Predictor-feedback stabilization of multi-input nonlinear systems

Constrained Stabilization of Multi-Input Linear Systems with Distinct Input Delays

The stabilization of nonlinear stochastic systems with input and output delays is investigated in this article. First, an observer that includes input delay information is designed to estimate the state information after eliminating the effect of input delay. Second, a predictive scheme with delay information is utilized to predict the system state affected by output delay, and an event‐triggered predictive controller is established to eliminate the effects of input and output delays simultaneously. Subsequently, the convergence of the prediction error is analyzed by utilizing a recursive method, and a sufficient condition to ensure the global exponential stability is given by constructing a new closed‐loop system. Finally, numerical simulations of two nonlinear stochastic systems with different delay sizes are utilized to verify the impact of the event‐triggered predictive control in addressing the stabilization of nonlinear stochastic systems with time delays.

We consider the problem of enforcing safety in multi‐input systems with distinct input delays via the use of Control Barrier Functions (CBFs). For systems with input delay(s), a popular approach for enforcing safety is the combination of a CBF designed for the delay‐free system with state‐predictors that compensate the input delays. Typically, this comes with the assumption that the system does not violate safety constraints before all input delays have been compensated and the system is fully under control. In this paper, we introduce two control approaches that enforce safety before all input delays have been compensated, whenever it is possible to do so. We do this by utilizing a robust CBF formulation that treats longer‐delayed inputs as known disturbances when determining control effort for shorter delayed inputs. This formulation ensures that, whenever possible, a subset of input channels with shorter delays will be utilized for keeping the system in the admissible safe set before longer input delays have been compensated. The effectiveness of our approaches is demonstrated with two numerical examples.

This paper addresses the design and analysis of a multivariable gradient-based stochastic extremum-seeking control method for multi-input systems with arbitrary input delays. The approach accommodates systems with distinct time delays across input channels. It achieves local exponential stability of the closed-loop system, guaranteeing convergence to a small neighborhood around the extremum point. By incorporating phase compensation for dither signals and a novel predictor- feedback mechanism with averaging-based estimates of the unknown gradient and Hessian, the proposed method overcomes traditional challenges associated with arbitrary, distinct input delays. Unlike previous work on deterministic multiparameter extremum-seeking with distinct input delays, this stability analysis is achieved without using backstepping transformations, simplifying the predictor design and enabling a more straightforward implementation. Specifically, the direct application of Artstein's reduction approach yields delay- and system-dimension-independent convergence rates, thus enhancing its practical applicability. A numerical example and a source-seeking application illustrate the robust performance and advantages of the proposed delay-compensated stochastic extremum-seeking method.

Predictor-based stabilisation results are provided for nonlinear systems with input delays and a compact absorbing set. The control scheme consists of an inter-sample predictor, a global observer, an approximate predictor, and a nominal controller for the delay-free case. The control scheme is applicable even to the case where the measurement is sampled and possibly delayed. The input and measurement delays can be arbitrarily large but both of them must be constant and accurately known. The closed-loop system is shown to have the properties of global asymptotic stability and exponential convergence in the disturbance-free case, robustness with respect to perturbations of the sampling schedule, and robustness with respect to measurement errors. In contrast to existing predictor feedback laws, the proposed control scheme utilises an approximate predictor of a dynamic type that is expressed by a system described by integral delay equations. Additional results are provided for systems that can be transformed to systems with a compact absorbing set by means of a preliminary predictor feedback.

SummaryThis paper is concerned with the stabilization of linear systems with both state and distinct input delays. Nested predictor feedback controllers are designed to predict the future states such that the distinct input delays that can be arbitrarily large yet bounded are compensated completely. It is shown that the compensated closed‐loop system possesses the same characteristic equation as the closed‐loop system without distinct input delays. Both continuous‐time and discrete‐time time‐delay systems are studied in this paper. Moreover, the safe implementation problem for the continuous‐time nested predictor feedback controller is solved via adding input filters. Three numerical examples show the effectiveness of the proposed approaches.

In this paper, we address the design and analysis of multivariable extremum seeking for unknown static maps subject to arbitrarily long time delays. Gradient-based method is considered. Multi-input systems with different time delays in each individual input channel are dealt with. The phase compensation of the dither signals and the inclusion of predictor feedback with a perturbation-based (averaging-based) estimate of the Hessian allow to obtain local exponential convergence results to a small neighborhood of the optimal point, even in the presence of delays. Unlike previous publications considering multiparameter extremum seeking and delays, the stability analysis is carried without using backstepping transformation, which also eliminates the complexity of the controller. In a nutshell, a simpler implementation scheme and direct analysis without invoking successive backstepping transformation can be assured. In particular, the delays in our approach are independent of the dither frequency and system's dimension such that fast convergence rates are still guaranteed. A numerical example illustrates the performance of the new delay-compensated extremum seeking scheme and its simplicity.

The Sixth International Symposium on Neural Networks (ISNN 2009)

Based on the working principle of a ship with a controllable pitch propeller (CPP), a mathematical model of the ship-propeller subsystem, the engine-propeller subsystem, and the ship main engine subsystem was established. The model was represented by a set of differential equations, in which the ship speed, the rotational speed of the propeller, and the rotational speed of the main engine were used as the state variables, and the angle of pitch and the fuel injection quantity were used as the control input. The equation set was a multi-input nonlinear speed control system with the characteristics of coupling and nonlinearity. In order to design a sliding mode variable structure (SMVS) controller, appropriate coordinates were selected for the nonlinear state transformation, and the multi-input nonlinear system was decomposed into two single-input systems to solve the equivalent control through the manifold calculation. MATLAB simulation results showed that under the control of the angle of pitch and the fuel injection quantity, the ship speed, the rotational speed of the propeller, and the rotational speed of the main engine could quickly approach the given values, verifying the advantages of the SMVS control for complex multi-input nonlinear systems.

In this brief, we investigate the control of linear systems with input delay with applications to active magnetic bearings (AMBs). This study is motivated by emerging energy explorations that require compressors and other rotating machines to operate in remote locations. The separation between the machine and the control site in these applications, which may extend for several kilometers, introduces a time delay in the transmission of the control signals. Here, we study the effect of input delay in the control of AMBs in remote rotating machines and we present a truncated predictor feedback (TPF) control design to maximize the input delay that the stable closed-loop system can tolerate. The controller corresponding to the maximum input delay is obtained iteratively from the solutions to local linear matrix inequality problems. The capabilities of the designed TPF controller are first verified through simulation and then experimentally validated on an AMB test rig.

New results of global robust exponential stability of neural networks with delays

In this paper, the global exponential stability and stabilization problems for a class of nonlinear systems are investigated. Some sufficient conditions to guarantee global exponential stable and estimate the minimum admissible value of the control width are presented in virtue of time-dependent width Lyapunov functions. Furthermore, a periodically intermittent smooth controller with variant control width is introduced and theoretical analysis is provided. The smooth index function of periodically intermittent smooth control inputs is defined and the supremum (or least upper bound) of smooth index function set can be solved. On the basis of the analysis, the designed periodically intermittent smooth controller not only can globally exponentially stabilize the nonlinear systems, but also can control the exponential convergence rate of the nonlinear systems. Finally, numerical simulations are given to verify the obtained theoretical results.

<abstract><p>This paper studies the global asymptotic stabilization problem of strict-feedforward nonlinear systems with state and input delays. We will first transform the considered system into an equivalent system by constructing the novel parameter-dependent state feedback controller and introducing the appropriate coordinate transformation. After that, the global asymptotic stability of the closed system is proved by giving the proper Lyapunov-Krasovskii functional and using the stability criterion of time-delay system.</p></abstract>

In this article, the adaptive fixed‐time prescribed performance (FTPP) regulation is investigated for a class of time‐varying state constrained switched stochastic systems with input delay. The time‐varying barrier Lyapunov function and a compensation system are presented, respectively, to deal with the design problems caused by the existence of both time‐varying state constraints and input delay. Some radial basis function neural networks are used to approximate unknown functions, and the common Lyapunov function method is displayed to handle the switched signals. Besides, by designing a fixed‐time prescribed performance function, the desired adaptive neural controller is constructed. Compared with the existing works for state constrained control problem, the FTPP regulation control scheme is first proposed for time‐varying state constrained stochastic switched systems under input delay, and the adaptive dynamic surface control scheme with the nonlinear filter is designed to solve the problem of “explosion of complexity.” Based on the stochastic stability theory, the FTPP of system output is achieved, other system state variables are restricted in the predefined regions, and all signals of this closed‐loop system remain bounded in probability. Finally, the availability of the proposed control scheme is illustrated via two simulation examples.

This brief addresses an output feedback control design for Lipschitz nonlinear systems in the presence of input delay. A nonlinear observer is introduced to estimate the system states. A truncated predictor state feedback, which is the conventional predictor state feedback law with the distributed term dropped, is implemented with the estimated states. Lyapunov-Krasovskii functionals are constructed to establish conditions under which this observer-based truncated predictor feedback law asymptotically stabilize the system at the origin. The effectiveness of the proposed method is demonstrated on the Chua's circuit.