Lyapunov Stability Conditions Research Articles

Nonlinear Robust Adaptive Control of Universal Manipulators Based on Desired Trajectory

The introduction of a dynamic model in robot trajectory tracking control design can significantly improve its trajectory tracking accuracy, but there are many uncertainties in the robot dynamic model which can be dealt with through robust control and adaptive control. The prevailing robust control as well as adaptive control methods require real-time computation of robot dynamics, but the extreme complexity of the robot dynamics equations makes it difficult to apply these methods in real industrial systems. To this end, this article proposes a robust adaptive control method based on the desired trajectory, which uses the desired trajectory to compute most of the control terms offline, including the robot’s nominal dynamics and regression matrices, and substantially reduces the need for real-time computation of the feedback signals. The robust term modifies the perturbation of the inertial parameters of the links, the adaptive term learns the friction coefficients of the joints online, and an additional compensation term is designed to satisfy the Lyapunov stability condition of the system. Finally, taking a universal manipulator as the experimental platform, the control performances of different control methods are compared to show the feasibility of the controller and the effective reduction in real-time computational complexity.

Boundary tracking control of a structure supporting a vibrational moving-mass-carrying string with unknown prescribed tension

This paper contributes to the problem of tracking control for nonlinearly vibrating structure supporting a vibrational mass-carrying string with unknown string tension. The control scheme is constructed based on optimally designed reaching law considering actuator force limit, parametric uncertainties and Lyapunov stability conditions. The reaching law is designed through real-time detection of stability margins and analyzing actuator capabilities such that system dynamics are either driven into reaching or attraction phase in finite time. The control algorithm boasts robustness against parametric uncertainty by narrowing stability bounds and features an online string tension estimation engine based on data analysis obtained from acceleration and proximity sensors measurements. In order to provide smooth tracking, string vibration suppression is achieved by incorporating corresponding dynamical terms in construction of control algorithm. The closed-loop system is simulated alongside conventional discrete sliding mode technique, the results of which are compared and analyzed to pinpoint viable advantages.

Zero-sum stochastic games in continuous-time with risk-sensitive average cost criterion on a countable state space

We consider zero-sum stochastic games in continuous time with controlled Markov chains and with risk-sensitive average cost criterion. Here the transition and the cost rates may be unbounded. We prove the existence of the value of the game and a saddle-point equilibrium in the class of all stationary strategies under a Lyapunov stability condition. This is accomplished by establishing the existence of a principal eigenpair for the corresponding Hamilton-Jacobi-Isaacs (HJI) equation. This, in turn, is established by using a nonlinear version of Krein-Rutman theorem. We then obtain a characterization of the saddle-point equilibrium in terms of the corresponding HJI equation. Finally, we use a controlled population system to illustrate our results.

A fixed-time converging neurodynamic approach with time-varying coefficients for l1-minimization problem

In this paper, we propose a novel neurodynamic network to deal with l1-minimization problem. In the framework of the fixed-time converging neurodynamic network (FxNN), time-varying coefficients are introduced to design the time-varying fixed-time converging neurodynamic network (TFxNN). It is shown that the fixed-time stability of the proposed TFxNN via the Lyapunov stability conditions. It is further shown that the proposed TFxNN trajectories from any initial points quickly converge to the unique equilibrium solution in fixed-time. A distinctive feature of the proposed TFxNN is its flexibility to choose time-varying coefficients to accelerate convergence. Simulation results based on signal and image reconstruction are presented that the proposed neurodynamic network is feasible and effective.

State Feedback H∞ Control for Trajectory Tracking of an Unmanned Surface Vehicle

Aiming at the sway velocity and uncertainties such as wind, wave, and current, a state feedback H∞ control method is proposed to improve the dynamic and steady-state performance with line of sight (LOS) law for trajectory tracking of an unmanned surface vehicle (USV). First, a novel LOS is proposed for trajectory tracking with the position error to obtain the references of surge velocity and heading angle. Second, the LPV model is established with varying parameters of the sway velocity for the USV. Third, the closed-loop system for the LPV model is established by the state feedback H∞ control method to design the controller. Since the coupling term of system coefficients in the Lyapunov stability condition is difficult to solve directly, a novel variable is introduced to decouple the stability condition, which can be solved with MATLAB LMI tools for the controller parameters. Finally, the simulation result shows that the proposed method has super performance on the dynamic and steady-state indices.

SIRC epidemic model with cross-immunity and multiple time delays.

Multi-strain diseases lead to the development of some degree of cross-immunity among people. In the present paper, we propose a multi-delayed SIRC epidemic model with incubation and immunity time delays. Here we aim to examine and investigate the effects of incubation delay [Formula: see text] and the impact of vaccine which provides partial/cross-immunity with immunity delay parameter ([Formula: see text]) on the disease dynamics. Also, we study the impact of the strength of cross-immunity [Formula: see text] on the disease prevalence. The positivity and boundedness of the solutions of the epidemic model have been established. Two different types of equilibrium points (disease-free and endemic) have been deduced. Expression for basic reproduction number has been derived. The stability conditions and Hopf-bifurcation about both the equilibrium points in the absence and presence of both delays have been discussed. The Lyapunov stability conditions about the endemic equilibrium point have been established. Numerical simulations have been performed to support our analytical results. We quantitatively demonstrate how oscillations and Hopf-bifurcation allow time delays to alter the dynamics of the system. The combined impacts of both the delays on disease prevalence has been studied. Through parameter sensitivity analysis, we observe that the infected population decreases with an increase in vaccination rate and the system starts to stabilize early with the increase in cross-immunity rate. Global sensitivity analysis for the basic reproduction number has been performed using Latin hypercube sampling and partial rank correlation coefficients techniques. The combined effect of vaccination rate with transmission rate and vaccination rate with re-infection probability (i.e. strength of cross-immunity) on [Formula: see text] have been discussed. Our research underlines the need to take cross-immunity and time delays into account in the epidemic model in order to better understand disease dynamics.

Distributed Event-Triggered Secondary Control for Islanded Microgrids With Disturbances: A Hybrid Systems Approach

This paper focuses on the event-triggered secondary control problem of islanded microgrid with nonlinear dynamics and unknown external disturbances. Distributed secondary control law is proposed to compensate for frequency and voltage deviation caused by primary control, and to realize accurate active power sharing. In order to save communication resources, a novel hybrid event-triggering mechanism (ETM) is proposed to schedule the information transmission for each distributed generator. A hybrid model is constructed to describe the closed-loop dynamics and incorporate the ETM into flow and jump sets. Based on this model, Lyapunov stability conditions and ETM design are developed. Compared with the existing results, the proposed hybrid event-triggered control strategy not only has potential to reduce triggering number, but also guarantee a strictly positive minimum event-triggering interval even in the presence of external disturbances, that is important for physical implementation. Finally, simulation results are provided to illustrate the proposed method.

Neural Networks Based Lyapunov Functions for Transient Stability Analysis and Assessment of Power Systems

This article presents the Lyapunov functions based on the artificial intelligence (AI) method for transient stability assessment and the determination of stability region (SR). First, Lyapunov stability theory and the definition of SR are introduced. Then, the characteristics of neural networks as a general function approximator are employed as the Lyapunov function learner, and the Lyapunov function is constructed combined with stochastic gradient descent (SGD). Then, the falsifier's task is to find the state vectors that violate Lyapunov stability conditions, and the counterexamples would be added to the training set for the function learner to accelerate convergence. After obtaining the Lyapunov function of power system, the estimation of SR boundary can be represented by the maximum level set of Lyapunov function. Finally, the IEEE 9-bus 3-machine system is used as test system to demonstrate the validity and effectiveness of the proposed construction method of Lyapunov functions for power system transient stability analysis.

Nonlinear Tracking Control Algorithm for Dynamical Output Systems Manipulated by the Hardly Constrained Oscillatory Actuator

In this article, a nonlinear control algorithm has been presented, in order to reduce vibrational movements of a flexible beam cantilevered in a cart supported by nonlinear stiffness, actuated by a hardly constrained second-order dynamic system. In the presented control algorithm, the interactive model of the solid body and a flexible beam has been extracted and utilized to design an algorithm, based on which the nonlinear vibrations of the tip point of the beam whose function is smooth and nonoscillatory tracking of the predefined desired trajectory are diminished. To design a nonlinear vibration control system of the beam, it has been assumed that the feedback system includes three modules: a vibrational beam cantilevered on a base, a base affected by nonlinear stiffness and actuator’s control force, and a force imposing system with constraints of bandwidth frequency limitation and actuation saturation boundary. In order to design this control system, the generalized tracking error function has been defined to nonlinearly amplify the vibrationally induced error and its rate, which by properly adjusting them, a new quantity can be extracted that satisfies the control system design constraints. To design a tracking control system for a nonlinear vibrating beam for the aforementioned three-modulus system, first, the generalized tracking error for the three modules is separately defined and parametrized, and then, by simultaneously adjusting all the defined parameters systematically, the predominant system presence in Lyapunov stability conditions is guaranteed. To illustrate the multimodule imitative (MMI) controller design algorithm, a moving support connected to a nonlinear spring and damper is considered, which carries a flexible cantilevered beam. The applied actuation system has second-order linear dynamics with the presence of command control saturation boundaries. For each of the abovementioned modules, a generalized tracking error is defined, and then, it is explained to how simultaneously adjust the parameters based on the stability and actuation constraints. The MMI controller is applied to the mentioned mechanical system modeled in the ANSYS® Mechanical APDL environment, and then, the necessary conclusions are discussed about the performance of the control system in eliminating the vibrations of the flexible arm, considering the actuation constraints while possessing the dominant Lyapunov stability.

Robust control and flexible operation for commercial-scale coal-fired power plant with solvent-based post-combustion carbon capture

Coal-fired power plant integrated with post-combustion carbon capture (CFPP-PCC) process exhibits strong nonlinearity and multi-variable connections. These dynamics along with unknown disturbances make it a challenging task to realize robust control and flexible operation in CFPP-PCC system. For this reason, this paper develops an extended state observer based stable model predictive control (ESOSMPC) in order to reject unknown disturbances and to achieve flexible operation with closed-loop stability. In the proposed control structure, an extended state observer is utilized to give accurate estimation for plant behavior change and unknown disturbances. This information will be used in active disturbance rejection, which aims to alleviate negative effects of disturbances and to enhance control performance. Stable model predictive control with infinite input-output based objective function is developed in order to fulfill Lyapunov stability condition, guaranteeing a stable operation for the integrated process. The proposed controller is compared with conventional MPC/PID controller under varying working conditions. Simulation results show that the presented ESOSMPC is able to achieve robust control in a multi-disturbance situation and to attain flexible operation in a broad-range load variation scenario.

Velocity Tracking Control Algorithm for Semiactive Control of Building Frames

A new control algorithm called velocity tracking control (VTC) is presented for the semi-active control of partially observed building frames using magneto-rheological (MR) dampers. The novelty of the control algorithm relies on the use of special features of the semi active control using MR dampers in which the time history of voltage to be applied to the MR damper can be generated with the help of feedback information from the observed states only; it does not require the estimation of the full state from the measured ones which introduces some errors in the estimated control force generated in the MR damper. The algorithm is developed based on a physical reasoning, which is validated by the Lyapunov stability condition of the first order filter used in the modified Bouc-wen model that models the MR damper. The efficiency of the control algorithm is compared with the passive on control for a number of earthquakes. The response quantities of interest, for which percentage reductions in responses are compared, include the peak top floor displacement, peak inter-story drift and maximum base shear. The results of the study show that the proposed control algorithm provides the maximum percentage reduction in peak inter-story drift in comparison to passive on control. Further, the efficiency of the proposed control algorithm, defined by the percentage reduction of responses per unit control force, is found to be more as compared to the passive on control.

Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control

A novel terminal sliding mode control is introduced to control a class of nonlinear uncertain systems in finite time. Having command on the definition of the final time as an input control parameter is the goal of this work. Terminal sliding mode control is naturally a finite-time controller though the time cannot be set as input, and the convergence time is not exactly known to the user before execution of the control loop. The sliding surface of the introduced controller is equipped with a finite-time gain that finishes the control task in the desired predefined time. The gain is found by partitioning the state-dependent differential Riccati equation gain, then arranging the sub-blocks in a symmetric positive-definite structure. The state-dependent differential Riccati equation is a nonlinear optimal controller with a final boundary condition that penalizes the states at the final time. This guides the states to the desired condition by imposing extra force on the input control law. Here the gain is removed from standard state-dependent differential Riccati equation control law (partitioned and made symmetric positive-definite) and inserted into the nonlinear sliding surface to present a novel finite-time terminal sliding mode control. The stability of the proposed terminal sliding mode control is guaranteed by the definition of the adaptive gain of terminal sliding mode control, which is limited by the Lyapunov stability condition. The proposed approach was validated and compared with state-dependent differential Riccati equation and conventional terminal sliding mode control as independent controllers, applied on a van der Pol oscillator. The capability of the proposed approach of controlling complex systems was checked by simulating a flapping-wing flying robot. The flapping-wing flying robot possesses a highly nonlinear model with uncertainty and disturbance caused by flapping. The flight assumptions also limit the input law significantly. The proposed terminal sliding mode control successfully controlled the illustrative example and flapping-wing flying robot model and has been compared with state-dependent differential Riccati equation and conventional terminal sliding mode control.

Model Predictive Control Strategy for Induction Motor Drive Using Lyapunov Stability Objective

This article presents a novel Lyapunov-based model predictive control strategy for squirrel-cage induction motor fed by a voltage source inverter. The model predictive control method has received enormous attention thanks to its rapid response to load perturbations. However, the traditional model predictive control method may suffer from instability due to the poor choice of the objective function, weighting factors, or other design parameters. In particular, the selection of the objective function may not be sufficient to ensure global stability. Since the performance of the model predictive control method highly relies on the explicit model of the system, a simple penalization term in the objective function can lead to system instability. As a result, the transient and steady-state performances are negatively affected. The objective function is reformulated as the Lyapunov energy function to deal with this ambiguous situation in this article. The control input constraints are defined in the formulated optimal control problem, and the optimal solution is explored by assessing the Lyapunov stability criterion. The proposed method ensures asymptotic stability since the control action that satisfies the Lyapunov stability condition is selected. The experimental work proves the theoretical concepts. Moreover, the proposed method does not require the weighting factors to control the multiple control goals. The experimental results demonstrate that the proposed control strategy improves the steady-state performance. The machine torque and speed are well regulated, and the quality of the stator current is improved.

How to stimulate Chinese energy companies to comply with emission regulations? Evidence from four-party evolutionary game analysis

Based on the current organization of China's energy regulatory system, we have established a four-party evolutionary game model involving the central energy regulatory agencies, local energy regulatory agencies, energy companies and whistleblowers. The game strategies and behavior evolution of each party have been analyzed. There are at least 25 equilibrium points in the four-party game system, including 16 pure-strategy equilibrium solutions, 6 dual-strategy equilibrium solutions, 2 three-strategy equilibrium solutions, and 1 mixed strategy equilibrium solution. According to the Lyapunov stability condition, we obtain 8 possible conditional stability points and their evolutionarily stable strategy, which represent the possible strategic equilibrium of various stakeholders in the supervision of energy pollution and reporting. We also simulated and analyzed the behavior differences and optimization paths of various stakeholders, and the final evolution results and influencing factors have also been deduced. The main conclusions are: (1) Public supervision and whistleblowing could reduce the opportunity cost of inspection by central energy regulatory agencies. (2) There is a significant positive correlation between energy supervision and the penalty cost on energy companies for non-compliant pollution emission. (3) There is a significant positive correlation between energy supervision and long-term social welfare. Finally, policy implications and corresponding recommendations are provided.

Specified QoS based networked observer and PI controller design with disturbance and noise rejection under random packet dropout

In this paper, a novel observer based networked PI controller has been designed by considering bounded disturbance and measurement noise under specified probability of random packet dropouts in networked control systems (NCSs). Lyapunov stability condition of the NCS with an observer and PI controller has been derived using an asynchronous dynamical system (ADS) approach as a linear matrix inequality (LMI) with nonconvex constraints to improve its quality-of-service (QoS) and quality-of-control (QoC). In order to improve QoC and QoS for NCS under process and measurement disturbance, we have used H∞ norm bound, and random packet drop rate in the NCS design. The algorithm has been tested and validated on four test-bench models of energy conversion systems. Confidence intervals and the mean responses of the state variables, state estimation errors of the observer and control signals have been reported using 100 Monte Carlo simulations. Our proposed NCS with PI controller and observer design method can be employed on a wide range of state space models of energy conversion systems. The specified probability of packet dropouts in the communication network between the sensor, observer, PI controller and actuator has been considered in the design which allows design of the NCS with guaranteed Lyapunov stability. We consider a switched linear dynamical system structure of the NCS with bounded noise and disturbance inputs while jointly searching for PI controller and observer gains. Using the controller and observer designed with 90% packet drop rate, we show the system performance degradation at 97%–99% QoS using phase portraits of the error dynamical system around the equilibrium as well as through univariate and multivariate hypothesis tests for equal dispersion around the equilibrium of the error dynamical system.

Synthetic Lyapunov stabilization technique for designing actuation-constrained multi-input multi-output control systems

The main objective of this paper is to present a non-predictive method in the design of nonlinear multi-input multi-output (MIMO) control systems with the presence of constraints that are determinant in practical conditions, namely, the frequency bandwidth limitation of the actuation system and saturation boundaries in control commands. If these constraints are applied in the non-predictive control design problem, it is not possible to simultaneously satisfy Lyapunov stability and actuation constraints, analytically. Instead of model-predictive-based algorithms, which in most cases are computationally expensive, this paper proposes an algorithm based on synthetic Lyapunov stability. In this technique, by defining an intelligent filter applied to the system desired trajectories, defining intelligent proximity coefficients in decoupled inequalities resulting from Lyapunov stability, and determining the admissible boundaries of control commands, a space of regulatory parameters is generated. By appropriately adjusting these parameters based on statistical analysis conducted on the overall dynamics of the system, the Lyapunov stability is guaranteed, and the mentioned control constraints are not violated. In summary, the proposed control algorithm includes the formulation of discrete-time dynamics of sliding functions, the presentation of the procedure of defining and adjusting the control algorithm parameters with the proposed synthetic stability criterion, and the calculation of control inputs based on constraints imposed on the problem. Finally, the algorithm is applied to a cart moving in the X-Y plane, including two rigid cooperative arms that are carrying a load. The most important features of synthetic Lyapunov stability compared to the model predictive-based method are its small computational load and its acceptable performance in satisfying both the Lyapunov stability conditions and determinant control constraints in more realistic situations.

Active Lane-Changing Control of Intelligent Vehicle on Curved Section of Expressway

In order to improve the intelligent vehicle lane-changing performance, an active lane-changing control algorithm is proposed considering the changes of road curvature and vehicle speed. Firstly, the vehicle dynamics model considering vehicle speed variation and lane-changing safety distance is established, and the expected lane-changing trajectory model under the curved road is designed simultaneously. Then, taking the yaw rate and longitudinal speed as the control objectives of lateral and longitudinal motions, respectively, the sliding-mode variable structure control method based on Lyapunov stability condition is adopted, and the trajectory tracking controller is designed by combining the inverted method to track the desired lane-changing trajectory. Finally, the lane-changing trajectory model and trajectory tracking controller are verified in simulation platform of CarSim/Simulink and hardware-in-the-loop (HIL) test bench. The results show that the proposed trajectory tracking control method can perform the lane-changing behavior well under different road curvatures and vehicle speeds while maintaining high trajectory tracking control accuracy.

Event-Triggered Mechanism for Multiple Frequency Services of Electric Vehicles in Smart Grids

This paper considers an event-triggered mechanism (ETM) for multiple frequency services of electric vehicles (EVs) in smart grids (SGs). A new model of a SG with an ETM and multiple agents of EVs is introduced. ETM is deployed to reduce the load on network bandwidth. EVs are incorporated in both secondary and primary frequency services to suppress frequency fluctuations in the system. The behaviours of EV owners are also considered by using batteries SOC controls. Then, a mathematical model of the system is derived which contains parametric uncertainties due to the incorporation of SOC control, an ETM variable arising from ETM control, and a time-varying delay in both the state and control input vectors. We derive new Lyapunov stability and stabilizability conditions to ensure robust <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> performance of the closed-loop system. The conditions are derived based on various mathematical tools including relaxed Jensen inequality, free weighting matrix technique, reciprocally convex approach, and matrix inequality for uncertainties. An effective procedure is proposed based on an approach using the state transformation and tractable linear matrix inequalities to synthesis robust static output feedback controller gains. Extensive simulations of a SG with three agents of EVs are undertaken to validate of our proposed control scheme.

Discrete-Time Takagi-Sugeno Stabilization Approach Applied in Autonomous Vehicles

This paper deals with a new robust control design for autonomous vehicles. The goal is to perform lane-keeping under various constraints, mainly unknown curvature and lateral wind force. To reach this goal, a new formulation of Parallel Distributed Compensation (PDC) law is given. The quadratic Lyapunov stability and stabilization conditions of the discrete-time Takagi–Sugeno (T-S) model representing the autonomous vehicles are discussed. Sufficient design conditions expressed in terms of strict Linear Matrix Inequalities (LMIs) extracted from the linearization of the Bilinear Matrix Inequalities (BMIs) are proposed. An illustrative example is provided to show the effectiveness of the proposed approach.

A brief survey on stability and stabilization of impulsive systems with delayed impulses

&lt;p style='text-indent:20px;'&gt;This survey addresses stability analysis for impulsive systems with delayed impulses, which constitute an important generalization of delayed impulsive systems. Fundamental issues such as the concept of a solution to an impulsive system with delayed impulses and methods to determine impulse instants are revisited and discussed. In view of the types of delays in impulses, impulsive systems with delayed impulses are classified into two categories including systems with time-dependent delayed impulses and systems with state-dependent delayed impulses. Then more efforts are devoted to the stability analysis of these two classes of impulsive systems, where corresponding Lyapunov-function-based sufficient conditions for Lyapunov stability, asymptotic stability, exponential stability, input-to-state stability and finite-time stability are presented, respectively. Moreover, the double effects of time-dependent delayed impulses on system performance are reemphasized, and recent applications of delayed impulses in synchronization control are discussed in detail. Several challenges are suggested for future works.&lt;/p&gt;