Input-to-output Stability Research Articles

Quadratic Programming for Continuous Control of Safety-Critical Multiagent Systems Under Uncertainty

This paper studies the control problem for safety-critical multi-agent systems based on quadratic programming (QP). Each controlled agent is modeled as a cascade connection of an integrator and an uncertain nonlinear actuation system. In particular, the integrator represents the position-velocity relation, and the actuation system describes the dynamic response of the actual velocity to the velocity reference signal. The notion of input-to-output stability (IOS) is employed to characterize the essential velocity-tracking capability of the actuation system. The standard QP algorithms for collision avoidance may be infeasible due to uncertain actuator dynamics. Even if feasible, the solutions may be non-Lipschitz because of possible violation of the full rank condition of the active constraints. Also, the interaction between the controlled integrator and the uncertain actuator dynamics may lead to significant robustness issues. Based on the current development of nonlinear control theory and numerical optimization methods, this paper first contributes a new feasible-set reshaping technique and a refined QP algorithm for feasibility, robustness, and local Lipschitz continuity. Then, we present a nonlinear small-gain analysis to handle the inherent interaction for guaranteed safety of the closed-loop multi-agent system. The proposed method is illustrated by numerical simulation and a physical experiment.

Distributed Hierarchical Shared Control for Flexible Multirobot Maneuver Through Dense Undetectable Obstacles.

When teleoperating a multirobot system (MRS) in outdoor environments, human operators can often detect obstacles that are not detected by robots and spot emergencies faster than robots do. However, the lack of efficient methods for operators to manipulate an MRS has limited the number of robots in a human-robot team. To handle this problem, a distributed hierarchical shared control scheme is proposed, aiming to provide a safe and flexible control interface for a few human operators to interact with a large MRS. The proposed hierarchical control scheme employs a two-layered structure. In the upper layer, intention field networks are designed to generate virtual human control signals. Two functionalities for human teleoperation, called: 1) group management and 2) motion intervention, are realized using intention fields, allowing the operators to split the robot formation into different groups and steer individual robots away from immediate danger. In parallel, a blending-based shared control algorithm is designed in the lower layer to resolve the conflict between human intervention inputs and autonomous formation control signals. The input-to-output stability (IOS) of the proposed distributed hierarchical shared control scheme is proved by exploiting the properties of weighting functions. Results from a usability testing experiment and a physical experiment are also presented to validate the effectiveness and practicability of the proposed method.

Improved Multiple Lyapunov Functions of input-to-output Stability and input-to-state Stability for Switched Systems

This paper investigates the Lyapunov characterizations of input-to-output stability (IOS) and input-to-state stability (ISS) for nonlinear switched systems. The restriction on the negativity of derivatives of Lyapunov functions is relaxed; additionally, subsystems are not restricted to be ISS/IOS throughout the state space. These problems are overcome by a newly proposed method, which extends multiple Lyapunov functions (MLFs) to an indefinite form. Indefinite multiple Lyapunov functions (iMLFs) are proposed on the basis of an inequality to estimate the upper boundary of the system state. Moreover, we apply the iMLFs method to verify the extended notions of IOS in a switched system. More relaxed sufficient conditions for ISS and IOS are proposed, a simulation experiment is carried out on a numerical example of the system, which demonstrates their effectiveness and superiority.

A Leader-Following Consensus Problem Via a Distributed Observer and Fuzzy Input-to-Output Small-Gain Theorem

This article presents a composite fuzzy distributed control law to solve a cooperative control problem of fuzzy multiagent systems with a nonlinear leader system based on the distributed observer. The contributions of this article are three-fold. First, the small-gain theorem is extended to fuzzy systems, and input-to-output stability (IOS) and output-Lagrange input-to-output stability (OL-IOS) fuzzy small-gain theorem are proposed for fuzzy systems. Second, the OL-IOS small-gain theorem of fuzzy systems with multiple loop is given. Third, based on fuzzy small-gain theorem, the fuzzy distributed control law is composed of a purely fuzzy distributed control law and a distributed observer for the leader system. Finally, the effectiveness of the proposed method is illustrated by using a numerical example and the leader-following consensus problem of the quadrotor.

Input-to-state stability and sliding mode control of the nonlinear singularly perturbed systems via trajectory-based small-gain theorem

This paper presents the trajectory-based input-to-state stability (ISS) and input-to-output stability (IOS) small-gain theorem, and the finite-time ISS (FTISS) and finite-time IOS (FTIOS) of nonlinear singularly perturbed systems. The contribution of this paper is threefold. Firstly, a novel idea is proposed to analyze the stability of the nonlinear singularly perturbed system, which is regarded as an interconnected system by using two-time-scale decomposition. Secondly, the trajectory-based approach is applied to establish ISS and IOS small-gain theorem for singularly perturbed systems and the FTISS and FTIOS properties are proposed. Thirdly, a novel sliding mode controller is developed for a class of nonlinear singularly perturbed systems. Finally, the effectiveness of proposed method is illustrated by using a numerical example, a DC motor simulation and a multi-agent singularly perturbed system.

On the relation of IOS-gains and asymptotic gains for linear systems

This paper presents a fundamental relation between Output Asymptotic Gains (OAG) and Input-to-Output Stability (IOS) gains for linear systems. For any Input-to-State Stable, strictly causal linear system the minimum OAG is equal to the minimum IOS-gain. Moreover, both quantities can be computed by solving a specific optimal control problem and by considering only periodic inputs. The result is valid for wide classes of linear systems (including delay systems or systems described by PDEs). The characterization of the minimum IOS-gain is important because it allows the non-conservative computation of the IOS-gains, which can be used in a small-gain analysis. The paper also presents a number of cases for finite-dimensional linear systems, where exact computation of the minimum IOS-gain can be performed.

Extremum Seeking for Nonlinear Uncertain Systems: A Small-Gain Synthesis

This paper studies the extremum seeking problem for static maps with the inputs of the maps generated by a nonlinear uncertain system. A new small-gain approach is developed which uses an extremum seeking strategy to generate a reference signal, and employs a control law for reference-tracking of the nonlinear uncertain systems. The notions of input-to-state stability (ISS) and input-to-output stability (IOS) are used to characterize the interconnection between the extremum seeking strategy and the reference-tracking controller, and the nonlinear small-gain theorem is employed to guarantee the stability of the closed-loop extremum seeking system. With the proposed approach, the extremum seeking problem for a complex nonlinear system is solvable as long as one can design a proper reference-tracking controller for the system. Examples are given to show the feasibility of the proposed approach, and a numerical simulation is employed to show the effectiveness of the proposed design.

Sampled-data observers for 1-D parabolic PDEs with non-local outputs

The present work provides a systematic approach for the design of sampled-data observers to a wide class of 1-D, parabolic PDEs with non-local outputs. The studied class of parabolic PDEs allows the presence of globally Lipschitz nonlinear and non-local terms in the PDE. Two different sampled-data observers are presented: one with an inter-sample predictor for the unavailable continuous measurement signal and one without an inter-sample predictor. Explicit conditions on the upper diameter of the (uncertain) sampling schedule for both designs are derived for exponential convergence of the observer error to zero in the absence of measurement noise and modeling errors. Moreover, explicit estimates of the convergence rate can be deduced based on the knowledge of the upper diameter of the sampling schedule. When measurement noise and/or modeling errors are present, Input-to-Output Stability (IOS) estimates of the observer error hold for both designs with respect to noise and modeling errors. The main results are illustrated by two examples which show how the proposed methodology can be extended to other cases (e.g., boundary point measurements).

An IOS Small-Gain Theorem for Large-Scale Hybrid Systems

This paper presents a small-gain result for large-scale interconnected systems composed of hybrid subsystems. It is assumed that the subsystems are input-to-output practically stable (IOpS) with offset and unboundedness observability (UO), and the interconnected system can be proved to be IOpS with offset and UO as long as the proposed small-gain condition is satisfied. If, moreover, the assumption of IOpS on the subsystems is reduced to input-to-output stability (IOS), then the interconnected system is IOS.

Computing compositional proofs of Input-to-Output Stability using SOS optimization and [formula omitted]-decidability

We present BFComp, an automated framework based on Sum-Of-Squares (SOS) optimization and δ-decidability over the reals, to compute Bisimulation Functions (BFs) that characterize Input-to-Output Stability (IOS) of dynamical systems. BFs are Lyapunov-like functions that decay along the trajectories of a given pair of systems, and can be used to establish the stability of the outputs with respect to bounded input deviations.In addition to establishing IOS, BFComp is designed to provide tight bounds on the squared output errors between systems whenever possible. For this purpose, two SOS optimization formulations are employed: SOSP 1, which enforces the decay requirements on a discretized grid over the input space, and SOSP 2, which covers the input space exhaustively. SOSP 2 is attempted first, and if the resulting error bounds are not satisfactory, SOSP 1 is used to compute a Candidate BF (CBF). The decay requirement for the BFs is then encoded as a δ-decidable formula and validated over a level set of the CBF using the dReal tool. If dReal produces a counterexample containing the states and inputs where the decay requirement is violated, this pair of vectors is used to refine the input-space grid and SOSP 1 is iterated.By computing BFs that appeal to a small-gain theorem, the BFComp framework can be used to show that a subsystem of a feedback-composed system can be replaced–with bounded error–by an approximately equivalent abstraction, thereby enabling approximate model-order reduction of dynamical systems. The BFs can then be used to obtain bounds on the error between the outputs of the original system and its reduced approximation. To this end, we illustrate the utility of BFComp on a canonical cardiac-cell model, showing that the four-variable Markovian model for the slowly activating Potassium current IKs can be safely replaced by a one-variable Hodgkin–Huxley-type approximation. In addition to a detailed performance evaluation of BFComp, our case study also presents workarounds for systems with non-polynomial vector fields, which are not amenable to standard SOS optimizers.

Integral Input-to-Output Stability for one Class of Discontinuous Dynamical Systems

It is well known that integral input-to-output stability(IIOS) is weaker than input-to -output stability(IOS).In this paper,IIOS problems for one class of discontinuous dynamical systems are considered.Piecewise smooth IIOS-Lyapunov functions are adopted.Furthermore,interconnecti ons of discontinuous dynamical systems are studied.The results for a class of discontinuous dynam ical systems and interconnection of discontinuous dynamical systems' IIOS character are shown,res pectively.

A Small Gain Framework for Networked Cooperative Teleoperation

A small gain approach for stability analysis of cooperative network-based force reflecting teleoperator systems is presented. The approach is based on a new version of input-to-output stability (IOS) nonlinear small gain theorem that is applicable to stability analysis of large-scale network-based interconnections. Using the small-gain theorem presented, we show that stability of the cooperative teleoperator system can always be achieved by choosing some of the system’ gains appropriately.

A Multichannel IOS Small Gain Theorem for Systems With Multiple Time-Varying Communication Delays

A version of the input-to-output stability (IOS) small gain theorem is derived for interconnections where the subsystems communicate asynchronously over multiple channels, and the communication is subject to multiple time-varying possibly unbounded communication delays. It is shown that a ldquomulti-dimensionalrdquo version of the small gain condition guarantees the stability of the interconnection of IOS subsystems under certain mild assumptions imposed on communication process. The fulfillment of these assumptions does not depend on characteristics of the communication channels, but can always be guaranteed by implementation of certain features of the underlying communication protocol. This result is applicable to a wide range of dynamical systems whose parts communicate over networks, such as Internet-based teleoperators.

Marine Vehicle Path Following Using Inner-Outer Loop Control

Abstract This paper addresses the problem of marine vehicle path following using inner-outer loop control, with due account for the vehicle dynamics. We propose an inner-outer control structure that lends itself to a simple intuitive interpretation and exhibits three key advantages: i) it allows for decoupling in the design of the inner and outer control loops, ii) the design of the outer loop controller does not require in depth knowledge of the internal vehicle dynamics, and iii) it affords practitioners a very convenient method to effectively implement path following controllers on a wide range of vehicles. The mathematical tools used for systems characterization and stability analysis borrow from Input to Output Stability (IOS) theory and small gain theorems. In this paper, formal system analysis is done only for the case where the paths to be followed are straight lines. However, the technical machinery developed and the results obtained shed light into the process of designing path following controllers with an inner-outer loop structure for arbitrary paths. The efficacy of the path following control structure developed has been proven during tests with an AUV and an ASV at sea.

Control schemes for stable teleoperation with communication delay based on IOS small gain theorem

The problem of stabilization of a force-reflecting telerobotic system in presence of time delay in the communication channel is addressed. We introduce an approach that is based on application of the input-to-output stability (IOS) small gain theorem for functional differential equations (FDEs). A version of the stabilization algorithm as well as its two adaptive extensions are proposed. For all these control schemes, the input-to-state stability (ISS) of the overall telerobotic system is guaranteed in the global, global practical, or semiglobal practical sense for any constant communication delay under the assumption that the environmental dynamics satisfy a weak form of finite-gain stability property. As an intermediate step, we formulate and prove a general IOS (ISS) small gain result for FDEs.

Non-uniform in time robust global asymptotic output stability for discrete-time systems

In this paper the notions of non-uniform in time robust global asymptotic output stability (RGAOS) and input-to-output stability (IOS) for discrete-time systems are studied. Characterizations as well as links between these notions are provided. Particularly, it is shown that a discrete-time system with continuous dynamics satisfies the non-uniform in time IOS property if and only if the corresponding unforced system is non-uniformly in time RGAOS. Necessary and sufficient conditions for the solvability of the robust output feedback stabilization (ROFS) problem are also given. Copyright © 2005 John Wiley & Sons, Ltd.

STABILIZATION SCHEME FOR FORCE REFLECTING TELEOPERATION WITH TIME-VARYING COMMUNICATION DELAY BASED ON IOS SMALL GAIN THEOREM

The problem of stabilization of force-reflecting teleoperators with time-varying delay in the communication channel is addressed. A control scheme is proposed which guarantees stability of the closed-loop telerobotic system in the presence of an arbitrary time-varying possibly unbounded transmission delay satisfying a set of technical assumptions. The proposed scheme also guarantees (in the “semiglobal” sense with respect to initial conditions and external forces) that the slave manipulator tracks the delayed trajectory of the master with error which is ultimately bounded by an arbitrarily small bound. The proof of this result is based on a special version of the IOS (input-to-output stability) small gain theorem.

The Non-Uniform in Time Small-Gain Theorem for a Wide Class of Control Systems with Outputs

The notions of non-uniform in time robust global asymptotic output stability and non-uniform in time input-tooutput stability (IOS) are extended to cover a wide class of control systemswith outputs that includes (finite or infinitedimensional) discrete-time and continuous-time control systems. A small-gain theorem,which makes use of the notion of non-uniform in time IOS property,is presented.

UNIVERSAL FORMULA FOR OUTPUT ASYMPTOTIC STABILIZATION

the work deals with universal control formula which provides output global asymptotic stability for affine in control nonlinear system. Development of this result to input-to-output stability (IOS) is also considered.

On Input-to-Output Stability for Systems not Uniformly Bounded

This paper generalizes a Lyapunov characterization for output-stable systems. A variant of input-to-state stability (ISS) known as input-to-output stability (IOS) is considered. A Lyapunov characterization of IOS under a uniform bounded input bounded state (UBIBS) condition has appeared in the literature. We present a more general characterization, which holds for systems which are not necessarily uniformly bounded.