A modified pseudospectral scheme for accurate solution of Bang‐Bang optimal control problems

Abstract

In the present contribution, a modified Legendre pseudospectral scheme for accurate and efficient solution of bang-bang optimal control problems is investigated. In this scheme control and state functions are considered as piecewise constant and piecewise continuous polynomials, respectively, and the switching points are also taken as decision variables. Furthermore, for simplicity in discretization, the integral formulation of the dynamical equations is considered. Thereby, the problem is converted into a mathematical programming problem which can be solved by the well-developed parameter optimization algorithms. The main advantages of the present method are that: (i) it obtains good results even by using a small number of collocation points and the rate of convergence is high; (ii) the switching times can be captured accurately; and (iii) the wrongly chosen number of switching points can be detected by the results of the method. These are illustrated through a numerical implementation of the method on three examples and the efficiency of the method is reported.A MODIFIED LEGENDRE PSEUDOSPECTRAL SCHEME 669 With this gradient, the switching times are corrected at each iteration. Lee et al. [11] presented the Control Parametrization Enhancing Transform technique (CPET), that maps a time optimal control problem to one with switching times at fixed locations, with the original switching times as parameters. This enables switching times and singular control values to be computed readily and precisely, by many existing techniques. Bertrand and Epenoy [12] developed a perturbation approach called continuation smoothing technique. They added a perturbed energy term in the objective function to yield a continuous optimal control. The perturbation parameter is then updated by a simple continuation procedure or through a homotopy method. Kaya and Noakes [13] gave the Switching Time Computation (STC) method. The STC method finds a feasible, but not necessarily optimal, concatenation of constant-input arcs from an initial point to a target point for a given number of switchings. Authors of [13] developed the Time-Optimal Switchings (TOS) algorithm [14], for time-optimal switching control of general nonlinear systems with a single control input. The TOS algorithm needs a feasible bang-bang solution to start, which is typically obtained using STC. More recently, Kaya et al. [15] gave a technique that can achieves what STC and TOS can achieve together, albeit requiring also the second-order variations for finding a stationary point of the Lagrangian. Also more general performance indices than just minimizing terminal time is considered. For other works placed to computation of bang-bang optimal control problems, see [16][17][18][19][20][21].The pseudospectral methods are powerful tools for numerical solution of ordinary and partial differential equations, which can be traced back to 1970s [22][23][24]. If one want to solve an ordinary or partial differential equation to high accuracy on a simple domain, and if the data definin...

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