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Methods which calculate state feedback matrices explicitly for uncontrollable systems are considered in this paper. They are based on the well-known method of the entire eigenstructure assignment. The use of a particular similarity transformation exposes certain intrinsic properties of the closed loop w-eigenvectors together with their companion z-vectors. The methods are extended further to deal with multi-input control systems. Existence of eigenvectors solution is established. A differentiation property of the z-vectors is proved for the repeated eigenvalues assignment case. Two examples are worked out in detail.

A study by [

As required by this method, the w-eigenvectors and companion z-vectors are extracted out of the null space of an augmented

Basically, the method in [

The procedure in [

For the single-input and multi-input cases, the study shows that calculations of the needed w-eigenvectors and the z-vectors are based on lower order matrices specifying the controllable part and the uncontrollable part of the system. Such approach simplifies the design process, and provides numerical advantages.

Finally, the two examples are worked out in Section 8 to illustrate the ease of use of the assignment process.

Consider the linear time-invariant system given by

where

or

Such setup as in (2.3) is associated in the control literature with the entire eigenstructure assignment method (see [

It is assumed that the open loop characteristic equation is given by

In the development of the explicit methods, a state transformation

resulting in the system

A similar transformation will be used in this paper, together with the following rearrangement of (2.3) as

Such rearrangement is preferable in order to avoid a mixture pluses and minuses in the resulting formulae.

The design procedure outlined in [

where q is the number of controllable eigenvalues and N is any

where

where

trix

It will now be shown that the calculation complexity can be eased through decomposing the closed loop eigenvectors into two vector parts. By doing so, reduced order matrices are dealt with, resulting in vector parts of dimension

Consider assignment of an eigenvalue

where

So

Since

Since

Equations in (4.5) are in the same format of as (3.3) where a solution always exists irrespective of

and

Note that both solutions of

Consider now reassignment of an uncontrollable eigenvalue

One choice for

Since

A second choice is that

Since

The arbitrariness in

This second choice is a must when using the entire eigenstructure assignment method. If

It is worth mentioning that an eigenvector corresponding to an uncontrollable eigenvalue can be tailored out of the two possible ones stemming from the two choices.

Finally, having obtained n independent eigenvectors

The explicit nature of the method can be extended to a multi-input case. This is possible in the case where matrices A and B have a particular structure which results in the following augmented matrix

To be an nxn square and nonsingular, where

To prove such assertion, use the same similarity transformation

where

The following proof is straightforward, achieved by substituting generalized matrix forms for the w-eigen- vectors and z-vectors in (5.3). It is presented for the case

and

Note that the z-vectors

The nested nature of the solutions is imminent, easily generalized for cases where

The extension of the assignment to multi-input uncontrollable systems is also straightforward. The number of the uncontrollable eigenvalues should be an integer multiple of

The same theory developed in Section 4 still applies. The uncontrollable eigenvalues are those of

In [

Consider the general setup of the entire eigenstructure assignment as formulated in (2.2). Let there be p identical eigenvalues

and

To facilitate the proof, a convenient rearrangement for

Differentiating (6.3) with respect to

Comparing (6.4) with (6.5), we infer

Similarly, differentiating (6.4) with respect to

or

Comparing (6.8) with (6.9)

We get

Repeating the same process, it can be shown that

or

Confirming what has been demonstrated in [

It’s worth considering the existence of the solutions when considering the controllable and uncontrollable subspaces. For the controllable subspace, we seek the solution

For the solution of

If

Alternatively,

i.e.

i.e. (7.4) holds. Hence, a solution always exists irrespective of

For the uncontrollable subspace

Since the right hand side is zero the condition in (7.4) always holds and the solution always exists given by a matrix representation of the null space of

Example 1

An uncontrollable system has the following system matrices

The system is unstable having eigenvalues 1, −1, −2, and −3. It is required to assign the eigenvalues −3, −4, −5, and of course to reassign the uncontrollable eigenvalue −2.

The similarity transformation used is

Leading to G and H matrices

As evident by the system after transformation, −2 is the uncontrollable eigenvalue, and that the controllable subspace has the matrix representation as that of (3.3).

Hence, the reduced order characteristic equation is

Utilizing explicit determination, the closed loop eigenvector corresponding to the −2 eigenvalue is calculated using (4.11), the remaining ones using (4.7), and the companion z-vector using (4.6) where

In order to have a nonsingular W matrix, the eigenvector

According to (4.12), the state feedback matrix in the original system representation is

N.B.; The state feedback matrix above assigns the four eigenvalues required according to the entire eigenstructure method. If the answer is to be checked using any other method like the Matlab place function, a different result for K may be obtained. This is due to the fact that K for uncontrollable systems is not unique.

Example 2

Consider an unstable multi-input system having the following A and B matrices

using the transformation

Hence,

Let the eigenvalues assigned be

Using the formulae given in (5.4) the closed loop eigenvectors are calculated pair-wise, i.e.

The companion z-vectors

According to (4.12) with

Note that Matlab calculates in double precision, however, format short of matlab has been used in the print out of the above results. So, to check the results, one may have to go through the calculations once more in case the precision of K provided in (8.14) is not adequate.

The study has shown that the explicit methods can be extended to uncontrollable systems just as easy with the benefit of dealing with lower order matrices, and consequently with reduced w-eigenvectors. The z-vectors are also determined using lower order characteristic equations and shown to bear a differentiation property for the repeated eigenvalues case. For the uncontrollable case, it turns out that the z-vectors have more degrees of freedom which can be used to shape the system response. The methods can also be extended to a special case of multi-input controllable and uncontrollable systems. The solutions of the w-eigenvectors and the z-vectors are always guaranteed. The two examples demonstrate the ease of application of the formulae in the design of state feedback matrices.