Synthesis Of Time-optimal Control Research Articles

Lyapunov and Minimum-Time Path Planning for Drones

In this paper, we study the problem of controlling an unmanned aerial vehicle (UAV) to provide a target supervision and/or to provide convoy protection to ground vehicles. We first present a control strategy based upon a Lyapunov-LaSalle stabilization method to provide supervision of a stationary target. The UAV is expected to join a predesigned admissible circular trajectory around the target which is itself a fixed point in the space. Our strategy is presented for both high altitude long endurance (HALE) and medium altitude long endurance (MALE) types of UAVs. A UAV flying at a constant altitude (HALE type) is modeled as a Dubins vehicle (i.e., a planar vehicle with constrained turning radiusand constant forward velocity). For a UAV that might change its altitude (MALE type), we use the general kinematic model of a rigid body evolving in \(\mathbb {R}^{3}\). Both control strategies presented are smooth, and unlike what is usually proposed in the literature, these strategies asymptotically track a circular trajectory of exact minimum turning radius. We also present the time-optimal control synthesis for tracking a circle by a Dubins vehicle. This optimal strategy is very rich, although much simpler than the point-to-point time-optimal strategy studied in the 1990s. Finally, we propose control strategies to provide supervision of a moving target, which are based upon theprevious ones.

Controlled pendulum on a movable base

A plane motion of a multilink pendulum hinged to a movable base (a wheel or a carriage) is considered. The control torque applied between the base and the first link of the pendulum is independent of the base position and velocity and is bounded in absolute value. The coordinate determining the base position is cyclic. The mathematical model of the system permits one to single out the equations describing the pendulum motion alone, which differ from the well-known equations of motion of a pendulum with a fixed suspension point both in the structure and in the parameters occurring in these equations. The phase portrait of motions of a control-free one-link pendulum suspended on a wheel or a carriage is obtained. A feedback control ensuring global stabilization of the unstable upper equilibrium of the pendulum is constructed. Time-optimal control synthesis is outlined.

Synthesis of time-optimal control of inverted rod

The problem of controlling a system consisting of a heavy rod fastened at its lower end on a hinge to a movable support is studied. In the linear statement and under the assumption that the support moves without inertia at a bounded velocity, a time-optimal motion of the rod from an arbitrary initial state with a nonzero velocity to a vertical unstable equilibrium position with the zero velocity of the rod at this position is found assuming that the point of support must travel a prescribed distance. The domain where a solution exists is found. Formulas for determining the switch times of the support velocity depending on the boundary conditions are obtained. A numerical example illustrating the procedure of choosing the control depending on the initial position of the rod in the absence of disturbances is considered.

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.

A time-optimal control synthesis for a nonlinear pendulum

A time-optimal control synthesis is obtained that steers a nonlinear pendulum to the bottom stable equilibrium position. The solution is based on the maximum principle and involves the analytical investigation in combination with numerical calculations. As a result, for different values of the maximum admissible control torque, switching curves and dispersal curves (or separation curves), which confine domains in the phase space that correspond to different values of the bang-bang optimal control, are constructed. The investigated patterns of synthesis reveal the mechanism of generation of numerous breaks on switching curves for small control torques.

Analytical synthesis of time-optimal control in a third-order system

A time-optimal control that steers the phase point for a third-order linear system to the origin is constructed in an explicit analytical form. It is assumed that the characteristic exponents are zero, and the constraints on the control function are non-symmetric. The system simulates the dynamics of a point mass driven by a force whose rate of change can be regulated. An optimal control is constructed both in the feedback and open-loop forms. In the latter case, the optimal control is a function of time. Relations are derived for the switching curve and surface and for the time intervals of the motion; optimal phase trajectories are constructed; the feedback control portrait is investigated. The influence of a parameter characterizing the degree of asymmetry of the constraints is studied. “Near-optimal” control modes, which are much simpler to implement, are constructed.

Analytic time-optimal control synthesis of fourth-order system and maneuvers of flexible structures

The analytic solution to the problem of minimum-time control of a fourth-order linear system near the origin is obtained by using Gulko's subsystem strategy. This system has two real eigenvalues (zeros) and two imaginary eigenvalues (vibrational frequency), which represents a flexible structure with one rigid-body mode and one flexible mode. The complete solution including the optimal switching times for the second-order subsystem is first obtained. Next, the geometry of the control switching surface near the origin for the third-order subsystem is analyzed through the phase-plane and phase-space technique. Then, the switching function for the full fourth-order system is constructed by using the roots of a quartic algebraic equation. This solution can be used to synthesize the feedback time-optimal control for the maneuvers of flexible structures. Numerical examples with one and more flexible modes are illustrated to show the applications of the solution.

Synthesis of time-optimal control of a third-order object with a phase constraint: PMM vol. 40, n≗ 3, 1976, pp. 446–454

We examine a problem, arising in engineering practice, of the time-optimal control of a third-order linear object with a constraint on the control and on the phase coordinate. The synthesis of the control is described.

Time-optimal control synthesis for a fourth-order nonlinear system: PMM vol. 39, n≗ 6, 1975, pp. 985–994

On the basis of Pontriagin's maximum principle we establish the structure of the optimal control and of the optimal trajectories, using the properties of the system being analyzed. We propose a rule for the construction of the program control satisfying the maximum principle. In the case when the terminal state lies outside some bounded region we prove that the rule mentioned determines the optimal control and permits us to solve the synthesis problem.

Synthesis of a tim-optimal control system†

Abstract A second-order system has been taken for the synthesis of time-optimal control. Important elements in the closed-loop system are the relay element, the controlled element and the controlling device performing time-optimal control. An equation for the switching boundary has been obtained by the phase-space technique and verified by a state-variable approach. The realized time-optimal system is found to give better response than any other non-optimal system.

Synthesis of time-optimal control by time interval adjustment

An iterative procedure for time-optimal control of linear plants with constrained control amplitudes is presented. It is assumed that the n th-order state-equation coefficient matrix has real eigenvalues, so that the time-optimal control is of the known bang-bang form with n switchings. The first step in the procedure is to arbitrarily choose n switching times (including the final time), and to calculate a precise constant control function which although not necessarily satisfying the amplitude constraints does bring the plant to the desired terminal state. In the following steps the switching times are systematically adjusted until the control function closely approximates the bang-bang form. The procedure is simple to implement, and experiments have shown fast convergence.

Synthesis of time-optimal control of a second-order nonlinear process

The time-optimal control of a soft spring is discussed. The domain of controllability and the switching locus are described. It is shown that the switching locus changes in a discontinuous manner as the nonlinearity varies continuously.

Synthesis of time-optimal control for linear processes

Synthesis of time-optimal control for linear processes