Application of recent nature-inspired meta-heuristic optimisation techniques to small permanent magnet DC motor parameters identification problem

: In this study, an attempt is made to find an effective solution on the small direct current (DC) motor's parameters identification problem by employing two recently introduced stochastic nature-inspired techniques. The first one is called ‘salp swarm algorithm’ (SSA) while the second one is called ‘ ant-lion optimiser ’ (ALO). These algorithms have been inspired by the interaction strategy between the ocean salps and the ants and ant-lions, respectively, in nature. To appraise the effectiveness of these algorithms, a DC motor model has been appropriately implemented and its performance is evaluated by using speed step responses, while all model's parameters are considered unknown and therefore search variables. Integral of the square of the error (ISE), integral absolute error and integral in time of absolute error have been adopted as objective functions for the algorithms’ evaluation. In order to judge the acceptability of these algorithms, the simulation results are compared with those of another one similar technique, namely the ‘grey wolf optimiser’ (GWO). The obtained results reveal very satisfactory performance and confirm that the examined SSA, ALO and GWO algorithms can identify accurately the DC motor parameters and can be applied effectively to the specific problem. Another finding is that SSA combined with ISE criterion seems to be the most appropriate technique among the three algorithms.


Introduction
It is known that there is a large number of every day's applications where small permanent magnet DC (PMDC) motors are used. These applications cover not only household use but also extend to many other areas such as medical or industrial equipment, whereas wide range of rotational speeds as well as high precision are required. Some examples include hand-held precision tools, portable battery-operated devices, radio-controlled models, pump equipment, fans, blowers, drills, robotic mechanisms/actuators, shakers and stirrers, security cameras' pan and tilt drives and many more. Moreover, the cost reduction on high energy rare earth magnets over the last few years has given even greater impetus to the use of the specific motors.
Additionally, in some cases there are low friction and low inertial mass requirements and therefore rapid acceleration/ deceleration and fast motor response times are exhibited. The thermal behaviour of these motors is also satisfactory and thus makes them the best choice especially when combined with a welldesigned control system. For the latter, however, it is necessary for the designer to know the motor parameters with as much as possible accuracy, since the design of the respective controllers is sensitive to high tolerances.
The parameters that fully cover the description of the mathematical model of a permanent magnet DC motor are six, namely (i) the armature resistance, (ii) the armature inductance, (iii) the inertia moment of the rotor, (iv) the coefficient of friction viscosity, (v) the torque coefficient and (vi) the coefficient of the back-electromotive force (back-emf). Most of the manufacturers either do not provide information about these parameters (considering them to be a sensitive industrial information), or give approximate values with a wide range of tolerance (especially in relatively inexpensive motor models).
On the other hand, there is no easy experimental way by which the parameters of a DC motor can be accurately evaluated, for example, in a laboratory using simple equipment. It therefore becomes apparent that the development and/or application of techniques and strategies/methods that would assist in this direction is always necessary. Moreover, since the values of these parameters in small DC motors are relatively small -in order of magnitude -to be measured accurately by an experimental apparatus, the usefulness of other approaches is evident. Once the motor parameters are determined, the next step in designing the appropriate controller for the application becomes easy and straightforward. Another feature which arises after the identification of the DC motor parameters is to analyse the motor performance over its full operational range through simulations and investigate its behaviour in extreme conditions, which would be hazardous if the actual motor was used.
General system identification methods have been applied to DC motor model identification, e.g. [1,2]. Other methods like the algebraic identification method [3,4] and the recursive least square method [5] have also been used. An expansion of the latter method has also been reported in parameter estimation of a permanent magnet synchronous motor [6]. A method based on block-pulse function series and neural networks was used to estimate the parameters of the continuous-time model of a PMDC [7], while adaptive Tabu-search method was adopted in [8] to identify the parameters of a separately excited DC motor. Another approach utilising no-load speed step responses have been proposed in [9]. Other methods include quantised speed measurements [10], and commercial software solutions like MATLAB parameter estimation/identification toolboxes [11,12]. At the same time, efforts employing evolutionary based techniques such as genetic algorithm (GA) [13] and non-dominated sorting genetic algorithm [14], reported results in the same problem. In [15], identification of PMDC motor parameters for an electric wheelchair application using GA was proposed.
In some cases, uncertain model were obtained from transfer functions identified in several working points chosen from the I/O steady characteristics under minimum and maximum adjustable loads, e.g. [16]. In [17], a methodology was described based on a variational approach combined with worst-case analysis and Monte Carlo simulation to determine the impact of a DC motor parameters variation on the system behaviour. Also, some works have focus in determining accurately only some of the required parameters. For example, in [18], a friction compensation and motor inertia estimation method to be used for small DC motors has been demonstrated.
Recent developments in meta-heuristic optimisation algorithms created many alternatives along with the need to explore their applicability in many engineering problems. Although these algorithms may be able to solve many real-world's challenging problems, the so-called 'no free lunch theorem' [19] allows researchers to try testing and validating these new algorithms and their performance. According to this theorem, all algorithms perform equally when solving all optimisation problems. Therefore, one algorithm can be very effective in solving one set of problems but ineffective on a different set of problems. Actually, this is the foundation of many works in this field. For example, the particle swarm optimisation (PSO) method has been used in induction motors parameter estimation in [20], while in [21] an improved PSO was proposed for the same problem in separately excited DC motors.
In many of the aforementioned works, there are two major issues that have not been addressed adequately. The first is related with the number of parameters under determination, while the second one has to do with the strategy employed. For example, in [8,14,21], the torque coefficient has been assumed equal to the back-emf coefficient (i.e. k t = k e ) and thus one less parameter had to be determined. Although this assumption can be quite acceptable in some cases (e.g. steady-state calculations), it is not true in real motors in practice. Moreover, it can affect the correct identification of the other motor parameters and may 'mislead' the corresponding controller design. Regarding the followed strategy, in [11] the mechanical related motor parameters (i.e. moment of inertia and viscous friction coefficient) failed to be identified in acceptable error ranges. Also, in [9,21] only the open-loop (no-load condition) speed characteristic has been used, leading to a great simplification concerning the frictional phenomena which greatly affects the motor performance in loading conditions.
In the above context, the paper contributes by considering the above issues while at the same time examines the performance of three nature-inspired meta-heuristic techniques to the DC motor parameter identification problem. These techniques -in chronological order -are the grey wolf optimiser (GWO) [22], the ant-lion optimiser (ALO) [23] and the salp swarm algorithm (SSA) [24], all of them proposed in the last 5 years. Their performance evaluation to the specific problem may be of great help to controller designers who need an accurate and sophisticated tool in order to extract crucial motor operational characteristics and consequently to embed this data to its control strategies.
It should be noted here that the paper does not intend to argue with the effectiveness of other parameter estimation techniques mentioned before, neither to propose a certain technique out of those examined here. Instead, the motivation in this work is clearly the investigation of the applicability of the examined natureinspired meta-heuristic algorithms as an alternative method for the specific problem, which may be used in conjunction with other well-established ones.
The work is organised as follows. The mathematical modelling of the PMDC motor as utilised here is given in Section 2. The description of the problem statement as well as the selection of the objective functions are presented in Section 3. The main features of the three nature-inspired techniques optimisation are given in Section 4. In Section 5, the identification strategy is analysed, the relevant results from the application of the three techniques along with the corresponding comments and discussion are given. Finally, the work concludes in Section 6.

General characteristics
Small PMDC motors differ from other types of motors in many ways. Fig. 1 depicts different views of a disassembled such motor. When the motor is assembled the only parts which are projected out of the frame (housing) are the shaft and the power supply terminals. In Fig. 1a, the brush cap has been removed and the two carbon brushes are shown which -under operation -are connected to the supply terminals. At the same figure, a three-pole rotor with its winding is shown installed inside the motor housing. Also, two permanent magnets surround the rotor in order to provide constant magnetic flux while the rotor is running. In Fig. 1b, the rotor has been pulled out, exposing the thin stator iron laminations and the necessary commutator. Metal pins are usually used to hold the permanent magnets in place. Finally, the rotor may or may not be supported by one or two ball bearings at the ends of its shaft so it can be rigidly affixed to the outside motor housing and to rotate freely with low friction.

Mathematical modelling
The typical equivalent circuit of a permanent magnet DC motor is shown in Fig. 2. The source voltage (V s ) is applied to the terminals of the armature winding which is characterised by its ohmic resistance (R a ) and its inductance (L a ). At any time instant during motor's operation, an armature current (i a ) is imposed to the winding and an induced electromotive force is developed (E a ), which is proportional to the rotational speed (ω m ) -or equivalently to the rate of change in shaft position (θ) -and the magnetic field's flux (Φ). Considering that the motor has been designed in such a way that its operating range is related to the linear portion of the stator/rotor ferromagnetic material's magnetisation curve, this proportionality is expressed through the back-emf coefficient (k e ). Thus, from this part of the equivalent circuit, the following wellknown equations are derived: where and In physical terms, the back-emf represents the transition between the electrical and the mechanical power. Since E a is less than V s due to the voltage drop across the armature winding, the  corresponding power, i.e. P a = E a i a is less than the input electrical power, i.e. P in = V s i a . This absorbing power P a , is 'reappearing' as gross (developed) mechanical power P em = T em ω m . In general, the net mechanical output power P m is less than P dev due to the no-load losses which are associated with the rotational (also called friction/ windage) losses. At the same time, the developed torque T em is proportional to the armature current and the magnetic flux with the proportionality coefficient k t . If the rotor's moment of inertia is J m , the viscous friction coefficient is B m and the load torque is T L , then the equations describing the mechanical part of the motor can be stated as and By combining (1)- (3) and (4)-(7), the following differential equations are obtained: Now, the Laplace transform of (8) and (9) yields to the following transfer function: where it can be seen that the DC motor is represented as a singleinput single-output system with input variable the supply voltage and output variable the rotor position or its derivative, i.e. speed. From (10) it is also clear that in order to fully identify the motor model, there are six parameters that need to be known. The block diagram representing the above transfer function is shown in Fig. 3, whereas an experimental (blue line) speed-step response is depicted in Fig. 4. Under constant input voltage and zero initial conditions, the first part of the curve in Fig. 4 is the no-load characteristic (the motor accelerates and reaches the no-load speed), while the second part is the load characteristic (a load torque is applied at t = 0.1 s and the motor slows down until reaches the load speed). Based on the above, the approach followed here comprises of three steps: (i) a step speed response under no-load (start-up) and under load torque disturbance is considered using the actual motor under determination, (ii) a model is built as the 'unknown' model to be fitted, either in transfer function form or in state-space form and (iii) the parameters R a , L a , J m , B m , k t and k e are estimated through the applied algorithms and the model is evaluated using certain criteria, performing a number of iterations for a given number of search agents. The procedure is repeated several times. The goal is to minimise at any time instant the error between the response to be fitted and the actual response. From the obtained results, mean absolute error and the standard deviation are mainly considered.

Optimisation problem formulation
Without loss of generality, the problem set out above is an optimisation problem under constraints [25,26]. Mathematically can be described as follows: let f be an objective function defined in the search space ℚ ⊆ ℝ n . Also, let ℂ and ℤ be the set of feasible solutions and the set of infeasible solutions, respectively, where ℂ ⊆ ℚ and ℤ = ℚ − ℂ. Additionally, the search variables domains are limited by their upper (ub) and lower (lb) bounds, i.e. the search space is defined as an n-dimensional rectangle in the set ℝ n . Each vector x ∈ ℂ is called a feasible solution, otherwise an infeasible one. Thus, the problem is denoted as whereas the feasible set ℂ can be defined also by a set of additional P ≥ 0 constraints.

Objective (fitness) function selection
Usually one fitness function to which an optimisation algorithm is applied is used in similar experiments. Here, three objective functions have been employed in order to evaluate the applied methods. Considering that the error (between the two signals according to Fig. 4) can be expressed as then, the first objective function is the integral of the square of the error (ISE) which penalises large errors more than the small ones. The second objective function is the integral absolute error (IAE) which penalises all errors equally regardless of 'direction' (sign). Finally, the third objective function is the integral in time of absolute error (ITAE), which penalises long duration transients and in some cases can be more selective than the ISE [27]. Thus, the formulation of the problem is now completed like find R a , L a , J m , B m , k t , k e so as to minimise: 4 Nature-inspired methods used

Grey wolf optimiser
In 2014, the GWO algorithm was presented by Mirjalili et al. [22]. His first application was to train a multi-layer perceptron network.
The results were compared with those of GAs, evolution strategy, ant-colony optimisation and PSO and it was found that GWO is superior to them. Recently, the first attempt of GWO application to the problem of determining DC motor parameters has been presented by Karnavas and Chasiotis [28]. The results were compared with GA and found that the GWO performed better.
Mirjalili et al. [22] claimed that the superiority of GWO is based on the fact that there is no need for specific input parameters along with the complexity-free inherent characteristic. A recent literature review about their application up to now can be found in [29]. The GWO algorithm mimics the leadership hierarchy as well as the mechanism by which the grey wolves (Canis lupus of Canidae family) hunt in nature (Fig. 5). Grey wolves usually live in 5-12 member packs and are at the top of the food chain in the area where they live. At the same time, they follow a strict social hierarchy. Four types of grey wolves were adopted to simulate this hierarchy: alpha (α), beta (β), delta (δ) and omega (ω). Alpha wolves correspond to the top of the hierarchy, i.e. leaders (they give the optimal solution to the optimisation problem). Beta wolves are the next level and actually help alphas in making decisions as well as commanding those of the lower levels. The omega wolves refer to the weakest subgroup of the pack and are submitted to the other subgroups. Finally, members who cannot be described as alpha, beta or omega are the delta wolves. The latter can be -for example -scouts or caretakers of the pack. Besides the social hierarchy, an additional characteristic of GWO is the group hunting. The hunting phases are (i) searching for and approaching the prey, (ii) encircling prey and (iii) attacking prey. The mathematical implementation of these phases is actual the mean of performing optimisation in a complex problem. The mathematical expressions used by GWO are formulated here as follows: where In (16), k is the current iteration, X wolf and X prey are the grey wolves' position vector and the prey's position vector, respectively. Moreover, A, C are coefficient vectors, r 1 , r 2 are arbitrary vectors r 1 , r 2 ∈ 0, 1 , a is a vector linearly reduced from 2 to 0 during iterations (approaching prey mechanism). Especially for the vector A, it can take values in the range [−a, a], so when A < 1 the wolves are forced to attack the prey, while when A > 1 the wolves are forced to stand off the prey and possibly find a fitter one.
The effect of (16) can be understood considering a threedimensional search space like in Fig. 6. If a grey wolf is considered in the position of (x, y, z) and the prey lie in the position (x′, y′, z′) then different positions around the prey can be reached with respect to the current position by adjusting the vectors A and C. In this adjustment, the random vectors r 1 and r 2 play a vital role in the algorithm performance. This is translated as the method's exploitation and exploration ability. In any iteration, the wolves' position update is performed by (see (17)) . In Fig. 7, the way by which a search agent (wolf) updates its position in a 2D search space is shown. It can be observed that the final position would be in a random place within a circle which is defined by the positions of alpha, beta and delta in the search space. Finally, an illustration of the GWO algorithm used here -in pseudo-code form -is depicted in Fig. 8.

Ant-lion optimiser
The next nature-inspired meta-heuristic technique adopted here was proposed in 2015 and it was inspired by the ant-lions and ants interaction [23]. In that study, 19 different mathematical problems were solved plus three engineering problems. Since then, there  were many applications of the ALO in several engineering problems (e.g. in [30,31]) which demonstrated its capabilities. However, ALO has never been used in the DC motor parameter identification problem up to now. Ant-lions (or doodlebugs) belong to the Myrmeleontidae family of the Neuroptera order. They undergo metamorphosis in a cocoon to become finally an adult. They mostly hunt in larvae stage (Fig. 9a), while their short adulthood period is for reproduction. The idea behind the ALO algorithm is to mimic the interaction between ant-lions and their favourite prey which is ants, in the trap made by the former. This interaction is modelled by considering that ants move in the search space, and ant-lions hunt them by setting traps (Fig. 9b) and thus become stronger (fitter). In this context, the main steps of the algorithm are as follows: (i) ants' random walk, (ii) trap setting by ant-lions, (iii) trapping of the ants, (iv) ant catch by the ant-lion and (v) trap rebuild for catching the next ant.
The location of the ants and the corresponding objective function are stored in matrices P ant = a 1, 1 a 1, 2 … a 1, d ⋮ ⋮ ⋮ ⋮ a n, 1 a n, 2 … a n, d 1 , a n, 2 , … a n, d ]) where d is the number of search variables, n is the ants population (search agents) and k is the current iteration. At the same time, the ant-lions position -in the search space -and their corresponding fitness are given by As the ants move randomly by nature, their movement ( Fig. 10) is modelled by a random walk technique which can be expressed by X k = 0, cs 2r 1 − 1 , cs 2r 2 − 1 , … …, cs 2r k − 1 , …, cs 2r N − 1 (20) where X is the vector of random walk positions, 'cs' stands for 'cumulative sum' and r k is a stochastic function which is given by (v is a uniformly distributed random value in the set [0, 1]). Actually, a random walk is a type of Markov chain, which is the random process where each step depends only on the previous step. Moreover, since ants are moving in the variables search space and the latter is -usually -limited by lower and upper bounds, the position update mechanism which ensures these boundaries is the following min-max normalisation method: where α i and β i are the minimum and maximum values, respectively, for the ith variable, γ i and δ i are the minimum and maximum value, respectively, of the ith variable at the iteration k, c and d are vectors with the minimum and maximum values of the search variable at iteration k, and R = wk/Iter is a ratio which models the increment/decrement of the ant's position distance from the trap (w a constant and Iter the total iterations). Continuing, the modelling for the entrapment of the ants is given by  Moreover, the elitism feature is modelled also in ALO. Elitism is an important aspect of the evolutionary algorithms which allows them to keep the best solution during the optimisation process. Thus, the mechanism which is considered in order to generate new positions for the ants is based on the elite ant-lion (the best ant-lion found up to a certain iteration) and the average of the elite and the roulette wheel selected ant-lion random walks, i.e.
At the final step of ant-lion hunting, an ant falls into the bottom of the trap and is caught by the ant-lion's jaws. The ant-lion pulls the ant into sand and consumes it. In the ALO algorithm, catching the prey occurs when an ant's fitness function becomes better than its corresponding ant-lion. In this situation, the ant-lion changes its position to the position of the hunted ant. This process can be mathematically expressed as Finally, an illustration of the ALO algorithm in pseudo-code form used here is depicted in Fig. 11 while a further detailed description can be found in [32].

Salp swarm algorithm
The last algorithm adopted in this work was shown in the literature in 2017 inspired by salps behaviour [24]. Despite the very short time since it was introduced, applications of SSA have already been reported (e.g. [33,34]). However, this is the first time where SSA is used in an electrical engineering problem. Salps belong to the family of Salpidae and have transparent barrel-shaped body. Their tissues are highly similar to jelly fishes. Essentially, salps are gelatinous zoo-plankton owing complex nervous and digestive systems. It is important species in the food chain since 200 other species eat them. Salps form chains of clones. These chains then break up and each salp gives birth to more salps. Their movement is accomplished, by pumping the water through their body as a propulsion mean to move forward. Fig. 12a depicts the shape of a salp, while in Fig. 12b a salp chain formation is shown.
In mathematical and modelling terms, the salp swarm behaviour is simpler compared to the previous techniques presented, as it will be shown next. Firstly, it is assumed that the salp chain has one leader (the first salp in the formation) while all the rest of the salps are followers. Then, if an n-dimension search space is considered, i.e. n is the number of search variables, the positions of the salps are stored in a matrix, let it be X. It is also considered that there is a target (let it be denoted F) which in nature is a food source. The relationship which SSA uses then is given by In essence, (27) implies that the leader salp only updates its position with respect to the food source. Additionally, X 1, j denotes the leader salp's position in the jth dimension, F j represents the food source's position, ub j and lb j indicate the upper and lower bound of the jth dimension, c 2 and c 3 are random numbers uniformly distributed in the interval of [0, 1] which are dictating if the next position will be towards negative or positive infinity and c 1 is a coefficient which depends on the current (k) and the total number of iterations (L), i.e.
Next, the Newton's law of motion is utilised for the modelling of the position updating mechanism of the follower salps where v 0 is the initial speed, a is a speed ratio As time (t) in optimisation is the iteration, by considering v 0 = 0, (29) can be written as The pseudo-code of the SSA algorithm implemented here is depicted in Fig. 13.  Fig. 4 shows the experimental speed response for this motor). Such a response can be obtained in a laboratory environment through a data acquisition system connected to a computer, which records the motor speed. Apart from that, the only equipment needed is a constant voltage power supply and a speed sensor (optical encoder). Also, Table 2 shows the common parameters of the three nature-inspired algorithms used here. Each algorithm ran 20 times for 500 iterations each with a population of 60 search agents. This was repeated for each of the three criteria that were adopted as objective functions. In each run, the best, worst and average values for each of the six search parameters were stored, as well as the algorithm's convergence curve. The above experiments were performed on a 64-bit PC with Intel i7 processor, 8 Gb memory and Matlab v. 9.1 software.

Results and discussion
The results obtained are presented and analysed next. In Fig. 14, the algorithm performance (in logarithmic scale) with respect to their ability to converge is presented. In particular, each sub-graph shows the best, worst and average convergence curves from the 20 independent algorithm executions. The classification was made based on the criterion examined ( Fig. 14a for ISE, Fig. 14b for IAE and Fig. 14c for ITAE), because the order of magnitude of the result of each criterion is -by mathematical definition -different. In addition, for easy comparison reasons, the scale of the y-axis in each criterion is the same. A general remark is that all algorithms exhibit a converging capability, even though the performance of some of them cannot be judged very satisfactory as it can be seen later.
With respect to the ISE criterion it is noted that SSA and ALO can minimise the objective function by up to an order of 10 −8 while the GWO marginally reaches a value in the order of 10 −2 .
Similarly, the average convergence curves present an analogous behaviour. The corresponding performances for the IAE criterion show that the objective function reaches (at best) a value at an order of 10 −3 and 10 −4 for SSA and ALO while for GWO this value is about 10 −1 . Regarding the ITAE criterion, 10 −4 and 10 −6 are the orders of magnitude for SSA and ALO, while for GWO is low again at an order of 10 −2 .
Another feature that is observed is as follows: the convergence of the SSA follows for a certain number of iterations a slight downward trend and after a point it decreases abruptly. On the other hand, the convergence of ALO follows a relatively constant rate downward trend across the whole range of iterations. Finally, GWO abruptly drops during a small number of iterations initially, and then almost gets its final value, indicating that it may be inadequate to explore further the possible solution space for the problem.
Results in tabular form are presented next. Tables 3-5 show the SSA performance regarding the DC motor parameters for each criterion examined, respectively. The second column of each table is the actual parameter value according to the manufacturer, while the third, fourth and fifth columns are the best, worst and average values found by the applied algorithm among the 20 independent runs ε % = P ave − P r P ave × 100, P ave = 1 The sixth column is the mean percentage of the error for each parameter -through the 20 runs -which has been calculated using (31) where P r is the exact value of the parameter to be found, P e is the estimated value of the parameter in a certain run and P ave is the mean value of all the P e values. Finally, the last column of the tables shows the standard deviation of the best estimated value for each parameter and for each independent run. The contents of the tables follow the same layout for the rest of them (Tables 6-11) for easy reference, as it will be presented later. Hence, it can be observed from Table 3-Table 5 (bold values) that for four out of the six search variables (i.e. R a , L a , J m and k t ), SSA gave the smallest error for ISE as objective function, while the same is given when IAE is used. Specifically, the accuracy range for motor winding resistance (R a ) is 3.2-5.6%, for the winding inductance (L a ) 2.2-4.9%, for the moment of inertia (J m ) (8 × 10 −6 )-0.2%, for the friction coefficient (B m ) 1.1-2.8%, for the torque coefficient (k t ) 3.2-5.6% and for the back-emf coefficient (k e ) 0.14-0.27%. It should be noted however here that all of the aforementioned accuracies (even the worst ones) are very satisfactory and considered fully acceptable from a technical point of view. Finally, it seems that ITAE maybe not suitable for performance criterion when SSA is adopted.
The corresponding results for ALO performance are given in Table 6-Table 8. Here, for three out of the six search variables (i.e. R a , L a , k t ) ITAE performed better as objective criterion, while for the mechanical parameters (J m , B m ) IAE seems to be better. Backemf coefficient (k e ) presented minimum error when ISE was used but it is comparable with this found using IAE, so it can be said that IAE performed equally well in estimating k e . The accuracy range for motor winding resistance (R a ) is 2.6-4.2%, for the winding inductance (L a ) 0.3-3.2%, for the moment of inertia (J m ) (8.9 × 10 −5 )-(2.3 × 10 −3 )%, for the friction coefficient (B m ) 2-5.3%, for the torque coefficient (k t ) 2.6-4.2% and for the back-emf coefficient (k e ) 0.09-0.14%. Also, it seems that ISE is not appropriate in case of ALO.
The last three tables (Tables 9-Table 11) refer to the GWO performance. Here it can be seen that four out of the six search variables (i.e. R a , J m , k t and k e ), are estimated with the minimum error using the ISE criterion again (as in SSA). Viscous friction coefficient (B m ) smallest error obtained by IAE, while winding inductance (L a ) by ITAE. The parameter estimation accuracy ranges obtained by GWO are: for motor winding resistance (R a ) is         1.8-6.7%, for the winding inductance (L a ) 0.2-8.9%, for the moment of inertia (J m ) (1.4 × 10 −4 )-0.02%, for the friction coefficient (B m ) 2.3-3.4%, for the torque coefficient (k t ) 1.8-6.7% and for the back-emf coefficient (k e ) 0.09-0.44%. It can be commented from the above that GWO does not perform quite 'stable', meaning that it may give small and also quite large errors as seen in the cases of R a , L a and k t . Also, IAE seems to be inappropriate choice as the problem's objective function when GWO is adopted. So far, the analysis was based solely according to the absolute mean percentage error of the estimation procedure for all the three examined techniques. In order to draw more robust conclusions, attention will be focused now at the standard deviation quantity. The reason for this is because it was observed that in some cases the estimated values were close enough and in some cases not so close to the exact motor parameter values. Thus, in some cases the deviation through the independent runs was small, while in the rest was be relatively larger. Also, because we face a comparative -and not a single -algorithm evaluation, the conclusions should be the more profound and the more objectively drawn. Thus, we present Fig. 15 as a visualisation mean where now a classification by each parameter separately is shown for the two performance measures, i.e. the absolute mean percentage error and the standard deviation which are now both normalised. If Fig. 15a is noticed carefully, it can be seen that regarding the error, none of the three techniques perform consistently, since (as analysed in the previous paragraphs) for some parameters give the less error while for the others do not. In other words, considering the obtained estimation mean error only, may not be a 'safe' way to argue about the robustness of the proposed meta-heuristic techniques to the specific problem. We remind here though, that all obtained errors are technically accepted and potentially a motor controller designer can use the results effectively.
On the other hand, the standard deviation measure helps significantly towards this direction. By observing Fig. 15b, it is seen that the SSA gives the smaller values for every each one of the motor parameters compared to ALO and GWO. This also happens if ISE is used as an objective function. The same is observed for the IAE and IAET with the exception of the motor inertia parameter (J m ).
Thus, it can be concluded that the SSA outperform ALO and GWO in terms of the standard deviation of the problem search parameters, when accompanied by the ISE criterion. For completing the comparison, it is also seen that neither of ALO and GWO can take the 'second' place, since the corresponding performances vary.

Conclusions
In this paper, a parameter identification approach of a PMDC motor was performed, by employing three recent nature-inspired meta-heuristic techniques. For this purpose, both the no-load and full-load speed responses were utilised and no simplifications made about the motor torque and back-emf coefficients which were treated unequal as this is true in practice. Three objective functions were examined and several simulations were conducted for each case. The general conclusion is that all of the algorithms can   potentially be successfully used in this or similar problems, because they succeeded to converge giving acceptable parameter values. It was also found that the percentage mean error cannot be safely used as an estimation metric. On the contrary, the estimated parameters standard deviation is proven as a more robust measure. Based on the total obtained results, the SSA presented the better performance compared to ALO and GWO, and seems to be a very promising optimisation technique for the DC motor parameter identification and relative problems. Nevertheless, the accurate characterisation of the DC motor allows, in turn, the ability to predict its performance over a wide range of operating conditions towards the easier design of relevant robust controllers.