New smoothing techniques for solving bang–bang optimal control problems—numerical results and statistical interpretation

In this paper, we investigate the solution of bang‐bang optimal control problems by shooting methods. We will show how modifying the performance index by a term depending on a small parameter ε yields more regular controls and shooting functions. A continuation procedure on ε will lead us to a good approximation of the initial solution. Then, a statistical interpretation of the method is given, providing us with a general framework for building new regular controls. Finally, two numerical examples are solved illustrating the interest of our method. Copyright © 2002 John Wiley & Sons, Ltd.     

Redundancy implies robustness for bang‐bang strategies

We develop in this paper a method ensuring robustness properties of bang‐bang strategies, for general nonlinear control systems. Our main idea is to add bang arcs in the form of needle‐like variations of the control. With such bang‐bang controls having additional degrees of freedom, steering the control system to some given target amounts to solving an overdetermined nonlinear shooting problem, what we do by developing a least‐square approach. In turn, we design a criterion to measure the quality of robustness of the bang‐bang strategy, based on the singular values of the end‐point mapping, and which we optimize. Our approach thus shows that redundancy implies robustness, and we show how to achieve some compromises in practice, by applying it to the attitude control of a 3d rigid body.     

On L1‐minimization in optimal control and applications to robotics

In this paper, we analyze optimal control problems with control variables appearing linearly in the dynamics. We discuss different cost functionals involving the L^p‐norm of the control. The case p = 0 represents the time‐optimal control, the case p > 1 yields a standard smooth optimal control problem, whereas the case p = 1 leads to a nonsmooth cost functional. Several techniques are developed to deal with the nonsmooth case p = 1. We present a thorough theoretical discussion of the necessary conditions. Two types of numerical methods are developed: either a regularization technique is used or an augmentation approach is applied in which the number of control variables is doubled. We show the precise relations between the L¹‐minimal control and the bang–bang or singular controls in the augmented problem. Using second‐order sufficient conditions (SSC) for bang–bang controls, we obtain SSC for L¹‐minimal controls. The different techniques and results are illustrated with an example of the optimal control for a free‐flying robot which is taken from Sakawa. Copyright © 2006 John Wiley & Sons, Ltd.     

On the computation of optimal singular and bang–bang controls

To avoid difficulties associated with the computation of optimal singular/bang–bang controls, a common approach is to add a perturbed energy term. The efficacy of this perturbation method is assessed here via a direct search iterative dynamic programming procedure. A potential limitation of the strategy is shown from a computational point of view, and some guidelines for selecting the perturbation parameter are provided using numerical examples. It is demonstrated that many gradient-based methods may not be well suited for computing singular/bang–bang controls when perturbation methods are used to solve optimal control problems in chemical process control. © 1998 John Wiley & Sons, Ltd.     

Using logarithmic penalties in the shooting algorithm for optimal control problems

The paper deals with optimal control problems of ordinary differential equations with bound control constraints. We analyse the logarithmic penalty method for converting the problem into an unconstrained one, the latter being solved by a shooting algorithm. Convergence of the value function and optimal controls is obtained for linear quadratic problems, and more generally when the control variable enters linearly in the state equation and in a quadratic way in the cost function. We display some numerical results on two examples: an aircraft maneuver, and the stabilization of an oscillating system. Copyright © 2003 John Wiley & Sons, Ltd.     

Closed‐form solutions of a fishery harvesting model with state constraint

We derive a closed‐form solution for a well‐known fisheries harvesting model with an additional state constraint. The problem is linear in the control and previous solutions appearing in the literature have been numerical in nature. The so‐called direct adjoining approach is used in our derivation and the optimal solutions turn out to be a mixture of bang‐bang and boundary arcs. Copyright © 2013 John Wiley & Sons, Ltd.     

Computations for bang–bang constrained optimal control using a mathematical programming formulation

An algorithm is proposed to solve the problem of bang–bang constrained optimal control of non‐linear systems with free terminal time. The initial and terminal states are prescribed. The problem is reduced to minimizing a Lagrangian subject to equality constraints defined by the terminal state. A solution is obtained by solving a system of non‐linear equations. Since the terminal time is free, time‐optimal control is given a special emphasis. Second‐order sufficient conditions of optimality are also stated. The algorithm is demonstrated by a detailed study of the switching structure for stabilizing the F–8 aircraft in minimum time, and other examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Optimal control of steer‐braking systems: Non‐existence of minimizing trajectories

In this paper, we investigate an optimal control problem in which the objective is to decelerate a simplified vehicle model, subject to input constraints, from a given initial velocity down to zero by minimizing a quadratic cost functional. The problem is of interest because, although it involves apparently simple drift‐less dynamics, a minimizing trajectory does not exist over the admissible input trajectories. This problem is motivated by a minimum‐time problem for a fairly complex car vehicle model on a race track. Numerical computations run on the car trajectory optimization problem provide evidence of convergence issues and of an apparently unmotivated ripple in the steer angle. Characterizing this ripple behavior is important to fully understand and exploit minimizing vehicle trajectories. We are able to isolate the key features of this chattering behavior in a very simple dynamics/objective setting. We show that the cost functional has an infimum, but an admissible minimizing input trajectory does not exist. We also show that the infimum can be arbitrarily approximated by bang‐bang inputs with a sufficiently large number of switches. We reproduce this phenomenon in numerical computations and characterize it by means of non‐existence of admissible minimizing trajectories. Copyright © 2015 John Wiley & Sons, Ltd.     

A variable time transformation method for mixed‐integer optimal control problems

The article discusses a variable time transformation method for the approximate solution of mixed‐integer non‐linear optimal control problems (MIOCP). Such optimal control problems enclose real‐valued and discrete‐valued controls. The method transforms MIOCP using a discretization into an optimal control problem with only real‐valued controls. The latter can be solved efficiently by direct shooting methods. Numerical results are obtained for a problem from automobile test‐driving that involves a discrete‐valued control for the gear shift of the car. The results are compared to those obtained by Branch&Bound and show a drastic reduction of computation time. This very good performance makes the suggested method applicable even for many discretization points. Copyright © 2006 John Wiley & Sons, Ltd.     

Bang‐bang optimal control for differentially flat systems using mapped pseudospectral method and analytic homotopic approach

The bang‐bang type optimal control problems arising from time‐optimal or fuel‐optimal trajectory planning in aerospace engineering are computationally intractable. This paper suggests a hybrid computational framework that utilizes differential flatness and mapped Chebyshev pseudospectral method to generate a related but smooth trajectory, from which the original non‐smooth solutions are achieved continuously by the analytic homotopic algorithm. The flatness allows for transcribing the original problem into an integration‐free flat outputs optimization problem with reduced number of decision variables. Chebyshev pseudospectral method is applied to parameterizing the flat outputs, and the numerical accuracy for the derivatives of flat outputs at collocation nodes, which are readily computed using differentiation matrices, is greatly enhanced by conformal map and barycentric rational interpolation techniques. Based on the obtained smooth trajectory, the analytic homotopic approach constructs an auxiliary optimal control problem whose costates are simply zero, avoiding the estimation of initial costates. The hybrid framework successfully addresses the difficulties of pseudospectral method and homotopic approach when they are applied separately. Numerical simulations of time‐optimal trajectory planning for spacecraft relative motion and attitude maneuver are presented, validating the performance of the hybrid computational framework. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal control of a magnetic bearing without bias flux using finite voltage

Conventional active magnetic bearings (AMB) are operated using a bias current (or flux) to achieve greater linearity and dynamic capability. Bias, however, results in undesirable rotating losses and consequent rotor heating. While control without bias flux is an attractive alternative, it is considerably more complex due to both force slew rate limitations and actuator non-linearity. In this paper, optimal control of a magnetic bearing without bias is investigated. A single-degree-of-freedom system consisting of a mass and two opposing electromagnets is considered. The optimal control problem is examined for a cost function that penalizes both poor regulation and rotational energy lost. Though a standard optimization procedure does not directly yield an analytical solution, it does show that the optimal control is always bang–bang including possibly a singular arc. First, the minimum time problem is solved for a simple switching law in three dimensional state space. A non-standard, physics-based approach is then employed to obtain an optimal solution for the general problem. The final result is an optimal variable structure feedback controller. This result provides a benchmark which can be used for evaluation of the performance of a practical feedback controller designed via other methods. The practical controller will be designed to support a flexible rotor and achieve robustness and optimally reject disturbance. This result may also be applied to many other applications which contain opposing quadratic actuators. © 1998 John Wiley & Sons Ltd.     

2,α‐norm optimal model reduction for optimal control problems subject to parabolic and hyperbolic evolution equations

This paper deals with a numerical solution method for optimal control problems subject to parabolic and hyperbolic evolution equations. Firstly, the problem is semi‐discretized in space with the boundary or distributed controls as input and those parts of the discretized state appearing in the cost functional as output variables. The corresponding transfer function is then approximated optimally with respect to the _2,α‐norm providing an optimally reduced optimal control problem, which is finally solved by a first‐discretize‐then‐optimize approach. To enable the application of this reduction method, a new constrained optimal model reduction problem subject to reduced systems with real system matrices is considered. Necessary optimality conditions and a transformation procedure for the reduced system to a canonical form of real matrices are presented. The method is illustrated with numerical examples where also complicated controls with many bang‐bang arcs are investigated. The approximation quality of the optimal control and its correlation to the decay rate of the Hankel singular values of the system are numerically studied. A comparison to the approach of using Balanced Truncation for model reduction is applied. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimal control of the atmospheric reentry of a space shuttle by an homotopy method

This paper deals with the optimal control problem of the atmospheric reentry of a space shuttle with a second‐order state constraint on the thermal flux. We solve the problem using the shooting algorithm combined with an homotopy method, which automatically determines the structure of the optimal trajectory (composed of one boundary arc and one touch point). Copyright © 2010 John Wiley & Sons, Ltd.     

Linear multi‐model time‐optimization

A linear optimization problem with unknown parameters from a given finite set is tackled. The problem is to find the robust time‐optimal control transferring a given initial point to a convex terminal compact set M for all unknown parameters in a shortest time. The robust maximum principle for this minimax problem is formulated. It gives a necessary and sufficient condition of robust optimality. Under natural conditions, the existence and uniqueness of robust optimal controls are proven when the resource set is a convex polytope. Several illustrating examples, including a bang–bang robust optimal control, are considered in detail. Copyright © 2002 John Wiley & Sons, Ltd.     

Nonlinear model predictive control strategy for low thrust spacecraft missions

In this paper, two nonlinear model predictive control (MPC) strategies are applied to solve a low thrust interplanetary rendezvous problem. Each employs a unique, nonclassical parameterization of the control to adapt the nonlinear MPC approach to interplanetary orbital dynamics with low control authority. The approach is demonstrated numerically for a minimum‐fuel Earth‐to‐Mars rendezvous maneuver, cast as a simplified coplanar circular orbit heliocentric transfer problem. The interplanetary transfer is accomplished by repeated solution of an optimal control problem over (i) a receding horizon with fixed number of control subintervals and (ii) a receding horizon with shrinking number of control subintervals, with a doubling strategy to maintain controllability. In both cases, the end time is left unconstrained. The performances of the nonlinear MPC strategies in terms of computation time, fuel consumption, and transfer time are compared for a constant thrust nuclear‐electric propulsion system. For this example, the ability to withstand unmodeled effects and control allocation errors is verified. The second strategy, with shrinking number of control subintervals, is also shown to easily handle the more complicated bounded thrust nuclear‐electric case, as well as a state‐control‐constrained solar‐electric case. Copyright © 2012 John Wiley & Sons, Ltd.     

Analysis of optimal controls for a mathematical model of tumour anti‐angiogenesis

Anti‐angiogenic therapy is a novel treatment approach for cancer that aims at preventing a tumour from developing its own blood supply system that it needs for growth. In this paper we consider a mathematical model where the endogenous stimulation term in the dynamics is taken proportional to the number of endothelial cells. This system is an example from a class of mathematical models for anti‐angiogenic treatment that were derived from a biologically validated model by Hahnfeldt, Panigrahy, Folkman and Hlatky. The problem how to schedule a given amount of angiogenic inhibitors to achieve a maximum reduction in the primary cancer volume is considered as an optimal control problem and it is shown that optimal controls are bang‐bang of the type 0a0 with 0 denoting a trajectory corresponding to no treatment and a a trajectory with treatment at maximum dose along that all inhibitors are being exhausted. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimization methods for the verification of second order sufficient conditions for bang–bang controls

It has been common practice to find controls satisfying only necessary conditions for optimality, and then to use these controls assuming that they are (locally) optimal. However, sufficient conditions need to be used to ascertain that the control rule is optimal. Second order sufficient conditions (SSC) which have recently been derived by Agrachev, Stefani, and Zezza, and by Maurer and Osmolovskii, are a special form of sufficient conditions which are particularly suited for numerical verification. In this paper we present optimization methods and describe a numerical scheme for finding optimal bang–bang controls and verifying SSC. A straightforward transformation of the bang–bang arc durations allows one to use standard optimal control software to find the optimal arc durations as well as to check SSC. The proposed computational verification technique is illustrated on three example applications. Copyright © 2005 John Wiley & Sons, Ltd.     

High‐order maximum principles for the stability analysis of positive bilinear control systems

We consider a continuous‐time positive bilinear control system, which is a bilinear control system with Metzler matrices. The positive orthant is an invariant set of such a system, and the corresponding transition matrix is entrywise nonnegative for all time. Motivated by the stability analysis of positive linear switched systems under arbitrary switching laws, we define a control as optimal if it maximizes the spectral radius of the transition matrix at a given final time. We derive high‐order necessary conditions for optimality for both singular and bang–bang controls. Our approach is based on combining results on the second‐order derivative of a simple eigenvalue with the generalized Legendre‐Clebsch condition and the Agrachev–Gamkrelidze second‐order optimality condition. Copyright © 2015 John Wiley & Sons, Ltd.     

Combination of optimal control approaches for aircraft conflict avoidance via velocity regulation

This paper deals with optimal control applied to one of the most crucial and challenging problems in Air Traffic Management that of aircraft conflict avoidance. We propose an optimal control model where aircraft separation is achieved by changing the speeds of aircraft, and the integral over a time window of their squared accelerations is minimized. Pairwise aircraft separation constraints constitute the main difficulties to be handled. We propose an original decomposition of the problem into 3 zones, in such a way that in two of them, no conflict occurs. Then, using the Pontryagin maximum principle, 2 new formulations of the original optimal control problem are proposed and solved via direct shooting methods. Thanks to our decomposition, these numerical methods are applied on subproblems having reduced size with respect to the original one, thus improving the efficiency of the solution process. Thirty problem instances are numerically solved, showing the effectiveness of the proposed approaches.     

Real-time computation of feedback controls for constrained optimal control problems. part 2: A correction method based on multiple shooting

On the basis of Part 1 of this paper, a numerical method is developed for the real-time computation of neighbouring optimal feedback controls for constrained optimal control problems. We use the idea of multiple shooting to develop a numerical method which has the following properties: 1. The method is applicable to optimal control problems with constraints (differential equations, boundary conditions, inequality constraints, problems with discontinuities, etc.). 2. The control variables and the switching points are computed for the remaining time interval of the process. 3. All constraints are checked. 4. The method is appropriate for real-time computations on onboard computers of space vehicles. 5. The scheme is robust in that controllable deviations from a precalculated flight path are much larger than deviations typical for perturbations occuring in space vehicles. The re-entry of a space vehicle is investigated as an example. One problem contains a control variable inequality constraint with a large variety of different switching structures, including problems with a corner. A second problem contains a state variable inequality constraint with one or two boundary points or one boundary arc. The different switching structures depend on the tightness of the constraints.