Controllability is not necessary for adaptive pole placement control

The key issue for adaptive pole placement control of linear time-invariant systems is the possible singularity of the Sylvester matrix corresponding to the coefficient estimate. The methods developed for modifying the estimates are either nonrecursive and with a high computation load or recursive but with random search involved. All the previous works are done under the assumption that the system is controllable. This paper gives a necessary and sufficient condition, which is weaker than controllability, for the system to be adaptively stabilizable. While proving the sufficient part, a nonrecursive algorithm is proposed to modify the estimates. It is proved that the algorithm terminates in finitely many steps. Further, under the same condition with the help of stochastic approximation a recursive algorithm is proposed for obtaining the modification parameters; it is proved that these modification parameters are convergent, and this leads to the convergence of the modified coefficient estimates. For both algorithms the Sylvester matrices corresponding to the modified coefficient estimates are asymptotically uniformly nonsingular; thus with these Sylvester matrices, the adaptive pole placement control problem can be solved, i.e. the system can be adaptively stabilized