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We know from the characteristic polynomial of <math>A</math> that the system is unstable since <math>det(sI-A)=(s-1)(s-2)-1=s^2-3s+1</math>, the matrix <math>A</math> will only have positive eigenvalues. Thus, to stabilize the system we shall put a feedback gain <math>K=\left[\begin{array}{cc} |
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We know from the characteristic polynomial of <math>A</math> that the system is unstable since <math>det(sI-A)=(s-1)(s-2)-1=s^2-3s+2</math>, the matrix <math>A</math> will only have positive eigenvalues. Thus, to stabilize the system we shall put a feedback gain <math>K=\left[\begin{array}{cc} |
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k_1 & k_2\end{array}\right]. </math> |
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k_1 & k_2\end{array}\right]. </math> |
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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
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 yieldsRewriting 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
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 L. 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
References
External links