
Such assertion is needed when using the Truncated method, which centers on the involvement of truncated polynomials extracted out of the open loop characteristic polynomial.
In utilization of the above facts, to settle down on a proper truncated polynomial one must observe the following:  If a system is open loop stable then every extracted truncated characteristic polynomial employed in the truncated method leads to a stabilized closed loop system.
 If the open loop system is unstable then stabilizability is still possible when employing particular truncated polynomial of certain orders. In this case, vigilance is needed to involve as many terms needed in the truncated polynomial as to ensure stability. That is, consider as many terms as the conditions in (52) permit.
When deciding to use the truncated method, if the trace of A is positive, the truncated method cannot be used at all as the open loop system will be unstable and the closed loop system will be unstable as well no matter what truncated polynomial is used. This is due to a change in sign in the first two highest powers of the truncated polynomial used.
However, an open loop system can be unstable, but a stabilized closed loop system is possible as long as the w can be any convenient column-vector ensuring a nonsingular W . Obviously, it may be the system input matrix b whenever the system is controllable.
However, a more efficient method is to use the iterative Faddeeva-Leverrier algorithm [22][23]. It is an efficient method for finding the coefficients of the characteristic polynomials. Furthermore, if A is nonsingular, an additional advantage is that the inverse of A is readily obtainable at no extra computational cost.
The decomposition given in (42) can lead to another structural property of the closed loop system as represented in (42). That is, the sum of the q eigenvalues assigned through the q-phase are given by the trace of Let tr stands for trace of a matrix . Hence, using (13), Referring to (32) and (41) in [1] , Note that neither of the two matrices give the q eigenvalues except in certain special cases as when using the spectral method as verified in (32).

VI. UNCONTROLLABLE SYSTEMS
Dealing with uncontrollable systems using Ackermann's method was made possible in [1] owing to a modification to the controllability matrix. The problem of uncontrollability was resolved by replacing the columns causing a singular controllability matrix C by columns of another matrix N to establish nonsingularity of the ensuing MCM.
As asserted in [1], there is no need to determine the uncontrollable eigenvalues (considered an advantage); only their number say  is needed in order to arrive at the dimension of N , which has to be  . The columns of N can be arbitrary; subject to ensuring a nonsingular MCM. Note that other available software methods like MATLAB place command [10] necessitates knowledge of the uncontrollable eigenvalues, otherwise, an error message is issued.
The feedback matrix used is u K given by 11 11 ( Where n  is the number of controllable eigenvalues. The remaining uncontrollable eigenvalues are thus inescapably re-assigned. Bear in mind that 1 () g Ab  is not unique. It depends on the N used , which has to satisfy the conditions in (12).
The non-uniqueness of N leads to a non-unique u K . This is fully justified in control theory as any feedback matrix for uncontrollable systems is not unique even for singleinput systems and even when using alternative approaches within the same method as in the IEA method . Depending on the choice of N , a different u K is obtained. This fact deserves the following warrant: you cannot validate the u K computed as in (56) using another different method such as MATLAB place command [10]. One better check validity of design by calculating the eigenvalues of The non-uniqueness of u K can be considered a design parameter fulfilling other system requirements such as minimizing energy, reducing certain norms, and modifying eigenvectors. Besides, it is possible to avoid feeding back certain states. This is made possible by settling on a u K with zero or almost zero coefficients.
Caution: The number of eigenvalues assigned by u K cannot exceed n . If tried inadvertently, the K obtained will result in incorrect assignment of the controllable eigenvalues.

Example 1
An unstable uncontrollable system has the following system matrices with a nonzero initial condition.
It is required to assign 8  and inescapably the uncontrollable eigenvalue 2  . The method used is IEA with different N matrices, see (10). In other words, N can be a design tuning parameter, not only in shifting the eigenvalues assigned, but also in shaping the response through their influence on the closed loop eigenvectors.
Example 2 An unstable controllable system has the following system matrices with a nonzero initial condition, 5.5 3 3 1 1 6 2.5 4 ; 2 ; (0) 0 The truncated method is to be used. Referring to the appendix, using w as simple as   Fig. 3. shows excessive overshoot in 1 () x t , followed by appreciable undershoot and a settling time of 3.5 seconds when 1 K and 2 K were used. A marked improvement in every respect was obtained when using 3 K . Although the controlled systems have the same set of eigenvalues they are structurally different due to the incorporation of three different choices for 2 v in the calculation of the feedback matrix.

VIII. CONCLUSION
The study expands what has been done in [1], culminating in two methods for eigenvalue assignment: the truncated and the spectral methods. They are based on IEA and MAM presented in [1]. The two methods contributed, merge the classical Ackermann's approach with newly proposed structures; rendering them of hybridized nature. Our study shows that the open loop system structure can be exploited in simplifying controller design subject to certain stability conditions. Besides, more control over the shape of system response can be exercised when using the spectral method. The two methods simplify the structure of the feedback matrix in certain cases of comlex eigenvalue assignment and especially when the system is uncontrollable. Besides, the study has shown that uncontrollable systems require no unnecessary special approaches and that they can simplify design and calculations to the full. The three graded numerical examples worked-out and simulated clarify the designs presented.
Lastly, the classical Ackermann's method can only be used when the system is controllable. When stabilizability and ease of design is sought, the truncated hybridized method is recommended. The spectral hybridized method should be used whenever more control on the entire spectrum and the shape of response are required.
Future work may tackle the problem of adapting the classical Ackermann's method and the hybridized methods to the eigenvalue assignment of controllable and uncontrollable multi-input systems. Furthermore, exploration of other matrix configurations and forms may result in even more simplified forms concerning the truncated and the spectral methods.