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The paper addresses a class of continuous-time, optimal control problems whose solutions are typically characterized by both bang-bang and "singular" control regimes. Analytical study and numerical computation of such solutions are very difficult and far from complete when only techniques from control theory are used. This paper solves optimal cont...
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Context 1
... an example, let us consider the construction of an arc that connects intervals n and m , 1 < n < m < K (see Figure 1). Let the singular dynamics at the intervals be xt t = X j n t j n t for n and xt t = X j m t j m t for m . ...
Context 2
... value of the slack function is ZZt out 1 = xxt in 1 − X j m t in 1 . For t out = t out 2 , System (6) is integrated from point t out 2 up to the end of interval m , because the stop condition never holds, i.e., tt never reaches j m t at this interval (see Figure 1). As a result, the slack function ZZt out 2 at t out = t out 2 is not defined. ...
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Citations
... To find a solution to binary optimal control problem, in addition to some traditional methods, such as dynamic programming [14,15] and the methods based on global maximum principle [16,17] , many other excellent methods have been proposed in recent years. Hahn et al. proposed a trust-region steepest descent method to solve binary optimal control problems by iteratively improving on existing solutions [18] . ...
... To find a solution to binary optimal control problem, in addition to some traditional methods, such as dynamic programming [14,15] and the methods based on global maximum principle [16,17], many other excellent methods have been proposed in recent years. Hahn et al. proposed a trust-region steepest descent method to solve binary optimal control problems by iteratively improving on existing solutions [18]. ...
This paper focuses on binary optimal control of fed-batch fermentation of glycerol by Klebsiella pneumoniae with pH feedback considering limited number of switches. To maximize the concentration of 1,3-propanediol at terminal time, we propose a binary optimal control problem subjected to time-coupled combinatorial constraint with the ratio of feeding rate of glycerol to that of NaOH as control variables. Based on time-scaling transformation and discretization, the binary optimal control problem is first transformed into a mixed binary parameter optimization problem consisting not only continuous variables but also binary variables, which is then divided into two subproblems via combinatorial integral approximation decomposition. Finally, a novel fruit fly optimizer with modified sine cosine algorithm and adaptive maximum dwell rounding are applied to solve the obtained subproblems numerically. Numerical results show the rationality and feasibility of the proposed method.
... In the literature, the concept of a junction has been introduced in the past [19,22], and used in indirect methods [6,7]. Most of them use junctions as shooting parameters to solve the Hamiltonian systems. ...
We propose a new strategy, called method of evolving junctions (MEJ), to compute the solutions for a class of optimal control problems with constraints on both state and control variables. Our main idea is that by leveraging the geometric structures of the optimal solutions, we recast the infinite dimensional optimal control problem into an optimization problem depending on a finite number of points, called junctions. Then, using a modified gradient flow method, whose dimension can change dynamically, we find local solutions for the optimal control problem. We also employ intermittent diffusion, a global optimization method based on stochastic differential equations, to obtain the global optimal solution. We demonstrate, via a numerical example, that MEJ can effectively solve path planning problems in dynamical environments.
... These necessary conditions are valid even for discrete control sets. In [6] a graph-based solution method exploiting the maximum principle was developed, which is however limited to single-state problems. The drawback of maximum principle based methods is that a very good initial guess of the switching structure is needed which is often not available for practical applications. ...
The article discusses a numerical approach to solve optimal control problems in discrete time that involve continuous and discrete controls. Special attention is drawn to the modeling and treatment of dwell time constraints. For the solution of the optimal control problem in discrete time, a dynamic programming approach is employed. A numerical example is included that illustrates the impact of dwell time constraints in mixed integer optimal control.
... Many results, which report progress on theoretical and computational issues for continuous-time or discrete-time versions of such problems, have appeared in the literature (see, e.g., [8,20,9,[21][22][23]19,24,[10][11][12][13][14][15][16][17]). Several successful families of algorithms have already been developed to seek the optimal control [22,24,[11][12][13]25,18]. ...
... Then the values of the derivatives of the cost functional with respect to the switching instants are obtained based on the solution of a two-point boundary-value differential algebraic equation which is obtained by the maximum principle [24]. For a class of problems whose solutions are typically characterized by both bang-bang and singular control regimes, Khmelnitsky [25] solved the globally optimal solutions by reducing them to the combinatorial search for the shortest path in a specially constructed graph. Moreover, Teo and Jennings et al. [26][27][28] proposed a control parameterization technique and the time scaling transform method [28] to find the approximate optimal control inputs and switching instants, which have been used extensively in [18,[29][30][31][32][33][34][35]. ...
... ϕ(t) = H p (t, ϕ(t),ũ(t),ṽ(t), λ 0 , λ(t)), (25) λ(t) = −H x (t, ϕ(t),ũ(t),ṽ(t), λ 0 , λ(t)), ...
In this paper, a class of optimal switching control problems is considered in which the mode sequence of active subsystems and the number of mode switchings are not pre-specified, and both the switching sequence and the control inputs are to be chosen such that the cost functional is minimized. For solving this problem, we discuss the necessary conditions for optimality and the construction method for suboptimal solutions, and develop a few sufficient conditions of judgment on the optimal or suboptimal solutions of the switched system. According to the sufficient conditions, a parallel computational algorithm is constructed to find optimal or suboptimal solutions. For illustration, two examples are solved using the proposed algorithm.
... References [1, Theorem 1, p. 234] and [2]. It is important to point out that the global maximum principle is valid even for discrete control sets V: Khmelnitsky [3] developed a graph-based solution method for single-state problems which is based on the maximum principle. Unfortunately, these approaches usually turn out to be cumbersome and complex for more complicated 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.
... Reference [3]. One of the amazing features of the maximum principle is that it is valid even for discrete control sets U: Based on the maximum principle Khmelnitsky [4] develops a graph-based solution method which is limited to single-state problems. Unfortunately, for more complicated problems these approaches usually turn out to be cumbersome and complex. ...
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.
... These techniques will result in a problem of finding an optimal set of switching parameters of a control input. These parameter optimization techniques for the design of time-optimal controllers are well addressed in earlier research151617. For verification of the optimality, the necessary condition for optimality is also included for the proposed techniques. ...
This paper presents a technique for the determination of time-optimal control profiles for rest-to-rest maneuvers of a mass–spring system, subject to Coulomb friction. A parameterization of the control input that accounts for the friction force, resulting in a linear analysis of the system is proposed. The optimality condition is examined for the control profile resulting from the parameter optimization problem. The development is illustrated on a single input system where the control input and the friction force act on the same body. The variation of the optimal control structure as a function of final displacement is also exemplified on the friction benchmark problem. Copyright © 2007 John Wiley & Sons, Ltd.
The main theme of this chapter is to derive gradient formulae for the cost and constraint functionals of several types of optimal parameter selection problems with respect to various types of parameters. An optimal parameter selection problem can be regarded as a special type of optimal control problems in which the controls are restricted to be constant functions of time. Many optimization problems involving dynamical systems can be formulated as optimal parameter selection problems. Examples include parameter identification problems and controller parameter design problems.
Suppose that one party proposes to another a contract for sharing an uncertain profit which maximizes the former’s expected utility, with respect to its beliefs, subject to a constraint on the latter’s expected utility, with respect to the latter’s beliefs. It turns out that the optimal contract, which we find, can be nonmonotone, as well as nonlinear, in the realized profit. To avoid the implausible lack of monotonicity, we formulate and solve a model constrained to have monotone increasing profits for both partners. If beliefs are identical, the (unconstrained) contract is shown to be monotone, and under certain conditions, linear. That might explain one famous contract from the history of jazz. If the other party can be assumed risk neutral, the linear contract reduces to the former receiving a constant amount, and the latter the residual net profit, as in the case of another famous contract from the history of jazz. Since in the type of partnerships, we have in mind the partners are always motivated to exert high effort due to other factors like reputation, our setting has no moral hazard or adverse selection, and the partnerships do not involve a large initial investment.
