Pole specification

The pole specification or eigenvalue specification is a straightforward method of designing a controller in the state space . It consists of three steps that can be repeated several times if necessary:

1. Specification of the eigenvalues ​​or poles of the closed control loop
2. Calculation of the state feedback
3. Check whether the transient behavior or the manipulated variable curve have the desired behavior

The basis for this procedure is the close connection between the eigenvalues ​​of a system (here the closed control loop) and its step response.

The process is particularly simple when designing complete state feedbacks for single-variable systems , since there is usually a clear feedback. It can be determined with the so-called formula from Jürgen Ackermann . For systems with more than one manipulated variable ( multi-variable systems ), there are methods of modal pole specification and the decoupling according to Falb-Wolovich.

If not all state variables can be measured, the non- measurable variables can be calculated from the measurable variables by an observer .

When designing output feedbacks, not all state variables are fed back. This is the case when, on the one hand, not all state variables can be measured, but on the other hand, an observer cannot be used. Here you have to rely on numerical methods.

Another way to design state controllers is to use optimal control .

Specification of the poles

No general procedure can be specified for the selection of the poles or eigenvalues ​​of the control loop. Of course, all poles must be to the left of the imaginary axis to ensure stability. You will not be able to place them too far to the left, since this leads to the manipulated variable limitation in real systems.

A small additional calculation at this point is intended to demonstrate why the eigenvalues ​​of the system matrix correspond to the poles of the transfer function.

The starting point is the equation of state , which is transformed into the image space with the help of the Laplace transformation (neglecting initial values): ${\displaystyle {\dot {x}}(t)=Ax(t)+Bu(t)}$

${\displaystyle sX(s)=AX(s)+BU(s)}$

${\displaystyle (sI-A)X(s)=BU(s)}$

${\displaystyle X(s)=(sI-A)^{-1}BU(s)}$

Setting up the transfer function provides:

${\displaystyle G(s)={\frac {Y(s)}{U(s)}}={\frac {CX(s)}{U(s)}}={\frac {C(sI-A)^{-1}BU(s)}{U(s)}}={\frac {C\operatorname {adj} (sI-A)B}{\det(sI-A)}}\qquad {\text{mit}}\qquad (sI-A)^{-1}={\frac {\operatorname {adj} (sI-A)}{\det(sI-A)}}}$,

where is the adjunct of the matrix . ${\displaystyle \operatorname {adj} (sI-A)}$ ${\displaystyle (sI-A)}$

If one now wants to determine the poles of the transfer function, the equation must be solved, which also represents the equation for determining the eigenvalues ​​of the matrix . ${\displaystyle \det(sI-A)=0}$ ${\displaystyle A}$

Determination of the controller parameters

The system is through in the state space

${\displaystyle {\dot {x}}=Ax+Bu}$

described and it should be a return

${\displaystyle u=-Kx}$

to be determined. Then the closed loop is through

${\displaystyle {\dot {x}}=(A-BK)x}$

described. The eigenvalues ​​of the closed loop are the solution to the equation ${\displaystyle s_{i}}$

${\displaystyle \det \left[sI-\left(A-BK\right)\right]=0.}$

The essence of the pole specification is to determine the elements of in such a way that ${\displaystyle K}$

${\displaystyle \det \left[sI-\left(A-BK\right)\right]=\prod _{i=1}^{n}(s-s_{i})}$

applies. They are the target poles of the closed control loop. ${\displaystyle s_{i}}$

If the system is in order, a coefficient comparison leads to equations. In single-variable systems, these are opposed to the elements of the line vector . If the system is controllable, a clear solution can be given. In multi-variable systems with manipulated variables, elements of must be determined, i.e. H. the system of equations is underdetermined. On the other hand, the equations are non-linear, so no general solution can be given. The methods of modal control and decoupling according to Falb-Wolovich restrict the solution space in such a way that a solution can be specified. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle k^{T}=K}$ ${\displaystyle p}$ ${\displaystyle p\times n}$ ${\displaystyle K}$

Pole specification for single-variable systems

The controller parameters are determined according to the Ackermann formula in the following way:

Be it

${\displaystyle p(s)=\prod _{i=1}^{n}(s-s_{i})=s^{n}+p_{n-1}s^{n-1}+\dots +p_{1}s+p_{0}}$

the desired characteristic polynomial of the closed control loop. Then the controller matrix is ​​determined

${\displaystyle K=k^{T}=t_{1}^{T}\left(A^{n}+p_{n-1}A^{n-1}+\dots +p_{1}A+p_{0}I\right)=t_{1}^{T}\cdot p(A).}$

It is the last line of the inverse Steuerbarkeitsmatrix ${\displaystyle t_{1}^{T}}$

${\displaystyle Q_{S}^{-1}=\left[b,Ab,\dots ,A^{n-1}b\right]^{-1}.}$

This also makes the connection between the pole specification and the necessary controllability condition obvious. In fact, of course, one will not invert, but the system of linear equations ${\displaystyle Q_{S}}$

${\displaystyle t_{1}^{T}\cdot Q_{S}=\left[0,\dots ,0,1\right]}$

to solve.

After this connection was given for the first time in.

The connection between this design formula and the name Jürgen Ackermann can be found e.g. B. in

Pole specification for multi-variable systems

Modal regulation

The modal control allows the shifting of eigenvalues, whereby the number of manipulated variables is. However, by repeating this, all poles can ultimately be shifted. The controller matrix is ​​determined accordingly ${\displaystyle p}$ ${\displaystyle p}$

${\displaystyle K={\begin{pmatrix}w_{1}^{T}B\\\vdots \\w_{p}^{T}B\end{pmatrix}}^{-1}\cdot \operatorname {diag} (\lambda _{1}-s_{1},\dots ,\lambda _{p}-s_{p})\cdot {\begin{pmatrix}w_{1}^{T}\\\vdots \\w_{p}^{T}\end{pmatrix}}.}$

Here, the so-called. Links eigenvectors , so the lines of the inverse of the eigenvector matrix of the system matrix , the eigenvalues of the system and the desired eigenvalues. ${\displaystyle w_{i}^{T}}$ ${\displaystyle A}$ ${\displaystyle \lambda _{i}}$ ${\displaystyle s_{i}}$

The eigenvalues ​​that are not specifically shifted remain unchanged in this design process. If these are also to be shifted, the procedure must be applied again to the closed (inner) control loop.

According to the method comes from HH Rosenbrock (1962).

Decoupling according to Falb-Wolovich

The aim of the decoupling according to Falb - Wolowich is a management behavior in which a change in a reference variable only influences the associated controlled variable.

After he is presented in. It can be extended to time-variant and non-linear systems. Please refer to for details.

Literature and individual references

The article is based on Otto Föllinger: Control engineering, introduction to the methods and their application . 8th edition. Hüthig Verlag, Heidelberg 1994, ISBN 3-7785-2336-8 .

1. ↑ Section 14
2. ↑ Section 13.3.1
3. ↑ Section 13.3.3, 13.5
4. ↑ Section 13.3.2 Formula (13.32)
5. ↑ ^{a b} Section 13.3.2
6. ↑ ^{a b} Section 13.3.3
7. ↑ ^{a b} Section 13.5

This cites the following individual articles, which are given for historical reasons.

1. ↑ J. Ackermann: The design of linear control systems in the state space. Control engineering 20 (1972), pp. 297-300.
2. ^ HH Rosenbrock: Distinctive Problems of Process Control. Chemical Engineering Progress, 58, pp. 43-50 (1962).
3. ↑ PL fawn - WA Wolovich: Decoupling in the Design and Synthesis of Multivariable Control Systems. IEEE Trans. On Automatic Control 12 (1967), pp. 651-659.