What's SMAC?

SMAC is a Matlab-Simulink based toolbox for Systems Modeling, Analysis and Control. It is developed within the Systems Control and Flight Dynamics Department (DCSD) of ONERA The French Aerospace Lab.


The Final Workshop of the SMAC project was organised at ONERA in Toulouse on June 16th, 2016.

55 participants

Final Program

  • 08.45 Welcome (Ph. Bidaud, B. Lamiscarre)
  • 09.00 Overview & general presentation of the Civilian Aircraft Landing Challenge (J-M. Biannic)
  • 09.30 LFT Modeling session
    • 09.30 A brief introduction to LFT modeling (C. Roos)
    • 09.40 LFT modeling with the GSS library and its graphical interface (C. Roos)
    • 10.00 Sparse polynomial & rational approximations for reduced LFT models (G. Hardier)
    • 10.30 Application to the Aircraft Benchmark (J-M. Biannic)
  • 10.45 Coffee break
  • 11.15 Robustness analysis session
    • 11.15 A brief introduction to LFT-based robustness analysis (C. Roos)
    • 11.20 Enhanced $\mu$ analysis with the SMART library (C. Roos)
    • 11.50 Frequency-domain IQC analysis by a new algorithmic approach (F. Demourant)
    • 12.10 Linear & nonlinear robustness analysis with illustrations on the aircraft benchmark
      • 12.10 Back to modeling aspects and preliminary $\mu$ analysis (J-M. Biannic)
      • 12.20 State-space IQC analysis with a dedicated Sedumi-based solver (G. Ferreres)
      • 12.40 Application to the aircraft landing benchmark (G. Ferreres)
  • 12.45 Photos & Lunch break
  • 14.30 Introduction to the control design libraries (J-M. Biannic)
    • 14.40 Convex synthesis (G. Ferreres)
    • 15.00 Anti-windup design (J-M. Biannic)
    • 15.20 Generalized dynamic-inversion-based design (J-M. Biannic)
    • 15.30 Output to Input Saturation Transformations (L. Burlion)
  • 15.40 A high-order flexible satellite benchmark: from modeling to control law validation (T. Loquen)
  • 16.10 Discussions & conclusions (All)
  • 17.00 End of the workshop

Download slides here

A new Matlab class to model uncertain and nonlinear systems

The GSS library (Generalized State Space) of the SMAC toolbox implements a new Matlab class, which allows to model uncertain and nonlinear systems as Linear Fractional Representations. It replaces and extends the LFR toolbox, with a more intuitive way to describe LFR and a more user-friendly interface, including a Simulink library. Several tools are proposed to manipulate GSS objects (addition, multiplication, inversion, concatenation, feedback...), obtain GSS objects from symbolic models, convert GSS/LFR/USS objects, manipulate the uncertainties and the nonlinearities (normalization, reordering, random sampling), and perform order reduction or approximation. A large class of continuous- and discrete-time systems can be handled, with real or complex uncertain or varying parameters, polytopic-type uncertain or varying elements, linear time-invariant uncertainties, sector nonlinearities, saturations, deadzones, or more general nonlinear operators. Full compatibility is also ensured with other modeling, analysis and control libraries of the SMAC toolbox (APRICOT, SMART, IQC, SAW...).

Generation of low-order LFR from numerical data

The APRICOT library (Approximation of Polynomial and Rational-type for Indeterminate Coefficients via Optimization Tools) of the SMAC toolbox allows to convert a set of scalar, vector or matrix samples into sparse polynomial or rational expressions, for which the number of terms in the numerator and denominator is as low as possible. Simple yet accurate LFR are obtained, which are tractable both for analysis and design purposes.

Robustness analysis by a frequency-domain IQC based approach

To analyze a large class of stability problems, typically non-linear, uncertain, time-varying, etc.. closed loop, an IQC approach is involved. In the context of this toolbox, we use standard IQC description and focus on the algorithmic issue. Usually the Kalman-Yakubovitch-Popov lemma based resolution is involved but with the consequence to add an auxiliary matrix P whose the size increases with the closed loop order. Finally this kind of approach leads to a strong increase in the number of optimization variables, which makes it untractable for high order models. Here a specific technique has been developed to solve the stability problem directly in the frequency domain with the guarantee that the solution is valid on the whole frequency domain. Some examples are given to illustrate the approach and a detailed description of the tool is provided.

Robustness analysis by a $\mu$-analysis based approach

The SMART library (Skew Mu Analysis based Robustness Tools) of the SMAC toolbox contains a set of $\mu$-analysis based tools to evaluate the robustness properties of high-dimensional LTI plants subject to numerous LTI uncertainties. These tools allow to compute both upper and lower bounds on the (skewed) robust stability margin, the worst-case $\mathcal{H}_\infty$ performance level, as well as the worst-case gain, phase, modulus and time-delay margins.

A detailed comparative analysis of $\mu$ lower bound algorithms

This page presents a detailed comparison of the most significant methods to compute lower bounds on the structured singular value $\mu$, i.e. upper bounds on the robust stability margin $k_r$. The behavior of these robustness analysis tools is characterized on a wide set of various real-world benchmarks and all numerical data are available for download.


When using the SMAC toolbox, please mention the following references:

The general toolbox

  • SMAC
    J-M. Biannic, L. Burlion, F. Demourant, G. Ferreres, G. Hardier, T. Loquen and C. Roos, "The SMAC Toolbox: a collection of libraries for Systems Modeling, Analysis and Control", June 2016, online available at http://w3.onera.fr/smac/.

Systems modeling

  • GSS library
    J-M. Biannic and C. Roos, "Generalized State Space: a new Matlab class to model uncertain and nonlinear systems as Linear Fractional Representations", February 2016, available with the SMAC toolbox at http://w3.onera.fr/smac/gss.
  • LFR toolbox
    J-F. Magni, "Linear Fractional Representation toolbox for use with Matlab", February 2006, available with the SMAC toolbox at http://w3.onera.fr/smac/lfrt.
  • APRICOT library
    C. Roos, G. Hardier and J-M. Biannic, "Polynomial and rational approximation with the APRICOT library of the SMAC toolbox", in Proceedings of the IEEE Multiconference on Systems and Control, Antibes, France, October 2014, pp. 1473-1478, available with the SMAC toolbox at http://w3.onera.fr/smac/apricot.

Robustness analysis

  • SMART library
    C. Roos, "Systems Modeling, Analysis and Control (SMAC) toolbox: an insight into the robustness analysis library", in Proceedings of the IEEE Multiconference on Systems and Control, Hyderabad, India, August 2013, pp. 176-181, available with the SMAC toolbox at http://w3.onera.fr/smac/smart.
  • IQC library
    F. Demourant, "New algorithmic approach based on integral quadratic constraints for stability analysis of high order models", in Proceedings of the European Control Conference, Zurich, Switzerland, July 2013, pp. 359-364, available with the SMAC toolbox at http://w3.onera.fr/smac/iqc.


  • Convex Synthesis library
    G. Ferreres, "Convex design of a Youla parameter for LTI and LFT plant models", available with the SMAC toolbox at http://w3.onera.fr/smac/convex_design.
  • SAW library
    J-M. Biannic and C. Roos, "Introduction to anti-windup design with the SAW library", available with the SMAC toolbox at http://w3.onera.fr/smac/saw.
  • OISTeR library
    L. Burlion, "A new saturation function to convert an output constraint into an input constraint", in Proceedings of the 20th Mediterranean Conference on Control and Automation, Barcelona, Spain, July 2012, pp. 1217-1222, available with the SMAC toolbox at http://w3.onera.fr/smac/oister.


  • Aircraft benchmark
    J-M. Biannic and C. Roos, "Flare control law design via multi-channel $H_\infty$ synthesis: Illustration on a freely available nonlinear aircraft benchmark", in Proceedings of the American Control Conferencel, Chicago, IL, USA, July 2015, pp.1303-1308.

    J-M. Biannic, "Nonlinear Civilian Aircraft Landing Benchmark", Technical note included in the benchmark package available with the SMAC toolbox at http://w3.onera.fr/smac/aircraftModel.

    J-M. Biannic and J. Boada-Bauxell, "A Civilian Aircraft Landing Challenge (based on the benchmark package available with the SMAC Toolbox)", Submitted as an open-track invited session to the IFAC World Congress Toulouse 2017.
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