Abstract
Up to now several implicit self-tuning control (STC) schemes for a SISO ARMAX system have been proposed. The analysis for the STC scheme based on the least squares method appears in the works of Ljung, and it depends upon an ordinary differential equation. In the analysis, however, a problem which concerns with the boundedness of the system variables remains unsolved. The analyses for the other STC schemes overcome the problem by using the martingale approach, but they have been individually established. All the schemes are in the same form in respect to how to generate the control input, not only to estimate a set of control law parameters. From this point of view, taking into account of error equations of the schemes, it is found that they can be equivalently represented by a stochastic feedback system with a linear time-invariant feedforward block and a linear time-varying feedback block.
This paper gives a unified approach to establish a.s. global convergence for a class of the implicit STC schemes for the minimum variance control. To work this, the equivalent stochastic feedback representation (ESFR) corresponding to the implicit STC is shown and its global convergence is established by using the martingale approach and a positive real condition. From this result of the ESFR, it is found that an STC which belongs to a class presented here is guaranteed in a.s. convergence under suitable conditions. Furthermore, a new STC scheme based on the stochastic approximation method which belongs to the class is proposed, and it is different from that studied by Goodwin. Comparing two schemes, it is pointed out that the proposed STC is more effective method for a system containg long time delay than that studied by Goodwin.
