Proposed Stability Conditions Research Articles

Stability Analysis and Stabilization of Fractional‐Order Systems With Distributed Delay

ABSTRACTThis paper investigates the stability and stabilization problems of fractional‐order systems with distributed delays (FOSDDs). Firstly, with the frequency sweeping technique and generalized Kalman‐Yakubovi‐Popov (KYP) lemma, a novel stability condition for FOSDDs is proposed. Secondly, based on the proposed stability condition, a robust stability condition for the uncertain FOSDDs with norm‐bounded uncertainties is derived. Thirdly, based on the proposed robust stability condition, a novel stabilization condition for the uncertain FOSDDs with norm‐bounded uncertainties is obtained. Finally, several numerical examples are provided to show the effectiveness of the proposed results. Moreover, in the numerical examples, the comparison of the proposed results with the existing results for FOSDDs is shown to demonstrate that the obtained results are less conservative.

Enhanced Looped Lyapunov Functional for Sampled-Data Control for T-S Fuzzy Systems with Time Delay

This paper addresses the problem of sampled-data control in T-S fuzzy systems with time delays. Initially, a negative-definite criterion is developed for the matrix containing membership functions with the help of the idea of a switching mechanism. Building upon this criterion, a membership-function-based looped Lyapunov functional is introduced. Furthermore, a sampling-time-dependent looped Lyapunov functional is constructed, incorporating crucial information about the sampling instance. Notably, this functional matrix is not constrained to satisfy positive-definiteness, and it varies with each sampling point. By utilizing the enhanced looped Lyapunov functional, less conservative stability conditions and a new controller criterion for the T-S fuzzy system with time delays are derived. This demonstrates that our proposed stability conditions allow for a larger sampling interval compared to existing work. Finally, the proposed method’s effectiveness is demonstrated through the validation of two simulation examples.

Predefined time quasi-sliding mode control with fast convergence based on a switchable exponent for nonlinear systems

This paper proposed a new predefined time nonsingular sliding mode control (SMC) method for nonlinear systems. Firstly, based on the definition of predefined time stability (PTS), a new sufficient condition is designed to ensure that the system states converge to the origin within a predefined time. The design of a simple variable exponent not only guarantees PTS, but also enables adaptive adjustment when the system states are far away from and near the equilibrium point. And compared with traditional methods, the proposed Lemma 2 enhances the control effect and achieves faster convergence whether the system states are far from or near to the equilibrium point. Secondly, based on the proposed stability condition, a new nonsingular SMC method is designed to ensure that the tracking error converges to an arbitrarily small region within a predefined time. Finally, the proposed method is verified to have good performance through simulation and physical experiments.

Stability analysis of discrete-time systems with arbitrary delay kernels based on kernel-related summation inequality and model transformation

In the existing papers considering the stability analysis of discrete-time systems with distributed delays via the Lyapunov-Krasovskii functional (LKF) method, the delay kernels are normally restricted to be non-negative. In this paper, we aim to remove such a restriction and deal with the stability analysis of systems with arbitrary delay kernels. For this purpose, a kernel-related summation inequality is first constructed. Then, a stability condition is derived based on the proposed inequality and a model transformation. Finally, two numerical examples are presented to show that the proposed stability condition not only has a wider scope of application and is less conservative than the existing ones.

Bumpless transfer switched control of aircraft for heavy payload dropping missions

Heavy payload airdrops induce significant changes in flight dynamics, posing a formidable challenge in flight controller design. This paper is concerned with the flight control issue of aircraft for heavy payload dropping missions. A multi-equilibrium switched systems scheme is introduced to describe dramatic variations in flight dynamics and trim points, thus covering complex scenarios where the aircraft drops heavy payloads at non-central positions. Then, a bumpless transfer switched controller for the underlying systems is proposed, with the aim of ensuring system stability and suppressing undesirable control bumps. In contrast to existing studies on switched systems with a common equilibrium, the proposed stability conditions and bumpless transfer methods are extended to encompass the realm of multi-equilibrium switchings. Simulations and hardware-in-loop experiments demonstrate that the proposed method provides a theoretical assurance for the stability of the flight control system and exhibits superior bumpless transfer performance.

Output feedback control of interval type-2 T-S fuzzy fractional order systems subject to actuator saturation

This paper investigates the stabilization problem of interval type-2 (IT2) Takagi-Sugeno (T-S) fuzzy fractional order systems (FOSs). Firstly, IT2 fuzzy model is constructed to approximate the subject to actuator saturation of systems. Then, the sufficient conditions of output feedback control of IT2 fuzzy FOSs in terms of linear matrix inequalities (LMIs) are presented by using the method involving matrix singular value decomposition (SVD). Furthwemore, we solve optimization problems to estimate stable regions according to the proposed stability conditions. Finally, we proposed a numerical example to demonstrate the availability of theoretical analysis results.

On impulse-free solutions and stability of switched nonlinear differential–algebraic equations

In this paper, we investigate solutions and stability properties of switched nonlinear differential–algebraic equations (DAEs). We introduce a novel concept of solutions, called impulse-free (jump-flow) solutions, and provide a geometric characterization that establishes their existence and uniqueness. This characterization builds upon the impulse-free condition utilized in previous works such as Liberzon and Trenn (2009, 2012), which focused on linear DAEs. However, our formulation extends this condition to nonlinear DAEs. Subsequently, we demonstrate that the stability conditions based on common Lyapunov functions, previously proposed in our work (Chen and Trenn, 2022) (distinct from those in Liberzon and Trenn (2012)), can be effectively applied to switched nonlinear DAEs with high-index models. It is important to note that these models do not conform to the nonlinear Weierstrass form. Additionally, we extend the commutativity stability conditions presented in Mancilla-Aguilar (2000) from switched nonlinear ordinary differential equations to the case of switched nonlinear DAEs. To illustrate the efficacy of the proposed stability conditions, we present simulation results involving switching electrical circuits and provide numerical examples. These examples serve to demonstrate the practical utility of the developed stability criteria in analyzing and understanding the behavior of switched nonlinear DAEs.

Unified stability criteria of asynchronous discrete-time impulsive switched delayed systems: bounded admissible edge-dependent average dwell time method

This article investigates input-to-state stability (ISS) problem of asynchronous discrete-time impulsive switched delayed systems, in which switching and impulse may be asynchronous. A novel bounded admissible edge-dependent average dwell time (BAED-ADT) switching signal is designed. Based on this division rule of switching subsystems, we adopt the slow switching method for the stable subsystems and the fast switching method for the unstable subsystems, so that stable subsystems can compensate unstable subsystems. Combined with admissible edge-dependent average impulsive interval (AED-AII) impulsive signal, a new unified ISS stability result is established, which could be applied to impulsive switched systems with an arbitrary combination of unstable and stable subsystems. Without solving the linear matrix inequality, it is allowed to readjust the AED-ADT boundary of the newly proposed stability condition according to the actual impulsive and switching signal. Further, compared with the existing fruits, the new stability condition is an extension and improvement of the previous one, which is less conservative. Several numerical examples are provided to show the validity of the conditions and the advantages of the theoretic results.

Complete Large-Signal Stability Analysis of DC Distribution Network via Brayton-Moser’s Mixed Potential Theory

For a nonlinear RLC network, Brayton and Moser have proposed the so-called general mixed potential function (GMPF) whose time-derivative is negative semi-definite. Then, the equilibrium of the RLC network is stable if it is a local minimum of the GMPF. Therefore, Brayton-Moser’s mixed potential theory is a powerful methodology, which has been widely used in the stability analysis of DC microgrid. However, most of the results in existing references are flawed and incomplete. This paper carries out the complete stability analysis of the DC distribution network with constant power loads via Brayton-Moser’s mixed potential theory. Firstly, several critical points in this theory that are often mistaken are emphasized. Secondly, based on Brayton-Moser’s mixed potential theory, the condition that the equilibrium is a local minimum is proposed. All the low-voltage equilibria are proved to be unstable, and only the high-voltage equilibrium can be stabilizable and the complete stability conditions are provided. Thirdly, some unsolved problems about the stability issue of DC microgrid via Brayton-Moser’s mixed potential theory are presented. Finally, hardware-in-the-loop (HIL) experimental results verify the proposed stability conditions.

Design of finite-/fixed-time ISS-Lyapunov functions for mechanical systems

For a canonical form of mechanical systems defined through gradients of potential energy and dissipative terms, the conditions of finite-time and fixed-time (integral) input-to-state stability are derived by finding suitable Lyapunov functions. The proposed stability conditions are constructive, which is demonstrated in several applications.

On input‐to‐state stability Lyapunov functions for mechanical systems

AbstractFor two canonical models of mechanical systems with disturbances presented by Rayleigh‐ and Liénard‐type equations, several designs of Lyapunov functions are investigated. Under given restrictions, the existence of these functions implies global or local input‐to‐state stability property for the systems. The proposed stability conditions are constructive, which is demonstrated in applications.

Sampled‐data‐based H∞ load frequency control for power systems with wind power

AbstractLoad disturbances, the intermittent generation of wind power, and the uncertainty of sampling period will significantly affect the frequency stability of power system. This paper presents a sampled‐data‐based load frequency control (LFC) scheme for power systems with wind power (PSWPs), which aims to guarantee the stable operation of the systems under a sampling period as large as possible while retaining the desired robust performance. First, by considering the feature that power system plant operates in continuous‐time domain and LFC controller operates in discrete‐time domain, the closed‐loop PSWPs installed with a PI‐type LFC controller is modeled as sampled‐data control system with aperiodic samplings. Second, by utilizing an augmented two‐side looped Lyapunov functional and the linear matrix inequality (LMI), two sampling‐interval‐dependent stability criteria with less conservatism are derived for the studied systems, which guarantee that the presented LFC scheme operates in a large sampling period. Then, based on proposed stability condition, a sampled‐data‐based PI‐type LFC controller with the consideration of sampling period and performance index is obtained. Following this, an algorithm is given to obtain the gain of sampled‐data‐based PI‐type LFC controller and the admissible maximum sampling period while preserving the desired robust performance. Finally, case studies are carried out on the one‐area power system and three‐area PSWPs. The simulation results demonstrate the effectiveness of the presented LFC scheme, and robustness against load disturbances, wind power fluctuations and sampling period. It also shows that the designed LFC scheme can tolerate a larger sampling period while preserving the desired performance.

Stability of finite horizon optimisation based control without terminal weight

This paper presents a stability analysis tool for model predictive control (MPC) where control action is generated by optimising a cost function over a finite horizon. Stability analysis of MPC with a limited horizon but without terminal weight is a well-known challenging problem. We define a new value function based on an auxiliary one-step optimisation related to stage cost, namely optimal one-step value function (OSVF). It is shown that a finite horizon MPC can be made to be asymptotically stable if OSVF is a (local) control Lyapunov function (CLF). More specifically, by exploiting the CLF property of OSFV to construct a contractive terminal set, a new stabilising MPC algorithm (CMPC) is proposed. We show that CMPC is recursively feasible and guarantees stability under the condition that OSVF is a CLF. Checking this condition and estimation of the maximal terminal set are discussed. Numerical examples are presented to demonstrate the effectiveness of the proposed stability condition and corresponding CMPC algorithm.

Relaxed stability analysis and sampled-data controller design for networked control systems with network-induced delays

This paper is devoted to the stability analysis and sampled-data controller design problem for networked control systems subject to network-induced delays. The objective is to provide relaxed conditions in terms of linear matrix inequalities. Indeed, reducing the conservatism of such conditions allows to maximize the admissible range of the network-induced delays. To do so, an augmented Lyapunov–Krasovskii functional is proposed, which involves a novel augmented state vector to include as much as possible the information from the time-varying network-induced delay into the stability conditions, together with the use of extended Wirtinger-based inequalities, an extended reciprocal convexity approach and the Finsler’s lemma. Then, declined from the proposed stability conditions, new relaxed delay-dependent conditions for the design of networked sampled-data controllers are proposed. These allow to obtain simultaneously the controller gains and the maximal allowable bound of the network-induced delays with regards to its lower bound. Three illustrative examples are provided to show the effectiveness of the proposed networked control systems stability and controller design conditions, as well as to highlight the so raised conservatism improvements regarding the previous relevant results from the literature.

On stability of mechanical systems with homogeneous and delayed forces

The problem of delay-independent stability is studied for a class of mechanical systems under dissipative, non-conservative and potential forces, which are described by homogeneous terms. The obtained conditions are extended to the case of switched force model under arbitrary and restricted commutation laws. The proof is based on analysis of a complete Lyapunov–Krasovskii functional (common and multiple ones are considered for switched scenario). The proposed stability conditions are illustrated by simulation examples.

Stability of Multimicrogrids: New Certificates, Distributed Control, and Braess’s Paradox

This paper investigates the theory of resilience and stability in multi-microgrid networks. We derive new sufficient conditions to guarantee small-signal stability of multi-microgrids in both lossless and lossy networks. The new stability certificate for lossy networks only requires local information, thus leads to a fully distributed control scheme. Moreover, we study the impact of network topology, interface parameters (virtual inertia and damping), and local measurements (voltage magnitude and reactive power) on the stability of the system. The proposed stability certificate suggests the existence of Braess's Paradox in the stability of multi-microgrids, i.e. adding more connections between microgrids could worsen the multi-microgrid system stability as a whole. We also extend the presented analysis to structure-preserving network models, and provide a stability certificate as a function of original network parameters, instead of the Kron reduced network parameters. We provide a detailed numerical study of the proposed certificate, the distributed control scheme, and a coordinated control approach with line switching. The simulation shows the effectiveness of the proposed stability conditions and control schemes in a four-microgrid network, IEEE 33-bus system, and several large-scale synthetic grids.

Necessary and Sufficient Convex Condition for the Stabilization of Linear Sampled-Data Systems Under Poisson Sampling Process

This work presents a control design method for linear sampled-data systems whose random sampling intervals form a Poisson process. Unlike a previous result in the literature, the proposed stabilization conditions, based on linear feedbacks of both the state and the past input values, are necessary and sufficient for the mean exponential stability of the system. Moreover, such non-conservative conditions correspond to linear matrix inequalities, implying then that the stabilization problem can be efficiently addressed through semidefinite programming. As a second contribution, the characterization and optimization of the mean exponential convergence rate of the closed-loop system is given in form of a generalized eigenvalue problem. A numerical example illustrates the theoretical results.

Small signal stability analysis of VSC based DC systems using graph theory

This paper presents a new method to evaluate the small signal stability based on state space and graph theory, which not only reduces the computational burden of high-dimensional system, but also has the ability to cope with the uncertain parameters and system structure. The system is first represented by a directed graph. Each node in the directed graph denotes a multi-input multi-output subsystem represented by state, input, and output matrices, and constrained by norm bounded uncertainties. The subsystems are connected by edges through node-edge incidence matrix and edge connection matrices. Then we propose the small signal stability conditions associated with the certain node dynamics, edge connection matrices, the norm bounds of uncertainty, and the maximum degree of the system, independent of the system structure and the number of nodes and edges. We apply our methods to the VSC based DC systems with uncertain parameters and structure. The comparative analysis of simulations results of Bode diagram, state matrix eigenvalues, and the new method validates the effectiveness of the proposed stability conditions.

On the orbital stability of periodic trajectories of a class of discontinuous systems

AbstractThis paper focuses on the stability analysis of periodic trajectories for non‐autonomous systems whose right‐hand sides are time‐periodic discontinuous functions. Such systems arise, in particular, in optimal control problems for nonlinear control systems with periodic bang‐bang inputs. We obtain sufficient orbital stability conditions of the Andronov–Witt type for this class of systems. The proposed stability conditions are applied to the analysis of periodic extremal trajectories of the controlled nonlinear chemical reaction model.

Conditions for Regional Frequency Stability in Power System Scheduling—Part I: Theory

This paper considers the phenomenon of distinct regional frequencies recently observed in some power systems. First, a reduced-order mathematical model describing this behaviour is developed. Then, techniques to solve the model are discussed, demonstrating that the post-fault frequency evolution in any given region is equal to the frequency evolution of the Centre Of Inertia plus certain inter-area oscillations. This finding leads to the deduction of conditions for guaranteeing frequency stability in all regions of a power system, a deduction performed using a mixed analytical-numerical approach that combines mathematical analysis with regression methods on simulation samples. The proposed stability conditions are linear inequalities that can be implemented in any optimisation routine allowing the co-optimisation of all existing ancillary services for frequency support: inertia, multi-speed frequency response, load damping and an optimised largest power infeed. This is the first reported mathematical framework with explicit conditions to maintain frequency stability in a power system exhibiting inter-area oscillations in frequency.