Tracking Control of Linear Systems

Chapter 13 NTC versus LTC

13.1 General consideration

In order to compare Natural Tracking Control (NTC) with Lyapunov Tracking Control (LTC) we accept the tracking operator T(.) to determine the tracking algorithm (12.5) (Section 12.1),

$\begin{array}{l}T\left(t,\epsilon ,{\epsilon }^{(1)},\mathrm{...},{\epsilon }^{(k)},{\displaystyle {\int }_{t_0=0}^t\epsilon }dt\right)=T\left(t,{\epsilon }^k,{\displaystyle {\int }_{t_0=0}^t\epsilon }dt\right)=0_N,\forall t\in {\Im }_0,\\ k\in \left\{1,2,\mathrm{...}\right\}.\end{array}$ (13.1)

In view of Theorem 364 (Section 12.1) we accepted the modified tracking goal 359 (Section 12.1):

$\begin{array}{c}T\left(t^N,{\epsilon }^k,\underset{t_0^N=0_N}{\overset{t^N}{{\displaystyle \int }}}Ed{t}^N\right)=0_N,\forall t^N\in {\mathfrak{T}}_0^N\text{if}T\left(0_N;{\epsilon }_0^k,0_N\right)=0_N,\\ {\epsilon }^k\left(t^{(k+1)N};{\epsilon }_0^k\right)=-f^k\left(t^{(k+1)N};f_0^k\right),\left\{\begin{array}{l}\forall t{1}_{(k+1)N}\in {\mathfrak{T}}_0^N,\text{if}\hfill \\ T\left(0_N,{\epsilon }_0^k,0_N\right)\ne 0_N\hfill \end{array}\right\},\end{array}$

$k\in \left\{0,1,2,\dots \right\},$ (13.2)

where we refer to (12.19) through (12.25) (Section 12.1) so that:

(13.3)

This inspires us to introduce both $T\left(t,{ϵ}^k,\underset{t_0=0}{\overset{t}{{\displaystyle \int }}}ϵdt\right)$ and the time-varying ...