Ackermann's formula
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. 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. 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 formula_4 such that any state formula_5 can be transferred to any other state formula_6. With that in mind, a feedback loop can be added  to the system with the control input formula_7, such that the new dynamics of the system will be In this new realization, the poles will be dependent on the characteristic polynomial formula_10 of formula_11, 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 formula_13, such as where formula_16 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 formula_18 is the desired characteristic polynomial evaluated at matrix formula_19, and formula_20 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 formula_21 is given by Calculating the powers of formula_23 results in Replacing the previous equations into formula_25 yieldsformula_26Rewriting the above equation as a matrix product and omitting terms that formula_16 does not appear isolated yields From the Cayley–Hamilton theorem, formula_29, thus  formula_30 Note that formula_31 is the controllability matrix of the system. Since the system is controllable, formula_20 is invertible. Thus, To find formula_34, both sides can be multiplied by the vector formula_35 giving Thus, Example. Consider formula_38 We know from the characteristic polynomial of formula_19 that the system is unstable since formula_40, the matrix formula_19 will only have positive eigenvalues. Thus, to stabilize the system we shall put a feedback gain formula_42 From Ackermann's formula, we can find a matrix formula_43 that will change the system so that its characteristic equation will be equal to a desired polynomial. Suppose  we want formula_44. Thus, formula_45 and computing the controllability matrix yields Also, we have that formula_48 Finally, from Ackermann's formula