Input-to-state Practical Stability Research Articles

A Hierarchical Data-Driven Predictive Control of Image-Based Visual Servoing Systems With Unknown Dynamics.

In this article, a hierarchical predictive control (PC) algorithm is designed for visual servoing mobile robot systems. At the kinematic level, the image-based visual servoing model of a wheeled mobile robot is established. By defining the corresponding performance index of the PC, an iterative linear quadratic regulator (iLQR) is used to obtain the velocity controller and to provide reference velocity for dynamics. In dynamics, a data-driven PC controller based on the Gaussian process (GP) is proposed to obtain the torque controller with unknown dynamics. The input-to-state practical stability (ISpS) of the system based on the proposed data-driven PC method is proved by introducing reasonable assumptions. The corresponding theorem also analyzes the maximum upper bound of GP inference error. Finally, the effectiveness of the proposed hierarchical controller is verified by simulations and experiments.

Learning From Human Demonstrations for Wheel Mobile Manipulator: An Unscented Model Predictive Control Approach.

Industry 4.0 requires new production models to be more flexible and efficient, which means that robots should be capable of flexible skills to adapt to different production and processing tasks. Learning from demonstration (LfD) is considered as one of the promising ways for robots to obtain motion and manipulation skills from humans. In this article, a framework that enables a wheel mobile manipulator to learn skills from humans and complete the specified tasks in an unstructured environment is developed, including a high-level trajectory learning and a low-level trajectory tracking control. First, a modified dynamic movement primitives (DMPs) model is utilized to simultaneously learn the movement trajectories of a human operator's hand and body as reference trajectories for the mobile manipulator. Considering that the auxiliary model obtained by the nonlinear feedback is hard to accurately describe the behavior of mobile manipulator with the presence of uncertain parameters and disturbances, a novel model is established, and an unscented model predictive control (UMPC) strategy is then presented to solve the trajectory tracking control problem without violating the system constraints. Moreover, a sufficient condition guaranteeing the input to state practical stability (ISpS) of the system is obtained, and the upper bound of estimated error is also defined. Finally, the effectiveness of the proposed strategy is validated by three simulation experiments.

Periodic Event-Triggered Robust Distributed Model Predictive Control for Multiagent Systems With Input and Communication Delays

This paper investigates the problem of event-triggered distributed model predictive control (DMPC) for continuous-time nonlinear multi-agent systems (MASs) subject to bounded disturbances, input and communication delays simultaneously. The compensation schemes for input and communication delays are proposed, respectively. A new optimal control problem (OCP) to achieve consensus among all agents is formulated under the proposed delay compensation schemes. To alleviate the communication burden and sensing cost, a novel input delay-related periodic event-triggering condition is proposed based on the state error to determine when to calculate the new control input. Sufficient condition guaranteeing the feasibility of the OCP is achieved with the proposed event-triggered control scheme. In stability analysis, a novel time-varying Lyapunov function is constructed, and the input-to-state practical stability (ISpS) of the MASs is derived by using the quasi-Lipschitz continuous property of the Lyapunov function. Numerical results related to vehicle platooning demonstrate the effectiveness of the proposed event-triggered DMPC.

Practical Input-to-State Stability of Nonlinear Systems with Time Delay

An investigation of the input-to-state practical stability (ISpS) and the integral input-to-state practical stability (iISpS) of nonlinear systems with time delays (NSWTDs) is presented in this paper. The ISpS and iISpS of the systems are obtained by using a continuously differentiable Lyapunov–Krasovskii functional (LKF) within-definite derivative, which generalizes the classical LKF with a positive definite derivative. As a result, the uniform practical asymptotic stability (UPAS) criteria of NSWTD were also given by using the proposed LKF.

Neuro-Based Consensus Seeking for Nonlinear Uncertainty Multi-Agent Systems Constrained by Dead-Zone Input

The topic about consensus target track seeking for high-order nonlinear multi-agent systems (MASs) with unmodeled dynamics, dynamic disturbances, and dead-zone input is considered in the paper. Using the strong nonlinear map characteristic of radial basis function neural networks (RBFNNs), the complex functions arising from recursive procedure are simplified. Also, inspired by input-to-state practical stability (ISpS), the authors construct a dynamical signal in order to counteract the impact of unmodeled dynamics and dynamic disturbances. The bounded inequality expression has been applied to tackle the unknown input of dead zone. Based on this, consensus control protocol suitable for nonlinear constraints has been constructed by using the recursive backstepping technique and adaptive neural network method. Theoretical analysis indicates not only the uniform boundary of all signals in the closed-loop under the neuro-based consensus controller, but uniform ultimate convergence of consensus tracking errors. The final simulations also confirmed the correctness of the theoretical analysis.

Stability properties of the adaptive horizon multi-stage MPC

This paper presents an adaptive horizon multi-stage model-predictive control (MPC) algorithm. It establishes appropriate criteria for recursive feasibility and robust stability using the theory of input-to-state practical stability (ISpS). The proposed algorithm employs parametric nonlinear programming (NLP) sensitivity and terminal ingredients to determine the minimum stabilizing prediction horizon for all the scenarios considered in the subsequent iterations of the multi-stage MPC. This technique notably decreases the computational cost in nonlinear model-predictive control systems with uncertainty, as they involve solving large and complex optimization problems. The efficacy of the controller is illustrated using three numerical examples that illustrate a reduction in computational delay in multi-stage MPC.

Input‐to‐state practical stability for nonautonomous nonlinear infinite‐dimensional systems

AbstractIn this article, we are concerned with the input‐to‐state practical stability (ISpS) of nonautonomous nonlinear infinite‐dimensional systems. Sufficient conditions of ISpS are provided based on Lyapunov functions and a nonlinear inequality. We characterize ISpS in terms of uniform practical asymptotic gain property. We show that for a class of admissible inputs the existence of an ISpS‐Lyapunov function implies the ISpS of a system in Banach spaces. These results are used to study the input‐to‐state practical stability of a certain class of nonautonomous semilinear evolution equations. Furthermore, we provide a small‐gain theorem for nonautonomous nonlinear interconnected systems. An example is used to illustrate the application of the developed method.

Input-to-state practical stability criteria of non-autonomous infinite-dimensional systems

We develop tools for the investigation of input-to-state practical stability (ISpS) and integral input-to-state practical stability (iISpS) of non-autonomous infinite-dimensional systems in Banach spaces. Sufficient conditions of ISpS and iISpS are given based on indefinite Lyapunov functions. The practical stability analysis is accomplished with the help of scalar practical stable functions. Then, a construction of ISpS Lyapunov function for a class of non-autonomous evolutions equations is provided in Hilbert spaces. We propose the ISpS Lyapunov methodology to make it suitable for the analysis of ISpS w.r.t. inputs from -spaces. Furthermore, we illustrate the theory with an example of a semi-linear reaction-diffusion equation.

Input‐to‐state practical stability analysis and controller design of switched affine systems: An observer‐based approach

AbstractThis paper addresses the problems of the input‐to‐state practical stability (ISPS) analysis and output feedback controller design for switched affine systems (SASs) subject to external disturbances. First, a switched affine observer is developed to estimate unmeasurable states. Then by combining the sampled‐data control approach, a less conservative mode‐dependent dynamic event‐triggered mechanism (ETM) is established. The proposed dynamic ETM cannot only avoid Zeno behavior but also reduce the network transmission burden effectively. Further, based on time‐dependent Lyapunov‐Krasovskii functional and state‐dependent switching laws, a set of feasible ISPS conditions are presented in the LMI forms by means of singular value decomposition. The designed switching law depends upon the sampled‐data information of the estimated state and gets rid of the chattering phenomenon. Finally, an application example of the DC‐DC flyback converter is given to verify the efficacy of the proposed algorithm.

An error gradient and accumulation‐type event‐driven model predictive control with relative thresholds for perturbed nonlinear systems

This work investigates the event-driven model predictive control problem for perturbed nonlinear systems with input and state constraints. Firstly, a new event-driven triggering condition, considering the gradient and accumulation of the error between the optimal and actual states at adjacent moments with a state-dependent relative threshold is designed with a guaranteed positive inter-event time to avoid Zeno behaviour. Secondly, an error gradient and accumulation type event-driven model predictive control framework is designed based on the new triggering condition and the dual-mode strategy, for effectively reducing the calculation and communication burden of the model predictive control controller while ensuring desired control performance. Furthermore, the feasibility and the input-to-state practical stability (ISpS) of gradient and accumulation-type event-driven model predictive control are rigorously guaranteed theoretically. At last, the simulation results are provided to validate the this algorithm.

Robust Min-Max Model Predictive Vehicle Platooning With Causal Disturbance Feedback

Platoon-based vehicular cyber-physical systems have gained increasing attention due to their potentials in improving traffic efficiency, capacity, and saving energy. However, external uncertain disturbances arising from mismatched model errors, sensor noises, communication delays and unknown environments can impose a great challenge on the constrained control of vehicle platooning. In this paper, we propose a closed-loop min-max model predictive control (MPC) with causal disturbance feedback for vehicle platooning. Specifically, we first develop a compact form of a centralized vehicle platooning model subject to external disturbances, which also incorporates the lower-level vehicle dynamics. We then formulate the uncertain optimal control of the vehicle platoon as a worst-case constrained optimization problem and derive its robust counterpart by semidefinite relaxation. Thus, we design a causal disturbance feedback structure with the robust counterpart, which leads to a closed-loop min-max MPC platoon control solution. Even though the min-max MPC follows a centralized paradigm, its robust counterpart can keep the convexity and enable the efficient and practical implementation of current convex optimization techniques. We also derive a linear matrix inequality (LMI) condition for guaranteeing the recursive feasibility and input-to-state practical stability (ISpS) of the platoon system. Finally, simulation results are provided to verify the effectiveness and advantage of the proposed MPC in terms of constraint satisfaction, platoon stability and robustness against different external disturbances.

A Gradient-Based Event-Driven MPC for Nonlinear Systems With Additive Disturbances Using State-Dependent Thresholds

In this article, we propose a gradient-based event-driven model predictive control (GEMPC) algorithm with a state-dependent threshold for nonlinear systems with additive disturbances and input and state constraints. Firstly, a novel gradient-based event-driven strategy is constructed in the light of the error gradient between the optimal prediction of the state and the real one, which could ensure the Zeno-free property via a positive triggering interval. Subsequently, the novel triggering mechanism and the dual-mode control are combined to establish a GEMPC framework, to further reduce the computing burden and communication transmission especially when the computational resources are limited. Additionally, the feasibility of the GEMPC algorithm and the input-to-state practical stability (ISpS) property of the considered system have been strictly proved in theory. Finally, the simulation comparison results on control of a perturbed nonlinear system are utilized to show the validity of the GEMPC algorithm.

Small-Gain Technique-Based Adaptive Neural Output-Feedback Fault-Tolerant Control of Switched Nonlinear Systems With Unmodeled Dynamics

In this article, the issue of adaptive neural fault-tolerant control (FTC) is addressed for a class of uncertain switched nonstrict-feedback nonlinear systems with unmodeled dynamics and unmeasurable states. In such a system, the uncertain nonlinear parts are identified by radial basis function (RBF) neural networks (NNs). Also, with the help of the structural characteristics of RBF NNs, the violation between the nontsrict-feedback form and backstepping method is tackled. Then, based on the small-gain technique, input-to-state practical stability (ISpS) theory, and common Lyapunov function (CLF) approach, an adaptive fault-tolerant tracking controller with only three adaptive laws is developed by designing an observer. It is shown that the designed controller can ensure that all the closed-loop signals are bounded under arbitrary switching, while the tracking error can converge to a small area of the origin. Finally, two simulation examples are provided to demonstrate the feasibility of the suggested control approach.

Event-triggered adaptive neural networks control for fractional-order nonstrict-feedback nonlinear systems with unmodeled dynamics and input saturation

The event-triggered adaptive neural networks control is investigated in this paper for a class of fractional-order systems (FOSs) with unmodeled dynamics and input saturation. Firstly, in order to obtain an auxiliary signal and then avoid the state variables of unmodeled dynamics directly appearing in the designed controller, the notion of exponential input-to-state practical stability (ISpS) and some related lemmas for integer-order systems are extended to the ones for FOSs. Then, based on the traditional event-triggered mechanism, we propose a novel adaptive event-triggered mechanism (AETM) in this paper, in which the threshold parameters can be adjusted dynamically according to the tracking performance. Besides, different from the previous works where the derivative of hyperbolic tangent function tanh(⋅) needs to have positive lower bound, a new type of auxiliary signal is introduced in this paper to handle the effect of input saturation and thus this limitation is released. Finally, two numerical examples and some comparisons are provided to illustrate our proposed controllers.

Input-to-state stability of the nonlinear singular systems via small-gain theorem

This paper presents the input-to-state stability (ISS) of nonlinear singular systems (NSSs) based on the small-gain theorem of singular system. The major contribution of this paper includes three aspects. Firstly, give and prove the existence and unique conditions for the solution of NSS based on the fast-slow subsystem. Secondly, the conditions of ISS property for the NSSs are obtained via the small-gain theorem, and a method for constructing ISS-Lyapunov function of NSSs is presented. Thirdly, the theorems for input-to-state practical stability (ISpS) and the trajectory-based small-gain theorem of interconnected NSSs are proposed. Finally, the effectiveness of proposed method is illustrated by using a circuit network model and a numerical example.

Stabilization of Perturbed Continuous-Time Systems Using Event-Triggered Model Predictive Control.

In this article, event-triggered model predictive control (EMPC) of continuous-time nonlinear systems with bounded disturbances is studied. Two novel event-triggered control schemes are proposed. In the first strategy, an event-triggering condition, designed based on the state error between the actual system state and the optimal one, with an absolute threshold is considered. In the second strategy, an event-triggering condition with a mixed threshold is designed to further save the computational resources. The minimal interevent times of both event-triggered control schemes are obtained to avoid the Zeno behavior. Sufficient conditions of recursive feasibility for these two triggering strategies, which refer to the prediction horizon, the triggering level, and the disturbance bound, are obtained, respectively. Input-to-state practical stability (ISpS) of both event-triggered control systems is established without requiring the system state entering the terminal set in finite time, respectively. Finally, the numerical simulation shows the effectiveness of the proposed methods.

Dissipativity-Based Sliding-Mode Control of Cyber-Physical Systems Under Denial-of-Service Attacks.

In this article, we investigate the problem of the dissipativity-based resilient sliding-mode control design of cyber-physical systems with the occurrence of denial-of-service (DoS) attacks. First, we analyze the physical layer operating without DoS attacks to ensure the input-to-state practical stability (ISpS). The upper bound of the sample-data rate in this situation can be identified synchronously. Next, for systems under DoS attacks, we present the following results: 1) combined with reasonable hypotheses of DoS attacks, the ISpS as well as dissipativity of the underlying system can be guaranteed; 2) the upper bound of the sample-data rate in the presence of DoS attacks can be derived; and 3) the sliding-mode controller is synthesized to achieve the desired goals in a finite time. Finally, two examples are given to illustrate the applicability of our theoretical derivation.

RMPC for Uncertain Nonlinear Systems With Non-Additive Dynamic Disturbances and Noisy Measurements

In this paper, we present a robust, model-predictive control scheme for the general class of uncertain and constrained discrete-time nonlinear systems subject to noisy measurements. The relationships between the system's dynamics, uncertainties, disturbances and the measurement noise are nonlinear and not necessarily additive. In particular, the disturbance is the output of an uncertain system with an unknown input. This study serves the threefold ultimate objective of ensuring robust satisfaction of the state constraints, recursive feasibility and stability. To satisfy state constraints, the proposed algorithms adopt a constraints tightening approach using the restricted constraint sets computed online. Several bounds on the prediction level and rate are derived and the size of the terminal region is maximized using polytopic linear differential inclusions (PLDI). An explicit bound on the maximum allowable disturbance for recursive feasibility is also derived based on optimization of the one-step ahead controllable set to the terminal region. The disturbance and uncertainties are non-vanishing and therefore only Input-to-state practical stability (ISpS) can be ensured. A simulation example demonstrates the efficacy of the mathematical framework and algorithms developed in this work.

State Measurement Error-to-State Stability Results Based on Approximate Discrete-Time Models

Digital controller design for nonlinear systems may be complicated by the fact that an exact discrete-time plant model is not known. One existing approach employs approximate discrete-time models for stability analysis and control design and ensures different types of closed-loop stability properties based on the approximate model and on specific bounds on the mismatch between the exact and approximate models. Although existing conditions for practical stability exist, some of which consider the presence of process disturbances, input-to-state stability (ISS) with respect to state-measurement errors and based on approximate discrete-time models has not been addressed. In this paper, we thus extend existing results in two main directions: 1) we provide ISS-related results, where the input is the state measurement error; and 2) our results allow for some specific varying-sampling-rate scenarios. We provide conditions to ensure semiglobal practical ISS, even under some specific forms of varying sampling rate. These conditions employ Lyapunov-like functions. We illustrate the application of our results on numerical examples, where we show that a bounded state-measurement error can cause a semiglobal practically stable system to diverge.

Adaptive fuzzy reliable control for switched uncertain nonlinear systems based on closed-loop reference model

This paper investigates an input-to-state practical stability (ISpS) problem for a class of switched strict-feedback nonlinear systems with closed-loop reference model. The considered systems are with unmodeled dynamics, dynamic disturbances and actuator failures. Fuzzy logical systems (FLSs) are utilized to approximate the unknown functions. Different from the open-loop model reference adaptive control, a feedback gain term is introduced into the reference model to improve the transient performance of the considered systems especially when actuator failures occur and switchings take place. By combining the small-gain approach and backstepping technique, an adaptive fuzzy control method is proposed. It is shown that under the developed control scheme, the tracking error converges to a small neighborhood of the origin, while all the signals of the closed-loop system are semi-globally uniformly ultimately bounded (SGUUB). Simulation study is given to illustrate the effectiveness of our results.