| Summary: | The optimal control in linear systems is a widely known problem that leads to the
solution of one or two equations of Ricatti. However, in non-linear systems is required
to obtain the solution of the Hamilton-Jacobi-Bellman equation (HJB) or variations,
which consist of quadratic first order and partial differential equations, that are really
difficult to solve.
On the other hand, many non-linear dynamical systems can be represented as polynomial
functions, where thanks to abstract algebra there are several techniques that
facilitate the analysis and work with polynomials. This is where the sum-of-squares
approach can be used as a sufficient condition to determine the positivity of a polynomial,
a tool that is used in the search for suboptimal solutions of the HJB equation
for the synthesis of a controller.
The main objective of this thesis is the analysis, improvement and/or extension of an
optimal control algorithm for polynomial systems by using the sum of squares approach
(SOS).
To do this, I will explain the theory and advantages of the sum-of-squares approach
and then present a controller, which will serve as the basis for our proposal. Next,
improvements will be added in its performance criteria and the scope of the controller
will be extended, so that rational systems can be controlled. Finally an alternative
will be presented for its implementation, when it is not possible to measure or estimate
the state-space variables of the system. Additionally, some examples that validated
the results are also presented. === Tesis
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