Proposed Lyapunov Function Research Articles

Asynchronous Quantized Control of Piecewise-Affine Systems

This technical note proposes a novel asynchronous control approach for discrete-time piecewise-affine (PWA) systems with logarithmic quantization of both multi-inputs and multi-state measurements. Since the actual system state and the quantized state may not always be in the same operating region due to quantization-induced uncertainties, the operating modes of the PWA system and the controller which depends on the quantized states may be asynchronous. Aiming at reducing the computational cost and the conservatism of the results, a mapping region-based algorithm is first proposed to determine the reachable dwelling regions for the quantized state. By using a convex combination model to approximate the quantization-induced uncertainties, a novel piecewise Lyapunov function taking into consideration the uncertainties is then proposed. It is shown that with the newly proposed Lyapunov function, the desired asynchronous controller can be obtained and the resulting closed-loop system is asymptotically stable. A simulated chemical reactor example is presented to illustrate the effectiveness and the superiority of the proposed asynchronous control approach.

Gain Calculation of 0-Degree Homogeneous Controllers

This paper presents sets of gains for the nested discontinuous controller family. The methodology for obtaining gains is based on a control Lyapunov function, this function determines the conditions for the value of each of the controller gains, the last gain being the one that is directly related to the boundaries of the disturbances and uncertainties. In this way the algorithm for obtaining gains can be summarized in three steps. The first step concerns about establishing the value of the weight vector, the degrees of homogeneity of the r-homogeneous functions that make up the proposed Lyapunov function. The second step consists of establishing the value of the first gain, which, for second and higher order controllers, is sufficient for it to be real positive. The third step is to determine the maximum value of an objective function. Moreover, it allows us to obtain the lower bound of the set of values for the gain that stabilize the system. The results show that the set of gains obtained stabilize the closed-loop system.

Generalized Lyapunov Functions for Discrete-Time Lurie Systems With Slope-Restricted Nonlinearities

A class of Lyapunov functions for discrete-time Lurie systems with monotonic non-linearities is proposed. The Lyapunov functions are composed of quadratic terms on the states and of the system's non-linearities as well as Lurie-Postnikov type integral terms. Crucially, positive definiteness of the matrix in the generalised quadratic form and positivity of the scaling terms of the Lurie-Postnikov integrals are relaxed in the stability conditions. Furthermore, they are used for regional stability analysis and performance assessment. Numerical examples show that the proposed Lyapunov function structure matches or outperforms existing ones for these systems.

Stability analysis of time-varying switched systems via indefinite difference Lyapunov functions

Asymptotic stability of time-varying switched systems is investigated in this paper. The less conservative sufficient criteria for asymptotic stability of time-varying discrete-time switched systems are proposed via common indefinite difference Lyapunov functions and multiple indefinite difference Lyapunov functions introduced in this note, respectively. Common indefinite difference Lyapunov functions can be used to analyze stability of a switched system with asymptotic stable subsystems and arbitrary switching signal. Multiple indefinite difference Lyapunov functions can be used to investigate stability of a switched system with unstable subsystems and a given switching signal. The difference of the proposed Lyapunov function may be positive at some instants for an asymptotically stable subsystem. We compare these main results and illustrate the effectiveness of the obtained theorems by three numerical examples.

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

This article investigates the stability analysis for a class of continuous-time switched systems with state constraints under pre-specified dwell time switchings. The state variables of the studied system are constrained to a unit closed hypercube. Firstly, based on the definition of set coverage, the system state under saturation is confined to a convex polyhedron and the saturation problem is converted into convex hull. Then, sufficient conditions are derived by introducing a class of multiple time-varying Lyapunov functions in the framework of pre-specified dwell time switchings. Such a dwell time is an arbitrary pre-specified constant which is independent of any other parameters. In addition, the proposed Lyapunov functions can efficiently eliminate the “jump” phenomena of adjacent Lyapunov functions at switching instants. The feature of this paper is that the definition of set coverage is utilized to replace the restriction on the row diagonally dominant matrices with negative diagonal elements to analyze stability. The other feature of the constructed time-varying Lyapunov functions is that there are two time-varying functions. One of the two time-varying functions contains the jump rate, which will present a certain degree of freedom in designing the dwell time switching signal. An iterative linear matrix inequality (LMI) algorithm is presented to verify the sufficient conditions. Finally, two examples are presented to show the validity of the method.

Variable-Parameter-Dependent Saturated Robust Control for Vehicle Lateral Stability

This article proposes a novel gain-scheduling control method to improve vehicle lateral stability based on a variable-parameter-dependent approach. A time-varying velocity-dependent model to describe the vehicle lateral dynamic characteristics is constructed under cornering stiffness uncertainties and controller saturation. A novel Lyapunov function with variable-parameter-dependent Lyapunov matrix is established to develop the conditions. Using the proposed Lyapunov function, a general condition of the variable-parameter-dependent saturated robust yaw moment controller is provided in terms of time-varying parameter-dependent matrix inequalities. In order to obtain a tractable solution, a condition is further developed with finite linear matrix inequalities. Moreover, an optimal distribution method is adopted to generate the desired yaw moment based on torque allocation. These torques are computed by optimizing the distribution errors and tire workloads. Simulations results under both J-turn and double-lane changing scenarios are used to illustrate the merits of the proposed method. The control synthesis approach is also applicable to other applications involving time-varying parameters.

Adaptive reinforcement learning in control design for cooperating manipulator systems

AbstractIn this paper, an optimal motion/force hybrid control strategy based on adaptive reinforcement learning (ARL) is proposed for cooperating manipulator systems. The optimal trajectory control and constraint force factor control, by using the Moore–Penrose pseudoinverse, are addressed to design the controller corresponding to the manipulator dynamic model. In addition, a frame of a different auxiliary term and an appropriate state‐variable vector are presented to address the non‐autonomous closed system with a time‐varying desired trajectory. The simultaneous actor/critic algorithm is implemented by optimizing the squared Bellman difference to be computed from the error of control policy and optimal control input. Moreover, the constraint force factor controller is discussed by a nonlinear technique after achieving the result of the constraint force factor. The tracking and convergence of ARL‐based optimal motion/force hybrid control strategy are validated in closed‐loop systems by a proposed Lyapunov function candidate. Finally, simulation results are implemented on a constrained system using three manipulators to verify the physical implementation of the presented optimal motion/force hybrid‐tracking control strategy.

Stabilization of a class of switched systems with state constraints via time-varying Lyapunov functions

This paper studies the stabilization problem of a class of continuous-time switched linear systems with state constraints under pre-specified dwell-time switchings. Such systems are defined on a closed hypercube as all state variables are constrained to the unit hypercube. The dwell time in this paper is an arbitrarily pre-specified rather than a calculated constant, which is independent of any parameters. First, a class of multiple time-varying Lyapunov functions is introduced to study the stability analysis, and sufficient conditions on stability of the studied switched systems without control input are derived in the framework of pre-specified dwell-time switchings. The distinguishing feature of the proposed Lyapunov functions is that this type of delicately constructed Lyapunov functions can efficiently eliminate the “jump” phenomena of adjacent Lyapunov functions at switching times. Second in the same framework of the dwell time, sufficient conditions for stabilization are proposed for that of the switched systems with state constraints by further designing state feedback controllers. Finally, two examples are provided to demonstrate the effectiveness of the proposed results. The results of this paper do not require to calculate the total time of each subsystem during which the state is saturated or non-saturated separately, which makes the pre-specified dwell-time switchings easy to apply.

Design of Quadrotor System for Transmission Line Inspection

A novel robust adaptive control is proposed for target detection and recognition in transmission line inspection, this paper proposes a quadrotor inspection system based on robust adaptive control of multi-sensor fusion. Robust adaptive control ensures the stability and anti-disturbance ability of the quadrotor during the flight. It is measurable for the distance between the quadrotor and cables by laser sensor and ultrasonic sensor. It can be used to detect and identify the target objects on cable lines by OpenMV. According to the proposed Lyapunov function, the asymptotically stable conditions is obtained, in which all trajectories converge in finite time to the desired location. Finally, the effectiveness of inspections is validated experimentally, and the system has strong anti-interference ability.

Fuzzy Estimator Indirect Terminal Sliding Mode Control of Nonlinear Systems Based on Adaptive Continuous Barrier Function

In this paper, the theory of control is considered on nonlinear systems. A closed-loop controller with a strong idea has been introduced to track system states and guarantee asymptotically stability. The proposed method is the indirect terminal sliding mode control technique based on adaptive and fuzzy rules, which has used the continuous barrier function as a new approach in its design to improve the performance of this controller. One of the significant challenges in the sliding mode control method is the chattering phenomenon due to the discontinuous sign function in the control law. In the proposed approach, the control law is obtained continuously and smoothly due to the mentioned continuous function and subsequently solves the chattering problem. Another feature of the proposed method is the asymptotical stability of the system dynamics within finite time, which is proved based on the Lyapunov function. The proposed Lyapunov function includes the function of fractional power of a sliding surface. On the other hand, the obtained control law using the sliding mode method is estimated using the fuzzy system. The adaptive approach adjusts the fuzzy law parameters and the unknown bound of external disturbance.

A Lyapunov analysis of the continuous-time adaptive Bellman–Ford algorithm

The shortest path problem, one of the most classical graph problems, has been addressed in many different ways suitable for various settings in the fields of computer science and artificial intelligence. In this paper, we revisit a distributed control solution, namely the continuous-time adaptive Bellman–Ford algorithm, to the shortest path problem. While previous work only concerned its global asymptotic stability, we not only prove its global asymptotic stability by formulating a Lyapunov function, but characterize the initial conditions under which the algorithm will converge exponentially, and show that the algorithm is globally ultimately bounded under persistent bounded perturbations based on the proposed Lyapunov function.

A novel fixed-time stability of nonlinear impulsive systems: a two-stage comparison principle method

In this article, we elaborated on the problem of fixed time stability (FxTS) of nonlinear impulsive systems. For handing this problem, we propose a new concept of fixed-time stable function set (FTSFS). By taking advantage of definitions of FTSFS and average impulsive interval, a unified FxTS criterion of nonlinear impulsive systems is presented, which is applicable to both destabilising impulses and stabilising impulses. Comparing with the available work, the proposed Lyapunov function is allowed to possess indefinite derivatives when deriving from the state trajectory of nonlinear impulsive systems. Finally, two numerical examples are provided to support the proposed criterion.

A Lyapunov-Function Based Controller for 3-Phase Shunt Active Power Filter and Performance Assessment Considering Different System Scenarios

Shunt active power filter (SAPF) belongs to the class of custom power devices (CPDs) and offers compensation to harmonics originated owing to customer side nonlinear loads, reactive power and unbalance in the distribution power networks functioning in current control mode (CCM). The performance of a SAPF as a harmonic compensator entirely relies on the control technique i.e. the precise detection of the harmonic current components of load that are necessary to be compensated. In the present work, a 3-phase SAPF, inspired by a Lyapunov function based control approach, has been designed for compensation of harmonics resulted in the feeder current owing to the customer side nonlinearity. A control law is determined in the proposed strategy which makes the derivative of the Lyapunov function consistently a negative one for an entire set of stable states. The DC-link capacitor voltage is regulated at constant reference through the proportional-integral (PI) controller. In this method rating of the shunt active power filter is considerably reduced than the other two broadly employed conventional methods. Furthermore, the harmonic compensation efficacy of the proposed Lyapunov function based SAPF is compared with the one based on other two conventional approaches under four different system scenarios namely a simple nonlinear load with and without utility side voltage distortion, a modified IEEE 13 bus test distribution system loaded with a 3-phase chopper fed direct current (DC) motor drive at a single bus and last especially for increasing the harmonic-constrained penetration level of renewable energy. Results obtained through simulation performed in MATLAB/Simulink shows that total harmonic distortion (THD) of source current and dynamic, as well as steady-state performance with Lyapunov function based controller, is significantly improved than the other two conventional methods. Also, the robust compensation performance of the SAPF empowers it to deal with the high penetration of renewable energy.

Lyapunov-Function-Based Controller for Single-Phase NPC Quasi-Z-Source Inverter with 2ω Frequency Ripple Suppression

This paper proposes a high-performance control technique based on Lyapunov’s stability theory for a single-phase grid-connected neutral-point-clamped quasi-impedance source inverter with LCL filter. The Lyapunov function based control is employed to regulate the inverter output current, whereas the proportional resonant controller is used to produce the reference of the inverter output current that is needed in the Lyapunov function based control. Use of proportional resonant controller ensures the zero steady-state error in the grid current. An important feature of the proposed Lyapunov function based control is the achievement of resonance damping without using a dedicated damping method. Furthermore, the modified simple boost control technique is proposed to eliminate the double-line frequency ripples in the quasi-impedance source inductor currents and minimize the double-line frequency ripples in the quasi-impedance source capacitor voltages. The proposed control technique considerably reduces the inverter size, weight, and cost as well as increases overall system efficiency since the required inductances and capacitances sizes are lower. Experimental results obtained from a 2.5 kW neutral-point-clamped quasi-impedance source inverter prototype are presented to validate the performance of the Lyapunov function based control technique.

Input-to-state stability and Lyapunov functions with explicit domains for SIR model of infectious diseases

<p style='text-indent:20px;'>This paper demonstrates input-to-state stability (ISS) of the SIR model of infectious diseases with respect to the disease-free equilibrium and the endemic equilibrium. Lyapunov functions are constructed to verify that both equilibria are individually robust with respect to perturbation of newborn/immigration rate which determines the eventual state of populations in epidemics. The construction and analysis are geometric and global in the space of the populations. In addition to the establishment of ISS, this paper shows how explicitly the constructed level sets reflect the flow of trajectories. Essential obstacles and keys for the construction of Lyapunov functions are elucidated. The proposed Lyapunov functions which have strictly negative derivative allow us to not only establish ISS, but also get rid of the use of LaSalle's invariance principle and popular simplifying assumptions.

Averaging technique in the problem of Lorentz attitude stabilization of an Earth-pointing satellite

An Earth-pointing satellite with Lorentz attitude control system is under consideration. The problem of the satellite attitude stabilization in the orbital reference frame in the presence of disturbances and complicated by underactuation is studied. The averaging technique is developed and applied in the problem. A rigorous mathematical justification is proposed, which makes it possible to consider a satellite with unequal moments of inertia and to obtain specific estimates for the control parameters that provide a solution to the problem of satellite attitude stabilization with the use of the Lorentz control torque. The octupole approximation of the Earth's magnetic field is used; the periodic functions of time arise in geomagnetic field induction. With the aid of nonlinear approach to the problem it is proved that Lorentz attitude control system ensures asymptotic stability of the satellite equilibrium under the perturbing action of disturbing torques provided dissipative torque is operating alongside with the restoring one. The proof is based on the Lyapunov direct method and the averaging technique. In the paper, original constructions of nonstationary Lyapunov functions are proposed. These constructions take into account the structure of perturbations acting on the considered system. With the aid of the proposed Lyapunov functions, sufficient conditions for the asymptotic stability of the satellite equilibrium position are derived. These conditions are formulated in terms of explicit inequalities on the control parameters. Thus, a constructive approach to the design of stabilizing controls is formulated.

An Adaptive Dual-Loop Lyapunov-Based Control Scheme for a Single-Phase UPS Inverter

This letter proposes an adaptive Lyapunov-based control strategy for a single-phase uninterruptible power supply inverter with inherent dual control loops. The control law is derived via a proposed Lyapunov function which takes both dual-loop requirements and the global large-signal stability demand into account. Besides, in the proposed control scheme, the load disturbance is suppressed adaptively without any additional load-current sensors or observers. The load voltage is regulated with better steady-state and dynamic performance, as well as with great robustness against the parametric variations. Finally, experimental results verify the effectiveness of the proposed control scheme.

A new continuous high order sliding mode controller for synchronizing perturbed Genesio–Tesi systems in finite time

This paper presents the solution for synchronizing of perturbed Genesio–Tesi systems in finite time. A new homogeneous high order sliding mode control approach is introduced ensuring the finite-time synchronization despite perturbations in the considered systems. By combining a proposed Lyapunov function and the homogeneity concept, the proposed controller possesses superior features such as requiring only one control input signal to obtain finite-time synchronization, resulting in a continuous control signal that significantly reduces the chattering effect without an additional differentiator, and compensating a wide class of perturbations. The computer simulation results are provided to demonstrate the effectiveness of the proposed control approach, compared with two existing control schemes.

Stability Analysis of Thermodynamic Systems: Heat Conduction in Solids

This paper addresses the fundamental difference between equilibria of thermodynamic systems and equilibria of autonomous state space systems. The notion of stability of thermodynamic equilibria is analyzed in terms of an entropy generating function that classifies as a Lyapunov function to prove asymptotic stability of thermal equilibria. The stability analysis is performed for both finite and infinite dimensional systems. It is shown how the proposed Lyapunov function naturally extends to assess stability of interconnected thermal systems. A number of examples is given to demonstrate the time evolution of the Lyapunov function.

Finite energy Lyapunov function candidate for fractional order general nonlinear systems

The construction of Lyapunov function candidates and the norm of infinite energy solutions remain among the most elusive unsolved problems of fractional order systems. This paper develops a new framework to construct finite energy Lyapunov function candidates for fractional order general nonlinear systems with randomness, uncertainty, time-delay or memory. Fundamentals of fractional order system and equilibrium are revisited to start up the investigation, where the pseudo and true states of fractional order systems are considered. The process of constructing fractional order Lyapunov function candidate is mainly divided into three steps: Firstly, converting the original system into an equivalent Volterra integro-differential equation, where the weak singularity of fractional order system is included. Secondly, the fractional order Lyapunov function candidate is derived by canceling out the weak singularity that acts as the catalyst, and is absorbed into the fractional finite energy terms. Lastly, the first order derivative of the proposed Lyapunov function candidate is negative definite and bounded by power-law relevant terms. From finite energy aspect, the proposed fractional order Lyapunov function candidate is composed of potential, kinetic and/or Riesz potential energies in terms of physics. The fractional order Lyapunov’s theorem and asymptotic stability of equilibrium points are discussed, and some non-Lp stable cases have been shown as fractional order finite energy ones. The impacts of fractional order, region of attraction and initialization state on the stability of equilibrium points are presented as well. Some classical integer order and fractional order results can be deduced from this work. A number of examples are illustrated to substantiate the effectiveness of the proposed unified framework.