Mayersches quality standard

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The mayersche quality measure is in the control theory for the optimal control and the optimum control employed quality measure. It also has an important role in the theory of optimal control, particularly with regard to the maximum principle of Pontryagin . The peculiarity of Mayer's quality criterion is that it only evaluates the final state, i.e. has no integral part.

definition

A measure of the quality of form

${\displaystyle J=\Phi (t_{f},x(t_{f}))}$

is referred to as Mayer's quality measure .

Conversion into other forms of quality measure

In addition to Mayer's quality standard, there are also Lagrangian and Bolza's quality standards . The mayersche quality measure is a special case of bolzaschen quality measure. The conversion to the Bolzano form is therefore trivial by setting . ${\displaystyle f_{0}(t,x(t),u(t))=0}$

For the conversion of Bolza's quality standard into Mayer's quality standard , we use the system

${\displaystyle {\dot {x}}(t)=f(t,x(t),u(t)),x(t_{0})=x_{0}}$

out. The quality functional has the form

${\displaystyle J=\Phi (t_{f},x(t_{f}))+\int _{t_{0}}^{t_{f}}f_{0}(t,x(t),u(t)){\text{d}}t}$.

We add a state with and to our system . So is . ${\displaystyle \zeta }$ ${\displaystyle {\dot {\zeta }}(t)=f_{0}(t,x(t),u(t))}$ ${\displaystyle \zeta (t_{0})=0}$ ${\displaystyle \zeta (t_{f})=\int _{t_{0}}^{t_{f}}f_{0}(t,x(t),u(t)){\text{d}}t}$

We denote the extended state vector with . With that we can summarize everything: ${\displaystyle {\tilde {x}}=\left(x^{T},\zeta \right)^{T}}$

${\displaystyle J=\Phi (t_{f},x(t_{f}))+\zeta (t_{f})={\tilde {\Phi }}(t_{f},{\tilde {x}}(t_{f}))}$

The quality measure is therefore only a function of the final state and thus has Mayer form . For the conversions to and from the Lagrangian quality measure, one goes over the intermediate step of the Bolzash quality measure.