The LQ controller problem: an example

We provide an example of LQ controller synthesis for a distributed plant, namely an RLCG transmission line without distortion. The characteristic feature of our example is that both control and observation are of boundary form, and thus this example is beyond the scope of the existing theory for the LQ problem. The formal spectral-factorization method is used to get an explicit analytic form of the optimal controller. It is shown that this control law has an extremely simple physical realization. The formally derived optimal controller is then confirmed by the Wiener-Hopf technique. Our example shows the need for some generalization of the existing LQ-problem theory as well as the spectral-factorization method. 1. Statement of the LQ problem Consider a controlled RLCG transmission line without distortion, i.e. R/L = G/C, loaded by a resistance R ^ and depicted in Fig. 1. The system is governed by the equations di(x, t) _ dv{x, t) v(0, t) = w(t) y{t)=v{\,t)\ (1.1b) 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 inde