Quadratic Lyapunov Function Research Articles

Prescribed-time practical stabilization via majorant systems

This paper provides a design methodology of simple control laws to solve the prescribed-time practical stabilization problem for a class of pseudo-linear systems with a single input, also unstable, subject to perturbations, nonlinear and discontinuous, but bounded. The perturbations on the dynamic matrix, such as a ratio of multi-affine polynomials of nonlinear functions or parameters, are not triangular, and a perturbation on the input of multiplicative type is also considered. The stated methodology is based on the majorant systems approach with optimized quadratic Lyapunov functions. The main advantages of the proposed control laws consist in the simplicity of their design and implementation and the reliable performance. Indeed, these control laws depend on a single design parameter, have constant gains, and guarantee the practical convergence with a prescribed relative error, also after a prescribed time. Some examples, also an engineering one, are provided to validate the obtained theoretical results and show the effectiveness and efficiency of the proposed control laws.

Stabilization of fractional nonlinear systems with disturbances via sliding mode control

AbstractIn this article, the sliding mode control (SMC) of fractional nonlinear systems (FNSs) with disturbances is studied. First, a useful fractional power‐rate inequality for is extended to a more general form . Based on the generalized inequality, the method of a modified fractional integral SMC is completed, in which the parameter of the symbolic function is extended to and . This increases the degree of freedom of the sliding mode surface (SMS). Then, the SMC of FNSs is studied for the cases of known and unknown disturbances. In the case of known disturbance, it is proved by the generalized inequality and quadratic Lyapunov function method that the state of the FNSs can converge asymptotically to zero on the SMS. For the case of unknown disturbance, a fractional disturbance observer is used to estimate the disturbance by introducing auxiliary variables . The disturbance estimation error of the proposed FNSs is proved to be bounded. An equivalent global control law is obtained and asymptotic convergence of the FNSs under the action of the controller is proved. Finally, two numerical simulation examples are given to verify the feasibility of the method proposed in this article.

On Special Quadratic Lyapunov Functions for Linear Dynamical Systems With an Invariant Cone

On Special Quadratic Lyapunov Functions for Linear Dynamical Systems With an Invariant Cone

On the local quadratic stability of T–S fuzzy systems in the vicinity of the origin

The main goal of this paper is to introduce new local stability conditions for continuous-time Takagi-Sugeno (T-S) fuzzy systems. These stability conditions are based on linear matrix inequalities (LMIs) in combination with quadratic Lyapunov functions. Moreover, they integrate information on the membership functions at the origin and effectively leverage the linear structure of the underlying nonlinear system in the vicinity of the origin. As a result, the proposed conditions are proved to be less conservative compared to existing methods using fuzzy Lyapunov functions. Moreover, we establish that the proposed methods offer necessary and sufficient conditions for the local exponential stability of T-S fuzzy systems. Discussions on the inherent limitations associated with fuzzy Lyapunov approaches are also given. To illustrate the theoretical results, we provide comprehensive examples that demonstrate the core concepts and validate the efficacy of the proposed conditions.

Distributed safe formation tracking control of multiquadcopter systems using barrier Lyapunov function.

Coordinating the movements of a robotic fleet using consensus-based techniques is an important problem in achieving the desired goal of a specific task. Although most available techniques developed for consensus-based control ignore the collision of robots in the transient phase, they are either computationally expensive or cannot be applied in environments with dynamic obstacles. Therefore, we propose a new distributed collision-free formation tracking control scheme for multiquadcopter systems by exploiting the properties of the barrier Lyapunov function (BLF). Accordingly, the problem is formulated in a backstepping setting, and a distributed control law that guarantees collision-free formation tracking of the quads is derived. In other words, the problems of both tracking and interagent collision avoidance with a predefined accuracy are formulated using the proposed BLF for position subsystems, and the controllers are designed through augmentation of a quadratic Lyapunov function. Owing to the underactuated nature of the quadcopter system, virtual control inputs are considered for the translational (x and y axes) subsystems that are then used to generate the desired values for the roll and pitch angles for the attitude control subsystem. This provides a hierarchical controller structure for each quadcopter. The attitude controller is designed for each quadcopter locally by taking into account a predetermined error limit by another BLF. Finally, simulation results from the MATLAB-Simulink environment are provided to show the accuracy of the proposed method. A numerical comparison with an optimization-based technique is also provided to prove the superiority of the proposed method in terms of the computational cost, steady-state error, and response time.

A Frequency-Domain Criterion for the Quadratic Stability of Discrete-Time Systems with Switching between Three Linear Subsystems

Connected systems with switching between three linear discrete-time subsystems are considered, and a new frequency-domain criterion for the existence of a quadratic Lyapunov function ensuring the stability of such systems under arbitrary switching is proposed. The application of this criterion is demonstrated on an example of a third-order system.

A simple approach for studying stability properties of an SEIRS epidemic model

Abstract In this work, we study stability properties of a well-known integer-order SEIRS model with nonlinear incidence and vertical transmission. Firstly, we introduce a simple approach to the analysis of global asymptotic stability (GAS) of the integer-order model. This approach is based on general quadratic Lyapunov functions and characteristic of quadratic forms associated with real matrices. The result is that the GAS of disease-free and disease-endemic equilibrium points is completely established. This provides an important improvement for results constructed in two previous works. Secondly, we generalize the integer-order SEIRS model by considering it in the context of the Caputo fractional-order derivative. After that, the present approach is utilized to investigate the GAS of the proposed fractional-order model. As an important consequence, not only the GAS but also the uniform stability of the fractional-order model are determined fully. Therefore, the applicability of the approach is shown. Finally, a series of numerical experiments is conducted to illustrate and support the theoretical findings.

Dynamic event-triggered mean-square consensus of positive multi-agent systems via impulsive control under false data injection attacks

This paper investigates the mean-square impulsive consensus problem of positive multi-agent systems (MASs) under stochastic sensor false data injection attacks (FDIA). False data injection attacks are mortal to a positive system since the system positivity could be destroyed with ease (by tampering with the sampled positive state value to a negative one, for instance). To deal with this issue, a data compensator is proposed which preprocesses deliberately distorted state values to reduce their adverse impacts on the positivity of the positive MASs. On the other hand, a novel sampled-data-based dynamic event-triggered mechanism (SDB-DETM) is tailored to generate impulsive control time sequences, save control costs, and promise a desired control performance. A maximum event-trigger time interval is introduced as a substitute for the assumption of the average time interval when considering the randomness of FDIA. Furthermore, the dynamics of the auxiliary dynamic variable are improved to adapt the novel SDB-DETM with mixed signals. A sufficient and necessary condition of the positivity of the positive MASs under FDIA is presented. The commonly used positive constraints on the error system are removed. By constructing a quadratic Lyapunov function, a sufficient condition of the mean-square consensus of the positive MASs is derived. Two numerical examples are provided to illustrate the effectiveness of theoretical results.

Active suspension hierarchical control with parameter uncertainty and external disturbance of electro-hydraulic actuators

In order to improve the ride comfort and handling stability of the electro-hydraulic active suspension system, a hierarchical control strategy is proposed. For the active suspension system with body mass uncertainty and safety constraints, an enhanced constraint adaptive backstepping controller is designed to generate the target force of body vertical motion. When dealing with constraints, nonlinear filters are introduced, backstepping technology and quadratic Lyapunov function are used to combine the control target and domain constraints, and the vertical displacement of the body and the suspension deflection control index are synthesized into a single controlled variable, so that the control variable converging to zero meets the suspension dynamic displacement limit. The ride comfort and safety of the active suspension system are improved. Secondly, stability analysis is conducted on the zero dynamic error system to ensure that all safety performance indicators are bounded. The effects of external disturbances, parameter uncertainties and modelling errors on the force tracking accuracy of asymmetric cylinder electrohydraulic systems are addressed. A backstepping dynamic surface control method based on a nonlinear perturbation observer is proposed, which estimates the composite perturbation terms such as modelling error and external perturbation, and uses the estimated value of the nonlinear observer to design a feedforward compensating control term to offset the effect of the composite perturbation on the system performance. In addition, the dynamic surface control method is used to calculate the derivative of the target force input, which avoids the derivative explosion phenomenon and reduces the computational complexity of the system. Simulation test results show that this method reduces the computational complexity of the system and improves the control performance. And under random and bumpy road conditions, the root-mean-square value of sprung mass acceleration performance index is reduced by 48.354 % and 72.677 % compared with passive suspension, which improves the ride comfort of the vehicle.

Uniform ultimate boundedness analysis for linear systems with asymmetric input backlash and dead-zone: A piecewise quadratic Lyapunov function approach

This paper deals with the interconnection between a linear system and a nonlinear operator consisting of asymmetric input backlash and asymmetric dead-zone. The uniform ultimate boundedness of the system is studied. A piecewise quadratic Lyapunov function, suitable with the polyhedral description of the nonlinear operator is proposed. The conservatism of existing results is therefore reduced. The effectiveness and improvement of the results are assessed using a numerical example.

Coordinated management of oxygen excess ratio and cathode pressure for PEMFC based on synthesis variable-gain robust predictive control

In this research endeavor, a novel synthesis variable-gain robust model predictive control (SVGRPC) method using the min-max technique with parameter dependent Lyapunov functions is introduced to achieve the desired trajectory tracking of oxygen excess ratio (OER) and cathode pressure in the polymer electrolyte membrane fuel cells (PEMFCs) system. Initially, a simplified control-oriented model of the air supply system is developed and then transformed into a polytopic form to accommodate the inherent uncertainties in the PEMFC. Subsequently, the polytopic model is integrated with the reference trajectory to construct an augmented state-space model for deriving an error state representation. In the framework of robust model predictive control (RPC) utilizing a parameter-dependent Lyapunov function, a series of variable feedback gains are employed to mitigate conservatism. Additionally, the online solution process imposes a significant computational burden, presenting challenges for the real-time implementation of RPC-based control strategy. Therefore, the offline computing is introduced to significantly reduce the computational burden, resulting in the development of the SVGRPC strategy proposed in this paper. Subsequently, the SVGRPC strategy computes explicit linear state-feedback control laws by solving linear matrix inequalities (LMIs) using an offline control algorithm. The effectiveness of the proposed controller is then validated through assessments conducted under three distinct operating conditions of the PEMFC system. The outcomes of this study affirm that the proposed controller improves the requirements of control performance when compared to the online RPC with a quadratic Lyapunov function while significantly alleviating the computational workload.

Robust H∞ adaptive model following control for fractional-order systems with polytopic and bounded uncertainties subject to input saturation

This paper investigates a type of robust observer-based tracking control for the well-known inverted pendulum as an uncertain fractional-order system with two-norm bounded uncertainties subject to input saturation along with external disturbance that focuses on the case of a fractional-order α such that [Formula: see text]. The main goal is to design a robust model following tracking control servomechanism by the side of stability analysis. This paper addresses the well-known quadratic Lyapunov function in addition to the Gronwall–Bellman lemma and the sector condition of the saturation function stability synthesis. This paper proposes an outstanding strategy to achieve the best model following robust observer in the cast of output-feedback control subject to input saturation in the framework of an uncertain fractional-order system based on the linear matrix inequality (LMI) solution. The LMI procedure can be used to achieve static output-feedback tracking control and observer gain. Several valuable theorems support this approach, and various simulation scenarios are available to showcase its effectiveness.

Composite quadratic Lyapunov function event-triggered control of vertical take-off and landing aircraft systems with actuator saturation

This paper deals with the problem of event-triggered control (ETC) for vertical take-off and landing (VTOL) aircraft systems under actuator saturation. By introducing the VTOL system with saturated actuators, a composite quadratic Lyapunov function (CQLF) approach is first proposed to describe the problem of the event triggering mechanism. The proposed transfer mechanism not only further saves communication resources, but also further suppresses the conservatism of the attraction domain estimation. Then, sufficient conditions are established via linear matrix inequality (LMI) by incorporating new event triggering mechanism condition and prescribed passive performance metrics under the closed-loop system that is asymptotically stable and strictly passive, respectively. Finally, the method proposed in this paper can effectively reduce the number of triggers and expand the domain of attraction. The validity and feasibility of the obtained results are demonstrated by two example simulations.

Parameterization to fault detection filtering for Itô stochastic T‐S fuzzy systems

AbstractThis article focuses on the fault detection filtering problem for Itô stochastic Takagi–Sugeno (T‐S) fuzzy systems. In terms of line integral Lyapunov function, a sufficient condition of exponential stability in mean square and extended dissipativity for the systems under consideration is established which has less conservatism than the one based on quadratic Lyapunov function. The obtained sufficient condition is nonlinear, which makes the synthesis of fault detection filter being difficult and challenging. By choosing general matrix variables and constructing the appropriate orthogonal complement matrices, the nonlinear sufficient condition can be converted into linear one ingeniously via utilizing Finsler's lemma twice. Thus, the fault detection filter can be developed by virtue of parameterization. It should be pointed out that our filter determined by the parameterization approach includes the one obtained by the equivalent transformation method as a special case. Finally, we demonstrate the feasibility and superiority of our proposed approach through three numerical examples.

New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation

In this paper, a simple, generally valid stability proof for an anti-windup method for PI-state controlled systems is presented, with which it is possible to directly conclude the stability of the PI-state controlled system from a stable P-state controlled system with constraints in the manipulated variables, i.e. without having to perform a separate stability investigation of the anti-windup measures. The technique presented is based on the system description by means of state equations and Lyapunov's Direct Method using quadratic Lyapunov functions. Furthermore, the PI-state controller is designed in such a way that it provides the same command response as the P-state controller, for which a stability statement is already available. Both continuous-time and discrete-time systems are considered, which, apart from the saturation of the manipulated variables, show linear, time-invariant behavior. In addition, a general stability proof is given for discrete-time systems, which makes it possible to establish stable anti-windup methods for P- and PI-state controlled systems, which contain dead time elements in the path of the manipulated variables, without having to carry out separate stability investigations for them. For this purpose, the state controller design for the system with dead time elements in the manipulated variable paths is based on the principle that the same characteristics of the control behavior should be achieved as for the system without such dead time elements, but delayed by the dead time. The effectiveness of the presented methods is illustrated by an example from the field of electrical drives.

Neural network-based robust adaptive super-twisting sliding mode fault-tolerant control for a class of tilt tri-rotor UAVs with unmodeled dynamics

Abstract Aiming at alleviating the adverse influence of coupling unmodeled dynamics, actuator faults and external disturbances in the attitude tracking control system of tilt tri-rotor unmanned aerial vehicle (UAVs), a neural network (NN)-based robust adaptive super-twisting sliding mode fault-tolerant control scheme is designed in this paper. Firstly, in order to suppress the unmodeled dynamics coupled with the system states, a dynamic auxiliary signal, exponentially input-to-state practically stability and some special mathematical tools are used. Secondly, benefiting from adaptive control and super-twisting sliding mode control (STSMC), the influence of the unexpected chattering phenomenon of sliding mode control (SMC) and the unknown system parameters can be handled well. Moreover, NNs are employed to estimate and compensate some unknown nonlinear terms decomposed from the system model. Based on a decomposed quadratic Lyapunov function, both the bounded convergence of all signals of the closed-loop system and the stability of the system are proved. Numerical simulations are conducted to demonstrate the effectiveness of the proposed control method for the tilt tri-rotor UAVs.

Stability analysis of hybrid integrator-gain systems: A frequency-domain approach

The hybrid integrator-gain system (HIGS) has been introduced recently with the aim to overcome fundamental limitations of linear time-invariant (LTI) control systems. To support the analysis and design of HIGS-based controllers, in this paper a novel frequency-domain condition for stability analysis of the feedback interconnection of an LTI system and HIGS is presented. Compared to existing frequency-domain stability conditions such as the one extending the circle-criterion, the condition presented in this paper exploits explicit knowledge regarding HIGS’ switching strategy, thereby potentially providing a significantly less conservative condition. In particular, the novel condition in this paper guarantees the existence of a quadratic Lyapunov function that does not need to be positive definite within the full state space. The proposed condition can be verified graphically in a manner that is reminiscent of the classical Popov plot, as will be illustrated in an experimental case-study.

Asynchronous event-triggered guaranteed cost control of uncertain 2-D Markov jump Roesser systems with actuator saturation

This paper investigates the asynchronous event-triggered guaranteed cost control problem for two-dimensional Markov jump Roesser systems with parameter uncertainties and actuator saturation. The hidden Markov model is constructed to characterize the asynchronous phenomenon induced by the mode mismatch between the system and the controller, and a hidden Markov model-based event-triggered mechanism is adopted in controller design to save communication resources. The convex hull representation is employed to process saturated inputs. By using the quadratic Lyapunov function methods and linear matrix inequality techniques, some sufficient criteria are developed to guarantee the asymptotic stability of the addressed system with a guaranteed cost under three different boundary conditions. Finally, a convex optimization algorithm with linear matrix inequality constraints is proposed to design the optimal asynchronous event-triggered guaranteed cost controller, and a numerical example is considered to demonstrate its validity.

Combined longitudinal and lateral dynamics regulation of autonomous vehicles via induced ℒ2$$ {\mathcal{L}}_2 $$‐gain linear parameter varying control strategies based on parameter‐dependent Lyapunov functions

AbstractThis article considers a path tracking control problem for autonomous vehicles constrained to maintain a safe distance from the next vehicle in the lane. The control design problem is solved by considering a mathematical model of the vehicle that jointly describes the lateral and longitudinal dynamics. Specifically, the optimal controller is designed aimed at minimizing the orientation and position tracking errors with respect to a given reference path. Moreover, the controller is able to ensure that the vehicle can both follow a speed profile and automatically adjusts the vehicle speed to maintain a proper distance from the next vehicle. The lateral dynamic is regulated by a 2‐DOF feedforward/feedback controller, where the feedforward law is implemented by exploiting an inverse kinematic model of the vehicle. On the other side, the feedback action, along with the action of the controller in charge to regulate the longitudinal dynamics, is the optimal gain‐scheduling state‐feedback controller. In particular, a parameter‐dependent quadratic Lyapunov function approach is used to reduce the conservativeness of the solution and a suitable convex LMI optimization problem is formulated for the synthesis. The vehicle is described by a seven‐degrees‐of‐freedom (7DOF) full‐car model, with the tire‐generated forces described by the classical Pacejka's Magic Formula. In order to assess the performance of the proposed control strategy, simulations have been undertaken in Matlab/Simulink by considering an autonomous vehicle driving into an urban scenario.

Tangential velocity tracking-based cross-coupled control method with variable feedrate for biaxial parametric curve following

To address the inaccurate contour error calculation problem in traditional cross-coupled control (CCC) methods for biaxial motion systems, this paper presents a novel CCC method based on a recently developed tangential velocity tracking (TVT) strategy. It has the advantage that existing high-precision algorithms for searching the foot point can be directly integrated to obtain the excellent estimation accuracy of contour error. The cumbersome parameter tuning for position controllers of each axis is unnecessary. Moreover, a velocity interpolator for parametric curves is developed to extend the TVT strategy to the variable feedrate case. The stability of the TVT-based CCC system is proved using a quadratic Lyapunov function. Comparative experiments are carried out, and the results indicate that the estimation deviation of contour error in the TVT-based CCC method can be constrained within 1 μm. The maximum contour error is significantly reduced by 65.32% and 50.10% compared with the traditional CCC methods based on tangential and circular approximations, respectively.