Computational optimal control of the terminal bunt manoeuvre—Part 2: minimum‐time case

This is the second part of a paper studies trajectory shaping of a generic cruise missile attacking a fixed target from above. The problem is reinterpreted using optimal control theory resulting in a minimum flight time problem; in the first part the performance index was time‐integrated altitude. The formulation entails non‐linear, two‐dimensional (vertical plane) missile flight dynamics, boundary conditions and path constraints, including pure state constraints. The focus here is on informed use of the tools of computational optimal control, rather than their development. The formulation is solved using a three‐stage approach. In stage 1, the problem is discretized, effectively transforming it into a non‐linear programming problem, and hence suitable for approximate solution with DIRCOL and NUDOCCCS. The results are used to discern the structure of the optimal solution, i.e. type of constraints active, time of their activation, switching and jump points. This qualitative analysis, employing the results of stage 1 and optimal control theory, constitutes stage 2. Finally, in stage 3, the insights of stage 2 are made precise by rigorous mathematical formulation of the relevant two‐point boundary value problems (TPBVPs), using the appropriate theorems of optimal control theory. The TPBVPs obtained from this indirect approach are then solved using BNDSCO and the results compared with the appropriate solutions of stage 1. The influence of boundary conditions on the structure of the optimal solution and the performance index is investigated. The results are then interpreted from the operational and computational perspectives. Copyright © 2007 John Wiley & Sons, Ltd.     

Minimizing the maximum heating of a re-entering space shuttle: An optimal control problem with multiple control constraints

This paper presents a numerical investigation of an optimal re-entry manoeuvre under several control and control-state constraints. The essential aim of the optimization is the minimization of the maximal skin temperature of an orbiter. It is demonstrated that the interaction of different solution techniques is indispensable in order to successfully treat such a highly constrained problem. The reduction of the skin temperature is significant. Moreover, the maximum heat flux and the integrated heat flux are also reduced considerably by the optimization.     

Time‐optimal control of automobile test drives with gear shifts

We present a numerical method and results for a recently published benchmark problem (Optim. Contr. Appl. Met. 2005; 26 :1–18; Optim. Contr. Appl. Met. 2006; 27 (3):169–182) in mixed‐integer optimal control. The problem has its origin in automobile test‐driving and involves discrete controls for the choice of gears. Our approach is based on a convexification and relaxation of the integer controls constraint. Using the direct multiple shooting method we solve the reformulated benchmark problem for two cases: (a) As proposed in (Optim. Contr. Appl. Met. 2005; 26 :1–18), for a fixed, equidistant control discretization grid and (b) As formulated in (Optim. Contr. Appl. Met. 2006; 27 (3):169–182), taking into account free switching times. For the first case, we reproduce the results obtained in (Optim. Contr. Appl. Met. 2005; 26 :1–18) with a speed‐up of several orders of magnitude compared with the Branch&Bound approach applied there (taking into account precision and the different computing environments). For the second case we optimize the switching times and propose to use an initialization based on the solution of (a). Compared with (Optim. Contr. Appl. Met. 2006; 27 (3):169–182) we were able to reduce the overall computing time considerably, applying our algorithm. We give theoretical evidence on why our convex reformulation is highly beneficial in the case of time‐optimal mixed‐integer control problems as the chosen benchmark problem basically is (neglecting a small regularization term). Copyright © 2009 John Wiley & Sons, Ltd.     

Solving mixed‐integer optimal control problems by branch&bound: a case study from automobile test‐driving with gear shift

The article discusses the application of the branch&bound method to a mixed integer non‐linear optimization problem (MINLP) arising from a discretization of an optimal control problem with partly discrete control set. The optimal control problem has its origin in automobile test‐driving, where the car model involves a discrete‐valued control function for the gear shift. Since the number of variables in (MINLP) grows with the number of grid points used for discretization of the optimal control problem, the example from automobile test‐driving may serve as a benchmark problem of scalable complexity. Reference solutions are computed numerically for two different problem sizes. A simple heuristic approach suitable for optimal control problems is suggested that reduces the computational amount considerably, though it cannot guarantee optimality anymore. Copyright © 2005 John Wiley & Sons, Ltd.     

Costate approximation in optimal control using integral Gaussian quadrature orthogonal collocation methods

Two methods are presented for approximating the costate of optimal control problems in integral form using orthogonal collocation at Legendre–Gauss (LG) and Legendre–Gauss–Radau (LGR) points. It is shown that the derivative of the costate of the continuous‐time optimal control problem is equal to the negative of the costate of the integral form of the continuous‐time optimal control problem. Using this continuous‐time relationship between the differential and integral costate, it is shown that the discrete approximations of the differential costate using LG and LGR collocation are related to the corresponding discrete approximations of the integral costate via integration matrices. The approach developed in this paper provides a way to approximate the costate of the original optimal control problem using the Lagrange multipliers of the integral form of the LG and LGR collocation methods. The methods are demonstrated on two examples where it is shown that both the differential and integral costate converge exponentially as a function of the number of LG or LGR points. Copyright © 2014 John Wiley & Sons, Ltd.     

Exact penalization of terminal constraints for optimal control problems

We study optimal control problems for linear systems with prescribed initial and terminal states. We analyze the exact penalization of the terminal constraints. We show that for systems that are exactly controllable, the norm‐minimal exact control can be computed as the solution of an optimization problem without terminal constraint but with a nonsmooth penalization of the end conditions in the objective function, if the penalty parameter is sufficiently large. We describe the application of the method for hyperbolic and parabolic systems of partial differential equations, considering the wave and heat equations as particular examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Short communication: A collocation-type method for linear quadratic optimal control problems

This communication presents a spectral method for solving time-varying linear quadratic optimal control problems. Legendre–Gauss–Lobatto nodes are used to construct the mth-degree polynomial approximation of the state and control variables. The derivative x ·(t) of the state vector x (t) is approximaed by the analytic derivative of the corresponding interpolating polynomial. The performance index approximation is based on Gauss–Lobatto integration. The optimal control problem is then transformed into a linear programming problem. The proposed technique is easy to implement, efficient and yields accurate results. Numerical examples are included and a comparison is made with an existing result.     

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.     

A Computational method for time-lag control problems with control and terminal inequality constraints

A computational algorithm for a class of time-lag optimal control problems involving control and terminal inequality constraints is presented. The convergence properties of the algorithm are also investigated. To test the algorithm, several examples are solved.     

On the optimal control of the Vidale-Wolfe advertising model

The Vidale-Wolfe advertising model is a singular optimal control problem with a non negative control. Sufficient conditions on a generalized time-varying market, for which a solution can be found, are given. Time is parametrized to describe impulsive optimal trajectories in a conventional manner. The solution is then found and verified by using the Hamilton-Jacobi-Bellman equation on the parametrized problem. © 1997 John Wiley & Sons, Ltd.     

A novel continuous genetic algorithm for the solution of optimal control problems

In this paper, the Continuous Genetic Algorithm (CGA), previously developed by the principal author, is applied for the solution of optimal control problems. The optimal control problem is formulated as an optimization problem by the direct minimization of the performance index subject to constraints, and is then solved using CGA. In general, CGA uses smooth operators and avoids sharp jumps in the parameter values. This novel approach possesses two main advantages when compared to other existing direct and indirect methods that either suffer from low accuracy or lack of robustness. First, our method can be applied to optimal control problems without any limitation on the nature of the problem, the number of control signals, and the number of mesh points. Second, high accuracy can be achieved where the performance index is globally minimized while satisfying the constraints. The applicability and efficiency of the proposed novel algorithm for the solution of different optimal control problems is investigated. Copyright © 2010 John Wiley & Sons, Ltd.     

The optimal feedback control of the linear‐quadratic control problem with a control inequality constraint

In this paper, we consider the linear‐quadratic control problem with an inequality constraint on the control variable. We derive the feedback form of the optimal control by the agency of the unconstrained linear‐quadratic control systems. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

Robust estimation of the normal to a curve using optimal control

We propose an optimal control problem whose optimal command approximates the normal vector field to a given curve. This problem is obtained by studying a partial differential equation satisfied by a map that jumps across the given curve. The gradient of the cost function is then estimated by an adjoint method, and an explicit algorithm is proposed to obtain the optimal command. Examples show that this numerical estimation of the normal is robust, in the sense that when the curve is not a simple closed curve, or when it is incomplete (dashed), the solution is still a good approximation of the normal. As applications, we show how the optimal state can help closing discontinuous curves and improve image restoration; it also provides a coloring of simple planar maps. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical solution of time-delayed optimal control problems with terminal inequality constraints

In this short communication we consider an approximation scheme for solving time-delayed optimal control problems with terminal inequality constraints. Time-delayed problems are characterized by variables x (t - τ) with a time-delayed argument. In our scheme we use a Páde approximation to determine a differential relation for y (t), an augmented state that represents x (t - τ). Terminal inequality constraints, if they exist, are converted to equality constraints via Valentine-type unknown parameters. The merit of this approach is that existing, well-developed optimization algorithms may be used to solve the transformed problems. Two linear/non-linear time-delayed optimal control problems are solved to establish its usefulness.     

Applications of the exterior penalty method in constrained optimal control problems

This paper gives a theoretical analysis of the applications of the exterior penalty method in continuous-time non-linear programs and constrained optimal control problems. As an example, the Cauchy Inequality in the continuous case is proved. Also, the exterior penalty method is used to treat constrained optimal control problems. It is proved, under suitable assumptions, that a constrained optimal control problem can be solved by performing a sequence of unconstrained optimal control problems, and the constrained solution to the constrained optimal control problem can be obtained as the limit of the solutions to the sequence of unconstrained optimal control problems. In using the exterior penalty method to solve constrained optimal control problems it is usually assumed that each of the modified unconstrained optimal control problems has at least one solution. An existence theorem for these problems is also given.     

Pointwise‐constrained optimal control of a semiactive vehicle suspension

This article presents an optimal control strategy (OCS) for semiactive vehicle suspensions with road profile sensors. The suspension is modeled as a quarter‐car model with a magnetorheological (MR) damper. The OCS main objective is to minimize the fourth‐power acceleration of the sprung mass. In addition, three pointwise constraints of the model are taken into account when the optimal control problem is solved: suspension travel limits (upper and lower) and tyre vertical force. In order to deal with a large number of constraints, we implement the gradient optimization method based on the method of moving asymptotes routine, which shows very good performance reaching optimal controls while satisfying the constrains. The solution has been compared with two passive MR damper configurations (low and high damping) as well as Skyhook and Balance control strategies for three different road inputs. Results show that OCS fulfills the constraints and reduces the sprung mass acceleration peak and the root‐mean‐quad acceleration up to 59% , in comparison to passive strategies.     

Non‐smooth monotonicity constraints in optimal control problems: Some economic applications

This paper presents a theorem on necessary conditions for optimal control problems containing monotonicity constraints that bear on a joint function of the control variable, the state variable, and the time. These constraints are often found, under continuity and piecewise smoothness assumptions for the endogenous variables of the problem, in various economic fields that include monopoly regulation, non‐uniform pricing, implicit contracts, and optimal taxation. After applying our theorem to a general incentive provision model, we show its usefulness in relaxing the standard continuity and smoothness assumptions, for the case of two screening problems among those that have received more attention in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

Singular perturbation analysis of the closed-loop discrete optimal control problem

The closed-loop optimal control of a linear, time-invariant, singularly perturbed discrete system is considered. The resulting matrix Riccati difference equation is formulated in the singularly perturbed structure. It is observed that the degeneration affects some of the final conditions of the Riccati equation. A singular perturbation method is developed to obtain approximate solutions in terms of an outer series and a final correction series. The outer series takes advantage of the order reduction associated with degeneration and the correction series takes care of the affected final conditions. Two examples are given to illustrate the proposed method.     

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.