Mathematica assistance in proving theorems in nonlinear control

Palle Kotta, Ülle Kotta

Institute of Cybernetics at TUT, Tallinn, Estonia

Abstract

The application of computer algebra system Mathematica is well-documented in control-related literature. Mostly, calculations are carried out to provide reliable solutions of problems whose theory is well understood. Our goal was to demonstrate that Mathematica can be used to formulate the conjectures and prove the theorems. 

The cornerstone of the modern nonlinear control is the system description by state equations. However, often the model obtained from experimental data results in the i/o equation. Therefore, modellers prefer to have the realizable (i.e. having state equations) model structures as templates. An important task is to transform the i/o equation into the state equations. Unlike the linear case, the problem is not always solvable. Checking the solvability  requires huge amount of symbolic calculations. Since simple polynomial systems, in particular bilinear and quadratic equations,  are popular in modeling of nonlinear systems,  we found the structures for low order bilinear and quadratic systems guaranteed to have state equations. We proved that if certain combinations of coefficients in the bilinear or quadratic i/o equation are zero, then the system is realizable. Moreover, we also found the explicit state equations for low order realizable input-output equations. 

We also suggested several subclasses of realizable input-output bilinear and quadratic systems of arbitrary order together with their state equations. Though the state equations of an arbitrary order input-output equations were suggested as a conjecture based on Mathematica research, their correctness can be proved (theoretically) directly by eliminating the state variables from the state equations.