Ackermann's formula explained

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

• x (t) = Ax(t) + Bu(t) y(t) = Cx(t)

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

G(s) = C(sI - A) ^{-1} B = C (\operatorname{Adj)/((sI - A)}{\det(sI - A)} B.)

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

u(t) such that any state x _{0} can be transferred to any other state x(t) . With that in mind, a feedback loop can be added to the system with the control input u(t) = r(t) - Kx(t) , such that the new dynamics of the system will be• x (t) = Ax(t) + B[r(t) - Kx(t)] = [A - BK] x(t) + Br(t) y(t) = Cx(t).

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

\Delta _{new} of A - BK , that is\Delta _{new(s)=\det(sI-(A-BK)).}

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

r , such asu(t) =- k ^{T} x(t) • x(t) = Ax(t) - Bk ^{T} x(t),

where

k ^{T} is a feedback vector of compatible dimensions. Ackermann's formula states that the design process can be simplified by only computing the following equation:k ^{T} = \left[ 0 0 … 0 1 \right] l{C} ^{-1} \Delta _{new(A),}

in which

\Delta _{new(A)} is the desired characteristic polynomial evaluated at matrix A , and l{C} 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

A _{CL} : = (A - Bk ^{T} ) is given by\Delta(A _{CL} ) = (A _{CL} ) ^{n} + \sum ^{n -1} _{k =0} \alpha _{k} A ^{k} _{CL}

Calculating the powers of

A _{CL} results in\begin{align} (A _{CL} ) ^{0} & = (A - Bk ^{T)} ^{0=I} \\ (A _{CL} ) ^{1} & = (A - Bk ^{T)} ^{1=A-Bk} ^{T} \\ (A _{CL} ) ^{2} & = (A - Bk ^{T)} ^{2=A} ^{2-ABk} ^{T-Bk} ^{T} A + (Bk ^{T)} ^{2=A} ^{2-ABk} ^{T-(Bk} ^{T)[A-Bk} ^{T]=A} ^{2-ABk} ^{T-Bk} ^{T} A _{CL} \\ \vdots \\ (A _{CL} ) ^{n} & = (A - Bk ^{T)} ^{n=A} ^{n-A} ^{n-1} Bk ^{T-A} ^{n-2} Bk ^{T} A _{CL} - … - Bk ^{T} A ^{n -1} _{CL} \end{align}

Replacing the previous equations into

\Delta(A _{CL} ) yieldsRewriting the above equation as a matrix product and omitting terms that k ^{T} does not appear isolated yields

$$\backslash Delta(A\_)=\backslash Delta(A)-\backslash left[B\backslash \; \backslash \; AB\backslash \; \backslash \; \backslash cdots\; \backslash \; \backslash \; A^\{n-1\}\; B\backslash right]\backslash left[\backslash begin\{array\}\{c\}\; \backslash star\; \backslash \backslash \; \backslash vdots\backslash \backslash \; k^T\; \backslash end\{array\}\backslash right]$$

From the Cayley–Hamilton theorem,

\Delta\left(A _{CL} \right) =0 , thus\left[B AB … A ^{n-1} B\right]\left[\begin{array}{c} \star \\ \vdots\\ k ^{T \end{array}\right]=\Delta(A)}

Note that

l{C}=\left[B AB … A ^{n-1} B\right] is the controllability matrix of the system. Since the system is controllable, l{C} is invertible. Thus,\left[\begin{array}{c} \star \\ \vdots\\ k ^{T} \end{array}\right] = l{C} ^{-1} \Delta(A)

To find

k ^{T} , both sides can be multiplied by the vector \left[\begin{array}{ccccc} 0 & 0 & 0 & … & 1 \end{array}\right] giving\left[\begin{array}{ccccc} 0 & 0 & 0 & … & 1 \end{array}\right]\left[\begin{array}{c} \star \\ \vdots\\ k ^{T} \end{array}\right] = \left[\begin{array}{ccccc} 0 & 0 & 0 & … & 1 \end{array}\right] l{C} ^{-1} \Delta(A)

Thus,

k ^{T} = \left[\begin{array}{ccccc} 0 & 0 & 0 & … & 1 \end{array}\right] l{C} ^{-1} \Delta(A)

Example

Consider^[4]

• x = \left[\begin{array}{cc} 1 & 1 \\ 1 & 2 \end{array}\right]x + \left[\begin{array}{c} 1 \\ 0 \end{array}\right]u

We know from the characteristic polynomial of

A that the system is unstable since det(sI - A) = (s -1 )(s -2 ) -1= s ^{2-3s+2} , the matrix A will only have positive eigenvalues. Thus, to stabilize the system we shall put a feedback gain K = \left[\begin{array}{cc} k _{1} & k _{2\end{array}\right].}

From Ackermann's formula, we can find a matrix

k that will change the system so that its characteristic equation will be equal to a desired polynomial. Suppose we want \Delta ^{2+11 s +30} _{desired (s) = s} .

Thus,

\Delta ^{2+11 A +30 I} _{desired (A) = A} and computing the controllability matrix yieldsl{C}=\left[\begin{array}{cc} B & AB\end{array}\right] = \left[\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right] and l{C} ^{-1} = \left[\begin{array}{cc} 1 & -1 \\ 0 & 1 \end{array}\right].

Also, we have that

A ^{2=\left[\begin{array}{cc} 2} & 3 \\ 3 & 5 \end{array}\right].

Finally, from Ackermann's formula

k ^{T} = \left[\begin{array}{cc} 0 & 1 \end{array}\right]\left[\begin{array}{cc} 1 & -1 \\ 0 & 1 \end{array}\right]\left[\left[\begin{array}{cc} 2 & 3 \\ 3 & 5 \end{array}\right] +11 \left[\begin{array}{cc} 1 & 1 \\ 1 & 2 \end{array}\right] +30 I\right] k ^{T=\left[\begin{array}{cc} 0} & 1 \end{array}\right]\left[\begin{array}{cc} 1 & -1 \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} 43 & 14 \\ 14 & 57 \end{array}\right] = \left[\begin{array}{cc} 0 & 1 \end{array}\right]\left[\begin{array}{cc} 29 & -43 \\ 14 & 57 \end{array}\right] k ^{T=\left[\begin{array}{cc} 14} & 57 \end{array}\right]

State observer design

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

\hat{x}(n +1 ) = A \hat{x}(n) + B u(n) + L [y(n) - \hat{y}(n)] \hat{y}(n) = C \hat{x}(n)

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

L ^{\top} = \left[ 0 0 … 0 1 \right]( l{O} ^{\top} ) ^{-1} \Delta ^{\top} _{new (A} ) l{O} . Here it is important to note, that the observability matrix and the system matrix are transposed: l{O} ^{\top} and A ^{\top} .

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

See also

• Full state feedback

External links

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

Notes and References

1. Ackermann. J.. 1972. Der Entwurf linearer Regelungssysteme im Zustandsraum. At - Automatisierungstechnik. 20. 1–12. 297–300 . 10.1524/auto.1972.20.112.297. 111291582. 2196-677X.
2. Modern Control System Theory and Design, 2nd Edition by Stanley M. Shinners
3. Book: Ackermann, J. E.. Control systems, robotics and automation. 2009. Eolss Publishers Co. Ltd. Unbehauen, Heinz.. 9781848265905. Oxford. Pole Placement Control. 703352455.
4. Web site: Topic #13 : 16.31 Feedback Control. Web.mit.edu . 2017-07-06.