]> 8.6 Physical realization of the optimal controller

8.6 Physical realization of the optimal controller

By (8.18), the optimal control law is
u = g#x = (1 p)x 2(0) + (b q)x1(r) (8.19)

Notice that (8.6) has been introduced to avoid the use of a non–linear nonautonomous operator

𝒜nx = x,D(𝒜 n) = {x W1,2(r, 0) W1,2(r, 0) : x(0) Ax(r) = b 0u(t)} .

Taking (8.19) into account, we get an abstract operator of the optimal closed–loop system in the form

𝒜cx = x , D(𝒜c) = x = x1 x2 W1,2(r, 0) W1,2(r, 0) : Mx(0) + Nx(r) = 0 (8.20)

where

M = 10 0p ,N = 0 1 q 0 .

Since our result (8.19) is partially based on some formal considerations, any possibility of checking its validity is of a great interest. Such a possibility arises by examining the eigenproblem of 𝒜c:

x(θ) =λx(θ), r θ 0 Mx(0) + Nx(r) =0 (8.21)

Clearly (8.21) has a solution iff det(M + eλrN) = p + qe2λr = 0. Hence the closed–loop characteristic function is confirmed to be the numerator of the spectral factor (8.13).

Now let us obtain the physical realization of the optimal feedback for the original system (8.1). To do this, using (8.12) we find, that the time–domain version of the optimal feedback in terms of system (8.5) is

u(t) = (1 p)z2(t) + (b q)z1(t r) = (1 p) z2(t) qz1(t r) (8.22)

Next, recalling (8.4) and again (8.19), we have

u(t) = 1 p p 1 2peαtΦ(νt) 1 2eαtΨ(νt) (8.23)

Now observe that, by (3.8),

ZI(0,t) =eαtΦ(νt) Ψ(νt) 2 V (0,t) =eαtΦ(νt) + Ψ(νt) 2 (8.24)

The quantities 1 2eαtΦ(νt) and 1 2eαtΨ(νt) can be uniquely evaluated from (8.24). Introducing them into (8.23), we get

u(t) = 1 p 2p (Zp + Z)I(0,t) + (p 1)V (0,t) (8.25)

But, from the original system (8.1), we know that u(t) = V (0,t), hence (8.25) yields the desired expression for the optimal feedback in terms of system (8.1), viz.

V (0,t) = Z1 p2 1 + p2I(0,t) (8.26)

The physical interpretation of (8.26) is astonishingly simple: the optimal feedback is realized by the impedance R0 = Z (1 p2) (1 + p2) plugged to the input of the distortionless RLCG–transmission line – see Figure 8.2.


PICT

Figure 8.2: The optimal control system