Similar Metzler m otrix determination using non-smooth optimization

Similar Metzler m otrix determination using non-smooth optimization

Emmanuel Chambon Laurent Burlion Pierre Apkarian 

ONERA BP74025 – 2, avenue Édouard Belin FR-31055 Toulouse Cedex 4, France

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The theory of interval observers requires the studied system to be cooperative. In the case of non-cooperative systems, the literature proposes to compute a state-coordinate change such that the dynamics is cooperative in the new coordinates. In this paper, a new numerical method is introduced to compute this state-coordinate change. It is based on the reformulation of the problem into a multiple dynamic linear systems stabilization problem. This problem is then solved using an existing nonsmooth technique. Computations and simulations are performed on two examples inspired by the literature.


interval observers, nonsmooth optimization, multi-model synthesis

1. Introduction
2. Définitions et notations
3. Observateurs par intervalles
4. Formulation du problème
5. Approche proposée
6. Exemples
7. Conclusion et perspectives

Apkarian P., Gahinet P., Buhr C. (2014, juin). Multi-model, multi-objective tuning of fixedstructure

controllers. In Proc. of the 13th European Control Conference, p. 856–861. Strasbourg,


Apkarian P., Noll D. (2006, janvier). Nonsmooth H1 synthesis. IEEE Transactions on Automatic

Control, vol. 51, no 1, p. 71–86.

Bartels R. H., Stewart G. W. (1972, septembre). Solution of the matrix equation AX+XB=C. Communications of the ACM, vol. 15, no 9, p. 820–826.

Burke J. V., Henrion D., Lewis A. S., Overton M. L. (2006, août). HIFOO – a MATLAB package for fixed-order controller design and H1 optimization. In Proc. of the 5th IFAC Symposium on Robust Control Design. Toulouse, France.

Cacace F., Germani A., Manes C. (2015, juin). A new approach to design interval observers for linear systems. IEEE Transactions on Automatic Control, vol. 60, no 6, p. 1665–1670.

Dinh T. N., Mazenc F., Niculescu S.-I. (2014, juin). Interval observer composed of observers for nonlinear systems. In Proc. of the European Control Conference, p. 660–665. Strasbourg, France.

Efimov D., Raïssi T., Chebotarev S., Zolghadri A. (2013, janvier). Interval state observer for nonlinear time-varying systems. Automatica, vol. 49, no 1, p. 200–205.

Efimov D., Raïssi T., Zolghadri A. (2013, mars). Control of nonlinear and LPV systems: interval observer-based framework. IEEE Transactions on Automatic Control, vol. 58, no 3, p. 773–778.

Gahinet P., Apkarian P. (2012, septembre). Frequency-domain tuning of fixed-structure control systems. In Proc. of the UKACC International Conference on Control, p. 178-183.

Gouzé J. L., Rapaport A., Hadj-Sadok M. Z. (2000). Interval observers for uncertain biological systems. Ecological Modelling, vol. 133, no 1-2, p. 45–56.

Mailleret L. (2004). Stabilisation globale des systèmes positifs mal connus - applications en biologie. Thèse de doctorat non publiée, Université de Nice Sophia-Antipolis, Nice.

MATLAB. (2014). Robust Control Toolbox version 5.2 (R2014b). Natick, Massachusetts, The MathWorks Inc.

Mazenc F., Bernard O. (2010, février). Asymptotically stable interval observers for planar systems with complex poles. IEEE Transactions on Automatic Control, vol. 55, no 2, p. 523–527.

Mazenc F., Bernard O. (2011). Interval observers for linear time-invariant systems with disturbances. Automatica, vol. 47, no 1, p. 140–147.

Raïssi T., Efimov D., Zolghadri A. (2012, janvier). Interval state estimation for a class of nonlinear systems. IEEE Transactions on Automatic Control, vol. 57, no 1, p. 260–265.

Smith H. L. (1995). Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems (vol. 41). Providence, Rhode Island, American Mathematical Society.