Balanced truncation for linear switched systems

In this paper, we present a theoretical analysis of the model reduction algorithm for linear switched systems. This algorithm is a reminiscence of the balanced truncation method for linear parameter varying systems. Specifically in this paper, we provide a bound on the approximation error in L2 norm for continuous-time and l2 norm for discrete-time linear switched systems. We provide a system theoretic interpretation of grammians and their singular values. Furthermore, we show that the performance of bal- anced truncation depends only on the input-output map and not on the choice of the state-space representation. For a class of stable discrete-time linear switched systems (so called strongly stable systems), we define nice controllability and nice observability grammians, which are genuinely related to reachability and controllability of switched systems. In addition, we show that quadratic stability and LMI estimates of the L2 and l2 gains depend only on the input-output map.Comment: We have corrected a number of typos and inconsistencies. In addition, we added new results in Theorem

Model Reduction of Multi-Dimensional and Uncertain Systems

We present model reduction methods with guaranteed error bounds for systems represented by a Linear Fractional Transformation (LFT) on a repeated scalar uncertainty structure. These reduction methods can be interpreted either as doing state order reduction for multi-dimensionalsystems, or as uncertainty simplification in the case of uncertain systems, and are based on finding solutions to a pair of Linear Matrix Inequalities (LMIs). A related necessary and sufficient condition for the exact reducibility of stable uncertain systems is also presented

A first cubic upper bound on the local reachability index for some positive 2-D systems

[EN] The calculation of the smallest number of steps needed to deterministically reach all local states of an nth-order positive 2-D system, which is called local reachability index (ILR) of that system, was recently tackled bymeans of the use of a suitable composition table. The greatest index ILR obtained in the previous literature was n+3 ([n/2]) 2 for some appropriated values of n. Taking as a basis both a combinatorial approach of such systems and the construction of suitable geometric sets in the plane, an upper bound on ILR depending on the dimension n for a new family of systems is characterized. The 2-D influence digraph of this family of order n = 6 consists of two subdigraphs corresponding to a unique source s. The first one is a cycle involving the first n(1) vertices and is connected to the another subdigraph through the 1-arc (2, n(1) +n(2)), being the natural numbers n(1) and n(2) such that n(1) > n(2) = 2 and n-n(1)-n(2) = 1. The second one has two main cycles, a cycle where only the remaining vertices n(1)+1,..., n appear and a cycle containing only the vertices n(1)+1, n(1)+n(2)-1. Moreover, the last vertices are connected through the 2-arc (n(1) +n(2)-1, n). Furthermore, if n > 12 and is a multiple of 3, for appropriate n(1) and n(2), the ILR of that family is at least cubic, exactly, it must be n(3)+9n(2)+45n+108/27, which shows that some local states can be deterministically reached much further than initially proposed in the literature.We are gratefully thankful to the reviewers for their valuable remarks. This work has been partially supported by the European Union [FEDER funds] and Ministerio de Ciencia e Innovacion through Grants MTM-2013-43678-P and DPI2016-78831-C2-1-R.Bailo Ballarín, E.; Gelonch, J.; Romero Vivó, S. (2019). A first cubic upper bound on the local reachability index for some positive 2-D systems. 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Frequency-Domain Analysis of Linear Time-Periodic Systems

In this paper, we study convergence of truncated representations of the frequency-response operator of a linear time-periodic system. The frequency-response operator is frequently called the harmonic transfer function. We introduce the concepts of input, output, and skew roll-off. These concepts are related to the decay rates of elements in the harmonic transfer function. A system with high input and output roll-off may be well approximated by a low-dimensional matrix function. A system with high skew roll-off may be represented by an operator with only few diagonals. Furthermore, the roll-off rates are shown to be determined by certain properties of Taylor and Fourier expansions of the periodic systems. Finally, we clarify the connections between the different methods for computing the harmonic transfer function that are suggested in the literature

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

Strong Analytic Controllability for Hydrogen Control Systems

The realization and representation of so(4,2) associated with the hydrogen atom Hamiltonian are derived. By choosing operators from the realization of so(4,2) as interacting Hamiltonians, a hydrogen atom control system is constructed, and it is proved that this control system is strongly analytically controllable based on a time-dependent strong analytic controllability theorem.Comment: 6 pages; corrected typo; added equations in section III for representation states of so(4,2). accepted by CDC 200

Enhanced LFR-toolbox for MATLAB and LFT-based gain scheduling

We describe recent developments and enhancements of the LFR-Toolbox for MATLAB for building LFT-based uncertainty models and for LFT-based gain scheduling. A major development is the new LFT-object definition supporting a large class of uncertainty descriptions: continuous- and discrete-time uncertain models, regular and singular parametric expressions, more general uncertainty blocks (nonlinear, time-varying, etc.). By associating names to uncertainty blocks the reusability of generated LFT-models and the user friendliness of manipulation of LFR-descriptions have been highly increased. Significant enhancements of the computational efficiency and of numerical accuracy have been achieved by employing efficient and numerically robust Fortran implementations of order reduction tools via mex-function interfaces. The new enhancements in conjunction with improved symbolical preprocessing lead generally to a faster generation of LFT-models with significantly lower orders. Scheduled gains can be viewed as LFT-objects. Two techniques for designing such gains are presented. Analysis tools are also considered

An efficient algorithm for positive realizations

We observe that successive applications of known results from the theory of positive systems lead to an {\it efficient general algorithm} for positive realizations of transfer functions. We give two examples to illustrate the algorithm, one of which complements an earlier result of \cite{large}. Finally, we improve a lower-bound of \cite{mn2} to indicate that the algorithm is indeed efficient in general