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This paper presents evolutionary approaches for designing rotational inverted pendulum (RIP) controller including genetic algorithms (GA), particle swarm optimization (PSO), and ant colony optimization (ACO) methods. The goal is to balance the pendulum in the inverted position. Simulation and experimental results demonstrate the robustness and effectiveness of the proposed controllers with regard to parameter variations, noise effects, and load disturbances. The proposed methods can be considered as promising ways for control of various similar nonlinear systems.

During the past decades, many modern control methodologies such as nonlinear control, optimal control, adaptive control, and variable structure control have been widely proposed for control approaches [

Over the past years, several evolutionary algorithms have been proposed to search for optimal PID controllers. Among them, GA has received great attention and PSO has been successfully applied to various fields [

In this paper, we compare the efficiency of three intelligent algorithms, that is, GA, PSO, and ACO methods. These evolutionary algorithms are used to adjust the PID controller parameters in order to ensure adequate servo and regulatory behavior of the closed-loop system. Also, we formulate the problem of designing PID controllers as an optimization dilemma which adjusts five performance indexes, that is, maximum overshoot, rise time, settling time, and steady-state error of the response and system control energy.

The rotational inverted pendulum system is a well-known test platform for evaluating various control algorithms. Also, it has some significant real-life applications such as position control, aerospace vehicles control, and robotics [_{p}_{p}_{b}

Schematic view of RIP system.

The plane of the pendulum is orthogonal to the radial arm. Figure

Built in-RIP system (advanced robotics research lab).

Block diagram of the whole system.

In this section, the dynamic equations of the RIP system considering backlash and friction effects are presented. The RIP dynamics are governed by [

The above nonlinear model can be found in the following equations:

The parameters of nonlinear model of the system are represented in Table

Parameters of the RIP system.

Parameters | Values | Parameters | Values |
---|---|---|---|

3.29 | 14.283 | ||

0.1252 | 1.4286 | ||

0.2369 | 1.72 | ||

6.052 | 141.32 | ||

0.0132 | 0.0012 |

Using (

Block diagram of RIP system.

System response without PID controller.

In the following, brief reviews of GA, PSO and ACO principle are illustrated.

Considering Darwin's original ideas, life in all its diverse forms is evolved by natural selection and adaptation processes controlled by the survivability of the fittest species. GA is an evolutionary optimizer that takes a sample of possible individuals and employs selection, crossover, and mutation as the primary operators for optimization [

Considering the social behavior of swarm of fish, bees, and other animals, the concept of the particle swarm optimization (PSO) is developed. The PSO is a robust stochastic evolutionary computation method based on the movement of swarms looking for the most fertile feeding location [

From the above statements, it is obvious that the theoretical bases of the two optimization methods rest upon two completely different structures. The GA is based on genetic encoding and natural selection, and the PSO method is based on social swarm behavior. PSO is based on the principle that all solutions can be represented as particles in a swarm. Each particle has a position and velocity vector, and each position coordinate represents a parameter value. Similar to GA, PSO requires a fitness evaluation function that takes the particle’s position and assigns a fitness value to it.

Here rand() and Rand() are two random numbers in the range [

The positions are updated based on their movement over a discrete time interval (

Then the fitness at each position is reevaluated. If any fitness is greater than

ACO is a relatively recent approach to solve optimization problems by simulating the behavior of ant colonies and modeling the behavior of ants, which are known to be able to find the shortest path from their nest to a food source [

All ants start their tours from source node and end up their tours in destination node. In each node, an ant chooses its path probabilistically, and the probability of choosing an edge is proportional to the pheromone on the edge, that is, roulette wheel selection.

All edges have an initial amount of pheromone,

The algorithms parameters have been chosen based on trial and error as follows. For GA method, population size = 50, crossover rate = 0.5, mutation rate = 0.01, and maximum generations = 20; for PSO algorithm, number of particles = 50; acceleration constants

Considered performance index includes the overshoot

First, specify the lower and upper bounds of controller parameters and initialize the particles of the population randomly. Each particle, that is,

The lower and upper bounds of the three controller parameters are shown in Table

range of three controller parameters.

Controller parameters | Lower bounds | Upper bounds |
---|---|---|

_{pi} | 0 | 4 |

_{di} | 0 | 4 |

_{po} | 0 | 4 |

_{do} | 0 | 4 |

In order to examine the dynamic behaviors and convergence characteristics of the proposed methods, two statistical indexes, namely, the mean value (

Several simulations are performed to investigate and compare controllers’ convergence characteristics. As it can be seen in simulations in Figures

Convergence tendency of mean values of cost function with

Convergence tendency of mean values of cost function with

Convergence tendency of standard deviation values of cost function with

Convergence tendency of standard deviation values of cost function with

Also, our simulation results demonstrate that ACO method is faster than GA and slower than PSO and the run time in 20 iterations for ACO is 2408.17 sec in comparison with 1805.42 sec for PSO and 3865.903 sec for GA.

In the following, the optimization procedure has been applied to the RIP system in reference tracking (servo behavior). For analysis of this behavior, the reference signal (

System response: (a) pendulum angle, (b) arm angle, and (c) energy signal, with

System response: (a) pendulum angle, (b) arm angle, and (c) energy signal, with

The simulation results of the best solution for various values of

Best results of PID controllers with different

cost | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

GA | 1 | 0.473 | 4.118 | 0.51 | 0.356 | 36.787 | 0.008 | 0.958 | 0.29 | 0.0008 | 18.213 | 20.908 |

1.5 | 0.666 | 3.276 | 0.444 | 0.248 | 28.34 | 0.008 | 1.776 | 0.44 | 0.0046 | 22.624 | 22.519 | |

PSO | 1 | 0.551 | 3.989 | 0.398 | 0.153 | 22.243 | 0.008 | 1.301 | 0.38 | 0.001 | 17.839 | 16.189 |

1.5 | 0.529 | 3.206 | 0.35 | 0.119 | 21.805 | 0.006 | 1.24 | 0.41 | 0.0008 | 18.242 | 17.728 | |

ACO | 1 | 0.35 | 4.091 | 0.281 | 0.125 | 16.528 | 0.035 | 1.864 | 0.32 | 0.0022 | 19.904 | 15.513 |

1.5 | 0.539 | 2.821 | 0.441 | 0.23 | 17.57 | 0.008 | 1.721 | 0.28 | 0.0022 | 19.095 | 16.571 |

Now, the servo performance is considered for the reference signal (

Pendulum angle (servo response) of the RIP control system with

Pendulum angle (servo response) of the RIP control system with

The control systems are always subject to external disturbances and internal noise which affect the system dynamics. If the nature of the disturbance is identified, it can be modeled mathematically. However, in practice, the nature of the disturbances is not clear and we may not be able to simulate them easily. In RIP system, the disturbance and noise effects can be applied by adding an additional load to the end of pendulum and adding noise to the position sensor, respectively.

The simulations are done subject to a band-limited white noise (noise power = 0,000523 and sampling time = 0.1 sec) and 10% parameter value changes. Simulation results shown in Figures

System response: (a) pendulum angle, (b) arm angle, and (c) energy signal, with

System response: (a) pendulum angle, (b) arm angle, and (c) energy signal, with

Servo and regulatory results motivate to consider the proposed procedure as a suitable tool for controller parameters design and also stimulate investigating the possibility of further research on design and development of other practical control systems.

We have performed experiments on the RIP system set at the University of Tabriz in the robotics research lab. The applied card in this project is PCI-6602 which creates the connection between computer and system and has A/D and D/A converters. Also the arm and pendulum links angles are measured using two E40S Autonics company encoders. The experimental results of the proposed methods on RIP system are shown in Figures

Experimental results of system responses with

Experimental results of system responses with

In order to study the stability of designed control system using ACO algorithm subject to parameters variations, we perform the following experiment. In this experiment, the adding effect of a 25 g body mass with the length of 13 cm and in the presence of disturbances is validated. Figure

Experimental results of system responses using ACO-PID by adding a body mass to the end of pendulum and in the presence of disturbances: (a) pendulum angle, (b) arm angle, and (c) energy signal.

In this paper, we present three evolutionary algorithms for designing of intelligent controllers of the RIP system. Each of the algorithms is tested in 50 independent runs involving 50 different initial solutions. The rotational inverted pendulum system is considered as a case study. Through the simulation results, the proposed controllers perform efficient search for proper PID parameters. To evaluate the controller performance, we tested the ability of the closed-loop system to follow set point changes (servo behavior) and the ability of the closed-loop system to reject disturbances (regulatory behavior). The work demonstrates that all methods can solve searching and tuning the controller parameters efficiently. The proposed methods could be considered as promising ways for nonlinear control systems in general. One of the important features of the system is using of xPC-Target toolbox and input-output card in Simulink environment which utilizes hardware in the loop (HIL), tele-lab implementation and fast-prototyping properties. The topic of our future researches is to employ other cognitive methods in order to achieve better results for designing controller and improving the performance in real time. Also, implementation, of heuristic algorithms for designing adaptive controllers will be our future challenging task. Furthermore, tele-operation control of RIP system using haptic device would be another challenge.

This paper was financially supported by the Research Affairs at University of Tabriz. The authors would like to appreciate Mr. Abbas Harifi for his guidance and assistance.