Control quality

The control performance is in control technology a measure for the control behavior of a control . It can be used to make a statement about the quality of the regulation. The quality measure must be adapted to the desired control behavior ( controlled variable , setpoint , manipulated variable ).

Standards such as the L ¹ standard (fast control behavior of the ITAE criterion), the L ² standard (quadratic quality criterion for minimum amplitudes ) or the maximum standard (maximum possible ratio of the energies or powers of error variables to input variables) or in particular are used for the quality measures for periodic signals the mean power pow. The standards give particular weight to certain deviations and must therefore be selected according to the task at hand.

Amount criterion ( L ¹ norm)

One possible measure of quality is the amount criterion. It takes into account positive and negative control differences equally:

${\displaystyle J_{abs}=\int _{0}^{\infty }|e(t)-e(\infty )|dt}$.

A special form of the amount criterion is the ITAE criterion , in which deviations over time are weighted more heavily:

${\displaystyle J_{ITAE}=\int _{0}^{\infty }|e(t)-e(\infty )|\cdot t\cdot dt}$.

Quadratic quality criterion ( L ² standard)

In contrast to the other standards presented here, the L ² standard (energy) is identical in the time and frequency domains.

It is the time-square integration of the control deviation of the actual value from the setpoint: ${\displaystyle e(t)}$

${\displaystyle J=\int _{0}^{\infty }e(t)^{2}dt}$

with   . ${\displaystyle e(t)=x_{\text{Soll}}(t)-x_{\text{Ist}}(t)}$

The smaller the value , the better the regulation. ${\displaystyle J}$

Since the resulting area would have an infinitely large value if the system deviation remained , the integral is often formed over the difference : ${\displaystyle e(\infty )}$ ${\displaystyle [e(t)-e(\infty )]}$

${\displaystyle J_{sqr}=\int _{0}^{\infty }[e(t)-e(\infty )]^{2}dt}$.

Quality criterion for the QL controller

In the previously considered quality criterion, it is not the manipulated variable u but only the controlled variable y that is considered. The quality criterion for the QL control also takes into account the relationship between these variables; the priorities can be determined using the and matrix ${\displaystyle Q_{y}}$ ${\displaystyle R}$

${\displaystyle J(x_{0},u(t))=\int _{0}^{\infty }(y'(t)Q_{y}y(t)+u'(t)Ru(t))dt.}$

The static optimization problem for this, which is solved by the QL control, is

${\displaystyle \min _{u(t)}\ J(x_{0},\ u(t))=\min _{u(t)=-Kx(t)}J(x_{0},\ u(t))=\min _{K}J(x_{0},\ -Kx(t)).}$

This static optimization problem is very often used for controller design, since the constant controller matrix K * does not depend on , i. H. depends on the initial state. The quality value J must be finite. The control for this is called LQ control because the quality function is quadratic and the distance is linear. This is a control through state feedback , which is why control is always set to 0. A setpoint sequence can only be implemented with an additional pre-filter . ${\displaystyle x_{0}}$

Medium performance

If the energy content of a signal ( L ² norm) is infinite, the mean power can be used for characterization.

The mean power is not a norm, since it can also become zero with signals other than zero. It is used for periodic signals (period ), since the above quality values ​​then do not become 0. ${\displaystyle T}$

${\displaystyle \operatorname {pow} (u):={\sqrt {\lim _{T\to \infty }{\frac {1}{2T}}\int _{-T}^{T}u^{2}(t)dt}}}$

Maximum norm

For the maximum norm only the maximum of the function in the interval [-∞, ∞] is decisive.

${\displaystyle ||u||_{\infty }:=\sup _{t}|u(t)|}$

The maximum norm of G ( ) is the greatest possible factor with which the "energy" of the input signal u is transferred to the output signal. ${\displaystyle ||G||_{\infty }}$

application

With these standards, precise specifications can be made that are to be met by the control (manipulated variable, controlled variable, control difference ). The degree of fulfillment can be checked by the result. If, for example, a standard x is minimized for a task, one speaks of an x-standard optimal control .

It is often checked what influence an unknown disturbance variable has on a controlled system. This is possible, for example, by calculating a norm for the transfer behavior from a disturbance to a control deviation and the disturbance was also previously characterized by a norm.

See also

• Regulator

swell

1. ↑ ^{a b} Kai Müller (1996)

literature

• Jan Lunze: Control engineering 2. Multi-variable systems. Digital regulation . 4., rework. Ed., Springer-Verlag, Heidelberg a. a. 2006 (= Springer textbook), ISBN 978-3-540-32335-8 .
• Jürgen Müller: Rules with SIMATIC. Practical book for controls with SIMATIC S7 and SIMATIC PCS 7 . Ed .: Siemens-AG Berlin u. Munich, 3rd edition, Publicis Corporate Publishing, Erlangen 2004, ISBN 3-89578-248-3 .
• Kai Müller: drafting robust regulations . Teubner-Verlag, Stuttgart 1996, ISBN 3-519-06173-2 .
• Serge Zacher, Manfred Reuter: Control technology for engineers. Analysis, simulation and design of control loops . 14th edition, Springer Vieweg Verlag, Wiesbaden 2014 (= Springer Fachmedien Wiesbaden), ISBN 978-3-8348-1786-0 , e-book ISBN 978-3-8348-2216-1
• Fritz Tröster: Control and regulation technology for engineers . 2., revised. and exp. Ed., R. Oldenbourg Verlag, Munich 2005, ISBN 3-486-57681-X . (P. 268f: "Control quality" )