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We introduce and solve the stabilization problem of a transport partial differential equation (PDE)/nonlinear ordinary differential equation (ODE) cascade, in which the PDE state evolves on a domain whose length depends on the boundary values of the PDE state itself. In particular, we develop a predictor-feedback control design, which compensates such transport PDE dynamics. We prove local asymptotic stability of the closed-loop system in the $C^1$ norm of the PDE state employing a Lyapunov-like argument and introducing a backstepping transformation. We also highlight the relation of the PDE–ODE cascade to a nonlinear system with input delay that depends on past input values and present the predictor-feedback control design for this representation as well. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the control of a transport partial differential equation/ nonlinear ordinary differential equation (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 and a Lyapunov-like argument. The relation of the PDE–ODE cascade with a state-dependent propagation speed to an ODE system with a state-dependent input delay, which is defined implicitly via an integral of past values of the ODE state, is also highlighted and the corresponding equivalent predictor-feedback design is presented together with an alternative proof of global asymptotic stability of the closed-loop system based on the construction of a Lyapunov functional. The practical relevance of our control framework is illustrated in an example that is concerned with the control of a metal rolling process. [ABSTRACT FROM PUBLISHER]

We develop a predictor-feedback control design for multi-input nonlinear systems with distinct input delays, of arbitrary length, in each individual input channel. Due to the fact that different input signals reach the plant at different time instants, the key design challenge, which we resolve, is the construction of the predictors of the plant's state over distinct prediction horizons such that the corresponding input delays are compensated. Global asymptotic stability of the closed-loop system is established by utilizing arguments based on Lyapunov functionals or estimates on solutions. We specialize our methodology to linear systems for which the predictor-feedback control laws are available explicitly and for which global exponential stability is achievable. A detailed example is provided dealing with the stabilization of the nonholonomic unicycle, subject to two different input delays affecting the speed and turning rate, for the illustration of our methodology. [ABSTRACT FROM PUBLISHER]

We consider the boundary stabilization problem of the non-uniform equilibrium profiles of a viscous Hamilton-Jacobi (HJ) Partial Differential Equation (PDE) with parabolic concave Hamiltonian. We design a nonlinear full-state feedback control law, assuming Neumann actuation, which achieves an arbitrary rate of convergence to the equilibrium. Our design is based on a feedback linearizing transformation which is locally invertible. We prove local exponential stability of the closed-loop system in the H^1 norm, by constructing a Lyapunov functional, and provide an estimate of the region of attraction. We design an observer-based output-feedback control law, by constructing a nonlinear observer, using only boundary measurements. We illustrate the results on a benchmark example computed numerically. [ABSTRACT FROM AUTHOR]

We introduce and solve stabilization problems for linear and nonlinear systems with state-dependent input delay. Since the state dependence of the delay makes the prediction horizon dependent on the future value of the state, which means that it is impossible to know a priori how far in the future the prediction is needed, the key design challenge is how to determine the predictor state. We resolve this challenge and establish closed-loop stability of the resulting infinite-dimensional nonlinear system for general non-negative-valued delay functions of the state. Due to an inherent limitation on the allowable delay rate in stabilization of systems with time-varying input delays, in the case of state-dependent delay, where the delay rate becomes dependent on the gradient of the delay function and on the state and control input, only regional stability results are achievable. For forward-complete systems, we establish an estimate of the region of attraction in the state space of the infinite-dimensional closed-loop nonlinear system and for linear systems we prove exponential stability. Global stability is established under a restrictive Lyapunov-like condition, which has to be a priori verified, that the delay rate be bounded by unity, irrespective of the values of the state and input. We also establish local asymptotic stability for locally stabilizable systems in the absence of the delay. Several illustrative examples are provided, including unicycle stabilization subject to input delay that grows with the distance from the reference position. [ABSTRACT FROM AUTHOR]

Compensation of distributed delays in MIMO, LTI systems is achieved using Artstein's reduction method. In this technical note, we construct a Lyapunov functional for the resulting closed-loop system and establish exponential stability. The key element in our work is the introduction of an infinite-dimensional forwarding-backstepping transformation of the infinite-dimensional actuator states. We illustrate the construction of the Lyapunov functional with a detailed example of a single-input system, in which the input is entering through two individual channels with different delays. Finally, we develop an observer equivalent to the predictor feedback design, for the case of distributed sensor delays and prove exponential convergence of the estimation error. [ABSTRACT FROM AUTHOR]

Compensation of infinite-dimensional input or sensor dynamics in SISO, LTI systems is achieved using the backstepping method. For MIMO, LTI systems with distributed input or sensor dynamics, governed by diffusion with counter-convection, we develop a methodology for constructing control laws and observers that compensate the infinite-dimensional actuator or sensor dynamics. The explicit construction of the compensators are based on novel transformations, which can be considered of “backstepping-forwarding” type, of the finite-dimensional state of the plant and of the infinite-dimensional actuator or sensor states. Based on these transformations we construct explicit Lyapunov functionals which prove exponential stability of the closed-loop system, or convergence of the estimation error in the case of observer design. Finally, we illustrate the effectiveness of our controller with a numerical example. [ABSTRACT FROM AUTHOR]