Hamilton-Jacobi Inequality Research Articles

Adaptive Reliable H∞ Control of Uncertain Affine Nonlinear Systems

For general input affine nonlinear systems, robust reliable control designs are commonly available that compensate the actuator faults in pure outage mode. In this paper, a more general and complex problem is considered and an adaptive reliable H∞ controller is designed for a class of uncertain input affine nonlinear systems in the presence of actuators fault. The key element of the work is the introduction of a novel adaptive mechanism that estimates the faults which are modeled as an outage or loss of effectiveness and stabilizes the overall system. Incorporating with the parameter projection algorithm and the solution of Hamilton-Jacobi-Inequality (HJI), the proposed method combines adaptive reliable control and robust H∞ control techniques. A numerical approach is developed based on the Taylor series expansion for solving the HJI. Various simulation examples are given to illustrate the effectiveness of the proposed adaptive reliable H∞ control scheme over the conventional H∞ control and reliable H∞ control method.

Numerical Solutions of Hamilton-Jacobi Inequalities by Constrained Gaussian Process Regression

This paper proposes numerical solutions of Hamilton-Jacobi inequalities based on constrained Gaussian process regression. While Gaussian process regression is a tool to estimate an unknown function...

Robust H∞ control for a class of uncertain nonlinear systems with mixed time-delays

This paper is concerned with the robust H∞ control problem for a general class of uncertain nonlinear systems with mixed time-delays. The mixed time-delays consist of both discrete and distributed delays. We aim to design a memoryless state feedback controller such that the closed-loop system is robustly stable for all admissible uncertainties with guaranteed H∞ disturbance rejection attenuation level. By introducing a new Lyapunov–Krasovskii functional that reflects the mixed delays, sufficient conditions are established for the closed-loop system ensuring the robust stability as well as the H∞ performance requirement. The controller design is facilitated in terms of the solvability of a Hamilton–Jacobi inequality. Two numerical examples are utilized to demonstrate the effectiveness of the proposed methods.

Hamilton–Jacobi inequality robust neural network control of a mobile wheeled robot

The work presented here presents a new approach to determine a robust neural network follow-up motion control of a mobile wheeled robot. The solution is a result of a Hamilton–Jacobi inequality, enabling synthesis of control of a non-linear object in terms of input to output stability. By applying Lyapunov’s theory of stability, it was demonstrated that all signals are limited, while the determined control provides a relatively high accuracy of actuated motion. The weights of the neural network were updated in real time and online. The produced simulation solutions confirmed the efficiency of the approach contemplated here.

Observer-based fault detection for uncertain nonlinear systems

This paper addresses L2 observer-based fault detection issues for a class of nonlinear systems in the presence of parametric and dynamic uncertainties, respectively. To this end, three different types of uncertain affine nonlinear system models studied in this paper are described first. Then, the integrated design schemes of L2 observer-based fault detection systems are derived with the aid of Hamilton–Jacobi inequalities (HJIs), respectively. Numerical examples are also provided in the end to demonstrate the effectiveness of the proposed results.

Robustness of DNA Strand Displacement Systems⋆

Abstract How to construct a reliable deoxyribonucleic acid (DNA) circuit is one of the most important issues in the field of molecular programming and computing. Such a circuit frequently suffers from various kinds of unintended binding reactions on account of a lower specificity of the molecular interaction emerging from base complementarity between DNA strands, which results in low reliability of the molecular circuit. In this study, we propose a method to quantitatively evaluate the robustness of a DNA circuit created by DNA strand displacement reactions. The highlight of our method is representing the circuit system contaminated by unintended reactions by a standard form of a disturbed nonlinear system and evaluating the robustness with the L2 gain by solving the Hamilton–Jacobi inequality.

Applying sum‐of‐squares decomposition technique to power system robust control problem

For the first time, we apply the sum‐of‐squares (SOS) decomposition technique for robust control of power systems. We propose an SOS L2 robust control scheme aiming at a power system characterized by exogenous disturbances. A set of states‐related inequalities are utilized to guarantee L2 gain disturbance attenuation performance of the system. It involves semidefinite programming relaxations based on the SOS decomposition technique, which is employed to solve the inequalities that bring about robust control of the system. This method is superior to other traditional approaches. It is able to construct a robust controller that guarantees L2 gain disturbance attenuation performance of the system featuring exogenous disturbance by the SOS algorithm without solving Hamilton–Jacobi inequalities or using the recursive design of back‐stepping. Then, the proposed approach is applied to derive a robust excitation controller for multimachine power systems. Moreover, a three‐machine power system model is employed to test the robust excitation controller, which finally verifies its effectiveness and superiority. © 2017 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Analysis for the Robust <inline-formula> <tex-math notation="LaTeX">$H_{\infty}$</tex-math> </inline-formula> Synchronization of Nonlinear Stochastic Coupling Networks Through Poisson Processes and Core Coupling Design

In this paper, we discuss a robust synchronization problem for nonlinear stochastic coupling networks through Poisson processes. The asymptotical synchronizability in probability is examined via the Hamilton-Jacobi inequality (HJI) criterion. Further, in order to effectively filter external disturbances of the nonlinear stochastic coupling networks, synchronization robustness can be guaranteed by solving an HJI. In other words, by solving two HJI criteria, both the asymptotical synchronizability and the robust synchronizability in probability are guaranteed. To simplify the HJI criteria, a linear matrix inequality criterion is imposed, allowing for robust $H_{\infty}$ synchronization based on the Takagi-Sugeno fuzzy model. Finally, we propose a recursive algorithm for the core coupling design by deleting the redundant Poisson couplings of the coupling networks while maintaining its asymptotical synchronizability in probability and synchronization robustness. A simulation example is provided to illustrate the core coupling design procedure and verify the robust synchronizability.

H∞ control for nonlinear stochastic Markov systems with time-delay and multiplicative noise

This paper is concerned with the H∞ control problem for a class of nonlinear stochastic Markov jump systems with time-delay and system state-, control input- and external disturbancedependent noise. Firstly, by solving a set of Hamilton-Jacobi inequalities (HJIs), the exponential mean square H∞ controller design of delayed nonlinear stochastic Markov systems is presented. Secondly, by using fuzzy T-S model approach, the H∞ controller can be designed via solving a set of linear matrix inequalities (LMIs) instead of HJIs. Finally, two numerical examples are provided to show the effectiveness of the proposed design methods.

Exponential stabilization of nonlinear switched systems with distributed time-delay: An average dwell time approach

In this paper, the exponential stabilization problem is discussed for nonlinear switched systems subject to the distributed time-delay. Attention is focused on the design of a state feedback controller such that, the closed-loop system is exponentially stable with a guaranteed weighted L2 gain. By resorting to the average dwell time method and the piecewise Lyapunov functional approach, sufficient conditions are derived for the solvability of the addressed problem in terms of the feasibility of certain Hamilton–Jacobi inequalities (HJIs). The explicit expression of the desired controller is formulated via solving the presented set of HJIs. Furthermore, within the proposed framework, the exponential stabilization problems are investigated, respectively, for nonlinear switched systems with mixed time-delays and linear switched systems with the distributed time-delay. Finally, a simulation example is given to illustrate the effectiveness and applicability of the proposed algorithm.

The Mayer and Minimum Time Problems with Stratified State Constraints

This paper studies optimal control problems with state constraints by imposing structural assumptions on the constraint domain coupled with a tangential restriction with the dynamics. These assumptions replace pointing or controllability assumptions that are common in the literature, and provide a framework under which feasible boundary trajectories can be analyzed directly. The value functions associated with the state constrained Mayer and minimal time problems are characterized as solutions to a pair of Hamilton-Jacobi inequalities with appropriate boundary conditions. The novel feature of these inequalities lies in the choice of the Hamiltonian.

Multiobjective Investment Policy for a Nonlinear Stochastic Financial System: A Fuzzy Approach

The financial market always suffers from continuous and discontinuous (jump) changes and can be regarded as a nonlinear stochastic jump diffusion system. Most investors expect their investment policies to be not only higher benefits but also lower risk as a multiobjective optimization problem (MOP). In this study, a multiobjective $H_{2}/H_{\infty }$ fuzzy investment is proposed for nonlinear stochastic jump diffusion financial systems to achieve the desired target with minimum investment cost and risk in Pareto optimal sense, simultaneously. The Takagi–Sugeno (T–S) fuzzy model is used to approximate the nonlinear stochastic jump diffusion financial system to simplify the multiobjective $H_{2}/H_{\infty }$ investment policy design procedure. By the help of the T–S fuzzy model, the multiobjective $H_{2}/H_{\infty }$ fuzzy investment policy problem of nonlinear stochastic financial system can be transformed to a linear-matrix-inequality-constrained (LMI-constrained) MOP to avoid solving the annoying Hamilton–Jacobi inequalities. Because the LMI-constrained MOP is not easy to directly calculate its Pareto optimal solutions, an indirect method is proposed to solve this MOP for the multiobjective $H_{2}/H_{\infty }$ fuzzy investment policy design of nonlinear stochastic jump diffusion financial systems. An LMI-constrained multiobjective evolution algorithm (LMI-constrained MOEA) is also developed to efficiently solve the Pareto optimal solutions of the LMI-constrained MOP for the multiobjective $H_{2}/H_{\infty }$ fuzzy investment policy design of nonlinear stochastic jump diffusion financial systems. When the Pareto optimal regulation solutions are solved by the proposed LMI-constrained MOEA, investors can select one investment policy to achieve their desired target with minimum investment cost and risk according to his/her own preference.

A note on guaranteed cost control for nonlinear stochastic systems with input saturation and mixed time‐delays

SummaryThisIn the system under investigation, the control input is subject to the saturation constraint, and both discrete and distributed time‐delays are taken into consideration. The objective of the addressed problem is to design a state feedback controller such that the resulting closed‐loop system is asymptotically stable in probability and an upper bound is guaranteed on a prespecified quadratic cost function. Sufficient conditions are established for the existence of the desired guaranteed cost control strategy in terms of the solvability of certain Hamilton–Jacobi inequalities. Simulation results demonstrate the correctness and applicability of the proposed control scheme. Copyright © 2017 John Wiley & Sons, Ltd.

H ∞ Filtering for General Delayed Nonlinear Stochastic Systems with Markov Jumps

This paper is concerned with the robust \(H_{\infty }\) filtering problem of nonlinear stochastic delayed systems with Markov jumps, where time delay exists in state equation, measurement equation and controlled output. Firstly, by introducing a nonlinear stochastic bounded real lemma, we present the \(H_{\infty }\) filter design via solving a set of Hamilton-Jacobi inequalities (HJIs). Secondly, by using fuzzy approach, the \(H_{\infty }\) filter can be designed via solving a set of Linear Matrix inequalities instead of HJIs. Finally, two numerical examples are given to show the effectiveness of our results.

Asymptotic behavior of a nonlocal KPP equation with an almost periodic nonlinearity

We consider a space-inhomogeneous KPP equation with a nonlocal diffusion and an almost-periodic nonlinearity. By employing and adapting the theory of homogenization, we show that solutions of this equation asymptotically converge to its stationary states in regions of space separated by a front that is determined by a Hamilton-Jacobi variational inequality.

Robust H 2/H ∞ global linearization filter design for nonlinear stochastic time-varying delay systems

One can design a robust H ∞ filter for a general nonlinear stochastic system with external disturbance by solving a second-order nonlinear stochastic partial Hamilton-Jacobi inequality (HJI), which is difficult to be solved. In this paper, the robust mixed H 2/H ∞ globally linearized filter design problem is investigated for a general nonlinear stochastic time-varying delay system with external disturbance, where the state is governed by a stochastic Ito-type equation. Based on a globally linearized model, a stochastic bounded real lemma is established by the Lyapunov–Krasovskii functional theory, and the robust H ∞ globally linearized filter is designed by solving the simultaneous linear matrix inequalities instead of solving an HJI. For a given attenuation level, the H 2 globally linearized filtering problem with the worst case disturbance in the H ∞ filter case is known as the mixed H 2/H ∞ globally linearized filtering problem, which can be formulated as a linear programming problem with simultaneous LMI constraints. Therefore, this method is applicable for state estimation in nonlinear stochastic time-varying delay systems with unknown exogenous disturbance when state variables are unavailable. A simulation example is provided to illustrate the effectiveness of the proposed method.

The Minimum Time Function for the Controlled Moreau's Sweeping Process

Let $C(t)$, $t\geq0$ be a Lipschitz set-valued map with closed and (mildly non-)convex values and $f(t, x,u)$ be a map, Lipschitz continuous w.r.t. $x$. We consider the problem of reaching a target $S$ within the graph of $C$ subject to the differential inclusion \[ (\star)\qquad \dot{x} \in -N_{C(t)}(x) + G(t,x) \] starting from $x_{0}\in C(t_{0})$ in the minimum time $T(t_{0},x_{0})$. The dynamics $(\star)$ is called a perturbed sweeping (or Moreau) process. We give sufficient conditions for $T$ to be finite and continuous and characterize $T$ through Hamilton-Jacobi inequalities. Crucial tools for our approach are characterizations of weak and strong flow invariance of a set $S$ subject to $(\star)$. Due to the presence of the normal cone $N_{C(t)}(x)$, the right hand side of $(\star)$ contains implicitly the state constraint $x(t)\in C(t)$ and is not Lipschitz continuous with respect to $x$.

NonlinearL2-Gain Analysis of Hybrid Systems in the Presence of Sliding Modes and Impacts

TheL2-gain analysis is extended towards hybrid mechanical systems, operating under unilateral constraints and admitting both sliding modes and collision phenomena. Sufficient conditions for such a system to be internally asymptotically stable and to possessL2-gain less than ana priorigiven disturbance attenuation level are derived in terms of two independent inequalities which are imposed on continuous-time dynamics and on discrete disturbance factor that occurs at the collision time instants. The former inequality may be viewed as the Hamilton-Jacobi inequality for discontinuous vector fields, and it is separately specified beyond and along sliding modes, which occur in the system between collisions. Thus interpreted, the former inequality should impose the desired integral input-to-state stability (iISS) property on the Filippov dynamics between collisions whereas the latter inequality is invoked to ensure that the impact dynamics (when the state trajectory hits the unilateral constraint) are input-to-state stable (ISS). These inequalities, being coupled together, form the constructive procedure, effectiveness of which is supported by the numerical study made for an impacting double integrator, driven by a sliding mode controller. Desired disturbance attenuation level is shown to satisfactorily be achieved under external disturbances during the collision-free phase and in the presence of uncertainties in the transition phase.

Robust Scheduling Filter Design for a Class of Nonlinear Stochastic Poisson Signal Systems

Nonlinear dynamic systems may suffer from both continuous Wiener noise and discontinuous Poisson noise. This paper studies a robust filter design for a class of nonlinear stochastic Poisson signal systems with external disturbances. Currently, there are no good filtering design methods to treat the discontinuous Poisson noise filter problem. Based on the Ito-Levy formula, a robust $H_{\infty}$ filter design is proposed for nonlinear stochastic Poisson signal systems by solving a Hamilton-Jacobi inequality (HJI) for the robust filter design of a nonlinear stochastic Poisson signal system. However, such an HJI is difficult to solve. Hence, this study employs the Polytopic Linear Model(PLM) scheduling scheme to approximate this HJI by a set of linear matrix inequalities(LMIs) so that the $H_{\infty}$ robust filter design problem for a nonlinear stochastic Poisson signal system can be simplified. The optimal $H_{\infty}$ robust scheduling filter design problem for the nonlinear stochastic Poisson signal system is also discussed. Since the PLM interpolation method transforms the HJI-constrained optimization problem into an LMI-constrained optimization problem which can be efficiently solved using the LMI toolbox in MATLAB, an optimal filtering level $\gamma^{\ast}$ (the minimum value of the filtering error-to-noise ratio in a mean square sense) can be achieved. Finally, a simulation example of a robust trajectory estimation problem in an anti-tactical ballistic missile radar system with discontinuous random maneuvering jets is given to illustrate the design procedure and to confirm the estimation performance of the proposed $H_{\infty}$ robust scheduling filter.

Finite-time H∞ filtering for non-linear stochastic systems

This paper describes the robust H∞ filtering analysis and the synthesis of general non-linear stochastic systems with finite settling time. We assume that the system dynamic is modelled by Itô-type stochastic differential equations of which the state and the measurement are corrupted by state-dependent noises and exogenous disturbances. A sufficient condition for non-linear stochastic systems to have the finite-time H∞ performance with gain less than or equal to a prescribed positive number is established in terms of a certain Hamilton–Jacobi inequality. Based on this result, the existence of a finite-time H∞ filter is given for the general non-linear stochastic system by a second-order non-linear partial differential inequality, and the filter can be obtained by solving this inequality. The effectiveness of the obtained result is illustrated by a numerical example.