Optimal backstepping controller for controlling DC motor speed

In this paper, squirrel search algorithm (SSA) is adapted to tune the optimal parameters of a backstepping controller for controlling speed in DC motor; Lyapunov stability theorem is applied to derive the control low to achieve system stability analysis. M-file and Simulink platform is used to simulate the response of the system, a comparison with conventional controllers is utilized, all gains are tuned using SSA method and integral time absolute error (ITAE) fitness function to test the efficient performance of the proposed controller also a comparison with other tuned different controllers is done. Transient response analysis is used to validating the proposed controller performance. The simulation results showed a stable response and efficint performance for the proposed controller when compared with the two conventional controllers used by 41.6% for PID controller and by 32% for PI controller in rise time while in settling time is superior by 36.97% for PID controller and by 41.82% for PI controlle these values led for achieving the desired speed in fast and stable response.


INTRODUCTION
Nowadays, DC motors achieves clear spread and prosperity in industrial and commercial markets like robotic manipulators, home appliances, medical devices, vacuum cleaners and electric vehicle industry due to many reasons like there flexibility, efficiency and low cost so due to this it must work in precise and stable response even when nonlinearities and disturbances are cognizable, for this issue conventional control methods like PID control are not suitable to achieve efficient performance [1]- [3].To avoid drawbacks of conventional control methods the nonlinear control techniques have been used such as, Backstepping control, predictive control, sliding mode control and fuzzy logic control, from these control methods, backstepping control method has achieved an efficient influence as a solution for control of nonlinear systems.The matter here is done by devising the plant into subsystems as a first order system and using Lyapunov relation to reach to the stability of the plant, for obtain the virtual control relation for the subsystem the control low will gained from different parts by solving it gradually step by step then the overall control low for the final subsystem will calculated [4]- [6].
Different control methods have been adopted from many researchers to reach to efficient response, some of them use the conventional controller scheme in two ways either by using it with an optimization method to enhance it response by finding the suitable values for its parameters that give the desired response like in [7] propose the Jaya optimization algorithm (JOA) for obtain best performance then compare results

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with particle swarm optimization (PSO), concluded that PSO method in steady state response is best than the JOA and JOA is more efficient in transient response as compared to PSO.In [8] adopts a stochastic fractal search (SFS) method, it depends on concept of Fractal, robustness analysis of suggested SFS/PID controller has been carried out when use different values in electrical resistance and motor torque, in [9] presents an improved weed optimization method based on chaotic theory, the issue of finding optimal controller gains values of a DC motor is done efficiently also robust performance achieved with a simplicity in the design of PID.In [10] optimal I-PD structure is used to enhance responses with minimal overshoot and good regulating time if it is compared with the conventional controller while in [11] a cascade and dual configuration is used by adopt an ANFIS with PID and ANFIS with PD controllers but these configurations have a rapid response, minimum overshoot, efficient dynamics and robustness also stable responses.In [12] TID controller is used and tuned by using Firefly algorithm (FA), TID's structure is a modified PID controller but the change is done in proportional part is which it modified to "(1/s) n " and n is a real value, different performance indices is tested such as ITSE, ISE, ITAE, IAE, the best response appeared clearly in ISE index that show best and efficient results.
The main contribution of the paper is the novel optimization method used (SSA) that have a smart search technique reflected on the backstepping controller behavior with the use of Lyapunov function as a control low to improve the performance of DC motor system to reach to the efficient performance with a high level of robustness to face the external disturbances that may happen.

DC MOTOR MODELLING
DC motors are divided into two types depending on the way of excitation either self exited or externally exited, here externally excited type will be used for controlling the speed through controlling the voltage of the armature, the circuit that represents DC motor using the control of armature voltage is shown in Figure 1 and the explanation of its variables will be in Table 1 and for evaluate the transfer function of the mathematical model Kirchhoff's law (KVL) is applied as shown [13]- [15]: Where R a  L a are the armature resistance and armature inductance, i a represents the armature current and e a and   represent the applied voltage and the induced emf in the armature conductors, then   is proportional to the motor angular speed w= K b is the back emf constant, the torque of the motor is obtained based on newton's low: K t is the torque constant, J is the rotor inertia and B is the friction constant.Then by applying Laplace transform to [1]- [3]: The transfer function will be written as in (7):

BACKSTEPPING CONTROLLER DESIGN
The backstepping control scheme is a recursive control approach used for achieve stability for highly nonlinear systems by dividing them into first order subsystems with the benefits of using virtual control parameters, to achieve stability for the system a Lyapunov function is used to stabilize the error generating and drive virtual control parameters, the design block diagram explain the DC motor controlled by an optimal backstepping controller and the steps for designing such controller are as shown [16]- [19].The transfer function for the system rearrange as shown: Now for simplify equations let LJ then it will be as shown: and can be written as (let w(s)=e) then ( 12) can be formulated as outlined (13) in a nonlinear dynamic equation: Where F(x)= −Bė− Ce, g(x)=A, d(t) is the external disturbance.Now for find the tacking error (z1) the desired value will be ed and the actual value of response is eb then The first L yapunov relation is: and V 1 derivative is: where k 1 is a constant value tuned by using SSA method, and virtual control is replaced by its desired value then (17) will be: Now the virtual control deviation from desired value is: Now the derivative of z 2 is: The second Lyapunov function is selected as: and is derivative is Substitute e b ̈ in ( 12) and ( 13) then the equation will be The disturbance signal effect d(t) here is considered as unknown, then for satisfying V 2 ̇(z 1 , z 2 ) ≤ 0, the system input u(t) will be: (28)

SQUIRREL SEARCH ALGORITHM
The squirrel search algorithm (SSA) was a novel optimization method proposed by Jain et al. [20], it assumes that (n) flying squirrels have (n) deciduous trees in the search space, each squirrel have one tree and the manner of searching is done individually, tree types is classified into three (normal, oak or acorn nuts and hickory) and the final assumptions is that three trees is classified as oak tree and one as hickory.The location vector for each ith flying squirrel is defined as follows [21]- [23]: Fs ij is dimension of each flying squirrel and the initial location is: Where Fs U and Fs L is the upper and lower search bounds and U(0,1) is a random value between [0, 1].The arrangement of the fitness values for the individuals will be in ascending order, the minimum value for hickory tree, the other three minimum values will be the acorn trees and the rest trees are the normal.The movements of the individuals for finding their food is affected by the predators in the forest, the probability d g is a random distance value for gliding and G c is a constant value=1.9used for maintaining balance to the exploration and exploitation states and R 1 is a random value between [0,1] 2-Flying squirrels in normal trees will move to acorn trees, the new position is: 3-Flying squirrels in normal trees may move to hickory trees, and then new position is: The gliding distance and angle can be expressed as in (34): Where h g = 8, sf = 8, V is the speed, S is the surface area of body, ρ is the density of air, C D = 0.6 and C L and are constant value between 0.675 and 1.5 then if all individuals have been updated, the season changes is calculated as in (38): At the end of winter season some individuals could not find their food so it must search in other paths, such squirrels will relocate as in (39): where Levy(n) is calculated as (41) Finally, for achieve an optimal system performance a good choice for the fitness criteria must be utilized [24], different fitness criteria is found like integral of absolute error (ITAE), integral of time square errors (ITSE), integral of square errors (ISE) and integral absolute errors (IAE).In this paper the ITAE in (42) is used as a fitness function [25] for the SSA algorithm to tune the best values for the backstepping controller gains to obtain optimal system response as shown in Figure 3.

SIMULATION RESULTS
The optimal backstepping controller is simulated using Matlab/Simulink environment and the parameters values of the DC motor system are given in Table 1 [26].The SSA tuning algorithm are used for finding the optimal gains of the controller to obtain the desired output for the system and the parameters used for achieve suitable control action by reducing the tracking error with a minimum value of objective function are listed in Table 2.
The ITAE function is considered as a fitness function to test the error continuously [27], the behavior of the ITAE fitness function for the optimal backstepping controller is is shown in Figure . 4. The system response using optimal backstepping controller is shown in Figure 5 and the gain of these two controllers is also tuned using SSA tuning algorithm and shown in Table 3, a comparison with two conventional controllers (PI &PID) is done to show the effectiveness of the proposed controller based on transient response analysis as indicated in Table 4.    From the simulation results, it can be seen that the backstepping controller has a smooth and fast settling time (0.087s) as compared to PI (0.208s) and PID controller (0.238s), that are optimized using the same SSA tuning algorithm, a comparison for these controllers is shown in terms of the main characteristics of response analysis'.It can be seen clearly that the backstepping controller has faster time for reaching to the desired value and without overshoot these criteria are reflected the efficient response of the backstepping controller to track the desired value and save the result with zero error for all the simulation time.

COMPARISON ANALYSIS WITH OTHER CONTROLLERS
The performance criteria in table 4 translates the behavior of classical controllers even if optimized with new optimization algorithms, it suffer from small overshoot and have long settling time to reach to desired speed with zero error value as compared with optimal backstepping controller, also a comparative analysis with other controllers tuned by new optimization is done in Table 5 to see the efficient response for the proposed method suggested in this paper.
The response of DC motor for different tuned controllers in Table 5 reflects the efficient response of the proposed controller as compared with conventional one even if it tuned with a new method, the issue is a tradeoff between the stability of the controller represented by the over shoot and the fast response represented by the fast settling time as appeared clearly in the obtained compared values.2.28 0.1663 0.5214 SFS-PID [8] 0 0.544 1.45 IWO-PID [9] 5.7644 0.1453 0.690 Cuckoo-I-PD [10] 4.2264 0.0841 0.2141 ANFIS-PI [11] 0.609 0.474 0.809 ANFIS-PD [11] 0.562 0.00966 0.0162 ANFIS-PID [11] 2.75 0.587 2.31

CONCLUSION
This paper presents an optimal backstepping controller for controlling speed in DC motor based on adopting a new swarm approach called SSA to obtain the optimal parameters and enhance system performance by reducing the error in the steady state analysis, tracking the desired output with less rising time and stable response without any overshoot.Results obtained are compared with the classical controllers (PI &PID) to show the effectiveness of the proposed controller, the proposed controller is faster than PID controller by nearly 41.6% and faster than PI controller by nearly 32% in rise time while in setting time faster than PID controller by nearly 36.97%andthan PI controller by nearly 41.82% also it is more stable than the two classical controllers (PI & PID) as shown in its response.The proposed controller and the smart tuning algorithm together led the system response to its optimal results

Figure 1 .
Figure 1. Circuit of DC motor

Figure 4 .Figure 5 .
Figure 4. ITAE function for the backstepping controller according to SSA tuning

Table 1 . DC motor parameters
̇ is considered as a virtual control.A stabilizing function that represents the desired value of the virtual control is expressed as shown: Optimal backstepping controller for controlling DC motor speed (Layla H. Abood) 2567 e b is Pdp, if there are no predators in the forest the way of finding food will be classified into three ways:  ISSN: 2302-9285 Bulletin of Electr Eng & Inf, Vol.11, No. 5, October 2022: 2564-2572 2568 of predators presence

Table 2 .
SSA algorithm parameter

Table 3 .
Optimal controllers gains by SSA

Table 4 .
Transient response analysis of the optimal three controllers

Table 5 .
Transient response comparison for other controllers Optimal backstepping controller for controlling DC motor speed (Layla H.Abood)2571