]> 8.2 Statement of the lq problem

8.2 Statement of the lq problem

Consider a controlled distortionless RLCG transmission line, i.e. RL = GC, loaded by a resistance R1 and depicted in Figure 8.1.

PICT

Figure 8.1: The control system under study

The system is governed by the equations

LI(η,t) t = V (η,t) η RI(η,t),0 η 1,t 0 CV (η,t) t = I(η,t) η GV (η,t),0 η 1,t 0 I(1,t)R1 =V (1,t), t 0 V (0,t) =u(t), t 0 y(t) =V (1,t), t 0 (8.1)

The control is an input voltage and its goal is to minimize the sum of energy of the input and output signals expressed by the performance index

J =0[y2(t) + u2(t)]dt (8.2)

The d’Alembert solutions of the first two equations take the form (3.8) with α = RL = GC. Substituting (3.8) into boundary conditions, we get the system of functional equations

Ψ(1 + νt) =κΦ(1 νt) 1 2eαt[Φ(νt) + Ψ(νt)] =u(t) y(t) =1 2eαt[Φ(1 νt) + Ψ(1 + νt)] (8.3)

where κ = (R1 Z)(R1 + Z) is the reflection coefficient. Introducing the new variables

z1(t) =1 2eαteαrΦ(1 νt) z2(t) =1 2eαtΦ(νt) (8.4)

one obtains

z(t) =Az(t r) + u(t)b0 y(t) =c0z(t r) (8.5)

here,

z(t) = z1(t) z2(t) ,A = 01 b0 ,b0 = 0 1 ,c0 = 0 a ,

where a := 1 + κ ρ 0, b := κ ρ2, b < 1, ρ := eαr.