Philipp Rostalski

Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals

Abstract For an ideal I⊆ℝ[x] given by a set of generators, a new semidefinite characterization of its real radical I(Vℝ(I)) is presented, provided it is zero-dimensional (even if I is not). Moreover, we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety Vℝ(I) as well as a set of generators of the real radical ideal. The latter is obtained in the form of a border or Gröbner basis. The algorithm is based on moment relaxations and, in contrast to other existing methods, it exploits the real algebraic nature of the problem right from the beginning and avoids the computation of complex components.

Parametric optimization and optimal control using algebraic geometry methods

We present two algebraic methods to solve the parametric optimization problem that arises in non-linear model predictive control. We consider constrained discrete-time polynomial systems and the corresponding constrained finite-time optimal control problem. The first method is based on cylindrical algebraic decomposition. The second uses Gröbner bases and the eigenvalue method for solving systems of polynomial equations. Both methods aim at moving most of the computational burden associated with the optimization problem off-line, by pre-computing certain algebraic objects. Then, an on-line algorithm uses this pre-computed information to obtain the solution of the original optimization problem in real time fast and efficiently. Introductory material is provided as appropriate and the algorithms are accompanied by illustrative examples.

A hybrid approach to modelling, control and state estimation of mechanical systems with backlash

Control of mechanical systems with backlash is a topic well studied by many control practitioners. This interest has been motivated by the fact that backlash in mechanical systems can cause severe performance degradation and lead to instability of the control system. Furthermore, high impact-forces in backlash-systems can lead to a lower durability of the components and to strokes and peaks in the output. In this paper a mechanical benchmark system is presented to provide facilities for testing the identification and control of systems with backlash. For controller design a hybrid model of the system was derived and used in a model predictive control (MPC) scheme. Observer-based state-estimation was used to recover unmeasured states, particularly the backlash angle. Explicit solutions of a tracking controller were computed to control the mechanical benchmark system in real-time. Simulation as well as experimental results are presented to show the applicability of this hybrid control approach.

Application of Model Predictive Control to a Cascade of River Power Plants

River power plants are important contributors to the over 19% of world electricity produced by hydro-electric plants. Built in the natural course of a river, they produce energy by manipulating the water discharge through their facilities. They therefore introduce fluctuations in the rivers natural water level and flow, which might conflict with various constraints imposed for environmental and operational purposes. Motivated by these issues, we present in this paper the application of Model Predictive Control for regulating the turbine discharge of river power plants, taking into account environmental, navigational and economical constraints and limitations. Large disturbances caused by the operation of locks are particularly investigated, as well as the issue of reducing abrasion by keeping the frequency of turbine discharge adjustments modest.

Moment matrices, border bases and real radical computation

In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming its complex (resp. real) variety is finite. The aim is to combine approaches for solving a system of polynomial equations with dual methods which involve moment matrices and semi-definite programming. While the border basis algorithms of Mourrain and Trebuchet (2005) are efficient and numerically stable for computing complex roots, algorithms based on moment matrices (Lasserre et al., 2008) allow the incorporation of additional polynomials, e.g., to restrict the computation to real roots or to eliminate multiple solutions. The proposed algorithm can be used to compute a border basis of the input ideal and, as opposed to other approaches, it can also compute the quotient structure of the (real) radical ideal directly, i.e., without prior algebraic techniques such as Grobner bases. It thus combines the strength of existing algorithms and provides a unified treatment for the computation of border bases for the ideal, the radical ideal and the real radical ideal.

Numerical Algebraic Geometry for Optimal Control Applications

A new technique for solving polynomial nonlinear constrained optimal control problems is presented. The problem is reformulated into a parametric optimization problem, which in turn is solved in a two-step procedure. First, in a precomputation step, the equation part of the corresponding first order optimality conditions is solved for a generic value of the parameter. Relying on the underlying algebraic geometry, this first solution makes it possible to solve efficiently and in real time the corresponding optimal control problem at the measured parameter value for each subsequent time step. This approach has a probability one guarantee of finding the global optimal solution at each step. Controller synthesis for two applications from the area of power electronics featuring a dc-ac converter and a dc-dc converter are discussed to motivate the proposed approach.

The approach of moments for polynomial equations

In this chapter we present the moment based approach for computing all real solutions of a given system of polynomial equations. This approach builds upon a lifting method for constructing semidefinite relaxations of several nonconvex optimization problems, using sums of squares of polynomials and the dual theory of moments. A crucial ingredient is a semidefinite characterization of the real radical ideal, consisting of all polynomials with the same real zero set as the system of polynomials to be solved. Combining this characterization with ideas from commutative algebra, (numerical) linear algebra and semidefinite optimization yields a new class of real algebraic algorithms. This chapter sheds some light on the underlying theory and the link to polynomial optimization.

An algebraic geometry approach to nonlinear parametric optimization in control

We present a method for nonlinear parametric optimization based on algebraic geometry. The problem to be studied, which arises in optimal control, is to minimize a polynomial function with parameters subject to semialgebraic constraints. The method uses Grobner bases computation in conjunction with the eigenvalue method for solving systems of polynomial equations. In this way, certain companion matrices are constructed off-line. Then, given the parameter value, an on-line algorithm is used to efficiently obtain the optimizer of the original optimization problem in real time

Deciding Polyhedrality of Spectrahedra

Spectrahedra are linear sections of the cone of positive semidefinite matrices which, as convex bodies, generalize the class of polyhedra. In this paper we investigate the problem of recognizing when a spectrahedron is polyhedral. We generalize and strengthen results of [M. V. Ramana, Polyhedra, spectrahedra, and semidefinite programming, in Topics in Semidefinite and Interior-Point Methods, Fields Inst. Commun. 18, AMS, Providence, RI, 1998, pp. 27--38] regarding the structure of spectrahedra, and we devise a normal form of representations of spectrahedra. This normal form is effectively computable and leads to an algorithm for deciding polyhedrality.

A numerical algebraic geometry approach to nonlinear constrained optimal control

A new method for nonlinear constrained optimal control based on numerical algebraic geometry is presented. First, the optimal control problem is formulated as a parametric optimization program. Then, certain structural information related to the optimization problem is computed off-line. Afterwards, given this information, numerical algebraic geometry techniques are used to efficiently obtain the optimal control input (i.e. optimal solution) of the original control problem in real time. By using homotopy continuation over the field of complex numbers, this approach has a probability-one guarantee of finding the global optimal solution to the problem at hand.