Using Bond Graphs to Synthesize Tree-structured Transfer Functions: Improved Frequency-Domain Analysis of Uncertain Systems

Various robust control applications require the value set of an uncertain transfer function, G (s, q) , which is the complex-plane region {G (jω, q) : q ∈ Q} , where q is a vector of uncertain parameters. Although a value set can be synthesized by gridding the uncertain parameters, this is only an option when a small number of the parameters are uncertain. When the transfer function can be decomposed into a tree structure with disjoint sets of uncertain parameters, the value set synthesis process is simplified. Bond graphs provide a means to synthesize tree-structured transfer functions, but the disjoint parameter property has not been guaranteed except for collocated input–output variables. In this paper this property is extended to include a much broader class of systems, and the paper provides a means to determine by inspection if a given input–output variable pair will result in a transfer function with disjoint parameters. Furthermore, the tree structures generated here provide certain vertex-type results that simplify value set synthesis. The ability to know a priori that a given input–output variable pair will lead to a tree-structured transfer function with disjoint parameters and the associated algorithms for synthesizing value sets expedite a normally problematic aspect of robust controller design.