Murad Abu-Khalaf

Bio: Murad Abu-Khalaf is an academic researcher from Massachusetts Institute of Technology. The author has contributed to research in topics: Optimal control & Nonlinear system. The author has an hindex of 18, co-authored 34 publications receiving 3401 citations. Previous affiliations of Murad Abu-Khalaf include University of Texas at Arlington & MathWorks.

Discrete-Time Nonlinear HJB Solution Using Approximate Dynamic Programming: Convergence Proof

Abstract: Convergence of the value-iteration-based heuristic dynamic programming (HDP) algorithm is proven in the case of general nonlinear systems. That is, it is shown that HDP converges to the optimal control and the optimal value function that solves the Hamilton-Jacobi-Bellman equation appearing in infinite-horizon discrete-time (DT) nonlinear optimal control. It is assumed that, at each iteration, the value and action update equations can be exactly solved. The following two standard neural networks (NN) are used: a critic NN is used to approximate the value function, whereas an action network is used to approximate the optimal control policy. It is stressed that this approach allows the implementation of HDP without knowing the internal dynamics of the system. The exact solution assumption holds for some classes of nonlinear systems and, specifically, in the specific case of the DT linear quadratic regulator (LQR), where the action is linear and the value quadratic in the states and NNs have zero approximation error. It is stressed that, for the LQR, HDP may be implemented without knowing the system A matrix by using two NNs. This fact is not generally appreciated in the folklore of HDP for the DT LQR, where only one critic NN is generally used.

Model-free Q-learning designs for discrete-time zero-sum games with application to H-infinity control

Abstract: In this paper, the optimal strategies for discrete-time linear system quadratic zero-sum games related to the H-infinity optimal control problem are solved in forward time without knowing the system dynamical matrices. The idea is to solve for an action dependent value function Q(x,u,w) of the zero-sum game instead of solving for the state dependent value function V(x) which satisfies a corresponding game algebraic Riccati equation (GARE). Since the state and actions spaces are continuous, two action networks and one critic network are used that are adaptively tuned in forward time using adaptive critic methods. The result is a Q-learning approximate dynamic programming model-free approach that solves the zero-sum game forward in time. It is shown that the critic converges to the game value function and the action networks converge to the Nash equilibrium of the game. Proofs of convergence of the algorithm are shown. It is proven that the algorithm ends up to be a model-free iterative algorithm to solve the (GARE) of the linear quadratic discrete-time zero-sum game. The effectiveness of this method is shown by performing an H-infinity control autopilot design for an F-16 aircraft.

Neurodynamic Programming and Zero-Sum Games for Constrained Control Systems

Abstract: 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, suboptimal Hinfin control, of nonlinear systems affine in input with the control policy having saturation constraints. The result is a closed-form representation, on a prescribed compact set chosen a priori, of the feedback strategies and the value function that solves the associated Hamilton-Jacobi-Isaacs (HJI) equation. The closed-loop stability, L2-gain disturbance attenuation of the neural network saturated control feedback strategy, and uniform convergence results are proven. Finally, this approach is applied to the rotational/translational actuator (RTAC) nonlinear benchmark problem under actuator saturation, offering guaranteed stability and disturbance attenuation.

Reinforcement learning and adaptive dynamic programming for feedback control

Abstract: Living organisms learn by acting on their environment, observing the resulting reward stimulus, and adjusting their actions accordingly to improve the reward. This action-based or reinforcement learning can capture notions of optimal behavior occurring in natural systems. We describe mathematical formulations for reinforcement learning and a practical implementation method known as adaptive dynamic programming. These give us insight into the design of controllers for man-made engineered systems that both learn and exhibit optimal behavior.

Discrete-Time Nonlinear HJB Solution Using Approximate Dynamic Programming: Convergence Proof

Abstract: Convergence of the value-iteration-based heuristic dynamic programming (HDP) algorithm is proven in the case of general nonlinear systems. That is, it is shown that HDP converges to the optimal control and the optimal value function that solves the Hamilton-Jacobi-Bellman equation appearing in infinite-horizon discrete-time (DT) nonlinear optimal control. It is assumed that, at each iteration, the value and action update equations can be exactly solved. The following two standard neural networks (NN) are used: a critic NN is used to approximate the value function, whereas an action network is used to approximate the optimal control policy. It is stressed that this approach allows the implementation of HDP without knowing the internal dynamics of the system. The exact solution assumption holds for some classes of nonlinear systems and, specifically, in the specific case of the DT linear quadratic regulator (LQR), where the action is linear and the value quadratic in the states and NNs have zero approximation error. It is stressed that, for the LQR, HDP may be implemented without knowing the system A matrix by using two NNs. This fact is not generally appreciated in the folklore of HDP for the DT LQR, where only one critic NN is generally used.

Reinforcement Learning and Feedback Control: Using Natural Decision Methods to Design Optimal Adaptive Controllers

Abstract: This article describes the use of principles of reinforcement learning to design feedback controllers for discrete- and continuous-time dynamical systems that combine features of adaptive control and optimal control. Adaptive control [1], [2] and optimal control [3] represent different philosophies for designing feedback controllers. Optimal controllers are normally designed of ine by solving Hamilton JacobiBellman (HJB) equations, for example, the Riccati equation, using complete knowledge of the system dynamics. Determining optimal control policies for nonlinear systems requires the offline solution of nonlinear HJB equations, which are often difficult or impossible to solve. By contrast, adaptive controllers learn online to control unknown systems using data measured in real time along the system trajectories. Adaptive controllers are not usually designed to be optimal in the sense of minimizing user-prescribed performance functions. Indirect adaptive controllers use system identification techniques to first identify the system parameters and then use the obtained model to solve optimal design equations [1]. Adaptive controllers may satisfy certain inverse optimality conditions [4].