Sufficient Conditions For Robust Stability Research Articles

Robust stability and stabilization conditions for uncertain switched positive systems under mode‐dependent dwell‐time constraints

AbstractThe article addresses the problems of robust stability and stabilization in exponential sense for uncertain switched positive systems (SPSs) under mode‐dependent dwell‐time constraints including mode‐dependent constant dwell time (MDCDT) constraint, mode‐dependent minimum dwell time (MDMDT) constraint, and mode‐dependent ranged dwell time (MDRDT) constraint. At first, the mixed time‐varying delay case is studied, and sufficient conditions of robust stability and stabilization in exponential sense are provided for uncertain SPSs with MDMDT, MDCDT, and MDRDT constraints, respectively. Next, the robust stability and stabilization issues are also studied for SPSs in delay‐free case. With mode‐dependent dwell‐time constraints, a new discretized linear copositive Lyapunov–Krasovskii functional technique is introduced in this article. Meanwhile, for some specific situations, the proposed conditions of robust stability and stabilization can be degenerated into the ones for SPSs in mode‐independent dwell‐time constraint case and in non‐uncertainty case, respectively. At last, we give five examples to explain the importance and effectiveness of the results.

An explicit robust stability condition for uncertain time-varying first-order plus dead-time systems

First-order plus dead-time (FOPDT) models are broadly used in process control to represent damped dynamic processes with time delays. An explicit condition for parameter- and delay-dependent robust stability of FOPDT systems with varying uncertain parameters and delay is derived in this paper. An internal model control (IMC) approach is proposed to parameterize stabilizing controllers that satisfy the output tracking objective in time-varying FOPDT systems represented by an uncertain first-order dynamic model with a time-varying delay in the control input. The small-gain theorem is used to derive an explicit necessary and sufficient parameter-dependent robust stability condition as a function of the nominal system gain, nominal varying delay, nominal time constant, and the bounds of the parameter uncertainties. An equivalent proportional-integral-derivative (PID) controller is then extracted to facilitate the implementation of the proposed IMC-based robust control. The application of the proposed explicit robust stability condition is studied in the context of air-fuel ratio (AFR) control in lean-burn spark ignition (SI) engines with a large time-varying transport delay in the control loop due to the placement of the universal exhaust gas oxygen (UEGO) sensor downstream the catalytic converter.

Stability analysis of deterministic and Markovian switching functional difference systems described by Stieltjes integrals

This paper studies the stability analysis of general classes of deterministic and Markovian switching functional difference systems (DFDSs and MSFDSs) described by Stieltjes integrals. A new general Lyapunov–Krasovskii (L‐K) functional stability theorem based on the Stieltjes integrals form of DFDSs is presented. In order to obtain a uniform form of stability conditions, the multiple Jensen inequalities are generalized to the case of Stieltjes integrals. With the help of these inequalities, two novel L‐K functionals with Stieltjes integrals are established to obtain sufficient stability and robust stability conditions based on linear matrix inequalities (LMIs). The results are then generalized to mean square exponential stability of MSFDSs. Finally, numerical examples are given to illustrate the effectiveness of the proposed methods.

Robust Finite-time Boundedness of Discrete-time Neural Networks with Time-varying Delays

This paper is concerned with the problem of robust finite-time boundedness for the discrete-time neural networks with time-varying delays. By constructing an appropriate Lyapunov-Krasovskii functional, we propose the sufficient conditions which ensure the robust finite-time boundedness of the discrete-time neural networks with time-varying delay in terms of linear matrix inequalities. Then the sufficient conditions of robust finite-time stability for the discrete-time neural networks with time-varying delays are given. Finally, a numerical example is presented to illustrate the efficiency of proposed methods.

Robust stability and stabilization of hybrid fractional-order multi-dimensional systems with interval uncertainties: An LMI approach

The hybrid fractional-order multi-dimensional (N-D) systems described by Roesser model are introduced in this paper. The N-D systems include discrete-time dimensions and fractional-order continuous-time dimensions with fractional-order 0<αi<1. Firstly, some novel sufficient stability conditions for nominal hybrid fractional-order N-D systems are presented. Then, against interval uncertainties, the sufficient conditions for robust stability and stabilization of hybrid fractional-order N-D systems are derived. All the results are in the form of linear matrix inequalities. Finally, illustrative examples are given to verify the validity of our results, and demonstrate that our results are less conservative than the existing ones.

M-Matrix-Based Robust Stability and Stabilization Criteria for Uncertain Switched Nonlinear Systems with Multiple Time-Varying Delays

This paper focuses on the robust stability and the memory feedback stabilization problems for a class of uncertain switched nonlinear systems with multiple time-varying delays. Especially, the considered time delays depend on the subsystem number. Based on a novel common Lyapunov functional, the aggregation techniques, and the Borne and Gentina criterion, new sufficient robust stability and stabilization conditions under arbitrary switching are established. Compared with existing results, the proposed criteria are explicit, simple to use, and obtained without finding a common Lyapunov function for all subsystems through linear matrix inequalities, considered very difficult in this situation. Moreover, compared with the memoryless one, the developed controller guarantees the robust stability of the corresponding closed-loop system with more performance by minimizing the effect of the delays in the system dynamics. Finally, two numerical simulation examples are shown to prove the practical utility and the effectiveness of the proposed theories.

LMI-Based Robust Stability Analysis of Discrete-Time Fractional-Order Systems With Interval Uncertainties

Robust stability problem of discrete-time fractional-order systems (DTFOSs) with interval uncertainties is investigated in this paper. Firstly, a new theorem for matrix root-clustering in union-region is established. Based on this theorem, the stability regions of DTFOSs are described as the union-region of closed sub-regions, and sufficient conditions for stability of DTFOSs are presented. Then, new sufficient conditions for robust stability of DTFOSs with interval uncertainties are derived. All the results are obtained in terms of linear matrix inequalities (LMIs) which are more tractable than the existing ones. Finally, numerical examples are given to show that our results are valid and less conservative than the existing ones.

A new approach based on the discriminant system of polynomial for robust stability and stabilization of two-dimensional systems

In this paper, we present necessary and sufficient stability and robust stability conditions for two-dimensional (2D) systems described by the Fornasini-Marchesini (FM) second model in terms of the discriminant systems of polynomial. This paper simplifies the traditional method of stability into a tractable method by the fractional linear transformation (FLT). More specifically, we reduce the stability analysis to a easy issue whether some polynomials are positive definite. Then we use the same idea to consider the stabilization and robust stabilization issues. Finally, the effectiveness of the proposed results is demonstrated by a practical example and two numerical examples.

Robust Current Control of Voice Coil Motor in Tip-Tilt Mirror Based on Disturbance Observer Framework

In this paper, a control methodology based on the disturbance observer (DOB) framework is proposed to improve the current dynamic characteristic and the robustness of the voice coil motor (VCM) in the tip-tilt mirror. It is shown that the open-loop current characteristic is a third-order plant, which is influenced obviously by both the electrical and mechanical uncertainties under different environment temperatures. Although the traditional PID control is helpful to improve the current dynamics, it is not able to provide enough robustness. Therefore, a new control methodology based on DOB is proposed. The plant model is simplified from third-order to first-order by considering the back-electromotive force (EMF) as the disturbance. The perturbation of the electrical parameters is compensated by the inner-loop of the DOB framework, and the perturbation of the mechanical parameters is compensated by suppressing the back-EMF. Moreover, new sufficient conditions for robust stability are presented in the control method based on H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> algorithm. Compared with the traditional researches, the upper bound of the uncertainties is calculated directly in this paper. Therefore, the ranges of the electrical parameters are very clear according to the Q-filter. Based on the stability conditions, new design guidelines for the DOB-based robust current control for induction motor are also presented. Experiment results are given to show the design details and the superiority of the proposed control method.

Time-Varying Tube-Based Output Feedback MPC for Constrained Linear Systems with Intermittently Delayed Data

Abstract This paper proposes a time-varying tube-based output feedback model predictive control (MPC) design for constrained linear systems in the presence of intermittently delayed observations, where the delayed/missing data patterns for each period satisfy a finite-length language. The design consists of a dynamic state estimator whose estimation errors satisfy equalized recovery (a weaker form of invariance with time-varying finite bounds), as well as an output feedback control law that extends existing tube-based output feedback MPC approaches to allow time-varying tubes for tightening the original state and input constraints. The resulting time-varying tube-based output feedback MPC design is robust to time-varying disturbances and errors, including when the observations are intermittently delayed. Further, we provide sufficient conditions for recursive feasibility and robust exponential stability of the proposed design. Simulation results demonstrate that the proposed approach is able to robustly stabilize and control a constrained linear system despite disturbances, noise and missing/delayed data.

Sufficient Conditions for Robust Frequency Stability of AC Power Systems

This paper analyses the frequency stability of ac grids in the presence of non-dispatchable generation and stochastic loads. Its main goal is to evaluate conditions in which the system is robust to large, persistent active power disturbances without recurring to time-domain simulations. Considering the ongoing energy transition to more renewable sources, defining robustness boundaries is a key topic for power system planning and operation. However, much of the research on long-term studies has not dealt with robust dynamic constraints, while short-term analyses usually depend on time-consuming simulations to evaluate nonlinearities. To bridge this gap, the authors derive an algebraic equation that provides sufficient conditions for robust frequency stability in ac power systems and a relationship among four key quantities: the maximum active power perturbation, the minimum system damping, the steady-state and the transient frequency limits. To achieve this goal, it uses a nonlinear average-model of the ac grid and Lyapunov's direct method extended by perturbation analysis requiring only limited knowledge of the system parameters. The algebraic calculations are validated using time-domain simulations of the IEEE 39-bus test system and results are compared to the traditional Swing Equation model.

Strong Stability Analysis of Linear Delay-Difference Equations With Multiple Time Delays

This article is concerned with the strong stability problem of linear continuous-time delay-difference equations with multiple time delays. A family of linear matrix inequalities (LMIs), indexed by a positive integer k, is derived to assess strong stability. A time-domain interpretation of the proposed LMI-based condition is given in terms of a quadratic integral Lyapunov functional, which allows us to reveal relations with an existing result. The LMI condition can easily be reformulated in a form where the dependence on the coefficients of the delay-difference equation is affine, which is instrumental to establishing a sufficient LMI condition for robust strong stability of delay-difference equations with norm-bounded uncertainty. A necessary and sufficient condition for robust strong stability is also given, in the form of a structured singular value characterization. Examples are given to illustrate the effectiveness of the presented results.

Online learning constrained model predictive control based on double prediction

SummaryA data‐based predictive controller is proposed, offering both robust stability guarantees and online learning capabilities. To merge these two properties in a single controller, a double‐prediction approach is taken. On the one hand, a safe prediction is computed using Lipschitz interpolation on the basis of an offline identification dataset, which guarantees safety of the controlled system. On the other hand, the controller also benefits from the use of a second online learning‐based prediction as measurements incrementally become available over time. Sufficient conditions for robust stability and constraint satisfaction are given. Illustrations of the approach are provided in a simulated case study.

Data-driven robust stabilization with robust domain of attraction estimate for nonlinear discrete-time systems

Nonlinear robust control is pursued by overcoming the drawback of linear robust control that it ignores available information about existing nonlinearities and the resulting controllers may be too conservative, especially when the nonlinearities are significant. However, most existing nonlinear robust control approaches just consider the affine nonlinear nominal model and thereby ignore available information about existing non-affine nonlinearities. When the general nonlinear nominal model is considered, the robust domain of attraction (RDOA) of closed-loops requires extensive investigation because it is hard to achieve the global stabilization. In this paper, we propose a new nonlinear robust control method based on Lyapunov function to stabilize a discrete-time uncertain system and to estimate the RDOA of closed-loops. First, a sufficient condition for robust stabilization of all plants in a plant set and estimation of the RDOA of all closed-loops is proposed. Then, to tackle the non-affine nonlinearities, a data-driven method of estimating the robust negative-definite domains (RNDD) is presented, and based on it the estimation of the RDOA of closed-loops and the resulting controller design are also given.

Robust finite-time stability of uncertain neutral nonhomogeneous fractional-order systems with time-varying delays

This article addresses the problem of finite-time stability for uncertain neutral nonhomogeneous fractional-order systems with time-varying delays where a stability test procedure is suggested. Based on the extended form of the generalized Gr?nwall inequality, a new sufficient condition for robust finite-time stability of such systems is established. Finally, a numerical example is given to show the effectiveness of the obtained result.

Hybrid Systems With Delayed Jumps: Asymptotic Stability via Robustness and Lyapunov Conditions

Hybrid systems subject to delayed jumps form a class of dynamical systems with broad applications. This article develops sufficient conditions for robust asymptotic stability of hybrid systems in the presence of delayed jumps. More precisely, given a delay-free hybrid system, we introduce a higher order delayed system parametrized by the length of delays. We show that when the delay parameter is set to zero, the higher order model captures the solutions of the delay-free system. Under mild conditions, it is shown that when the delay-free system has an asymptotically stable compact set, for small enough delays, solutions of the delayed system converge to a neighborhood of a set of interest related to the aforementioned compact set. Then, Lyapunov functions for the delay-free system are used to develop sufficient conditions for asymptotic stability in the presence of delays. Unlike prior work in the literature, the results pertaining to these notions of stability hold for systems possessing Zeno solutions, with time-varying delays. Importantly, the required conditions are expressed in finite-dimensional space, and depend primarily on the data of the delay-free system. The practical stability result is validated numerically through the hybrid system model of a controlled boost converter circuit with state-triggered switches and Zeno solutions. The higher order model and the derived Lyapunov conditions are utilized to obtain quantitative bounds on maximum allowable delays for switched systems and sampled-data control.

Fault-Tolerant Resilient Control For Fuzzy Fractional Order Systems

This paper investigates the problem of resilient fault-tolerant control for a class of Takagi–Sugeno fuzzy model which is described by fractional order differential equations. In particular, the system input under consideration is subject to actuator saturations, actuator faults, and nonlinearities. The main purpose of this paper is to put forward a generalized nonlinear resilient fault-tolerant controller that can make the considered system robustly asymptotically stable. Utilizing the concept of continuous frequency distribution, a new set of linear matrix inequalities, which are the sufficient conditions for robust asymptotic stability of the closed-loop system, are derived by using the indirect Lyapunov approach. Two numerical examples are included to illustrate the feasibility and effectiveness of the proposed method.

Robust local and string stability for a decentralized car following control strategy for connected automated vehicles

This paper presents a robust car-following control strategy under uncertainty for connected and automated vehicles (CAVs). The proposed control is designed as a decentralized linear feedback and feedforward controller with a focus on robust local and string stability under (i) time-varying uncertain vehicle dynamics and (ii) time-varying uncertain communication delay. The former uncertainty is incorporated into the general longitudinal vehicle dynamics (GLVD) equation that regulates the difference between the desired acceleration (prescribed by the control model) and the actual acceleration by compensating for nonlinear vehicle dynamics (e.g., due to aerodynamic drag force). The latter uncertainty is incorporated into acceleration information received from the vehicle immediately ahead. As a primary contribution, this study derives and proves (i) a sufficient and necessary condition for local stability and (ii) sufficient conditions for robust string stability in the frequency domain using the Laplacian transformation. Simulation experiments verify the correctness of the mathematical proofs and demonstrate that the proposed control is effective for ensuring stability against uncertainties.

Robust stability of impulsive switched neural networks with multiple time delays

A class of switched neural networks (SNNs) in the presence of parameter uncertainties, impulsive effects, and both variable and continuously distributed delays are investigated. Firstly, the existence and uniqueness conditions of equilibrium point of each subsystem are proposed, which implies that the addressed SNNs has a unique equilibrium point. Secondly, we derive several sufficient conditions for robust exponential stability under average dwell time (ADT) switching and arbitrary switching, respectively. These conditions are formulated in terms of algebraic inequalities and M-matrices and are derived by employing inequality technique incorporated with the idea of vector Lyapunov function. Lastly, the less conservativeness and effectiveness of the obtained results over some existing literature are verified by simulation examples.

Stability of an Interval Family of Differential-Algebraic Equations with Variable Coefficients

We consider a linear nonstationary system of ordinary differential equations with interval coefficients which is not solvable with respect to the derivative of the unknown vector-valued function for any matrix coefficients in a given interval family. We obtain conditions sufficient for the preservation of the internal structure of the system. Under conditions guaranteeing the structure preservation, we obtain sufficient and necessary robust stability conditions. A variable rank of matrix coefficients and an arbitrary high unsolvability index are allowed.