Universal Singular Optimal Control: Affine Systems

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.

In this paper, 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. The approach is based on successive approximate solution of the Hamilton-Jacobi-Isaacs (HJI) equation, which appears in optimal control. Successive approximation approach for updating control and disturbance inputs for DT nonlinear affine systems are proposed. Moreover, sufficient conditions for the convergence of the approximate HJI solution to the saddle point are derived, and an iterative approach to approximate the HJI equation using a neural network (NN) is presented. Then, the requirement of full knowledge of the internal dynamics of the nonlinear DT system is relaxed by using a second NN online approximator. The result is a closed-loop optimal NN controller via offline learning. A numerical example is provided illustrating the effectiveness of the approach.

In this paper, we present an optimal control approach using Linear Matrix Inequalities (LMIs) for trajectory tracking control of a three-wheeled omnidirectional mobile robot in the presence of external disturbances on the robot's actuators and noise in the robot's sensor measurements. First, a state-space representation of the omnidirectional robot dynamics is derived using a point-mass dynamic model. Then, we propose an LMI-based full-state feedback H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -optimal controller for the tracking problem. The robot's tracking performance with the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -optimal controller is compared to its performance with a classical full-state feedback tracking controller in simulations with circular and bowtie-shaped reference trajectories. In order to evaluate our proposed controller in practice, we also implement the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -optimal and classical controllers for these reference trajectories on a three-wheeled omnidirectional robot. The H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -optimal controller guarantees stabilization of the robot motion and attenuates the effects of frictional disturbances and measurement noise on the robot's tracking performance. Using the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -optimal controller, the robot is able to track the reference trajectories with up to a 47.8% and 45.8% decrease in the maximum pose and twist errors, respectively, over a full cycle of the trajectory compared to the classical controller. The simulation and experimental results show that our LMI-based H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -optimal controller is robust to undesired effects of disturbances and noise on the dynamic behavior of the robot during trajectory tracking and can outperform the classical controller in attenuating their effects.

This paper studies the problem of optimal parallel tracking control for continuous-time general nonlinear systems. Unlike existing optimal state feedback control, the control input of the optimal parallel control is introduced into the feedback system. However, due to the introduction of control input into the feedback system, the optimal state feedback control methods can not be applied directly. To address this problem, an augmented system and an augmented performance index function are proposed firstly. Thus, the general nonlinear system is transformed into an affine nonlinear system. The difference between the optimal parallel control and the optimal state feedback control is analyzed theoretically. It is proven that the optimal parallel control with the augmented performance index function can be seen as the suboptimal state feedback control with the traditional performance index function. Moreover, an adaptive dynamic programming ( ADP ) technique is utilized to implement the optimal parallel tracking control using a critic neural network ( NN ) to approximate the value function online. The stability analysis of the closed-loop system is performed using the Lyapunov theory, and the tracking error and NN weights errors are uniformly ultimately bounded ( UUB ) . Also, the optimal parallel controller guarantees the continuity of the control input under the circumstance that there are finite jump discontinuities in the reference signals. Finally, the effectiveness of the developed optimal parallel control method is verified in two cases.

In this study, a finite‐time online optimal controller was designed for a nonlinear wheeled mobile robotic system (WMRS) with inequality constraints, based on reinforcement learning (RL) neural networks. In addition, an extended cost function, obtained by introducing a penalty function to the original long‐time cost function, was proposed to deal with the optimal control problem of the system with inequality constraints. A novel Hamilton‐Jacobi‐Bellman (HJB) equation containing the constraint conditions was defined to determine the optimal control input. Furthermore, two neural networks (NNs), a critic and an actor NN, were established to approximate the extended cost function and the optimal control input, respectively. The adaptation laws of the critic and actor NN were obtained with the gradient descent method. The semi‐global practical finite‐time stability (SGPFS) was proved using Lyapunov's stability theory. The tracking error converges to a small region near zero within the constraints in a finite period. Finally, the effectiveness of the proposed optimal controller was verified by a simulation based on a practical wheeled mobile robot model.

Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria

In this paper, near optimal tracking of a class of nonlinear systems is addressed. Adaptive (approximate) dynamic programming approach is used to calculate the optimal control in closed form. ADP (Adaptive (approximate) dynamic programming) has been widely used to resolve optimal regulation and tracking problems of nonlinear control systems. Despite advances in the so called supervised and unsupervised ADP techniques for optimal tracking, they have a main draw back. That is, the optimal controller needs to be recalculated for every particular reference trajectory. The main goal of this work is to address this issue for a class of nonlinear systems. Finally, this approach is applied on a Delta robot and the performance of the method is analyzed experimentally.

In this paper, we study the optimal control problem for Takagi–Sugeno (T-S) fuzzy systems with disturbances. Due to the presence of disturbances in the T-S fuzzy systems, a fuzzy optimal state feedback control approach is presented by employing a two-player zero-sum game. Since the analytical optimal control solutions and the worst-case disturbance policies of T-S fuzzy systems can be boiled down to solving the algebraic Riccati equations (AREs), which are difficult to be obtained directly, a model-free policy iteration (PI) learning algorithm is proposed to obtain their approximation solutions. It is proved that the developed fuzzy optimal state feedback controller can ensure the fuzzy systems to be asymptotically stable and satisfy the disturbance attenuation condition simultaneously. Also, the designed PI learning algorithm can converge to their optimal solutions. Finally, we apply the developed fuzzy optimal state feedback control method to the truck-trailer system and the simulation results demonstrated the effectiveness of the developed scheme.

Optimal yaw moment control law for improved vehicle handling

Design of an intelligent optimal neural network-based tracking controller for nonholonomic mobile robot systems

SummaryThis paper proposes an optimal control method of an unmanned aerial helicopter (UAH) with unknown disturbances. Solving the Hamilton‐Jacobi‐Bellman (HJB) equation is considered as the common approach to design an optimal controller under a meaningful cost function when facing the nonlinear optimal control problem. However, the HJB equation is hard to solve even for a simple problem. The inverse optimal control method that avoids the difficulties of solving the HJB equation has been adopted. In this inverse optimal control approach, a stabilizing optimal control law and a particular cost function that are obtained by a control Lyapunov function are required. An integrator backstepping method is used in designing the optimal control law of the UAH. Furthermore, a disturbance‐observer–based control (DOBC) approach has been adopted in the optimal control law for dealing with the unknown disturbances of the UAH system. Simulation results have been given to certify the stability of the nonlinear UAH system and the validity of this developed control method.

Optimal control of interconnected systems with time-correlated noises: Application to vehicle platoon

This paper presents a data-driven method for designing optimal controllers and robust controllers for unknown nonlinear systems. Mathematical models for the realization of the control are difficult to develop owing to a lack of knowledge regarding such systems. The proposed multidisciplinary method, based on optimal control theory and machine learning with kernel functions, facilitates designing appropriate controllers using a data set. Kernel-based system models are useful for representing nonlinear systems. An optimal and an H-infinity controller can be designed by solving Hamilton-Jacobi (HJ) equations, which unfortunately, are difficult to solve owing to the nonlinearity and complexity of the kernel-based models. The objective of this study consists of overcoming two challenges. The first challenge is to derive exact solutions to the HJ equations for a class of kernel-based system models. A key technique in overcoming this challenge is to reduce the HJ equations to easily solvable algebraic matrix equations, from which optimal and H-infinity controllers are designed. The second challenge is to control an unknown system using the obtained controllers, wherein the system is identified as a kernel-based model. Additionally, this study analyzes probabilistic stability of the feedback system with the proposed controllers. Numerical simulations demonstrate control performances of both the derived optimal and H-infinity controllers and stability of the feedback system.

We present a new design methodology for the optimal control of nonlinear systems described by the Takagi-Sugeno(TS) fuzzy model. After classifying the TS fuzzy systems into two families based on how diverse their input matrices are, controller synthesis procedure based on the inverse optimal design approach is given for each of these families. Also, it is shown that to find the parameters of the optimal controller can be represented as an LMI(linear matrix inequality) problem. Designed optimal controllers have the robustness with respect to certain input uncertainties. The proposed method is applied to a regulation problem of a rotating rigid spacecraft for verification.

The optimal control of nonlinear time-varying systems, particularly when the mathematical model of the system is unavailable or inexact, is an interesting and difficult control problem. This paper outlines a methodology for the design of an intelligent controller to perform optimal control of a nonlinear system adaptively, using emerging technologies of fuzzy logic (FL) and artificial neural networks (ANN). FL is utilized to incorporate the available knowledge into the control system, and ANN technology is applied to adaptively provide an optimal control strategy based on some performance criteria. The technique is tested on a system that consists of a DC motor (a linear time-invariant (LTI) system) and a thermal system (a time-varying nonlinear system). Performance criteria such as tracking accuracy, cost, robustness, are considered, and the results are presented in this paper. >