This paper addresses the problem of exact generic symbolic formulation and solution of Parameters Varying Systems (PVS) problems in control, systems, and optimization. Comparisons between the generic symbolic mathematical approach versus various techniques such as the probabilistic (or random), fuzzy, chaotic and other approaches are thoroughly investigated. The comparisons revealed the generality and flexibility of the generic symbolic concept compared to the other approaches. New classifications of PVS symbolic derivations mechanisms and solution methods, as well as the operations symbolic coding are proposed. The PVS are addressed from the physical, mathematical, and representation perspectives. For expediting the productivity of the derivation of the symbolic (or algebraic) solutions, both combined manual and interactive symbolic derivations manipulation are considered, together with fully programming (one-shot program) symbolic computation. The new concept is then demonstrated by solving four different applications of symbolic problems. These problems are the stabilization of inverted pendulum in control theory, the HIV/AIDS modeling and analysis in life sciences/medicine systems and solving unconstrained and constrained nonlinear optimization problems. For control applications, the notion of symbolic PVS control strategy realization is carried out through the incorporation of embedded configurable function units. It is shown that the new approach provides very powerful tools specially when handling formulations (even with high dimension) that could be represented and solved through basically matrix-based manipulations. Finally, it is highlighted that using the proposed symbolic-based framework, the parameters varying systems derivations could be equivalently handled and implemented in the form of parameters invariant systems where most of their very wide scope of methodologies could be applied. |
