State-Space Transformations of Uncertain Systems with Purely Real and Conjugate-Complex Eigenvalues into a Cooperative Form

Cooperativity of uncertain dynamic systems can be exploited to simplify several tasks such as the computation of guaranteed state enclosures, the design of interval observers, forecasting worst-case bounds for selected system outputs in predictive control, and the identification of unknown parameters. Although many system models in biological, chemical, and medical applications are naturally cooperative, there is also a great number of systems (typically from the fields of electric, magnetic, and mechanical applications) which do not show this property if the state equations are derived using first-principle techniques. Hence, it is often desired to transform such system models into an equivalent cooperative form. Unfortunately, these transformations are often not straightforward, especially, if linear systems and nonlinear ones with state-dependent system matrices are subject to bounded parameter uncertainty. This paper presents two approaches for the transformation of state equations into a cooperative form for which the original system models do not fulfill sufficient criteria for cooperativity in their basic formulation. These are a time-invariant transformation for systems with purely real eigenvalues and a time-varying transformation in the case of conjugate-complex eigenvalues. Both procedures are tested on real-life application scenarios with interval parameters.