Input-to-state Stability Research Articles

Event-Triggered Prescribed Performance Fuzzy Fault-Tolerant Control for Unknown Euler–Lagrange Systems With Any Bounded Initial Values

This paper investigates the tracking problem of event-triggered prescribed performance fuzzy fault-tolerant control (FTC) for unknown Euler-Lagrange systems with actuator faults and external disturbances. Firstly, the barrier Lyapunov functions (BLFs) and prescribed performance functions are synthesized to guarantee that the tracking errors satisfy the preset transient performance. Different from existing prescribed performance control methods, which require the initial values of the tracking errors to be within the prescribed performance functions, an error transformation method is introduced to ensure that the tracking errors with any bounded initial values can enter the preset boundaries within a preset time. Then, considering the unavailability of system parameters, the fuzzy logic systems (FLSs) are used to approximate unknown parameters of the system. What's more, to solve the problem of limited communication and computing resources in practical systems, an improved event-triggered control (ETC) scheme is proposed, which can reduce the communication and computation burden without satisfying the input-to-state stability (ISS) assumption. Meanwhile, the Zeno phenomenon can be avoided. Furthermore, the effects of actuator faults and the event-triggered mechanism are handled by Nussbaum gain technology. Finally, the superiority of the proposed control algorithm is verified by simulation results.

Higher Order Barrier Certificates for Leader–Follower Multiagent Systems

This paper presents control strategies based on time-varying convergent higher order control barrier functions for a class of leader-follower multi-agent systems under signal temporal logic (STL) tasks. Each agent is assigned a local STL task which may be dependent on the behavior of agents involved in other tasks. We consider one or more than one leader in the multi-agent system. The leader has knowledge on the associated tasks and controls the performance of the subgroup involved agents. The followers are not aware of the tasks, and do not have any control authority to reach them. They follow the leader commands indirectly, according to their dynamics interconnections, for the task satisfaction. We further assume that the input-to-state stability (ISS) property for the multi-agent system is fulfilled. First, robust solutions for the task satisfaction, based on the leader's accessibility to the follower agents' states are suggested. In addition, using the notion of higher order barrier functions, individual barrier certificates for each agent evolving in a formation dynamic structure are proposed. For the case of presence of more leaders in the subgroups, we provide decentralized barrier certificates. Our approach finds solutions to guarantee the satisfaction of STL tasks independent of the agents' initial conditions.

Input-to-state stability of positive delayed neural networks via impulsive control

This paper is concerned with the positivity and impulsive stabilization of equilibrium points of delayed neural networks (DNNs) subject to bounded disturbances. With the aid of the continuous dependence theorem for impulsive delay differential equations, a relaxed positivity condition is derived, which allows the neuron interconnection matrix to be Metzler if the activation functions satisfy a certain condition. The notion of input-to-state stability (ISS) is introduced to characterize internal global stability and disturbance attenuation performance for impulsively controlled DNNs. The ISS property is analyzed by employing a time-dependent max-separable Lyapunov function which is able to capture the positivity characterization and hybrid structure of the considered DNNs. A ranged dwell-time-dependent ISS condition is obtained, which allows to design an impulsive control law via partial state variables. As a byproduct, an improved global exponential stability criterion for impulse-free positive DNNs is obtained. The applicability of the achieved results is illustrated through three numerical examples.

Input-mapping based data-driven model predictive control for unknown linear systems with bounded disturbances

The data-driven model predictive control (MPC) approach has been an effective tool for unknown constrained systems. However, most of the existing designs rely on the prior collected data sequence through offline trials, which may be affected by time-varying disturbances, or the online estimated system model with the non-trivial computation cost. This limits the applications of these designs. To alleviate these restrictions, in this paper, an input-mapping data-driven scheme is developed. This scheme online directly maps the future control policy and the predicted state to the past online noisy input/state data which are updated once new data come. This overcomes the limitations of previous designs. To ensure the system constraints, this scheme is combined with a tube MPC method and the proposed input-mapping data-driven tube MPC can guarantee the recursive feasibility and the input-to state stability. Two examples illustrate the advantages of the proposed method.

Cyber-Physical System-Based Path Tracking Control of Autonomous Vehicles Under Cyber-Attacks

As one of the typical applications of cyber-physical system (CPS), autonomous vehicles (AVs) are vulnerable to malicious disturbance from cyber-attacks while tracking the desired path. This paper focuses on CPS-based path tracking control problem of AVs under cyber-attacks. Firstly, the nonlinear state and measurement equations of AVs under cyber-attacks are established based on vehicle dynamics model. Secondly, to improve the robustness of AVs against cyber-attacks, sensor redundancy is introduced. A cyber-attack detection method is designed by using extended Kalman filter (EKF), and the computational complexity of the cyber-attack detection method is analyzed. In addition, sensor switching rules are developed to isolate the disturbance of cyber-attacks. Then, the control problem of the AVs is formulated based on model predictive control (MPC). Input-to-state stability (ISS) of the control system under cyber-attacks is established, and a link between the tolerable attack intensity and the detection thresholds is clearly revealed. Finally, the simulation results demonstrate the effectiveness of the proposed control strategy.

Virtual Sensor: Simultaneous State and Input Estimation for Nonlinear Interconnected Ground Vehicle System Dynamics.

This paper proposes a new observer approach used to simultaneously estimate both vehicle lateral and longitudinal nonlinear dynamics, as well as their unknown inputs. Based on cascade observers, this robust virtual sensor is able to more precisely estimate not only the vehicle state but also human driver external inputs and road attributes, including acceleration and brake pedal forces, steering torque, and road curvature. To overcome the observability and the interconnection issues related to the vehicle dynamics coupling characteristics, tire effort nonlinearities, and the tire-ground contact behavior during braking and acceleration, the linear-parameter-varying (LPV) interconnected unknown inputs observer (UIO) framework was used. This interconnection scheme of the proposed observer allows us to reduce the level of numerical complexity and conservatism. To deal with the nonlinearities related to the unmeasurable real-time variation in the vehicle longitudinal speed and tire slip velocities in front and rear wheels, the Takagi-Sugeno (T-S) fuzzy form was undertaken for the observer design. The input-to-state stability (ISS) of the estimation errors was exploited using Lyapunov stability arguments to allow for more relaxation and an additional robustness guarantee with respect to the disturbance term of unmeasurable nonlinearities. For the design of the LPV interconnected UIO, sufficient conditions of the ISS property were formulated as an optimization problem in terms of linear matrix inequalities (LMIs), which can be effectively solved with numerical solvers. Extensive experiments were carried out under various driving test scenarios, both in interactive simulations performed with the well-known Sherpa dynamic driving simulator, and then using the LAMIH Twingo vehicle prototype, in order to highlight the effectiveness and the validity of the proposed observer design.

Control Lyapunov function method for robust stabilization of multistable affine nonlinear systems

AbstractIn this paper, we study the problem of robust stabilization of affine nonlinear multistable systems in the presence of exogenous disturbances. The results are based on the theory of input‐to‐state stability (ISS) and integral input‐to‐state stability (iISS) for systems with multiple invariant sets. The notions of ISS and iISS control Lyapunov functions (CLFs) and the small control property are extended within the multistability framework. Such properties are also complemented by the concept of a weak iISS CLF and corresponding small control property. It is verified that the universal control formula can be applied to yield the ISS (iISS) property for the closed‐loop system. The efficiency of the extended CLF framework in the multistable sense is illustrated for a Duffing system and in application to a noise‐induced transition in a semiconductor‐gas‐discharge gap system.

Distance-Based Elliptical Circumnavigation Control for Non-Holonomic Robots With Event-Triggered Unknown System Dynamics Estimators

How to efficiently collect multi-dimensional information of traffic accidents with high dangers has triggered intensive concerns in intelligent transportation community. This paper provides a viable and continuous observation solution using low-budgeted mobile robots. At the kinematic level, a relative position estimator utilizing available distance data is constructed with an exponential error convergence, removing the dependency of using global position. To proceed, resorting to estimation results, an elliptical encircling guidance rule with a time-varying radius is established to preserve circumnavigation concerning targets. At the kinetic level, a new fuel-saving uncertainty mitigation scheme, i.e., event-triggered unknown system dynamics estimators (ETUSDEs) with the feature of a reduced transmission load and a concise structure are respectively developed in velocity and angular rate subsystems to reconstruct uncertainties with a prescribed decaying rate, where event-triggering conditions are enforced to schedule updating frequency of actuators and measurements in an aperiodic manner instead of a fixed time interval. Then, an event-triggered robust kinetic control protocol is synthesized to achieve an accurate command tracking without incurring Zeno behaviors. Finally, the convergence of entire system is illustrated through input-to-state stable (ISS) criterion. Simulation results are delivered to testify the effectiveness of proposed method.

Stability of networked nonlinear systems: Generalization of small-gain theorem and distributed testing

This paper studies the stability problem for networked control systems. A general result is presented to determine either global uniform boundedness (GUB), global asymptotic stability (GAS) or input-to-state stability (ISS), for interconnected nonlinear systems. This result checks stability in terms of a scalar called the network gain, hence we call the result the network gain theorem. The result generalizes the previously known matrix small-gain theorem and cyclic small-gain theorem for ISS. As in these results, our theorem does not assess the stability of a given networked system, but of a whole family of networked systems satisfying certain common assumption. An advantage of our stability condition is that it is not only sufficient, but also necessary, in the sense that, if not met, there exists an unstable networked system within that family. To complement our theoretical result, we propose a fully distributed algorithm to compute the network gain. We present simulation results to illustrate the proposed algorithm.

A controller management scheme for active fault-tolerant tracking: Linear discrete-time systems

This paper proposes an active fault-tolerant control (FTC) approach based on the controller management and virtual actuator idea for linear discrete-time systems subject to unknown L2-bounded disturbances, input constraint, and time-varying additive actuator faults. The closed-loop faulty system, which includes the modified nominal controller, the fault and state estimator, and the virtual actuator, suppresses the effects of disturbances and faults, while ensuring input-constraint satisfaction. The management of the nominal controller is performed through an online optimization method – in the form of a standard quadratic programming problem – by manipulating the reference input and intervening in the nominal controller evolution. The proposed method proves the input-to-state stability (ISS) criterion of the overall closed-loop faulty system. The problem of minimizing the ultimate bound of the ISS criterion is formulated in terms of tractable linear matrix inequality (LMI) conditions that allow the fault and state estimation errors to converge to a small neighborhood of the origin. To illustrate the capabilities and advantages of the proposed control strategy, comparative simulation results are presented for a flexible joint robotic system, tracking control of a DC motor’s angular velocity, and the multivariable VTOL aircraft.

Lie-brackets-based averaging of affine systems via a time-delay approach

In this paper, we study input-to-state stability (ISS) of affine systems with a small parameter ɛ>0 and additive disturbances in the presence of state-delays. We present a time-delay approach to Lie-brackets-based averaging, where we transform the system to a time-delay (neutral type) one. The latter has a form of perturbed Lie brackets system. The ISS of the time-delay system guarantees the same for the original one. We present a direct Lyapunov–Krasovskii (L–K) method for the time-delay system and provide sufficient conditions for regional ISS. Further we apply the results to stabilization of linear uncertain systems under unknown control directions using the bounded extremum seeking controller with measurement delay. In contrast to the existing results that are all qualitative, we derive constructive linear matrix inequalities for finding quantitative upper bounds on ɛ and the time-delay that ensure regional ISS of the original system and on the resulting ultimate bound. Numerical examples illustrate the efficiency of our method.

Exponential input-to-state stability for neutral stochastic delay differential equations with Lévy noise and Markovian switching

This paper mainly focuses on the pth ()-moment input-to-state stability (ISS) of neutral stochastic delay differential equations (NSDDEs) with Lévy noise and Markovian switching. By using the generalized integral inequality and the Lyapunov function methodology, the ISS, integral input-to-state stability (iISS), and stochastic input-to-state stability (SISS) of such equations are obtained. When the input signal is a constant signal and a zero signal, the pth ()-moment ISS reduces to the pth ()-moment practical exponential stability and the pth ()-moment exponential stability, respectively. Finally, an example of the mass–spring–damping (MSD) model under the stochastic perturbation is given to verify the validity of the results.

Input-to-State Stability for Time-Delay Systems With Large Delays

In this article, we consider the input-to-state stability (ISS) problem for a class of time-delay systems with intermittent large delays, which may cause the invalidation of traditional delay-dependent stability criteria. The topic of this article features that it proposes a novel kind of stability criterion for time-delay systems, which is delay dependent if the time delay is smaller than a prescribed allowable size. While if the time delay is larger than the allowable size, the ISS can be preserved as well provided that the large-delay periods satisfy the kind of duration condition. Different from existing results on similar topics, we present the main result based on a unified Lyapunov-Krasovskii function (LKF). In this way, the frequency restriction can be removed and the analysis complexity can be simplified. A numerical example is provided to verify the proposed results.

Input-to-State Stability of Time-Delay Systems With Hybrid Impulses and Continuous Subdynamics Based on Vector Lyapunov Function

This article focuses on input-to-state stability (ISS) of impulsive time-delay systems where the hybrid effect of impulses with time-dependent multiple jump maps in different subsystems is fully considered. By virtue of M-matrix and vector Lyapunov function, some theorems for ISS are established for Krasovskii-type conditions which avoid the common threshold of impulses in subsystems and allow the simultaneous existence of stable and unstable continuous subdynamics. If the time intervals between impulses are bounded, the ISS of time-delay systems can be ensured even though hybrid impulses exist in each subsystem, which determines the robustness of stable time-delay systems with respect to hybrid impulses. The unstable time-delay systems with different subdynamics can be stabilized in input-to-state-stability sense by involving hybrid impulses. In addition, several extended criteria are presented to bridge the results derived via vector Lyapunov function, scalar Lyapunov function, and comparison principle. At last, the theoretical results are validated by numerical example and a simplified model of sewage treatment tank.

A Lyapunov-Based ISS Small-Gain Theorem for Infinite Networks of Nonlinear Systems

In this article, we show that an infinite network of input-to-state stable (ISS) subsystems, admitting ISS Lyapunov functions, itself admits an ISS Lyapunov function, provided that the couplings between the subsystems are sufficiently weak. The strength of the couplings is described in terms of the properties of an infinite-dimensional nonlinear positive operator, built from the interconnection gains. If this operator induces a uniformly globally asymptotically stable (UGAS) system, a Lyapunov function for the infinite network can be constructed. We analyze necessary and sufficient conditions for UGAS and relate them to small-gain conditions used in the stability analysis of finite networks.

Distributed Economic MPC for Dynamically Coupled Linear Systems: A Lyapunov-Based Approach

This article develops a distributed economic model predictive control (EMPC) method which is applied in a group of interconnected linear subsystems subject to unknown bounded disturbances. Multiple subsystems are coupled through the dynamics, and the control objective is to optimize some general performance criteria of the whole system which may take economic considerations into account. First, a two operation modes EMPC optimization problem is formulated, which incorporates the constraints derived from the Lyapunov technique. In the first mode, each subsystem focuses on the optimization of the economic performance while maintaining the state in a certain region. In the second mode, the system states are steered to a neighborhood of a steady state by making use of the Lyapunov-based constraints. Furthermore, a consensus alternating direction method of multipliers (ADMM) is adopted to solve the model predictive control optimization problems with a coupled predicted model constraint in a distributed way. By introducing consensus constraints, the resulting local optimization problem does not depend on real-time optimal solutions from neighboring subsystems and allows subsystems to solve it in parallel. Moreover, the closed-loop system is ensured to be input-to-state stable (ISS) with respect to the disturbances. To demonstrate the effectiveness of the algorithm, we conduct numerical simulations on a thermal power interconnected system.

Adaptive Neural Network Control Using Nonlinear Information Gain for Permanent Magnet Synchronous Motors.

In this study, an adaptive neural network (NN) control using nonlinear information (NI) gain for permanent magnet synchronous motors (PMSMs) is proposed to improve control and estimation performance. The proposed method consists of a nonlinear controller, a three-layer NN approximator, and NI gain. The nonlinear controller is designed via a backstepping procedure for position tracking. The commutation scheme is designed to implement the PMSM control without the direct-quadrature (DQ) transform. The three-layer NN approximator is designed to estimate the unknown complex function generated by the recursive backstepping process. The NI gains are designed to enhance the control and estimation performance according to the increased tracking errors owing to the load torque and the desired position variations. All of signals in the closed-loop system guarantee the semiglobal uniformly ultimately boundness (UUB) using the Lyapunov stability theorem and the input-to-state stability (ISS) property. The performance of the proposed method was validated by experiments performed using a PMSM testbed.

Distributed Event-Triggered Formation Control of Planar Vehicles Without Velocity Sensors

In this paper, an event-triggered distributed formation control method is proposed for multiple planar vehicles which are modeled as uncertain Euler-Lagrangian systems (ELSs). Firstly, a position and orientation (PAO)-based event-triggered mechanism is designed to decrease the frequency of data transmission through the wireless network. According to the PAO information at the event-triggered time, an event-triggered high-gain extended state observer (EHGESO) is developed to remove the demand of velocity information in conventional backstepping technique and eliminate the requirement of continuous PAO information in traditional HGESO design. Under the developed EHGESO and the PAO-based event-triggered mechanism, planar vehicles can cooperatively move along the parameterized path, and can transmit information with their neighbor vehicles only if the designed event-triggered mechanism is violated. The stability is proved by employing input-to-state stability (ISS) and semi-global uniformly ultimately bounded (SGUUB) stability theory. Eventually, a simulation of three-degree-of-freedom surface vehicle models is offered to illustrate the effectiveness of the developed control method.

Data Driven prediction of forced nonlinear vibrations using stabilised Autoregressive Neural Networks

AbstractIn this work, we propose a novel approach to the data‐driven prediction of vibration responses of nonlinear systems. The main idea is based on Autoregressive Neural Networks (ARNN) to model the nonlinear transfer behaviour between an external excitation and the system response. We propose an autoregressive network architecture with embedded symmetry using bias‐free tanh activation and guarantee Input‐to‐State‐Stability (ISS) by enforcing a special penalty term to the weights. The resulting training procedure is analysed for the example of a DUFFING oscillator with white noise excitation. In a BAYESian optimisation, it is found that beyond enforcing input‐to‐state‐stability, the stabilising penalty term also decreases sensitivity with respect to other training parameters compared to other classical techniques. Furthermore, we show that the stabilised ARNN is able to give excellent approximations of the nonlinear response of the DUFFING oscillator for a wide range of excitation intensities. In contrast, linear models, such as autoregressive models with exogenous input (ARX) in time domain or linear transfer functions in frequency domain, will only find some linear approximation. In particular, by construction, they will not be able to capture nonlinear effects for arbitrary amplitudes and excitation levels.

High‐order control barrier functions‐based optimization control for time‐varying nonlinear systems with full‐state constraints: A dynamic sub‐safe set approach

SummaryThis study focuses on the problems of safety control and stabilization for time‐varying nonlinear systems in the presence of time‐varying full‐state constraints. High‐order control barrier functions (HoCBFs) without the assumption of forward completeness are introduced to prevent high‐relative degree constraint violations. The activated HoCBFs, which determine the dynamic sub‐safe sets, are proposed to indicate that the state constraints are already or about to be violated. Subsequently, the HoCBFs‐based controller is designed to obtain the admissible control set by utilizing the quadratic programming (QP) with activated HoCBFs conditions to synthesize the nominal controller. An explicit analytical expression is provided for the HoCBFs‐based controller. The input‐to‐state stable (ISS) property of the closed‐loop system under the proposed controller is investigated to ensure that all closed‐loop states remain bounded and achieve asymptotic tracking without violation of the constraint. Moreover, a Lyapunov‐like constraint is added to the HoCBFs‐based QP problem such that the solvability of the QP problem is a sufficient condition for the system ISS. The effectiveness of the proposed HoCBFs‐based control method for time‐varying nonlinear systems is illustrated using a numerical example.