Elsevier

Applied Soft Computing

Volume 13, Issue 1, January 2013, Pages 58-75
Applied Soft Computing

Variable feedback gain control design based on particle swarm optimizer for automatic fighter tracking problems

https://doi.org/10.1016/j.asoc.2012.07.032Get rights and content

Abstract

The main focus of this paper is to develop an optimization method for the automatic fighter tracking (AFT) problem. The AFT problem is similar to a general evader–pursuer maneuvering automation problem between the dynamic systems of two highly interactive objects. This paper proposes a particle swarm optimizer-based variable feedback gain controller (PSO-based VFGC) for dealing with AFT problems. 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 can be obtained through PSO, and the actual control values can be derived from the obtained values. Simulation results confirm the capabilities of the proposed method by comparing the results against two other methods in the field: the weight matrix value defined Ricatti equation, and the linear matrix inequality (LMI) based linear quadratic regulator (LQR). The performance of the proposed method is superior to that of its alternatives.

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:(a)x˙=Vcosγsinψ(b)y˙=Vcosγcosψ(c)z˙=Vsinγwhere 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 x˙,y˙ and z˙. 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

References (50)

  • K.H. Hsia et al.

    A first approach to fuzzy differential game problem: guarding a territory

    Fuzzy Sets and Systems

    (1993)
  • Y.S. Lee et al.

    A strategy for a payoff-switching differential game based on fuzzy reasoning

    Fuzzy Sets and Systems

    (2002)
  • R. Issacs

    Differential Games

    (1965)
  • P.K.A. Menon et al.

    Time-optimal aircraft pursuit evasion with a weapon envelope constraint

    Journal of Guidance, Control and Dynamics

    (1992)
  • K. Virtanen et al.

    Modeling pilot's sequential maneuvering decisions by multistage influence diagram

    Journal of Guidance, Control and Dynamics

    (2004)
  • B.S. Chen et al.

    Fuzzy differential games for nonlinear stochastic systems: suboptimal approach

    IEEE Transactions on Fuzzy Systems

    (2002)
  • J. Paul

    Nahin Chases and Escapes: The Mathematics of Pursuit and Evasion

    (2007)
  • H. Kwakernaak et al.

    Linear Optimal Control Systems

    (1972)
  • E. Jones

    On the existence of optimal stabilizing controls

    IEEE Transactions on Automatic Control

    (1979)
  • L. Shaw et al.

    Asymptotic pole locations and nonlinear controllers

    IEEE Transactions on Automatic Control

    (1980)
  • R. Reid et al.

    Design of the steering controller of a supertanker using linear quadratic control theory: a feasibility study

    IEEE Transactions on Automatic Control

    (1982)
  • I. Khalifa et al.

    A note on trajectory sensitivity reduction using a three-term controller

    IEEE Transactions on Automatic Control

    (1984)
  • H.M. Al-Rahmani et al.

    A new optimal multirate control of linear periodic and time-invariant systems

    IEEE Transactions on Automatic Control

    (1990)
  • Y.Y. Wang et al.

    The robustness properties of the linear quadratic regulators for singular systems

    IEEE Transactions on Automatic Control

    (1993)
  • Y.B. Shtessel

    Principle of proportional damages in a multiple criteria LQR problem

    IEEE Transactions on Automatic Control

    (1996)
  • P.O.M. Scokaert et al.

    Constrained linear quadratic regulation

    IEEE Transactions on Automatic Control

    (1998)
  • W.C. Jae et al.

    LQR design with eigenstructure assignment capability and application to aircraft flight control

    IEEE Transactions on Aerospace and Electronic Systems

    (1999)
  • D.E. Miller et al.

    Simultaneous stabilization with near optimal LQR performance

    IEEE Transactions on Automatic Control

    (2001)
  • G. Marro et al.

    Geometric insight into discrete-time cheap and singular linear quadratic Riccati (LQR) problems

    IEEE Transactions on Automatic Control

    (2002)
  • T.A. Johansen et al.

    Gain-scheduled wheel slip control in automotive brake systems

    IEEE Transactions on Control Systems Technology

    (2003)
  • K. El-Awady et al.

    Programmable thermal processing module for semiconductor substrates

    IEEE Transactions on Control Systems Technology

    (2004)
  • H.N. Wu et al.

    Finite-dimensional constrained fuzzy control for a class of nonlinear distributed process systems

    IEEE Transactions on Systems, Man, and Cybernetics. Part B: Cybernetics

    (2007)
  • F. Liao et al.

    Reliable robust flight tracking control: an LMI approach

    IEEE Transactions on Control Systems Technology

    (2002)
  • G. Feng et al.

    A new digital control algorithm to achieve optimal dynamic performance in DC-to-DC converters

    IEEE Transactions on Power Electronics

    (2007)
  • M. Vasak et al.

    Hybrid theory-based time-optimal control of an electronic throttle

    IEEE Transactions on Industrial Electronics

    (2007)
  • Cited by (20)

    • Multi-objective design of optimal higher order sliding mode control for robust tracking of 2-DoF helicopter system based on metaheuristics

      2019, Aerospace Science and Technology
      Citation Excerpt :

      In addition, conventional optimization methods such as the gradient search method are restricted to the eigenvalues of the linear system matrix not merely increases the difficulty but also consumes long time to reach a global optimum solution. On the other hand, metaheuristics, such as Particle Swarm Optimization (PSO) [5], Ant Colony Optimization (ACO) [6] and Genetic Algorithms (GA) [7] are applied to determine the optimal weighting matrices of the LQR strategy for improving the control performances. However, it is worth mentioning that the majority of applications considers a Single-Objective Optimization (SOO) while neglecting to mention that the majority of systems requires the task of simultaneously optimizing two or more conflicting objectives with respect to a set of certain constraints.

    • Adaptive PSO for optimal LQR tracking control of 2 DoF laboratory helicopter

      2016, Applied Soft Computing Journal
      Citation Excerpt :

      Even if all of the control strategies are optimal in nature, different values of Q and R will ultimately end up with a different system response, which indicates that the response is non optimal in true sense. Conventional optimization methods, such as the gradient search method, used for designing the state feedback controller are restricted to the eigen values of the linear system matrix that not only increases the difficulty but also consumes long time to find the global optimum solution [21]. Hence, evolutionary computation (EC) can be considered an alternative method to solve this type of optimization problem.

    • Multi-objective design of state feedback controllers using reinforced quantum-behaved particle swarm optimization

      2016, Applied Soft Computing Journal
      Citation Excerpt :

      They exploited a conventional weighted aggregation of control objectives which can only provide the designer with one solution. More studies on PSO-based LQR design can be found in [4,17,41–44]. They are a few studies that exploit multi-objective design for tuning LQR controllers.

    • A probabilistic approach for designing nonlinear optimal robust tracking controllers for unmanned aerial vehicles

      2015, Applied Soft Computing Journal
      Citation Excerpt :

      This technology can be used to solve many social problems, and this explains the great interest that the area has received from different research groups and organizations throughout the world. Examples of research related to unmanned aerial vehicles are path planning [27,52], squadrons formation reconfiguration [16], wireless networks [28], autopilot design [5], multiple unmanned aircraft coordination [30], formation [22], task assignment [35], trajectories generation [34], patrolling or surveillance [2], searching [11], tracking [48], flight control [24] and source seeking [53]. Despite using unmanned aerial vehicles, the aircraft has to be capable of following references, which are commands that determine the motion of the aircraft.

    • Estimating parameters of muskingum model using an adaptive hybrid PSO algorithm

      2014, International Journal of Pattern Recognition and Artificial Intelligence
    View all citing articles on Scopus
    View full text