Quadratic Lyapunov Function Research Articles

Extended bounded real lemma based sum of squares for static output feedback H∞ heading control

AbstractAim at the fore and aft asymmetry of unmanned surface vehicle (USV) and the loss of linear velocity, a static output feedback (SOF) control method is proposed by the sum of squares (SOS) based on two types of extended bounded real lemmas (EBRL) for heading regulation to guarantee the robustness and improve the rapidity. First, an asymmetric nonlinear heading error model with hydrodynamic parameters is established by the consideration of lack of linear velocity. Second, SOF controller with performance is presented for the asymmetric nonlinear USV model, and it's SOS conditions to solve controller parameters is deduced from EBRL by homogeneous Lyapunov function (HERBL) to maintain the system robustness. Further, the solution of SOF controller relies on the state feedback gain matrix, EBRL is deduced by quadratic Lyapunov function (QEBRL) to obtain the SOS conditions without the addition of an extended matrix to cut computing cost for the solution of the state feedback gain matrix. Finally, the heading control simulations for USV indicate that, QEBRL method takes less time for SOS conditions to solve the state feedback gain matrix, and SOF system of HERBL has a superior performance on the transient response and the robustness.

Development of autonomous vehicle lateral control using time-varying asymmetric barrier Lyapunov function via ADRC approach

In a surface vehicle ride, the passenger’s comfort is desirable, ensuring safety under various tight constraints. To consider this requirement, this article has proposed a lateral control for autonomous vehicle considering the lateral offset error constraints. A reduced-order dynamics of the vehicle has been considered for improving and easy implementation of backstepping. A nonlinear control has been developed by utilizing the Time-varying Asymmetric Barrier Lyapunov Function via the Active Disturbance Rejection Control (ADRC) approach. The Extended State Observer, a component of applied ADRC, estimates all unknown system states as well as the unknown disturbance. The desired trajectories (smooth and non-smooth) based on the yaw rate profile have been synthesized to test the performance of the vehicle. Simulation results are presented to show the effectiveness of the proposed closed-loop control strategy. The vehicle performance has been compared using the proposed method, the conventional Quadratic Lyapunov Function and Symmetric Barrier Lyapunov Function based control. The proposed control provides a smoother control signal, small tracking error and robustness in presence of hard road constraints for smooth/non-smooth vehicle motion.

Verifying Provable Stability Domains for Discrete-Time Systems Using Ellipsoidal State Enclosures

Stability contractors, based on interval analysis, were introduced in recent work as a tool to verify stability domains for nonlinear dynamic systems. These contractors rely on the property that - in case of provable asymptotic stability - a certain domain in a multi-dimensional state space is mapped into its interior after a certain integration time for continuous-time processes or after a certain number of discretization steps in a discrete-time setting. However, a disadvantage of the use of axis-aligned interval boxes in such computations is the omnipresent wrapping effect. As shown in this contribution, the replacement of classical interval representations by ellipsoidal domain enclosures reduces this undesirable effect. It also helps to find suitable ratios for the edge lengths if interval-based domain representations are investigated. Moreover, ellipsoidal domains naturally represent the possible regions of attraction of asymptotically stable equilibrium points that can be analyzed with the help of quadratic Lyapunov functions, for which stability criteria can be cast into linear matrix inequality (LMI) constraints. For that reason, this paper further presents possible interfaces of ellipsoidal enclosure techniques with LMI approaches. This combination aims at the maximization of those domains that can be proven to be stable for a discrete-time range-only localization algorithm in robotics. There, an Extended Kalman Filter (EKF) is applied to a system for which the dynamics are characterized by a discrete-time integrator disturbance model with additive Gaussian noise. In this scenario, the measurement equations correspond to the distances between the object to be localized and beacons with known positions.

Robust Adaptive Control Laws for a Winged Re-entry Vehicle

The flight control system for reusable launch vehicles has to work for an extensive flight envelope ranging from hypersonic to subsonic Mach numbers. Since the plant dynamics are highly nonlinear, time-varying, and coupled, classical controllers are designed for large stability margins about the linearised points. This limits the performance of the classical controller, and the design process is iterative and time-consuming. Hence adaptive controllers are designed to handle such complex systems using a standard quadratic Lyapunov function. The existing parameter update laws of the model reference adaptive control system are not robust to bounded disturbances and unmodelled dynamics. Hence this paper proposes modified controller parameter update laws using a rectangular projection operator and barrier Lyapunov function, ensuring robustness to structured and unstructured uncertainties. The rectangular projection-based control law provides that the controller parameters remain within the specified upper and lower limits, whereas the barrier Lyapunov-based controller maintains the trajectory also within the specified error limit. The effectiveness of the proposed update laws is demonstrated for the automatic landing phase of a reusable launch vehicle.

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.

Fixed-Time Fractional-Order Global Sliding Mode Control for Nonholonomic Mobile Robot Systems under External Disturbances

The present study addresses the problem of fixed-time stabilization (FTS) of mobile robots (MRs). The study’s distinguishing aspects are that the system under examination is subjected to external disturbances, and the system states are pushed to zero in a finite time. This paper suggests new control techniques for chained-form nonholonomic systems (CFNS) subjected to disturbances. First, a switching fractional-order (FO) control approach is proposed for a first-order subsystem (FOS) of an MR under complex disturbances. Secondly, an FO generic global sliding mode control approach is designed for the second-order system (SOS) of the MR in the presence of disturbances. The suggested sliding manifold for the SOS of the MR guarantees global system stability and reduces the chattering problem during control operations. A conventional quadratic Lyapunov function (QLF) is used to converge to the origin in a finite time (FnT). Through this study, a stabilizer for an MR in the presence of disturbances based on an FO switching time-varying controller that can stabilize immeasurable states in a fixed time is proposed. Finally, three case simulations are provided to demonstrate the efficacy of the control strategy proposed in this work against external disturbances.

Stability for generalized Caputo proportional fractional delay integro-differential equations

A scalar nonlinear integro-differential equation with time-variable and bounded delays and generalized Caputo proportional fractional derivative is considered. The main goal of this paper is to study the stability properties of the zero solution. Results are given concerning stability, exponential stability, asymptotic stability, and boundedness of solutions. The investigations are based on an application of a quadratic Lyapunov function, its generalized Caputo proportional derivative, and a modification of the Razumikhin approach. Some auxiliary properties of the generalized Caputo proportional derivative are proved. Five illustrative examples are included.

Stability of Gene Regulatory Networks Modeled by Generalized Proportional Caputo Fractional Differential Equations

A model of gene regulatory networks with generalized proportional Caputo fractional derivatives is set up, and stability properties are studied. Initially, some properties of absolute value Lyapunov functions and quadratic Lyapunov functions are discussed, and also, their application to fractional order systems and the advantage of quadratic functions are pointed out. The equilibrium of the generalized proportional Caputo fractional model and its generalized exponential stability are defined, and sufficient conditions for the generalized exponential stability and asymptotic stability of the equilibrium are obtained. As a special case, the stability of the equilibrium of the Caputo fractional model is discussed. Several examples are provided to illustrate our theoretical results and the influence of the type of fractional derivative on the stability behavior of the equilibrium.

Stability Analysis and Synthesis for 2-D Switched Systems with Random Disturbance

In this paper, the stability analysis and controller design of two-dimensional (2-D) switched system with random disturbances are studied. First, based on the Fornasini–Marchesini local state space model, a random discrete 2-D switched system model is proposed. Secondly, based on the method of switching quadratic Lyapunov function, Schur complement and linear matrix inequality the criterion of the sufficiency of the asymptotically stable under any switching signal is established. The extended average residence time method is used to obtain a sufficient criterion of the mean square exponential stability of the random discrete 2-D switched system under restricted switching conditions. According to the stability analysis results, the conditions for the existence of the state feedback controller are given and the corresponding state feedback controller is designed. Finally, two examples are given to illustrate the effectiveness of the proposed method.

Fault Estimation for Discrete-Time T–S Fuzzy Systems With Unmeasurable Premise Variables Based on Fuzzy Lyapunov Functions

In this work, the fault estimation (FE) problem of discrete-time nonlinear systems subject to fault is researched by resorting to the powerful Takagi-Sugeno (T-S) fuzzy model. In contrast with the existing FE methods where the premise variables (PVs) are required to be measurable, the case of unmeasurable PVs is our primary concern. Then, a fuzzy FE observer with the estimated PVs is well-presented to achieve the reconstructions of fault and state simultaneously. Combining the introduction of a slack matrix with the application of a noncommon quadratic Lyapunov function, some stability criteria of the observer synthesis are provided against unmeasurable PVs, and completely expressed as linear matrix inequalities (LMIs). Finally, the performance of the achieved FE scheme is verified through a tunnel diode circuit.

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.

Virtual-clock-dependent [formula omitted] controller design for discrete-time switched interval type-2 fuzzy systems with intermittent control inputs

This paper is devoted to the study of intermittent H∞ control problem for discrete-time switched interval type-2 fuzzy systems (DSIT2FSs) under minimum dwell time switching. A novel virtual-clock-dependent H∞ intermittent interval type-2 (IT2) fuzzy controller design scheme is proposed. Firstly, we transform the N-mode DSIT2FS with intermittent control inputs into a new one with N controller-busy modes and N controller-free modes. Then, by constructing a class of non-increasing virtual-clock-dependent multiple quadratic Lyapunov functions (VMQLFs), sufficient conditions on the existence of the H∞ intermittent IT2 fuzzy controllers are obtained for DSIT2FSs with dwell time constraints. The controller gains can be gotten by solving a set of strict linear matrix inequalities (LMIs). The obtained results are also applicable to non-switched IT2 fuzzy systems. It is shown that the designed IT2 fuzzy controllers work intermittently but still can guarantee stability and H∞ performance of the close-loop systems. Finally, a tunnel diode circuit system model and a single link robot arm model are provided to illustrate the effectiveness and advantages of the results.

Adaptive constrained control for automotive electronic throttle control system with experimental analysis

Abstract This article proposes an adaptive constrained control strategy with state constraints and uncertain parameters for an electronic throttle control (ETC) system. Compared with the current control strategies for an ETC system, state constraints and parameter uncertainties are adequately considered in the proposed control strategy. First, the nonlinear dynamic model for control of an ETC is described. Second, the asymmetric Barrier Lyapunov Function (BLF) and a backstepping control algorithm are used to ensure that the throttle opening does not exceed the constrained boundary. A parameter adaptive law is given to estimate the unknown parameter and external disturbances with an ETC system. Third, the proposed BLF controller is compared with the existing Quadratic Lyapunov Function (QLF) controller by simulation and experiment. The results show that the proposed control algorithm not only ensure fast transient performance in the control response, but also avoid the out of bounds of the throttle opening. The proposed constrained control strategy can provide excellent control performance for an ETC system.

Observer design for a flexible structure with distributed and point sensors

The paper is devoted to the observability study of a dynamic system, which describes the vibrations of an elastic beam with an attached rigid body and distributed control actions. The mathematical model is derived using Hamilton's principle in the form of the Euler-Bernoulli beam equation with hinged boundary conditions and interface condition at the point of attachment of the rigid body. It is assumed that the sensors distributed along the beam provide output information about the deformation in neighborhoods of the specified points of the beam. Based on the variational form of the equations of motion, the spectral problem for defining the eigenfrequencies and eigenfunctions of the beam oscillations is obtained. Some properties of the eigenvalues and eigenfunctions of the spectral problem are investigated. Finite-dimensional approximations of the dynamic equations are constructed in the linear manifold spanned by the system of eigenfunctions. For these Galerkin approximations, observability conditions for the control system with incomplete information about the state are derived. An algorithm for observer design with an arbitrary number of modal coordinates is proposed for the differential equation on a finite-dimensional manifold. Based on a quadratic Lyapunov function with respect to the coordinates of the finite-dimensional state vector, the exponential convergence of the observer dynamics is proved. The proposed method of constructing a dynamic observer makes it possible to estimate the full system state by the output signals characterizing the motion of particular point only. Numerical simulations illustrate the exponential decay of the norm of solutions of the system of ordinary differential equations that describes the observation error.

Exponential stability and L2 gain analysis of uncertain fractional reset control systems

Abstract This paper considers the stability problem of a class of uncertain fractional reset control systems that undergo the ${L}_2$ gain performance improvement via the conformable fractional calculus. To remove the Zeno phenomenon in the system’s response, a new reset law based on the time regularization technique is designed. By developing a theory to design a new reset control, the stability of the proposed fractional order (FO) linear system with model uncertainties and bounded exogenous input is proved via an extended quadratic Lyapunov function and some efficient linear matrix inequalities. Likewise, the impulsive effect in the control input is removed by employing an appropriate low-pass filter (LPF). In the end, two numerical examples show that when the plant is modelled by the FO differential equations, by utilizing the constructive role of fractional reset controller, the robustness of FO reset system can also be attained.

Regional T-S Model-based Attitude Tracking Control of a Quadrotor with Input Saturation and External Disturbances

This paper presents a model reference attitude tracking controller design for a quadrotor unmanned aerial vehicle subject to saturated actuators and external disturbances. The dynamical nonlinear model of the quadrotor's attitude is represented by an uncertain-like Takagi-Sugeno model, with exact matching on a compact subset of the state space. The generalized sector condition is employed to deal with the saturated inputs of the quadrotor. LMI conditions are derived for the design of the proposed tracking controller from a quadratic Lyapunov function, together with a performance index used to minimize the L2-norm transfer between the disturbances and the state tracking errors. Because of the input saturation and the validity domain of the Takagi-Sugeno model, the obtained results only hold regionally and an optimization procedure to estimate the closed-loop tracking domain of attraction is proposed. Simulation results are provided to illustrate the effectiveness of the proposed design methodology.

A Neural Lyapunov Approach to Transient Stability Assessment of Power Electronics-Interfaced Networked Microgrids

This paper proposes a novel Neural Lyapunov method-based transient stability assessment framework for power electronics-interfaced networked microgrids. The assessment framework aims to determine the large-signal stability of the networked microgrids and to characterize the disturbances that can be tolerated by the networked microgrids. The challenge of such assessment is how to construct a behavior-summary function for the nonlinear networked microgrids. By leveraging strong representation power of neural network, the behavior-summary function, i.e., a Neural Lyapunov function, is learned in the state space. A stability region is estimated based on the learned Neural Lyapunov function, and it is used for characterizing disturbances that the networked microgrids can tolerate. The proposed method is tested and validated in a grid-connected microgrid, three networked microgrids with mixed interface dynamics, and the IEEE 123-node feeder. Case studies suggest that the proposed method can address networked microgrids with heterogeneous interface dynamics, and in comparison with conventional methods that are based on quadratic Lyapunov functions, it can characterize the stability regions with much less conservativeness.

Asymptotically Necessary and Sufficient Quadratic Stability Conditions of T-S Fuzzy Systems Using Staircase Membership Function and Basic Inequality

Asymptotically necessary and sufficient quadratic stability conditions of Takagi-Sugeno (T-S) fuzzy systems are obtained by utilizing staircase membership functions and a basic inequality. The information of the membership functions is incorporated in the stability analysis by approximating the original continuous membership functions with staircase membership functions. The stability of the T-S fuzzy systems was investigated based on a quadratic Lyapunov function. The asymptotically necessary and sufficient stability conditions in terms of linear matrix inequalities were derived using a basic inequality. A fuzzy controller was also designed based on the stability results. The derivation process of the stability results is straightforward and easy to understand. Case studies confirmed the validity of the obtained stability results.

Barrier Adaptive Iterative Learning Control for Tank Gun Control Systems Under Nonzero Initial Error Condition

In this paper, a barrier adaptive iterative learning control scheme is proposed to solve the trajectory-tracking problem for tank gun control systems under nonzero initial error condition. A novel construction method of rectified reference trajectory is presented for dealing with the initial position problem of iterative learning control for tank gun control systems. With a quadratic form barrier Lyapunov function adopted to controller design, the quadratic form of system error is constrained within the preset range during each iteration. Adaptive iterative learning control technique and robust control technique are jointly used to compensate for the parametric/nonparametric uncertainties and nonsymmetric deadzone nonlinearity. As the iteration number increases, the system state of tank gun control systems may accurately track the rectified reference trajectory, which leads to a excellent tracking performance during the part operation interval of tank gun control systems. Simulation results are presented to verify the effectiveness of the proposed barrier adaptive iterative control scheme.

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.