Relaxed Stability Conditions Research Articles

Dependent delay stability characterization for a polynomial T-S Carbon Dioxide model

<p style='text-indent:20px;'>By extending some linear time delay systems stability techniques, this paper, focuses on continuous time delay nonlinear systems (TDNS) dependent delay stability conditions. First, by using the Takagi Sugeno Fuzzy Modeling, a novel relaxed dependent delay stability conditions involving uncommon free matrices, are addressed in Linear Matrix Inequalities (LMI). Then, as application a Nonlinear Carbon Dioxide Model is used and rewritten by a change of coordinate to the interior equilibrium point. Next, by using the non-linearity sector method the model is transformed to a corresponding Fuzzy Takagi Sugeno (TS) multi-model. Also, the maximum delay margin to which the model is stable, is identified. Finally, to prove the analytic results a numerical simulation is also performed and compared to other methods.</p>

Adaptive control systems with unstructured uncertainty and unmodelled dynamics: a relaxed stability condition

As it is well-known, system uncertainties and unmodeled dynamics can deteriorate stability properties of model reference adaptive control systems. Motivated by this standpoint, we first analyse stability conditions of model reference adaptive control architectures in the presence of unstructured system uncertainties and unmodeled dynamics. We then synthesise adaptive robustifying terms to relax the aforementioned stability condition, which presents our main contribution. Specifically, these terms in the feedback loop guarantee overall system stability even in the presence of significant system uncertainties when unmodeled dynamics satisfy a condition. We further demonstrate our theoretical findings in an experiment involving an inverted pendulum on a cart (modeled dynamics) coupled with another cart through a spring (unmodeled dynamics).

Relaxed Stabilization and Disturbance Attenuation Control Synthesis Conditions for Polynomial Fuzzy Systems.

This article presents both stabilization and disturbance attenuation control synthesis conditions for a class of polynomial fuzzy systems. The sum of squares conditions is relaxed by restricting the domain of the polynomial variables that represent the membership functions as well as by using the line integral polynomial fuzzy Lyapunov function. For the disturbance attenuation problem, we consider the transformation of the Hamilton-Jacobi-Isaacs equation into a set of inequalities and a policy iteration algorithm to approximate its value function. Furthermore, we propose an alternative path following to find the initial L2 gain stabilizing control policy. Five examples are provided to demonstrate the effectiveness of the proposed conditions.

Static Output-Feedback Tracking Control for Positive Polynomial Fuzzy Systems

Nonlinear positive control system can be found in many real-world applications but the positivity requirements lead to challenges in system analysis and control design. In this article, we approach the problem by fuzzy-model-based control techniques and overcome some challenges including transforming the nonconvexity conditions when both positive and stability conditions exist into convexity conditions that can be solved by the convex programming techniques. This article focuses on the static output-feedback tracking control issue of positive polynomial fuzzy-model-based systems. The purpose of the tracking control is to design an appropriate static output feedback polynomial fuzzy controller which can drive the system states of the nonlinear plant to follow those of a stable reference model subject to 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> performance. The concept of imperfectly matched premises is employed to enhance the design and implementation flexibility. To circumvent the problem of nonconvex stability conditions, an approach is employed to transform the nonconvex stability conditions into convex ones by introducing a novel scalar implantation transformation technique. Besides, the partition approximation of membership functions with local information of membership functions is used to promote stability analysis and synthesis of controllers. The positive and relaxed stability conditions for static output-feedback tracking control with <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> performance being taken into account are obtained in terms of sum-of-squares. Finally, a simulation example is presented to verify the effectiveness of the proposed tracking control approach.

Relaxed Stabilization Conditions for Interconnected Nonlinear Systems

This paper deals with the conservatism reduction of stability and stabilization conditions for nonlinear continuous-time interconnected systems. Based on Takagi–Sugeno modeling, the interconnected system is described by a convex combination of interconnected linear systems. Line integral fuzzy Lyapunov functions are considered to develop stability and stabilizability criteria for the considered class of interconnected nonlinear systems. All analysis and design conditions are formulated in LMI terms. The effect of the coupling due to interconnection terms is also investigated and taken into account to obtain less conservative conditions. Examples are given to illustrate the merit of the obtained results.

Relaxed Stabilization Conditions for TS Fuzzy Systems With Optimal Upper Bounds for the Time Derivative of Fuzzy Lyapunov Functions

This paper initially proposes an optimization problem and after presents its optimal solution. Then, this result is applied to obtain relaxed conditions to design controllers for nonlinear plants described by Takagi-Sugeno (TS) models, based on fuzzy Lyapunov function (FLF) and Linear Matrix Inequalities (LMI). The FLF is given by $V(x(t)) = x(t)^{T}P(\alpha (x(t)))x(t)$ , where $x(t)$ is the plant state vector, $P(\alpha (x(t))) = \alpha _{1}(x(t))P_{1} + \alpha _{2}(x(t))P_{2} + \cdots + \alpha _{r}(x(t))P_{r}$ , $P_{i}=P_{i}^{T} > 0$ and $\alpha _{i}(x(t))$ is the weight related to the local model $i$ in the representation of the plant by TS fuzzy models, for $i=1,2,\cdots,r$ . When one calculates the time derivative of this $V(x(t))$ , it appears the term $x(t)^{T}\dot {P}(\alpha (x(t)))x(t)$ , that is usually handled using conservative upper bounds, supposing that the bounds of the time derivative of $\alpha _{i}(x(t))$ , $i=1,2,\cdots,r$ , are available. The main result of this paper is a procedure to obtain optimal upper bounds for the term $x(t)^{T}\dot {P}(\alpha (x(t)))x(t)$ , such that they contemplate the maximum value and are always smaller than or equal to the maximum value. It is a relevant result on this subject, because these optimal upper bounds do not add any constraint. With these optimal upper bounds, a relaxed design method for stabilization of TS fuzzy models is proposed. Two numerical examples illustrate the effectiveness of this procedure.

Event-Based Consensus Tracking for Nonlinear Multi-Agent Systems Under Semi-Markov Jump Topology

This paper studies the event-triggering leader-follower consensus with the strictly dissipative performance for nonlinear multi-agent systems (MASs) with semi-Markov changing topologies. First, a polynomial fuzzy model is established to describe the error nonlinear multi-agent system that is formed by one virtual leader and followers. Then, a new event-triggering transmission strategy is proposed to mitigate communication and computational load. By utilizing the event-triggering mechanism and modeling the switching topologies by semi-Markov process, a sampled-data based consensus protocol is designed. Compared with traditional Markov jump topologies, the transition rate is time-varying for semi-Markov switching topologies. By mode-dependent Lyapunov-Krasovskii functional, the sum of square based relaxed stabilization conditions for fuzzy MASs are obtained to guarantee event-triggering consensus with strict dissipativity in an even-square sense, i.e., the derived conditions take into account the joint effects of event-triggering control, semi-Markov jump topologies and external disturbance. An illustrative example is provided to verify the proposed consensus design schemes.

An efficient two-dimensional heat transfer model for building envelopes

A two-dimensional model is proposed for energy efficiency assessment through the simulation of heat transfer in building envelopes, considering the influence of the surrounding environment. The model is based on the Du Fort–Frankel approach that provides an explicit scheme with a relaxed stability condition. The model is first validated using an analytical solution and then compared to three other standard schemes. Results show that the proposed model offers a good compromise in terms of high accuracy and reduced computational efforts. Then, a more complex case study is investigated, considering non-uniform shading effects due to the neighboring buildings. In addition, the surface heat transfer coefficient varies with wind velocity and height, which imposes an addition non-uniform boundary condition. After showing the reliability of the model prediction, a comparison over almost 120 cities in France is carried out between the two- and the one-dimensional approaches of the current building simulation programs. Important discrepancies are observed for regions with high magnitudes of solar radiation and wind velocity. Last, a sensitivity analysis is carried out using a derivative-based approach. It enables to assess the variability of the solution according to the modeling of the two-dimensional boundary conditions. Moreover, the proposed model computes efficiently the solution and its sensitivity to the modeling of the urban environment.

Finite-Time Lyapunov Functions and Impulsive Control Design

In this paper, we introduce finite-time Lyapunov functions for impulsive systems. The relaxed sufficient conditions for asymptotic stability of an equilibrium of an impulsive system are given via finite-time Lyapunov functions. A converse finite-time Lyapunov theorem for controlling the impulsive system is proposed. Three examples are presented to show how to analyze the stability of an equilibrium of the considered impulsive system via finite-time Lyapunov functions. Furthermore, according to the results, we design an impulsive controller for a chaotic system modified from the Lorenz system.

Membership-function-dependent stability analysis and local controller design for T–S fuzzy systems: A space-enveloping approach

For Takagi–Sugeno (T–S) fuzzy systems, relaxed stability conditions are obtained by more effectively enveloping the trajectory of the membership functions (MFs) in a unified space of MF. Considering an open-loop T–S fuzzy system, the system premise variables operation domain can be divided into a series of subdomains. Based on an MF unified space extremum calculation technique, the MFs extremum values in each premise variable corresponding to the unified space subdomain are calculated. With these extremes, a tight local convex polyhedron enveloping the MF trajectory is constructed in each subdomain. Thus, linear matrix inequality (LMI) stability conditions are derived via a piecewise Lyapunov function. Then, a state-feedback local controller is designed to close the system loop. From a geometric viewpoint, MF extremum enveloping and piecewise linear-approximation methods are both utilized to achieve relaxed stability and robustness conditions. Finally, several examples are adopted to illustrate the metrics of the proposed approaches.

Event-based fuzzy control for T-S fuzzy networked systems with various data missing

This paper is concerned with the design of fuzzy controller for networked control systems (NCSs). T-S fuzzy system approach is adopted to study the problem. In account of NCSs, various data missing on both sides of the controller caused by event-triggered scheme, data losses and event-driven controller are involved. For handling this issue, an auxiliary random series method is presented to describe the data transferring in the network. Based on this method, the closed-loop NCSs is constructed and then analyzed via Lyapunov stability theory. Additionally, the probability of random data losses at each instant is supposed to be different which is more practical. For the purpose of obtaining more relaxed stabilization conditions, the fuzzy Lyapunov function obtained on the basis of both plant and controller fuzzy rules is employed. Furthermore, variations of membership functions are taken into account, and consequently, slack matrices are introduced to obtain less conservative result. At last, by considering a cart-damper-spring system, the effectiveness of the proposed method is illustrated.

Event-triggered [formula omitted] control for active seat suspension systems based on relaxed conditions for stability

An event-triggered H∞ controller is designed for an active seat suspension in this paper, where the continuous event-trigger scheme is applied to transfer the dynamic system states to the controller only at event-triggered time instants. Delay-dependent stability criteria in the form of linear matrix inequality (LMI) are presented to guarantee the asymptotic stability of the seat suspension system. One Lyapunov function is chosen where some matrices are introduced with relaxed conditions. Two tight inequalities are applied to prove the positive definiteness of the Lyapunov function and stability of the system, which reduces the conservatism of the system for the time delay to the controller. The proposed control method can reduce the workload of data transmission of the seat suspension system and work as a filter to remove the effect of noise, so it can decrease the precision requirement of the actuator, which can help to reduce the cost of the seat suspension. Both simulation and experimental results are presented to show the effectiveness of the proposed control method.

Stability and stabilisation of switched time-varying delay systems: a multiple discontinuous Lyapunov function approach

In this paper, stability and stabilisation of switched time-varying delay systems are investigated. By developing, a multiple discontinuous Lyapunov function approach and using the mode-dependent average dwell time (MDADT) switching signal, sufficient conditions are first proposed to guarantee the exponential stability of the switched system with stable and unstable subsystems where the time-varying delay is considered in the states. Relaxed stability conditions are also obtained by using a form of the Bessel–Legendre inequality. Then stabilisation of the proposed system is presented by designing a sliding mode controller. Sufficient conditions and sufficient relaxed conditions of a reduced-order sliding mode dynamics are derived, and desired sliding surfaces in the form of linear matrix inequalities (LMIs) are also given. The obtained conditions are checked by using the combined genetic algorithm and convex programming techniques. Moreover, some numerical and practical examples are given to illustrate the effectiveness of the obtained theoretical results.

Dispersion‐relationship‐preserving seismic modelling using the cross‐rhombus stencil with the finite‐difference coefficients solved by an over‐determined linear system

ABSTRACTFinite‐difference modeling with a cross‐rhombus stencil with high‐order accuracy in both spatial and temporal derivatives is a potential method for efficient seismic simulation. The finite‐difference coefficients determined by Taylor‐series expansion usually preserve the dispersion property in a limited wavenumber range and fixed angles of propagation. To construct the dispersion‐relationship‐preserving scheme for satisfying high‐wavenumber components and multiple angles, we expand the dispersion relation of the cross‐rhombus stencil to an over‐determined system and apply a regularization method to obtain the stable least‐squares solution of the finite‐difference coefficients. The new dispersion‐relationship‐preserving based scheme not only satisfies several designated wavenumbers but also has high‐order accuracy in temporal discretization. The numerical analysis demonstrates that the new scheme possesses a better dispersion characteristic and more relaxed stability conditions compared with the Taylor‐series expansion based methods. Seismic wave simulations for the homogeneous model and the Sigsbee model demonstrate that the new scheme yields small dispersion error and improves the accuracy of the forward modelling.

Robust Controller Design for Uncertain Linear Systems with Finite-frequency Specifications: A Polynomially Parameter-dependent Approach

This paper investigates the robust controller design for polytopic-type uncertain linear systems disturbed by external disturbances in restricted frequency ranges. The primary goal is to design a state-feedback controller, ensuring that the closed-loop system is robustly stable and keeps a prescribed finite-frequency (FF) disturbanceattenuation performance. For this purpose, a robust generalized Kalman-Yakubovich-Popov (KYP) lemma is applied to describe the finite-frequency specification, aiming at improving the disturbance-attenuation performance over the given frequency range. In this setting, more relaxed analysis conditions for robust stability and finitefrequency specifications are derived by introducing additional slack variables, which contain some existing conditions as special cases. Based on the homogeneous polynomially parameter-dependent (HPPD) technique, new controller design conditions in terms of linear matrix inequalities (LMIs) are developed. It is shown that the proposed finite-frequency design scheme can achieve a better disturbance- suppression performance in restricted frequency ranges than the existing ones, which is illustrated by an example about the satellite system.

Fixed/Preassigned-Time Synchronization of Complex Networks via Improving Fixed-Time Stability.

This article is concerned with the problem of fixed-time (FXT) and preassigned-time (PAT) synchronization for discontinuous dynamic networks by improving FXT stability and developing simple control schemes. First, some more relaxed conditions for FXT stability are established and several more accurate estimates for the settling time (ST) are obtained by means of some special functions. Based on the improved FXT stability, FXT synchronization for discontinuous networks is discussed by designing a simple controller without a linear feedback term. Besides, the PAT synchronization is also explored by developing several nontrivial control protocols with finite control gains, where the synchronized time can be prespecified according to actual needs and is irrelevant with any initial value and any parameter. Finally, the improved FXT stability and the synchronization for complex networks are confirmed by two numerical examples.

New versions of Bessel–Legendre inequality and their applications to systems with time-varying delay

This paper is concerned with the stability of linear time-delay systems by applying the Lyapunov–Krasovskii (L–K) functional method. Two versions of Bessel–Legendre (B-L) inequality are developed that are more suitable to deal with the stability problem of systems with a time-varying delay. Meanwhile, in order to take full advantage of the interest of the new versions of B-L inequality, a novel L–K functional is properly tailored by integrating the integral information of the state into quadratic and integral terms. As a result, more relaxed stability conditions are obtained, whose effectiveness is illustrated by two commonly-used numerical examples.

An efficient numerical model for the simulation of coupled heat, air, and moisture transfer in porous media

AbstractThis article proposes an efficient explicit numerical model with a relaxed stability condition for the simulation of heat, air, and moisture transfer in porous material. Three innovative approaches are combined to solve the system of two differential advection‐diffusion equations coupled with a purely diffusive equation. First, the DU FORT‐FRANKEL scheme is used to solve the diffusion equation, providing an explicit scheme with an extended stability region. Then, the two advection‐diffusion equations are solved using both the SCHARFETTER‐GUMMEL numerical scheme for the space discretisation and the two‐step RUNGE‐KUTTA method for the time variable. This combination enables to relax the stability condition by one order. The proposed numerical model is evaluated on three case studies. The first one considers quasi‐linear coefficients. The theoretical results of the numerical schemes are confirmed by our computations. Indeed, the stability condition is relaxed by a factor of 40 compared to the standard EULER explicit approach. The second case provides an analytical solution for a weakly nonlinear problem. A very satisfactory accuracy is observed between the reference solution and the one provided by the numerical model. The last case study assumes a more realistic application with nonlinear coefficients and ROBIN‐type boundary conditions. The computational time is reduced 10 times by using the proposed model in comparison with the explicit EULER method.

New relaxed stability and stabilization conditions for differential linear repetitive processes

The paper develops new results on the stability analysis of differential linear repetitive processes. These processes are a distinct class of two-dimensional systems that arise in the modelling of physical processes and also the existing systems theory for them can be used to effect in solving control problems for other classes of systems, including iterative learning control design. This paper uses a version of the Kalman-Yakubovich-Popov Lemma to develop relaxed conditions for the stability property in terms of linear matrix inequalities. The main result is reduced conservatism in applying tests for the stability property with an extension to control law design. A numerical example to illustrate the application of the new results is also given.

Relaxed stability conditions for linear systems with time-varying delays via some novel approaches

&lt;abstract&gt; &lt;p&gt;In this paper, the stability problem with considering time-varying delays of linear systems is investigated. By constructing new augmented Lyapunov-Krasovskii (L-K) functionals based on auxiliary function-based integral inequality (AFBI) and considering Finsler's lemma, a stability criterion is derived. Based on the previous result, a less conservative result is proposed through the augmented zero equality approach. Finally, numerical examples are given to show the effect of the proposed criteria.&lt;/p&gt; &lt;/abstract&gt;