Small Gain Theory Research Articles

Secure control for remote networked stochastic systems via integral sliding mode

This paper deals with the secure control problem for a class of networked stochastic systems with false data injection attacks via an integral sliding mode control technique. The networked control system is with a hierarchical structure, and the main controller and a remote controller are considered to realize the secure control against false data injection attacks on the network between a main controller and the plant. A mode-shared event-triggering controller is designed as the main controller, by utilizing a time delay approach. An input–output model based on a two-term approximation is applied to cope with the formulated time-varying delay. Then, the scaled small gain theory is employed to analyze the stability of the resulting system. Sufficient conditions on ensuring the desired system performance are derived and then the controller parameters are synthesized. Moreover, an elaborated sliding mode control law is proposed for the desired secure control action. Finally, two simulation examples are permitted to verify the effectiveness of the theoretical derivations.

A Small-Gain Theory for Abstract Systems on Topological Spaces

We develop a small-gain theory for systems described by set-valued maps between topological spaces. We introduce an abstract notion of stability unifying the continuity properties behind different existing concepts, such as Lyapunov stability of equilibria, sets, or motions, (incremental) input-output stability, asymptotic gain properties, and continuity with respect to fast-switching inputs. Then, we prove that a feedback interconnection enjoying a given abstract small-gain property is stable. While, in general, the proposed small-gain property cannot be decomposed as the union of stability of the subsystems and a contractiveness condition, we show that it is implied by standard assumptions in the context of input-to-state stable systems. Finally, we provide application examples illustrating how the developed theory can be used for the analysis of interconnected systems and synthesis of control systems.

H∞ Robust Control for ICPT System With Selected Weighting Function Considering Parameter Perturbations

H∞ control can guarantee the performance of a nominal ICPT system. However, the dynamic performance may suffer degradation when the parameter deviates from its nominal value. Thus, to enhance the overall dynamic performance of the ICPT system under parameter perturbations, an H∞ control method with a selected weighting function considering parameter perturbations is proposed in this article. First, a generalized state-space model is established to analyze the influence of parameter perturbations on the open-loop system in detail. After that, an uncertain model is established via frequency-domain analysis, generating a numerical solution of a weighting function used to describe the parameter perturbations. Then, the generalized plant with uncertainty is obtained and transferred into a standard H∞ optimal problem via small gain theory. Based on the proposed controller, the dynamic performance and robustness of the closed-loop system are analyzed and verified theoretically. Finally, simulation and experimental results further verify that the proposed H∞ control method can make the closed-loop system obtain good dynamic performance and strong robustness against parameter perturbations. The settling time is less than 6.8 ms without any overshoot when the system suffers parameter perturbations. Besides, the controller can maintain good dynamic performance in the startup and reference tracking experiments.

From Small-Gain Theory to Compositional Construction of Barrier Certificates for Large-Scale Stochastic Systems

This article is concerned with a compositional approach for the construction of control barrier certificates for large-scale interconnected stochastic systems while synthesizing hybrid controllers against high-level logic properties. Our proposed methodology involves decomposition of interconnected systems into smaller subsystems and leverages the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">control</i> <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sub-barrier certificates</i> of subsystems, enabling one to construct control barrier certificates of interconnected systems by employing some <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\max$</tex-math></inline-formula> -type small-gain conditions. The main goal is to synthesize hybrid controllers enforcing complex logic properties, including the ones represented by the accepting language of deterministic finite automata, while providing probabilistic guarantees on the satisfaction of given specifications in bounded-time horizons. To do so, we propose a systematic approach to first decompose high-level specifications into simple reachability tasks by utilizing automata corresponding to the complement of specifications. We then construct control sub-barrier certificates and synthesize local controllers for those simpler tasks and combine them to obtain a hybrid controller that ensures satisfaction of the complex specification with some lower bound on the probability of satisfaction. We finally apply our proposed techniques to a fully-interconnected Kuramoto network composed of 100 nonlinear oscillators.

Intelligent Control for Pressurizer System in Nuclear Power Plants

Fine control of output power for nuclear power plants is the essential goal for safe operation. In this work, a Fuzzy analytical proportional-integral-derivative (FPID) controller with different configurations is designed to adjust and control the pressure of the PZR system. The stability analysis of the FPID controller with variable gains is established, and conditions for bounded-input bounded-output stability conditions (BIBO) are derived using the small gain theory. Two scenarios are applied for evaluating the dynamic response of applied controllers. In addition, performance indices are compared between the PZR model and data measured from the PCtran VVER-1200 simulator. Finally, a simulation platform is developed for MATLAB / Simulink to implement the three-region nonlinear non-equilibrium PZR model and the designed pressure controllers. The analysis and evaluation results showed good durability of the designed controllers and satisfactory performance of the control. These results further show that the nonlinear PZR model is accurate, feasible, and valuable for dynamic simulation and control unit design.

Composite Velocity-Tracking Control for Flexible Gimbal System With Multi-Frequency-Band Disturbances

In this paper, a composite velocity-tracking control scheme is presented for improving dynamics response and control accuracy of the flexible gimbal system in the control moment gyro (CMG). Specifically, an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${H}_{\infty } $ </tex-math></inline-formula> optimal vibration controller (HOVC) is developed to weigh multi-performance of the gimbal system, such as stability, fast-tracking performance, and attenuation ability of multi-frequency-band disturbances. Meanwhile, a robust disturbance observer (RDO) designed on optimization technology is utilized to improve the nonlinear friction rejection ability of the gimbal system. Moreover, rigorous theoretical analysis based on the small-gain theory is provided to demonstrate the stability of the closed-loop gimbal system under multiple disturbances. Finally, experimental studies are presented to verify the effectiveness of the proposed method.

Set stability of infinite networks: ISS small-gain theory and its applications

Motivated by a paradigm shift towards a hyper-connected world, we develop a computationally tractable small-gain theorem for networks of infinitely many subsystems, termed as infinite networks. The proposed small-gain theorem addresses exponential input-to-state stability with respect to closed sets, which enables us to analyze diverse stability problems in a unified manner. The small-gain condition, expressed in terms of the spectral radius of a gain operator collecting all the information about the internal Lyapunov gains, can be numerically checked efficiently for a large class of systems. To demonstrate broad applicability of our small-gain theorem, we apply it to consensus of infinitely many agents, and to the design of distributed observers for infinite networks.

Distributed optimization for multi-agent system over unbalanced graphs with linear convergence rate

Distributed optimization over unbalanced graphs is an important problem in multi-agent systems. Most of literatures, by introducing some auxiliary variables, utilize the Push-Sum scheme to handle the widespread unbalance graph with row or column stochastic matrix only. But the introduced auxiliary dynamics bring more calculation and communication tasks. In this paper, based on the in-degree and out-degree information of each agent, we propose an innovative distributed optimization algorithm to reduce the calculation and communication complexity of the conventional Push-Sum scheme. Furthermore, with the aid of small gain theory, we prove the linear convergence rate of the proposed algorithm.

Fault tolerant control using integral sliding modes with control allocation along the null-space

In this paper, a new fault tolerant control scheme with control allocation is presented. The pseudo-inverse along the null-space control allocation is applied to the fault tolerant control system to handle the actuator constraints. The stability of the overall closed-loop system is proved via the small gain theory. The null-space vector is viewed as uncertainty, and is disposed by an integral sliding mode controller and a robust controller. The simulation results show that the new method can solve both failure scenarios and actuator saturation problems well.

Research of Mapping Element Space Method for Uncertain Lurie System Stability Diagnosis

A new diagnosis method of Lurie system stability by using Chaotic time series data was proposed in order to solve stability analysis of flight control system with fault and uncertain. Firstly, the reasons for the instability of the nonlinear system of Ruri leaf was analyzed by using the small gain theory and linear matrix inequalities in the robust control theory. And the stability conditions of Rui leaf system under the condition of uncertainty and failure were proved theoretically. In order to quantify the stability of nonlinear systems in complex Ruri leaves, based on the theory of phase space reconstruction, the nonlinear Ruri system with continuous discrete characteristics was converted into an approximate time discrete equation, mapping it to a low dimensional primitive space, by introducing Q Gauss function into Kernel function, the generalization ability of neural networks are enhanced, realizing the stability analysis method based on the characteristic of primitive migration, which is suitable for various fault and uncertainty conditions, and the relative quantized stability norm can be given. The simulation shows that the present method can effectively solve the stability analysis and determination of flight control system under various factors.

A novel observer-predictor control for uncertain systems with unknown time-varying input and output delays

In this paper, a novel observer-predictor control is proposed to stabilise uncertain systems subject to unknown but bounded arbitrarily-fast time-varying input and output delays. Due to the presence of time-varying model uncertainties and delay mismatches, the separation principle cannot be used to design the observer and control gains separately, which becomes a challenging problem. By applying small gain theory and Lyapunov-Krasovskii functionals (LKF), in combination with Bessel-Legendre inequalities and the Projection Lemma, we develop sufficient conditions based on Linear Matrix Inequalities (LMIs) for control design. As a result, the closed-loop robust performance can be improved whilst satisfying a prescribed decay rate in the closed-loop dynamics at the expense of introducing more LMI decision variables without increasing the complexity of the control scheme. Also, the Extended State Observer (ESO) is integrated in the proposed control scheme to improve the rejection performance of mismatched disturbances. The effectiveness of the proposed method is illustrated by simulation results.

Analysis of the Small Signal Gain of a Sheet Beam BWO With a Partially Dielectric-Loaded Nonuniform Grating

A partially dielectric-loaded (PDL) nonuniform grating with lower ohmic losses for a sheet beam backward wave oscillator (BWO) and a clinotron is proposed in this paper. The expressions of the electromagnetic fields for the PDL nonuniform-grating-based slow-wave structure (SWS) are derived by using Maxwell’s equations and Floquet’s theorem. The dispersion equation is derived using the field-matching method and is verified by simulation. The expression of the Pierce small signal gain of a BWO with a PDL nonuniform grating is obtained by using the electromagnetic field expressions and the small signal gain theory. The effects of different parameters on the gain of a BWO with a nondielectric-loaded (NDL) nonuniform grating and those of a BWO with a PDL nonuniform grating are discussed, and the mode competition of this kind of BWO is analyzed in terms of gain. The oscillation frequencies obtained from the mode competition analysis are verified by a particle-in-cell (PIC) simulation, confirming the validity of the analysis.

Data-driven global robust optimal output regulation of uncertain partially linear systems

In this paper, a data-driven control approach is developed by reinforcement learning ( RL ) to solve the global robust optimal output regulation problem ( GROORP ) of partially linear systems with both static uncertainties and nonlinear dynamic uncertainties. By developing a proper feedforward controller, the GROORP is converted into a global robust optimal stabilization problem. A robust optimal feedback controller is designed which is able to stabilize the system in the presence of dynamic uncertainties. The closed-loop system is assured to be input-to-output stable regarding the static uncertainty as the external input. This robust optimal controller is numerically approximated via RL. Nonlinear small-gain theory is applied to show the input-to-output stability for the closed-loop system and thus solves the original GROORP. Simulation results validates the efficacy of the proposed methodology.

Stabilization of Uncertain Feedforward Nonlinear Systems With Application to Underactuated Systems

This note presents saturated controllers to stabilize uncertain feedforward nonlinear systems whose nominal dynamics contains uncertain “gains” and is subject to linear perturbations. We assign a class of saturated controllers within which multiplicative coefficients appear before both states and saturation functions, and the convergence analysis is conducted in the following way: to verify the reduction of saturated terms, we calculate the time derivative of a boundary surface in a small domain when the states reach the boundary surface; to prove the asymptotic stability of the corresponding reduced system, we use an $M$ -matrix-based comparison principle. In this way, the suggested algorithm does not depend on the small-gain theory, and is suitable for dealing with strongly nonlinear underactuated systems including the cart-pendulum system, the overhead crane system, and the uncertain vertical takeoff and landing vehicle.

Data-Driven Shared Steering Control of Semi-Autonomous Vehicles

This paper presents a cooperative/shared framework of the driver and his/her semi-autonomous vehicle in order to achieve desired steering performance. In particular, a copilot controller and the driver together operate and control the vehicle. Exploiting the classical small-gain theory, our proposed shared steering controller is developed independent of the unmeasurable internal states of the human driver, and only relies on his/her steering torque. Furthermore, by adopting data-driven adaptive dynamic programming and an iterative learning scheme, the shared steering controller is studied from the measurable data of the driver and the vehicle. Meanwhile, the accurate knowledge of the driver and the vehicle dynamics is unnecessary, which settles the problem of their potential parametric variations/uncertainties in practice. The effectiveness of the proposed method is validated by rigorous analysis and demonstrated by numerical simulations.

A Robust Resonant Controller for High-Speed Scanning of Nanopositioners: Design and Implementation

This brief presents a novel damping control scheme for piezoactuated nanopositioning platforms with robust resonant control (RRC). The RRC is developed to attenuate the resonant-vibrational modes of the lightly damped dynamics of the stage in an inner positive feedback loop. The parameters in the proposed RRC are determined through an analytical approach. Indeed, the controller gains constrained by both the robustness and the damping ratio of the inner loop are tuned based on the small gain theory. Then, a high gain integral tracking controller is applied in the outer loop to minimize the tracking errors due to unmodeled nonlinearity and uncertainties. To validate the effectiveness of the proposed RRC, comparative experiments with conventional positive position feedback (PPF) and integral resonant control (IRC) are conducted on a piezoactuated nanopositioning stage. Results demonstrate that the proposed RRC improves the closed-loop bandwidth from 67 Hz with PPF and 135 Hz with IRC to 176 Hz. Moreover, better robustness against load variations with a range of 0–1000-g loading under 0–20-Hz input raster scanning signals are obtained by RRC compared with PPF as well as IRC.

Enhanced Predictor-Based Control Synthesis for Discrete-Time TS Fuzzy Descriptor Systems With Time-Varying Input Delays

We present a novel predictor-feedback gain-scheduled control synthesis method for discrete-time Takagi-Sugeno (TS) fuzzy descriptor systems with unknown time-varying input delays. The goal is to reduce the computational complexity in the control design by keeping the descriptor form whilst improving the closed-loop robust performance. It is worth mentioning the following aspects: 1) our approach easily allows dealing with parametric uncertainties on the descriptor matrix, which are generally difficult to handle via matrix inversion; and 2) the control design problem is casted into linear matrix inequality conditions by using nonquadratic Lyapunov functions, the Projection Lemma and small gain theory. Finally, three examples are provided to show the effectiveness of the proposed method.

Novel consistency control strategy for jet dispensing

A new consistency control method for jet dispensing is proposed. First, the working parameters, namely, viscosity, supply pressure and supply time, are experimentally investigated. Then, the glue viscosity is approximated by a polynomial model using the least square method. Based on this model and temperature control implemented using the Dahlin principle, the viscosity of the glue can be maintained at a constant value. Then, the viscosity model of the glue is applied to deriving the droplet mass as the nominal model of the temperature controller. The robustness of the temperature controller is analyzed by applying the small gain theory. The glue supply pressure controller is designed using the consistency control strategy, and the robustness is analyzed. Finally, simulations and experiments are conducted using a jet dispensing control system. The results show that the proposed control strategy can significantly improve the droplet consistency.

Nonconservative Discrete-Time ISS Small-Gain Conditions for Closed Sets

This paper presents a unification and a generalization of the small-gain theory subsuming a wide range of existing small-gain theorems. In particular, we introduce small-gain conditions that are necessary and sufficient to ensure input-to-state stability (ISS) with respect to closed sets. Toward this end, we first develop a Lyapunov characterization of $\omega$ ISS via finite-step $\omega$ ISS Lyapunov functions. Then, we provide the small-gain conditions to guarantee $\omega$ ISS of a network of systems. Finally, applications of our results to partial ISS, ISS of time-varying systems, synchronization problems, incremental stability, and distributed observers are given.

Stochastic and adaptive optimal control of uncertain interconnected systems: A data-driven approach

This paper provides a novel non-model-based, data-driven stochastic H∞ control design for linear continuous-time stochastic interconnected systems with unknown dynamics. Our contributions are three-fold. First, we develop a tool to show how to assign an arbitrarily small input-to-output stochastic L2 gain of the closed-loop system, by combining the gain assignment technique with the zero-sum dynamic game-based H∞ control design. Second, robustness to dynamic uncertainties is tackled using the small-gain theory. Third, we develop a non-model-based stochastic robust adaptive dynamic programming (RADP) algorithm for adaptive optimal controller design. In sharp contrast to the existing methods, the obtained algorithm is based on value iteration (VI), and the knowledge of an initial stabilizing control policy is no longer needed. An example of a power electronic system is adopted to illustrate the obtained results.