Compensation of input delay that depends on delayed input

Abstract

For nonlinear systems, we develop a PDE-based predictor-feedback control design, which compensates actuator dynamics, governed by a transport PDE with outlet boundary-value-dependent propagation velocity. Global asymptotic stability under the predictor-feedback control law is established assuming spatially uniform strictly positive transport velocity. The stability proof is based on a Lyapunov-like argument and employs an infinite-dimensional backstepping transformation that is introduced. An equivalent representation of the transport PDE/nonlinear ODE cascade via a nonlinear system with an input delay that is defined implicitly through an integral of the past input is also provided and the equivalent predictor-feedback control design for the delay system is presented. The validity of the proposed controller is illustrated applying a predictor-feedback “bang–bang” boundary control law to a PDE model of a production system with a queue. Consistent simulation results are provided that support the theoretical developments.

Introduction

Cascades of partial and ordinary differential equations are widely used for modeling of complex dynamics in various engineering applications, such as screw extrusion processes in 3D printing (Diagne & Krstic, 2015), metal cutting processes (Otto & Radons, 2013), moisture in convective flows (Bresch-Pietri & Coulon, 2015), populations (Smith, 1993), transport phenomena in gasoline engines Bresch-Pietri et al. (2014), Detwiler and Wang (2006), Guzzella and Onder (2009), Jankovic and Magner (2011), Kahveci and Jankovic (2010), crushing-mills (Richard, 2003), production of commercial fuels by blending (Chebre, Creff, & Petit, 2010), and of stick–slip instabilities during oil drilling Bekiaris-Liberis and Krstic (2014), Cai and Krstic (2015), Cai and Krstic (2016), Krstic (2009), to name only a few. Depending on the application, the PDE state may evolve on a time-varying domain Cai and Krstic (2015), Cai and Krstic (2016), Diagne and Krstic (2015), Diagne et al. (2016a), Diagne et al. (2016b) or its transport coefficient may vary with time Bresch-Pietri et al. (2014), Otto and Radons (2013).

The nonlinear predictor-feedback concept, which enables one to design efficient feedback laws that compensate constant input delays arising in nonlinear systems was originally introduced in Krstic (2010a), Krstic (2010c) where the PDE backstepping methodology combined with a Lyapunov analysis was exploited to establish stability results. For nonlinear systems with time-varying and state-dependent delays, the analogous control design methodology was developed in Bekiaris-Liberis, Jankovic, and Krstic (2012) and Bekiaris-Liberis and Krstic (2012), Bekiaris-Liberis and Krstic (2013a), Bekiaris-Liberis and Krstic (2013b), . For linear systems Karafyllis, Malisoff, de Queiroz, Krstic and Yang (2015) and Mazenc and Malisoff (2015) proposed alternative prediction-based approaches. Later, the method was extended to deal with the stabilization problem of nonlinear systems with actuator dynamics governed by a wave PDE with moving boundary that depends on the ODE state Cai and Krstic (2015), Cai and Krstic (2016).

However, the problem of design of predictor-feedback controllers for compensation of input delays that depend on the control input itself is left out in most of the existing contributions. As described in Richard (2003), the design of delay-compensating control laws for such systems of transport PDE/ODE cascades with input-dependent transport coefficient (that appear for example when describing the dynamics of crushing-mill processes (Richard, 2003), recycling CSTR (Albertos & Garcia, 2012), and single-phase marine cooling systems (Hansen, Stoustrup, & Bendtsen, 2013)) remains an open problem. To our knowledge, the result in Bresch-Pietri et al. (2014), which is motivated by the dynamical model of fuel to air ratio (FAR) in gasoline engines, is perhaps the only contribution that covers this particular subject on delay compensation. Due to the dependency of the prediction horizon on the future input values a design that completely compensate the input delay does not seem possible.

The present work deals with the problem of compensation of transport PDE actuator dynamics with boundary-value-dependent propagation speed in nonlinear systems. Equivalently, the nonlinear ODE’s actuator dynamics are described as a  delayed-input-dependent input delay. Here, the delay function is implicitly given by an integral equation, similarly to Bresch-Pietri et al. (2014), but is dependent on the delayed rather than the current input.

The predictor-feedback control law for both the PDE and the delay system representations of the PDE–ODEcascade system is developed. Our contribution stands as the first one in which actual compensation of a delayed-input-dependent input delay is achieved. A global stability result of the closed-loop system is established. The designed compensator is employed for control of PDE models of production systems with a finite buffer size at the end of the production chain Borsche et al. (2010), Herty et al. (2007), Sun and Dong (2008).

The paper is organized as follows. In Section 2 the general problem is described and the main result together with a global stability proof, based on a PDE representation of the predictor-feedback control law, is presented in Section 3. An alternative representation of the actuator dynamics as an implicitly defined delayed-input-dependent input delay and the associated delay compensator are given in Section 4. Section 5 is dedicated to the application of the designed control law to a PDE model of production systems enabling a delay-compensating “bang–bang” feedback law. Concluding remarks are stated in Section 7.