Abstract
This paper considers a dual control problem for stochastic linear MIMO (Multiple Input Multiple Output) systems with parameter uncertainty. A novel dual adaptive control law for MIMO systems is proposed. The design is based on the innovation of the dual control cost function, which was originally developed for conventional adaptive control of linear systems. However, the design process is modified and developed to cater to the stochastic MIMO case. This is a more challenging problem because the superposition of parameter uncertainty and the MIMO property makes the problem more intractable. As in all dual adaptive strategies, it leads to a control law that balances out the need for caution, due to parameter uncertainty, with the conflicting requirement of probing that acts to quickly reduce parameter uncertainty, which is the nature of dual control. The proposed control law has two parts. One reflects the goal of regulating the output, and the second reflects the ability to handle uncertain parameters. A learning factor is introduced to balance these two parts to obtain a control law that can be applied to the original system. In the simulation examples, the uncertain parameters can be estimated quickly and accurately for the unknown but constant case and the abrupt parameter case. Furthermore, it is shown that the novel dual control law is superior to other control strategies by comparing the performance of the cost function in a statistical sense.
Highlights
In all control problems, there are certain degrees of uncertainty with respect to the process to be controlled
In this paper, a novel multiple-input multiple-output (MIMO) dual control law for a MIMO stochastic linear system with parameter uncertainty is proposed as an extension of a single-input single-output (SISO) stochastic linear system
The key to this paper is that the learning factor can automatically change at any time and is closely related to the estimation covariance matrix, which balances the control target and parameter estimation
Summary
There are certain degrees of uncertainty with respect to the process to be controlled. F. Qian et al.: Dual Control for Stochastic Linear MIMO Systems With Parameter Uncertainty probing aims to continuously excite the system using control with a large amplitude so that the system generates richer information to improve the estimation quality effectively. IEEE Control Systems Magazine listed this challenging topic with significant theoretical implications and practical value as one of the top 25 questions that had a significant impact on Control theory in the last century [2] Several such control laws have been proposed for standard adaptive control cases involving an externally predefined reference input [6]–[9], none of them addresses the dual adaptive control problem with the superposition of the MIMO case and two uncertainties. The expression ξ ∼ N (m, S) means that the variable ξ has a normal distribution with mean m and variance S, and the quadratic form ξ T Qξ will be denoted by |ξ |2Q
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