Design of a system in controller canonical form

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

with

This canonical form has the following properties:

- The characteristic polynomial can be directly determined from the last line of
, which is

- A system of this structure is always controllable as its
controllability matrix according to
Eq. (12.16) has always full rank.
- The transfer function of the system is immediately given by

The feedback is now defined as

with

For the closed-loop system the system matrix is

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

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

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

Finally, the feedback vector is transformed back to

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

and

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

With the row vectors of the matrix this equation is

and after multiplication:

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

and when is known the remaining rows of the matrix are

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:

or in transposed form:

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

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:

- 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
.
- Row-wise calculation of the remaining rows of the
transformation matrix
using Eq. (13.46).
- Choice of the eigenvalues .
- Calculation of the coefficients of the desired
polynomial according to Eqs. (13.29) and (13.30).
- 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.
- Calculation of the coefficients of the feedback vector
according to Eq. (13.38).
- Back transformation of the feedback vector according to
Eq. (13.41).