Ackermann's formula. In control theory, Ackermann's formula provides a method for designing controllers to achieve desired system behavior by directly calculating the feedback gains needed to place the closed-loop system's poles (eigenvalues) at specific locations (pole allocation problem). 

These poles directly influence how the system responds to inputs and disturbances. Ackermann's formula provides a direct way to calculate the necessary adjustments—specifically, the feedback gains—needed to move the system's poles to the target locations. This method, developed by Jürgen Ackermann, is particularly useful for systems that don't change over time (time-invariant systems), allowing engineers to precisely control the system's dynamics, such as its stability and responsiveness.

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

where  is the state vector,  is the input vector, and  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

where  is the determinant and  is the adjugate.
Since the denominator of the right equation is given by the characteristic polynomial of , the poles of  are eigenvalues of  (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 , 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  that will feed the state variable  into the input .

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.  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

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

Also, we have that 

Finally, from Ackermann's formula

State observer design 
Ackermann's formula can also be used for the design of state observers. Consider the linear discrete-time observed system

with observer gain . Then Ackermann's formula for the design of state observers is noted as

with observability matrix . Here it is important to note, that the observability matrix and the system matrix are transposed:  and .

Ackermann's formula can also be applied on continuous-time observed systems.

See also 

 Full state feedback

References

External links 

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

Engineering concepts
Control engineering
Control theory