Integral action controllers for systems with time delays

Consider a stabilizing controller C 1 for a given plant P. If C 1 and P do not have any zeros at the origin, then one can use a cascade connected PI (proportional plus integral) controller C pi with C 1 and keep the feedback system stable. In this work we examine the allowable range of the integral action gain in C pi , and discuss how C 1 should be chosen to maximize this range for systems with time delays. © 2009 Springer-Verlag Berlin Heidelberg

Modified Schur-Cohn Criterion for Stability of Delayed Systems

A modified Schur-Cohn criterion for time-delay linear time-invariant systems is derived. The classical Schur-Cohn criterion has two main drawbacks; namely, (i) the dimension of the Schur-Cohn matrix generates some round-off errors eventually resulting in a polynomial of s with erroneous coefficients and (ii) imaginary roots are very hard to detect when numerical errors creep in. In contrast to the classical Schur-Cohn criterion an alternative approach is proposed in this paper which is based on the application of triangular matrices over a polynomial ring in a similar way as in the Jury test of stability for discrete systems. The advantages of the proposed approach are that it halves the dimension of the polynomial and it only requires seeking real roots, making this modified criterion comparable to the Rekasius substitution criterion

A Continuum Framework and Homogeneous Map Based Algorithms for Formation Control of Multi Agent Systems

In this dissertation, new algorithms for formation control of multi agent systems (MAS) based on continuum mechanics principles will be suggested. For this purpose, agents of the MAS are considered as particles in a continuum, evolving in R^n, whose desired configuration is required to satisfy an admissible deformation function. Considered is a specific class of mappings that are called homogenous where the Jacobian of the mapping is only a function of time and is not spatially varying. The primary objectives of this dissertation are to develop the necessary theory and its validation on a mobile-agent based swarm test bed that includes two primary tasks: 1) homogenous transformation of MAS and 2) deployment of a random distribution of agents on a desired configuration. Developed will be a framework based on homogenous transformations for the evolution of an MAS in an n-dimensional space (n=1,2, and 3), under1) no inter-agent communication (predefined motion plan), 2) local inter-agent communication, and 3) intelligent perception by agents. In this dissertation, different communication protocols for MAS evolution that are based on certain special features of a homogenous transformation will be developed. It is also aimed to deal with the robustness of tracking of a desired motion by an MAS evolving in R^n. Furthermore, the effect of communication delays in an MAS evolving under consensus algorithms or homogenous maps is investigated. In this regard, the maximum allowable communication delay for MAS evolution is formulated on the basis of eigen-analysis.Ph.D., Mechanical Engineering and Mechanics -- Drexel University, 201

Das Spektrum zeitverzögerter Differentialgleichungen: numerische Methoden, Stabilität und Störungstheorie

Three types of problems related to delay-differential equations (DDEs) are treated in this thesis. We first consider the problem of numerically computing the eigenvalues of a DDE. Here, we present an application of a projection method for nonlinear eigenvalue problems (NLEPs). We compare this projection method with other methods, suggested in the literature, and used in software packages. The projection method is computationally superior to all of the other tested method for the presented large-scale examples. We give interpretations of methods based on discretizations in terms of rational approximations. Some notes regarding a special case where the spectrum can be explicitly expressed with a formula containing a matrix version of the are Lambert W function are presented. We clarify its range of applicability, and, by counter-example, show that it does not hold in general. The second part of this thesis is related to exact stability conditions of the DDE. All those combinations of the delays such that there is a purely imaginary eigenvalue (called critical delays) are parameterized. In general, an evaluation of the parameterization map consists of solving a quadratic eigenvalue problem of squared dimension. We show how the computational cost for one evaluation of the map can be reduced by exploiting a relation to a Lyapunov equation. The third and last part of this thesis is about generalizations of perturbation results for NLEPs. A sensitivity formula for the movement of the eigenvalues extends to NLEPs. We introduce a fixed point form for the NLEP, and show that some methods in the literature can be interpreted as set-valued fixed point iterations for which asymptotic convergence can be established. We also show how the Bauer-Fike theorem can be generalized to the NLEP under special conditions.In dieser Arbeit werden drei verschiedene Problemklassen im Bezug zu delay-differential equations (DDEs) behandelt. Als erstes gehen wir auf die Berechnung der Eigenwerte von DDEs ein. In dieser Arbeit wenden wir eine Projektionsmethode für nichtlineare Eigenwertprobleme (NLEPe) an. Wir vergleichen diese mit anderen bereits bekannten Verfahren, wobei die hier vorgestellte Methode bedeutend bessere numerische Eigenschaften für die verwendeten Beispiele hat. Zusätzlich treffen wir Aussagen über Diskretisierungsmethoden zur rationalen Approximation. Desweiteren betrachten wir einen Spezialfall, bei welchem das Spektrum explizit mit Hilfe einer Matrix-Version der Lambert W-Funktion dargestellt werden kann. Für diese Formel bestimmen wir einen möglichen Anwendungsbereich. Im zweiten Teil der Arbeit werden exakte Stabilitätsbedingungen von DDEs betrachtet. Die Menge der Delays, für welche die DDE einen imaginären Eigenwert hat (sogenannte kritische Delays), wird parameterisiert. Im Allgemeinen ist zur Auswertung der Parametrisierungsabbildung das Lösen eines quadratischen Eigenwertproblems nötig, dessen Größe dem Quadrat der Dimension der DDE entspricht. Wir zeigen wie der Rechenaufwand durch Ausnutzung einer Lyapunov-Gleichung reduziert werden kann. Der letzte Teil dieser Arbeit befasst sich mit der Verallgemeinerung der Störungstheorie auf NLEPe. Unter anderem lässt sich eine Sensitivitätsformel auf NLEPe erweitern. Desweiteren wird eine Fixpunktform für NLEPe vorgestellt, und gezeigt dass einige Methoden aus der Literatur als mengenwertige Fixpunktiterationen dargestellt werden können, für welche wir asymptotische Konvergenz feststellen. Wir zeigen zusätzlich, dass das Bauer-Fike Theorem unter bestimmten Bedingungen auf NLEPe verallgemeinert werden kann

Yalta: A Matlab Toolbox For The H∞-stability Analysis Of Classical And Fractional Systems With Commensurate Delays

In this paper we describe YALTA, a Matlab toolbox dedicated to the H ∞-stability analysis of classical and fractional systems with commensurate delays given by their transfer function. Delay systems of both retarded and neutral type are considered. The asymptotic position of high modulus poles is given. For a fixed known delay, poles of small modulus of standard delay systems are approximated through a Padé-2 scheme. For a delay varying from zero to a prescribed positive value, stability windows as well as root loci are given. We deeply describe how we have circumvented the numerical issues of algorithms developed in Fioravanti et al. [2010a, 2012] as well as the limitations of this toolbox. Finally, several examples are given. © 2013 IFAC.839844IFAC - Technical Committee on Linear Control Systems,Technical Committee on Control Design,Technical Committee on Non-Linear Control Systems,Technical Committee on Robust Control,Technical Committee on Distributed Parameter SystemsBellman, R., Cooke, K., (1963) Differential-Difference Equations, , Academic PressBonnet, C., Partington, J.R., Analysis of fractional delay systems of retarded and neutral type (2002) Automatica, (38), pp. 1133-1138Bonnet, C., Fioravanti, A., Partington, J.R., Stability of neutral systems with multiple delays and poles asymptotic to the imaginary axis (2011) SIAM J. Control Opt., 49 (2), pp. 498-516Engelborghs, K., Luzyanina, T., Samaey, G., DDE-BIFTOOL v. 2.00: A matlab package for bifurcation analysis of delay differential equations (2001) Technical Report TW-330, Department of Computer Science, , K.U. Leuven, Leuven, BelgiumFioravanti, A.R., Bonnet, C., Özbay, H., Niculescu, S.-I., A numerical method to find stability windows and unstable poles for linear neutral time-delay systems (2010) 9th IFAC Workshop on Time Delay Systems, , Prague Czech Republic, June 7-9Fioravanti, A.R., Bonnet, C., Özbay, H., Stability of fractional neutral systems with multiple delays and poles asymptotic to the imaginary axis (2010) 49th IEEE Conference on Decision and Control, , Atlanta, USA, December 15-17Fioravanti, A.R., Bonnet, C., Ozbay, H., Niculescu, S.I., A numerical method for stability windows and unstable root-locus calculation for linear fractional time-delay systems (2012) Automatica, 48 (11), pp. 2824-2830Gu, K., Kharitonov, V.L., Chen, J., (2003) Stability of Time-delay Systems, , BirkhauserGumussoy, S., Michiels, W., Root-locus for SISO dead-time systems: A continuation based approach (2012) Automatica, 43 (3), pp. 480-489Hilfer, R., (2000) Applications of Fractional Calculus in Physics, , World ScientificHwang, C., Cheng, Y.C., A note on the use of the lambert W function in the stability analysis of time-delay systems (2005) Automatica, 41 (11), pp. 1979-1985Hwang, C., Cheng, Y.C., A numerical algorithm for stability testing of fractional delay systems (2006) Automatica, 42 (5), pp. 825-831Lee, E.B., Gu, G., Khargonekar, P.P., Misra, P., Finite-dimensional approximation of infinite dimensional systems (1990) Conference on Decision and ControlLee, E.B., Gu, G., Khargonekar, P.P., Misra, P., Approximation of infinite dimensional systems (1989) IEEE Trans. Automatic Control, 34, pp. 610-618Mäkilä, M., Partington, J.R., Shift operator induced approximations of delay systems (1999) SIAM J. Control Optimiz., 37 (6), pp. 1897-1912Maset, S., Vermiglio, R., Pseudospectral differencing methods for characteristic roots of delay differential equations SIAM J. Sci. Comput., 27 (2), pp. 482-495Matignon, D., Stability properties for generalized fractional differential systems ESAIM: Proceedings, 5, pp. 145-158Nguyen, L.H.V., Fioravanti, A.R., Bonnet, C., Stability of neutral systems with commensurate delays and many chains of poles asymptotic to same points on the imaginary axis (2012) 10th IFAC Workshop on Time Delay Systems, , TDS 12, Boston, JuneNiculescu, S.-I., (2001) Delay Effects on Stability. A Robust Control Approach, , Springer-VerlagNiculescu, S.-I., Gu, K., (2004) Advances in Time-Delay Systems, , SpringerOlgac, N., Sipahi, R., An exact method for the stability analysis of time delayed LTI systems (2002) IEEE Transactions on Automatic Control, 47 (5), pp. 793-797Olgac, N., Sipahi, R., A practical method for analyzing the stability of neutral type LTI-time delayed systems (2004) Automatica, 40, pp. 847-853Partington, J.R., (2004) Linear Operators and Linear Systems: An Analytical Approach to Control Theory, , Cambridge University PressPontryagin, L.S., On the zeros of some elementary transcendental functions (1955) Amer. Math. Soc. Transl, 2 (1), pp. 95-110Richard, J.P., Time-delay systems: An overview of some recent advances and open problems (2003) Automatica, 39, pp. 1667-1694Seydel, R., (2010) Practical Bifurcations and Stability Analysis, , SpringerVyhlídal, T., Zítek, P., Quasipolynomial mapping based rootfinder for analysis of time delay systems (2003) Proc. of IFAC Workshop on Time-Delay Systems, , TDS'03, RocquencourtWalton, K., Marshall, J.E., Direct method for TDS stability analysis (1987) IEE Proceedings, (134 PART D), pp. 101-10