Neural-network-based online HJB solution for optimal robust guaranteed cost control of continuous-time uncertain nonlinear systems.

In this paper, an infinite horizon optimal robust guaranteed cost control scheme of a class of continuous-time uncertain nonlinear systems is established based on adaptive dynamic programming. The main idea lies in that the optimal robust guaranteed cost control problem can be transformed into an optimal control problem. Actually, the optimal cost function of the nominal system is nothing but the optimal guaranteed cost of the original uncertain system. A critic neural network is constructed to help solving the modified Hamilton-Jacobi-Bellman equation corresponding to the nominal system. Then, an additional stabilizing term is introduced to reinforce the updating process of the weight vector and reduce the requirement of an initial stabilizing control. An example is provided to illustrate the effectiveness of the present control approach.

1 A Survey on Computational Optimal Control.- Issues in the Direct Transcription of Optimal Control Problems to Sparse Nonlinear Programs.- Optimization in Control of Robots.- Large-scale SQP Methods and their Application in Trajectory Optimization.- Solving Optimal Control and Pursuit-Evasion Game Problems of High Complexity.- 2 Theoretical Aspects of Optimal Control and Nonlinear Programming.- Continuation Methods In Boundary Value Problems.- Second Order Optimality Conditions for Singular Extremals.- Synthesis of Adaptive Optimal Controls for Linear Dynamic Systems.- Control Applications of Reduced SQP Methods.- Time Optimal Control of Mechanical Systems.- 3 Algorithms for Optimal Control Calculations.- Second Order Algorithm for Time Optimal Control of a Linear System.- An SQP-type Solution Method for Constrained Discrete-Time Optimal Control Problems.- Numerical Methods for Solving Differential Games, Prospective Applications to Technical Problems.- Construction of the Optimal Feedback Controller for Constrained Optimal Control Problems with Unknown Disturbances.- Repetitive Optimization for Predictive Control of Dynamic Systems under Uncertainty.- Optimal Control of Multistage Systems Described by High-Index Differential-Algebraic Equations.- A New Class of a High Order Interior Point Method for the Solution of Convex Semiinfinite Optimization Problems.- A Structured Interior Point SQP Method for Nonlinear Optimal Control Problems.- 4 Software for Optimal Control Calculations.- Automated Approach for Optimizing Dynamic Systems.- ANDECS: A Computation Environment for Control Applications of Optimization.- Application of Automatic Differentiation to Optimal Control Problems.- OCCAL: A mixed symbolic-numeric Optimal Control CALculator.- 5 Applications of Optimal Control.- A Robotic Satellite with Simplified Design.- Nonlinear Control under Constraints of a Biological System.- An Object-Oriented Approach to Optimally Describe and Specify a SCADA System Applied to a Power Network.- Near-Optimal Flight Trajectories Generated by Neural Networks.- Performance of a Feedback Method with Respect to Changes in the Air-Density during the Ascent of a Two-Stage-To-Orbit Vehicle.- Linear Optimal Control for Reentry Flight.- Steady-State Modelling of Turbine Engine with Controllers.- Shortest Paths for Satellite Mounted Robot Manipulators.- Optimal Control of the Industrial Robot Manutec r3.

This article addresses the optimal tracking control problem with prescribed performance for uncertain nonlinear systems subject to input constraint and unknown disturbances. First, a fixed‐time monotonic convergence function is introduced to restrain tracking error, and a nonlinear mapping technique is employed to transform the constrained error into an unconstrained variable, then the fixed‐time output tracking issue is boiled down to the boundedness problem of the transformed variable. With the aid of a nonquadratic cost function, the input constraint is encoded into the optimization problem. To solve the unknown disturbances, an auxiliary system and an auxiliary disturbance policy are constructed, and the optimal control problem is formulated as a two‐player zero‐sum game. Moreover, a Hamilton–Jacobi–Isaacs (HJI) equation associated with this nonquadratic zero‐sum game is established to give the optimal control and the worst‐case disturbance policy solution. Subsequently, to avoid using knowledge of the system dynamics, three neural network approximators, namely, actor, critic, and disturbance, which are tuned online and simultaneously for approximating the solution of HJI, are constructed based on the integral reinforcement learning algorithm. Theoretical analysis shows that the reconstructed error system states and the weight estimation errors are semi‐globally uniformly ultimately bounded. Finally, the simulation study further tests the availability of the proposed control strategy.

In general, optimal control problems rely on numerically rather than analytically solving methods, due to their nonlinearities. The direct method, one of the numerically solving methods, is mainly to transform the optimal control problem into a nonlinear optimization problem with finite dimensions, via discretizing the objective functional and the forced dynamical equations directly. However, in the procedure of the direct method, the classical discretizations of the forced equations will reduce or affect the accuracy of the resulting optimization problem as well as the discrete optimal control. In view of this fact, more accurate and efficient numerical algorithms should be employed to approximate the forced dynamical equations. As verified, the discrete variational difference schemes for forced Birkhoffian systems exhibit excellent numerical behaviors in terms of high accuracy, long-time stability and precise energy prediction. Thus, the forced dynamical equations in optimal control problems, after being represented as forced Birkhoffian equations, can be discretized according to the discrete variational difference schemes for forced Birkhoffian systems. Compared with the method of employing traditional difference schemes to discretize the forced dynamical equations, this way yields faithful nonlinear optimization problems and consequently gives accurate and efficient discrete optimal control. Subsequently, in the paper we are to apply the proposed method of numerically solving optimal control problems to the rendezvous and docking problem of spacecrafts. First, we make a reasonable simplification, i.e., the rendezvous and docking process of two spacecrafts is reduced to the problem of optimally transferring the chaser spacecraft with a continuously acting force from one circular orbit around the Earth to another one. During this transfer, the goal is to minimize the control effort. Second, the dynamical equations of the chaser spacecraft are represented as the form of the forced Birkhoffian equation. Then in this case, the discrete variational difference scheme for forced Birkhoffian system can be employed to discretize the chaser spacecraft's equations of motion. With further discretizing the control effort and the boundary conditions, the resulting nonlinear optimization problem is obtained. Finally, the optimization problem is solved directly by the nonlinear programming method and then the discrete optimal control is achieved. The obtained optimal control is efficient enough to realize the rendezvous and docking process, even though it is only an approximation of the continuous one. Simulation results fully verify the efficiency of the proposed method for numerically solving optimal control problems, if the fact that the time step is chosen to be very large to limit the dimension of the optimization problem is noted.

An efficient method is introduced for solving optimal control and optimal parameter selection problems of nonlinear differential-algebraic systems involving general constraints. These infinite dimensional problems are converted into the finite dimensional optimization problems by using the control parametrization technique. Moreover, the transformed time locations for the control variables are included in such problems. As a result, optimal control problems become acausal optimal parameter selection problems. A modified version of differential evolution is introduced to solve these problems. The integration of the penalty functions is used to ensure the solution onto the feasible domain of the problems. Two industrial systems, a chemical reactor and a robotics arm, are used to illustrate the robustness and effectiveness of the method. The control effects can be improved by selecting the proper time locations as observed from the simulated results.

In this paper, the problem of the robust optimal control (ROC) for a class of nonlinear systems with uncertainties and external disturbances is studied. For this purpose, the State Dependent Riccati Equation (SDRE) is considered as a solution of the ROC. The proposed method is to transform a robust control problem into an optimal control problem, where the uncertainties and disturbances are reflected in the performance index. Hence, the performance index is selected as an energy function and using the proposed constraints in this paper the Lyapunov stability for this class of nonlinear systems is proved and the robust optimal controller is designed. In order to illustrate the high capability of the ROC method in optimal control of complex systems such as Hyper-chaotic system, the time response of state variables in the proposed method and the Global Robust Optimal Sliding Mode Control (GROSMC) method are compared together. The simulation results show that the time response of state variables in the ROC method is more sufficient than the GROSMC method.

This article presents a unified approach to simultaneous solution of optimization and optimal control problems in batch distillation, operating under different modes of operation: variable, constant, or optimal reflux. The simplified, computationally efficient short‐cut method and a novel algorithm to solve the optimal control problems in batch distillation is the basis of this unified approach. The short‐cut method identifies the feasible region of operation essential for optimization and optimal control problems, and provides analytical partial derivatives of the model parameters crucial to the solution.The new algorithm for the solution of optimal combination of the maximum principle and NLP optimization techniques. It circumvents the problems associated with the maximum principle approach (iterative solution of a two‐point boundary value problem, undounded control variables, and inability to handle the simultaneous optimization and optimal control problem), and the coupled ODE discretization‐NLP optimization scheme for nonlinear models (higher system nonlinearities, multiplicity of solutions, sensitivity of convergence to initial guesses). This algorithm reduces the dimensionality of the problem, and the nature of the algorithm allows a common platform to optimal solutions of different operating conditions. This article also shows that different categories of the optimal control problems in batch distillation essentially involve the solution of the maximum distillate problem.

In this paper, an iterative adaptive dynamic programming (ADP) algorithm is developed to solve the optimal cooperative control problems for residential multi-battery systems. To avoid solving high-dimensional optimal control problems, we first constrain all the batteries at their worst performance, which transforms the multi-input optimal control problem into a single-input one. Based on the worst-performance optimal control law, the optimal cooperative control law for the residential multi-battery systems is obtained, where in each iteration, only a single-input optimization problem is implemented. Finally, numerical results are given to illustrate the performance of the developed algorithm.

This paper builds up two equivalence theorems for different kinds of optimal control problems of internally controlled Schrodinger equations. The first one is an equivalence of the minimal norm and the minimal time control problems where the target sets are the origin of the state space. (The minimal time control problems are also called the first type of optimal time control problems.) The second one is an equivalence of three optimal control problems: optimal target control problems, optimal norm control problems, and the second type of optimal time control problems.

The paper presents an output feedback optimal guaranteed cost control result for a new class of uncertain linear systems. The cost function considered is a quadratic cost function defined over a finite time interval. The new uncertainty class introduced in the paper involves structured uncertainties which are required to satisfy a certain averaged integral quadratic constraint. The introduction of this new structured uncertainty description enables us to solve a previously unsolved problem of optimal guaranteed cost control via output feedback for an uncertain system with structured uncertainties. The solution is obtained by solving a pair of parametrized Riccati differential equations of the game type. >

Solutions to optimal control problems obtained via direct methods are usually of low resolution. As the desired resolution becomes higher, the number of parameters in the resulting parameter optimization problem increases nonlinearly. A concatenated approach to trajectory optimization previously introduced by the authors to obtain high resolution optimal solutions with low storage requirements is applicable only to a specific class of problems, such as minimum time and minimum energy optimal control problems. The hodograph ballooning approach introduced in this paper generalizes the concatenated approach to the most general optimal control problem. Introduction The development of powerful nonlinear programming software [1, 2] has made direct approaches to trajectory optimization the method of choice for most practical applications [3-5]. The discretization of an optimal control problem to a finite dimensional parameter optimization problem avoids many of the troublesome pathologies that are typical for the optimization over infinite dimensional function spaces [6-8]. Furthermore, direct approaches usually do not. require guessing the switching structure, and the dynamical system is devoid of the nonintuitive costate variables. However, to achieve good resolution it is usually necessary to refine some type of discretization mesh, and, ultimately, to use a large number of design parameters. For typical gradient search methods, this increase in the dimension of the optimization problem has nearly always a negative effect on the convergence rate and the robustness. Additionally, CPU time and RAM requirements for each iteration increase with the second power of the number of parameters (third power for non-sparse solvers) so that direct optimization approaches can become quite expensive. In [9], a method was introduced that achieves excellent resolution without solving high-dimensional nonlinear programming problems. This was achieved by lo'Senior Engineer, Member AIAA. t Supervising Engineer, Member AIAA. 117 Research Drive, Hampton, VA 23666. Copyright ©1996 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved cally optimizing the trajectory on an iterative sequence of short, partly overlapping subarcs. However, this approach is not applicable to the most general problem formulation. Loosely sneaking, it was required in [9] that the cost state does not appear explicitly in the right-hand side of the equations of motion. The present paper introduces a method through which the approach introduced in [9] becomes applicable to the most general problem formulation. Application of this method, however, requires a priori knowledge of the optimal cost. In practice it is observed that the optimal cost can be obtained with very high precision with reasonably low order discretizations. Once a good guess for the optimal cost is obtained presented in this paper can be applied to refine the trajectory, or, if the optimal cost value is not known a priori with sufficient precision, it can be applied iteratively with the prescribed cost value adjusted in an outer loop. Problem Formulation Let us consider the following simple optimal control problem stated in Mayer form: ^ 4 ( x ( t f ) , t f ) (1 )

Max–Min Optimal Control of Constrained Discrete-time Systems

Optimal control for multistage uncertain random dynamic systems with multiple time delays

This paper addresses the robust guaranteed cost tracking control problem of nonlinear matched uncertain systems using event-based adaptive dynamic programming (ADP). By constructing an augmented system and using a discounted cost function, the guaranteed cost control problem is transformed into an optimal control problem of the nominal augmented system. The ADP framework is utilized to approximate the solution of the Hamilton-Jacobi-Bellman (HJB) equation associated with the optimal control problem. The event-based guaranteed cost is derived, and its relation with the time-based one is discussed. Furthermore, it is proven that the derived event-based controller can make the tracking error uniformly ultimately bounded (UUB). The applicability of the suggested technique is exhibited using a simulation study.

This paper focuses on the event-triggered guaranteed cost control design of nonlinear systems via a self-learning technique. In brief, an event-based guaranteed cost control strategy of nonlinear systems subjects to matched uncertainties is developed, thereby balancing the performance of guaranteed cost and the actuality of limited communication resource. The original control design is transformed into an optimal control problem with an event-based mechanism, where the relationship of guaranteed cost performance compared to the time-based formulation is discussed. A critic neural network is constructed for implementing the event-based optimal control design with stability guarantee. Simulation experiments are carried out to verify the theoretical results in detail.