Hamilton-Jacobi Inequality Research Articles

Fixed-time stability of ODE and fixed-time stability of neural network

We give the sufficient conditions to fixed-time stability for nonautonomous ODEs for which the right-hand side is only Lebesgue measurable with respect to time t. Using dual tools, we present completely new approach to fixed-time stability by defining new class of functions and consideration of dual Hamilton-Jacobi inequality with the use of Lyapunov function. Duality means that we move all main notions as trajectories, Lyapunov function, Hamiltion-Jacobi inequality to dual space which is determined by us. Then we study stability of the dual trajectories and next we come back to our primal space and infer suitable stability for the original problem. As an application of the main theorems we give numerical examples of fixed-time stable neural network.

Guaranteed cost control for impulsive nonlinear Itô stochastic systems with mixed delays

This paper mainly investigates the problem of guaranteed cost control for a class of impulsive nonlinear Itô stochastic systems with mixed delays. Several sufficient conditions are proposed to ensure the closed-loop systems are asymptotically stable in probability and the given cost function is guaranteed to have an upper bound by using Lyapunov–Krasovskii functional, Hamilton–Jacobi inequalities, Dynkin formula and Razumikhin technique. Finally, a simulation result shows the effectiveness of the proposed method. Different the previous works, the employed method in this paper is novel and the obtained results are less conservative.

On Hamilton-Jacobi Approaches to State Reconstruction for Dynamic Systems

We investigate the use of Hamilton-Jacobi approaches for the purpose of state reconstruction of dynamic systems. First, the classical formulation based on the minimization of an estimation functional is analyzed. Second, the structure of the resulting estimator is taken into account to study the global stability properties of the estimation error by relying on the notion of input-to-state stability. A condition based on the satisfaction of a Hamilton-Jacobi inequality is proposed to construct estimators with input-to-state stable dynamics of the estimation error, where the disturbances affecting such dynamics are regarded as input. Third, the so-developed general framework is applied to the special case of high-gain observers for a class of nonlinear systems.

Dissipative control for nonlinear Markovian jump systems with mixed time‐delays: The discrete‐time case

SummaryThis article is concerned with the dissipative control problem for discrete‐time nonlinear Markovian jump systems subject to both discrete and distributed time‐delays. The purpose is to design a state feedback controller that is capable of guaranteeing the required closed‐loop stability and dissipativity performances simultaneously. By resorting to Lyapunov functional methodology and completing square technique, sufficient conditions are established for the existence of the desired state feedback controller in terms of certain Hamilton‐Jacobi inequalities (HJIs). Within the provided framework, the required controller parameters can be obtained by solving the corresponding HJIs. Finally, two numerical simulation examples are presented to demonstrate the correctness and effectiveness of the developed control paradigm.

Lyapunov Functions for State Observers of Dynamic Systems Using Hamilton–Jacobi Inequalities

Lyapunov functions enable analyzing the stability of dynamic systems described by ordinary differential equations without finding the solution of such equations. For nonlinear systems, devising a Lyapunov function is not an easy task to solve in general. In this paper, we present an approach to the construction of Lyapunov funtions to prove stability in estimation problems. To this end, we motivate the adoption of input-to-state stability (ISS) to deal with the estimation error involved by state observers in performing state estimation for nonlinear continuous-time systems. Such stability properties are ensured by means of ISS Lyapunov functions that satisfy Hamilton–Jacobi inequalities. Based on this general framework, we focus on observers for polynomial nonlinear systems and the sum-of-squares paradigm to find such Lyapunov functions.

Sliding Mode Robust Control of a Wire-Driven Parallel Robot Based on HJI Theory and a Disturbance Observer

A sliding mode robust control law based on Hamilton-Jacobi Inequality (HJI) theory and a disturbance observer is proposed for a wire-driven parallel robot (WDPR) used in a wind- tunnel test. First, the wire-driven parallel robot is described, and its kinematics model established. Second, according to the uncertainty, external disturbance, and redundant drive of the system, the dynamic model of the end-effector and drive system, and the overall dynamic model of the system are established. Hamilton-Jacobi Inequality theory and the designed disturbance observer are applied to the designed sliding mode robust control law, and the anti-interference ability of the WDPR is verified. The stability of the closed-loop system is analyzed by Lyapunov’s second method, and the results show that the closed-loop system tends to be asymptotically stable. Finally, taking the dynamic trajectory simulation of compound motion and six-degree-of-freedom motion as examples, the designed sliding mode robust control law is verified by simulation, and the contrastive simulation analysis shows that the disturbance observer can effectively reduce the switching gain, thus effectively reducing chattering and improving the control accuracy of the system. The simulation results show that the designed sliding mode robust control law can effectively suppress the influence of external disturbance on the control error. The control input and the length of the wire change in a certain range. They also prove that the pose error is small, and the control accuracy is high. All of the foundings lay a theoretical foundation and technical support for the practical application of the prototype in a wind-tunnel test.

Stochastic Robust Team Formation Tracking Design of Multi-VTOL-UAV Networked Control System in Smart City Under Time-Varying Delay and Random Fluctuation

In this study, the robust H∞ event-triggered team formation tracking control design of multi-VTOL-UAVs in networked system is investigated. To describe the realistic networked system and UAV model, the intrinsic continuous Wiener random fluctuation, discontinuous Poisson random fluctuations, external disturbances and time-varying delays of wireless network are formulated in the proposed nonlinear stochastic jump diffusion system structure. By combining the event-triggered multi-UAV dynamic models and reference model into an augmented system, the robust H∞ event-triggered multi-UAV networked team tracking problem can be transformed to a Hamilton-Jacobi inequality(HJI)-constraint optimization problem. Due to the difficulties in solving HJI-constraint optimization problem, for practical application, the T-S fuzzy techniques are adopted to efficiently approximate the nonlinear multi-UAVs system by a set of local linearized networked systems. Thus, the HJI-constraint optimization problem for the H∞ event-triggered robust formation team tracking control can be transformed to a linear matrix inequality(LMI)-constraint optimization problem and can be easily solved by the convex optimization techniques. Finally, a simulation example is given to validate the effectiveness of the proposed event-triggered robust H∞ team formation tracking control for the multi-VTOL-UAV system.

Security-Enhanced Filter Design for Stochastic Systems under Malicious Attack via Smoothed Signal Model and Multiobjective Estimation Method

In this study, a novel security-enhanced filter (SEF) is proposed for the system state and malicious attack signal estimation of the stochastic jump-diffusion systems with the external disturbance, measurement noise and malicious attack signal on system and sensor. To efficiently estimate the system state and malicious attack signal by the traditional Luenberger-type filter, a novel smoothed signal model of malicious attack signals is embedded in the system model so that the attack signals in the augmented system do not corrupt the augmented states estimation of SEF again. For the optimal filtering robustness and security, the stochastic multi-objective (MO) $H_{2}/H_{\infty }$ SEF scheme is proposed to achieve optimal disturbance and noise filtering performance and the optimal security enhancement under malicious attack. By using the suboptimal method, the stochastic MO $H_{2}/H_{\infty }$ SEF design could be equivalently transformed to linear matrix inequalities (LMIs)-constrained multi-objective optimization problem (MOP). In the case of nonlinear stochastic system, the MO $H_{2}/H_{\infty }$ SEF design problem could be converted to a Hamilton-Jacobi inequalities (HJIs)-constrained MOP. In order to overcome the difficulty in solving the HJIs-constrained MOP, based on the global linearization technique, the HJIs-constrained MOP for SEF design of nonlinear stochastic systems could be transformed to an LMIs-constrained MOP. Further, an LMIs-constrained multi-objective evolution algorithm (MOEA) is proposed to efficiently solve the LMIs-constrained MOP for the design of SEF. Two simulation examples including the missile trajectory estimation problem by ground radar system under the malicious attack signals and estimation of netwoked-based mass spring system are given to validate the effectiveness of the proposed method.

Leader-follower formation suboptimal control for quadrotors

In this work, a leader-follower formation control problem of quadrotors is investigated. The quadrotor dynamic model is represented by unit quaternion with the consideration of external disturbance. Nonlinear H∞ design approach is introduced and robust controllers for both leader and follower robots are derived by solving a Hamilton-Jacobi inequality following from a result for general nonlinear affine systems. Then some robustness conditions of the proposed controllers are obtained by selecting appropriate parametrised Lyapunov functions. The resultant state feedback controllers establish the asymptotically stability of the closed-loop nonlinear system. In addition, integral backstepping (IB) controllers are also derived for the leader-follower formation control problem. The purpose of designing IB controllers is to evaluate the robustness of H∞ controllers by comparison. The simulation results from both types of controllers are compared and robustness performance of the H∞ controllers over the IB controllers are demonstrated.

Leader-Follower Formation Suboptimal Control for Quadrotors

In this work, a leader-follower formation control problem of quadrotors is investigated. The quadrotor dynamic model is represented by unit quaternion with the consideration of external disturbance. Nonlinear H∞ design approach is introduced and robust controllers for both leader and follower robots are derived by solving a Hamilton-Jacobi inequality following from a result for general nonlinear affine systems. Then some robustness conditions of the proposed controllers are obtained by selecting appropriate parametrised Lyapunov functions. The resultant state feedback controllers establish the asymptotically stability of the closed-loop nonlinear system. In addition, integral backstepping (IB) controllers are also derived for the leader-follower formation control problem. The purpose of designing IB controllers is to evaluate the robustness of H∞ controllers by comparison. The simulation results from both types of controllers are compared and robustness performance of the H∞ controllers over the IB controllers are demonstrated.

Fault detection filtering for Itô‐type affine nonlinear stochastic systems

AbstractThis paper studies the mixed H−/H∞ fault detection filtering of Itô‐type nonlinear stochastic systems. Mixed H−/H∞ filtering combines system robustness to external disturbance and the sensitivity to the fault of the residual signal. First, for Itô‐type affine nonlinear stochastic systems, some sufficient criteria are obtained for the existence of H−/H∞ filter in terms of Hamilton‐Jacobi inequalities (HJIs). Second, for a class of quasi‐linear Itô systems, a sufficient condition is given for the existence of the H−/H∞ filter by means of linear matrix inequalities (LMIs). Finally, a numerical example is presented to illustrate the effectiveness of the proposed results.

Robust $${{\cal H}_\infty}$$ Control for Switched Nonlinear Port-controlled Hamiltonian Systems

In the paper, the robust $${{\cal H}_\infty}$$ control for switched nonlinear port-controlled Hamiltonian (SNPCH) systems with external disturbances is investigated via the multiple Lyapunov functions method. Under two cases that the SNPCH system is subject to actuator saturation (AS) and the SNPCH system is without AS, a design approach for the switching robust $${{\cal H}_\infty}$$ feedback controller is developed to attenuate the external disturbances. The sufficient conditions are established for $${{\cal H}_\infty}$$ control of the corresponding closed-loop systems based on Hamilton-Jacobi inequality techniques. Furthermore, via the Hamiltonian realization method, the obtained results are applied to solve the $${{\cal H}_\infty}$$ control for switched nonlinear affine systems with external disturbances. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed $${{\cal H}_\infty}$$ control methods.

Multiobjective $H_{2}/H_{\infty }$ Control Design of the Nonlinear Mean-Field Stochastic Jump-Diffusion Systems via Fuzzy Approach

In this study, the multiobjective $H_{2}/H_{\infty }$ fuzzy control design is investigated for nonlinear mean-field jump diffusion (MFSJD) systems for concurrently minimizing both $H_{2}$ and $H_{\infty }$ performance. Since $H_{2}$ and $H_{\infty }$ performance usually conflict with one another, the optimization problem that concurrently minimizes $H_{2}$ and $H_{\infty }$ performance can be regarded as a dynamically constrained multiobjective optimization problem (MOP). Because Hamilton–Jacobi inequalities of the nonlinear MFSJD systems are difficult to derive, multiobjective $H_{2}/H_{\infty }$ control design problems of nonlinear MFSJD system are difficult to solve. The Takagi–Sugeno fuzzy interpolation scheme and an indirect method are introduced to help transform the dynamically constrained MOP into linear matrix inequalities (LMIs) constrained MOP. Thus, one can accomplish the multiobjective $H_{2}/H_{\infty }$ fuzzy control design via LMI-constrained multiobjective evolutionary algorithms (MOEAs). To efficiently solve the multiobjective $H_{2}/H_{\infty }$ control design problem, we propose a novel LMI-constrained MOEA called fronts-squeezing. The fronts-squeezing LMI-constrained MOEA can concurrently search Pareto front from both sides of feasible and infeasible regions and narrow the search region down to increase efficiency. Finally, we present a simulation example about the multiobjective regulation of nonlinear MFSJD financial system to illustrate the design procedure and verify the proposed theories.

Incremental passivity-based H∞ output tracking control for switched nonlinear systems

This paper is concerned with the problem of H∞ output tracking control for a class of switched nonlinear systems with external disturbances using incremental passivity, even if the problem of H∞ output tracking control for none of subsystems is solvable. First, an incremental passivity concept of switched nonlinear systems without external disturbances is proposed. This incremental passivity property requires each active subsystem is incrementally passive. Then, sufficient conditions to be incrementally passive are given. Second, by resorting to the established incremental passivity theory, a state-dependent switching law and a set of feedback controllers are designed to solve the problem of H∞ output tracking control for a class of switched nonlinear systems with external disturbances. This avoids solving Hamilton-Jacobi (HJ) inequality. Third, a composite state-dependent switching law and the state feedback controllers are designed to solve the H∞ output tracking problem for a class of cascaded switched nonlinear systems with external disturbances. The designed composite switching law allows the driven switched system and the driving switched system switch asynchronously. Finally, two examples are provided to verify the effectiveness of the proposed method.

Multiobjective control for nonlinear stochastic Poisson jump-diffusion systems via T-S fuzzy interpolation and Pareto optimal scheme

Unlike the conventional mixed H2/H∞ control design method, this study provides a multiobjective fuzzy control design method for nonlinear stochastic Poisson jump-diffusion systems to simultaneously achieve optimal cost and robustness performance in the Pareto optimal sense via the proposed evolutionary algorithm. For a nonlinear stochastic Poisson jump-diffusion system, the Poisson jumps cause its system behaviors to change intensely and discontinuously. To design an efficient controller for a nonlinear stochastic jump-diffusion system is much more difficult. On the other hand, the H2 and H∞ performance indices generally conflict with each other and can be regarded as a multiobjective optimization problem (MOP). It is not easy to directly solve this MOP, owing to (i) the Pareto front of the MOP is difficult to obtain through direct calculation; (ii) the MOP is a Hamilton-Jacobi-Inequalities (HJIs)-constrained MOP. To address these issues, we use Takagi-Sugeno (T-S) interpolation scheme to transform the HJIs-constrained MOP into a linear matrix inequality (LMI)-constrained MOP. Then, we employ the proposed LMI-constrained multiobjective optimization evolutionary algorithm (LMI-constrained MOEA) to efficiently search for the Pareto optimal solution, from which the designer can select one kind of design according to their preference. Finally, a design example is given to illustrate the design procedure and to verify our results.

${H_\infty}$ Robust Control of Permanent Magnet Synchronous Motor Based on PCHD

The surface permanent magnet synchronous motor (SPMSM) speed regulation system is easily affected by the inner parameter perturbation and the external load disturbance at run time. To solve this problem, the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> robust control strategy is proposed in this paper. First, given the systematic uncertainties, the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> robust current controller based on the Hamilton-Jacobi Inequality is designed to ensure the robustness of current control under the SPMSM nominal mathematical model. This model is expressed as the port-controlled Hamiltonian with the dissipation form; second, the linear matrix inequality-based H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> sliding surface and the sliding control law are designed under the extended state space expression of the SPMSM motion equation. Thereby, the robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> sliding mode speed controller is acquired, thus realizing the robustness of speed control and improving the dynamic characteristics of the system. Finally, the effectiveness and availability of the proposed control strategy are verified by the hardware-in-the-loop simulation experiment.

IEEE Access Special Section Editorial: Special Section on New Developments on Reliable Control and Filtering of Complex Nonlinear Systems

During the past three decades, the problems of reliable control and filtering have attracted considerable attention due to the growing demands for system safety, reliability, maintainability, and survivability. The main aim of reliable control and filtering is to design a suitable controller and filter with guaranteed stability and performance, respectively; not only when all the system components are in operation, but also when some component failures occur. To guarantee a higher reliability level and better control performance, reliable control systems depending on various control strategies have tried to achieve these critical requirements, i.e., the pole region assignment method, coprime factorization method, the algebraic Riccati equation (ARE)-based method, the Hamilton-Jacobi inequality (HJI)-based method, and linear matrix inequality (LMI)-based method. Although many researchers have investigated the reliable control/filtering problems for linear systems with different scenarios for many years, the topic of reliable control/filtering in complex nonlinear systems is still in the early stages of development and many critical issues remain to be further investigated.

H∞ Control for Nonlinear Infinite Markov Jump Systems

In this paper, we discuss the infinite horizon H∞ control problem for a class of nonlinear stochastic systems with state, control, and disturbance dependent noise. The jumping parameters are modelled as an infinite-state Markov chain. Based on the solvability of a set of coupled Hamilton-Jacobi inequalities (HJIs), the exponential mean square H∞ controller for the considered nonlinear stochastic systems is obtained. A numerical example is given to show the effectiveness of the proposed design method.

Robust neural networks control of omni-mecanum wheeled robot with hamilton-jacobi inequality

This paper presents a novel approach to the problem of controlling mechanical objects of unspecified description, considering variable operating conditions. The controlled object is a mobile robot with mecanum wheels (MRK M). To solve the control task, taking into account compensation for nonlinearity and the object variable operating conditions, the Lyapunov stability theory is applied, including the Hamilton-Jacobi (HJ) inequality. A neural network with basic sigmoid functions is used to compensate for the nonlinearity and variable operating conditions of the robot. A simulation example is provided in order to evaluate the analytical considerations. The simulation results obtained confirmed high accuracy of the predicted robot motion in variable operating conditions.

Dissipative control for nonlinear Markovian jump systems with actuator failures and mixed time-delays

This paper addresses the dissipative control problem for nonlinear Markovian jump systems subject to actuator failures and mixed time-delays, where the mixed time-delays consist of both discrete and distributed time-delays and are mode-dependent. The purpose of the problem under investigation is to design a state feedback controller such that, in the presence of actuator failures and mixed time-delays, the closed-loop system is asymptotically stable in the mean square sense while achieving the pre-specified dissipativity. By constructing a Lyapunov–Krasovskii functional and using a completing square approach, sufficient conditions are proposed for the existence of the desired controller in terms of the solvability of certain Hamilton–Jacobi inequalities. Finally, an illustrative numerical example is provided to demonstrate the effectiveness of the developed control scheme.