Lagrangian quality measure

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The Lagrangian quality measure is a frequently used quality measure in control theory for optimal control and optimal regulation . It has the peculiarity that it only consists of an integral term.

definition

A measure of the quality of form

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

is called the Lagrangian quality measure.

Conversion into other forms of quality measure

In addition to the Lagrangian quality standard, there is also the Mayer and the Bolzash quality standard . The Lagrangian quality measure is a special case of Bolza's quality measure. The conversion to the Bolza form is therefore trivial by setting . ${\displaystyle \Phi (t_{f},x(t_{f}))=0}$

The conversion of the Bolza's quality measure into the Lagrangian quality measure is done by differentiating.

We assume that the system is the form

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

Has. The quality measure has the bolt ash shape

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

Assuming sufficient continuity, we can write

${\displaystyle \Phi (t_{f},x(t_{f}))=\Phi (t_{0},x(t_{0}))+\int _{t_{0}}^{t_{f}}\left({\frac {\partial \Phi }{\partial t}}+{\frac {\partial \Phi }{\partial x}}{\frac {{\text{d}}x}{{\text{d}}t}}\right)dt=\Phi (t_{0},x(t_{0}))+\int _{t_{0}}^{t_{f}}\left({\frac {\partial \Phi }{\partial t}}+{\frac {\partial \Phi }{\partial x}}f(t,x(t),u(t))\right){\text{d}}t}$.

With this we can rewrite the standard of quality

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

The term only depends on the initial conditions. It can therefore be viewed as a constant additive component and therefore does not have to be taken into account in further calculations. This changes the numerical value of the quality measure, but this has no influence on the calculation of an optimal control . ${\displaystyle \Phi (t_{0},x(t_{0}))}$

The conversion from or into the Mayer measure of quality is usually done using the Bolas measure of quality.