Stabilization by linear feedback. Pole assignment.

We saw at the end of the previous section that, if a linear control system with one control variable 16.1 $$x = Ax + bu$$ is controllable, then by the using linear feedback u = cTx the system becomes 16.2 $$x = \left( {A + b{c^T}} \right)x;$$ in this special case linear feedback can be used to stabilize the system; in fact, by the choice of cT we have complete control of the spectrum of (A+bcT) (see Proposition 10.4). Stabilization by linear feedback is the oldest method for the analysis and design of feedback controls and dates back at least to the early part of the 19th century (see Fuller [1]). It was almost the only method used up to the 1950’s and remains of importance up to the present time. The result we present here is of more recent origin. It had been looked at by Langenhop [1] in 1964 over the complex field and was discovered independently by a number of engineers (see Wonham [1] and Padulo and Arbib [1, pp. 596-601]).