Compensation of State-Dependent Input Delay for Nonlinear Systems

Abstract

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 nonnegative-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.

Time- and State-Dependent Input Delay-Compensated Bang-Bang Control of a Screw Extruder for 3D Printing

“…The transformation of the coupled PDE-ODE system into a statedependent input delay system, which describes the dynamics of the material convection in the extruder chamber, is achieved after solving the PDE by the Method of Characteristics (MC) [17], [19]. In order to also account for potential periodic fluctuations of the materials transport speed when processing granular pellets [21], due to the thermal energy that is supplied into the system from the heater of the extruder and due to the mechanical shearing effect by the rotation of the screw, the state-dependent input delay model is extended to a nonlinear system with an input delay that depends simultaneously on the state and the time variable (see [20], [23] and [22] for the treatment of systems with time-and state-dependent delays).…”

“…By combining the nominal, piecewise exponential feedback controller [19] with nonlinear predictor feedback, which is extended from the state-dependent input delay case [20] to the case in which the vector field and the delay function depend explicitly on time, the compensation of the time-and state-dependent input delay of the non-isothermal screw extrusion model is achieved. GAS of the closed-loop system under the delay-compensated Bang-Bang controller is established when certain conditions, related to the extruder design and the material properties, as well as to the periodic fluctuations, are satisfied.…”

Stability analysis and fuzzy smith compensation control for semi-active suspension systems with time delay

“…Recently, intensive research has been conducted on addressing the problem of the smith predictor [16,25,27] and time delay compensation control [19,21,24,26]. The authors in [17] investigated the effect of the time delay on stability and investigated a method to compensate for the time delay effect in order to ensure galloping suppression.…”

“…In [24] and [26], the authors adopted the Smith predictor to achieve time delay compensation and presented a stability analysis. In addition, the authors in [21] established a closed-loop stability of the resulting infinite-dimensional nonlinear system for a general non-negative-valued delay function of the state and local asymptotic stability for locally stabilizable systems in the absence of the delay. The authors in [1,3,33,35] investigated a novel modified particle swarm optimization (MPSO) algorithm to identify nonlinear and showed that the proposed algorithm was successful in tracking time-varying parameters systems.…”

Sampled-Data Stabilization of Nonlinear Delay Systems With a Compact Absorbing Set

“…Almost all available results rely on delay-dependent conditions for the existence of stabilizing feedback and in most cases the stability domain depends on the sampling interval/ delay. Predictive feedback seems to be the only possible choice for handling large delays (see [3,4,5,11,12,13,14,15,18,19,26]). Global stabilization of control systems with large delays by means of sampled-data output feedback with positive sampling rate remains a challenging problem.…”