Efficient ST Techniques for Nonlinear Optimal Control Analyses Using a Pseudospectral Framework

Most direct methods solve finite time horizon optimal control problems with nonlinear programming solver. In this paper, we propose a numerical method for solving nonlinear optimal control problem with state and control inequality constraints. This method used quasilinearization technique and Haar wavelet operational matrix to convert the nonlinear optimal control problem into a quadratic programming problem. The linear inequality constraints for trajectories variables are converted to quadratic programming constraint by using Haar wavelet collocation method. The proposed method has been applied to solve Optimal Control of Multi-Item Inventory Model. The accuracy of the states, controls and cost can be improved by increasing the Haar wavelet resolution.

Here, a two-phase algorithm based on integrating particle swarm optimization (PSO) with modified hybrid genetic algorithm (MHGA) is proposed for solving the associated nonlinear programming problem of a nonlinear optimal control problem. In the first phase, PSO starts with a completely random initial swarm of particles, where each of them contains two random matrices in time nodes. After phase 1, to achieve more accurate solutions, the number of time nodes is increased. The values of the associated new control inputs are estimated by linear or spline interpolations using the curves computed in the phase 1. In addition, to maintain the diversity in the population, some additional individuals are added randomly. Next, in the second phase, MHGA, starts by the new population constructed by the above procedure and tries to improve the obtained solutions at the end of phase 1. MHGA combines a GA with a successive quadratic programming, SQP, as a local search. Finally, we implement the proposed algorithm on some well-known nonlinear optimal control problems. The numerical results show that the proposed algorithm can find almost better solution than other proposed algorithms. Keywords: Optimal control problem, Hybrid genetic algorithm, ?particle swarm optimization, Spline interpolation.

Category A certification is required for multi-engine rotorcrafts with over 20,000lbs. Normally Category A performance is proofed with actual flight tests. But federal aviation regulations also state that simulation methods also could be applied as ancillary method but authorization must be needed from airworthiness authorities before used. In this study, firstly formulate the Non-linear Optimal Control Problems (NOCP) to analysis the 4 types of Category A takeoff and landing procedures found in regulations. Pseudo-spectral method and robust Sequential Quadratic Programming (rSQP) methods are used to solve the NOCP formulation. The rotorcraft mathematical model suggested by the Chen is used in this study and the BO-105 rotorcraft is implemented to the model. After then, results are reviewed whether they satisfy the requirements of Category A and compared with Category A takeoff and landing procedures showed at the rotorcraft flight manual of BO-105 rotorcraft. As a result, the proposed methods could be used for the rotorcraft Category A certification programs.

An algorithm, based on quasi-linearization and state variable parametrization by using Chebyshev polynomials, is proposed to solve the nonlinear optimal control problem subject to terminal state constraints and control saturation constraints. By the virtue of the second method of quasi-linearization and state parametrization, the difficult nonlinear optimal control problem is converted into a sequence of standard quadratic programming problems. An example is given to show the effectiveness of the proposed algorithm.

A pseudospectral collocation method for the brachistochrone problem

Owing to the dynamic imaging system, the trajectory model plays a very important role in the geometric processing of high resolution satellite imagery. However, establishing a trajectory model is difficult when only discrete and noisy data are available. In this manuscript, we proposed a general robust trajectory model, the penalized spline model, which could fit trajectory data well and smooth noise. The penalized parameter λ controlling the smooth and fitting accuracy could be estimated by generalized cross-validation. Five other trajectory models, including third-order polynomials, Chebyshev polynomials, linear interpolation, Lagrange interpolation and cubic spline, are compared with the penalized spline model. Both the sophisticated ephemeris and on-board ephemeris are used to compare the orbit models. The penalized spline model could smooth part of noise, and accuracy would decrease as the orbit length increases. The band-to-band misregistration of ZiYuan-3 Dengfeng and Faizabad multispectral images is used to evaluate the proposed method. With the Dengfeng dataset, the third-order polynomials and Chebyshev approximation could not model the oscillation, and introduce misregistration of 0.57 pixels misregistration in across-track direction and 0.33 pixels in along-track direction. With the Faizabad dataset, the linear interpolation, Lagrange interpolation and cubic spline model suffer from noise, introducing larger misregistration than the approximation models. Experimental results suggest the penalized spline model could model the oscillation and smooth noise.

In this paper, a new method is proposed to solve a nonlinear optimal control problem and determine the Dynamic Load-Carrying Capacity (DLCC) of fixed and mobile manipulators in point-to-point motion. Solution methods for designing nonlinear optimal controller in closed loop form are usually based on indirect methods, but the proposed method is a combination of direct and indirect methods. The optimal control law with state feedback form, for nonlinear dynamic systems, is given by the solution to the nonlinear Hamilton–Jacobi–Bellman (HJB) equation. The Galerkin procedure and a nonlinear optimization algorithm are used to solve this equation numerically. Another innovation of this paper is optimal trajectory planning, which is done simultaneously with the controller design procedure. Finally, a new algorithm is developed to find DLCC of manipulators and the related optimal trajectory using proposed method. The validity of the method is demonstrated via simulation and experimental tests for a fixed manipulator and two-link wheeled mobile manipulator named Scout.

A new fractional orthogonal basis and its application in nonlinear delay fractional optimal control problems

A EROSPACE engineering applications greatly stimulated the development of optimal control theory during the 1950s and 1960s, where the objective was to drive the system states in such a way that some defined cost was minimized. This turned out to have very useful applications in the design of regulators (where some steady state is to be maintained) and in tracking control strategies (where some predetermined state trajectory is to be followed). Among such applications was the problem of optimal flight trajectories for aircraft and space vehicles. Linear optimal control theory in particular has been very well documented and widely applied, where the plant that is controlled is assumed linear and the feedback controller is constrained to be linear with respect to its input. However, the availability of powerful low-cost microprocessors has spurred great advantages in the theory and applications of nonlinear control. The competitive era of rapid technological change, particularly in aerospace exploration, now demands stringent accuracy and cost requirements in nonlinear control systems. This has motivated the rapid development of nonlinear control theory for application to challenging, complex, dynamical real-world problems, particularly those that bear major practical significance in aerospace, marine, and defense industries. Infinite-time horizon nonlinear optimal control (ITHNOC) presents a viable option for synthesizing stabilizing controllers for nonlinear systems by making a state-input tradeoff, where the objective is to minimize the cost given by a performance index. The original theory of nonlinear optimal control dates from the 1960s. Various theoretical and practical aspects of the problem have been addressed in the literature over the decades since. In particular, the continuous-time nonlinear deterministic optimal control problem associated with autonomous (time-invariant) nonlinear regulator systems that are affine (linear) in the controls has been studied by many authors. The long-established theory of optimal control offers quite mature and well-documented techniques for solving this control-affine nonlinear optimization problem, based on dynamic programming or calculus of variations, but their application is generally a very tedious task. Bellman’s dynamic programming approach reduces to solving a nonlinear first-order partial differential equation (PDE), expressed by the Hamilton–Jacobi–Bellman (HJB) equation. The solution to the HJB equation gives the optimal performance/cost value (or storage) function and determines an optimal control in feedback form under some smoothness assumptions. Alternatively, in the classical calculus of variations, optimal control problems can be characterized locally in terms of the Hamiltonian dynamics arising from Pontryagin’s minimum principle. These are the characteristic equations of the HJB PDE, which result in a nonlinear, constrained two-point boundary value problem (TPBVP) that, in general, can only be solved by successive approximation of the optimal control input using iterative numerical techniques for each set of initial conditions. Numerically, even though the nonlinear TPBVP is somewhat easier to solve than the HJBPDE, control signals can only be determined offline and are thus best suited for feedforward control of plants for which the state trajectories are known a priori. Therefore, contrary to the dynamic programming approach, the resultant control law is not generally in feedback form. Open-loop control, however, is sensitive to random disturbances and requires that the initial state be on the optimal trajectory. In contrast, nonlinear optimal feedback has inherent robustness properties (inherent in the sense that it is obtained by ignoring uncertainty and disturbances). The potential difficulty with the HJB approach is that no efficient algorithm is available to solve the PDE when it is nonlinear and the problem dimension is high, making it impossible to derive exact expressions for optimal controls for most nontrivial problems of interest. The optimal can only be computed in special cases, such as linear dynamics and quadratic cost, or very low-dimensional systems. In particular, if the plant is linear time invariant (LTI) and the (infinite-time) performance index is quadratic, then the corresponding HJB equation for this infamous linear-quadratic regulator (LQR) problem reduces to an algebraic Riccati equation (ARE). Contrary to the well-developed and widely applied theory and computational tools for theRiccati equation (for example, see [1]), theHJB equation is difficult, if not impossible, to solve for most practical applications. The exact solution for the optimal control policies is very complex

Using the Approximate Dynamic Programming (ADP) approach, nonlinear optimal control problems can be solved by using a dual neural network architecture called the Adaptive Critic (AC). A Single Network Adaptive Critic (SNAC) architecture has been shown to have much faster convergence than AC. Even though the SNAC offers a systematic optimal control design method for optimal state regulation for a class of nonlinear control systems, it lacks the generality necessary for command tracking as it is practically difficult to anticipate a proper training domain in state space when the command sequence is not known a priori. Dynamic Inversion (DI), on the other hand, is a very popular nonlinear control design method for both output regulation and command tracking that offers a closed form expression for control. However, it lacks the interpretation and advantages of optimal control design. In this paper we propose a systematic technique for selecting the DI gains with the help of a pre-synthesized SNAC network in order that the resulting DI solution closely mimics the SNAC solution for output regulation and can be extended to command tracking problems. Moreover, it retains all the benefits of DI design. The potential of this new methodology of near-optimal dynamic inversion design is demonstrated in this paper by considering a benchmark nonlinear system.

A unified numerical method for computing the quadrature weights, integration matrix, and differentiation matrix is newly developed in this study. For this purpose, a spline-like interpolation using piecewise continuous polynomials is converted into a global spline interpolation formula, with which the quadrature formulas can be derived from integration and differentiation of the transformed function in an exact manner. To prove the usefulness of the suggested approach, both the Lagrange and tension spline interpolations are represented in exactly the same form as global spline interpolation. The applicability of the proposed method on arbitrary nodes is illustrated using two different sets of nodes. A series of validations using three test functions is conducted to show the flexibility in selecting computational nodes with the present method.

This study develops discrete mechanics for 1-dimensional distributed parameter mechanical systems under free boundary conditions, and a control method for the systems based on a blending method of discrete mechanics and nonlinear optimization. First, under the assumption on free boundary conditions, discrete Euler-Lagrange equations and boundary equations are derived by discrete Hamilton's principle. Next, for the discrete Euler-Lagrange equations with control inputs and boundary equations, a nonlinear optimal control problem with constraints by setting an objective function, and initial and boundary conditions is formulated. Then, a vibration suppression control problem for a free-fixed string is considered as a physical example. As a result, it turns out that vibration of the string is suppressed and the whole of the system is stabilized by the proposed method.

Within the headway distance constraints, the potential for reduction of energy consumption by hybrid electric vehicles (HEVs) with connectivity could be achieved by optimizing the ego vehicle motion. This paper proposes a look-ahead traffic information-based real-time model predictive control scheme to minimize total monetary cost of HEVs. A chain Gaussian process approach is employed to estimate the probability distribution of future increments of vehicle number over a look-ahead horizon from vehicle-to-vehicle and vehicle-to-infrastructure information. The future motion of preceding vehicles could be predicted by the evolution of the traffic density model and velocity tracking model. The above problem is formulated as a nonlinear optimal control problem with predicted disturbance input and dynamic constraints. Optimal solutions are derived through Pontryagin's maximum principle. The effectiveness of the proposed control scheme is evaluated on a traffic-in-the-loop powertrain simulation platform by integrating a commercial traffic platform and an enterprise-level powertrain simulator.

In this paper, an iterative computational method is proposed to handle constrained nonlinear quadratic optimal control problems. with terminal state constraints, saturation state and saturation control variables constraints. The method is based on replacing the original constrained nonlinear optimal control problem by a sequence of constrained linear time-varying quadratic optimal control problems. To this end, an iterative technique is used to replace the original constrained nonlinear dynamic system by a sequence of constrained linear time-varying systems. Then each of constrained linear time-varying quadratic optimal control problems is approximated by a quadratic programming problem by parameterizing each of the state variable by a finite length Legendre polynomials with unknown parameters. To show the effectiveness of the proposed method, simulation results of a constrained nonlinear optimal control problem are presented.

In this paper, the nonlinear optimal control problem is investigated for spacecraft formation flying with collision avoidance. Based on a nonlinear model of formation flying, two optimal tracking control laws are proposed to ensure the formation pattern converges to the predetermined configuration. The first optimal control law which combines Lyapunov optimizing control with a trajectory-following optimization technique is developed to solve the finite-time nonlinear optimal control problem. For the second controller, an extended [Formula: see text]-[Formula: see text] method is applied to design a closed-form feedback control scheme for nonlinear control problem of spacecraft formation flying with non-standard cost functions. By taking into account the flying safety, a repulsive control scheme is incorporated in the optimal controller to ensure collision avoidance and minimize the performance index. Finally, a numerical example is performed to demonstrate the effectiveness of the proposed approaches.