Control of transport PDE/nonlinear ODE cascades with state-dependent propagation speed

In this paper, we deal with the control of a transport PDE/nonlinear ODE cascade system in which the transport coefficient depends on the ODE state. We develop a PDE-based predictor-feedback boundary control law, which compensates the transport dynamics of the actuator and guarantees global asymptotic stability of the closed-loop system. The stability proof is based on an infinite-dimensional backstepping transformation that is introduced, with the aid of which, a Lyapunov functional is constructed. The relation of the PDE-ODE cascade to an ODE system with a state-dependent input delay, which is defined implicitly via an integral of the ODE state, is also highlighted and the corresponding equivalent predictor-feedback design is presented. The practical relevance of our control framework is illustrated in an example that is concerned with the control of a metal rolling process.

[1]  Miroslav Krstic,et al.  State-dependent input delay-compensated Bang-Bang control: Application to 3D printing based on screw-extruder , 2015, 2015 American Control Conference (ACC).

[2]  Kornel Ehmann,et al.  Review of chatter studies in cold rolling , 1998 .

[3]  Miroslav Krstic,et al.  Robustness of nonlinear predictor feedback laws to time- and state-dependent delay perturbations , 2012, Autom..

[4]  Stephen Magner,et al.  Disturbance attenuation in time-delay systems — A case study on engine air-fuel ratio control , 2011, Proceedings of the 2011 American Control Conference.

[5]  Jean-Pierre Richard,et al.  Time-delay systems: an overview of some recent advances and open problems , 2003, Autom..

[6]  Günter Radons,et al.  Transformations from Variable Delays to Constant Delays with Applications in Engineering and Biology , 2017 .

[7]  Iasson Karafyllis,et al.  Numerical schemes for nonlinear predictor feedback , 2014, Math. Control. Signals Syst..

[8]  Qingwen Hu.  A Model of Cold Metal Rolling Processes with State-Dependent Delay , 2016, SIAM J. Appl. Math..

[9]  Miroslav Krstic,et al.  Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch , 2008, 2008 American Control Conference.

[10]  Delphine Bresch-Pietri,et al.  Prediction-Based Stabilization of Linear Systems Subject to Input-Dependent Input Delay of Integral-Type , 2014, IEEE Transactions on Automatic Control.

[11]  Nicolas Petit,et al.  Feedback control and optimization for the production of commercial fuels by blending , 2010 .

[12]  Kenneth J. Hunt,et al.  Neural control of a steel rolling mill , 1993 .

[13]  I. Karafyllis.  Stabilization by Means of Approximate Predictors for Systems with Delayed Input , 2009, SIAM J. Control. Optim..

[14]  Miroslav Krstic,et al.  Nonlinear stabilization through wave PDE dynamics with a moving uncontrolled boundary , 2016, Autom..

[15]  Delphine Bresch-Pietri,et al.  Practical Delay Modeling of Externally Recirculated Burned Gas Fraction for Spark-Ignited Engines , 2013, TDS.

[16]  Miroslav Krstic,et al.  Nonlinear control under delays that depend on delayed states , 2013, Eur. J. Control.

[17]  S. Foda,et al.  Control of the Metal Rolling Process: A Multidimensional System Approach , 1989, 1989 American Control Conference.

[18]  Nazli E. Kahveci,et al.  Adaptive controller with delay compensation for Air-Fuel Ratio regulation in SI engines , 2010, Proceedings of the 2010 American Control Conference.

[19]  Miroslav Krstic,et al.  Compensating a string PDE in the actuation or sensing path of an unstable ODE , 2009, 2009 American Control Conference.

[20]  Hugh Ford,et al.  The Calculation of Roll Force and Torque in Cold Strip Rolling with Tensions , 1948 .

[21]  Miroslav Krstic,et al.  Nonlinear control under wave actuator dynamics with time- and state-dependent moving boundary , 2015 .

[22]  Miroslav Krstic,et al.  Compensation of Wave Actuator Dynamics for Nonlinear Systems , 2014, IEEE Transactions on Automatic Control.

[23]  Delphine Bresch-Pietri,et al.  Robust compensation of a chattering time-varying input delay , 2014, 53rd IEEE Conference on Decision and Control.

[24]  Miroslav Krstic,et al.  Compensation of Time-Varying Input and State Delays for Nonlinear Systems , 2012 .

[25]  Eduardo Sontag,et al.  On characterizations of the input-to-state stability property , 1995 .

[26]  Günter Radons,et al.  Application of spindle speed variation for chatter suppression in turning , 2013 .

[27]  John Pittner,et al.  Control of a continuous tandem cold metal rolling process , 2008 .

[28]  Miroslav Krstic,et al.  Input Delay Compensation for Forward Complete and Strict-Feedforward Nonlinear Systems , 2010, IEEE Transactions on Automatic Control.

[29]  Delphine Bresch-Pietri,et al.  Prediction-based control of moisture in a convective flow , 2015, 2015 European Control Conference (ECC).

[30]  Kai Zenger,et al.  Modelling and control of a class of time-varying continuous flow processes☆ , 2009 .

[31]  M. Mackey,et al.  Age-structured and two-delay models for erythropoiesis. , 1995, Mathematical biosciences.

[32]  R. Johnson,et al.  Functional equations, approximations, and dynamic response of systems with variable time delay , 1972 .

[33]  Thomas Parisini,et al.  Adaptive task-space metal strip-flatness control in cold multi-roll mill stands , 2013 .

[34]  H L Smith,et al.  Reduction of structured population models to threshold-type delay equations and functional differential equations: a case study. , 1993, Mathematical biosciences.

[35]  Lino Guzzella,et al.  Introduction to Modeling and Control of Internal Combustion Engine Systems , 2004 .

[36]  Z. Artstein.  Linear systems with delayed controls: A reduction , 1982 .

[37]  Michael Malisoff,et al.  Reduction model approach for systems with a time-varying delay , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[38]  Eric J. Detwiler,et al.  Exhaust oxygen sensor dynamic study , 2006 .

[39]  M. Krstic.  Lyapunov Stability of Linear Predictor Feedback for Time-Varying Input Delay , 2010, IEEE Trans. Autom. Control..

[40]  Fu Zhang,et al.  Modeling and simulation of time-varying delays , 2012, SpringSim.

[41]  Nikolaos Bekiaris-Liberis,et al.  Compensation of state-dependent input delay for nonlinear systems , 2011, IEEE Conference on Decision and Control and European Control Conference.