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This paper proposes new metaheuristic algorithms for an identification problem of nonlinear friction model. The proposed cooperative algorithms are formed from the bacterial foraging optimization (BFO) algorithm and the tabu search (TS). The paper reports the search comparison studies of the BFO, the TS, the genetic algorithm (GA), and the proposed metaheuristics. Search performances are assessed by using surface optimization problems. The proposed algorithms show superiority among them. A real-world identification problem of the Stribeck friction model parameters is presented. Experimental setup and results are elaborated.

AI-based algorithms have been successfully applied to solve optimal solutions of complex and NP-hard problems in engineering. For some examples, dated back to 1995, the genetic algorithm was proposed to solve line-balancing problem to minimize the cycle time of the line for a given number of workstations [

An identification task can be formulated as an optimization problem solvable by available optimization algorithms. Artificial intelligence- (AI-) based methods are efficient candidates of the present technology. There is a wide range of algorithms which has been applied to solve identification problems. For some instances, the genetic algorithm has been applied to various problems including the modelling of a laboratory scale process involving a coupled water tank system and the identification of a helicopter rotor speed controller [

Among those metaheuristics, evolutionary and bioinspired algorithms have gained major interests since they are not hard to understand, and programming according to the procedural lists is not so strict compared with conventional scientific programming. It also opens a new route to effectively obtain an optimal or suboptimal solution for a complex system. This paper proposes the use of the tabu search (TS) and the bacterial foraging optimization (BFO) algorithm in a cooperative manner. Due to the dominant explorative property of the BFO algorithm, the modified TS with the BFO built-in is able to start searching with an elite initial solution. While the sharp focusing property of the TS remains, the proposed algorithms can move towards the solution very rapidly. Section

Tabu search (TS) originated by Glover [

The ATS consists of two major additional strategies made to the conventional TS. These are back-tracking (BT) and adaptive search radius (AR) mechanisms, respectively. The former assists the TS to release itself from being locked by a local solution. It looks up the tabu list (TL), that is, short-term memory, for a visited elite solution, and uses this solution for starting a new search move. The later enhances the focusing characteristic of the TS. This strategy decreases the search radius gradually when the search comes close to a solution of high quality having a potential of being the optimal one. However, too short of the search radius could result in a slow search. Recommendations for selection of search parameters are in [

In 2002, Passino developed a new bioinspired optimization algorithm called bacterial foraging optimization (BFO) [

This mechanism imitates the swimming movement of a bacterium. The position of a bacterium is denoted as

When one bacterium presents itself in an elite position, that is, a local hill or valley, it attracts the others. Simultaneously, each bacterium tries to repel the others nearby. The attractive and the repellent effects are modeled as weighted summation of exponential terms presenting the objective function,

The bacteria are classified during the computing process as healthy and unhealthy due to their cost values. Only the healthy ones reproduce by duplicating themselves at the same positions.

The mechanism allows the unhealthy bacteria to be discarded. The healthy ones are dispersed randomly over the search space with the probability

The original BFO and the ABFO algorithms run iteratively and terminate on the maximum iteration criterion. The solutions obtained from search are stored in a memory and eventually sorted to find the optimal solution. From testing the ABFO algorithm, it demonstrates a strong explorative (or diversification) property. This property is commonly found in population-based algorithms, and the ABFO algorithm is one of them. In contrast, single-solution-based algorithms, such as the TS, have strong exploitative property [

As mentioned, the TS has a dominant focusing characteristic, while the ABFO is strong in explorative operation. Such properties can complement each other. Since the TS has straightforward procedures, and moves rapidly towards a local solution, the method forms the hunting steps for a satisfied solution to the problem. The two algorithms are combined to form new metaheuristics working in a cooperative manner. The new algorithms are referred to as bacterial foraging-tabu search or BTS in short. In this new algorithmic form, it is unnecessary to employ the reproduction mechanism of the ABFO part because ranking the available solutions to single out one with the minimum cost is an important step. This specific solution is transferred to the TS part as an initial solution. The procedural list of the BTS algorithms is as follows.

Initialize search parameters:

Randomly or heuristically select an initial solution

Compute objective functions

Generate randomly

If

If

Do minimum sorting of the objective functions

Generate a neighbourhood around

Evaluate the objective function of each member belonging to

If

Invoke the

look back in the

define _{0} = best_neighbor

If the termination criterion based on the

Invoke the

…

Updated count. If

Referring to Step

This section presents the performance comparison studies among the following algorithms: adaptive tabu search (ATS), adaptive bacterial foraging optimization (ABFO), bacterial foraging-tabu search metaheuristics (BTS), and genetic algorithm (GA). Review of the GA is omitted since the algorithm is well known. Good sources that readers may refer to are [

This approach is commonly referred to as multiple-points-single-strategy (MPSS) in metaheuristic contexts. The test functions adapted are well-known unconstrained problems for testing optimization algorithms. These include Bohachevsky function (BF), Rastrigin function (RF), Shekel’s fox-holes function (SF), Schwefel function (SchF), and Shubert function (ShuF), respectively. Table

Summary of the unconstrained problems used for performance test.

Test functions | Equations | Surfaces in 3D |
---|---|---|

BF | ||

RF | ||

SF | ||

SchF | ||

ShuF |

ATS parameters.

Test functions |
BT, | AR | |||||

Stage I | Stage II | Stage III | |||||

BF | 30 | 10,000 | 0.2 | 5 | — | ||

RF | 30 | 10,000 | 0.2 | 5 | — | ||

SF | 30 | 10,000 | 0.8 | 5 | — | ||

SchF | 30 | 10,000 | 50 | 5 | |||

ShuF | 30 | 10,000 | 1.0 | 5 | — |

ABFO parameters.

Test functions | ABFO parameters | ||||||||||

BF | 30 | 20 | 4 | 4 | 2 | 0.25 | 10 | 0.1 | 0.1 | 0.2 | 1 |

RF | 30 | 200 | 4 | 4 | 2 | 0.25 | 100 | 0.1 | 0.1 | 0.2 | 1 |

SF | 30 | 500 | 4 | 4 | 2 | 0.25 | 0.01 | 0.1 | 0.1 | 0.2 | 10 |

SchF | 30 | 2000 | 4 | 4 | 2 | 0.25 | 1 | 0.1 | 0.1 | 0.2 | 10 |

ShuF | 30 | 1000 | 4 | 4 | 2 | 0.25 | 1 | 0.1 | 0.1 | 0.2 | 10 |

Table

Summary of the results (averaged over 50 trials).

Test functions | Average search time (seconds) | Average search rounds | ||||||

ATS | ABFO | BTS | GA | ATS | ABFO (_{C} | BTS | GA | |

BF | 11.66 | 5.75 | 6.83 | 49.12 | 616.48 | 20 | 151.20 | 1177.18 |

RF | 14.60 | 48.63 | 5.81 | 11.78 | 868.28 | 200 | 323.30 | 225.54 |

SF | 4.18 | 146.67 | 3.69 | 8.53 | 139.36 | 500 | 25.70 | 141.28 |

SchF | 408.58 | 728.87 | 172.52 | 868.32 | 6889.42 | 2000 | 1469.14 | 354.44 |

ShuF | 3.28 | 182.83 | 2.80 | 3.39 | 68.06 | 1000 | 55.28 | 34.24 |

Convergence curves—(a) BF, (b) RF, (c) SF, (d) SchF, and (e) ShuF.

Table

Solutions obtained from different approaches.

Objective function | ATS | ABFO | BTS | GA |
---|---|---|---|---|

BF: | ||||

Average | 4.5090 | 1.0833 | 5.30112 | 6.0074 |

Min | 7.3184 | 2.2204 | 3.569 | 1.1830 |

Max | 9.4151 | 8.8759 | 9.97229 | 2.9855 |

Std. | 2.7476 | 2.0742 | 3.13515 | 4.2216 |

RF: | ||||

Average | 5.3478 | 1.1534 | 5.2460 | 4.9642 |

Min | 2.9051 | 3.5527 | 5.9587 | 1.1486 |

Max | 9.3771 | 6.7094 | 9.6620 | 9.5235 |

Std. | 2.4328 | 1.4942 | 2.6721 | 2.6740 |

SF: | ||||

Average | 0.9982 | 0.9981 | 0.9983 | 0.9983 |

Min | 0.9980 | 0.9980 | 0.9980 | 0.9980 |

Max | 0.9989 | 0.9988 | 0.9990 | 0.9990 |

Std. | 0.0002 | 0.0002 | 0.0003 | 0.0003 |

SchF: | ||||

Average | 0.0456 | 0.0139 | 0.0273 | 0.0439 |

Min | 0.0006 | 2.5455 | 0.0002 | 0.0008 |

Max | 0.0998 | 0.0768 | 0.0944 | 0.0982 |

Std. | 0.0278 | 0.0240 | 0.0341 | 0.0281 |

ShuF: | ||||

Average | −186.7305 | −186.7305 | −186.7305 | −186.7304 |

Min | −186.7309 | −186.7309 | −186.7309 | −186.7309 |

Max | −186.7300 | −186.7300 | −186.7301 | −186.7300 |

Std. | 0.0003 | 0.0003 | 0.0003 | 0.0002 |

Figure

Bacterial search movements—(a) BF, (b) RF, (c) SF, (d) SchF, and (e) ShuF.

The BTS has been applied to a constrained parametric search problem, that is, an identification of the nonlinear friction model. In the next section, experimental setup, and identification results are presented.

A closed loop position control system is a necessary test bed for monitoring stick-slip phenomenon. The diagram in Figure

Circuit diagram representing the experimental setup.

For the reflector to follow a ramp command, a closed loop position control has been built. The hardware components consist of a PC as a P-controller, a 12-bit ADC, a 2Q-drive circuit, a current sensor, an ultrasonic transducer, a 2nd-order differentiator producing a speed signal from a position signal, and a few signal conditioning circuits including zero-span circuits and a bipolar voltage generator, respectively, and a dc power supply. In control mode, the motion follows an up-down ramp command directing the reflector to move rightward (positive direction, ramp-up command) and leftward (negative direction, ramp-down command). The reflector moves in the range of 50–350 mm in the control mode. A desired speed can be set via the keyboard of the PC functioning as a P-controller.

When two solid materials translating over one another at very low velocity, a stick-slip phenomenon occurs. This phenomenon is caused by nonlinear friction characteristics also known as Stribeck’s effect [

Stribeck friction curve.

This situation is referred to as stick mode, and described by the stick-friction force

During the search process to identify the friction model parameters, an objective function

Calculate an average displacement

For ramping-up motion, calculate an approximated force

If

If

If

Calculate

If

If

Calculate the following forces:

externally applied force—

Calculate velocity and displacement of the mass:

If

Calculate the objective function:

Return to main search.

Regarding this identification problem, the mass (

Initialization: search parameters:

Randomly assign real values to the parameters to be searched for

Calculate the objective functions,

Random the value of

Evaluate the objective functions: if (

Repeat Steps

Evaluate the objective functions, _{0}_{0}

In the neighborhood of _{1}(r)

Based on the objective functions, do minimum sorting for the solutions in _{1}(r)_{1}

If (

If the frequency of solution cycling occurrence is equal to BT, do minimum sorting for the previous solutions stored in the TL, retrieve the 5th backward solution set, and assign it as the initial solution set for the next search move.

If (

If (

If (

If (

Go to Step

Due to the strong nonlinearity in friction force, it is necessary to identify two sets of model parameters corresponding to rightward and leftward motions. Referring to Figure

Identification results of ramp-up command at 5 mm/s—(a) convergence curve, (b) displacement, and (c) force exerted by motor. (Note: positions in the range of 112–295 mm).

For the leftward motion, that is, ramp-down command of −5 mm/s, the graphical displays of identification results are shown in Figure

Identification results of ramp-down command at −5 mm/s—(a) convergence curve, (b) displacement, and (c) force exerted by motor. (Note: positions in the range of 325–127 mm).

Model validation was conducted for both directions of motion. Figure

Validation results of ramp-up command—(a) displacement (44–112 mm), (b) force exerted by motor (44–112 mm), (c) displacement (295–352 mm), and (d) force exerted by motor (295–352 mm).

Validation results of ramp-down command—(a) displacement (352–325-mm), (b) force exerted by motor (352–325 mm), (c) displacement (127–68 mm), and (d) force exerted by motor (127–68 mm).

Furthermore, the friction curves based on model plots are shown against the experimental data in Figure

Plots of friction force curves (ramp command of ±5 mm/s).

This paper has proposed new metaheuristics denoted as bacterial foraging-tabu search (BTS), which are formed from the adaptive bacterial foraging optimization algorithm (ABFO) and the adaptive tabu search (ATS). The paper has elaborated the search performance assessment among the ABFO, ATS, GA, and BTS. The proposed BTS algorithms provide superior search performances as the presentation appears in Section

Maximum iteration

Coefficient representing the depth of attractant released

Coefficient representing the height of the repellant effect

Number of parameters to be optimized

Coefficient representing the width of the attractant signal

Coefficient representing the width of the repellant by the cell

Adaptive radius

Frequency of solution cycling

Step size taken in random direction specified by the tumble

Cost value of

Number of the neighbourhood

Number of iterations to be carried out in a chemotactic loop

Maximum number of elimination and dispersal events

Number of reproduction loop

Swimming length after which tumbling of bacteria in a chemotactic loop

Probability with which the elimination and dispersal continues

Search radius

Number of bacteria in the population

A half of number of bacteria (

Tabu list

A positive constant

Random vector on

Position of

Bohachevsky function

Rastrigin function

Schwefel function

Shubert function

Shekel’s fox-holes function.

Gear ratio = 5.9

Motor current (A)

Gravity constant (N/mm)

Ball screw lead = 5 mm

Mass (kg)

Number of data

Velocity (mm/s)

Crossover velocity (mm/s)

Displacement of spring (mm)

Displacement of mass (mm)

Viscous friction coefficient

External input force (N)

Friction force (N)

Friction force of motor (N)

Internal input force (N)

Force equivalent to the inertia of motor (Nm)

Inertia of motor ^{2}

Inertia to force conversion factor =

Linear to angular velocity conversion factor =

Torque constant of motor

Torque to ball screw force conversion factor =

Coulomb friction (N)

Static friction (N)

Viscous friction (Ns/mm)

Moment (Nm)

Displacement from measured (mm)

Velocity band around zero velocity

Notation for the term

Gear box efficiency = 0.81

Ball screw efficiency = 0.925

Proportional controller gain.

The authors are thankful to the Royal Golden Jubilee Ph.D. Program under Grant PHD/0091/2551 for the research grants as well as some partial fundings available from Suranaree University of Technology.