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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. [ABSTRACT FROM AUTHOR]

We consider Ponomarev’s recent predictor-based control design for nonlinear systems with distributed input delays and remove certain restrictions to the class of systems by performing the stability analysis differently. We consider nonlinear systems that are not necessarily affine in the control input and whose vector field does not necessarily satisfy a linear growth condition. Employing a nominal feedback law, not necessarily satisfying a linear growth restriction, which globally asymptotically, and not necessarily exponentially, stabilizes a nominal transformed system, we prove global asymptotic stability of the original closed-loop system, under the predictor-based version of the nominal feedback law, utilizing estimates on solutions. We then identify a class of systems that includes systems transformable to a completely delay-free equivalent for which global asymptotic stability is shown employing similar tools. For these two classes of systems, we also provide an alternative stability proof via the construction of a novel Lyapunov functional. Although in order to help the reader to better digest the details of the introduced analysis methodology we focus on nonlinear systems without distributed delay terms, we demonstrate how the developed approach can be extended to the case of systems with distributed delay terms as well. [ABSTRACT FROM AUTHOR]

The problem of compensation of arbitrary large input delay for nonlinear systems was solved recently with the introduction of the nonlinear predictor feedback. In this paper we solve the problem of compensation of input delay for nonlinear systems with simultaneous input and state delays of arbitrary length. The key challenge, in contrast to the case of only input delay, is that the input delay-free system (on which the design and stability proof of the closed-loop system under predictor feedback are based) is infinite-dimensional. We resolve this challenge and we design the predictor feedback law that compensates the input delay. We prove global asymptotic stability of the closed-loop system using two different techniques—one based on the construction of a Lyapunov functional, and one using estimates on solutions. We present two examples, one of a nonlinear delay system in the feedforward form with input delay, and one of a scalar, linear system with simultaneous input and state delays. [ABSTRACT FROM AUTHOR]

Abstract: Much recent progress has been achieved for stabilization of linear and nonlinear systems with input delays that are long and dependent on either time or the plant state—provided the dependence is known. In this paper we consider the delay variations as unknown and study robustness of nominal constant-delay predictor feedbacks under delay variations that depend on time and the state. We show that when the delay perturbation and its rate have sufficiently small magnitude, the local asymptotic stability of the closed-loop system, under the nominal predictor-based design, is preserved. For the special case of linear systems, and under only time-varying delay perturbations, we prove robustness of global exponential stability of the predictor feedback when the delay perturbation and its rate are small in any one of four different metrics. We present two examples, one that is concerned with the control of a DC motor through a network and one of a teleoperation-like system. [Copyright &y& Elsevier]

SUMMARY We solve stabilization problems of LTI systems with unknown parameters and distributed input delay. The key challenge is that the infinite-dimensional input dynamics are distributed, which makes traditional infinite-dimensional backstepping inapplicable. We resolve this challenge by employing backstepping-forwarding transformations of the finite-dimensional state of the plant and of the infinite-dimensional actuator state. These transformations enable us to design Lyapunov-based update laws. We also design an adaptive controller for rejecting a constant disturbance in the input of the LTI plant, in the case where the parameters of the plant are known. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: We extend the technique for compensating state-dependent delays from systems with delayed inputs to systems with delayed states. We focus on predictor-feedback design for nonlinear systems in the strict-feedback form, having a state-dependent state delay on the virtual input. The two key challenges are the definition of the predictor state and the fact that the predictor design does not follow immediately from the delay-free design. We resolve these challenges and we establish asymptotic stability of the resulting infinite-dimensional nonlinear system for general nonnegative-valued delay functions of the state. Due to an inherent limitation on the delay rate, and since the delay rate depends on the state, we obtain only regional stability results. However, for forward-complete systems, we provide an estimate of the region of attraction in the state space of the infinite-dimensional system. We finally provide two examples, including an example of stabilization of a cooling system. [Copyright &y& Elsevier]

Abstract: We present a tutorial introduction to methods for stabilization of systems with infinite-dimensional input dynamics including delays, diffusion, counter-convection and wave propagation. The methods are based on techniques originally developed for boundary control of partial differential equations. We consider multi-input linear time-invariant systems with input dynamics governed by distributed delays, and diffusion with counter-convection or wave PDEs. For the special case of single-input linear time-invariant systems with a single discrete delay we prove robustness of the control law to a small uncertainty in the delay and in the case of completely unknown delay we present an adaptive control approach. For this special case, we also present a method for compensating arbitrarily large but known time-varying delays. Finally, we consider nonlinear control problems in the presence of arbitrarily long input delays. [ABSTRACT FROM AUTHOR]

Abstract: The problem of compensation of input delays for unstable linear systems was solved in the late 1970s. Systems with simultaneous input and state delay have remained a challenge, although exponential stabilization has been solved for systems that are not exponentially unstable, such as chains of delayed integrators and systems in the ‘feedforward’ form. We consider a general system in strict-feedback form with delayed integrators, which is an example of a particularly challenging class of exponentially unstable systems with simultaneous input and state delays, and design a predictor feedback controller for this class of systems. Exponential stability is proven with the aid of a Lyapunov–Krasovskii functional that we construct using the PDE backstepping approach. [Copyright &y& Elsevier]