Tracking Control of Linear Systems

Appendix B

From IO system to ISO system

What follows is well known in the control theory, but is not available in many books on the first course of control systems. In order to transform the IO system (2.1) (Section 2.1),

${{\displaystyle \sum _{k=0}^{k=v}{A}_k{Y}^{\left(k\right)}\left(t\right)={\displaystyle \sum _{k=0}^{k=\mu }{B}_{k}I}}}^{\left(k\right)}\left(t\right),det{A}_V\ne 0,t\in 𝔗,v\ge 1,0\le \mu \le v,$ B.1

into the equivalent ISO system (2.29), (2.30) (Section 2.2),

$\frac{\text{d}X\left(t\right)}{dt}=AX\left(t\right)+BI\left(t\right),t\in 𝔗$ B.2

$Y\left(t\right)=CX\left(t\right)+DI\left(t\right),t\in 𝔗$ B.3

let mathematically, without any physical justification or meaning, subsidiary vector variables X_ı, X₂, … X_v be

$X_1=Y-B_{v}I,$ B.4

$X_2={\stackrel{•}{X}}_1+A_{v-1}Y-B_{v-1}I,$ B.5

$X_3={\stackrel{•}{X}}_2+A_{v-2}Y-B_{v-2}I,$ B.6

$X_4={\stackrel{•}{X}}_3+A_{v-2}Y-B_{v-3}I,$ B.7

$\dots .$ B.8

$X_{v-2}={\stackrel{•}{X_v}}_{-3}+A_{3}Y-B_{3}I,$ B.9

$X_{v-1}={\stackrel{•}{X_v}}_{-2}+A_{2}Y-B_{2}I$ B.10

$X_v=$