Common Quadratic Lyapunov Function Research Articles

On the boundedness of solutions of some fuzzy dynamical control systems

<abstract> <p>The asymptotic behavior of solutions of fuzzy control systems is a component of the study of fuzzy control theory. The study of stability for T-S (Takagi-Sugeno) fuzzy systems, which process qualitative data through linguistic expressions, is the subject of this paper. Asymptotic stability is conservative in many real-world applications due to measurement noise and other disruptions. The ultimate limit, which indicates that the mistakes stay in a specific area close to the origin after a long enough amount of time, is a crucial characteristic that is frequently defined for such systems. We are interested with the problem of the state feedback controller for T-S fuzzy models with uncertainties where the global exponential ultimate boundedness of solutions is studied for certain fuzzy control systems. We use common quadratic Lyapunov function and parallel distributed compensation controller techniques to study the asymptotic behavior of the solutions of fuzzy control system in presence of perturbations. An example demonstrating the validity of the main result is discussed.</p> </abstract>

Common Quadratic Lyapunov Functions for Sets of Second-Order Linear Systems: A Simple Graphical Criterion

Common Quadratic Lyapunov Functions for Sets of Second-Order Linear Systems: A Simple Graphical Criterion

Observer-based leader-following consensus for positive multi-agent systems over time-varying graphs

This paper addresses the leader-following consensus problem for discrete-time positive multi-agent systems over time-varying graphs. We assume that the followers may have mutually different positive dynamics that can also be different from the leader. Compared with most existing positive consensus works for homogeneous multi-agent systems, the formulated problem is more general and challenging due to the interplay between the positivity requirement and high-order heterogeneous dynamics. To solve the problem, we present an extended version of existing observer-based design for positive multi-agent systems. By virtue of the common quadratic Lyapunov function technique, we show the followers will maintain their state variables in the positive orthant and finally achieve an output consensus specified by the leader. A numerical example is used to verify the efficacy of our algorithms.

Switching Control Strategy for DC–DC Converters Based on Polynomial Lyapunov Function and Sum-of-Squares Approach

The introduction of switched system theory opens new perspectives for the control design of dc/dc converters. Regarding the classical two-level converters, two switching control design methodologies have been developed based on the common quadratic Lyapunov function (CQLF) or piecewise quadratic Lyapunov function (PQLF). However, the CQLF-based design method is not applicable for the multilevel dc/dc converter (MC). As for the PQLF-based design method, it encounters the problem that the control design is very complex, involving a nonlinear optimization problem. To further advance the application of the switching system theory in power converters, a design framework based on the polynomial Lyapunov function and the sum-of-squares approach is proposed in this article. The proposed design method generalizes the CQLF-based one. Moreover, compared with the PQLF-based design method, the mathematical design procedure can be significantly simplified. Experimental validations on an MC, i.e., the three-level flying capacitor converter and a two-level Buck-Boost converter illustrate the versatility and the effectiveness of the proposed design framework. Moreover, comparative tests show the good control performance of the designed switching control strategy. The currently presented works are preliminary but could constitute an interesting direction for developing control solutions of power converters.

Disturbance observer-based fuzzy event-triggered ISMC design: Tracking performance

This article presents a disturbance observer (DO)-based event-triggered integral sliding mode tracking control (ISMTC) for continuous-time Takagi–Sugeno (T–S) fuzzy model (TSFM) subject to external disturbances. Merging the event-triggered control (ETC) with integral sliding mode control (ISMC) approach is led to reach the better accommodate the features of ISMC. To do this, two forms of fuzzy integral sliding surfaces (FISSs) are proposed. The first is the periodic-time-based FISS that is presented to provide the robustness of the tracking performance from the initial moment. The second is the event-triggered-based FISS which is proposed to obtain the fuzzy event-triggered ISMTC law. To more decrease the chattering effects and compensate the tracking performance against the disturbances, a fuzzy disturbance observer (FDO) is proposed to estimate the mismatched disturbances. Compared with the existing works, a more practical controller is proposed based on the asynchronous premise variables. Utilizing the common quadratic Lyapunov function, appropriate conditions are derived to verify that the tracking error is robust from the initial moment. Furthermore, by adopting a decaying triggering threshold, it is guaranteed that the system is Zeno-free during the process. To verify the effectiveness of the suggested event-triggered ISMTC, a practical system including continuous stirred tank reactor (CSTR) is simulated and the results are compared.

Constrained Control for a Class of TS Fuzzy Systems

This article investigates handling control constraints in the controller design phase for a class of Takagi–Sugeno (TS) fuzzy systems. Line integral Lyapunov function (LILF) is used to reduce the conservativeness compared to common quadratic Lyapunov function (CQLF) and fuzzy Lyapunov function (FLF). Two cases, namely unsaturating and saturating control, are studied. First, an inclusion condition is set to guarantee the behavior in a region where control constraints violation does not occur. For the second case, the saturation may take effect and is allowed. Thanks to the saturation function, convex writing to outrun the nonlinearity inconvenience. For both cases, stabilizing design procedures are proposed such that closed-loop stability and its linked performances are not affected. Furthermore, the proposed conditions enlarge the domain of stability as possible. Hence, comparisons are made with other approaches to show the effectiveness of our approach.

Common Lyapunov Functions for Switched Linear Systems: Linear Programming-Based Approach

We study the stability of an equilibrium of arbitrarily switched, autonomous, continuous-time systems through the computation of a common Lyapunov function (CLF). The switching occurs between a finite number of individual subsystems, each of which is assumed to be linear. We present a linear programming (LP) based approach to compute a continuous and piecewise affine (CPA) CLF and compare this approach with different methods in the literature. In particular we compare it with the prevalent use of linear matrix inequalities (LMIs) and semidefinite optimization to parameterize a quadratic common Lyapunov function (QCLF) for the linear subsystems.

An enhanced control strategy for an ultra-fast EV charging station in a DC microgrid

Uncoordinated and fast electric-vehicle (EV) charging schemes have significant impacts on the dynamic operation of a microgrid. This paper proposes an enhanced control method for an ultra-fast EV charging station in an islanded dc microgrid (DCMG). Ultra-fast charging stations, which utilize a high dc charging current, create high transients at the common bus voltage. Such transients are higher in islanded microgrids where the dc-bus voltage is more vulnerable due to the absence of the ac grid. Conventionally, the transients are managed by complicating the hardware such as by using a large and costly super-capacitor or superconducting magnetic energy storage. This paper proposes an enhanced software-based solution for EV chargers in a DCMG to significantly reduce the transients of the dc-bus voltage via intelligently controlling the rising time and settling time of the charging current in each sampling time based on the dc-bus voltage variations. Compared with the conventional control methods, the proposed method exhibits higher robustness against system uncertainties and unexpected disturbances, and higher resiliency against out-of-range variations of the charging current. The effectiveness of the proposed control system is validated in MATLAB Simulink, and its stability is verified using both root locus analysis and common quadratic Lyapunov function.

On one inner point algorithm for common Lyapunov functions

In this paper, we consider the problem of existence and evaluation of common quadratic Lyapunov functions for a finite number of matrices. For this purpose, an inner point algorithm is proposed. The technique is based on Khalil (1982), where an algorithm is given which determines the diagonal stability of a matrix A . We obtain a necessary and sufficient condition for the existence and give an efficient algorithm which gives common solution in the case of existence and negative answer after a few iterations in the case of non existence. A number of examples are presented verifying the fast speed and efficiency of the obtained results.

Adaptive tracking control of time‐varying delay switched nonstrict‐feedback nonlinear systems with unknown backlash‐like hysteresis

SummaryIn this article, the adaptive tracking control issue is addressed for uncertain switched nonstrict‐feedback nonlinear systems with unknown time‐varying delay and backlash‐like hysteresis under neural networks (NNs) estimation framework. In contrast to most existing results of nonlinear systems with time‐delay, the time‐varying delay we consider depends entirely on the number of switching subsystems. Based on the inherent properties of radial basis function NNs, the nonstrict‐feedback structure is effectively compensated. Meanwhile, with the help of the combination of Razumikhin function method, adaptive backstepping scheme, and dynamic surface control approach, the quadratic common Lyapunov function is established for all subsystems, and then an adaptive common control strategy is developed. For arbitrary switching signals, it is shown that all the signals in the closed‐loop system can be remained bounded and the system output can follow a desired signal under the designed controller. Finally, the validity of the presented strategy is verified via two simulation examples.

COMMON QUADRATIC LYAPUNOV FUNCTIONS FOR TWO STABLE MATRICES

In this paper we consider the problem of existence and evaluation of common solutions to Lyapunov equations for switched systems consisting of two real or complex matrices. Conditions for existence and solution algorithm based on the gradient of a matrix function are given. Number of examples are provided.

Stability of Discrete-Time Switched Systems With Multiple Equilibria Using a Common Quadratic Lyapunov Function

This letter studies stability of a class of discrete-time switched systems under arbitrary switching in which each subsystem has a different equilibrium. We extend the concept of a common quadratic Lyapunov function in order to characterize asymptotic stability of a set that contains all the equilibria. The results are applied to a model of visuomotor adaptation to analyze boundedness of solutions when the visual error measurement is intermittent.

Global Practical Stabilization of Discrete-Time Switched Affine Systems via a General Quadratic Lyapunov Function and a Decentralized Ellipsoid

This paper addresses the problem of global practical stabilization of discrete-time switched affine systems via state-dependent switching rules. Several attempts have been made to solve this problem via different types of a common quadratic Lyapunov function and an ellipsoid. These classical results require either the quadratic Lyapunov function or the employed ellipsoid to be of the centralized type. In some cases, the ellipsoids are defined dependently as the level sets of a decentralized Lyapunov function. In this paper, we extend the existing results by the simultaneous use of a general decentralized Lyapunov function and a decentralized ellipsoid parameterized independently. The proposed conditions provide less conservative results than existing works in the sense of the ultimate invariant set of attraction size. Two different approaches are proposed to extract the ultimate invariant set of attraction with a minimum size, i.e., a purely numerical method and a numerical-analytical one. In the former, both invariant and attractiveness conditions are imposed to extract the final set of matrix inequalities. The latter is established on a principle that the attractiveness of a set implies its invariance. Thus, the stability conditions are derived based on only the attractiveness property as a set of matrix inequalities with a smaller dimension. Illustrative examples are presented to prove the satisfactory operation of the proposed stabilization methods.

Simultaneous and Sequential Control Design for Discrete-Time Switched Linear Systems Using Semi-Definite Programming

The control of switched linear discrete-time systems occurs in multiple engineering fields, where it has been used to deal with complex and non-linear systems. This letter presents two strategies to design control laws for discrete-time switched linear systems, whilst guaranteeing asymptotic stability of the closed loop. Firstly, an arbitrary switching signal is considered. In this scenario a common quadratic Lyapunov function is used for stability, but subsystem Lyapunov functions are employed to improve local subsystem performance. Secondly, a constrained switching signal, associated with subsystem lower dwell time bounds is studied. In this case, a decrease in Lyapunov cost is achieved by design, based on dwell time constraints only, thus removing the need for both a common quadratic Lyapunov function or direct stable switches. It is shown in both cases that the control design problems can be formulated as one or a sequence of semi-definite programming problems, and therefore can be solved efficiently. Finally, two examples are provided in order to illustrate the different techniques presented.

Filtered Split-Path Nonlinear Integrator: A Hybrid Controller for Transient Performance Improvement

The filtered split-path nonlinear integrator (F-SPANI) is a generic nonlinear controller designed to improve the transient performance of linear (motion) systems in terms of overshoot. The main idea underlying F-SPANI is that the amplitude and phase of an integrator can be tuned using independent filters, resulting in more efficient use of the buffer of the integrator. In this article, a general description of F-SPANI is presented. In addition, a stability analysis result is presented that provides sufficient conditions in the form of linear matrix inequalities (LMIs) for closed-loop stability analysis on the basis of construction of a common quadratic Lyapunov function (CQLF). The ease of the design, implementation, and the potential of the proposed controller are illustrated both in simulations and in experiments on an industrial pick-and-place machine.

Data-driven stability analysis of switched linear systems with Sum of Squares guarantees

Abstract We present a new data-driven method to provide probabilistic stability guarantees for black-box switched linear systems. By sampling a finite number of observations of trajectories, we construct approximate Lyapunov functions and deduce the stability of the underlying system with a user-defined confidence. The number of observations required to attain this confidence level on the guarantee is explicitly characterized. Our contribution is twofold: first, we propose a novel approach for common quadratic Lyapunov functions, relying on sensitivity analysis of a quasi-convex optimization program. By doing so, we improve a recently proposed bound. Then, we show that our new approach allows for extension of the method to Sum of Squares Lyapunov functions, providing further improvement for the technique. We demonstrate these improvements on a numerical example.

On the global practical stabilization of discrete-time switched affine systems: application to switching power converters

This paper presents new sufficient conditions as a set of Bilinear Matrix Inequalities (BMIs) for the global practical stabilization of discrete-time switched affine systems. The main contribution is on proposing the stability conditions based on a common quadratic Lyapunov function that can be used to stabilize the discrete-time switched affine systems around a desired equilibrium point for which it is not required to find any Schur stable convex combination of operating modes as a pre-processing stage, that needs special algorithms and is an NP-hard problem. The result is that the existing two-stage stabilization methods based on a pre-calculation of a Schur stable convex combination of operating modes are simplified to a single-stage method by which a high degree of applicability is obtained. The proposed stability conditions are developed in a way the size of the convergence ellipsoid is minimized. Moreover, it is not required the equilibrium point, around which the invariant set of attraction is constructed, be inside a predetermined set of attainable equilibrium points. The satisfactory operation of the proposed stability conditions is illustrated by an academic example and application on various DC-DC converters.

Line Integral Approach to Extended Dissipative Filtering for Interval Type-2 Fuzzy Systems

This article is concerned with the problems of extended dissipativity analysis and filter design for interval type-2 (IT2) fuzzy systems. Based on the line integral Lyapunov function, a sufficient condition of asymptotic stability and extended dissipativity of the systems under consideration is established. A LMI-based equivalent condition to the obtained one in a nonlinear form is provided by combining congruence transformation with change of variables. This LMI condition obtained is more general than the one which is based on the common quadratic Lyapunov function. Meanwhile, in terms of parameterization, the extended dissipative filter is developed which guarantees the asymptotic stability and extended dissipativity for the filtering error system. Furthermore, our filter obtained by the parameterization method includes the one obtained by the equivalent transformation method as a special case. Two simulation examples are provided to show the merits and effectiveness of the proposed approach.

Stability Evaluation of Interpolation, Extrapolation, and Numerical Oscillation Damping Methods Applied in EMT Simulation of Power Networks With Switching Transients

For Electro-Magnetic-Transient (EMT) simulations of power networks with switches, techniques such as linear interpolation and Critical Damping Adjustment (CDA) are widely used for improving numerical robustness. This paper analyzes the numerical stability of simulations with these techniques. Firstly, it is mathematically shown that the interpolation or CDA step is equivalent to the introduction of additional switching states. Subsequently, Common Quadratic Lyapunov Function (CQLF) theory is used to investigate the numerical stability of the whole simulation considering these new switching states. It is proved that the widely used strategies like linear interpolation and CDA always result a stable simulation if the original switched system is strictly passive in all switching states. Finally, it is shown that the developed approach can be used to determine the stability of other practical interpolation methods. Examples are provided to verify the proposed technique.

A new design of output feedback sliding mode controller for T‐S fuzzy‐affine systems

AbstractThis work focuses on output feedback sliding mode control (SMC) for nonlinear systems via T‐S fuzzy affine models. Associated with the sliding surface, a singular system is firstly proposed to describe the dynamical features of the sliding motion. Adopting some convexification techniques and based on piecewise quadratic Lyapunov functions (PQLFs) and a common quadratic Lyapunov function (CQLF), sufficient criteria for the asymptotic stability analysis of the sliding mode dynamical system are attained. Two output feedback SMC synthesis schemes are developed to drive the closed‐loop system states to the sliding surface in finite time. A simulation example is presented to show the validity of the proposed methods.