QFT template generation for time-delay plants based on zero-inclusion test

Abstract

Plant template generation is the key step in applying quantitative feedback theory (QFT) to design robust control for uncertain systems. In this paper we propose a technique for generating plant templates for a class of linear systems with an uncertain time delay and affine parameter perturbations in coefficients. The main contribution lies in presenting a necessary and sufficient condition for the zero inclusion of the value set f( T , Q )={f(τ, q ) : τ∈ T ≔[τ −,τ +], q ∈ Q ≔∏ k=0 m−1[q k −,q k +]} , where f(τ, q )=g( q )+h( q ) e − jτω ∗ , g( q ) and h( q ) are both complex-valued affine functions of the m-dimensional real vector q , and ω ∗ is a fixed frequency. Based on this condition, an efficient algorithm which involves, in the worst case, evaluation of m algebraic inequalities and solution of m2 m−1 one-variable quadratic equations, is developed for testing the zero inclusion of the value set f( T , Q ) . This zero-inclusion test algorithm allows one to utilize a pivoting procedure to generate the outer boundary of a plant template with a prescribed accuracy or resolution. The proposed template generation technique has a linear computational complexity in resolution and is, therefore, more efficient than the parameter gridding and interval methods. A numerical example illustrating the proposed technique and its computational superiority over the interval method is included.