Hamilton-Jacobi Inequality Research Articles

Robust Approximate Dynamic Programming for Nonlinear Systems With Both Model Error and External Disturbance.

Model error and external disturbance have been separately addressed by optimizing the definite H∞ performance in standard linear H∞ control problems. However, the concurrent handling of both introduces uncertainty and nonconvexity into the H∞ performance, posing a huge challenge for solving nonlinear problems. This article introduces an additional cost function in the augmented Hamilton-Jacobi-Isaacs (HJI) equation of zero-sum games to simultaneously manage the model error and external disturbance in nonlinear robust performance problems. For satisfying the Hamilton-Jacobi inequality in nonlinear robust control theory under all considered model errors, the relationship between the additional cost function and model uncertainty is revealed. A critic online learning algorithm, applying Lyapunov stabilizing terms and historical states to reinforce training stability and achieve persistent learning, is proposed to approximate the solution of the augmented HJI equation. By constructing a joint Lyapunov candidate about the critic weight and system state, both stability and convergence are proved by the second method of Lyapunov. Theoretical results also show that introducing historical data reduces the ultimate bounds of system state and critic error. Three numerical examples are conducted to demonstrate the effectiveness of the proposed method.

Multi-Player Non-Cooperative Game Strategy of a Nonlinear Stochastic System with Time-Varying Parameters

This paper discusses the multi-player non-cooperative game of nonlinear stochastic time-varying systems described by Itô-type differential equations in a finite time interval. Multi-player non-cooperative game problems are represented by multi-objective Pareto (MOP) control problems to describe the fact that each player has their own goals. By applying Hamilton–Jacobi inequalities (HJIs), the criterion of upper bounds of the MOP boundary is obtained for nonlinear stochastic systems, and the corresponding strategies are designed for such games, so the MOP problem is transformed into a HJI-constrained MOP problem. In order to overcome the difficulty of solving HJIs, a global linearization method is proposed to approximate the nonlinear systems. By the proposed global linearization method, multi-player non-cooperative game problems are transformed into Riccati equation-constrained MOP problems, and the approximate solutions of HJI-constrained MOP problems are obtained. Finally, a practical example is given to illustrate the effectiveness of the proposed method.

Sharp log-Sobolev inequalities in [formula omitted] spaces with applications

Given p,N>1, we prove the sharp Lp-log-Sobolev inequality on noncompact metric measure spaces satisfying the CD(0,N) condition, where the optimal constant involves the asymptotic volume ratio of the space. This proof is based on a sharp isoperimetric inequality in CD(0,N) spaces, symmetrization, and a careful scaling argument. As an application we establish a sharp hypercontractivity estimate for the Hopf–Lax semigroup in CD(0,N) spaces. The proof of this result uses Hamilton–Jacobi inequality and Sobolev regularity properties of the Hopf–Lax semigroup, which turn out to be essential in the present setting of nonsmooth and noncompact spaces. Moreover, a sharp Gaussian-type L2-log-Sobolev inequality and a hypercontractivity estimate are obtained in RCD(0,N) spaces. Our results are new, even in the smooth setting of Riemannian/Finsler manifolds. In particular, an extension of the celebrated rigidity result of Ni (2004) [55] on Riemannian manifolds will be a simple consequence of our sharp log-Sobolev inequality.

Hamilton–Jacobi Inequality Adaptive Robust Learning Tracking Controller of Wearable Robotic Knee System

A Wearable Robotic Knee (WRK) is a mobile device designed to assist disabled individuals in moving freely in undefined environments without external support. An advanced controller is required to track the output trajectory of a WRK device in order to resolve uncertainties that are caused by modeling errors and external disturbances. During the performance of a task, disturbances are caused by changes in the external load and dynamic work conditions, such as by holding weights while performing the task. The aim of this study is to address these issues and enhance the performance of the output trajectory tracking goal using an adaptive robust controller based on the Radial Basis Function (RBF) Neural Network (NN) system and Hamilton–Jacobi Inequality (HJI) approach. WRK dynamics are established using the Lagrange approach at the outset of the analysis. Afterwards, the L2 gain technique is applied to enhance the control motion solutions and provide the main features of the designed WRK control systems. To prove the stability of the controlled system, the HJI approach is investigated next using optimization techniques. The synthesized RBF NN algorithm supports the easy implementation of the adaptive controller, as well as ensuring the stability of the WRK system. An analysis of the numerical simulation results is performed in order to demonstrate the robustness and effectiveness of the proposed tracking control algorithm. The results showed the ability of the suggested controller of this study to find a solution to uncertainties.

Robust Control for Stochastic Nonlinear Delay Systems with Jumps

The problem of infinite horizon H∞ control for general delayed nonlinear stochastic Markov jump systems with the infinite jumping parameters is considered in this paper, in which the noise is dependent on the state, control, and external disturbance. The coupled Hamilton–Jacobi inequalities (HJIs)‐based sufficient condition is given to ensure the existence of the H∞ controller. As a corollary, infinite horizon H∞ controllers are designed for nonlinear stochastic time‐delay systems without jumps by solving a series of coupled HJIs. Besides, the effectiveness of the proposed method is verified by a numerical example.

Robust control of a hub-spoke tethered formation system of microsatellites using Hamilton-Jacobi inequality

The problem of controlling a rotating hub-spoke tethered formation system in low Earth orbit is considered, in which microsatellites are located radially around the central spacecraft (hub) and connected to it by tethers (spokes). To analyze the dynamics of the tethered system, a mathematical model is developed in the orbital coordinate system by Lagrange method, in which the central spacecraft is regarded as a rigid body. In the proposed control scheme, the spin motion of the central body is regulated by the control moment, and tether deployment control law is proposed by robust approach, which is carried out by regulating the tether tensions and low thrusts acting on the microsatellites. The robustness and stability of the system are investigated using Lyapunov theory and Hamilton-Jacobi inequality, which is used to determine the robustness index of the control system. The results of numerical calculations are presented, which confirm that the proposed control scheme is effective when taking into account periodic gravitational perturbations, external perturbations and perturbations associated with uncertainty in the initial states of the system and with the rotation of the central body.

Finite-time and fixed-time stability analysis for time-varying systems: A dual approach

In this paper we present a new dual approach to investigating the stability of the system of two nonautonomous ODE with the right-hand side being only Lebesgue measurable with respect to time t. We define a new dual Lapunov function S which depends on three variables, and we give sufficient conditions for finite and fixed-time stability for the system using a dual Hamilton-Jacobi inequality and taking advantage of a class of functions M and MB. This method allows simultaneous investigation of the finite and fixed-time stability of the primary and the dual equation. We give some examples to illustrate the results obtained. As an application of this dual approach, we define the region of attraction and illustrate with an example.

Critical Review on Robust Speed Control Techniques for Permanent Magnet Synchronous Motor (PMSM) Speed Regulation

The permanent magnet synchronous motor (PMSM) is a highly efficient energy saving machine. Due to its simple structural characteristics, good heat radiation capability, and high efficiency, PMSMs are gradually replacing AC induction motors in many industrial applications. The PMSM has a nonlinear system and lies on parameters that differ over time with complex high-class dynamics. To achieve the excessive performance operation of a PMSM, it essentially needs a speed controller for providing accurate speed tracking, slight overshoot, and robust disturbance repulsion. Therefore, this article provides an overview of different robust control techniques for PMSMs and reviews the implementation of a speed controller. In view of the uncertainty factors, such as parameter perturbation and load disturbance, the H∞ robust control strategy is mainly reviewed based on the traditional control techniques, i.e., robust H∞ sliding mode controller (SMC), and H∞ robust current controller based on Hamilton–Jacobi Inequality (HJI) theory. Based on comparative analysis, this review simplifies the development trend of different control technologies used for a PMSM speed regulation system.

Robust Collaborative Team Formation Control of Hybrid Teams of Biped Robots and Wheeled Vehicles Under External Disturbance and Communication Interactions

In this study, a collaborative team formation system of hybrid teams of biped robots and tractor-trailers is designed for some common task. The collaborative team formation system consists of a number of hybrid team formation subsystems. Each hybrid team formation subsystem is composed of a tractor-trailer as the leader and a group of biped robots as followers. For the convenience of collaborative team formation design, novel reference models are proposed to generate the desired collaborative time-varying leader-follower team formation for the cooperative tractor-trailers and biped robots in each hybrid team. Accordingly, the collaborative team formation design problem is simplified to a set of independent robust model reference tracking design problems for each agent under external disturbances and communication couplings from other agents. Subsequently, the robust <i>H</i>&#x221E; decentralized collaborative team formation tracking control strategy is proposed to achieve the independent team formation tracking control for each agent and efficiently attenuate the effect of external disturbance and communication coupling from other agents on the team formation tracking performance of each agent. In order to avoid solving the difficult Hamilton-Jacobi inequality (HJI) of the robust <i>H</i>&#x221E; decentralized collaborative team formation tracking control for each agent, the numerical linear parameter-varying (LPV) modeling method is employed to approximate the nonlinear team formation tracking error dynamic system of each agent such that the HJI can be equivalently transformed to a set of linear-matrix inequalities (LMIs) which can be efficiently solved by MATLAB LMI TOOLBOX. Since the robust <i>H</i>&#x221E; decentralized collaborative team formation tracking control scheme is employed based on local information, the collaborative team formation system can be extended to very large-scale biped robots and tractor-trailers. Finally, a simulation example of robust <i>H</i>&#x221E; decentralized collaborative team formation tracking control design of three collaborative teams, in which each hybrid team consists of one tractor-trailer as the leader and six biped robots as followers, is given to compare with the optimal <i>H</i>2 decentralized method to verify the effectiveness of the proposed scheme.

Homogeneous Lp Stability for Homogeneous Systems

The motivation of this paper comes from the fact that <i>L_p</i>&#x2014;stability and <i>L_p</i>&#x2014;gain, using the classical signal norms, is not well-defined for arbitrary continuous weighted homogeneous systems. However, using homogeneous signal norms it is possible to show that every internally stable homogeneous system has a globally defined finite homogeneous <i>L_p</i>&#x2014;gain, for <i>p</i> sufficiently large. If the system has a homogeneous approximation, the homogeneous <i>L_p</i>&#x2014;gain is inherited locally. Homogeneous <i>L_p</i>&#x2014;stability can be characterized by a homogeneous dissipation inequality, which in the input affine case can be transformed to a homogeneous Hamilton-Jacobi inequality. An estimation of an upper bound for the homogeneous <i>L_p</i>&#x2014;gain can be derived from these inequalities. Homogeneous <i>L</i>_&#x221E;&#x2014;stability is also considered and its strong relationship to Input-to-State stability is studied. These results are extensions to arbitrary homogeneous systems of the well-known situation for linear time-invariant systems, where the Hamilton-Jacobi inequality reduces to an algebraic Riccati inequality. A natural application of finite-gain homogeneous <i>L_p</i>&#x2014;stability is in the study of stability for interconnected systems. An extension of the small-gain theorem for negative feedback systems and results for systems in cascade are derived for different homogeneous norms. Previous results in the literature use classical signal norms, hence, they can only be applied to a restricted class of homogeneous systems. The results are illustrated by several examples.

Позиционный принцип минимума: вариационное усиление понятий экстремальности в оптимальном управлении

Existing maximum principles of Pontryagin’s type and related optimality conditions, such as, e.g., the ones derived by F. Clarke, B. Kaskosz and S. Lojasiewicz Jr., and H.J. Sussmann, can be strengthened up to global necessary optimality conditions in the form of so-called feedback minimum principle. This is possible for both classical and non-smooth optimal control problems without terminal constraints. The formulation of the feedback minimum principle (or related extremality conditions) remains within basic constructions of the mentioned maximum principles (the Hamiltonian or Pontryagin function, the adjoint differential equation or inclusion, and its solutions –– co-trajectories). At the same time, the actual maximum condition –– maximization of the Hamiltonian –– takes a variational form: any optimal trajectory of the addressed problem should be optimal for a specific “accessory” problem of dynamic optimization. The latter is stated over all tubes of Krasovskii-Subbotin constructive motions generated by feedback strategies, which are extremal with respect to a certain supersolution of the Hamilton-Jacobi equation. Such a supersolution can be represented explicitely in terms of the co-trajectory of a reference control process and the terminal cost function. In a general version, the feedback minimum principle operates with generalized solutions of the proximal Hamilton-Jacobi inequality for weakly decreasing (𝑢-stable) functions

Research on Underwater Manipulator Control Based on HJI Theory and RBF Neural Network

Aiming at the influence of the underwater complex working environment on the control performance of the manipulator, and the traditional manipulator control method is not effective, a new method based on HJI (Hamilton-Jacobi Inequality) theory and RBF (radial basis function) neural network is proposed. The intelligent controller to realize the stable control of the underwater manipulator. Considering the influence of various additional forces on the manipulator in the underwater environment, the Lagrange method is used to establish the dynamic model of the underwater manipulator. Then the robust conditions of HJI differential inequality are used to ensure the stability of the control system, and the stability of the underwater manipulator system is verified by the Lyapunov stability theory. Finally, MATLAB/Simulink is used to simulate the control of the manipulator. The simulation results verify the effectiveness of the method and can realize the stable control of the underwater manipulator system.

Robust Decentralized Formation Tracking Control for Stochastic Large-Scale Biped Robot Team System Under External Disturbance and Communication Requirements

In this study, a robust decentralized stochastic H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> tracking control design is proposed to deal with the team formation tracking problem for a large-scale biped robot team system with external disturbance and communication requirement on biped robots and leader. Under the concept of virtual leader and the decentralized control approach, biped robots walking in a large-scale team formation are controlled to track their own reference paths with the robust stochastic H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> tracking performance. Consequently, the original robust decentralized stochastic H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> large-scale biped robot team formation tracking control design problem can be transformed to a set of independent Hamilton-Jacobi inequality (HJI)-constrained optimization problems for each biped robot. To cope with the difficulties in solving the HJIs for the robust decentralized stochastic H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> team formation tracking control design, a numerical framework is introduced to approximate the walking biped robot systems. Based on the tensor-product model transformation technique, the set of independent HJI-constrained optimization problems for the robust decentralized stochastic H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> team formation tracking control design is converted into a set of independent linear-matrix-inequality-constrained optimization problems, which can be solved efficiently by the convex optimization methods. Since the robust decentralized stochastic H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> team formation tracking control can be designed individually, the magnitude of a biped robot team can be easily increased to a very large scale. Finally, a simulation example of decentralized H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> formation tracking control design for a 30-scale biped robot team is given to validate the effectiveness of the proposed scheme.

Incremental dissipative control for nonlinear stochastic Markovian jump systems

This paper investigates the incremental dissipative control problem for nonlinear stochastic Markovian jump systems (MJSs) driven by the Markov process, Brownian motion, and disturbance-dependent noise. A stochastic version of incremental dissipativity for Markovian jump processes is presented, where incremental Lyapunov functions and a state dissipation inequality in expectation are used. Furthermore, the sufficient conditions of stochastic incremental dissipativity for nonlinear stochastic MJSs are given in terms of a set of coupled incremental Hamilton–Jacobi inequalities (iHJIs). Also, the incrementally globally asymptotically stable in the mean (IGASiM) conditions for nonlinear stochastic MJSs are derived by using stochastic incremental dissipativity. The sufficient conditions for the stochastic state-feedback incremental H∞ control problem are obtained by solving a set of coupled iHJIs and by resorting to multiple Lyapunov functions. Two numerical examples demonstrated the effectiveness of the proposed method.

Hamiltonian‐driven adaptive dynamic programming for mixedH2/H∞performance using sum‐of‐squares

AbstractIn this article, the mixedH2/H∞performance optimization is first formulated as a nonzero‐sum game, of which the sufficient condition guaranteeing the existence of the Nash equilibrium is derived using the Hamilton–Jacobi (HJ) theory. Then, Hamiltonian‐driven inequalities are presented to evaluate theH2andH∞performances. Using this Hamiltonian‐inequality driven approach, the coupled HJ equations arising from finding the Nash equilibrium are relaxed to the HJ inequality constraints. A novel mixed policy iteration (PI) algorithm is developed that uses sum‐of‐squares (SOS) program in policy evaluation step, and consists of anH2performance improvement step and anH∞performance guarantee step. This constrained‐driven approach allows us to present a PI algorithm that takes into account both robustness and performance objectives. Finally, a numerical simulation is carried out to highlight the efficacy of the proposed framework.

Robust Stochastic Observer-Based Attack-Tolerant Missile Guidance Control Design Under Malicious Actuator and Sensor Attacks

In this study, the robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> stochastic observer-based attack-tolerant guidance control strategy is designed for the nonlinear stochastic missile guidance control system under the external disturbance and measurement noise as well as actuator attack signal and sensor attack signal. To simplify the attack signal estimation, a novel nonsingular smoothed dynamic model is introduced to efficiently describe the actuator and sensor attack signals. Consequently, the state/attack signal estimation can be easily achieved by using conventional Luenberger observer. Next, to attenuate the effect of external disturbance, measurement noise and approximation errors of attack signal on the missile guidance control system, the robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> attack-tolerant guidance control performance is considered and the design condition is derived in terms of nonlinear Hamilton-Jacobi inequality (HJI) constrained problem. Since HJI cannot be easily solved analytically or numerically, the Takagi-Sugeno (T-S) fuzzy modelling method is utilized to facilitate the robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> attack-tolerant guidance control strategy design. Thereafter, the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> observer-based attack-tolerant control design problem is transformed into linear matrix inequalities problem (LMIP) which can be solved very efficiently by using the convex optimization techniques. Simulation example, with the comparison between the proposed method and conventional robust missile guidance strategy, is given to illustrate the design procedures and validate the performance of the proposed method.

Robust [formula omitted] filtering for nonlinear discrete-time stochastic systems

This paper mainly discusses the H∞ filtering of general nonlinear discrete time-varying stochastic systems. A nonlinear discrete-time stochastic bounded real lemma is firstly obtained by means of the smoothness of the conditional mathematical expectation, and then, based on the given stochastic bounded real lemma and a stochastic LaSalle-type theorem, a sufficient condition for the existence of the H∞ filtering of general nonlinear discrete time-varying stochastic systems is presented via a newly introduced Hamilton–Jacobi inequality (HJI), which is easily verified. When the worst-case disturbance {vk∗}k∈N is considered, the suboptimal H2∕H∞ filtering is studied. Three examples including a practical engineering example show the effectiveness of our main results.

Hamilton-Jacobi inequalities on a metric space

In some applied models (of flocking or of the crowd control) it is more natural to deal with elements of a metric space (as for instance a family of subsets of a vector space endowed with the Hausdorff metric) rather than with vectors in a normed vector space. We consider an optimal control problem involving the so called morphological control system whose trajectories are time dependent tubes of subsets of RN and show that the theory of Hamilton-Jacobi-Bellman inequalities can be extended to this framework.

Weighted $$\mathbf{H}_\infty$$ Performance Analysis of Nonlinear Stochastic Switched Systems: A Mode-Dependent Average Dwell Time Method

In this paper, the issues about weighted $$H_\infty$$ performance analysis and $$H_\infty$$ control for the stochastic switched nonlinear systems (SSNSs) with multiplicative noise are investigated. The mode-dependent average dwell time (MDADT) method is used to deal with the switching in different modes. Firstly, we get a sufficient condition to show that the considered system with all stable subsystems achieves the exponential stability in mean square sense and a specified weighted $$H_\infty$$ performance, which is extended to the case that both stable and unstable subsystems coexist in terms of second-order Hamilton-Jacobi inequalities (HJIs). Then, by using Takagi-Sugeno (T-S) fuzzy approach, the $$H_\infty$$ controller for SSNSs is designed via solving a set of linear matrix inequalities (LMIs). Finally, an example is supplied to illustrate the effectiveness of our results.

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.