Optimal Periodic Control Research Articles

Optimal control of a spinning double-pyramid Earth-pointing tethered formation

The dynamics and control of a tethered satellite formation for Earth-pointing observation missions is considered. For most practical applications in Earth orbit, a tether formation must be spinning in order to maintain tension in the tethers. It is possible to obtain periodic spinning solutions for a triangular formation whose initial conditions are close to the orbit normal. However, these solutions contain significant deviations of the satellites on a sphere relative to the desired Earth-pointing configuration. To maintain a plane of satellites spinning normal to the orbit plane, it is necessary to utilize “anchors”. Such a configuration resembles a double-pyramid. In this paper, control of a double-pyramid tethered formation is studied. The equations of motion are derived in a floating orbital coordinate system for the general case of an elliptic reference orbit. The motion of the satellites is derived assuming inelastic tethers that can vary in length in a controlled manner. Cartesian coordinates in a rotating reference frame attached to the desired spin frame provide a simple means of expressing the equations of motion, together with a set of constraint equations for the tether tensions. Periodic optimal control theory is applied to the system to determine sets of controlled periodic trajectories by varying the lengths of all interconnecting tethers (nine in total), as well as retrieval and simple reconfiguration trajectories. A modal analysis of the system is also performed using a lumped mass representation of the tethers.

Periodic Optimal Search Control for a Target Appearing Randomly

This paper investigates an optimal search problem under the assumption that a target to be found appears randomly and stays for a fixed time interval. We first formulate the search problem as an optimization problem to maximize the probability of finding the target while reducing the control energy consumption, and provide a solution to this problem. In this paper, we moreover derive the expectation value of how many times the target appears without being found and a necessary and sufficient condition for the agent’s behavior to converge to a periodic trajectory. The effectiveness of the proposed method is demonstrated through a numerical simulation.

Improved clipped periodic optimal control for semi-active harmonic disturbance rejection

This paper presents a new approach for harmonic disturbance rejection using semi-active vibration control. The approach is illustrated through application to the problem of maximizing the energy dissipated by a semi-active damper under harmonic excitation. In order to establish a baseline for the evaluation of the performance of the semi-active damper, the effectiveness of the optimal passive and active cases are first presented. The study then examines the ability of the clipped optimal control (or clipping control) approach to improve the energy dissipation capacity of the semi-active damper over the optimal passive damper. An approximate solution to the nonlinear dynamic problem, obtained using the method of averaging, and a time integration based numerical method indicate that this approach improves the energy dissipated by the semi-active damper over the optimal passive damper. The approach presented in this paper intends to further “improve”, or “fine tune”, the control parameters given by the clipped optimal control approach. This is done using an approximated solution of the problem and an appropriate optimization algorithm. Results clearly indicate that this new approach provides significant improvement on energy dissipation over the clipped optimal control approach for the semi-active damper.

Applying a finite-horizon numerical optimization method to a periodic optimal control problem

Computing a numerical solution to a periodic optimal control problem can be difficult, especially when the period is unknown. A method of approximating a solution to a stochastic optimal control problem using Markov chains was developed in [Krawczyk, J. B. (2001). A Markovian approximated solution to a portfolio management problem. Information Technology for Economics and Management, 1, http://www.item.woiz.polsl.pl/issue/journal1.htm]. This paper describes the application of that method to a periodic optimal control problem formulated in [Gaitsgory, V. & Rossomakhine, S. (2006). Linear programming approach to deterministic long run average problems of optimal control. SIAM Journal on Control and Optimization, 44(6), 2006–2037]. As a result, approximately optimal feedback rules are computed that can control the system both on and off the optimal orbit.

On some general almost periodic Optimal Control problems: links with periodic problems and necessary conditions

In this paper, we are concerned with periodic, quasi-periodic (q.p.) and almost periodic (a.p.) Optimal Control problems. After defining these problems and setting them in an abstract setting by using Abstract Harmonic Analysis, we give some structure results of the set of solutions, and study the relations between periodic and a.p. problems. We prove for instance that for an autonomous concave problem, the a.p. problem has a solution if and only if all problems (periodic with fixed or variable period, q.p. or a.p.) have a constant solution. After that, we give some first order necessary conditions (weak Pontryagin) in the space of Harmonic Synthesis and we also give in this space an existence result.

Optimal periodic control of a drug delivery system

Administration of certain drugs at a steady rate results in deterioration of drug effect, also known as drug tolerance. Periodic delivery is an attractive option for minimizing tolerance and maximizing the desired effect of such drugs. In this paper, periodic drug infusion strategies for maximizing a time-averaged measure of drug effect are investigated. A simple pharmacokinetic–pharmacodynamic (PKPD) model of a system exhibiting tolerance is considered and optimal periodic control theory is employed. First, regions of PKPD parameter space in which periodic infusion provides a locally improved average effect compared to steady infusion are characterized using the so-called π -test. Then, optimal drug delivery strategies, obtained using two different computational approaches, are presented for a representative set of parameter values, and insight is provided into the results. The first method, proposed by the authors, is based on the notion of differential flatness, while the second is based on a standard shooting method for dynamic optimization problems.

On‐Line Feedback Control for Optimal Periodic Control Problems

AbstractIn this paper a method for generating periodic orbits for a class of non‐linear systems is presented. A periodic Hamiltonian system is defined as the target equilibrium for non‐linear systems in strict feedback form. A parameter dependent non‐linear periodic system description is introduced for a two‐dimensional subsystem and extended to higher dimensions using backstepping. Extremum seeking is used as a parameter update law that guarantees the convergence of the system to the optimal periodic orbit with respect to a given parameterization. The procedure resulted in an on‐line feedback control implementation that typically steers the system to a suboptimal solution that nevertheless improves the static optimization solution. Typical examples from optimal periodic control literature are used to illustrate the application of this approach.

Open‐loop stable solutions of periodic optimal control problems in robotics

AbstractWe present a numerical method that optimizes the open‐loop stability of solutions of periodic optimal control problems. We consider general periodic processes that may have several phases, each characterized by its own set of differential equations, and discontinuities of the state variables and the right hand side between phases. Stability is measured in terms of the spectral radius of the monodromy matrix which results in a nonsmooth optimization criterion. We have applied this method to design walking robots that can perform stable periodic gaits without any sensors or feedback; three such examples are presented in this paper.

Nonlinear Periodic Optimal Control: A Pseudospectral Fourier Approach

A pseudospectral method for generating optimal trajectories of the class of periodic optimal control problems is proposed. The method consists of representing the solution of the periodic optimal control problem by an mth degree trigonometric interpolating polynomial, using Fourier nodes as grid points, and then discretizing the problem using the trapezoidal rule as the quadrature formula for smoothly differentiable periodic functions. The periodic optimal control problem is thereby transformed into an algebraic nonlinear programming problem. Due to its dynamic nature, the pseudospectral Fourier approach avoids many of the numerical difficulties typically encountered in solving standard periodic optimal control problems. An illustrative example is provided to demonstrate the applicability of the proposed method.

OPTIMAL DISCRETE-TIME MAGNETIC ATTITUDE CONTROL OF SATELLITES

The problem of designing discrete-time attitude controllers for magnetically actuated spacecraft is considered. Several methods are discussed and a novel approach to the tuning of ”projection based” controllers is proposed relying on periodic optimal output feedback control techniques. The main advantages of the proposed method are discussed and illustrated in a simulation study.

Preliminary Design of Earth-Mars Cyclers Using Solar Sails

Motivated by the concept of using cycler trajectories for Mars return missions, an optimal Earth-Mars cycler is designed for spacecraft that use solar sails. The design is preliminary in the sense that the trajectories are assumed to be coplanar and the orbits of Earth and Mars are assumed to be circular. The optimality criterion is the vehicle's hyperbolic excess speed V ∞ with respect to Mars. This trajectory optimization problem is formulated using the framework of optimal periodic control with interior point constraints that include jump discontinuities that arise as a result of gravity assists. The problem is numerically solved using the concept of pseudospectral knots (PSK) by way of a general-purpose dynamic optimization software. Feasibility of the PSK-derived trajectories is demonstrated using numerical propagation. Results show that solar sails can be used to reduce significantly V ∞ at Mars. For example, it is shown that a sail with a lightness number of 0.17 can achieve a V ∞ with respect to Mars of 2.5 km/s at every synodic encounter.

Optimal Periodic Control of a Drug Delivery System

Administration of certain drugs at a steady rate results in a deterioration of the drug effect due to a phenomenon known as tolerance. Periodic drug delivery is an attractive option for maximizing the effect of drugs exhibiting tolerance. In this paper, periodic drug infusion strategies for maximizing an averaged measure of the drug effect are investigated. A simple pharmacokinetic-pharmacodynamic model of a system exhibiting tolerance is considered and optimal periodic control theory is employed. The regions in the parametric space where periodic infusion gives better drug effect than steady infusion are characterized using the scalled π test. The optimal drug delivery strategy obtained using two different computational approaches are presented for a representative set of parameter values and insight is provided into the results. The first method, proposed by the authors, is based on the notion of differential flatness and the second, is based on the standard shooting method for dynamic optimization problems.

Numerical Solution of the Optimal Periodic Control Problem Using Differential Flatness

Optimal periodic control (OPC) is of interest in many engineering applications. In practice, the numerical solution of the OPC problem has been found to be quite challenging. In this note, we present a method which uses differential flatness for the solution of OPC problems. The OPC problem is reformulated using the flatness of the underlying dynamical system to eliminate the differential equations and the periodicity constraints, resulting in simpler and generally more efficient computation. The effect of point-wise constraints and the analytical computation of the Jacobian matrix are also discussed. The approach is demonstrated using two examples.

Optimization of cyclically operated reactors and separators

An efficient version of a numerical gradient optimization procedure for the computation of solutions to periodic optimal control problems is presented. This procedure consists of a first order gradient method in combination with the Newton–Picard shooting method. The first order gradient method is able to compute the optimal control of a periodic process under one or more nonlocal constraints, such as a minimum purity constraint. The Newton–Picard method computes very efficiently periodic solutions to the state and adjoint equations. The presented numerical procedure is used to optimize a rapid pressure swing reactor and a rapid pressure swing adsorber. It is shown that the optimal cycle scheme consists of four steps: a no-inflow pre-pressurization step, a pressurization step, a no-inflow post-pressurization step, and a depressurization step.

The Carleman Inequality and Its Application to Periodic Optimal Control Governed by Semilinear Parabolic Differential Equations

This paper deals with optimal control problems for semilinear parabolic differential equations, which may be governed by nonmonotone operators and have no global solution, with periodic inputs. The Pontryagin maximum principle is obtained and the Carleman inequality for the backward linearized adjoint system associated with the state system is established.

The second variation test for vector-valued optimization of periodic control

An existing scalar second-order condition for the optimal periodic control is adapted to test the possibility of performance improvement in the case of vector-optimality. The π-test is applied to the decomposed vector-valued optimization problem. Some simulations of a chemical reactions are presented as applications of a proposed vector-valued test.

The Second Variation Test for Vector-valued Optimization of Periodic Control

An existing scalar second-order condition for the optimal periodic control is adapted to test the possibility of performance improvement in the case of vector-optimality. The @ -test is applied to the decomposed vector-valued optimization problem. Some simulations of a chemical reactions are presented as applications of a proposed vector-valued test.

Periodic attitude control techniques for small satellites with magnetic actuators

The problems of attitude stabilization and disturbance torque attenuation for a small spacecraft using magnetic actuators is considered and a solution to the problem in terms of optimal periodic control is proposed, based on an optimal estimation and compensation scheme for external disturbances for linear time-periodic systems.

Optimal Controls of 3-Dimensional Navier--Stokes Equations with State Constraints

This work is concerned with the maximum principles for optimal control problems governed by 3-dimensional Navier--Stokes equations. Some types of state constraints (time variables) are considered.

Autonomous Orbit Control for Spacecraft on Elliptical Orbits Using a Non-Inertial Coordinate Frame

A method for designing an autonomous orbit controller for spacecraft on elliptical orbits is presented. For the derivation of the linearized differential and difference equations of motion, a non-inertial coordinate frame is utilized which coincides with the one which is applied in the well-known Clohessy-Wiltshire equations for small excentricities. The derivation takes atmospheric drag into account. The resulting time-discrete periodic system is augmented by an integrator state. A control law is obtained by applying linear optimal periodic control. The orbit controller is tested and verified in simulation runs.