157 research outputs found
On second-order cone positive systems
Internal positivity offers a computationally cheap certificate for external
(input-output) positivity of a linear time-invariant system. However, the
drawback with this certificate lies in its realization dependency. Firstly,
computing such a realization requires to find a polyhedral cone with a
potentially high number of extremal generators that lifts the dimension of the
state-space representation, significantly. Secondly, not all externally
positive systems posses an internally positive realization. Thirdly, in many
typical applications such as controller design, system identification and model
order reduction, internal positivity is not preserved. To overcome these
drawbacks, we present a tractable sufficient certificate of external positivity
based on second-order cones. This certificate does not require any special
state-space realization: if it succeeds with a possibly non-minimal
realization, then it will do so with any minimal realization. While there exist
systems where this certificate is also necessary, we also demonstrate how to
construct systems, where both second-order and polyhedral cones as well as
other certificates fail. Nonetheless, in contrast to other realization
independent certificates, the present one appears to be favourable in terms of
applicability and conservatism. Three applications are representatively
discussed to underline its potential. We show how the certificate can be used
to find externally positive approximations of nearly externally positive
systems and demonstrated that this may help to reduce system identification
errors. The same algorithm is used then to design state-feedback controllers
that provide closed-loop external positivity, a common approach to avoid over-
and undershooting of the step response. Lastly, we present modifications to
generalized balanced truncation such that external positivity is preserved
where our certificate applies
Rank Reduction with Convex Constraints
This thesis addresses problems which require low-rank solutions under convex constraints. In particular, the focus lies on model reduction of positive systems, as well as finite dimensional optimization problems that are convex, apart from a low-rank constraint. Traditional model reduction techniques try to minimize the error between the original and the reduced system. Typically, the resulting reduced models, however, no longer fulfill physically meaningful constraints. This thesis considers the problem of model reduction with internal and external positivity constraints. Both problems are solved by means of balanced truncation. While internal positivity is shown to be preserved by a symmetry property; external positivity preservation is accomplished by deriving a modified balancing approach based on ellipsoidal cone invariance.In essence, positivity preserving model reduction attempts to find an infinite dimensional low-rank approximation that preserves nonnegativity, as well as Hankel structure. Due to the non-convexity of the low-rank constraint, this problem is even challenging in a finite dimensional setting. In addition to model reduction, the present work also considers such finite dimensional low-rank optimization problems with convex constraints. These problems frequently appear in applications such as image compression, multivariate linear regression, matrix completion and many more. The main idea of this thesis is to derive the largest convex minorizers of rank-constrained unitarily invariant norms. These minorizers can be used to construct optimal convex relaxations for the original non-convex problem. Unlike other methods such as nuclear norm regularization, this approach benefits from having verifiable a posterior conditions for which a solution to the convex relaxation and the corresponding non-convex problem coincide. It is shown that this applies to various numerical examples of well-known low-rank optimization problems. In particular, the proposed convex relaxation performs significantly better than nuclear norm regularization. Moreover, it can be observed that a careful choice among the proposed convex relaxations may have a tremendous positive impact on matrix completion. Computational tractability of the proposed approach is accomplished in two ways. First, the considered relaxations are shown to be representable by semi-definite programs. Second, it is shown how to compute the proximal mappings, for both, the convex relaxations, as well as the non-convex problem. This makes it possible to apply first order method such as so-called Douglas-Rachford splitting. In addition to the convex case, where global convergence of this algorithm is guaranteed, conditions for local convergence in the non-convex setting are presented. Finally, it is shown that the findings of this thesis also extend to the general class of so-called atomic norms that allow us to cover other non-convex constraints
Analysis of Control Systems on Symmetric Cones
It is well known that exploiting special structure is a powerful way to extend the reach of current optimization tools to higher dimensions. While many linear control systems can be treated satisfactorily with linear matrix inequalities (LMI) and semidefinite programming (SDP), practical considerations can still restrict scalability of general methods. Thus, we wish to work with high dimensional systems without explicitly forming SDPs. To that end, we exploit a particular kind of structure in the dynamics matrix, paving the way for a more efficient treatment of a certain class of linear systems. We show how second order cone programming (SOCP) can be used instead of SDP to find Lyapunov functions that certify stability. This framework reduces to a famous linear program (LP) when the system is internally positive, and to a semidefinite program (SDP) when the system has no special structure
Differential Dissipativity Theory for Dominance Analysis
High-dimensional systems that have a low-dimensional dominant behavior allow
for model reduction and simplified analysis. We use differential analysis to
formalize this important concept in a nonlinear setting. We show that dominance
can be studied through linear dissipation inequalities and an interconnection
theory that closely mimics the classical analysis of stability by means of
dissipativity theory. In this approach, stability is seen as the limiting
situation where the dominant behavior is 0-dimensional. The generalization
opens novel tractable avenues to study multistability through 1-dominance and
limit cycle oscillations through 2-dominance
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