Engine Controller Optimal Design based on Grey Wolf Optimizer

Abstract: This paper describes a controller parameter optimization algorithm for a high-bypass turbofan engine controller. The turbofan engine is a nonlinear system working under extreme environment. Under such circumstances the engine system should ensure high reliability, stability and transient response performance. The contribution of the research lies in providing a new off-line optimization algorithmthe Grey Wolf Optimizer (GWO) to optimize the parameters of turbofan engine’s Proportional Integral controller to provide the engine a better transient performance under certain working condition. The simulation results shows that compared with the original controller, the new controller provides better transient performance.


Introduction
Gas turbine engines (GTE), especially high bypass turbofan engine have played a significant role in the expansion of the flight capabilities of modern aircraft [1].The high bypass turbofan engine installed in commercial aircraft must be operated by means of feedback control.The objective of the controller is to achieve nice thrust response performance while maintaining the engine output inside safety intervals [2].To improve the transient performance of the turbofan engine, many advanced methods have been applied to the turbofan its control system.
Generally speaking, the turbofan engine is a complex system with strong nonlinearity.The performance of certain turbofan engine will change considerably with working condition, aging and maintenance.So, designing its controller to fulfill the requirements is a very difficult task.Classical linear compensation has been widely applied on the turbofan engine control to govern the engine close to a certain operating point [2].Since the thrust of the turbofan engine installed on the aircraft cannot be measured in real time, Low Pressure Turbine (LPT) rotation speed or Engine Pressure Ratio (EPR) measured is usually designed to be the controlled variable.And, although many advanced control methods appear [2].
Since the thrust of the turbofan engine installed on the aircraft cannot be measured in real time, Low Pressure Turbine (LPT) rotation speed or Engine Pressure Ratio (EPR) measured is usually designed to be the controlled variable.And, although many advanced control methods appear [2], the classical Proportional Integral (PI) controller is still used in Turbofan engine control because of its robustness and adaption.Optimizing the PI controller design is still a valuable method to improve the turbofan engine performance.
If the controlled variable of the turbofan engine is the LPT rotation speed, the control problem is similar to the Load Frequency Control (LFC) problem in stationary steam thermal power system.In Guha and Sharma's researches, an evolutionary algorithm (EA) known as grey wolf optimization (GWO) has been applied for optimal design of PI/PID controller to solve LFC problem [3,4].The simulated result shows that the optimized controller design shows pretty good performance.
In view of the above discussion, the main aim of the present research is to use the GWO algorithm to optimize the PI controller design of a high bypass turbofan engine.A dynamic model of P&W JT9D established in previous research is applied here [5].Based on the given model, the turbofan engine working under changing working condition is simulated.The parameter of the PI controller is optimized by the GWO under certain working condition.The simulated transient response of the optimal controller is evaluated.
The following part of the paper is organized as follows.Section 2 the system scheme of the certain turbofan engine.The GWO algorithm is briefly described in Section 3. Section 4 presents detailed information about the controller.Section 5 is about the result and the comparison.Section 6 shows the conclusion and future work.

Model Description
The components structure and the close-loop scheme of the certain turbofan engine is shown in Figure 1 [5,6].The whole dynamic model of the turbofan engine is built based on the example provided by NASA [6].The steady state nonlinear maps of the turbo machinery, including the Fan, Low Pressure Compressor (LPC), High Pressure Compressor (HPT), Low Pressure Turbine (LPT) are achieved from real data collected.The Dynamic Iterative Solver (DIS) inside the scheme can simulate the transient state performance of the engine under dynamic state condition.According to the previous research, the JT9D modeled shows nice accuracy compared with real data [5][6][7].The component system scheme of the turbofan engine is presented in Figure .1.The dynamic system scheme is presented in Figure 2.  3 Optimiazation method

Grey Wolf Optimizer
GWO, as a heuristic optimization technique, can find the candidate solution from very large solution space with no specific input parameters required.Such characteristics are very suitable to deal with the nonlinear problems like controller parameter tuning.The algorithm is designed to achieve its target through mimicking the social hierarchy and hunting mechanism of the wolf society.
The social hierarchy of the wolf society is a special one.The wolves at the top of the wolf pack are called alphas.They are responsible for making important decisions about hunting, sleeping place, wake time, etc.However, some kind of democratic behavior is also observed in which they follow other wolves in the wolf pack.In gatherings, the entire wolf pack acknowledges the alpha by holding their tails down.The alpha wolves are also called the dominant wolves.
In the second level of the hierarchy, the grey wolves are named as beta category wolves and they are subordinate of alpha category wolves.They help alphas in the decision-making process and/or other pack activities.They are probably the best candidate and may transform into the alpha category wolves in case one of the alpha wolves passes away or become very old.
The lowest stage of the hierarchy is occupied by the omega types of wolves.They are basically used as a scapegoat and always follow the decision made by other dominant wolves.They are the worst category of wolves those are all owed to eat.
The wolves which do not come under alpha, beta and omega categories are grouped under delta or subordinate category.Delta types of wolves always follow the alphas and betas but dominate omegas [3,9].
During hunting, the main steps are listed as follows and shown in Figure 3. (i) Tracking, chasing and approaching the prey.(ii) Pursuing, encircling and harassing the prey until it stops moving.(iii) Attack towards the prey.In the following sections, the hunting process is rewritten into three stages: encircling prey, hunting and attacking [9].The characteristics of the social hierarchy, and hunting are represented mathematically in the following section to show how the algorithm works.

Social Hierarchy
For modeling of the social hierarchy of the grey wolf, alpha is considered to be the fittest solution followed by beta and delta, respectively, and the rest of the solutions are group edunder omega.In GWO, the hunting (optimization) process is guided by alpha, beta and delta, whereas omega always follows these three wolves [3].The society structure of the wolf pack is presented in Figure 4.

Encircling Prey
In order to mathematically model encircling behavior the following equations are presented as: (1) (2) Where indicates the current iteration, and are coefficient vectors, is the position vector of the prey, and indicates the position vector of a grey wolf.and are defined as follows: (3) Where components of are linearly decreased from 2 to 0over the course of iterations and , are random vectors in [3,9].

Hunting
In order to mathematically simulate the hunting behavior of grey wolves, we suppose that the alpha (best candidate solution) beta, and delta have better knowledge about the potential location of prey.Therefore, we save the first three best solutions obtained so far and oblige the other search agents (including the omegas) to update their positions according to the position of the best search agents [9]. ( (11)

Attack
As the cost function converges to the neighborhood of a certain point, the cost function has reached the optimal value.The wolves begin to attack the prey.

Whole Optimization Process Overview
The whole optimization pseudocode is shown Figure 5.According to the dynamic scheme shown in Figure 1.This PI controller is used to govern the LPT rotation Speed (LPS).The parameters required to be optimized are proportional gain and integral gain .The PI controller output/plant input can be described as following equations: Here stands for the input control signal, which is the fuel signal.stands for the error signal, which is the difference between , the demanded, and the measured.

Objective Function
There exist many objective functions which are used to evaluate the performance of a controller.In this research, Integral Time Absolute Error (ITAE) is applied to evaluate the controller's performance. (14)

Implementation of GWO
In this paper, GWO algorithm is applied to solve turbofan engine control problem.The algorithm sequences of the proposed method are listed as below [3].
Step 1. Initialize input parameters of GWO algorithm such as search agents no, number of control variables according to the controller structure, upper and lower bounds of the search space, number of elitism parameters and total number of generations.
Step 2. In the initialization process, search agents or grey wolves (a two row vector ) are randomly generated between upper and lower bounds in the search space.
Step 3. Evaluate the fitness function using (14) and assign alpha, beta, delta wolves in the search space.
Step 4. Update the positions of alpha, beta and delta using pseudo code shown in Figure 6.
Step 5. Update the positions of alpha, beta and delta using pseudo code shown in Figure 6.

Simulation Result and Discussion
To test the effectiveness and superiority of proposed algorithm, the whole process is simulated under MATLAB condition and GWO is written in the.m file.ITAE criterion based objective function is minimized using GWO algorithm to find optimum gains of controller parameters.

Simulated Turbofan Working Condition
The turbofan engine is simulated working under certain outer condition, which is listed in the Table 1.In the30s simulation, the is a step change signal from 3667r/min to 3467r/min at 15s.

Algorithm Setting
Search agents (the capacity of the wolf pack), Iteration time and the boundary of searching are required to be set initially.In this research, the number of search agents is 30, and the max iteration time is 50.The original control parameters provided by the given model in previous research are , [6].To search for the optimal control parameters, specific searching boundary is selected.The boundary of the searching area is built around the original control parameters.The lower boundary of the searching area is .The higher boundary of the searching area is .For simplicity, searching of the controller parameter rsearching of the controller parameter and shares the same boundary in this research.

Result and Comparison
The simulation results can be seen in Table 2 and Figure 7.It can be seen that the performance of the controllers are well optimized.All of the optimized controllers show pretty good performance compared with the original controller.

Conclusion
From the research carried out, the PI controller parameters for LPS control of JT9D turbofan engine are optimized through GWO.The simulation work has been done to verify the effectiveness of the proposed method.The simulation result shows that compared with the controller parameters given in previous research, the optimized parameters show better performance under certain working condition.

Table 1 .
Initial condition

Table 2 .
Simulation result