Discrete-time neural net controller for a class of nonlinear dynamical systems

A family of two-layer discrete-time neural net (NN) controllers is presented for the control of a class of mnth-order multi-input-multi-output (MIMO) dynamical system, No initial learning phase is needed so that control action is immediate; in other words, the neural network (NN) controller exhibits a learning-while-functioning-feature instead of a learning-then-control feature. A two-layer NN is used which is linear in the tunable weights. However, this is a far milder assumption than the adaptive control requirement of linearity in the parameters, since the universal approximation property of the NN means that any smooth nonlinear function can be reconstructed. The structure of the neural net controller is derived using a filtered error approach. It is indicated that delta-rule-based tuning, when employed for closed-loop control, can yield unbounded NN weights if: 1) the net cannot exactly reconstruct a certain required function, or 2) there are bounded unknown disturbances acting on the dynamical system. Certainty equivalence is not used, overcoming a major problem in discrete-time adaptive control. In this paper, new on-line tuning algorithms for discrete-time systems are derived which are similar to E-modification for the case of continuous-time systems that include a modification to the learning rate parameter and a correction term to the standard delta rule. These improved weight-tuning algorithms guarantee tracking as well as bounded NN weights in nonideal situations so that persistency of excitation (PE) is not needed.