_{1}

^{*}

A state feedback method of reduced order for eigenvalue assignment is developed in this paper. It offers immediate assignment of m eigenvalues, with freedom to assign the remaining n-m eigenvalues. The method also enjoys a systematic one-step application in the case where the system has a square submatrix. Further simplification is also possible in certain cases. The method is shown to be applicable to uncontrollable systems, offering the simplest control law when having maximum uncontrollable eigenvalues.

The problem of eigenvalue assignment is well established in control theory where numerous methods have been proposed―each with certain advantages and disadvantages. However, a need still arises for methods which are simple in concept and can be easily implemented. A fulfillment to such need is contributed by this paper.

As compared with some previous methods for eigenvalue assignment, this method doesn’t require specific transformations, knowledge of the open loop eigenvalues or the determination of the closed loop eigenvectors. The method utilizes submatrices stemming from a particular state transformation. The transformation is only needed in the development of the method and not the actual assignment of the eigenvalues.

The proposed method tackles eigenvalue assignment by manipulating lower order matrices, hence enjoying some numerical advantages. Furthermore,

The method is also shown to apply to uncontrollable systems where certain features of some submatrices are pointed out, thus providing additional degrees of freedom in the control law. Furthermore, in the case of maximum number of uncontrollable eigenvalues, the controller is shown to exhibit its simplest form and offer arbitrariness which may be utilized in fulfilling a myriad of design objectives.

Finally, the systematic and straightforward nature of the method is demonstrated by two examples.

The assignment law considered is a state feedback law of the form

where

For the development of the simplified methods, a state transformation T is used where

where

Such requirement on

where

Using the terminology above, the submatrices become

In addition

With reference to the recursive method of Hassan et al. [

The recursive method [

According to [

where

i.e.

substituting

Substituting the values of

Using the fact that

The advantage of this feedback law as given in (2.10) is that assignment of n eigenvalues is split into independent assignment of

Although the previous development resulted in a controller which manipulates lower order matrices; the selection of

Due to the presence of identical terms within the parenthesis’s, we simplify one term in the state feedback matrix

where

Assuming the nonsingularity of

Substituting the value of

Using (2.7), and recalling

substituting this value for the two terms in the parenthesis’s in Equation (2.10) gives

Some remarks regarding the control law are stated below.

A necessary condition for the invertibility of

To see this, suppose

No need to do the state transformation. The determination of (2.4) is only needed to extract

Assignment of

As compared with other assignments laws the highest power of

Additional simplification can be done to the form of (3.4). By replacing

Ending up with a compact form for K as

If

The choice of

§ The selection of N is systematic.

§ Such choice gives the advantage of inverting an

§ Further computational advantages are gained if the Gram-Schmidt ortho-normalization procedure is used (can be easily programmed on a digital computer and is already within the MATLAB function library). In this case, if

A further simplification to (4.1) is possible in the case where

So, the design process now reduces to the selection of

The non-recursive feedback law can still be applied when the system is uncontrollable. In our case, and as has been shown by [

For the case

a) The matrix

b) Since

nations of each other. To see this, due to uncontrollability, the matrix

late

In the light of the above facts since a nonsingular

If the system has the maximum number of

However, although (3.4) cannot be used to get the final feedback matrix

The justification for this form stems from the fact that in our case all uncontrollable eigenvalues are those of

Seeing it differently, since in our case

Note that

Note that

Example 1: Consider the controllable system given by

It is required to assign the eigenvalues −2, −3 and −5 ± j4.

To extract F_{3}, MATLAB was used with

Hence, to five significant digits

The matrices

Using the control law given by (3.4) results in the following state feedback matrix

To check, the system closed loop matrix

Which has the eigenvalues −2, −3, −5 + j4 and −5 − j4.

Example 2: Consider the following system [

This system is uncontrollable with −1 and −4 being the uncontrollable eigenvalues. It is desired to assign the two eigenvalues −4 and −5.

So let

To expose the controllable and uncontrollable eigenvalues, we may take

Yielding

Which shows that

Besides, the inverse of T isn’t needed to extract

Using K as in (6.1) yields a state feedback K matrix, say

Another

Which results in a different state feedback K matrix, say

Both

The paper has considered a method for eigenvalue assignment based on a scheme of recursive nature. The method involves algebraic manipulation of lower order matrices with an advantage of not requiring state transformation or eigenvectors determination. The method is further simplified in the case where