Nikolaos Bekiaris-Liberis

Compensation of State-Dependent Input Delay for Nonlinear Systems

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.

Compensation of Time-Varying Input and State Delays for Nonlinear Systems

We consider general nonlinear systems with time-varying input and state delays for which we design predictor-based feedback controllers. Based on a time-varying infinitedimensional backstepping transformation that we introduce, our controller achieves global asymptotic stability in the presence of a time-varying input delay, which is proved with the aid of a strict Lyapunov function that we construct. Then, we “backstep” one time-varying integrator and we design a globally stabilizing controller for nonlinear strict-feedback systems with time-varying delays on the virtual inputs. The main challenge in this case is the construction of the backstepping transformations since the predictors for different states use different prediction windows. Our designs are illustrated by three numerical examples, including unicycle stabilization. [DOI: 10.1115/1.4005278]

Robustness of nonlinear predictor feedback laws to time- and state-dependent delay perturbations

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.

Lyapunov Stability of Linear Predictor Feedback for Distributed Input Delays

For multi-input, linear time-invariant systems with distributed input delays, Artsteins reduction method provides a predictor-based controller. In this paper, 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.

Nonlinear Control Under Nonconstant Delays

The authors have developed a methodology for control of nonlinear systems in the presence of long delays, with large and rapid variation in the actuation or sensing path, or in the presence of long delays affecting the internal state of a system. In addition to control synthesis, they introduce tools to quantify the performance and the robustness properties of the designs provided in the book. The book is based on the concept of predictor feedback and infinite-dimensional backstepping transformation for linear systems and the authors guide the reader from the basic ideas of the concept - with constant delays only on the input - all the way through to nonlinear systems with state-dependent delays on the input as well as on system states. Several examples are presented that help the reader digest some of the intricacies in the methodology. Readers will find the book useful because the authors provide elegant and systematic treatments of long-standing problems in delay systems, such as systems with state-dependent delays that arise in many applications. In addition, the authors give all control designs by explicit formulae, making the book especially useful for engineers who have faced delay-related challenges and are concerned with actual implementations and they accompany all control designs with Lyapunov-based analysis for establishing stability and performance guarantees. Audience: This book is intended for researchers working on control of delay systems, including engineers, mathematicians, and students. Contents: Chapter 1: Introduction; Part I: Chapter 2: Linear Systems with Input and State Delays; Chapter 3: Linear Systems with Distributed Delays; Chapter 4: Application: Automotive Catalysts; Chapter 5: Nonlinear Systems with Input Delay; Part II: Chapter 6: Linear Systems with Time-Varying Input Delay; Chapter 7: Robustness of Linear Constant-Delay Predictor Feedback to Time-Varying Delay Perturbations; Chapter 8: Nonlinear Systems with Time-Varying Input Delay; Chapter 9: Nonlinear Systems with Simultaneous Time-Varying Delays on the Input and the State; Part III: Chapter 10: Predictor Feedback Design When the Delay Is a Function of the State; Chapter 11: Stability Analysis for Forward-Complete Systems with Input Delay; Chapter 12: Stability Analysis for Locally Stabilizable Systems with Input Delay; Chapter 13: Nonlinear Systems with State Delay; Chapter 14: Robustness of Nonlinear Constant-Delay Predictors to Time- and State-Dependent Delay Perturbations; Chapter 15: State-Dependent Delays That Depend on Delayed States; Appendix A: Basic Inequalities; Appendix B: Input-to-Output Stability; Appendix C: Lyapunov Stability, Forward-Completeness, and Input-to-State Stability; Appendix D: Parameter Projection

Rejection of sinusoidal disturbance of unknown frequency for linear system with input delay

We present a new approach for rejection of a sinusoidal disturbance of unknown frequency, bias, amplitude, and phase for a linear unstable plant with a delay in the control channel. To solve the problem, we combine the well-known predictor feedback approach with the adaptive scheme that identifies the frequency of the disturbance. Compared to the existing results, the dynamic order of our adaptive scheme is low (equal to three) and the approach applies to plants that are unstable and have an arbitrary relative degree. The results are illustrated by the numerical example.

Output control algorithm for unstable plant with input delay and cancellation of unknown biased harmonic disturbance

Abstract We present a new stabilization approach for a linear plant with input delay, parametric uncertainties, and an unknown harmonic disturbance. To solve this problem, we combine the well-known predictor feedback approach with the state observer and adaptive scheme that identifies the frequency of the disturbance. Compared to the existing approaches, the dynamic order of our adaptive scheme is low (equal to three) and the results apply to plants that are unstable, non-minimum phase, and have an arbitrary relative degree.

Compensating the Distributed Effect of Diffusion and Counter-Convection in Multi-Input and Multi-Output LTI Systems

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.

Compensation of Wave Actuator Dynamics for Nonlinear Systems

The problem of stabilization of PDE-ODE cascades has been solved in the linear case for several PDE classes, whereas in the nonlinear case the problem has been solved only for the transport/delay PDE, namely for compensation of an arbitrary delay at the input of a nonlinear plant. Motivated by a specific engineering application in off-shore drilling, we solve the problem of stabilization of the cascade of a wave PDE with a general nonlinear ODE. Due to the presence of nonlinearities of arbitrary growth and the time-reversibility of the wave PDE, and due to the possibility of using arguments based on Lyapunov functionals or explicit solutions, several stability analysis approaches are possible. We present stability results in the H2 × H1 and C1 × C0 norms for general nonlinear ODEs, as well as in the H1 × L2 norm for linear ODEs. We specialize our general design for wave PDE-ODE cascades to the case of a wave PDE whose uncontrolled end does not drive an ODE but is instead governed by a nonlinear Robin boundary condition (a “nonlinear spring,” as in the friction law in drilling). This is the first global stabilization result for wave equations that incorporate non-collocated destabilizing nonlinearities of superlinear growth. We present two numerical examples, one with a nonlinear ODE and one with a nonlinear spring at the uncontrolled boundary of the wave PDE.

Compensating the distributed effect of a wave PDE in the actuation or sensing path of MIMO LTI systems

Abstract The problem of compensation of infinite-dimensional actuator or sensor dynamics of more complex type than pure delay was solved recently using the backstepping method for PDEs. In this paper we construct an explicit feedback law for a multi-input LTI system which compensates the wave PDE dynamics in its input and stabilizes the overall system. Our design is based on a novel infinite-dimensional backstepping–forwarding transformation. We illustrate the effectiveness of our design with a simulation example of a single-input second order system, in which the wave input enters the system through two different channels, each one located at a different point in the domain of the wave PDE. Finally, we consider a dual problem where we design an exponentially convergent observer that compensates the distributed effect of the wave sensor dynamics.