Parallel evolutionary programming optimisation for evader's trajectory in pursuit-evasion game
A number of impulse investigations on earthing systems have been performed in the field [1-6]. The studies were conducted on a few types of earth electrode: vertical rods, horizontal electrodes and the mesh and other geometries of electrodes. The type selected may depend on the type of soil encountered and the available depth or area available. Impulse voltages of up to 3MV with impulse currents of up to 30kA were applied on earth electrodes with different configurations. The experimental results from previous studies in the field [1-6] revealed that under high currents, a ‘non-linear’ soil behaviour would occur. However, so far the field measurements were conducted on the imitative earth electrodes, and not on the live operating systems. In this study, the performance and the electrical characteristics of earthing systems under impulse conditions were investigated. For the first time, the impulse tests were conducted on the earthing systems of operational stations, in which connected to the external paths and other equipments. Impulse voltages with increasing magnitudes up to 6kV were injected into the earth electrode. The results reveal that there is no flashover or damages on the equipments at the stations/sites under tests. It was also observed that the earthing systems are mainly inductive under impulse conditions of up to 6kV. It also includes the investigations on earthing systems using computer models [7]. The earth resistance values at power frequency and low-magnitude voltage and current of these selected sites were also obtained using Current Distribution Electromagnetic Interference Grounding and Soil Structure Analysis (CDEGS) [7]. The impedance values of the earthing systems were found to be ranging between 2.7? and 9.2 ?.
Some Practical Approaches to Pursuit-Evasion Dynamic Games
For problems of realistic dynamic games, some practical approaches are proposed to the construction of suboptimal strategies of the behavior of players for different data on the choice of controls by their opponents. All the approaches proposed employ only solvers of one-sided nonlinear optimal control problems and simulation programs. The approaches are as follows: (i) give both players some suboptimal feedback strategies and conduct a great number of Monte Carlo simulations, (ii) give one player a suboptimal feedback strategy and the other player an exact one-sided nonlinear optimal control, (iii) give both players suboptimal feedback strategies dependent on parameters. Each player tries to optimize his parameters with respect to the one-sided nonlinear optimal control of the opponent. Methods (i) and (ii) can be used to solve multi-player problems. The efficiency of the method of choice of a suboptimal strategy is illustrated by examples important for actual practice.
Modified quasilinearization algorithm for optimal control problems with nondifferential constraints
This paper considers the numerical solution of optimal control problems involving a functionalI subject to differential constraints, nondifferential constraints, and terminal constraints. The problem is to find the statex(t), the controlu(t), and the parameter π so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy.A modified quasilinearization algorithm is developed. Its main property is the descent property in the performance indexR, the cumulative error in the constraints and the optimality conditions. Modified quasilinearization differs from ordinary quasilinearization because of the inclusion of the scaling factor (or stepsize) α in the system of variations. The stepsize is determined by a one-dimensional search on the performance indexR. Since the first variation δR is negative, the decrease inR is guaranteed if α is sufficiently small. Convergence to the solution is achieved whenR becomes smaller than some preselected value.In order to start the algorithm, some nominal functionsx(t),u(t), π and nominal multipliers λ(t), ρ(t), μ must be chosen. In a real problem, the selection of the nominal functions can be made on the basis of physical considerations. Concerning the nominal multipliers, no useful guidelines have been available thus far. In this paper, an auxiliary minimization algorithm for selecting the multipliers optimally is presented: the performance indexR is minimized with respect to λ(t), ρ(t), μ. Since the functionalR is quadratically dependent on the multipliers, the resulting variational problem is governed by optimality conditions which are linear and, therefore, can be solved without difficulty.To facilitate the numerical solution on digital computers, the actual time θ is replaced by the normalized timet, defined in such a way that the extremal arc has a normalized time length Δt=1. In this way, variable-time terminal conditions are transformed into fixed-time terminal conditions. The actual time τ at which the terminal boundary is reached is regarded to be a component of the parameter π being optimized.The present general formulation differs from that of Ref. 3 because of the inclusion of the nondifferential constraints to be satisfied everywhere over the interval 0⩽t⩽1. Its importance lies in that (i) many optimization problems arise directly in the form considered here, (ii) there are problems involving state equality constraints which can be reduced to the present scheme through suitable transformations, and (iii) there are some problems involving inequality constraints which can be reduced to the present scheme through the introduction of auxiliary variables.Numerical examples are presented for the free-final-time case. These examples demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.