Petteri Laakkonen

Frequency Domain Robust Regulation of Signals Generated by an Infinite-Dimensional Exosystem

This paper deals with frequency domain robust regulation of signals generated by an infinite-dimensional exosystem. The problem is formulated and the stability types are chosen so that one can generalize the existing finite-dimensional theory to more general classes of infinite-dimensional systems and signals. The main results of this article are extensions of the internal model principle, of a necessary and sufficient solvability condition for the robust regulation problem, and of Davisons simple servo compensator for stable plants in the chosen algebraic framework.

Directed structure at infinity for infinite-dimensional systems

In this article the structure at infinity of infinite-dimensional linear time invariant systems with finite-dimensional input and output spaces is discussed. It is shown that a diagonal form describing behaviour near infinity can be found. This diagonal form is a generalisation of the Smith–McMillan form at infinity for rational matrices. It is then used to simplify certain solvability conditions of a regulation problem. Examples on time-delay and distributed parameter systems are given.

NP-completeness results for partitioning a graph into total dominating sets

Abstract A total domatic k-partition of a graph is a partition of its vertex set into k subsets such that each intersects the open neighborhood of each vertex. The maximum k for which a total domatic k-partition exists is known as the total domatic number of a graph G, denoted by d t ( G ) . We extend considerably the known hardness results by showing it is -complete to decide whether d t ( G ) ≥ 3 where G is a bipartite planar graph of bounded maximum degree. Similarly, for every k ≥ 3 , it is -complete to decide whether d t ( G ) ≥ k , where G is split or k-regular. In particular, these results complement recent combinatorial results regarding d t ( G ) on some of these graph classes by showing that the known results are, in a sense, best possible. Finally, for general n-vertex graphs, we show the problem is solvable in 2 n n O ( 1 ) time, and derive even faster algorithms for special graph classes.

Robust regulation of SISO systems : the fractional ideal approach

We solve the robust regulation problem for single-input single-output plants by using the fractional ideal approach and without assuming the existence of coprime factorizations. In particular, we are able to formulate the famous internal model principle for stabilizable plants which do not necessarily admit coprime factorizations. We are able to give a necessary and sufficient solvability condition for the robust regulation problem, which leads to a design method for a robustly regulating controller. The theory is illustrated by examples.

Solvability of the output regulation problem with a feedforward controller

In this paper the solvability of the output regulation problem with an infinite-dimensional exosystem by using a linear feedforward controller is considered. New sum conditions that are necessary and sufficient for the solvability are found. In addition, the required smoothness properties of the reference signals are discussed in detail.

NP-completeness Results for Partitioning a Graph into Total Dominating Sets.

A total domatic k-partition of a graph is a partition of its vertex set into k subsets such that each intersects the open neighborhood of each vertex. The maximum k for which a total domatic k-partition exists is known as the total domatic number of a graph G, denoted by \(d_t(G)\). We extend considerably the known hardness results by showing it is \(\textsc {NP}\)-complete to decide whether \(d_t(G) \ge 3\) where G is a bipartite planar graph of bounded maximum degree. Similarly, for every \(k \ge 3\), it is \(\textsc {NP}\)-complete to decide whether \(d_t(G) \ge k\), where G is a split graph or k-regular. In particular, these results complement recent combinatorial results regarding \(d_t(G)\) on some of these graph classes by showing that the known results are, in a sense, best possible. Finally, for general n-vertex graphs, we show the problem is solvable in \(2^n n^{O(1)}\) time, and derive even faster algorithms for special graph classes.

A simple controller with a reduced order internal model in the frequency domain

We use frequency domain methods to study robust output regulation of a stable plant in a situation where the controller is only required to be robust with respect to a predefined class of perturbations. We present a characterization for the solvability of the control problem and design a minimal order controller that achieves robustness with respect to a given class of uncertainties. The construction of the controller is illustrated with an example.

Robust Regulation Theory for Transfer Functions With a Coprime Factorization

Classical frequency domain results of robust regulation are extended by requiring only a right or a left coprime factorization of a plant, but not both. The famous internal model principle is generalized first, which leads to a necessary and sufficient solvability condition of the robust regulation problem and to a parametrization of all robustly regulating controllers. In addition, a procedure for constructing robustly regulating controllers is proposed.

Polynomial Input-Output Stability for Linear Systems

We introduce the concept of polynomial input-output stability for infinite-dimensional linear systems. We show that this stability type corresponds exactly to the recent notion of P-stability in the frequency domain. In addition, we show that on a Hilbert space a regular linear system whose system operator generates a polynomially stable semigroup is always polynomially input-output stable, and present additional conditions under which the system is input-output stable. The results are illustrated with an example of a polynomially input-output stable one-dimensional wave system.

A fractional representation approach to the robust regulation problem for SISO systems

The purpose of this article is to develop a new approach to the robust regulation problem for plants which do not necessarily admit coprime factorizations. The approach is purely algebraic and allows us dealing with a very general class of systems in a unique simple framework. We formulate the famous internal model principle in a form suitable for plants defined by fractional representations which are not necessarily coprime factorizations. By using the internal model principle, we are able to give necessary and sufficient solvability conditions for the robust regulation problem and to parameterize all robustly regulating controllers.