# Policy Iterations on the Hamilton–Jacobi–Isaacs Equation for $H_{\infty}$ State Feedback Control With Input Saturation

@article{AbuKhalaf2006PolicyIO,
title={Policy Iterations on the Hamilton–Jacobi–Isaacs Equation for \$H\_\{\infty\}\$ State Feedback Control With Input Saturation},
author={Murad Abu-Khalaf and Frank L. Lewis and Jie Huang},
journal={IEEE Transactions on Automatic Control},
year={2006},
volume={51},
pages={1989-1995}
}
• Published 2006
• Mathematics, Computer Science
• IEEE Transactions on Automatic Control
An Hinfin suboptimal state feedback controller for constrained input systems is derived using the Hamilton-Jacobi-Isaacs (HJI) equation of a corresponding zero-sum game that uses a special quasi-norm to encode the constraints on the input. The unique saddle point in feedback strategy form is derived. Using policy iterations on both players, the HJI equation is broken into a sequence of differential equations linear in the cost for which closed-form solutions are easier to obtain. Policy… Expand
166 Citations

#### Topics from this paper

Neurodynamic Programming and Zero-Sum Games for Constrained Control Systems
• Mathematics, Computer Science
• IEEE Transactions on Neural Networks
• 2008
In this paper, neural networks are used along with two-player policy iterations to solve for the feedback strategies of a continuous-time zero-sum game that appears in L2-gain optimal control,Expand
Neural Network Based Online Simultaneous Policy Update Algorithm for Solving the HJI Equation in Nonlinear $H_{\infty}$ Control
• Mathematics, Computer Science
• IEEE Transactions on Neural Networks and Learning Systems
• 2012
A neural network (NN)-based online simultaneous policy update algorithm (SPUA) is developed to solve the HJI equation, in which knowledge of internal system dynamics is not required. Expand
Online Adaptive Policy Learning Algorithm for $H_{\infty }$ State Feedback Control of Unknown Affine Nonlinear Discrete-Time Systems
• Computer Science, Medicine
• IEEE Transactions on Cybernetics
• 2014
An online adaptive policy learning algorithm (APLA) based on adaptive dynamic programming (ADP) is proposed for learning in real-time the solution to the Hamilton-Jacobi-Isaacs (HJI) equation, which appears in the H∞ control problem. Expand
Ternary Policy Iteration Algorithm for Nonlinear Robust Control
• Computer Science, Engineering
• ArXiv
• 2020
The proposed ternary policy iteration (TPI) algorithm can converge to the optimal solution for the linear plant, and has high resistance to disturbances for the nonlinear plant. Expand
Model-Free Adaptive Control for Unknown Nonlinear Zero-Sum Differential Game
• Computer Science, Medicine
• IEEE Transactions on Cybernetics
• 2018
The online learning algorithm is proposed based on the GDHP method to solve the Hamilton–Jacobi–Isaacs equation associated with optimal regulation control problem and three neural networks are established to approximate the optimal saddle point feedback control law, the disturbance law, and the performance index. Expand
Zero-sum two-player game theoretic formulation of affine nonlinear discrete-time systems using neural networks
• Mathematics, Computer Science
• The 2010 International Joint Conference on Neural Networks (IJCNN)
• 2010
The nearly optimal solution for discrete-time (DT) affine nonlinear control systems in the presence of partially unknown internal system dynamics and disturbances is considered and an iterative approach to approximate the HJI equation using a neural network is presented. Expand
Stable value iteration for two-player zero-sum game of discrete-time nonlinear systems based on adaptive dynamic programming
• Computer Science, Mathematics
• Neurocomputing
• 2019
A stable value iteration (SVI) algorithm is developed to solve discrete-time two-player zero-sum game (TP-ZSG) for nonlinear systems based on adaptive dynamic programming (ADP) and it is proved that if the iteration number reaches a given number, then the iterative control inputs make the closed-loop system asymptotic stable. Expand
Computationally efficient simultaneous policy update algorithm for nonlinear H∞ state feedback control with Galerkin’s method
• Mathematics
• 2013
SUMMARY The main bottleneck for the application of H∞ control theory on practical nonlinear systems is the need to solve the Hamilton–Jacobi–Isaacs (HJI) equation. The HJI equation is a nonlinearExpand
Nonlinear $$H_\infty$$H∞ control via measurement feedback using neural network
• Mathematics
• 2017
In this paper, we solve the nonlinear $$H_\infty$$H∞ optimal control with output feedback via the neural network (NN)–least squares method for the affine nonlinear system. The approach is based onExpand
Robust Control of Unknown Observable Nonlinear Systems Solved as a Zero-Sum Game
• Computer Science, Mathematics
• IEEE Access
• 2020
An optimal robust control solution for general nonlinear systems with unknown but observable dynamics is advanced here and it is shown that controlling the former implies controlling the latter. Expand

#### References

SHOWING 1-10 OF 44 REFERENCES
Nearly optimal control laws for nonlinear systems with saturating actuators using a neural network HJB approach
• Mathematics, Computer Science
• Autom.
• 2005
It is shown that the constrained optimal control law has the largest region of asymptotic stability (RAS) and the result is a nearly optimal constrained state feedback controller that has been tuned a priori off-line. Expand
Neural network H/sub /spl infin// state feedback control with actuator saturation: the nonlinear benchmark problem
• Mathematics
• 2005
In this paper, we describe a constrained H/sub /spl infin// state feedback controller to stabilize the Rotational/Translational Actuator (RTAC) benchmark problem with L/sub 2/ disturbanceExpand
Role of performance functionals in control laws design
• S. Lyshevski
• Mathematics
• Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148)
• 2001
The research in optimal control has been concentrated on design of control laws in order to optimize the dynamic performance of systems. These optimization problems were explicitly formulated andExpand
Parameterized LMIs in Control Theory
• Mathematics, Computer Science
• SIAM J. Control. Optim.
• 2000
This paper investigates relaxations of parameterized L MI problems into standard LMI problems using techniques relying on directional convexity concepts, and discusses the impact of the proposed techniques in quadratic programming, Lyapunov-based stability and performance analysis, $\mu$ analysis, and linear parameter-varying control. Expand
Successive Galerkin approximation algorithms for nonlinear optimal and robust control
Nonlinear optimal control and nonlinear H infinity control are two of the most significant paradigms in nonlinear systems theory. Unfortunately, these problems require the solution of Hamilton-JacobiExpand
L/sub 2/-gain analysis of nonlinear systems and nonlinear state-feedback H/sub infinity / control
Previously obtained results on L2-gain analysis of smooth nonlinear systems are unified and extended using an approach based on Hamilton-Jacobi equations and inequalities, and their relation toExpand
Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations
• Mathematics
• 1997
Preface.- Basic notations.- Outline of the main ideas on a model problem.- Continuous viscosity solutions of Hamilton-Jacobi equations.- Optimal control problems with continuous value functions:Expand
On parameter-dependent Lyapunov functions for robust stability of linear systems
• Mathematics
• 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601)
• 2004
For a linear system affected by real parametric uncertainty, this paper focuses on robust stability analysis via quadratic-in-the-state Lyapunov functions polynomially dependent on the parameters.Expand
New robust stability and performance conditions based on parameter dependent multipliers
• Mathematics
• Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187)
• 2000
This paper introduces a new robust stability and performance characterization for parameter dependent systems. The main idea is to jointly use parameter dependent Lyapunov functions and parameterExpand
Disturbance attenuation and H/sub infinity /-control via measurement feedback in nonlinear systems
• Mathematics
• 1992
A solution to the problem of disturbance attenuation via measurement feedback with internal stability is presented for an affine nonlinear system. It is shown that the concept of disturbanceExpand