In order to get the optimal performance of controller and improve the design efficiency, artificial bee colony (ABC) algorithm as a metaheuristic approach which is inspired by the collective foraging behavior of honey bee swarms is considered for optimal linear quadratic regulator (LQR) design in this paper. Furthermore, for accelerating the convergence speed and enhancing the diversities of population of the traditional ABC algorithm, improved solution searching approach is proposed creatively. The proposed approach refers to the procedure of differential mutation in differential evolutionary (DE) algorithm and produces uniform distributed food sources in employed bee phase to avoid local optimal solution. Meanwhile, during the onlooker bees searching stage where the solution search area has been narrowed by employed bees, new solutions are generated around the solution with higher fitness value to keep the fitness values increasing monotonously. The improved ABC algorithm is applied to the optimization of LQR controller for the circular-rail double inverted pendulum system, and the simulation results show the effect on the proposed optimization problem.
Linear quadratic regulator, as a representative optimal control, has been widely used for complex system control. The objective of LQR controller is to minimize the quadratic cost function with weighting matrices selected by engineers. Often this means that the controller synthesis will be an iterative process where an engineer should judge the produced “optimal” controller through simulation and then adjusts the weighting matrices to get a controller more in line with the specified design goals. This tedious process limits the application of the LQR-based controller synthesis [
Fortunately, computational intelligence (CI), as a set of nature-inspired computational methodologies and approaches including artificial neural networks, genetic algorithm, swarm intelligence, and artificial immune algorithm, has become a remarkably developing research area because of its ability of intelligent reasoning and decision making. Relative researches have been deployed in solving complex problem in diverse fields as well as in LQR controller optimization [
As one of the most recent swarm intelligence approaches, artificial bee colony (ABC) algorithm inspired by the foraging behavior of honey bees has been presented by Karaboga [
In this paper, in order to optimize the performance of LQR controller designed for balance control of rotary double inverted pendulum system, the ABC algorithm is introduced; meanwhile, for preventing the local optimization and accelerating convergence speed simultaneously, the solution search phase as the key of the algorithm is improved and applied without using the complex algorithms mentioned above. On one hand, differential mutation in differential evolutionary (DE) algorithm is learned and adopted in the employed bees searching phase to enhance the diversity of the population. On the other hand, in onlooker bees searching phase, the comparison of fitness values between the current and the neighboring solutions is introduced to keep the fitness values increasing monotonously and accelerate the convergence speed. The major contributions of this paper are as follows: (i) different from the traditional ABC algorithm mentioned in [
The rest of the paper is arranged as follows. In Section
LQR is a MIMO design theory with observation-based control which is concerned with operating a dynamic system at minimum cost. The structure of LQR control system is shown in Figure
The structure of LQR control system.
For a linear, time-invariant system,
Obviously, the cost function as well as the control performance of LQR controller is affected by the weighting matrices
In order to simplify the LQR design procedure, meanwhile obtaining the optimal performance, the artificial bee colony (ABC) algorithm is proposed.
The ABC algorithm is inspired by the collective foraging behavior of honey bee swarms. The artificial bee colony can be separated into three groups as in the real world: employed bees, onlooker bees, and scouts bees. The employed bees are responsible for exploiting the food sources, bring loads of nectar from the food source to the hive, and share the information about food source with onlooker bees through waggle dance. “Onlookers” are those bees that are waiting in the hive for the information to be shared by the employed bees such as distance, direction, and profitability; then further search around the selected food source based on the probability will be done. “Scouts” are those bees which are currently searching for new food sources in the vicinity of the hive [
In ABC algorithm, the food sources represent the possible solutions. And the numbers of food sources SN are equal to the employed bees.
Each of the employed bees searches in its vicinity and generates new solution
The nectar amount of the food source as well as the quality of each solution is represented by the fitness which could be calculated by
After all employed bees finish the searching process, they share the fitness of each food source with the onlookers, each of whom selects a food source according to the probability as shown in (
Then, better food source around its chosen food source will be searched randomly based on the fitness.
If a solution does not improve for a number of iterations, the food source will be abandoned, and the associated employed bee becomes scout bee. Random search will be performed and new solutions will be generated as
As mentioned above, the flow chart of ABC algorithm is presented in Figure
For the problems of slow convergence speed and premature convergence existing in the standard ABC algorithm, improved ABC algorithm is proposed mainly on search mechanism.
DE [
As the result of vector operation, distance and direction information could be extracted and taken into account during the process of generating the new vector; thus, a broader solution space and the new vector in wider range could be obtained compared with Figure
The new generated solution in ABC algorithm.
Procedure of ABC algorithm.
In order to maintain population diversity especially in employed bee phase which is the first stage in ABC algorithm, motivated by DE and based on the property of the ABC algorithm, the
Therefore, from Figure
New solution generation as described in (
As mentioned above, the improved search mechanism will raise the possibility to get the global optimal solution; but on the other hand, the new solutions are randomly selected and the fitness of new candidate food source
The new candidate food source is determined based on the comparison of fitness information between the current and neighboring food sources as shown in
From (
New solution generation as described in (
As analysed above, the solution search mechanism as described in (
Thus the main steps of the improved ABC algorithm could be concluded as follows.
Initialize the population of solutions; define the parameters including SN and MCN (maximum cycle number), as well as the boundary of the solution; what should be noted is the selection of
Generate new solutions
Calculate the fitness value of new candidate food source.
Obtain the probabilities
Produce the new solutions
Determine the abandoned solution; if it exists, replace it with a new randomly produced solution
Record the best food source position achieved so far.
Judge whether the termination conditions (
In this section, in order to prove the optimization performance, the improved ABC algorithm and the traditional ABC algorithm are applied to the LQR controller optimization for the same circular-rail double inverted pendulum system and the results are compared and analyzed.
As a typical underactuated system, the circular-rail inverted pendulum system consists of the horizontal driving arm driven by torque motor and two-stage pendulum rods as shown in Figure
The structure of circular-rail double inverted pendulum.
Based on the Lagrange equation, the state-space equation of the inverted pendulum system could be deduced as (
As an unstable but controllable, observable system, the closed loop system and LQR control algorithms could be introduced to improve the performance of inverted pendulum system.
As mentioned above, the purpose of LQR control is to design the state feedback controller
Here, the matrices
Based on the optimization objective, the improved ABC algorithm is introduced. According to the procedure mentioned above, considering both the search quality and computational efforts, the algorithm parameters, including the population size SN and MCN, are fixed to 10 and 300, respectively. The upper and lower boundary of the solution are set to be [1000, 1000, 1000, 50, 50, 50] and [0.6, 0.6, 0.6, 0.6, 0.6, 0.6].
In order to determine the values of parameters
Relationship between the two parameters and the fitness of the optimization problem.
Based on the same initial parameter settings, with the traditional ABC algorithm and the improved ABC algorithm, the optimization procedure has been carried out, respectively. As shown in Figure
The procedure of optimization.
To verify the effectiveness of the improved ABC algorithm and the optimized LQR controller, the simulations have been done on the circular-rail double inverted pendulum system. The step input whose amplitude is 0.01 rad is added to the horizontal rod, and the other two inputs are set to be zero. As shown in Figure
(a) The step response of horizontal rod. (b) The step response of lower pendulum. (c) The step response of upper pendulum.
Furthermore, in order to verify the correctness of the improved ABC algorithm, the relationship between the selected fitness values corresponding to different values of
(a) The relationship between fitness values and the amplitude of the output with noise. (b) The relationship between the fitness and setting time of lower pendulum. (c) The relationship between the fitness and overshoot of lower pendulum.
The foregoing results indicate that the improved ABC algorithm is a very effective method for parameters optimization in LQR controller design which is proper for rotary double inverted pendulum system control.
This paper focuses on the issue of the LQR controller optimization for the rotary double inverted pendulum system control. Aiming at accelerating the convergence speed meanwhile enhancing the diversity of the population, a novel improved ABC algorithm which incorporates the ideas of DE and fitness comparison has been creatively introduced and applied to the optimization problem. The simulations have been done and the results show the validity of improved ABC algorithm that outperforms the traditional ABC algorithm in optimizing the parameters of LQR controller for inverted pendulum system.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (NSFC) (Grant nos. 61074022 and 61304115) and the Program for International Science Cooperation and Communication (2010DFA22770). The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.