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
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.
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
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