Page - (000021) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

2.1 TransferFunctionsandStateSpace 11 The simplemultiplication of the signal by a processmeans that we can easily cascade multiple input–output processes. For example, Fig.2.1b shows a system with extended input processing.Thecascadebeginswith an initial reference input, r, which is transformed into the command input, u, by a preprocessing controller, C, and thenfinally into theoutput, y, by the intrinsicprocess,P. The input–output calculation for the entire cascade follows easily bynoting thatC(s)=U(s)/R(s), yielding Y(s)= R(s)C(s)P(s)= R(s)U(s) R(s) Y(s) U(s) . (2.3) These functionsof s arecalled transfer functions. Each transfer function in a cascade can express any general systemof ordinary linear differential equations for vectors of state variables, x, and inputs, u, with dynamicsgivenby x(n)+a1x(n−1)+···+an−1x(1)+anx =b0u(m)+b1u(m−1)+···+bm−1u(1)+bmu, (2.4) inwhich parenthetical superscripts denote the order of differentiation.By analogy withEq.2.2, theassociatedgeneral expression for transfer functions is P(s)= b0s m +b1sm−1+···+bm−1s+bm sn +a1sn−1+···+an−1s+an . (2.5) The actual biological or physical process does not have to include higher-order derivatives. Instead, thedynamicsofEq.2.4and its associated transfer functioncan alwaysbeexpressedbyasystemoffirst-orderprocessesof the form x˙i = ∑ j aijxj + ∑ j bijuj, (2.6) which allows formultiple inputs, uj. This system describes the first-order rate of change in thestatevariables, x˙i, in termsof thecurrent statesand inputs.This state- spacedescription for thedynamics isusuallywritten invectornotationas x˙=Ax+Bu y=Cx+Du, whichpotentiallyhasmultiple inputs andoutputs,uandy. Forexample, the single input–outputdynamics inEq.2.1 translate into thestate- spacemodel