Deep reinforcement learning based finite-horizon optimal tracking control for nonlinear system

Approximate dynamic programming (ADP) aims to obtain an approximate numerical solution to the discrete-time Hamilton-Jacobi-Bellman (HJB) equation. Heuristic dynamic programming (HDP) is a two-stage iterative scheme of ADP by separating the HJB equation into two equations, one for the value function and another for the policy function, which are referred to as the critic and the actor, respectively. Previous ADP implementations have been limited by the choice of function approximator, which requires significant prior domain knowledge or a large number of parameters to be fitted. However, recent advances in deep learning brought by the computer science community enable the use of deep neural networks (DNN) to approximate high-dimensional nonlinear functions without prior domain knowledge. Motivated by this, we examine the potential of DNNs as function approximators of the critic and the actor. In contrast to the infinite-horizon optimal control problem, the critic and the actor of the finite horizon optimal control (FHOC) problem are time-varying functions and have to satisfy a boundary condition. DNN structure and training algorithm suitable for FHOC are presented. Illustrative examples are provided to demonstrate the validity of the proposed method.

This paper investigates finite-horizon optimal consensus control problem for unknown multiagent systems with state delays. It is well known that optimal consensus control is the solutions to the coupled Hamilton-Jacobi-Bellman (HJB) equations. An off-policy reinforcement learning (RL) algorithm is developed to learn the two-stage optimal consensus solutions to the coupled time-varying HJB equations using the measurable state data instead of the knowledge of the state-delayed system dynamics. Subsequently, for each agent, a single critic neural network (NN) is utilized to approximate the time-varying cost function and help to calculate optimal consensus control policy. Based on the method of weighted residuals, adaptive weight update laws for the critic NNs are proposed. Finally, the simulation results are provided to illustrate the effectiveness of the proposed off-policy RL method.

The finite-horizon optimal control problem with input constraints consists in controlling the state of a dynamical system over a finite time interval (possibly unknown) minimising a cost functional, while satisfying hard constraints on the input. In this framework, the minimum-time optimal control problem and some related problems are of interest for both theory and applications. For linear systems, the solution of the problem often relies upon the use of bang-bang control signals. For nonlinear systems, the “shape” of the optimal input is in general not known. The control input can be found solving a Hamilton–Jacobi–Bellman (HJB) partial differential equation (PDE): it typically consists of a combination of bang-bang controls and singular arcs. In this paper, a methodology to approximate the solution of the HJB PDE is proposed. This approximation yields a dynamic state feedback law. The theory is illustrated by means of two examples: the minimum-time optimal control problem for an industrial wastewater treatment plant and the Goddard problem, i.e. a maximum-range optimal control problem.

For the optimal tracking control problem of affine nonlinear systems, a general value iteration algorithm based on adaptive dynamic programming is proposed in this paper. By system transformation, the optimal tracking problem is converted into the optimal regulating problem for the tracking error dynamics. Then, general value iteration algorithm is developed to obtain the optimal control with convergence analysis. Considering the advantages of echo state network, we use three echo state networks with levenberg-Marquardt (LM) adjusting algorithm to approximate the system, the cost function and the control law. A simulation example is given to demonstrate the effectiveness of the presented scheme.

In this paper, finite horizon optimal control and communication co-design problem has been investigated for uncertain networked control system (NCS) with limited transmit power. Firstly, the mathematical relation between network imperfections (e.g. network-induced delays and packet dropouts) and practical wireless communication channel quality has been studied. Then, a novel networked control system model has been represented that includes both physical system dynamics and network dynamics. Next, adopting the emerging neuro dynamics programming (NDP) technique and actor-critic-identifier (ACI) architecture, a novel time-based finite horizon optimal control and communication co-design has been proposed. The proposed scheme cannot only learn the optimal co-design within fixed final time, but also relax the requirement of NCS dynamics and satisfy the practical transmit power constraint. The Lyapunov Theorem is used to validate the effectiveness of proposed scheme. With the proper NN weight update law, proposed scheme can ensure all closed-loop signals and NN weights are uniformly ultimately bounded (UUB). Moreover, simulation results are also included to demonstrate the effectiveness of the proposed scheme.

A neural-network-based finite-horizon optimal tracking control scheme for a class of unknown nonlinear discrete-time systems is developed. First, the tracking control problem is converted into designing a regulator for the tracking error dynamics under the framework of finite-horizon optimal control theory. Then, with convergence analysis in terms of cost function and control law, the iterative adaptive dynamic programming algorithm is introduced to obtain the finite-horizon optimal controller to make the cost function close to its optimal value within an ε-error bound. Furthermore, in order to implement the algorithm via dual heuristic dynamic programming technique, three neural networks are employed to approximate the error dynamics, the cost function, and the control law, respectively. In addition, a numerical example is given to demonstrate the validity of the present approach.

This paper presents a novel approach for developing tracking controllers for nonlinear systems. The approach involves the numerical solution, by Galerkin approximation, of the time-varying Hamilton-Jacobi-Bellman (HJB) equation and results in a nonlinear controller approximating the optimal tracking control law for a specified desired trajectory and cost function. Experimental results are presented for a nonlinear pendulum system.

This paper introduces an optimal tracking controller for robot manipulators with asymmetrically saturated torques and partially - unknown dynamics based on a reinforcement learning method using a neural network. Firstly, the feedforward control inputs are designed based on the backstepping technique to convert the tracking control problem into the optimal tracking control problem. Secondly, a cost function of the system with asymmetrically saturated input is defined, and the constrained Hamilton-Jacobi-Bellman equation is built, which is solved by the online reinforcement learning algorithm using only a single neural network. Then, the asymmetric saturation optimal control rule is determined. Additionally, the concurrent learning technique is used to relax the demand for the persistence of excitation conditions. The built algorithm ensures that the closed-loop system is asymptotically stable, the approximation error is uniformly ultimately bounded (UUB), and the cost function converges to the near-optimal value. Finally, the effectiveness of the proposed algorithm is shown through comparative simulations.

In this paper, the output feedback based finite-horizon near optimal regulation of nonlinear affine discrete-time systems with unknown system dynamics is considered by using neural networks (NNs) to approximate Hamilton-Jacobi-Bellman (HJB) equation solution. First, a NN-based Luenberger observer is proposed to reconstruct both the system states and the control coefficient matrix. Next, reinforcement learning methodology with actor-critic structure is utilized to approximate the time-varying solution, referred to as the value function, of the HJB equation by using a NN. To properly satisfy the terminal constraint, a new error term is defined and incorporated in the NN update law so that the terminal constraint error is also minimized over time. The NN with constant weights and time-dependent activation function is employed to approximate the time-varying value function which is subsequently utilized to generate the finite-horizon near optimal control policy due to NN reconstruction errors. The proposed scheme functions in a forward-in-time manner without offline training phase. Lyapunov analysis is used to investigate the stability of the overall closed-loop system. Simulation results are given to show the effectiveness and feasibility of the proposed method.

Deep neural networks (DNNs) are constructed by considering highly complicated configurations including network structure and several tuning parameters (number of hidden states and learning rate in each layer), which greatly affect the performance of speech processing applications. To reach optimal performance in such systems, deep understanding and expertise in DNNs is necessary, which limits the development of DNN systems to skilled experts. To overcome the problem, this paper proposes an efficient optimization strategy for DNN structure and parameters using evolutionary algorithms. The proposed approach parametrizes the DNN structure by a directed acyclic graph, and the DNN structure is represented by a simple binary vector. Genetic algorithm and covariance matrix adaptation evolution strategy efficiently optimize the performance jointly with respect to the above binary vector and the other tuning parameters. Experiments on phoneme recognition and spoken digit detection tasks show the effectiveness of the proposed approach by discovering the appropriate DNN structure automatically.

The problem of resilient optimal bipartite tracking control for heterogeneous multi-agent systems with multiple targets under denial-of-service (DoS) attacks is investigated in this paper. A bipartite tracking mechanism is devised in which the agents track the targets under bipartite consensus control, which becomes non-autonomous. This is owing to DoS attacks, as the agents cannot obtain real-time information on the tracked targets and neighboring agents. Consequently, A target observer with a storage module has been developed for the efficient estimation of agent states and the storage of observed information as historical data. By recalling the historical data of the observed state, a new type of distributed resilient optimal controller is formulated, which can achieve the control objective in the case of communication blockage, while minimizing the performance index function of the system. Numerical simulations are performed to verify the proposed secure control design.

Learning-based optimal cooperative control using edge-based triggering strategy

Q-learning solution for optimal consensus control of discrete-time multiagent systems using reinforcement learning

The optimal output tracking control problem for nonlinear large-scale dynamic systems is considered. A series-based approximate approach for designing the optimal controller is proposed with respect to quadratic performance indexes. By using the approach, the high order, coupling, nonlinear two-point boundary value problem is transformed into a sequence of linear decoupling two-point boundary value problems. By intercepting the frontal finite terms of the solution series, we obtained a suboptimal tracking control law. A simulation example illustrates the efficiency of the optimal output tracking control law designed.

Over the last couple of decades the demand for high precision and enhanced performance of physical systems has been steadily increasing. This demand often results in miniaturization and complex design, thus increasing the need for complex nonlinear control methods. Some of the state of the art nonlinear methods are stymied by the requirement of full state information, model and parameter uncertainties, mathematical complexity, etc. For many scenarios it is nearly impossible to consider all the uncertainties during the design of a feedback controller. Additionally, while designing a modelbased nonlinear control there is no standard mechanism to incorporate performance measures. Some of the mentioned issues can be addressed by using online learning. Animals and humans have the ability to share, explore, act or respond, memorize the outcome and repeat the task to achieve a better outcome when they encounter the same or a similar scenario. This is called learning from interaction. One instance of this approach is reinforcement learning (RL). However, RL methods are hindered by the curse of dimensionality, non-interpretability and non-monotonic convergence of the learning algorithms. This can be attributed to the intrinsic characteristics of RL, as it is a modelfree approach and hence no standard mechanism exists to incorporate a priori model information. In this thesis, learning methods are proposed which explicitly use the available system knowledge. This can be seen as a new class of approaches that bridge model-based and model-free methods. These methods can address some of the hurdles mentioned earlier. For example, i) a prior system information can speed up the learning, ii) new control objectives can be achieved which otherwise would be extremely difficult to attain using only model-based methods, iii) physical meaning can be attributed to the learned controller. The developed approach is as follows: themodel of the given physical system is represented in the port-Hamiltonian (PH) form. For the system dynamics in PH form a passivity-based control (PBC) law is formulated, which often requires the solution to a set of partial differential equations (PDEs). Instead of finding an analytical solution, the PBC control law is parameterized using an unknown parameter vector. Then, by using a variation of the standard actor-critic learning algorithm, the unknown parameters can be learned online. Using the principles of stochastic approximation theory, a proof of convergence for the developed method is shown. The proposedmethods are evaluated for the stabilization and regulation ofmechanical and electro-mechanical systems. The simulation and experimental results show comparable learning curves. In the final part of the thesis a novel integral reinforcement learning approach is developed to solve for the optimal output tracking control problem for a set of linear heterogeneous multi-agent systems. Unlike existing methods, this approach does not need to solve either the output regulator equation or requires a p-copy of the leader’s dynamics in the agent’s control law. A detailed numerical evaluation has been conducted to show the feasibility of the developed method.