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Control theory from an algebro-geometric viewpoint

Control theory is a branch of science bridging mathematics with a branch of engineering, known as control engineering. Almost every single branch of mathematics finds applications in Control Theory, and problems in Systems and Control introduce new challenges for Mathematics. This project has used techniques from exterior algebra, algebraic geometry and optimization to find critical frequencies such as poles and zeros under compensation (system transformation) for linear systems. Such problems are reduced to finding real solutions of non-linear equations expressed as intersection of varieties. The project has introduced a methodology to define exact and approximate solutions.
Control theory from an algebro-geometric viewpoint
The input of a dynamical system is defined by a controller with stimulus and commands in order to obtain the user's desired response. The main objective of control theory is to take the desired outcome and determine the action necessary to correct for differences in the actual outcome by deploying feedback action of an appropriate type.

The EU-funded project A-DAP (Approximate solutions of the determinantal assignment problem and distance problems) was centred on such an abstract problem formulation. The DAP problem can be decomposed into a linear and a multilinear subproblems, where the linear problem introduces a linear variety and the multilinear problem is defined by quadratics characterising the Grassmann variety of a projective space. The solvability of the determinantal assignment problem (DAP) is reduced to finding real solutions of the linear and non-linear equations, or otherwise real intersections of the two varieties. The solutions define the compensators that assign critical frequencies of the system such as the poles and zeros.

The new methodology introduced in A-DAP uses a matrix representation of the Grassmann variety, distance problems between varieties and notions of approximate decomposition of tensors to produce exact solutions when the problem has a solution and approximate solutions when exact intersection of varieties do not exist. This problem was to solve the linear sub-problem. The approach makes a qualitative change to the classical algebraic geometry approach looking for real intersections by examining problems of distance between the relevant varieties.

In mathematics, a Grassmann variety in a projective space is a variety that characterises all linear subspaces of a vector space of a given dimension. The scientists introduced a series of new tools enabling the distance calculation of a given vector that describes the variety implied by the linear sub-problem from the Grassmann variety. Specifically, constrained optimisation and decomposition of tensor techniques were used for addressing the problem of distance between linear and Grassmann variety. If the distance is zero, then the solution to the corresponding intersection problem is found. Otherwise, the approach could lead to an approximate solution of DAP.

The approximate DAP could be completely solved in the 2D case and a closed form solution has been derived. In addition, a dedicated numerical algorithm for DAP approximations in higher dimensions has been derived using the new representation of the Grassmann variety and its dual.. In this case, the solution is obtained by decomposing the parametrised tensor derived from the system of linear equations.

A-DAP concluded with important extensions of this methodology in the redesign of electrical circuits consisting of resistor, inductor and a capacitor. A new formulation as a decision-making problem was also proposed, opening the way for practical applications in management and finance.

Related information


Control theory, non-linear equations, A-DAP, determinantal assignment problem, constrained optimisation, computational algebraic geometry
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