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Subsections


Design of a system in controller canonical form

A state-space system in controller canonical form has the following structure:

    initial condition (13.32)

(13.33)

with

(13.34)

This canonical form has the following properties:

The feedback is now defined as

(13.36)

with

   

For the closed-loop system the system matrix is

(13.37)

and its characteristic polynomial is

   

Equating this polynomial with the polynomial with the desired poles from Eqs. (13.30) one obtains by comparison of the corresponding terms of both polynomials the controller parameters as

   for (13.38)

In the controller canonical form the calculation of the controller feedback parameters is reduced to the calculation of a simple difference between the coefficients of two polynomials.


Design of a system not in a canonical form

In general, when a system is not given in the controller canonical form, one has to transform it by a regular transformation

(13.39)

which brings the system into the desired canonical form according to Eqs. (13.32) to (13.34). The determination of the controller parameters in is performed according to Eq. 13.38. The feedback law in the original state is, using Eq. (13.39) given by

(13.40)

Finally, the feedback vector is transformed back to

(13.41)

The main task in the pole-placement design for systems that are not in controller canonical form, is the determination of the transformation matrix . When the original state equation from Eq. (13.1) is transformed by Eq. (13.39), one obtains the transformed entities

(13.42)

and

(13.43)

Eq. (13.42) will be further analysed in the form by right-multiplying with

   

With the row vectors of the matrix this equation is

(13.44)

and after multiplication:

(13.45)

From the first rows of both sides one obtains the recursive relationship

   for    

and when is known the remaining rows of the matrix are

   for (13.46)

The first row is obtained from Eq. (13.43), which is also valid. Using the results from Eq. (13.46), the right-hand side of Eq. (13.43) has the form:

(13.47)

or in transposed form:

(13.48)

The matrix is the controllability matrix from Eq. (12.16), which has full rank if the system is completely controllable. Under this condition one can obtain the first row by

(13.49)

which is the last row of the inverse controllability matrix, because is the -th unit vector.

Summarising the procedure of the pole-placement design for a system that is not in a canonical form, the following steps are necessary:

  1. Calculation of the controllability matrix from Eq. (12.16), its inverse and extracting the last row according to Eq. (13.49). This is the first row of the transformation matrix .

  2. Row-wise calculation of the remaining rows of the transformation matrix using Eq. (13.46).

  3. Choice of the eigenvalues .

  4. Calculation of the coefficients of the desired polynomial according to Eqs. (13.29) and (13.30).

  5. Transformation of the system matrix according to Eq. (13.42) and extracting the coefficients of the characteristic polynomial of the open-loop system from the last row of the transformed matrix.

  6. Calculation of the coefficients of the feedback vector according to Eq. (13.38).

  7. Back transformation of the feedback vector according to Eq. (13.41).



Next: Design using Ackermann's formula Up: Design of state-feedback controllers Previous: Design of state-feedback controllers   Contents
Christian Schmid 2005-05-09