The balanced realization truncation is adopted to deduce the internal balanced model,and then to obtain the reduced model based on internal balance transform for quadratic characteristic indexes function is obtained.

By minimizing the quadratic performance index, a simple adaptive control algorithm is obtained.

In this paper, based on the optimal control thought of the quadratic performance index, system performance index incremental-contribution method is put forward to make it apply the elastic-plastic state for viscous dampers.

The quadratic index is introduced in order to realize the optimal control of the single-neuron adaptive PSD controller.

From the start of analysing the stability of the airconditioning control system according to linear system the calculating formulas of the optimal PID parameters is inferred by the quadratic form performance index to solve the parameters optimal question of the controller and the weightel optimal method.

At first,a linear quadratic performance function is defined in iteration domain,then an optimal iterative learning algorithm is presented for linear discrete-time systems,and a guaranteed cost iterative learning algorithm and its optimization are developed for linear discrete-time systems with uncertainties.