Variable feedback gain control design based on particle swarm optimizer for automatic fighter tracking problems
Graphical abstract
Highlights
► A PSO-based variable feedback gain controller (PSO-based VFGC) to deal with the automatic fighter tracking (AFT) problem. ► The PSO-based VFGC is designed to obtain the control value of a pursuer through an error-feedback gain controller. ► Once conditions of system closed-loop stability have been satisfied, the optimal feedback gains and control values can be obtained through PSO. ► The performance of the proposed method is superior to that of its alternatives.
Introduction
This paper proposes the particle swarm optimizer-based variable feedback gain controller (PSO-based VFGC) to deal with automatic fighter tracking (AFT) problems. The proposed PSO-based VFGC provides on-line searching abilities within a control period for the optimal feedback gain to offer a fighter the best control strategy for tracking an enemy target based on its trajectory maneuvering capability. The AFT problem is similar to a general evader–pursuer maneuvering automation problem between the dynamic systems of two highly interactive objects. The majority of approaches to the problems of maneuvering automation rely heavily on optimization techniques using the differential game theory [1], [2], [3], [4], [5], [6], [7]. For the purpose of keeping the AFT problem mathematically tractable and solvable using differential game, some limitations are necessary. The dynamic behaviors of the pursuer and the evader must be clearly defined mathematically as part of the limitations. Due to this limitation, the solution of the AFT problem is deviated from the reality in a real-life pursuit-evasion situation. This paper assumes that the fighter could not predict future behaviors of the targeted enemy; therefore the fighter would not know the characteristics and dynamic range of the regulated system states in advance. Under these uncertain conditions, the states’ parameters selection and the input weighting matrices could affect the system performance. Fixed gain controllers using the off-line linear quadratic regulator (LQR) optimization methods based on the Ricatti equation and linear matrix inequality (LMI) constraints have difficulties tackling these problems.
In the Riccati equation based LQR optimization method of designing the optimal feedback gain [8], three items must be defined first, such as the characteristics of the regulated system states, the final time, and the time step of the adjacent system states variation. Then the optimal feedback gain which is variable in respect to time can be obtained using the Riccati equation from the final time backwards to the initial time in the time backward manner. This kind of method to obtain the optimal feedback gain can be viewed as an off-line approach. In a real time situation, however, the final time is too hard to confirm, such that the variable feedback gain becomes difficult to obtain. In order to solve the problem, the final time can be set as an infinitely large value, and then the variable feedback gain is turned into a fixed feedback gain, which is obtained by the Riccati equation [8]. In recent years, the Riccati equation based LQR methods have performed well in many control problems [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. Since then, emerging tracking techniques based on LMI constraints have been developed to achieve better results than Riccati equation based LQR optimization, such as the linear parameter varying (LPV) LMI technique, to obtain an optimal LPV controller [23]. In addition, there is the alternate method of utilizing the three optimal gain parameters of proportional–integral–derivative (PID) controllers to solve regulation control problems [20], [21], [22], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33].
The algorithms mentioned above for obtaining fixed optimal feedback gain and the parameters of a PID controller are performed in an off-line environment. These methods need to know the characteristics of the regulated system states. In the AFT problem discussed in this paper, the same characteristics are unobtainable beforehand. A careful selection of the proper parameters, such as the states and the input weighting matrices will affect the effectiveness of the system. In the LPV LMI technique, an optimal feedback gain can be obtained for dealing with the system uncertainty status. In the AFT problem discussed in this paper, the linear model obtained by the input–output linearization technique will be given only one optimal feedback gain using the LPV LMI based technique. A single optimal feedback gain cannot properly deal with different fighter tracking statuses as discussed in the paper since the future dynamic behaviors of the targeted enemy are assumed to be unknown. Fixed gain controllers designed using the off-line LQR optimization methods based on the Ricatti equation and LMI constraint have difficulties dealing with the problem. Because the variable feedback gains of system are unobtainable beforehand, and the selection of parameters for the states and input weighting matrices would affect the efficiency of the system. Hence, this paper proposes an on-line method to obtain and supply variable gains to the feedback controller as a solution.
There is a non-linear model predictive control (NLMPC) approach that uses on-line optimization technique for fighter tracking problems [34]. Although the NLMPC approach can be used to control the non-linear model directly, the stability analysis of close-loop system is also important. Since in the discussed AFT problems, the feedback gain values of control systems could not be efficiently obtained from general parameters optimization methods for control problems. The process of obtaining feedback gain values should be regarded as searching-for-optimal-parameters problems. The faster a search speed is the more advantageous it is for the fighter in AFT problems. Hence, the proposed control system must adjust to the best feedback gains for the fighter's automatic tracking as quickly as possible in response to the changes in the enemy target's flying trajectory. It is discovered during the analysis of the states feedback gains controller that conventional optimization methods, such as the gradient search method, are restricted to the eigenvalues of the linear system matrix that increases the difficulty and time-consumption of finding the global optimum solution. Evolutionary computation (EC) is the most advantageous method for this issue.
When using EC, this restriction caused by the eigenvalues of the linear system matrix can be eliminated. Several different methods exist for EC, including genetic algorithm (GA) and PSO [36]. In this case, the faster a method is, the more advantageous it is for the fighter. GA simulates a biological chromosome, encoding parameters in problems like a chromosome. The more parameters a problem has, the longer the genetic codes, causing a direct increase in computation time to encode and decode. In comparison to GA, the PSO contains a simpler architecture, more effective searching methods, and more accurate results [37]. There are also several successful PSO applications for control problems [38], [39], [40], [41], [42], [43]. In this paper, the proposed method incorporates PSO to search for the optimal state feedback gains for the controller. Once the optimal state feedback gain is obtained, the pseudo control value is obtained as well, and then the actual control value can be obtained through a decoupling matrix.
In order to develop the PSO-based VFGC, the on-line design of the feedback gain controller must be analyzed. The input–output feedback linearization is employed for its characteristics of state transformation [35], and utilized to transform the fighter's original non-linear system equation into a linear system equation. Contained within the linear system is a set of pseudo control variables; there is a transformation relationship through the decoupling matrix between the fighter's actual control variable and the pseudo control variable. The pseudo control variable can be designated as the states feedback gain controller, which will give the fighter closed loop stability when controlling actions to pursue the enemy target; the fighter's best control strategy will be used to obtain the best states feedback gains for the controller.
The simulation environments discussed in this paper are based in Matlab; simulations are run for three different enemy flight patterns (of) non-varying flight pattern (non-maneuvering), varying flight pattern (maneuvering), and dramatically varying flight pattern (heavy-maneuvering). The PSO-based VFGC is compared to the Ricatti equation based [44] and the LMI constraint based LQR methods. In the situation that the enemy fighter exhibits less frequent movement, as in non-maneuvering and maneuvering flight patterns, the Ricatti equation and LMI constraint based LQR methods can track the enemy with a minor relative position error. When the enemy exhibits more dynamic flight, as in heavy-maneuvering patterns, the same LQR methods failed to track the enemy with the same accuracy. The method proposed in this paper was able to track the enemy with minor relative position errors in all three enemy flight pattern simulations. The results reveal that the proposed algorithm out performs the Ricatti equation and LMI constraint based LQR methods, and accomplishes the expectations set by the research. The results also show that PSO-based VFGC can obtain the optimal feedback gain value on-line. The feedback gain value does not require information on the characteristics of the regulated system states in advance and is adjustable according to the behavior variations of the enemy.
This paper is divided into five sections, the first section is the introduction, the second contains the flight dynamic equation, the design and analysis of the fighter's variable feedback gain controller, the third is the structure of the PSO-based VFGC algorithm, the fourth contains the simulation results, and the fifth contains the conclusion.
Section snippets
Flight dynamic equation of fighter
The posture and position of the fighter in a three-dimensional (3D) environment is shown in Fig. 1. The fighter's dynamic behaviors are listed below:where x, y and z are the fighter's positions in the inertia coordinate system with units in meters, and there first order differential are denoted as and . The path angle γ with unit in degrees represents the angle between the velocity vector and the horizon. The heading angle Ψ with units
Structure of PSO-based VFGC for fighter
This section explores the structure of the PSO-based VFGC for the pursuer. At first the algorithm of PSO is introduced. And then, we present the methods of using PSO to find the optimal feedback gain values for the controller.
Simulations
The proposed PSO-based VFGC structure can perform an on-line search for the nine optimal variable feedback gain values, obtain the best control command, chase and target the enemy, and enter optimal attack situations under reasonable and realistic mechanical restrictions. The PSO algorithm is a method of random search; in this paper, there are 100 different functions with randomly generated initial parameters for the PSO algorithm, such as r1 and r2 in (43). Lastly, the root mean square error
Conclusions and future works
The goal of the PSO-based VFGC algorithm developed from this research is to ensure a fighter could maintain optimal air combat status by following the optimal strategy created by the PSO-based VFGC system.
In this research, a system controller is designed by incorporating a linear system model developed by putting a non-linear flight equation through the input–output linearization process. Then, the best feedback gains are obtained through the PSO process. Results from simulations show that the
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