Forming an approximating construction for calculation and implementation of optimal control in real time

A new approach to realization of the time-optimal control in real time for linear systems under control with a constraint is proposed. It is based on dividing the computer costs into those made in advance of the control process and those carried out as it proceeds. The preliminary computations do not depend on a particular initial condition and rely on approximation of attainability sets in different periods of time by a union of hyperplanes. Methods of their construction and identification of the support hyperplane are given. Methods of approximate finding and subsequent refining of the normalized vector of the initial conditions of the adjoint system as well as switching times and instants of switching of time-optimal control are proposed. Results of modeling and numerical calculations are presented.     

A method of optimal real-time computation of a linear system with retarded control

A new method of solving time-optimal control problems in real time is developed. The method is based on the following: 1) approximating the attainability sets with a family of hyperplanes; 2) subdividing the whole computational process into the computations performed beforehand and those that are carried out while the control takes place; 3) integrating the differential equations only over the displacement intervals of the final time point and the switching time points. The computational cost of the method is evaluated. The peculiarities of calculating the optimal control of a linear system with retarded control in real time are considered. The results of simulation and numerical calculations are presented.     

Computational Method for Time-Optimal Switching Control

An efficient algorithm, called the time-optimal switching (TOS) algorithm, is proposed for the time-optimal switching control of nonlinear systems with a single control input. The problem is formulated in the arc times space, arc times being the durations of the arcs. A feasible switching control, or as a special case bang-bang control, is found using the STC method previously developed by the authors to get from an initial point to a target point with a given number of switchings. Then, by means of constrained optimization techniques, the cost being considered as the summation of the arc times, a minimum-time switching control solution is obtained. Example applications of the TOS algorithm involving second-order and third-order systems are presented. Comparisons are made with a well-known general optimal control software package to demonstrate the efficiency of the algorithm.     

Sequential synthesis of time-optimal control for linear systems with disturbances

A method of sequential synthesis of time-optimal control for a linear system with unknown disturbances is considered. A system of linear algebraic equations is obtained which relates the increments of phase coordinates to the increments of initial conditions of a normalized adjoint system and to the increment of control completion time. Evaluations consist in solving repeatedly a system of linear algebraic equations and integrating a matrix differential equation on the displacement intervals of control switching times and on the displacement interval of final control time. A procedure of correcting the switching times and the completion time in moving along the phase trajectory of a controllable object is examined. Simple and constructive conditions are specified for a discontinuous mode to occur, for a representation point to move along the switching manifolds, and for the optimal control structure to transform in moving along the phase trajectory of a system with uncontrollable disturbance. A computational algorithm is presented. It is proved that a sequence of controls converges locally at a quadratic rate and globally to a time-optimal control.     

Cell-to-cell mapping method for time-optimal trajectory planning of multiple robot arm systems

This paper presents the results obtained by applying the cell-to-cell mapping method to solve the problem of the time-optimal trajectory planning for coordinated multiple robotic arms handling a common object along a specified geometric path. Based on the structure of the time-optimal trajectory control law, the continuous dynamic model of multiple arms is first approximated by a discrete and finite cell-to-cell mapping on a two-dimensional cell space over a phase plane. The optimal trajectory and the corresponding control are then determined by using the cell-to-cell mapping and a simple search algorithm. To further improve the computational efficiency and to allow for parallel computation, a hierarchical search algorithm consisting of a multiple-variable optimization on the top level and a number of cell-to-cell searches on the bottom level is proposed and implemented in the paper. Besides its simplicity, another distinguishing feature of the cell-to-cell mapping methods is the generation of all optimal trajectories for a given final state and all possible initial states through a single searching process. For most of the existing trajectory planning methods, the planning process can be started only when both the initial and final states have been specified. The cell-to-cell method can be generalized to any optimal trajectory planning problem for a multiple robotic arms system.     

Time-optimal control of multivariable systems near the origin

This article deals with bang-bang solutions of linear time-optimal control problems. Linear multivariable systems are considered which have one or more control components. It is shown in which way the control components act together to make the system achieve the ultimate aim (namely, the origin in the state space) as quickly as possible. The theory only applies to initial positions sufficiently near the origin. Criteria are given which give the number of switches per control component.Asymptotic dependences of the switching times and the final time on the distance of the initial position from the origin are established. The theory provides a numerical procedure to calculate the time-optimal control. These calculations are very simple. Basic to the proof of these results is a generalized implicit function theorem due to Artin (Ref. 1).     

An algorithm for the asymptotic solution of a singularly perturbed linear time-optimal control problem

An algorithm for the approximate solution (in the asymptotic sense) of a singularly perturbed linear time-optimal control problem is proposed. A computational procedure is outlined, which permits the use of the resulting asymptotic approximation for. the exact solution of the problem with a prescribed value of the small parameter.     

Resource-optimal control of linear systems

A numerical method for minimizing the resource consumption for linear dynamical systems is proposed. It is based on forming a finite-time control that steers the linear system from an arbitrary initial state to the desired terminal state in a given fixed time; this control gives an approximate solution of the problem. It is shown that the structure of the finite-time control makes it possible to determine the structure of the resource-optimal control. A method for determining an initial approximation is described, and an iterative algorithm for calculating the optimal control is proposed. A system of linear algebraic equations relating the deviations of the initial conditions in the adjoint system to the deviations of the phase coordinates from the prescribed terminal state at the terminal point in time is obtained. A computational algorithm is described. The radius of local convergence is found and the quadratic rate of convergence is established. It is proved that the computational procedure and the sequence of controls converge to the resource-optimal control.     

Optimal control problems for linear fractional-order systems defined by equations with Hadamard derivative

Two optimal control problems are studied for linear stationary systems of fractional order with lumped variables whose dynamics is described by equations with Hadamard derivative, a minimum-norm control problem and a time-optimal problem with a constraint on the norm of the control. The setting of the problem with nonlocal initial conditions is considered. Admissible controls are sought in the class of functions p-integrable on an interval for some p. The main approach to the study is based on the moment method. The well-posedness and solvability of the moment problem are substantiated. For several special cases, the optimal control problems under consideration are solved analytically. An analogy between the obtained results and known results for systems of integer and fractional order described by equations with Caputo and Riemann–Liouville derivatives is specified.

     

Fractional Order Systems Time-Optimal Control and Its Application

This paper deals with the time-optimal control problem for a class of fractional order systems. An analytic solution of the time-optimal problem is proposed, and the optimal transfer route is provided. Considering it is usually adopted in the discrete situation for actual control system, the sampling date may induce chattering phenomenon, an alternative sub-optimal solution is constructed. Additionally, the special and meaningful application of fractional order tracking differentiator is introduced to explain our main results. The effectiveness and advantages of the proposed method have been illustrated by numerical examples.     

A numerical technique for small-noise stochastic control problems

A numerical technique is described for solving approximately certain small-noise stochastic control problems. The method uses quantities computable from the optimal solution to the corresponding deterministic control problem. Numerical results are given for a two-dimensional linear regular problem with saturation and a time-optimal problem.     

Decomposition and suboptimal control in dynamical systems

A non-linear controllable dynamical system described by Lagrange equations is considered. The problem of constructing bounded controlling forces which steer the system to a given state in a finite time is investigated. Sufficient conditions are indicated for the problem to be solvable. Under these conditions, the initial system splits into subsystems, each with the degree of freedom. On the basis of this decomposition, using a game-theoretic approach, a feedback control law is proposed which solves the problem posed above and is nearly time-optimal. It is shown that the control must be constructed with proper allowance for the maximum values of the non-linear terms and perturbations in the equations of motion. The perturbations may be ignored only if the ratio of the maximum level of the perturbation to that of the control does not exceed the “golden section”.     

Numerical method for solving a nonlinear time-optimal control problem with additive control

Nonlinear systems whose right-hand sides are divided by the state and control and are linear in control are considered. An iterative method is proposed for solving time-optimal control problems for such systems. The method is based on constructing finite sequences of adjacent simplexes with their vertices lying on the boundaries of reachability sets. For a controllable system, it is proved that the minimizing sequence converges to an ?-optimal solution in a finite number of iterations.     

Multi-objective Optimization of Zero Propellant Spacecraft Attitude Maneuvers

The zero propellant maneuver (ZPM) is an advanced space station, large angle attitude maneuver technique, using only control momentum gyroscopes (CMGs). Path planning is the key to success, and this paper studies the associated multi-objective optimization problem. Three types of maneuver optimal control problem are formulated: (i) momentum-optimal, (ii) time-optimal, and (iii) energy-optimal. A sensitivity analysis approach is used to study the Pareto optimal front and allows the tradeoffs between the performance indices to be investigated. For example, it is proved that the minimum peak momentum decreases as the maneuver time increases, and the minimum maneuver energy decreases if a larger momentum is available from the CMGs. The analysis is verified and complemented by the numerical computations. Among the three types of ZPM paths, the momentum-optimal solution and the time-optimal solution generally possess the same structure, and they are singular. The energy-optimal solution saves significant energy, while generally maintaining a smooth control profile.     

Constrained optimization problems using multiplier methods

A modified multiplier method for optimization problems with equality constraints is suggested and its application to constrained optimal control problems described. For optimal control problems with free terminal time, a gradient descent technique for updating control functions as well as the terminal time is developed. The modified multiplier method with the simplified conjugate gradient method is used to compute the solution of a time-optimal control problem for a V/STOL aircraft.     

The quickest transfer of a multidimensional dynamic object onto the surface of an ellipsoid

The time-optimal control of motion of a multidimensional dynamic object is investigated. Both open-loop and feedback controls are considered. It is assumed that the force vector is constrained by a non-degenerate ellipsoid. The terminal set is described by the surface of an ellipsoid that allows degeneracy, and the terminal velocity is not fixed. The initial position of the object can be either outside or inside the ellipsoid. Using the maximum principle, the necessary and sufficient conditions for the optimality of the control are established in the form of a system of polynomial equations, the orders of which depend on the dimension of the problem and the degree of degeneracy of the terminal ellipsoid. Iterative procedures and a technique for continuation with respect to the parameters of the approximating ellipsoid in the admissable region of change of the phase vector are proposed. Control problems for the limiting shapes of the terminal ellipsoid are investigated. The qualitative effect of the discontinuity of the functional and the control defined as functions of the phase vector is established.     

A NEW INVERSION METHOD OF TIME-LAPSE SEISMIC

Time-Lapse Seismic improves oil recovery ratio by dynamic reservoir monitoring. Because of the large number of seismic explorations in the process of time-lapse seismic inversion, traditional methods need plenty of inversion calculations which cost high computational works. The method is therefore inefficient. In this paper, in order to reduce the repeating computations in traditional, a new time-lapse seismic inversion method is put forward. Firstly a homotopy-regularization method is proposed for the first time inversion. Secondly, with the first time inversion results as the initial value of following model, a model of the second time inversion is rebuilt by analyzing the characters of time-lapse seismic and localized inversion method is designed by using the model. Finally, through simulation, the comparison between traditional method and the new scheme is given. Our simulation results show that the new scheme could save the algorithm computations greatly.     

An approximate eigensolver for self-consistent field calculations

In this paper, we give a comprehensive error analysis for an approximate solution method for the generalized eigenvalue problems arising for instance in the context of electronic structure computations based on density functional theory. The solution method has been demonstrated to excel as compared to established solvers in both computational effort and scaling for parallelization. Here we estimate the improvement provided by our proposed subspace method starting from the initial approximations for instance provided in the course of the self-consistent field iteration, showing that in general the approximation quality is improved by our method to yield sufficiently accurate eigenvalues.     

Application of optimal control to a biomechanics model

A model of sport biomechanics describing short-distance running (sprinting) is developed by applying methods of optimal control. In the considered model, the motion of a sportsman is described by a second-order ordinary differential equation. Two interconnected optimal control problems are formulated and solved: the minimum energy and time-optimal control problems. Based on the comparison with real data, it is shown that the proposed approach to sprint modeling provides realistic results.     

Numerical solution of a nonlinear time-optimal control problem

Nonlinear systems with a stationary (i.e., explicitly time independent) right-hand side are considered. For time-optimal control problems with such systems, an iterative method is proposed that is a generalization of one used to solve nonlinear time-optimal control problems for systems divided by phase states and controls. The method is based on constructing finite sequences of simplices with their vertices lying on the boundaries of attainability domains. Assuming that the system is controllable, it is proved that the minimizing sequence converges to an ɛ-optimal solution after a finite number of iterations. A pair {T, u(·)} is called an ɛ-optimal solution if |T − T _opt| − ɛ, where T _opt is the optimal time required for moving the system from the initial state to the origin and u is an admissible control that moves the system to an ɛ-neighborhood of the origin over the time T.