A fuzzy approach to reduced order variance constrained control design

Constraints on the variances of the system inputs and selected system outputs have been shown to have practical applications. One way of designing control laws that achieve these objectives is by designing an LQG (or optimal ℋ₂) controller with an appropriate choice of the input and output weighting matrices. However, one of the well known limitations of practical implementation of an LQG controller is that it is of the same order as the plant model. The two primary methods of designing reduced order controllers are the truncation (or indirect) methods and parameter optimization (or direct methods). These two methods are often used simultaneously since the truncated controllers are often used to initialize the direct methods. This paper considers the design of ℋ₂ optimal reduced order controllers to meet a set of variance constraints. As with full order control, this problem involves the proper choice of the weighting matrices in the cost function. Although the behavior of the variances with weight variations in reduced order control design are experimentally seen to be more erratic than in optimal full order design, a fuzzy algorithm previously developed for the full order variance constrained problem is shown to be applicable to the reduced order variance constrained problem. Three reduced order schemes are developed and compared. The first two schemes involve direct reduced order design while the third scheme involves reduced order design using modified balanced truncation. The first two schemes differ only in how they are initialized with the first approach using the weights from the full order variance constrained problem and the second approach using unity weights for initialization. The three schemes are compared using numerical experiments. The results clearly demonstrate the feasibility of reduced order variance constrained control design