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The problem of an adaptive Lyapunov redesign is revisited for a class of systems without a priori knowledge of the function majoring nonlinear uncertainties and disturbances. An adaptive barrier function-based gain for unit control is proposed, ensuring an arbitrary a priori predefined uniform ultimate bound for solutions despite of the presence of uncertainties and disturbances. The usage of positive semi-definite barrier function generates a continuous control signal adjusting the chattering, when the perturbations are decreasing to zero.

This article deals with the robust control design for linear systems with input and state delays, subject to nonvanishing matched disturbances. The controller is synthesized with the Lyapunov redesign technique in two stages: first, an optimal predictor compensates the input delay of the nominal system while a quadratic performance index is minimized. Then, the robust component guarantees that the solutions are ultimately bounded. The theoretical results are experimentally validated through a dehydration process, with the controller digital implementation on an industrial programmable automation controller cDAQ-9132.

The paper studies the control problem for nonlinear uncertain systems with multiple channel uncertainties, and presents the manifold of a class of second-order systems including unmatched and match...

Hepatitis C is the liver disease caused by the Hepatitis C virus (HCV) which can lead to serious health problems such as liver cancer. In this research work, the non‐linear model of HCV having three state variables (uninfected hepatocytes, infected hepatocytes and virions) and two control inputs has been taken into account, and four non‐linear controllers namely non‐linear PID controller, Lyapunov Redesign controller, Synergetic controller and Fuzzy Logic‐Based controller have been proposed to control HCV infection inside the human body. The controllers have been designed for the anti‐viral therapy in order to control the amount of uninfected hepatocytes to the desired safe limit and to track the amount of infected hepatocytes and virions to their reference value which is zero. One control input is the Pegylated interferon (peg‐IFN‐α) which acts in reducing the infected hepatocytes and the other input is ribavirin which blocks the production of virions. By doing so, the uninfected hepatocytes increase and achieve the required safe limit. Lyapunov stability analysis has been used to prove the stability of the whole system. The comparative analysis of the proposed nonlinear controllers using MATLAB/Simulink have been done with each other and with linear PID. These results depict that the infected hepatocytes and virions are reduced to the desired level, enhancing the rate of sustained virologic response (SVR) and reducing the treatment period as compared with previous strategies introduced in the literature.

Integrated-control solutions will play a significant role in future tokamaks, in which a variety of coupled control problems will need to be solved simultaneously by means of a limited number of actuators. In this article, the problem of simultaneously regulating the global toroidal rotation, $\Omega _\phi $ , and total plasma energy, $W$ , is tackled. These two 0-D variables, $\Omega _\phi $ and $W$ , depend on the ion toroidal rotation and electron temperature profiles, respectively. Both $\Omega _\phi $ and $W$ also depend on the electron density and safety factor profiles. The actuation methods considered in this article are co-current and counter-current neutral beam injection. A nonlinear, robust controller that makes use of Lyapunov redesign techniques is synthesized based on 0-D, control-oriented models of the $\Omega _\phi $ and $W$ dynamics. In addition, an actuator management scheme is designed to handle variations in the control priorities and availability of the neutral beam injectors. The actuator manager solves an optimization problem in real time in order to find the most appropriate course of action when unexpected changes occur. The integrated control architecture is tested for a DIII-D scenario by means of the 1-D code Control-Oriented Transport Simulator (COTSIM), which predicts the time evolution of the electron temperature, electron density, ion toroidal rotation, and safety factor profiles.

Within the framework of the classical problem of Lyapunov redesign, the barrier function-based adaptation control is presented. The proposed approach does not require the knowledge of an upper bound of the uncertainty. However, producing a continuous control signal adjusts the chattering problem and ensures that the solution will converge in finite time to a region with a priori prescribed ultimate bound.

The problem control of fully actuated of mechanical systems under uncertainties is considered. With this aim a concept of Lyapunov redesign is revisited. The derivative of the Lyapunov function for a nominal model of the robot is used for the sliding surface design. This surface permits to design a super-twisting controller allowing to compensate the Lipschitz uncertainties, providing theoretically exact convergence of the states of uncertain system to the origin by means of a continuous control signal. The proposed result is illustrated for simulation example controlling an uncertain planar robot.

The importance of microgrids has been acknowledged with the increasing amount of research in direct current (DC) microgrids. The main reason for this is the straightforward structure and efficient performance. In this research article, double integral sliding mode controllers (DISMCs) have been proposed for energy harvesting and DC microgrid management involving renewable sources and a hybrid energy storage system (HESS). DISMC offers a better dynamic response and reduced amount of chattering than the traditional sliding mode controllers. In the first stage, the state differential model for the grid was derived. Then, the nonlinear control laws were proposed for the PV system and hybrid energy storage system to achieve the main objective of voltage regulation at the DC link. In the later part, the system’s asymptotic stability was proven using Lyapunov stability criteria. Finally, an energy management algorithm was provided to ensure the DC microgrid’s smooth operation within the safe operating limit. The proposed system’s effectiveness was validated by implementing on MATLAB/Simulink software and comparing against sliding mode control and Lyapunov redesign. Moreover, to ensure the proposed controller’s practical viability for this scheme, it has been tested on real-time hardware-in-the-loop test bench.

In this paper, a robust guaranteed cost control is designed for linear systems with multiple pointwise and distributed state delays, when it is considered the presence of a perturbation that satisfies the matching condition. This result is carried out by combining a complete type Lyapunov-Krasovskii functional, a quadratic cost function, and the Lyapunov redesign technique. The obtained nonlinear state feedback is composed by three control laws: the first one stabilizes the nominal system (a new algorithm is presented), the second one is such that an upper bound of a cost function is guaranteed, and the last one compensates the perturbation effects, ensuring the robust stability of the closed loop system. In order to validate the proposal, numerical simulations are presented, and the controller is applied to regulate a real time dehydration process temperature, implemented on an industrial Programable Automatic Control NI cDAQ − 9132.

In this paper, robust control of nonholonomic mobile robots is considered which is more complex with respect to holonomic types, due to fewer degrees of freedom in their model. In addition, side slipping is considered which disturbs the nonholonomic constraint adversely. In previous researches, the slip is either estimated by an estimator or is measured using additional sensors. In this paper, the Kanayama transformation is modified such that the model of the robot only includes matched disturbances. Then, a robust tracking controller is designed using the Lyapunov redesign method when only the upper bound of the side slip is known. Moreover, in order to compare the performance of the proposed controller, the nonlinear H∞ control is also developed for this robot. Finally, the performance of these two controllers is compared and the simulation results show the appropriate efficiency of the proposed controllers.

In this research work, a Three-Dimensional Cancer model (TDCM) has been used and nonlinear controllers; Lyapunov Redesign, Synergetic and Sliding Mode based controllers have been designed in order to reduce the growth of tumor cells to a level where they become harmless, to maintain the hunting predator cells to their maximum possible value and to retain resting predator cells to 40% of the hunting predator cells. The proposed controllers have been developed for chemotherapy and their effect on different cells has been studied. Asymptotic stability of the proposed controllers has been studied using Lyapunov stability theory. Theoretical analysis has been supported by simulation results using MATLAB/Simulink. Under the proposed controllers, the system behaves very nicely even in the presence of un-modeled disturbances and noise.

In this paper, we study a new current control method for Permanent Magnet Synchronous Motor (PMSM). First, we introduce a modified torque modulation for a velocity control loop. Then, we design a nonlinear current controller based on Lyapunov redesign in the stationary reference (α,β) frame. We show that the nonlinear current control guarantees uniform convergence of current tracking errors in finite-time. Further, uniform convergence of velocity tracking error in finite-time is also proved using the input-to-state stability (ISS). The proposed method is applied to velocity control of PMSM. We present simulation results using co-simulation with AMESim PMSM model and MATLAB/Simulink to validate the performance of the proposed method.

This paper investigates the output regulation problem for a class of Autonomous Underwater Vehicles (AUVs) having non-minimum phase nonlinear dynamics, via conditional servocompensators in steering plane. The 6DOF dynamic model of AUV when decoupled in steering plane and transformed into strict normal form results in unstable zero dynamics. A robust output feedback Sliding Mode Control (SMC) is then designed employing Extended High Gain Observer to achieve semi-global asymptotic stabilization of the internal state. The regulator design is inspired from the pioneering work of Isidori [1] and Khalil [2] where controller is constructed using internal model dynamics and a robust globally bounded output feedback controller. With the inherent leverage of incorporating any stabilizing controller in the design framework, the servocompensator is introduced in the Lyapunov redesign framework for the closed loop system retaining the asymptotic robustness properties of the overall feedback design. Furthermore the controller ensures that servocompensation is only provided in a neighborhood of the zero-error manifold, while forcing the trajectory to the manifold outside it, which tends to effectively improve the transient response, while achieving zero steady-state tracking error and rejecting matched uncertainties. Analytical results and simulations are included to shown that the proposed controller ensures zero error convergence with tight stability margins and paves the way for further advancements to achieve exponential stability in output regulation problems.

In this paper, an adaptive nonlinear control strategy for the energy management of a polymer electrolyte membrane fuel cell and supercapacitor-based hybrid electric vehicle is proposed. The purpose of this work was to satisfy: (i) tight DC bus voltage regulation, (ii) good fuel cell reference current tracking, (iii) better supercapacitor reference current tracking (iv) global asymptotic stability of the closed-loop control system, and (v) better vehicle performance by catering to slowly-varying parameters. We have selected the power stage schematic of a hybrid electric vehicle and utilized adaptive backstepping and adaptive Lyapunov redesign-based nonlinear control methods to formally derive adaptive parametric update laws for all slowly-varying parameters. The performance of the proposed system has been tested under varying load conditions using experimental data from the “Extra Urban Driving Cycle.” Mathematical analysis and Matlab/Simulink results show that proposed controllers are globally asymptotically stable and satisfy all the design requirements. The physical effectiveness of proposed system has been verified by comparing simulation results with the real-time controller hardware in the loop experimental results. Results show that proposed system shows satisfactory performance and caters for the time-varying parametric variations and the load requirements.

In the monitoring of structural systems, the use of multiple high end sensors may prove to be economically prohibitive. The alternative approach would be to use fewer devices that move across the span of the structural system. In the proposed approach, the dynamic of a one-dimensional system is evaluated using a velocity sensor that is able to move across the domain and obtains pointwise velocity measurements at the desired locations. Based on the measured velocities, a state estimator is developed. The gain of the filter depends on the motion of the sensor. The motion of the sensor is defined using Lyapunov redesign methods and depends only on the estimation error at the current sensor position. The guidance policy is performance-based and steers the sensor to spatial regions of the structure with larger estimation errors. The proposed approach is validated with a one dimensional flexible structure, described by an Euler-Bernoulli partial differential equation. The moving sensor is simulated with the use of a laser scanning vibrometer, that provides both the moving measurements and additional measurements against which the proposed approach will be validated. Once a large number of locations is measured, the experimental results are fed to the algorithm that selects the instantaneous sensor location. Experimental results for a linear cantilever are presented that show the benefit of using a state estimator with a moving sensor. Analysis on how the state observer gains are optimally chosen will also be presented. The approach is demonstrated to be feasible and robust.

In this chapter, the effectiveness of the Lyapunov redesign is investigated side by side for the adaptive identification of linear time-delay and distributed parameter systems using Razumikhin and Krasovskii extensions. Identifiability analysis and model reference adaptive control synthesis are proposed for linear time-delay systems with delayed states, control inputs and measured outputs and for parabolic distributed parameter systems with a scalar spatial variable. Sufficiently rich inputs, necessary to identify unknown plant parameters, are specified for both time-delay and distributed parameter systems in question. The theory is supported by numerical studies of engine transient fuel identification and that of heat process with boundary sensing and actuation.

This work presents fault-tolerant control of two-time-scale systems in both linear and nonlinear cases. An adaptive approach for fault-tolerant control of singularly perturbed systems is used in linear case, where both actuator and sensor faults are examined in the presence of external disturbances. For sensor faults, an adaptive controller is designed based on an output-feedback control scheme. The feedback controller gain is determined in order to stabilize the closed-loop system in the fault-free case and vanishing disturbance, while the additive gain is updated using an adaptive law to compensate for the sensor faults and the external disturbances. To correct the actuator faults, a state-feedback control method based on adaptive mechanism is considered. The both proposed controllers depend on the singular perturbation parameter \(\varepsilon \) leading to ill-conditioned problems. A well-posed problem is obtained by simplifying the Lyapunov equations and subsequently the controllers using the singular perturbation method and the reduced subsystems yielding to an \(\varepsilon \)-independent controller. In the nonlinear case, an additive fault-tolerant control for nonlinear time-invariant singularly perturbed system against actuator faults based on Lyapunov redesign principle is presented. The full-order two-time-scale system is decomposed into reduced slow and fast subsystems by time-scale decomposition using singular perturbation method. The time-scale reduction is carried out by setting the singular perturbation parameter to zero, which permits to avoid the numerical stiffness due to the interaction of two different dynamics. The fault-tolerant controller is computed in two steps. First, a nominal composite controller is designed using the reduced subsystems. Second, an additive part is appended to the nominal controller to compensate for the effect of an actuator fault. In both cases, the Lyapunov stability theory is used to prove the stability provided the singular perturbation parameter is sufficiently small. The designed control schemes guarantee asymptotic stability in the presence of additive faults. Finally, the effectiveness of the theoretical results is illustrated using numerical examples.

This paper addresses the issue of the regulation of the dehydration air temperature in a dryer. The classical optimization approach is used to solve the optimal control problem for this kind of system because the linearized mathematical model involves the delay state. However, the presence in the model of nonlinear uncertainties, which satisfy the matching condition, is a reason to apply the Lyapunov redesign approach. It gives robustness to the optimal linear control, improving the closed-loop performance. Furthermore, the optimal–robust control algorithm was tested in a dehydration process. This is the first time that the optimal and the proposed optimal–robust controller has been applied in a real-time process.

This paper investigates the problem of synchronization for nonlinear systems. Following a Lyapunov approach, we first study the global synchronization of nonlinear systems in the canonical control form with both distributed proportional-derivative and proportional-integral-derivative control actions of any order. To do so, we develop a constructive methodology and generate in an iterative way inequality constraints on the coupling matrices that guarantee the solvability of the problem or, in a dual form, provide the nonlinear weights on the coupling links between the agents such that the network synchronizes. The same methodology allows us to include a possible distributed integral action of any order to enhance the rejection of heterogeneous disturbances. The considered approach does not require any dynamic cancellation, thus preserving the original nonlinear dynamics of the agents. The results are then extended to linear and nonlinear systems admitting a canonical control transformation. Numerical simulations validate the theoretical results.

This paper studies stability of stochastic impulsive switched systems. Different from exponential Lyapunov function and average dwell-time in the previous works, general Lyapunov function and fixed dwell-time are implemented in this paper to analyze input-to-state stability and global stability of stochastic impulsive switched systems. Two cases are investigated, that is, the case that the continuous dynamics is stable and the case that the discrete dynamics is stable. To implement multiple Lyapunov functions, there are two subcases considered in this paper: the subcase that the estimates on the derivatives of the multiple Lyapunov functions are the same and the subcase that the growths of the multiple Lyapunov functions by the jumps are the same. For aforementioned different cases, sufficient stability conditions are established for stochastic impulsive switched systems. Finally, the developed theory is illustrated through examples from networked control systems and synchronization problem of 3-D novel chaotic circuit systems.

This paper presents a unification and a generalization of the small-gain theory subsuming a wide range of existing small-gain theorems. In particular, we introduce small-gain conditions that are necessary and sufficient to ensure input-to-state stability (ISS) with respect to closed sets. Toward this end, we first develop a Lyapunov characterization of $\omega$ ISS via finite-step $\omega$ ISS Lyapunov functions. Then, we provide the small-gain conditions to guarantee $\omega$ ISS of a network of systems. Finally, applications of our results to partial ISS, ISS of time-varying systems, synchronization problems, incremental stability, and distributed observers are given.

This paper investigates the distributed adaptive control problem for synchronization of multi-agent systems where the dynamics of the agents are nonlinear, nonidentical, unknown and subject to external disturbances. In our recent work, we solved this problem for general higher order systems under two types of communication topologies, represented, respectively, by a fixed strongly connected directed graph and by a switching connected undirected graph. A common Lyapunov function technique is employed to establish the results. In this paper, we solve the problem for a more general communication topology which is represented by a switching strongly-connected directed graph. We construct a sequence of different Lyapunov functions and use them to establish our results. Simulation study verifies the effectiveness of our theoretical results.

In wake of frequency stability challenges enhanced by the volatile renewable energy based generation, there is a renewed interest in the methods that can provide stability insights for broad range of operating conditions of the oscillating multi-machine systems, such as power system. With this perspective in sight, this work proposes an analysis of transient stability with emphasis on the Gram–Schmidt and covariant Lyapunov vectors. Numerical accuracy of these Lyapunov vectors is validated by comparison with Floquet exponents obtained from the Jacobian matrix leading to an increased confidence in their potential application. In addition, a study of global stability using Lyapunov exponents and of local stability by Lyapunov vectors is performed. The assumptions necessary for the derivation of energy functions are avoided resulting in a broader stability perspective that is also applicable on the operating point outside the stability manifold. The methods are evaluated for the IEEE 9 bus system.

In this paper, an admittance control scheme for a user-in-charge exoskeleton is presented. The controller basically consists of a composite adaptive controller implementing a feedback law to estimate the structured uncertainties and to modify the apparent dynamics of the robot, and an LWPR estimator which tries to give an appropriate approximation of unmodeled uncertainty along with a robust term aiming to overcome the approximation residue. The control scheme offers a unified general control structure that explains the effect of each control component on the others. It is proved that based on the developed controller, the tracking and estimation errors converge to small boundaries with ultimate boundedness property due to the presence of the unstructured uncertainty. Based on simulations of a 2-DOF leg, the effectiveness of the controller is investigated. The results show the effectiveness of employing a universal approximator alongside a robust adaptive control and the success of the recommended...

We propose a Lyapunov theory based linguistic reinforcement learning (RL) framework for stable tracking control of robotic manipulators. In particular, we employ Lyapunov theory to constrain fuzzy rule consequents for ensuring stability of the designed controller. Proposed fuzzy RL controller employs Lyapunov theory dictated rules for discovering an optimal yet stable control strategy for robotic manipulators. Furthermore, our proposed linguistic RL controller handles payload variations and external disturbances quite effectively. We validate linguistic Lyapunov RL controller on two benchmark control problems: (i) a standard two-link robotic arm manipulator, and (ii) a two link selective compliance assembly robotic arm (SCARA). Simulation results and comparison against (a) baseline fuzzy Q learning (FQL) controller, and (b) a recently proposed Lyapunov theory based Markov game controller showcases our controller's superior tracking performance and lower computational complexity. Furthermore, our controller exhibits high stability with disturbances and payload variations.

In this note we introduce a direct approach for the Lyapunov functional method to study the global stability of the unique equilibrium solution of reaction diffusion systems and partially degenerate reaction diffusion systems permitting with delays. We first provide the abstract framework and results. Then we give an example as the application.

We study the global asymptotic stability in probability of the zero solution of linear stochastic differential equations with constant coefficients. We develop a sum-of-squares program that verifies whether a parameterized candidate Lyapunov function is in fact a global Lyapunov function for such a system. Our class of candidate Lyapunov functions are naturally adapted to the problem. We consider functions of the form \begin{document} $V(\mathbf{x}) = \|\mathbf{x}\|_Q^p: = (\mathbf{x}^\top Q\mathbf{x})^{\frac{p}{2}}$ \end{document} , where the parameters are the positive definite matrix \begin{document} $Q$ \end{document} and the number \begin{document} $p>0$ \end{document} . We give several examples of our proposed method and show how it improves previous results.

The Lyapunov equation is widely employed in the engineering field to analyze stability of dynamic systems. In this paper, based on a new evolution formula, a novel finite-time recurrent neural network (termed finite-time Zhang neural network, FTZNN) is proposed and studied for solving a nonstationary Lyapunov equation. In comparison with the original Zhang neural network (ZNN) model for a nonstationary Lyapunov equation, the convergence performance has a remarkable improvement for the proposed FTZNN model and can be accelerated to finite time. Besides, by solving the differential inequality, the time upper bound of the FTZNN model is computed theoretically and analytically. Simulations are conducted and compared to validate the superiority of the FTZNN model to the original ZNN model for solving the nonstationary Lyapunov equation. At last, the FTZNN model is successfully applied to online tracking control of a wheeled mobile manipulator.

This study presents a linear matrix inequality (LMI) approach to the sampled-data control of Takagi-Sugeno fuzzy systems, based on the intelligent digital redesign (IDR) technique. The objective of the IDR is to design a digital control system whose trajectory closely matches that of a given well-constructed analogue control system by minimising the state-matching error. In this study, state-matching performance is enhanced by using a continuous-time state-matching criterion, which guarantees that the state-matching error is minimised through the entire time interval. Unlike previous studies, mismatched information of membership functions for both analogue and digital control systems is directly manipulated. Moreover, the authors introduce an improved fuzzy Lyapunov functional that consists of both membership functions for analogue and digital control systems, which relaxes the conservativeness of LMI conditions. Finally, two examples demonstrating the effectiveness of the authors' method are provided.