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Loss of Control (LOC) is the primary factor responsible for the majority of fatal air accidents during past decade. LOC is characterized by the pilot’s inability to control the aircraft and is typically associated with unpredictable behavior, potentially leading to loss of the aircraft and life. In this work, the minimum time dynamic optimization problem to LOC is treated using Pontryagin’s Maximum Principle (PMP). The resulting two point boundary value problem is solved using stochastic shooting point methods via a differential evolution scheme (DE). The minimum time until LOC metric is computed for corresponding spatial control limits. Simulations are performed using a linearized longitudinal aircraft model to illustrate the concept.

Air crash analyses during the past decade have concluded that about 40 percent of fatal air accidents in civil aviation occur due to Aircraft Loss of Control (LOC), the most contributing factor amongst others [

Quantifying LOC boundaries is a first step toward addressing the larger issue of LOC prevention. While there exist envelope protection features on an aircraft, they are of little use during LOC flight regimes due to degradation of normal control modes [

This remainder of the paper is structured as follows. Section 2 provides an overview of the minimum time problem. Section 3 treats the optimal control problem using Pontryagin Maximum Principle and describes the resulting two point boundary value problem (TP-BVP). The solution to the TPBVP using differential evolution (DE) based methods is described in Section 4. Simulation results based on linear longitudinal model from [

We consider the problem of obtaining the minimum time to exit the flight envelope for a linearized longitudinal aircraft model. Let the operating envelope be defined in the

where,

where,

The optimal control problem is posed as follows: Given an initial point in state space

Consider the linear dynamical system given by:

where,

The cost functional for the minimum time problem then becomes,

where,

In the framework of PMP, the existence of an optimal control also mandates the existence of co-states (denoted by

The optimal control law is the one that maximizes the Hamiltonian at every time step from

and so, the control law can be expressed as:

where,

subject to the boundary conditions:

where,

boundary value problem (TPBVP) is solved by matching the initial conditions of the unknown variables

In other words, the solution to the TPBVP is obtained by solving for the zeros of

Differential Evolution (DE) is a metaheuristic iterative optimization strategy that tries to improve candidate solutions over generations. Since its inception in the mid 90s, there has been a growing interest in using DE due to its simple yet powerful approach to solve several engineering optimization problems [

During the first generation, the population of guess vectors was initialized. Mutation was applied to each agent to generate the mutants. During the mutation step, a “local best” candidate was used along with the “global best” and “random” candidates to update the search direction, similar to that of Particle Swarm Optimization (PSO). These variants were crossed over with the existing agents based on a cross over probability (CR). In this implementation, a binomial crossover was performed to generate trial solutions which were compared to their counterparts from the original population to populate the next generation.

The objective function to be minimized is the final error, which depends on the initial conditions^{th} agent ^{th} agent’s best (minimum error) historical position as of current generation be represented by _{max} denote the parameters corresponding to mutation weight, cross over probability, number of agents in population, maximum allowable generations respectively, then the flow chart depicting differential evolution solution is shown in

The solution to the TPBVP yields the initial conditions (IC) for forward simulation of the minimum time trajectories. The optimal control problems from an initial state _{1}-BP) overlapping with the optimal trajectory of another IC (IC_{2}-BP) leverages information from the optimal trajectory computation of the latter.

It can also be observed from

The convergence of the optimal solution using the Differential Evolution algorithm is shown in

This work investigated the issue of LOC prediction using tools from optimal control theory to develop spatio-

Point | A | B | C | D | E | F | G | H | I | J | K | L | C_{1} | C_{2} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Min Time | 1.87 | 1.89 | 2.06 | 2.41 | 1.68 | 1.19 | 0.69 | 0.24 | 0.10 | 0.21 | 0.79 | 1.33 | 0.20 | 2.08 |

temporal pilot aids. The time optimal problem to violate the Loss-of-control boundary was considered. The minimum time to reach any point on the boundary was computed using PMP. The resulting TPBVP was solved using Differential Evolution. Simulation of the linear longitudinal model was carried out in MATLAB and the optimal trajectories were found to be not necessarily the minimum phase space distance paths. The minimum time to envelope was computed as the infimum of minimum times to various boundary points. Future work could investigate minimum time trajectory generation over a space of initial conditions (IC), which would expedite optimal trajectory generation by leveraging the principle of optimality. Alternative strategies for solving TPBVP like hybrid optimization schemes could be explored to reduce computational loads and restricted control bounds could be considered to obtain practically viable minimum times. Further, the solution methodology employed here (using PMP-DE) can be readily extended to nonlinear models, which better characterize dynamics near LOC boundaries away from local trim conditions. In conclusion, this work provides an initial step to augment spatial pilot aids with minimum time temporal information aimed at LOC prevention.

The authors thank the NASA Ames research center for their support.

ChaitanyaPoolla,Abraham K.Ishihara, (2015) Temporal Prediction of Aircraft Loss-of-Control: A Dynamic Optimization Approach. Intelligent Control and Automation,06,241-248. doi: 10.4236/ica.2015.64023