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In this paper, we address the group controllability issue for discrete-time multi-agent systems (MASs) with two-time-scale feature. Since the existence of singular perturbation parameter in modeling the two-time-scale MASs will cause ill-posedness problems, we decouple the MASs into fast and slow subsystems by using the iterative and approximate methods, and then eliminate the singular perturbation parameter. By using matrix theory, we propose a lot of necessary and/or sufficient conditions of group controllability, where some easy-to-use criteria are determined only by eigenvalues of matrices. Lastly, a simulation example is given to illustrate the effectiveness of the proposed theoretical results.

In this paper, a feedback control law based on two time‐scale assignment and state extension is implemented for the nonlinear non‐minimum phase dynamics of three degree‐of‐freedom autonomo...

A two-time-scale system involves both fast and slow dynamics. This paper studies the design of nonlinear observers for general nonlinear two-time-scale systems and presents a reduced-order observer design approach. The reduced-order observer is derived based on a lower dimensional model to reconstruct the slow states, along with the algebraic slow-motion invariant manifold function to reconstruct the fast states. Through an error analysis, it is shown that even though the observer is designed based on the reduced model by neglecting the fast dynamics, it is capable of providing accurate estimation of the states of the original detailed system. It will render a vanishing estimation error, with exponential convergence rate governed by the subsystem of fast dynamics and the chosen observer design parameters.

In this study, the controllability of multi-agent systems with second-order dynamic under discrete-time model, which evolve on two time scales. The authors first characterise the systems by difference equations with a singular perturbation parameter. Then, in order to get rid of the singular perturbation parameter, they split the systems into a slow subsystem and fast subsystem based on the iterative method and approximate approach. Moreover, according to matrix theory, they deduce a lot of necessary and/or sufficient controllability criteria of the systems. In particular, a second-order controllability criterion only depending on the eigenvalues of matrices is proposed, which is more easy-to-use and efficient. A simulation example is lastly given to clarify the validity of all theoretical results.

This paper studies the distributed consensus problem of a new class of multi-agent systems, where each agent is represented by the two-time-scale system. Two fully distributed, ϵ-dependent adaptive...

This paper is concerned with robustly input-to-state stable (ISS) and Robust ISS by feedback of uncertain discrete-time singularly perturbed systems (SPSs) with disturbances. Meanwhile, robust stab...

Motivated by broad applications within engineering and sciences, we study distributed consensus-based gradient methods for solving optimization problems over a network of nodes. A fundamental challenge for solving this problem is the impact of finite communication bandwidth, so information that is exchanged between the nodes must be quantized. In this paper, we utilize the dithered (random) quantization and study the distributed variant of the well-known two-time-scale methods for solving the underlying optimization problems under the constraint of finite bandwidths. In addition, we provide more insight and an explicit formula of how to design the step sizes of these two-time-scale methods and their impacts on the performance of the algorithms.

As a basic concept in modern control theory, controllability reveals the fundamental structural characteristics of a dynamic system, and it also plays an important part in the analysis and control of a dynamic system. With the increasing complexity of multi-agent systems, the multi-agent networks can be divided into some subnetworks in terms of time scales. This paper concentrates on the group controllability of two-time-scale multi-agent networks, establishes the necessary and sufficient criterion of group controllability based on singular perturbation methods, and deduces easy-to-use group controllability criteria by using matrix theory and graph theory. Lastly, a simulation example is presented to illustrate the effectiveness of the proposed methods.

This paper presents a novel multivariable two-time scale control method for flexible joint robots. The fundamental limitations of the conventional two-time scale control are clearly explained based on a singular perturbation analysis with a linearized dynamic model. Unlike the conventional methods, the bandwidth of the slow control can be made high and the derivatives of the rigid control are not required. The closed-form solution of a slow model that is more accurate than the rigid model is presented. It is shown that the modeling error of the proposed two-time scale control can be made arbitrarily small by the control gain only. A real-world experiment with a six degrees-of-freedom commercial industrial robot was performed to validate the presented approach.

This short communication is a discussion of the paper entitled “Adaptive sliding-mode-observer for sensorless induction motor drive using two-time-scale approach” by A. Mezouar, M.K. Fellah and S. Hadjeri published in the Simulation Modelling Practice and Theory 16 (2008) 1323–1336. In the discussed paper the authors present a current and flux sliding mode observer for the induction motor that also incorporates an adaptive laws in order to estimate the rotor speed and the inverse of the rotor time constant. However the proposed design for the observer and for the adaptive laws, employs the real value of the rotor time constant and the real value of the rotor speed, which are unknown, and therefore cannot be used in the observer design nor in the adaptive laws design.

This paper considers the problem of robust multi-objective

In this paper, a non-linear controller for linear switched reluctance motors (LSRMs) is developed by utilizing their properties and the energy dissipation theory. As the electrical time constant is much smaller than the mechanical time constant, the whole LSRM driving system can be treated as a two-time-scale system. It is then decomposed into an electrical subsystem and a mechanical subsystem, which are interconnected by negative feedbacks. Controllers for these two subsystems are designed to guarantee that both systems are passive. Because a system consisting of two passive subsystems connected through negative feedbacks is still passive as a whole, stability of an LSRM driving system can be achieved at system level. The proposed control strategy is characterized by a simple structure and easy implementation. Experimental results are also provided to prove the effectiveness and robustness of the proposed position control for LSRM.

This paper deals with the problem of H ? control of linear two-time scale systems. The authors' attention is focused on the robust regulation of the system based on a new modeling approach under the assumption of norm-boundedness of the fast dynamics. In the proposed approach, the fast dynamics are treated as a norm-bounded disturbance (dynamic uncertainty). In this view, the synthesis is performed only for the certain dynamics of the two-time scale system, whose order is less than that of the original system. It should be noted, however, that this scheme is significantly different from the conventional approaches of order reduction for linear two-time scale systems. Specifically, in the present work, explicitly or implicitly, all the dynamics of the system are taken into consideration. In other words, the portion that is treated as a perturbation is incorporated in the design by its maximum possible gain --- in the L 2 sense --- over different values of the inputs. One of the advantagesof this approach is that --- unlike in the conventional approaches of the order reduction --- the reduced-order system still keeps some information of the `deleted' subsystem. Also, we consider the robust stability analysis and stability bound improvement of perturbed parameter (?) in the two-time scale systems by using linear fractional transformations and structured singular values (?) approach. In this direction, by introducing the parametric uncertainty and dynamic uncertainty in the two-time scale systems, we represent the system as a standard ?-interconnection framework by using linear fractional transformations, and derive a set of new stability conditions for the system in the frequency domain. The exact solution of ?-bound is characterized. It is shown that, in spite of the coupling between the dynamic uncertainties and certain dynamics, the designed H ? controller stabilizes the overall closed-loop system, in the presence of norm-bounded disturbances. To show the effectiveness of the approach, the modeling of the single-link flexible manipulator and control of the Tip-position of the manipulator by utilizing the mentioned method are presented in the case study.

Two-time-scale (TTS) distributions are introduced. For a class of stable systems, it is shown that every TTS distribution has a two-frequency-scale (TFS) Laplace transform. Conversely, it is shown that the impulse response of any stable TFS transfer function, and hence any stable (standard) singularly perturbed system, can be characterized in terms of a stable TTS distribution. A time domain decomposition for TTS distributions is obtained which parallels the slow and fast decomposition of singularly perturbed systems and also the frequency domain decomposition of TFS transfer functions. It is shown that every stable TTS distribution can be decomposed in terms of two simpler distributions represented in two different time scales. A composite distribution is constructed from these two which approximates the TTS distribution arbitrarily closely in the L 1 norm.

This note is concerned with the robust stability problem of uncertain two-time-scale systems, where the only information available for system uncertainties is their norm upper bounds. The stability bound /spl epsiv//sup */ is derived via a state transformation and the constructive use of a Lyapunov function. Finally, examples are given to illustrate the applications of the proposed results.

The present paper has two main purposes. The first is to provide an explicit state-space solution of the robust stabilization problem with respect to the NLCF for two-time scale systems. In this construction only the solutions of the control and of the filtering Riccati equations associated with the slow and with the fast dynamics of the nominal plant are used. These solutions do not depend upon the singular perturbation and therefore all computations are well-conditioned. The second purpose of the paper is to investigate the case when an k -independent robust controller can be used for the robust stabilization of a two-time scale nominal system. Two constructions of such controllers are developed under some additional assumptions. Numerical examples illustrate the theoretical developments.

New regulations for emission control require the improvement of the system composed by spark ignition internal combustion engine and three-way catalytic converter (TWC). In particular, an important problem is to minimize harmful emissions during the transient warm-up phase where the TWC is not working yet and, hence, a large amount of pollutants are emitted in the air. Toward this goal we present a dynamical thermo-chemical TWC model simple enough for the design and test of warm-up control strategies. The model is obtained through an asymptotic approximation of a more detailed model, i.e., by letting the adsorption coefficient between gas and substrate tend to infinity. Further, we present a fast integration algorithm based partly on a “method of lines” space-discretization, partly on the “method of characteristics” for “quasi linear” hyperbolic partial differential equations, the separation being allowed by a two time scale analysis of the system. The model has been identified, through a purposely designed genetic algorithm, and validated on experimental data.

This note considers the problem of robustness of discrete two-time-scale systems. Robustness is first analyzed for a nominal system as the general e-bound problem which characterizes an upper bound of the singular perturbation parameter. The stability robustness is then extended to handle the problem of systems subject to parameter variations. For any singular perturbation parameter bounded above by that obtained from the e-bound problem, a stability robustness bound with respect to parameter variations is derived. The results include both unstructured and structured variations.

This paper focuses on the synthesis of well-conditioned multivariable robust controllers for a broad class of multi-input multi-output two-time-scale nonlinear systems with time-varying uncertain variables, modeled within the framework of singular perturbations. The proposed controller stabilizes the fast dynamics, guarantees boundedness of trajectories, and ensures that the ultimate discrepancy between the outputs and the external reference inputs in the closed-loop system can be made arbitrarily small by an appropriate choice of controller parameters, provided that the singular perturbation parameter is sufficiently small. The developed control method is tested, through simulations, on a fluidized catalytic cracking reactor.

In this paper a reduced-order model for control design applications of a seventh-order two-time-scale discrete-time model of synchronous machines is developed. The numerical simulation is performed on synchronous machines, and the simulation results obtained using the iterative separation of time-scales provide improved accuracy of the reduced-order models, compared with that of the same obtained via the classical quasi-steady-state (qss) technique.

A two-stage observer-based controller design of discrete two-time-scale (DTTS) systems is presented in this paper. The necessary and sufficient conditions for this kind of controlled system are all addressed. The robust slow-observer-based (SOB) controller is investigated in detail. For a fast sampling rate model, prediction type observer-based controller and SOB controller are proposed. Both prediction type and current estimation type observer-based controller and SOB controllers are developed for a slow sampling rate model of DTTS systems. A twin-engine aviation aircraft is utilized to illustrate the derived schemes.

International audience; This paper addresses the state estimation of two-time scale nonlinear systems with an unknown input multi-observer (UIMO). In order to design such an observer, the nonlinear system presented in a singular perturbed form is transformed into an equivalent multiple model with unmeasurable premise variables (UPV) affected by unknown inputs (UI). Then an observer is built and the stability analysis of the state reconstruction error is performed by using the Lyapunov method that leads to the resolution of linear matrix inequalities (LMIs). The performances of the proposed estimation method are highlighted through the application to a wastewater treatment plant (WWTP) model.

Multi-rate control design of two-time-scale systems is considered via digital redesign method. After the multi-rate control design procedures via singular perturbation theory is reviewed and the relationships between slow and fast subproblems are clarified, the design method via the bi-linear transformation method is investigated. Furthermore, in order to verify the validity of this method, an application to the longitudinal control problem of an aircraft is given.

Asymptotic behaviour of a two time scale stochastic approximation algorithm is analysed in terms of a related singular ordinary differential equation.

Output feedback design of discrete-time decentralized systems with slow and fast modes is considered. Conditions for the complete separation of slow and fast subsystems are given. The slow and fast subsystem outputs, which are obtained by applying the slow and fast subcontrollers to the corresponding subsystems, will be shown to approximate those of the original system. Also, the composite control, when being applied to the original system, will place the eigenvalues sufficiently close to the desired locations.

Output feedback control of linear time-invariant singularly perturbed systems is studied. The set of all compensators that stabilize a singularly perturbed system while preserving its two-time-scale structure is parameterized. The parameterization is used to show that any two-frequency-scale stabilizing compensator can be asymptotically approximated by a compensator designed via a sequential procedure. In this procedure, a fast (high-frequency) compensator is designed first to stabilize the fast model of the system. Then, a strictly proper slow (low-frequency) compensator is designed to stabilize a modified slow model. The parallel connection of the two compensators forms a two-frequency-scale stabilizing compensator for the singularly perturbed system.

The block processing technique is applied to a class of two-time-scale linear discrete-time systems. Block state-space structures for slow and fast subsystems have been obtained. Based upon these models, blocked slow and fast controls are designed. In addition, an extra control is designed to exactly compensate for the effects of the parasitics.

The design of stabilizing feedback control of singularly perturbed diserete-time systems is decomposed into the design of slow and fast controllers which are combined to form the composite control. Composite control strategies are developed for the case of single rate measurements (all variables are measured at the same rate) as well as for the case of multirate measurements (slow variables are measured at a rate slower than that of fast variables).

Linear shift-invariant discrete-time systems satisfying the two time scale property-are considered. Conditions for a complete separation of slow and fast subsystems are given. Subsystem regulator problems are investigated and the controllability property of the system is established based on the subsystem controllability properties. A subsystem controller design method is proposed depending on the separability of the slow and fast controls.

The stabilization of a class of singularly perturbed linear time-varying systems is considered through the separate stabilization of two lower dimensional subsystems in two different time-scales. A composite stabilizing controller is synthesized from the separate stabilizing controllers of the two subsystems, the mutually independent gains of which do not require knowledge of the small singular perturbation parameter.

Linear shift-invariant discrete-time systems satisfying the two-time-scale property are considered. A transformation matrix is formed and applied to the system to separate its slow and fast modes. Two subsystems are then defined, one for the slow modes and the other for the fast modes. An eigenvalue placement technique is proposed that will place the eigenvalues of the closed loop sub-systems at desired locations. The composite controller will also be formed for the original system.

The design of a state feedback controller for a two-time-scale discrete-time system is considered via a two-stage design approach. After discrete-time modelling via slow and fast sampling rates has been discussed, state feedback design approaches for the fast sampling model (i.e. the two-stage design approach and the direct approach) are studied, and the relation between them is clarified. Following these results, the eigenvalue assignment and the stabilizing problems are investigated.

For a class of nonlinear singularly perturbed systems, feedback stabilizing and near-optimal controls are designed using two separate lower order subsystems. The two-time-scale properties greatly simplify the stability analysis and the nonlinear controller design. Two electric machine examples (a dc motor and a synchronous generator) illustrate the proposed design procedure.

A linear, shift-invariant discrete system exhibiting two-time-scale (slow and fast) character is addressed. An iterative algorithm based on linear programming for the computation of time-optimal control is presented by decoupling the original system into slow and fast subsystems. It is found that the computation time and storage capacity for the computation of time-optimal control of the slow and fast subsystems is less than that required for the original system. A numerical example is provided to illustrate the proposed algorithm.

Two-time-scale decoupling is adopted to obtain slow and fast subsystems. Based on these subsystems, a multirate composite control strategy is derived. It is shown how a recently presented multirate scheme can be modified to handle slow, nonzero mean disturbances, by introducing time incremental models. The improvements with the given multirate control law are illustrated by an example. >

International audience; The paper presents some results on global exponential stability of linear time invariant systems with different time scales. The full system is decomposed in two reduced ones by means of singular perturbations and a smooth approximation of a Variable Structure Control is synthesized for each of them using Reaching Law approach. The global closed loop stability is then proved for the whole system using Lyapunov methods and a particular state space decomposition. Moreover a literal form for \epsilon* (a parameter which measures the minimum separation between time scales) is derived.