Ackermann's formula

Summary

In control theory, Ackermann's formula is a control system design method for solving the pole allocation problem for invariant-time systems by Jürgen Ackermann.[1] One of the primary problems in control system design is the creation of controllers that will change the dynamics of a system by changing the eigenvalues of the matrix representing the dynamics of the closed-loop system.[2] This is equivalent to changing the poles of the associated transfer function in the case that there is no cancellation of poles and zeros.

State feedback control

Consider a linear continuous-time invariant system with a state-space representation

where x is the state vector, u is the input vector, and A, B and C are matrices of compatible dimensions that represent the dynamics of the system. An input-output description of this system is given by the transfer function

Since the denominator of the right equation is given by the characteristic polynomial of A, the poles of G are eigenvalues of A (note that the converse is not necessarily true, since there may be cancellations between terms of the numerator and the denominator). If the system is unstable, or has a slow response or any other characteristic that does not specify the design criteria, it could be advantageous to make changes to it. The matrices A, B and C, however, may represent physical parameters of a system that cannot be altered. Thus, one approach to this problem might be to create a feedback loop with a gain K that will feed the state variable x into the input u.

If the system is controllable, there is always an input such that any state can be transferred to any other state . With that in mind, a feedback loop can be added to the system with the control input , such that the new dynamics of the system will be

In this new realization, the poles will be dependent on the characteristic polynomial of , that is

Ackermann's formula

Computing the characteristic polynomial and choosing a suitable feedback matrix can be a challenging task, especially in larger systems. One way to make computations easier is through Ackermann's formula. For simplicity's sake, consider a single input vector with no reference parameter , such as

where is a feedback vector of compatible dimensions. Ackermann's formula states that the design process can be simplified by only computing the following equation:

in which is the desired characteristic polynomial evaluated at matrix , and is the controllability matrix of the system.

Proof

This proof is based on Encyclopedia of Life Support Systems entry on Pole Placement Control.[3] Assume that the system is controllable. The characteristic polynomial of is given by

Calculating the powers of results in


Replacing the previous equations into yields

Rewriting the above equation as a matrix product and omitting terms that does not appear isolated yields

From the Cayley–Hamilton theorem, , thus

Note that is the controllability matrix of the system. Since the system is controllable, is invertible. Thus,

To find , both sides can be multiplied by the vector giving

Thus,

Example

Consider[4]

We know from the characteristic polynomial of that the system is unstable since , the matrix will only have positive eigenvalues. Thus, to stabilize the system we shall put a feedback gain

From Ackermann's formula, we can find a matrix that will change the system so that its characteristic equation will be equal to a desired polynomial. Suppose we want .

Thus, and computing the controllability matrix yields

and

Also, we have that

Finally, from Ackermann's formula

See also

References

  1. ^ Ackermann, J. (1972). "Der Entwurf linearer Regelungssysteme im Zustandsraum" (PDF). At - Automatisierungstechnik. 20 (1–12). doi:10.1524/auto.1972.20.112.297. ISSN 2196-677X. S2CID 111291582.
  2. ^ Modern Control System Theory and Design, 2nd Edition by Stanley M. Shinners
  3. ^ Ackermann, J. E. (2009). "Pole Placement Control". Control systems, robotics and automation. Unbehauen, Heinz. Oxford: Eolss Publishers Co. Ltd. ISBN 9781848265905. OCLC 703352455.
  4. ^ "Topic #13 : 16.31 Feedback Control" (PDF). Web.mit.edu. Retrieved 2017-07-06.

External links

  • Chapter about Ackermann's Formula on Wikibook of Control Systems and Control Engineering