CONTROLLABILITY IS NOT NECESSARY FOR ADAPTIVE POLE-PLACEMENT CONTROL

The key issue for adaptive pole-placement control of linear time-invar iant systems is the possible singularity of the Sylvester matrix corre sponding to the coefficient estimate. However, to overcome the difficu lty, the estimate is modified by several methods which are either nonr ecursive and with high computational load or recursive but with random search involved. All of the previous works are done under the assumpt ion that the system is controllable. This paper gives the necessary an d sufficient condition, which is weaker than controllability, for the system to be adaptively stabilizable. First, a nonrecursive algorithm is proposed to modify the estimates, and the algorithm is proved to te rminate in finitely many steps. Then, with the help of stochastic appr oximation, a recursive algorithm is proposed for obtaining the modific ation parameters; it is proved that these modification parameters turn out to be a constant vector in a finite number of steps. This leads t o the convergence of the modified coefficient estimates. For both algo rithms the Sylvester matrices corresponding to the modified coefficien t estimates are asymptotically uniformly nonsingular; thus, the adapti ve pole-placement control problem can be solved, i.e., the system can be adaptively stabilized.