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

**56 participants **

**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

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**...).

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.

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.

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.

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.

**Control**

- 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.

**Benchmarks**

- 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.