, On the contrary, the two applications presented in this chapter are representative of the category of systems for which using a data-driven control technique. The first one is a continuous crystallizer, see Section 8.1, and the second one is an open-channel for hydroelectricity generation, see Section 8.2. Both systems are described by Partial Differential Equations (PDE), resulting in irrational transfer functionss. Contents 7.1 Preliminary: data-driven closed-loop stability assessment and enforcement

, Controller reduction under data-driven closed-loop stability condition

. .. , 113 7.2.2 Stability-preserving reduction of the controller model

, It is a separation process which goal is to produce high-purity solids from liquids. The system is SISO: its input is the solute feed concentration c f (t) and its output is the solute concentration in the crystallizer c(t). The state of the system is x(t) =, p.129

. .. , 135 8.2 Application to an hydroelectricity generation channel

. Bibliography-[ahn, Iterative learning control: Brief survey and categorization, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews, vol.37, issue.6, pp.1099-1121, 2007.

A. , Comparing global input-output behavior of frozen-equivalent LPV state-space models, 20th IFAC World Congress, 2017.

. Antoulas, A tutorial introduction to the Loewner framework for model reduction. Model Reduction and Approximation for Complex Systems, 2015.

. Apkarian, P. Noll-;-apkarian, and D. Noll, Structured H ? control of infinite dimensional systems, International Journal of Robust and Nonlinear Control, vol.28, issue.9, pp.3212-3238, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01868497

H. Åström, K. Åström, T. Hägglund, and . Åström, Instrument society of America Research Triangle Park, NC, Automatica, vol.2, issue.5, pp.457-476, 1977.

[. Battistelli, Direct control design via controller unfalsification, International Journal of Robust and Nonlinear Control, vol.28, issue.12, pp.3694-3712, 2018.

[. Battistelli, Stability of unfalsified adaptive switching control in noisy environments, IEEE Transactions on Automatic Control, vol.55, issue.10, pp.2424-2429, 2010.

[. Bazanella, Datadriven controller design: the H2 approach, 2011.

[. Bontempi, Lazy learning for local modelling and control design, International Journal of Control, vol.72, issue.7-8, pp.643-658, 1999.

[. Bonvin, Control and optimization of batch processes, IEEE Control Systems Magazine, vol.26, issue.6, pp.34-45, 2006.

[. Brozenec, Controller validation, International Journal of Adaptive Control and Signal Processing, vol.15, issue.5, pp.431-444, 2001.

[. Campestrini, Virtual reference feedback tuning for non-minimum phase plants, Automatica, vol.47, issue.8, pp.1778-1784, 2011.

[. Campi, Virtual reference feedback tuning: a direct method for the design of feedback controllers, Automatica, vol.38, issue.8, pp.1337-1346, 2002.

[. Campi, M. C. Savaresi-;-campi, and S. M. Savaresi, Direct nonlinear control design: the virtual reference feedback tuning (vrft) approach, IEEE Transactions on Automatic Control, vol.51, issue.1, pp.14-27, 2006.

[. Chilali, Robust pole placement in lmi regions, IEEE transactions on Automatic Control, vol.44, issue.12, pp.2257-2270, 1999.

[. Cooman, Estimating unstable poles in simulations of microwave circuits, 2018 IEEE/MTT-S International Microwave Symposium-IMS, pp.97-100, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01958795

[. Cooman, Model-free closed-loop stability analysis: A linear functional approach, IEEE Transactions on Microwave Theory and Techniques, vol.66, issue.1, pp.73-80, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01381731

. Dalmas, From infinite dimensional modelling to parametric reduced order approximation: Application to open-channel flow for hydroelectricity, 15th European Control Conference (ECC'16), 2016.
URL : https://hal.archives-ouvertes.fr/hal-01341086

D. Bruyne, F. Kammer-;-de-bruyne, and L. C. Kammer, Iterative feedback tuning with guaranteed stability, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251), vol.5, pp.3317-3321, 1999.

[. Dehghani, Validating controllers for internal stability utilizing closed-loop data, IEEE Transactions on Automatic Control, vol.54, issue.11, pp.2719-2725, 2009.

, System identification and time series analysis: Past, present, and future, Stochastic Theory and Control, pp.97-109, 2002.

[. Demourant, Falsification of an aircraft autopilot, Proceedings of the 2002 American Control Conference, vol.1, pp.803-808, 2002.

P. Demourant, F. Demourant, and C. Poussot-vassal, A new frequency-domain subspace algorithm with restricted poles location through lmi regions and its application to a wind tunnel test, International Journal of Control, vol.90, issue.4, pp.779-799, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01385393

D. Hamer and A. , Data-driven optimal controller synthesis: a frequency domain approach, 2010.

. [den and . Hamer, , 2009.

, Model-free norm-based fixed structure controller synthesis, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, pp.4030-4035

G. Duan, Analysis and design of descriptor linear systems, vol.23, 2010.

[. Eckhard, Virtual disturbance feedback tuning, IFAC Journal of Systems and Control, vol.3, pp.23-29, 2018.

, Model-free control and intelligent pid controllers: towards a possible trivialization of nonlinear control?, IFAC Proceedings Volumes, vol.42, pp.1531-1550, 2009.

M. Fliess and C. Join, Model-free control, International Journal of Control, vol.86, issue.12, pp.2228-2252, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00828135

. Formentin, A data-driven approach to control of batch processes with an application to a gravimetric blender, IEEE Transactions on Industrial Electronics, vol.61, issue.11, pp.6383-6390, 2014.

. Formentin, Vrft for lpv systems: Theory and braking control application, Robust Control and Linear Parameter Varying Approaches, pp.289-309, 2013.

. Formentin, Nonparametric lpv data-driven control, IFAC-PapersOnLine, vol.48, issue.26, pp.146-151, 2015.

. Formentin, Non-iterative direct data-driven controller tuning for multivariable systems: theory and application, IET control theory & applications, vol.6, issue.9, pp.1250-1257, 2012.

. Formentin, S. Savaresi-;-formentin, and S. M. Savaresi, Virtual reference feedback tuning for linear parameter-varying systems, IFAC Proceedings Volumes, vol.44, pp.10219-10224, 2011.

. Formentin, A comparison of model-based and data-driven controller tuning, International Journal of Adaptive Control and Signal Processing, vol.28, issue.10, pp.882-897, 2014.

M. Gevers, Identification for control: From the early achievements to the revival of experiment design, European journal of control, vol.11, issue.4-5, pp.335-352, 2005.

, A personal view of the development of system identification: A 30-year journey through an exciting field. IEEE Control systems magazine, vol.26, pp.93-105, 2006.

M. Gevers and X. Bombois, Input design: From open-loop to control-oriented design, IFAC Proceedings, vol.39, issue.1, pp.1329-1334, 2006.

. Goodwin, G. C. Payne-;-goodwin, and R. L. Payne, Dynamic system identification: experiment design and data analysis, vol.136, 1977.

I. Gosea and A. Antoulas, Stability preserving post-processing methods applied in the loewner framework, 20th Workshop on Signal and Power Integrity, 2016.

. Havre, K. Skogestad-;-havre, and S. Skogestad, Effect of rhp zeros and poles on performance in multivariable systems, UKACC International Conference on Control, vol.2, pp.930-935, 1996.

. Havre, K. Skogestad-;-havre, and S. Skogestad, Achievable performance of multivariable systems with unstable zeros and poles, International Journal of Control, vol.74, issue.11, pp.1131-1139, 2001.

[. Hildebrand, Optimal prefiltering in iterative feedback tuning, IEEE Transactions on Automatic Control, vol.50, issue.8, pp.1196-1200, 2005.

H. Hjalmarsson, Control of nonlinear systems using iterative feedback tuning, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No. 98CH36207), vol.4, pp.2083-2087, 1998.

H. Hjalmarsson-;-hjalmarsson, Efficient tuning of linear multivariable controllers using iterative feedback tuning, International journal of adaptive control and signal processing, vol.13, issue.7, pp.553-572, 1999.

H. Hjalmarsson, Iterative feedback tuning-an overview, International journal of adaptive control and signal processing, vol.16, issue.5, pp.373-395, 2002.

H. Hjalmarsson, From experiment design to closed-loop control, Automatica, vol.41, issue.3, pp.393-438, 2005.

[. Hjalmarsson, Iterative feedback tuning: theory and applications, IEEE control systems magazine, vol.18, issue.4, pp.26-41, 1998.

[. Hjalmarsson, A convergent iterative restricted complexity control design scheme, 33rd IEEE Conference on Decision and Control, vol.2, pp.1735-1740, 1994.

[. Hjalmarsson, , 1995.

, Model-free tuning of a robust regulator for a flexible transmission system, European Journal of Control, vol.1, issue.2, pp.148-156

[. Hori, Datadriven controller tuning for sensitivity minimization, International Conference on Advanced Mechatronic Systems, 2016.

I. Horowitz, Quantitative feedback theory, IEE Proceedings D (Control Theory and Applications), vol.129, pp.215-226, 1982.

Z. Hou-;-hou, Nonparametric models and its adaptive control theory, 1999.

[. Hou, An overview of dynamiclinearization-based data-driven control and applications, IEEE Transactions on Industrial Electronics, vol.64, issue.5, pp.4076-4090, 2017.

W. ;. Hou, Z. Hou, and Z. Wang, From model-based control to data-driven control: Survey, classification and perspective, Information Sciences, vol.235, pp.3-35, 2013.

K. Huang, B. Huang, R. Kadali, A. C. Ionita, and A. C. Antoulas, Dynamic modeling, predictive control and performance monitoring: a data-driven subspace approach, SIAM Journal on Scientific Computing, vol.36, issue.3, pp.984-1007, 2008.

M. Jun and M. G. Safonov, Controller parameter adaptation algorithm using unfalsified control theory and gradient method, IFAC Proceedings Volumes, vol.35, pp.283-287, 2002.

C. Kammer, Frequency-domain control design in power systems, 2018.

[. Karimi, . Galdos, A. Karimi, and G. Galdos, Fixed-order h controller design for nonparametric models by convex optimization, Automatica, vol.46, issue.8, pp.1388-1394, 2010.

A. Karimi and C. Kammer, A data-driven approach to robust control of multivariable systems by convex optimization, 2017.

[. Karimi, Convergence analysis of an iterative correlation-based controller tuning method, 15th IFAC World Congress, p.1546, 2002.

[. Karimi, Robust H ? controller design using frequency-domain data via convex optimization, International Journal of Robust and Nonlinear Control, vol.28, issue.12, pp.3766-3783, 2018.

[. Karimi, Noniterative data-driven controller tuning using the correlation approach, European Control Conference (ECC), pp.5189-5195, 2007.

[. Karlsson, The inverse problem of analytic interpolation with degree constraint and weight selection for control synthesis, IEEE Transactions on Automatic Control, 2010.

. Kergus, Identification of parametric models in the frequency-domain through the subspace framework under lmi constraints, International Journal of Control, pp.1-12, 2018.
URL : https://hal.archives-ouvertes.fr/hal-02061484

. Kergus, Data-driven control design in the Loewner framework: Dealing with stability and noise, 2018.
URL : https://hal.archives-ouvertes.fr/hal-02099590

. Kergus, Data-driven reference model selection and Application to Loewner Data-Driven Control, 2019.

. Kergus, From reference model selection to controller validation: Application to Loewner Data-Driven Control, IEEE Control Systems Letters, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02181447

. Kergus, Frequency-domain data-driven control design in the loewner framework, IFAC-PapersOnLine, vol.50, issue.1, pp.2095-2100, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01850582

[. Khadraoui, A measurement-based approach for designing reducedorder controllers with guaranteed bounded error, International Journal of Control, vol.86, issue.9, pp.1586-1596, 2013.

M. Köhler-;-köhler, On the closest stable descriptor system in the respective spaces rh2 and rh, Linear Algebra and its Applications, vol.443, pp.34-49, 2014.

D. Kostic, Data-driven robot motion control design, 2004.

[. Landau, A flexible transmission system as a benchmark for robust digital control, European Journal of Control, 1995.

A. Lecchini-;-lecchini and M. C. Campi, , 2001.

S. S. , Sensitivity shaping via virtual reference feedback tuning, Proceedings of the 40th IEEE Conference on Decision and Control, vol.1, pp.750-755, 2001.

J. H. Lee-;-lee, Model predictive control: Review of the three decades of development, International Journal of Control, Automation and Systems, vol.9, issue.3, p.415, 2011.

[. Lefteriu, Modeling systems based on noisy frequency and time domain measurements, Perspectives in Mathematical System Theory, Control, and Signal Processing, pp.365-378, 2010.

[. Liu, Frequency domain structural system identification by observability range space extraction, Journal of Dynamic Systems, Measurement, and Control, vol.118, issue.2, pp.211-220, 1996.

L. Ljung, Perspectives on system identification, Annual Reviews in Control, vol.34, issue.1, pp.1-12, 2010.

M. Lovera, M. Lovera, and G. Mercere, Identification for gainscheduling: a balanced subspace approach, IEEE American Control Conference, 1967.
URL : https://hal.archives-ouvertes.fr/hal-00164365

D. Luenberger, Dynamic equations in descriptor form, IEEE Transactions on Automatic Control, vol.22, issue.3, pp.312-321, 1977.

A. Macfarlane and N. Karcanias, Poles and zeros of linear multivariable systems: a survey of the algebraic, geometric and complex-variable theory, International Journal of Control, vol.24, issue.1, pp.33-74, 1976.

[. Markusson, Iterative learning control of nonlinear non-minimum phase systems and its application to system and model inversion, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No. 01CH37228), vol.5, pp.4481-4482, 2001.

J. Mårtensson and H. Hjalmarsson, Variance error quantifications for identified poles and zeros, Automatica, 2009.

[. Martínez, Convex optimisation method for matching filters synthesis, 7th International Workshop on Microwave Filters, 2018.

[. Mayo, A. Mayo, and A. Antoulas, A framework for the solution of the generalized realization problem, Linear algebra and its applications, vol.425, issue.2, pp.634-662, 2007.

T. Mckelvey, Identification of state-space models from time and frequency data, 1995.

[. Mckelvey, Subspace-based multivariable system identification from frequency response data, IEEE Transactions on Automatic Control, vol.41, issue.7, pp.960-979, 1996.

[. Miller, D. Callafon, D. Miller, and R. De-callafon, Subspace identification with eigenvalue constraints, Automatica, 2013.

K. Moore-;-moore, Iterative learning control for deterministic systems, Automatica, vol.6, issue.32, pp.948-949, 1996.

[. Mor-toolbox-;-mor and . Toolbox, , 2018.

[. Nicoletti, A data-driven approach to power converter control via convex optimization, IEEE Conference on Control Technology and Applications (CCTA), pp.1466-1471, 2017.

[. Panzani, Active motorcycle braking via direct data-driven load transfer scheduling, IFAC Proceedings Volumes, vol.45, pp.1257-1262, 2012.

[. Passenbrunner, Direct data-driven control of internal combustion engine test benches using closed-loop experiments, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), pp.3765-3770, 2012.

A. Phillips, A. W. Phillips, and R. J. Allemang, An overview of mimo-frf excitation/averaging/processing techniques, Journal of Sound and vibration, vol.262, issue.3, pp.651-675, 2003.

[. Piga, Direct datadriven control of constrained systems, IEEE Transactions on Control Systems Technology, vol.26, issue.4, pp.1422-1429, 2018.

R. Pintelon and J. Schoukens, System identification: a frequency domain approach, 2012.

[. Pintelon, Estimation of nonparametric noise and frf models for multivariable systems-part i: Theory. Mechanical Systems and Signal Processing, vol.24, pp.573-595, 2010.

[. Pintelon, Estimation of nonparametric noise and frf models for multivariable systems-part ii: Extensions, applications, Mechanical Systems and Signal Processing, vol.24, issue.3, pp.596-616, 2010.

. Inria-sophia-antipolis,

C. Poussot-vassal, Large-scale dynamical model approximation and its applications, 2019.
URL : https://hal.archives-ouvertes.fr/tel-02188802

L. Pronzato, Optimal experimental design and some related control problems, Automatica, vol.44, issue.2, pp.303-325, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00259532

[. Rachah, A mathematical model for continuous crystallization, Mathematical Methods in the Applied Sciences, vol.39, issue.5, pp.1101-1120, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01620286

G. Rallo and F. S. ,

S. S. , Data-driven H ? -norm estimation via expert advice, Proceedings of the 56th IEEE Conference on Decision and Control, pp.1560-1565, 2017.

P. Rapisarda and A. C. Antoulas, A bilinear differential forms approach to parametric structured state-space modelling, Systems & Control Letters, vol.92, pp.14-19, 2016.

[. Saeki, Low-order h controller design on the frequency domain by partial optimization, International Journal of Robust and Nonlinear Control: IFAC-Affiliated Journal, vol.20, issue.3, pp.323-333, 2010.

[. Safonov, M. Tsao-;-safonov, and T. Tsao, The unfalsified control concept: A direct path from experiment to controller, Feedback Control, Nonlinear Systems, and Complexity, pp.196-214, 1995.

[. Safonov, M. G. Tsao-;-safonov, and T. Tsao, The unfalsified control concept and learning, Proceedings of 1994 33rd IEEE Conference on Decision and Control, vol.3, pp.2819-2824, 1994.

. Sala, A. Esparza-;-sala, and A. Esparza, Extensions to "virtual reference feedback tuning: A direct method for the design of feedback controllers, Automatica, vol.41, issue.8, pp.1473-1476, 2005.

[. Selvi, Towards direct datadriven model-free design of optimal controllers, European Control Conference, pp.2836-2841, 2018.

[. Seron, Fundamental limitations in filtering and control, 1997.

J. Sjöberg, Descriptor systems and control theory, 2005.

[. Sjöberg, Iterative controller optimization for nonlinear systems, Control Engineering Practice, vol.11, issue.9, pp.1079-1086, 2003.

[. Sjöberg, Nonlinear controller tuning based on a sequence of identifications of linearized timevarying models, Control Engineering Practice, vol.17, issue.2, pp.311-321, 2009.

R. Tóth, Modeling and identification of linear parameter-varying systems, 2010.

. Van-helvoort, Unfalsified control using an ellipsoidal unfalsified region applied to a motion system, IFAC Proceedings Volumes, vol.38, pp.157-162, 2005.

. Van-helvoort, Direct data-driven recursive controller unfalsification with analytic update, Automatica, vol.43, issue.12, pp.2034-2046, 2007.

[. Van-helvoort-;-van-helvoort and J. J. , Unfalsified control: data-driven control design for performance improvement, 2007.

. Heusden and K. Van-heusden, Non-iterative data-driven model reference control, 2010.

. Van-heusden, Datadriven estimation of the infinity norm of a dynamical system, 46th IEEE Conference on Decision and Control, pp.4889-4894, 2007.

. Van-heusden, Datadriven controller tuning with integrated stability constraint, 47th IEEE Conference on Decision and Control, pp.2612-2617, 2008.

. Van-heusden, Datadriven controller validation, Proceedings of the 15th IFAC Symposium on System Identification, pp.1050-1055, 2009.

. Van-heusden, , 2011.

, Data-driven model reference control with asymptotically guaranteed stability, International Journal of Adaptive Control and Signal Processing, vol.25, issue.4, pp.331-351

. Van-heusden, On identification methods for direct data-driven controller tuning, International Journal of Adaptive Control and Signal Processing, vol.25, issue.5, pp.448-465, 2011.

G. Vinnicombe, Frequency domain uncertainty and the graph topology, IEEE Transactions on Automatic Control, vol.38, issue.9, pp.1371-1383, 1993.

R. Vollmer, U. Vollmer, and J. Raisch, H-control of a continuous crystallizer, Control Engineering Practice, vol.9, issue.8, pp.837-845, 2001.

[. Wahlberg, Nonparametric methods for l2-gain estimation using iterative experiments, Automatica, vol.46, issue.8, pp.1376-1381, 2010.

S. Wang, R. Wang, and M. G. Safonov, The comparison of unfalsified control and iterative feedback tuning, 2002.

J. Xu and Z. Hou, Notes on data-driven system approaches, Acta Automatica Sinica, vol.35, issue.6, pp.668-675, 2009.

[. Yin, Data-driven control and process monitoring for industrial applications-part i, IEEE Transactions on Industrial Electronics, vol.61, issue.11, pp.6356-6359, 2014.

[. Yin, Data-driven control and process monitoring for industrial applications-part ii, IEEE Transactions on Industrial Electronics, vol.62, issue.1, pp.583-586, 2015.

D. ;. Zhou, K. Zhou, and J. C. Doyle, Essentials of robust control, vol.104, 1998.

[. Zhou, Robust and optimal control, vol.40, 1996.

N. ;. Ziegler, J. Ziegler, and N. Nichols, Optimum settings for automatic controllers, trans. ASME, issue.11, p.64, 1942.