Determination of Nonlinear Optimal Feedback Law for Satellite Injection Problem Using Neighboring Optimal Control

A highly computational efficient method to solve nonlinear 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

This paper considers the global optimal control for nonlinear interconnected large-scale systems with respect to linear quadratic functional. A successive approximation procedure is proposed to design the feedforward feedback nearly optimal control law. By introducing the Successive Approximation Approach of the nonlinear differential equation theory, a nonhomogeneous linear two-point boundary value problem sequence is constructed to approximate the nonlinear large-scale two-point boundary value problem, which is the necessary condition of the nonlinear control problem. By iteratively solving this nonhomogeneous linear approximating sequence, a global nearly optimal control law that composed of a linear closed-loop feedback compensatory term and an open-loop feedforward compensatory term is designed. An algorithm is presented to design certain qualified nearly optimal control law by truncating an infinite iteration result. An illustrative example shows the validity of the algorithm.

An efficient parallel processing optimal control scheme for a class of nonlinear composite systems

Solution of nonlinear boundary value problems — I method of third order convergence for solution of nonlinear two-point boundary value problems single second order equation

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.

This paper concerns state constrained optimal control problems, in which the dynamic constraint takes the form of a differential inclusion. If the differential inclusion does not depend on time, then the Hamiltonian, evaluated along the optimal state trajectory and the co-state trajectory, is independent of time. If the differential inclusion is Lipschitz continuous, then the Hamitonian, evaluated along the optimal state trajectory and the co-state trajectory, is Lipschitz continuous. These two well-known results are examples of the following principle: the Hamiltonian, evaluated along the optimal state trajectory and the co-state trajectory, inherits the regularity properties of the differential inclusion, regarding its time dependence. We show that this principle also applies to another kind of regularity: if the differential inclusion has bounded variation with respect to time, then the Hamiltonian, evaluated along the optimal state trajectory and the co-state trajectory, has bounded variation. Two applications of these newly found properties are demonstrated. One is to derive improved conditions which guarantee the nondegeneracy of necessary conditions of optimality, in the form of a Hamiltonian inclusion. The other application is to derive new, less restrictive, conditions under which minimizers in the calculus of variations have bounded slope. The analysis is based on a new, local, concept of differential inclusions that have bounded variation with respect to the time variable, in which conditions are imposed on the multifunction involved, only in a neighborhood of a given state trajectory.

A collocation method for solving a class of singular nonlinear two-point boundary value problems

Iterated deferred correction is a widely used approach to the numerical solution of first order systems of nonlinear two-point boundary value problems. Normally the orders of accuracy of the various methods used in a deferred correction scheme differ by 2, and, as a direct result, each time a deferred correction is applied the order of the overall scheme is increased by a maximum of 2. In this paper we consider the construction of mono-implicit Runge--Kutta (MIRK) methods where an increase of four orders of accuracy is obtained for each deferred correction. We develop a very powerful yet rather straightforward theory which allows us to identify the appropriate Runge--Kutta formulae for inclusion in such schemes. In particular, we will focus on the construction of pairs of MIRK formulae of order 4 and 8 which will allow this superconvergence to be realized. We will further show that it is possible to derive formulae of this type for which high order interpolants and accurate error estimates are readily available.

Numerov's method for strongly nonlinear two-point boundary value problems

In this paper, the Half-Sweep Successive Over-Relaxation (HSSOR) iterative method together with Newton scheme namely Newton-HSSOR is investigated in solving the nonlinear systems generated from the fourth-order half-sweep finite difference approximation equation for nonlinear two-point boundary value problems. The Newton scheme is proposed to linearize the nonlinear system into the form of linear system. On top of that, we also present the basic formulation and implementation of Newton-HSSOR iterative method. For comparison purpose, combinations between the Full-Sweep Gauss-Seidel (FSGS) and Full-Sweep Successive Over-Relaxation (FSSOR) iterative methods with Newton scheme, which are indicated as Newton-FSGS and Newton-FSSOR methods respectively have been implemented numerically. Numerical experiments of two problems are given to illustrate that the Newton-HSSOR method is more superior compared with the tested methods.

High accuracy variable mesh method for nonlinear two-point boundary value problems in divergence form

The extrapolation of Numerov's scheme for nonlinear two-point boundary value problems

The global warming and the environmental destruction are caused from the mass consumption of energy in a factory in which robotic manipulators are used. It is necessary to take the trajectory planning into considerations in order to save the consumption energy of the manipulator controlled with Point-to-Point (PTP) motion. An optimal trajectory that minimizes the consumption energy can be obtained by solving a two-point boundary value problem. The optimal trajectory can be analyzed easily if the system is linear. However, the dynamical equation of the multi-link manipulator with Coulomb friction has a strong non-linearity. Then the Genetic Algorithm (GA) is applied to obtain the solution of two-point boundary value problem with the strong non-linearity. The GA is the method for searching discrete solution space by genes. However, the trajectory function is continuous with respect to time. In this paper, the coefficients of the approximation functions are determined as the gene’s phenotype. This method has disadvantages that are long computation time and increase the possibility of lapsing into the local optimal solution. So, a new method is proposed in order to shorten the computation time and to lessen the possibility of lapsing into the local optimal solution. It is seen from application to the minimization of the consumption energy of a manipulator with Coulomb friction that a proposed method is effective.

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.