Combined Methods

A given polynomial system whose degree and/or number of equations is relatively modest can nowadays typically be solved, given the right choice of method. Whether or not the polynomials are dense or sparse, or whether the problem is ill-conditioned, for example, are factors which influence the best choice. The task remains to automate the selection of the optimal method, and to improve the speed and extend the scope of such solvers.

The theory of solving polynomial systems is generally segregated into the three main categories (Subsections 5.3.4, 5.3.5, and 5.3.6), with little crossover. However, given the strengths and weaknesses of each type of solver — Newton-based methods don’t always work, whereas symbolic methods tend to have an inherently high structural complexity, for example — it seems sensible to combine methods, if in some way possible. There have been a couple of notable attempts: Jäger and Ratz [JR95] combined the method of Gröbner bases with interval computations. A NAG-led collaborative project,Framework for Integrated Symbolic-Numeric Computation (FRISCO)[FRI99], aimed to integrate a number of techniques (both symbolic and numeric) in the development of a fast black-box solver suitable for multi-purpose industrial applications.

There are more solvers for general systems of equations, cf. Subsections 5.3.2 and 5.3.3, than for polynomial systems alone. The drawback of general-purpose solvers is precisely their generality: a single method that could succesfully tackle any nonlinear system would seem to be all but impossible to achieve. The Newton and interval Newton methods and their variants remain the most widely used general nonlinear solvers, although it appears there is no consensus on an optimal choice of method. Given the wide diversity of problem types (both polynomial and non-polynomial), it seems doubtful that there isa universally good or optimal approach.

6 Problems Involving Polynomial Inequalities

We conclude the first part of this thesis with a brief overview of systems of polynomial inequalities, and problems which may include polynomial inequalities, such as global op-timisation problems. The Bernstein expansion (introduced in Chapter 3) has previously been applied to robust stability problems, cf. [Gar00], and to the solution of systems of polynomial inequalities [GG99b]. Constraint satisfaction problems typically consist of both equations and inequalities. In [Gra07], for example, interval analysis (introduced in Chapter 2) is applied within a branch-and-bound scheme for their solution.

6.1 Systems of Polynomial Inequalities

Dynamical systems arising in control theory are commonplace, occurring in application areas such as manufacturing plant control, digital control, guidance systems for rockets and missiles, and automobile cruise control. Numerous problems in control system design and analysis, often problems of robust control and robust stability, may be recast and reduced to systems of inequalities involving multivariate polynomials in real variables. This corresponds broadly to the following problem:

Let p₁, . . . , p_m be polynomials in x = (x₁, . . . , x_n) and let a box X in Rⁿ be given. We wish to determine the solution setΣof the system of polynomial inequalities, given by

Σ := {x∈X | p_i(x)>0, i= 1, . . . , m}. (6.1) In general, it is not possible to describe the solution setΣexactly; instead a good approxi-mation to it is sought. A subdivision scheme (i.e. branch-and-bound scheme, cf. Subsection 2.3.1), in which the boxXis repeatedly partitioned into sub-boxes and bounds for the ranges of the component polynomials thereupon are sought, may typically be employed. A number of techniques from the theory of interval analysis (cf. Chapter 2), or else the Bernstein ex-pansion (cf. Chapter 3), may be employed. In the latter case, the Bernstein coefficients of each pi,i= 1, . . . , m, can be computed for each sub-box, and the range enclosing property (3.17) used to determine whether each polynomial is strictly negative, strictly positive, or neither over the box. Sub-boxes should be subdivided until either strict positivity for all polynomials or non-positivity for at least one polynomial is satisfied, or until the sub-box volume or maximum sub-box edge length falls below a certain minimumε >0, which spec-ifies a satisfactory level of ‘resolution’ for the final box-union approximation of the solution set.

An inner approximation (underestimation) ofΣ, and for its complement, and an approx-imation of the boundary ofΣ, respectively, may be labelled and defined as follows:

6 Problems Involving Polynomial Inequalities

• Σi: an inner approximation (underestimation) of Σ, consisting of the union of sub-boxes ofX on which all polynomialsp_i are positive.

• Σe: an inner approximation of the exterior (complement) of Σ, given by the union of sub-boxes of X with the property that on each there is at least one non-positive polynomialpi^∗.

• Σ_b: an outer approximation (overestimation) of the boundary ∂Σof Σ, consisting of the union of sub-boxes ofXon which all polynomialspi attain positive values, but on which at least one polynomial also attains non-positive values.

The union ofΣi and Σb forms an outer approximation (an overestimation, or guaranteed enclosure) forΣ. An example of these three sets and corresponding sub-boxes is depicted in Figure 6.1.

x₁

Σ

x₂

Σ_i Σe Σb

X

Figure 6.1: Approximations of the solution setΣand its boundary∂Σover a boxXinR². Details of a branch-and-bound algorithm utilising the Bernstein expansion and its ap-plication to the computation of D-stability regions (see below) are given in [GG99b] and [Gar00].

In the context of robust control problems, a common formulation is as follows: we are presented with the problem of determining theD-stability region of a family of polynomials within a given parameter box Q. For a given polynomial family p in x with a tuple of parametersq (6.3), this is the set

{q∈Q | p(x, q)6= 0, ∀x /∈ D}. (6.2)

6 Problems Involving Polynomial Inequalities

This problem can be reformulated as a system of polynomial inequalities. We may consider a family of polynomials

p(x, q) = a0(q)x^m+. . .+am−1(q)x+am(q), (6.3) where the coefficients depend polynomially on n parameters q = (q1, . . . , qn), i.e., for k= 0, . . . , m,

a_k(q) =

l

X

i1,...,in=0

a^(k)_i

1...inq₁ⁱ¹·. . .·qⁱ_nⁿ. (6.4) The uncertain parametersq_i belong to specified intervals

qi ∈[q_i, q_i], i= 1, . . . , n. (6.5) These parameter intervals are represented in the form of a boxQ:= [q₁, q₁]×. . .×[q_l, q_l].

A polynomial pmay be termedD-stable, whereD is a set in the complex plane, if all the zeros ofp are insideD. TherobustD-stability problemconsists of determining whether the family of polynomials p(q) are robustly D-stable for Q, i.e. whether the polynomials p(q) areD-stable for all q∈Q. Three problems of interest of this type are

• Hurwitz stability, whereD is the open left half of the complex plane,

• Schur stability, whereD is the open unit disc, and

• damping, whereD is a sector centered around the negative real axis with its vertex at the origin.

Testing of determinants for positivity (determinantal criteria) may be used in order to reduce the problem to one of strict inequalities involving multivariate polynomials, which may then be solved with a subdivision-based scheme, as outlined above. The use of deter-minants is generally restricted to systems depending on only a small number of parameters.

Several other types of control problems, e.g. static output feedback stabilisation, can also be reduced to the solution of systems of polynomial inequalities. Apart from stability, other performance specifications of control systems may also be modelled in the frequency domain as polynomial inequalities. In [ZG98] a method based on Bernstein expansion is developed which is capable of treating robust Hurwitz stability problems with a larger number of parameters; an application to robust Schur stability is given in [GG99a].