Is Input-to-state Stable Research Articles

Delayed feedback implementation of decentralized derivative-dependent control of large-scale systems with input delays and disturbed measurements

We study decentralized derivative-dependent control of large-scale nth-order systems with input delays via delayed feedback implementation. The unavailable derivatives can be approximated by finite differences giving rise to a time-delayed feedback. In the centralized case, an efficient simple linear matrix inequalities (LMIs)-based method for designing of such static output-feedback and its sampled-data implementation was recently suggested. In the present paper, we extend this design to large-scale systems in the presence of input delays and disturbed measurements. Under the assumption of the stabilizability of the system with small enough input delays and small enough interactions by a state-feedback that depends on the output and its derivatives, a delayed static output-feedback that stabilizes the system is presented by using the current and past disturbed measurements. To compensate the errors due to the input delays, we add the appropriate terms to the corresponding Lyapunov–Krasovskii functional that lead to LMIs conditions. The efficient bounds on the delays preserving that the resulting system is input-to-state stable (ISS) are found by verifying the LMIs. In addition, we employ the vector Lyapunov functional method that may allow larger couplings compared with the existing method. Finally, the effectiveness of the proposed methods is illustrated by numerical examples.

Safety-Critical Game-Based Formation Control of Underactuated Autonomous Surface Vehicles

Dear Editor, This letter addresses the noncooperative formation control of multiple underactuated autonomous surface vehicles (ASVs) in an obstacle-laden environment. Each ASV is subject to external disturbances and fully unknown model parameters. A safety-critical game-based formation control method is proposed such that the multiple ASVs are able to track a reference trajectory, and accomplish each individual interest in safety behaviors simultaneously. The stability and safety analyses show that the closed-loop system is input-to-state stable (ISS), and the multi-ASV system is guaranteed for input-to-state safety (ISSf). Simulation results substantiate the proposed safety-critical game-based formation control method.

Existence of an equilibrium for delayed neural fields under output proportional feedback

Recently, Chaillet et al. (2017) proved that the closed-loop system resulting from the output proportional feedback stabilization of a class of delayed neural fields is input-to-state stable (ISS) for sufficiently high gain, subject to the existence of an equilibrium point for the closed-loop system. In the present paper, we show that a sufficient condition for such an equilibrium to exist is that the activation functions are bounded.

Singular Perturbation: When the Perturbation Parameter Becomes a State-Dependent Function

This paper introduces a new class of singularly perturbed systems in which the small, but constant, perturbation coefficient in standard singular perturbation theory is replaced by a state-dependent function. This generalization is aimed at broadening the applicability of singular perturbation theory in practice. For this class of singularly perturbed systems, it is assumed that the boundary-layer subsystem is globally asymptotically stable (GAS) at the origin and the reduced subsystem is input-to-state stable (ISS) with respect to the state of the boundary-layer subsystem. Under a mild monotonicity condition, sufficient conditions on the perturbation functions are given under which the singularly perturbed system is GAS at the origin. ISS and nonlinear small-gain techniques are exploited in the stability analysis. The efficacy of the proposed theoretical result is validated via its applications to tackling integral control and feedback optimization problems.

L₁-Robust Interval Observer Design for Uncertain Nonlinear Dynamical Systems

This paper presents a novel interval observer design for uncertain locally Lipschitz continuous-time (CT) and discrete-time (DT) systems with noisy nonlinear observations that is input-to-state stable (ISS) and minimizes the L1-gain of the observer error system with respect to the uncertainties. Using mixed-monotone decompositions, the proposed observer is correct and positive by construction without the need for additional constraints/assumptions. This, in turn, allows us to directly leverage techniques for positive systems to design an ISS and L1-robust interval observer via mixed-integer (linear) programs instead of semi-definite programs with linear matrix inequalities. Further, our observer design offers additional degrees of freedom that may serve as a surrogate for coordinate transformations. Finally, we demonstrate the effectiveness of our proposed observer on some CT and DT systems.

Nonlinear integral control with event-triggered feedback: Unknown decay rates, zeno-freeness, and asymptotic convergence

This paper studies the event-triggered implementation of integral control laws for nonlinear uncertain systems. It is assumed that the plant already admits an integral control law such that the closed-loop system is input-to-state stable (ISS) with respect to the state sampling errors and admits an ISS-Lyapunov function. The unknown offset between the plant equilibrium and the set point means that one may not find a uniform lower bound for all the possible decay rates of the ISS-Lyapunov function, which however is usually required in the existing results. The design in this paper is based on a sequence of lower bound estimates of the decay rate, and the proposed dynamic event trigger guarantees that the intersampling intervals are lower bounded by a positive constant, and the asymptotic convergence property of integral control is retained even with event-triggered feedback. The theoretical result is physically validated by a pendulum control system.

Periodic event‐triggered control for incrementally quadratic nonlinear systems

AbstractPeriodic event‐triggered control (PETC) evaluates triggering conditions only at periodic sampling times, based on which it is decided whether the controller needs to be updated. This article investigates the global stabilization of nonlinear systems that are affected by external disturbances under PETC mechanisms. Sufficient conditions are provided to ensure the resulting closed‐loop system is input‐to‐state stable (ISS) for the state feedback and the observer‐based output feedback configurations separately. The sampling period and the triggering functions are chosen such that the ISS‐Lyapunov function of continuous dynamics is also the ISS‐Lyapunov function of the overall system. Based on that, sufficient conditions in the form of linear matrix inequalities are provided for the PETC design of incrementally quadratic nonlinear systems. Two simulation examples are provided to illustrate the effectiveness of the proposed method.

State observer with Round-Robin aperiodic sampled measurements with jitter

A sampled-data observer is proposed for linear continuous-time systems whose outputs are sequentially sampled via non-uniform sampling intervals repeating a prescribed Round-Robin sequence. With constant sampling intervals (jitter-free case) we provide constructive necessary and sufficient conditions for the design of an asymptotic continuous–discrete observer whose estimation error is input-to-state stable (ISS) from process disturbances and measurement noise. We use a time-varying gain depending on the elapsed time since the last measurement. With non-constant sampling intervals (jitter-tolerant case), our design conditions are only sufficient. A suspension system example shows the effectiveness of the proposed approach.

Trajectory Tracking Control of Autonomous Ground Vehicles Using Adaptive Learning MPC.

In this work, an adaptive learning model predictive control (ALMPC) scheme is proposed for the trajectory tracking of perturbed autonomous ground vehicles (AGVs) subject to input constraints. In order to estimate the unknown system parameter, we propose a set-membership-based parameter estimator based on the recursive least-squares (RLS) technique with the ensured nonincreasing estimation error. Then, the estimated system parameter is employed in MPC to improve the prediction accuracy. In the proposed ALMPC scheme, a robustness constraint is introduced into the MPC optimization to handle parametric and additive uncertainties. For the designed robustness constraint, its shape is decided off-line based on the invariant set, whereas its shrinkage rate is updated online according to the estimated upper bound of the estimation error, leading to further reduced conservatism and slightly increased computational complexity compared with the robust MPC methods. Furthermore, it is theoretically shown that the proposed ALMPC algorithm is recursively feasible under some derived conditions, and the closed-loop system is input-to-state stable (ISS). Finally, a numerical example and comparison study are conducted to illustrate the efficacy of the proposed method.

Nonlinear Model Predictive Trajectory Tracking Control of Underactuated Marine Vehicles: Theory and Experiment

The article studies the trajectory tracking control problem of underactuated marine vehicles via the nonlinear model predictive control (NMPC) strategy, where practical control and state constraints present. It is a well-known challenging issue that the conventional NMPC is not applicable for underactuated marine vehicles, due to the fact that there does not exist a local static continuous state-feedback controller to stabilize the underactuated dynamics. To resolve this issue, this article proposes to construct an auxiliary time-varying tracking controller to aid terminal constraint design in the NMPC framework, where the time-varying tracking controller borrows the ideas from Lyapunov's direct method and backstepping approach. Based on this, a novel NMPC algorithm is designed to ensure trajectory tracking control of underactuated marine vehicles. Furthermore, a systematic parameter design approach is developed. Under the designed parameters, we show that the tracking error system is input-to-state stable (ISS). Finally, the effectiveness of the designed algorithm is verified by thorough simulation and hardware experiments.

Switched Systems With Multiple Equilibria Under Disturbances: Boundedness and Practical Stability

This paper addresses robustness to external disturbances of switched discrete and continuous systems with multiple equilibria. First, we prove that if each subsystem of the switched system is input-to-state stable (ISS), then under switching signals that satisfy an average dwell-time bound, the solutions are ultimately bounded within a compact set. The size of this set varies monotonically with the supremum norm of the disturbance signal. These results generalize existing ones in the common equilibrium case to accommodate multiple equilibria. Then, we relax the (global) ISS conditions to consider equilibria that are locally exponentially stable (LES), and we establish practical stability for such switched systems under disturbances. Our motivation for studying this class of switched systems arises from certain motion planning problems in robotics, where primitive movements, each corresponding to an equilibrium point of a dynamical system, must be composed to obtain more complex motions. As a concrete example, we consider the problem of realizing safe adaptive locomotion of a three-dimensional biped under persistent external force by switching among motion primitives characterized by LES limit cycles. The results of this paper, however, are relevant to a much broader class of applications, in which composition of different modes of behavior is required.

Event-triggered input-to-state stabilization of nonlinear systems subject to disturbances and dynamic uncertainties

This paper studies the design of event-triggered control schemes for nonlinear systems subject to both external disturbances and dynamic uncertainties. To avoid Zeno behavior, this paper proposes an event-triggering mechanism (ETM) that uses not only the measured system state but also an estimation of the influence of the external disturbances and dynamic uncertainty. It is shown that the proposed ETM guarantees that the inter-sampling intervals are bounded below by a positive constant, leading to the absence of Zeno phenomenon. Moreover, the closed-loop event-triggered system is input-to-state stable (ISS) with the external disturbance as the input. Nonlinear small-gain tools are refined for stability analysis.

Feedback stabilization of discrete time singularly perturbed systems with communication constraints

AbstractIn this paper, the quantized feedback problem for a class of discrete time singularly perturbed systems with information constraints is considered. First, a proper coder‐decoder pair is presented so that the transmission errors tend to zero exponentially under information constraints. Next, linear matrix inequalities are constructed such that the resulting closed‐loop system is input‐to‐state stable (ISS) with respect to the transmission error, and the asymptotic stability of the closed‐loop system also can be guaranteed. The theoretical results have shown that the presented method is simple and easy to operate. Moreover, the upper bound of the small perturbed parameter of the stable system can be explicitly estimated using this feasible method. Finally, two numerical examples are given to illustrate the effectiveness of the proposed method.

Minimum-Rank Dynamic Output Consensus Design for Heterogeneous Nonlinear Multi-Agent Systems

In this paper, we propose a new and systematic design framework for output consensus in heterogeneous Multi-Input Multi-Output (MIMO) general nonlinear Multi-Agent Systems (MASs) subjected to directed communication topology. First, the input-output feedback linearization method is utilized assuming that the internal dynamics is Input-to-State Stable (ISS) to obtain linearized subsystems of agents. Consequently, we propose local dynamic controllers for agents such that the linearized subsystems have an identical closed-loop dynamics which has a single pole at the origin whereas other poles are on the open left half complex plane. This allows us to deal with distinct agents having arbitrarily vector relative degrees and to derive rank-$1$ cooperative control inputs for those homogeneous linearized dynamics which results in a minimum rank distributed dynamic consensus controller for the initial nonlinear MAS. Moreover, we prove that the coupling strength in the consensus protocol can be arbitrarily small but positive and hence our consensus design is non-conservative. Next, our design approach is further strengthened by tackling the problem of randomly switching communication topologies among agents where we relax the assumption on the balance of each switched graph and derive a distributed rank-$1$ dynamic consensus controller. Lastly, a numerical example is introduced to illustrate the effectiveness of our proposed framework.

Self-triggered fault estimation and fault tolerant control for networked control systems

A framework of self-triggered fault diagnosis (FD) and fault tolerant control (FTC) for networked control systems (NCSs) is presented in this paper. The self-triggered scheduler is implemented in a smart sensor node. By means of uncertain polytopic theory, we first design a fault diagnosis observer, which has a similar structure with Kalman filter, to simultaneously estimate the fault and state using the self-triggered nonuniform sampled outputs. Then, based on the obtained fault and state information, an active fault tolerant controller with a state-estimate-dependent self-triggered scheduler is provided. We prove that the closed-loop faulty system is input-to-state stable (ISS) under the proposed self-triggered sampling mechanism. Finally, simulation results are provided to verify that the proposed self-triggered FD and FTC scheme can significantly reduce the sampling cost while preserving the desired fault estimation and FTC performance.

Predictor Observers for Proportional Navigation Systems Subjected to Seeker Delay

We propose predictor observers as viable tools to compensate for pure time delays in the seeker subsystem of missile guidance systems employing proportional navigation guidance (PNG). With the intention of use in terminal guidance in which the missile is already on a near-collision course with a target, we demonstrate that a classical predictor observer for linear time-delay systems is input-to-state stable (ISS) with respect to bounded target maneuver and measurement disturbances. We then show how the classical predictor observer can be implemented as a Kalman filter for the delayed state augmented with a predictor. Finally, we illustrate the practicality of the approach in missile guidance system design through two examples that exhibit seeker delay and random target maneuvers. For both examples, numerical simulations show that predictor observer guidance laws are able to substantially reduce miss distance versus baseline PNG laws that have no delay compensation, inviting further studies of the approach for designing real missile guidance systems in practice.

A novel architecture for cooperative remote rehabilitation system

In this paper, the problem of trilateral interaction between therapist, patient, and virtual environment is considered. Herein, a new architecture for robotic rehabilitation is investigated based on the robots together with patient and therapist, cooperating with each other to accomplish the therapy in the virtual reality, in which the rehabilitation system is divided into three subsystems: patient, therapist, and virtual environment. Each subsystem is controlled locally, ensuring that it is input-to-state stable (ISS). Besides, the associated interconnections are shown to be stable in the presence of communication time delays. First, the patient is capable of generating movements until losing his strength, i.e. interacting with the robot without help of the therapist. Afterwards, the therapist assists the patient to follow up the process, i.e. the patient is active in the former condition while becomes passive in the latter. Therefore, the controllers should be able to deal with not only active but also passive robots. Also, impedance reflection, adaptive projection method, and impedance control are utilized. Finally, simulation and experimental results are extracted in evaluation.

Adaptive control for an uncertain robotic manipulator with input saturations

In this paper, we address the control problem of an uncertain robotic manipulator with input saturations, unknown input scalings and disturbances. For this purpose, a model reference adaptive control like (MRAC-like) is used to handle the input saturations. The model reference is input to state stable (ISS) and driven by the errors between the required control signals and input saturations. The uncertain parameters are dealt with by using linear-in-the-parameters property of robotic dynamics, while unknown input scalings and disturbances are handled by non-regressor based approach. Our design ensures that all the signals in the closed-loop system are bounded, and the tracking error converges to the compact set which depends on the predetermined bounds of the control inputs. Simulation on a planar elbow manipulator with two joints is provided to illustrate the effectiveness of the proposed controller.

Inverse identification of inputs of a random utility based model using optimal control

The random utility based multiregional input–output (RUBMRIO) model is used by several integrated land use and transportation models (ILUTMs) to forecast commodity flows in a region and for spatial allocation of production activities. It makes use of multiregional input–output models which are based on random utility theory. In this work, we use optimal control techniques to find optimal final demand and transportation costs of a RUBMRIO model, that would lead to a desired level of commodity flow and production costs.At first, the RUBMRIO model is formulated as a discrete time dynamical system. It is shown, using Lyapunov argument, that the fixed point of the dynamical system is input-to-state stable (ISS). Then, a discrete time optimal control problem is formulated with states as commodity flows and production costs, and the final demand and transportation costs as control inputs. An optimization problem is then solved to obtain control inputs that would lead to desired commodity flows and production costs.Further, the proposed methodology is applied to a numerical example. It is shown that the optimal control based method can achieve user specified commodity flows and production costs up to an acceptable accuracy level.

Generalization of an input-to-state stability preserving Runge–Kutta method for nonlinear control systems

Given a nonlinear control system that is input-to-state stable (ISS) it is assured that the states of the system remain bounded if the input is bounded and the system is global asymptotic stable (GAS) in the sense of Lyapunov in the absence of inputs. Recently, Hu and Liu (2007) [7] studied under which condition ISS of a continuous time nonlinear control system implies ISS of the discrete time system obtained by an implicit Runge–Kutta (RK) method. In this contribution, we extend those results to explicit RK methods. This represents an important extension with respect to applications like system identification where explicit RK methods are presupposed or real-time applications where the computational burden of implicit RK methods is prohibitive.