Path-Complete Lyapunov Functions for Continuous-Time Switching Systems

Finite-time stability analysis and control of stochastic SIR epidemic model: A study of COVID-19

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In this paper, we develop a systematic Lyapunov approach to the regional stability and performance analysis of saturated systems in a general feedback configuration. The only assumptions we make about the system are well-posedness of the algebraic loop and local stability. Problems to be considered include the estimation of the domain of attraction, the reachable set under a class of bounded energy disturbances and the nonlinear L2 gain. The regional analysis is established through an effective treatment of the algebraic loop and the saturation/deadzone function. This treatment yields two forms of differential inclusions, a polytopic differential inclusion (PDI) and a norm-bounded differential inclusion (NDI) that contain the original system. Adjustable parameters are incorporated into the differential inclusions to reflect the regional property. The main idea behind the regional analysis is to ensure that the state remain inside the level set of a certain Lyapunov function where the PDI or the NDI is valid. With quadratic Lyapunov functions, conditions for stability and performances are derived as linear matrix inequalities (LMIs). To obtain less conservative conditions, we use a pair of conjugate non-quadratic Lyapunov functions, the convex hull quadratic function and the max quadratic function. These functions yield bilinear matrix inequalities (BMIs) as conditions for stability and guaranteed performance level. The BMI conditions cover the corresponding LMI conditions as special cases, hence the BMI results are guaranteed to be as good as the LMI results. In most examples, the BMI results are significantly better than the LMI results

Stability analysis for a class of switched systems with state constraints via time-varying Lyapunov functions

This paper presents the stability analysis for polynomial fuzzy systems by utilizing the equality constraints of sum-of-squares (SOS) Program. Recently, T-S fuzzy systems have been extended to polynomial fuzzy systems, in which the consequent parts could be polynomial sub-systems. Moreover, stability and stabilization analyses for polynomial fuzzy systems have been done based on polynomial Lyapunov functions that contain quadratic Lyapunov functions as a special case. Since polynomials exist in the sub-systems and the positive definite matrices of the Lyapunov functions, the linear matrix inequality (LMI) tools are not valid for the polynomial analyses. Instead of LMI tools, SOSTOOLS of Matlab toolbox is applied to solve solutions for the polynomial analyses. The SOS program of SOSTOOLS can solve two kinds of constraints that are equality constraints and inequality constraints. The existing stability and stabilization analyses for polynomial fuzzy systems represent their conditions only in the inequality constraints. This paper analyzes the stability of the polynomial fuzzy systems by polynomial Lyapunov functions, and represents the new stability conditions not only in terms of the inequality constraints but also in terms of the equality constraints.

This paper addresses stability analysis and stabilization for Takagi-Sugeno (T-S) fuzzy systems with parametric uncertainties via a so-called fuzzy Lyapunov function which is a multiple Lyapunov function. The fuzzy Lyapunov function is defined by fuzzily blending quadratic Lyapunov functions. First, the Takagi-Sugeno (T-S) fuzzy model with parametric uncertainties is used as the model for the uncertain nonlinear system. Based on the fuzzy Lyapunov function approach and a parallel distributed compensation (PDC) scheme, we give stabilization conditions for closed-loop fuzzy systems with parametric uncertainties. Second, all the conditions are formulated in the format of linear matrix inequalities (LMIs) and contain upper bounds of the time derivative of premise membership functions as LMI variables. Finally, the T-S fuzzy model of the chaotic Lorenz system, which has complex nonlinearity, is developed as a test bed. A numerical example of the chaotic Lorenz system is given to illustrate the utility of the fuzzy Lyapunov function approach.

On Absolute Stability of Lur’e Control Systems with Multiple Non-Linearities

This article concerns the finite-time robust guaranteed cost control problem for a class of linear continuous-time singular systems with norm-bounded uncertainties. In this study, the problem is to design a state feedback controller such that the closed-loop system is finite-time stable, and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainties. By constructing an appropriate Lyapunov function, a sufficient condition for the finite-time robust stability of the system based on linear matrix inequality (LMI) is established. Furthermore, the sufficient condition for the existence of the guaranteed cost controller is formulated in terms of LMIs, which can make the closed-loop uncertain singular system finite-time robust stable. Finally, two numerical examples are given for illustration of the proposed theoretical results.

This paper presents a constructive method for showing exponential stability of autonomous nonlinear systems consisting of state-dependent weighted linear systems. This kind of system representation is common in for instance fuzzy systems or is the result of an exact or approximative description of an arbitrary nonlinear vector field. Stability is shown by joining multiple local Lyapunov functions properly in the state-space. The overall Lyapunov function, consisting of the local ones, are allowed to be discontinuous at the states where the trajectory passes from one local region to another. By using local quadratic Lyapunov functions the stability conditions are formulated as linear matrix inequalities (LMIs), which can be solved efficiently by computerized methods.

The robust control for a class of distributed parameter switched systems with time-delay is studied. The sufficient condition for the existence of the hybrid state feedback robust controller for the closed-loop switched systems is obtained based on Green formula and Lyapunov function method. Using the linear matrix inequality (LMI) approach, the design of the robust controller is converted into finding feasible solutions of a group of LMIs, which can be efficiently carried out using Matlab LMI toolbox. Simultaneously, the method to reduce the conservativeness of the design for the robust control system is also investigated. Finally, a numerical example is presented to show the validity of the proposed design method.

In modern science technology and many motion process of engineering fields, such as neural network activity, movement of missile and spacecraft, control of robot and so on, there are many phenomena which are suddenly changed when their movement is disturbed in some time. The mathematical model of the instantaneous phenomena which we referred is impulsive differential equation. More and more control experts and mathematicians pay close attention to impulsive system, because the study of impulsive has the wide actual background and the value of the application. The robust control problems of the uncertain linear impulsive delay system are studied. Based on the stability of the impulsive delay system, we take advantage of the stability theory of Lyapunov, Lyapunov function and the technology of linear matrix inequality to design a robust feedback controller in order to eliminate the impact of pulse and time delay on the stability of the system. The controller can be got by the way of solving linear matrix inequality by using MATLAB toolbox. And some sufficient conditions of the robust exponential stability are proposed so that the system contains robust feedback control.

Observers and initial state recovering for a wave equation: an LMI approach

Nonlinear active disturbance rejection control is much more effective than linear active disturbance rejection control in tolerance to uncertainties and disturbances. However, it brings a great challenge for theoretical analysis, especially the stability analysis. This article proposes a linear matrix inequality method to analyze the absolute stability of generalized nonlinear active disturbance rejection control form which contains multiple nonlinearities with different parameters in both extended state observer and control law for single-input single-output systems. The generalized nonlinear active disturbance rejection control algorithm and the single-input single-output system are transformed into a direct multiple-input multiple-output Lurie system. A sufficient condition to determine its absolute stability based on linear matrix inequality method is given. The Lyapunov function of the Lurie system exists when the group of linear matrix inequalities is feasible. The free parameters and coefficients in Lyapunov function are given by the solution of these linear matrix inequalities. The electromagnetic valve actuator system in camless engine is presented as an application to illustrate how to perform the proposed method for absolute stability analysis and the stable region of parameter perturbations is obtained via the method. Simulation results show that the linear matrix inequality–based method is convenient and effective to determine whether the closed-loop system is absolutely stable.

This paper presents a nonlinear control design method for robust stabilization and robust performance of linear differential inclusions. A recently introduced non-quadratic Lyapunov function, the convex hull quadratic function, is used for the construction of nonlinear state feedback laws. Design objectives include stabilization with maximal convergence rate, disturbance rejection with minimal reachable set and least /spl Lscr//sub 2/ gain. Conditions for stabilization and performances are derived in terms of bilinear matrix inequalities (BMIs), which cover the existing linear matrix inequality (LMI) conditions as special cases. Optimization problems with BMI constraints are formulated and solved effectively by combining the path-following algorithm and the direct iterative algorithm. The design results are guaranteed to be at least as good as the existing results obtained from LMI conditions. In most examples, significant improvements on system performances have been achieved, which demonstrate the advantages of using nonlinear feedback control over linear feedback control for linear differential inclusions. It is also observed through numerical computation that nonlinear control strategies may help to reduce control effort substantially.

ABSTRACTThis paper provides a procedure for designing a sliding mode observer for a nonlinear system in the presence of Gaussian input disturbances and sensor noises. The paper first proposes a novel candidate for Lyapunov stability, termed a Time-averaged Lyapunov (TAL) function. The TAL averages the Lyapunov analysis over a small finite time interval, allowing for intuitive analysis of noises and disturbance. The paper then provides the necessary and sufficient condition for the design of the sliding observer gainsusing the TAL in the form of Linear Matrix Inequality (LMI). The LMIs can then be explicitly solved offline using commercial LMI solvers. The paper compares the LMIs for the two observer designs to demonstrate the design of the sliding mode observer using TAL can greatly enhance the scope of observer design for nonlinear systems.