Learning control theory for dynamical systems

Three types of learning control laws are proposed for mechanical or mechatronics systems with linear and nonlinear dynamics, which may be operated repeatedly at low cost. Given a desired output Yd over a finite time duration [0,T] and an appropriate input u0, these laws are formed by the following simple iterative processes: 1) uk+1 = uk + ??(yd - yk), 2) uk+1 = uk + ??d/dt(yd - yk), and 3) uk+1 = uk + (?? + ??d/dt)(yd - Yk), where uk(uk+1) denotes the kth(k+1th) input, Yk the measured output at the kth operation corresponding to uk, and ?? and ?? positive definite constant gain matrices. It is shown that the first law 1) with an appropriate gain matrix ?? is convergent in the sense that Yk(t) approaches Yd(t) as k ?? ?? in the meaning of L2[0,T] norm if the objective system is linear and strictly positive. The same conclusion is also proved when the system is subject to a linear time-invariant or time-varying mechanical system. In addition, a rough sketch of the convergency proof of the second and third learning control laws is presented for a class of linear and nonlinear dynamical systems. Finally some discussions on potential applicabilities of these learning methods for robot controls are given.