Tracking Control of Linear Systems

Chapter 2 Control Systems

2.1 IO control systems

2.1.1 General IO system description

What follows in general is the basis for the next subsections.

The mathematical modeling of many physical systems results (after a possible linearization) in the time-invariant linear vector Input-Output (IO) differential equation (2.1) to be called the IO system,

${\displaystyle \sum _{k=0}^{k=\nu }{A}_k}{Y}^{(k)}(t)={\displaystyle \sum _{k=0}^{k=\mu }{B}_k}{I}^{(k)}(t),det{A}_{\nu }\ne 0,\forall t\in \mathfrak{T},\nu \ge 1,0\le \mu \le \nu ,$

$Y^{(k)}(t)=\frac{d^{k}Y(t)}{d{t}^k},A_k\in {\Re }^{N\times N},B_k\in {\Re }^{N\times M},k=0,1,\mathrm{..},\nu ,$

$\mu <\nu \Rightarrow B_i=O,i=\mu +1,\mu +2,\dots ,\nu .$ (2.1)

This mathematical description can be the general IO mathematical description of an object/plant, of a controller and of a whole control system.

Let ℭ^k be the k-dimensional complex vector space, ℜ^k be the k-dimensional real vector space, O_M×N be the ...