Optimal Bang-Bang Control With Quadratic Performance Index

The following optimal regulator problem is considered: Find the scalar control function u = u(t) which minimizes the performance index J[u]= 12∫0T〈x(t),Qx(t)〉dt, subject to the conditions x˙=Ax+u(t)f,|u(t)|≦1x(0)=x0(x0isunrestricted)x(T)=0(Tisfree)Q, A are constant n × n-matrices; f is a constant n-vector. It is shown that optimal control includes both a bang-bang mode and a linear mode, the latter arising from the “singular” solutions of the Pontriagin canonical equations. Conditions are given under which nth-order systems are equivalent, for control purposes, to systems of first or second order. One example of a second-order system is worked in detail and some results of an analog computer study are presented.