Optimal Periodic Control Research Articles

Periodic H∞ Attitude Control for Satellites with Magnetic Actuators

The problem of attitude stabilisation for a small spacecraft using magnetic actuators is considered and a solution to the problem in terms of H∞ periodic control is proposed. An optimal estimation and compensation scheme for external disturbance torques is also presented, based on recent extensions to the optimal periodic control theory.

CONSTRAINED OPTIMAL CONTROL OF VIBRATION DAMPERS

Optimal control of dampers has been proposed to mitigate vibration effects in mechanical systems. In many cases, systems are subject to periodic forcing and the goal is to maximize the energy dissipated by the damper. In contrast to prior work utilizing instantaneous or infinite-time—horizon optimization, this paper employs periodic optimal control to maximize the energy dissipated per cycle. For single-degree-of-freedom systems in which the maximum allowable control effort is of the same order as the forcing magnitude, a state-dependent singular control law is shown to deliver maximum energy dissipation. Alternate control laws are proposed for situations when rattle space requirements dictate damper displacements other than that of the singular solution. Saturation of the damping force is also considered.

Homogenization of periodic optimal control problems via multi-scale convergence

The aim of this paper is to provide an alternate treatment of the homogenization of an optimal control problem in the framework of two-scale (multi-scale) convergence in the periodic case. The main advantage of this method is that we are able to show the convergence of cost functionals directly without going through the adjoint equation. We use a corrector result for the solution of the state equation to achieve this.

Solution techniques for periodic control problems: a case study in production planning

Two numerical techniques for solving optimal periodic control problems with a free period are developed. The first method uses shooting techniques for solving an appropriate boundary value problem associated with the necessary conditions of the minimum principle. A convenient form of the transversality condition for the free period is incorporated. The second method is a direct optimization method that applies non-linear programming techniques to a discretized version of the control problem. Both numerical methods are illustrated in detail by a non-convex economic production planning problem. In this model, the π-test reveals that the steady-state operation is not optimal. The optimal periodic control is computed such that a complete set of necessary conditions is verified. The solution techniques are extended to obtain the optimal periodic control under various state constraints. A sensitivity analysis of the optimal solution is performed with respect to a specific parameter in the model. © 1998 John Wiley & Sons, Ltd.

Gradient-Based Approach to Solve Optimal Periodic Output Feedback Control Problems

We discuss the numerical solution of the output feedback optimal periodic control problem by using a gradient search based optimization approach. Extension of this approach to the simultaneous stabilization of periodic multimodels is also considered. For the evaluation of the function and its gradient explicit expressions are derived which involve the numerical solution of a pair of discrete-time periodic Lyapunov equations. Numerically reliable algorithms based on the periodic Schur decomposition are used for the efficient solution of these equations. A satellite positioning example illustrates the effectiveness of the presented approach to solve a practical periodic control problem.

The heredity shift operator in descent approach to the optimal periodic hereditary control problem

A new computational approach to the optimal periodic hereditary control problem is proposed. The description of the periodic hereditary process in the equivalent integral form using the forward heredity shift operator is considered. The particular form of such a model reduced to a fixed point of a mapping defined on the control horizon is advantageously exploited in computational algorithms. The normal Lagrange functional approach is applied to derive the integral adjoint equation with the backward heredity shift operator, and to obtain the gradient formula of the goal functional with respect to the control. The descent optimization method, employing the Newton algorithm for solving the periodic integral state equation in the roof basis, and the projected gradient method in the control space with the step basis, is depicted. To verify the properness of the problem, and to estimate the best operation period the /7 test specialized to the case of many lumped and distributed delays is obtained. The computer implementation of the method is performed in the Mathematica system, and verified on examples of periodic control problems for ecological processes.

Asymptotic Solution of Periodic Control Problem Perturbed By Matrix

By the direct substitution of the postulated asymptotic expansion of the solution into the problem conditions the asymptotic expansion with respect to powers of the small parameter ε is constructed under some conditions for the solution of the classic nonlinear periodic optimal control problem with fixed period and with the linear operator A + εB before the derivative in the state equation, where det A = 0 and det ( A + εB ) ≢ 0. The closeness estimates between approximate and exact solution are given; the lack of increase of the performance index is established when the new terms of the control asymptotic are used.

Periodic Lyapunov equations: Some applications and new algorithms

The discrete-time periodic Lyapunov equation has several important applications in the analysis and design of linear periodic control systems. Specific applications considered are the solution of state- and output-feedback optimal periodic control problems, the stabilization by periodic state feedback and the square-root balancing of discrete-time periodic systems. Efficient and numerically reliable algorithms based on the periodic Schur decomposition are proposed for the solution of periodic Lyapunov equations. The proposed algorithms are extensions of the direct solution methods for standard discrete-time Lyapunov equations for the case of indefinite as well as non-negative definite solution.

Periodic optimal flight

For a host of dynamic systems, periodic motion is more efficient than steady-state operation. This work focuses on atmospheric fuel-efficient periodic flight. To understand some of the generic properties of periodic optimal flight, the theoretical aspects are presented, which form the basis for the numerical computation of periodic optimal paths. Furthermore, the simplest problem for which periodic optimal control can be induced is formulated and various solutions based on asymptotic analysis are given. Intuitive physical mechanisms that contribute to producing periodic optimal flight are discussed on the basis of various levels of approximation of the aircraft dynamic models and upon perturbation analysis about the steady-state cruise path. The performance improvement and the dramatic periodic optimal flight path of a hypersonic vehicle are computed. To mechanize this flight path, a neighboring optimum perturbation guidance law is developed and its performance is presented.

Periodic solutions and optimization problems for a class of semilinear parabolic control systems

This paper is devoted to the study of the existence of periodic solutions for a class of control problems described by a semilinear parabolic equation. Related optimization problems are also considered. Periodic control problems and optimal periodic control problems for evolution equations described both by ordinary differential equations and by parabolic equations arise in many different situations. Examples are found in the theory of chemical processes [7], biomedical models [13], competing species ([6], and the extensive references therein), thermostat problems [3], [5], [18] and [20] (and the references therein). Several authors have also treated the existence of periodic solutions of differential inclusions in Banach spaces. These differential inclusions can model periodic control problems. In fact, under suitable assumptions, a large class of control problems can be reduced to the problem of finding solutions of a differential inclusion. We mention here the following papers: [8], [9], [10], [15], [16] and the references therein. The variety of techniques used in the quoted papers is quite large. Referring only to those papers dealing with parabolic equations, we recall that in [6] the properties of the Poincare map and the theory of ordered spaces

Regulators for optimal periodic processes

A feedback controller called a periodic regulator is developed for implementing an optimal periodic process to a physical system under state deviations and constant parameter variation. The objective of the periodic regulator is to bring the perturbed plant back to a neighboring optimal periodic path in such a manner that the infinite-time second-variation of the cost is minimized. The transition matrix associated with the variational problem evaluated over one period is symplectic and generically possesses two unity eigenvalues coupled in the same Jordan box; the primary eigenvector associated with the unity eigenvalue corresponds to the orbital velocity direction. It is shown that there exists a locally stable subspace in the neighborhood of the periodic orbit. Moreover, a simple condition is given for defining the state variation which guarantees that it is contained in this stable subspace. A controller including feedback of the parameter variation is derived which provides first-order convergence to a new optimal periodic trajectory. Finally, The periodic regulator developed is applied to a simple two-state periodic optimal control problem to demonstrate its convergence properties.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Nonconcavity and proper optimal periodic control

This paper deals with optimal periodic control problems in an economic context. The main result is that nonconcavity is necessary for proper periodicity of optimal solution paths. The theoretical results are illustrated by an unemployment and inflation trade-off model.

Optimal periodic forcing of nonlinear chemical processes for performance improvements

AbstractUnsteady‐state periodic operations can improve the optimal steady‐state performance of nonlinear chemical processes. To examine if the optimal periodic operation is proper and to obtain the optimal forcing functions subject to various control and state constraints it is suggested in this paper to convert the problems into a form which is suitable for constrained nonlinear programming. The adopted numerical optimization method is based on employing the control parametrization technique and is thus capable of dealing with the problem of multiple input forcings and obtaining optimal forcing functions and/or parameters while subject to general constraints. Besides, it provides information about to what extent the process performance can be improved by adopting the optimal periodic control.

Optimal periodic control of the input into a heterogeneous catalytic reactor

The possibility of conversion increase by periodic control of chemical reactors is studied. Method of determination of parameters of the periodic input control (amplitude and period) is presented. As an optimality criterion the time integral of the difference of the conversion under periodic and constant operation of the reactor is chosen. The proposed method is of gradient type, the gradient components are computed with the help of the system of adjoint equations. The method is illustrated by the example of the following catalytic system: continuous stirred tank reactor, narrow active region catalyst and catalytic reaction described by Langmuir-Hinshelwood kinetics. In this example the improvement of the reactor performance is significant.

Optimal periodic control for spacecraft pointing and attitude determination

A new approach to autonomous magnetic roll/yaw control of polar-orbiting, nadir-pointing momentum bias spacecraft is considered as the baseline attitude control system for the next Tiros series. It is shown that the roll/yaw dynamics with magnetic control are periodically time varying. An optimal periodic control law is then developed. The control design features a state estimator that estimates attitude, attitude rate, and environmental torque disturbances from Earth sensor and sun sensor measurements; no gyros are needed. The state estimator doubles as a dynamic attitude determination and prediction function. In addition to improved performance, the optimal controller allows a much smaller momentum bias than would otherwise be necessary. Simulation results are given.

Enhancement of endurance performance by periodic optimal camber control

A new method for endurance maximization is proposed. This method consists of the periodic optimal control of variable camber in a coordinated process with a corresponding control of throttle and elevator. It is shown that the proposed method can significantly increase the endurance when compared with the best steady-state flight. Using camber as a control provides an efficient means for improving the lift/drag ratio at each flight condition during the unsteady phases of periodic optimal endurance cruise. Engine considerations include fuel consumption during idling phases. They also address a reduction of the number of engine cycles in regard to alternate operating at a high and low thrust setting. The optimal control technique used is based on the minimum principle and the method of multiple shooting. Necessary conditions for optimality, including optimal control of variable camber setting, are analyzed.

Optimal periodic control implemented as a generalized sampled-data hold output feedback control

A method for converting analog linear quadratic regulation control to a generalized sampled-data hold output-feedback control for a linear periodic system or a linear time-variant system is presented. It is shown that by using such a conversion, one can implement the optimal periodic control scheme in the presence of incomplete and delayed state measurements. >

Descent algorithms for optimal periodic output feedback control

Periodic output feedback is investigated in the context of linear-quadratic regulation for finite-dimensional time-invariant linear systems. Discrete output samples are multiplied by a periodic gain function to generate a continuous feedback control. The optimal solution is obtained in two steps by separating the continuous-time from the discrete-time structure. First, the optimal pole placement problem under periodic output feedback is solved explicitly under the assumption that the behavior at the sample times has been specified in terms of a gain matrix G. Then the minimum value, which depends on G, is substituted into the overall objective. This results in a finite-dimensional nonlinear programming problem over all admissible gain matrices G. The solution defines the optimal periodic output feedback control via the formulas of the optimal pole placement problem. A steepest descent and a direct iterative method for solving this problem are formulated and compared. Numerical examples show that the performance using periodic output feedback is almost equivalent to that using optimal continuous-state feedback.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Dynamic decrease of drag by optimal periodic control

Dynamic decrease of drag by optimal periodic control

Necessary Conditions for Optimal Control of Distributed Parameter Systems

The Ekeland variational principle is used to prove a maximum principle for the optimal controls of a general semilinear evolutionary control system in a Banach space with strictly convex dual. The boundary constraint is of general type which includes the optimal periodic control problem as a special case.