Input-to-state Stability Research Articles

Periodic Averaging of Discrete-Time Systems: A Time-Delay Approach

This article is concerned with the stability of discrete-time systems with fast-varying coefficients that may be uncertain. Recently, a constructive time-delay approach to averaging was proposed for continuous-time systems. In the present article, we develop, for the first time, this approach to discrete-time case. We first transform the system to a time-delay system with the delay being the period of averaging, which can be regarded as a perturbation of the classical averaged system. The stability of the original system can be guaranteed by the resulting time-delay system. Then under assumption of the classical averaged system being exponentially stable, we derive sufficient stability conditions for the resulting time-delay system, and find a quantitative upper bound on the small parameter that ensures the exponential stability. Moreover, we extend our method to input-to-state stability (ISS) analysis of the perturbed systems. Finally, we apply the approach to the practical stability of discrete-time switched affine systems, where an explicit ultimate bound in terms of the switching period is presented. Two numerical examples are given to illustrate the efficiency of results.

Reconfigurable-Based Fault-Tolerant Control for Continuous-Time Markov Jump Piecewise-Affine Systems with Allowable Switching Paths

In this paper, the reconfigurable control design problem is studied for a class of continuous-time Markov jump piecewise-affine (PWA) systems with sensor and actuator faults. A reconfiguration block is designed to recover mean-square input-to-state stability (ISS) of the faulty closed-loop system under virtual sensor and virtual actuator. Then, a novel algorithm for computing all the allowable switching paths (ASPs) is developed to overcome the issues of redundant empty paths and mismatch regions, and a S-procedure with the ellipsoidal outer approximation estimation is employed to cope with the affine term. Furthermore, a nonlinear approximation is introduced using the PWA technique to approximate the faulty Markov jump nonlinear systems. Finally, one example is illustrated to demonstrate the effectiveness and advantages of the designed reconfigurable control strategy with an ASPs algorithm.

Observer‐Based PID Security Control for Discrete‐Time T‐S Fuzzy Systems under Distributed Dynamic Event‐Triggered Mechanism

This study addresses the observer‐based proportional‐integral‐derivative (PID) security control problem for nonlinear discrete‐time fuzzy systems subject to deception attacks. In order to reduce the burden of communication, a distributed dynamic event‐triggered mechanism (DETM) is proposed, and each sensor node is equipped with an event‐trigger generator. A new discrete‐time Takagi–Sugeno (T‐S) fuzzy model is established, in which both DETM and deception attacks are taken into consideration. Then, based on Lyapunov stability theory, the sufficient conditions for the input‐to‐state stability (ISS) are proposed, and the desired fuzzy PID controllers are given. Finally, a simulation example is provided to show the usefulness of the proposed approach.

Robust Nonlinear Model Predictive Control With Model Predictive Sliding Mode for Continuous-Time Systems

Abstract This paper presents a robust, tube-based nonlinear model predictive controller for continuous-time systems with additive disturbances which cascades two sampled-data model predictive controllers: the first creates a desired path using nominal dynamics, and the second maintains the true state close to the nominal state by regulating a sliding variable designed on the error between the true and nominal states. The sampled-data model predictive approach permits easy incorporation of continuous-time sliding mode dynamics, allowing a dynamic boundary layer and tube design to be included. In this way, the control applied to the system capitalizes on the robustness properties of traditional sliding mode control (SMC) while incorporating system constraints. Stability analysis is presented in the context of input-to-state stability (ISS) for continuous-time systems. The proposed controller is implemented on two case studies, is compared to benchmark tube-based model predictive controllers, and is evaluated using average root-mean-square (RMS) values on the state and input variables, in addition to average integral square error (ISE) and integral absolute error (IAE) values on the position states. Results reveal that the proposed technique responds to higher levels of disturbance with significant increases in control effort, eliminates constraint violation by using of constrained SMC as the secondary controller, and maintains similar tracking performance to benchmark controllers at lower levels of control effort.

Input to state stabilization of networked systems under a specified packet dropout rate

This paper studies an input to state stabilizing control of networked control systems (NCSs) under a specified packet dropout rate. By considering packet dropouts in the NCSs, the transmission intervals are categorized by small delay intervals (packet-dropout-free case) and large delay intervals (packet-dropout case). Based on such classifications, we establish the concept of average packet dropout rate (ADR) to characterize the quality of service (QoS) for networks. Then, a switched systems approach is used to derive the ISS (input to state stability) conditions by exploiting Lyapunov theory and input delay approach for a specified ADR. In what follows, the controller design method for the NCSs under a specified ADR is reached by solving linear matrix inequalities (LMIs). According to the proposed results, a control and communication co-design method is developed such that one can design the controller gain according to QoS. Finally, simulations on self-steering control of autonomous vehicles are presented to verify the effectiveness of the proposed co-design method.

Impact of renewable energy penetration rate on power system transient voltage stability

Transient voltage stability of power system with high penetration of renewable energy faces challenges. This paper proposes a method for quantitatively evaluating the impact of renewable energy penetration rate on transient voltage stability. Based on input-to-state stability (ISS) theory, security and stability indices are constructed to evaluate the stability characteristics, and a quantitative transient voltage stability assessment framework for power system with high penetration of renewable energy is established. Case studies are performed on MATLAB/Simulink, and security/stability indices under different renewable energy penetration rates are calculated. Finally, the critical scale of renewable energy integration is obtained through the coupling relationship between stability characteristics and renewable energy penetration rate.

Predictive control based on event-triggering mechanism of cyber-physical systems under denial-of-service attacks

In this paper, an event-triggering mechanism (ETM) and a predictor compensation control strategy are proposed for cyber-physical systems (CPSs) with stochastic disturbance under malicious denial-of-service (DoS) attacks. First, a novel ETM excluded Zeno behaviour for CPSs is designed without considering DoS attacks, and the input-to-state stability (ISS) of the closed-loop systems is guaranteed. Second, given that DoS attacks could block the transmission of information, a predictor is used to compensate for the lost signals of CPSs. Then, a control strategy is proposed by combining the ETM and predictor, which can reduce the times of data transmission and control updates. Furthermore, a sufficient condition is derived to guarantee the ISS of the closed-loop system. Finally, the effectiveness of the proposed control strategy is verified by an example.

Distributed Inverse Optimal Control for Discrete-Time Nonlinear Multi-Agent Systems

This letter describes a robust distributed inverse optimal control framework for a multi-agent discrete-time nonlinear system, where the dynamics of each agent is directly affected by terms that depend on the state and input of the neighborhood agents and other disturbance signals. The individual local cost is formulated and a control solution for each agent is derived considering an inverse optimal control approach. To address the interaction between the agents, a coordination method based on a non-cooperative game is proposed. Using Lyapunov and Input-to-State Stability (ISS) arguments, we derive conditions under which the proposed game converges to a fixed point and the overall multi-agent system is ISS with respect to the disturbance signals. Simulation results for a coupled pendula system are presented.

Output-Based Tracking Control for a Class of Car-Like Mobile Robot Subject to Slipping and Skidding Using Event-Triggered Mechanism

This paper presents an output-based tracking controller for a class of car-like mobile robot (CLMR) subject to slipping and skidding. The slipping and skidding are regarded as external disturbances, and an event-triggered extended state observer (ET-ESO) is utilized to recover the velocities as well as to estimate the uncertainties and disturbances. The constrained longitudinal velocity is established, conforming to the traffic flow theory on the kinematic level. The velocity control law and heading angle control law are developed on the dynamic level, respectively. The input to state stability (ISS) of the closed-loop system is analyzed via cascade theory. Simulation results are given to demonstrate the effectiveness of the proposed tracking controller for CLMR subject to slipping and skidding.

Event-triggered consensus control of continuous-time stochastic multi-agent systems

This paper addresses the event-triggered consensus of stochastic multi-agent systems (MASs), the main breakthroughs are the avoidance of infinitely fast execution and the stochastic stability analysis. Toward this, first, a novel event-triggered mechanism (ETM) based on relative information is proposed to reduce the control effects, and the enforced fixed positive lower bound of the inter-execution times in the ETM can effectively exclude the infinitely fast execution behavior. Moreover, the input-to-state stability (ISS) is analyzed and the execution error is delicately estimated so that the constant control on each inter-execution time possesses enough feedback capability to dominate the effect of execution error. Based on this, the desired consensus is achieved in the almost sure sense. Particularly, the involved stochastic convergence strategy, without using the well-known Lyapunov theorem, is hopeful to provide a potential pattern to achieve an event-triggered consensus for more general stochastic linear/nonlinear MASs. Finally, a simulation example is presented to illustrate the effectiveness of the proposed results.

Input-to-state stability for large-scale stochastic impulsive systems with state delay

This article addresses a class of large-scale stochastic impulsive systems with time delay and time-varying input disturbance having bounded magnitude. The main interest is to develop sufficient conditions for the input-to-state stability (ISS) and stabilization in the presence of impulsive effects. The method of Razumikhin–Lyapunov function is used to develop the ISS and stabilization properties. Later, these results are applied to a class of control systems where the controller actuators are susceptible to failures. It should be noted that our results are delay independent, and the designed reliable controller is robust with respect to the actuator failures and to the system uncertainties. It is also observed that if the isolated continuous system is ISS and subjected to bounded impulsive effects, then the resulting impulsive system preserves the ISS property. Moreover, if the isolated continuous subsystems are all ISS and the interconnection amongst them is bounded from above, then the impulsive interconnected system is ISS provided that the degree of stability of each subsystem is larger than the magnitude of interconnection. If the underlying continuous system is unstable, then the input-to-state stabilization of the impulsive system is guaranteed if the stabilizing impulses are applied to the system frequently. As an implication to these results, if the input disturbance is zero, then the input-to-state stability (or stabilization) reduces to the stability (or stabilization) of the equilibrium state of the underlying disturbance-free system. A numerical example and simulations are provided to illustrate the proposed results.

On the stability properties of Gated Recurrent Units neural networks

The goal of this paper is to provide sufficient conditions for guaranteeing the Input-to-State Stability (ISS) and the Incremental Input-to-State Stability ({\delta}ISS) of Gated Recurrent Units (GRUs) neural networks. These conditions, devised for both single-layer and multi-layer architectures, consist of nonlinear inequalities on network's weights. They can be employed to check the stability of trained networks, or can be enforced as constraints during the training procedure of a GRU. The resulting training procedure is tested on a Quadruple Tank nonlinear benchmark system, showing satisfactory modeling performances.

Input-to-State Stability of Nonlinear Systems Using Observer-Based Event-Triggered Impulsive Control

In this article, we are concerned with the input-to-state stability (ISS) problem for a class of nonlinear systems with exogenous disturbances, where the states of the system are not fully available. Based on the idea of event-triggered control (ETC) strategy, a novel event-triggered mechanism (ETM) is designed to reduce the burden of the communication and controller updating with guaranteed performance requirement. Correspondingly, an observer-based impulsive controller coupled with sample control is proposed such that the controlled system is ISS under the designed ETM. Moreover, the possible accumulations of triggered instants (i.e., Zeno behavior) in the proposed control strategy are excluded. The controller gains and ETM parameters are co-designed by solving linear matrix inequalities (LMIs). Finally, a numerical example is provided to illustrate the effectiveness of the obtained theoretical results.

ISS small-gain criteria for infinite networks with linear gain functions

This paper provides a Lyapunov-based small-gain theorem for input-to-state stability (ISS) of networks composed of infinitely many finite-dimensional systems. We model these networks on infinite-dimensional ℓ ∞ -type spaces. A crucial assumption in our results is that the internal Lyapunov gains, modeling the influence of the subsystems on each other, are linear functions. Moreover, the gain operator built from the internal gains is assumed to be subadditive and homogeneous, which covers both max-type and sum-type formulations for the ISS Lyapunov functions of the subsystems. As a consequence, the small-gain condition can be formulated in terms of a generalized spectral radius of the gain operator. By an example, we show that the small-gain condition can easily be checked if the interconnection topology of the network has some sort of symmetry. While our main result provides an ISS Lyapunov function in implication form for the overall network, an ISS Lyapunov function in a dissipative form is constructed under mild extra assumptions.

Active vibration isolation performance of the bistable nonlinear electromagnetic actuator with the elastic boundary

This paper thoroughly investigates the broadband active vibration isolation of a bistable nonlinear electromagnetic actuator with an elastic boundary (simply named EB-bistable actuator). A new feedback control law is proposed to control the actuator's input current, to significantly attenuate the broadband vibration transmissibility from the base excitation to the actuator mover (supporting the payload). The control law comprises the polynomial function of the mover's absolute velocity. The mathematical models of the EB-bistable actuator and its application for vibration isolation are derived and experimentally validated. Then, based on the verified model, the paper comprehensively investigates the EB-bistable actuator with the proposed control law, which validates the broadband active vibration isolation performance for different system parameters. The input-to-state stability (ISS) of the control law for any non-negative control weights is proved, and thus it is a model-free control method. Results of one investigated case show that the maximum vibration transmissibility can be attenuated by over 90%. The lower bound of the effective vibration isolation bandwidth (where the vibration transmissibility is smaller than 0 dB) is reduced by 19.69%, i.e. the bandwidth is significantly broadened. Moreover, the study proves the effectiveness of the active vibration isolation for the structural variation and initial condition change. Finally, the paper thoroughly discusses the influence of the control law parameters on the active vibration isolation performance. The parametric studies develop useful guidelines for the active vibration isolation of the EB-bistable actuator.

ISS and integral-ISS of switched systems with nonlinear supply functions

The problem of input-to-state stability (ISS) and its integral version (iISS) are considered for switched nonlinear systems with inputs, resets and possibly unstable subsystems. For the dissipation inequalities associated with the Lyapunov function of each subsystem, it is assumed that the supply functions, which characterize the decay rate and ISS/iISS gains of the subsystems, are nonlinear. The change in the value of Lyapunov functions at switching instants is described by a sum of growth and gain functions, which are also nonlinear. Using the notion of average dwell-time (ADT) to limit the number of switching instants on an interval, and the notion of average activation time (AAT) to limit the activation time for unstable systems, a formula relating ADT and AAT is derived to guarantee ISS/iISS of the switched system. Case studies of switched systems with saturating dynamics and switched bilinear systems are included for illustration of the results.

Robust impulsive observer design for infinite‐dimensional cell population balance models

AbstractThe observer design problem for a class of cell population balance models, describing the time evolution of the cell mass density distribution function and the substrate concentration in a continuous‐stirred tank bioreactor with irregular discrete‐time measurements of the cell mass distribution is considered. The model consists of a partial integro‐differential equation coupled with an ordinary differential equation. Using the theory of impulsive systems sufficient conditions for the input‐to‐state‐stability (ISS) of the observation error in the state‐space with respect to the measurement uncertainty are derived in terms of the maximum time between successive measurements and the ISS gain. In absence of measurement uncertainty the convergence conditions imply exponential stability of the observation error dynamics. Besides these rigorous conditions, application‐oriented tuning guidelines are established. The theoretical results are illustrated with numerical simulations including a comparison with a continuous–discrete extended Kalman filter based on the numerical approximation, showing that a similar accuracy is achieved when using a finite‐dimensional approximation of the proposed impulsive observer scheme with a considerably lower computational effort.

Sampled-data finite-dimensional boundary control of 1D parabolic PDEs under point measurement via a novel ISS Halanay’s inequality

Recently, finite-dimensional observer-based controllers were introduced for 1D parabolic PDEs via the modal decomposition method. In the present paper we suggest a sampled-data implementation of a finite-dimensional boundary controller for 1D parabolic PDEs under discrete-time point measurement. We consider the heat equation under boundary actuation and point (either in-domain or boundary) measurement. In order to manage with point measurement, we employ dynamic extension and prove H1-stability. Due to dynamic extension, which leads to proportional–integral controller, we suggest a sampled-data implementation of the controller via a generalized hold device. We take into account the quantization effect that leads to a disturbed closed-loop system and input-to-state stability (ISS) analysis. We use Wirtinger-based piecewise continuous in time Lyapunov functionals which compensate sampling in the finite-dimensional state and lead to the simplest efficient stability conditions for ODEs. To compensate sampling in the infinite-dimensional tail, we introduce a novel form of Halanay’s inequality for ISS, which is appropriate for functions with jump discontinuities that do not grow in the jumps. Numerical examples demonstrate the efficiency of our method.

Input-to-state stability for system identification with continuous-time Runge–Kutta neural networks

Runge–Kutta neural networks (RKNN) bridge the gap between continuous-time advantages and the discrete-time nature of any digital controller. RKNN defines a neural network in a continuous-time setting and explicitly discretises it by a variable sample time Runge–Kutta method. As a result, a-priori model knowledge such as the well-known continuous-time state space model can easily be incorporated directly into the neural network weights. Additionally, RKNN preserves a long-term prediction accuracy and a reduced parameter precision requirement. In this contribution, we enhance RKNN with global-asymptotic stability (GAS) and input-to-state stability (ISS) criteria. Based on a Lyapunov function, we develop constraints on the neural network weights for an arbitrary sample time. The constraints are independent of any measurements, have only a few hyperparameters, can be combined with any standard constraint optimisation algorithm and guarantee stability already during training. We apply the algorithm on two public real-world nonlinear identification tasks, an electro-mechanical positioning system and a cascaded water tank benchmark. Both benchmarks are solved with the same hyperparameters and the presented method is competitive to nonlinear identification methods beyond neural networks in the literature. In prediction configuration, all other black-box nonlinear identification approaches are outperformed regarding root mean squared error (RMSE) by an order of magnitude on the cascaded tank benchmark.

Self-discipline predictive control against large-scale packet dropouts using input delay approach

This paper develops a novel self-discipline predictive control (SPC) scheme to compensate the missing control inputs of networked control systems (NCS) under arbitrary bounded packet dropouts. Since no feedback measurements are available when there are packet dropouts, the key idea of SPC scheme is that one can adjust its controller gains only based on the latest received state measurement rather than waiting the next available measurement. Then, the future controller gains are predicted by using input delay approach while input to state stability (ISS) is arrived under Lypunov–Krasovskii method and switched system framework. In what follows, the SPC scheme based on the predicted controller gains are used to update the control inputs during packet dropout intervals. A main advantage of this work lies in that the proposed SPC scheme realises a self-stabilisation with only limited feedback measurements under large-scale packet dropouts. At last, simulations on path following control of autonomous vehicles are carried out to show the validity of the proposed SPC scheme.