Electr Eng DOI 10.1007/s00202-017-0592-5 ORIGINAL PAPER Backstepping control of a separately excited DC motor Abdelkader Harrouz1 · Houcine Becheri2 · Ilhami Colak3 · Korhan Kayisli3 Received: 10 November 2016 / Accepted: 20 June 2017 © Springer-Verlag GmbH Germany 2017 Abstract In the last decades, researches on speed control of electric motors have been very important in industrial systems. On the other hand, the development of power electronics and semiconductor technologies has expanded the scope of the AC and DC machines. However, the separately excited DC motor is widely used in the field of variable speed applications due to its simplicity to control. In this paper, we present the control of a separately excited DC motor using an analogue PI controller, a backstepping method and an adaptive backstepping control techniques. Simulations are performed in MATLAB Simulink software, and improved results of adaptive backstepping controller have been obtained overall system performance compared to conventional PI and backstepping. Keywords Control · Modelling · Backstepping · PI · DC motor · Adaptive backstepping B Ilhami Colak ilhcol@gmail.com Abdelkader Harrouz harrouz@univ-adrar.dz Houcine Becheri houcine.becheri@gmail.com Korhan Kayisli korhankayisli@gmail.com 1 Department of Hydrocarbon and Renewable Energy,Faculty of Science and Tech, University of Adrar, Adrar, Algeria 2 Department of Electrical Engineering, Faculty of Technology, University of Bechar, Béchar, Algeria 3 Department of Electrical Electronics Eng., Faculty of Engineering Architecture, Nisantasi University, Istanbul, Turkey List of symbols DC AC PI PM Va Ra Ia La Lm if Vf Rf Lf ωr ω R Rh u1 u2 u max Kp θ Hc Vc s P K pω K iω Kθ e1 fc V1 V2 Direct current Alternating current Proportional–integral controller Power motor Armature voltage Armature resistance Armature current Armature inductance Mutual inductance Field current Field voltage Field resistance Field inductance Armature (rotor) speed Change of speed Nominal armature resistance Rheostat resistance First value of the voltage Second value of the voltage Maximal value of the voltage Static gain of the proportional controller Position Transfer function of chopper Control voltage of chopper Laplace variable Poles number Static gain of the proportional speed controller Static gain of the integral speed controller Static gain of position controller Error of rotor speed Coefficient of friction First Lyapunov function Second Lyapunov function 123 Electr Eng e2 k1 k2 e3 U L V3 Rf E ref Error of current and the armature current ia First coefficient of armature voltage Second coefficient of armature voltage Final signal error Voltage Inductance Third Lyapunov function Field resistance Set point weaken the flow ωref reference speed 1 Introduction DC motors are a type of devices that transform electrical energy to the mechanical energy. The usage of DC motors might be restricted since distribution systems are based on AC current but DC motors are used in a huge number of industrial applications. In particular, separately excited DC motors have many fields of applications [1–4]. However, for some applications, it may be an advantage to use DC current motors powered by static converters that convert AC to DC. DC motors have many benefits such as flexibility, continuous and instantaneous speed. Therefore, they have been widely used in many fields including electric traction, positioning a radar and handling robots [5]. In addition, the variable speed electrical motors have very important roles in the industrial and transportation systems. In recent years, the development in power electronics has expanded the scope of the DC machines [6,7]. For instance, the separate excitation DC machine is widely used in the field of variable speed applications with the simplicity of control. There are several existing solutions for speed control of a DC motor for separate excitation. The principle speed control of separately excited DC motors has developed from the equation of electromotive force of the motor [8,9]. The speed control of a DC motor can be achieved using the following methods: • varying excitation flux • varying the resistance of armature circuit • varying the supply voltage of armature circuit Because of its simplicity and stability, electric drives use conventional controller “proportional–integral” (PI) to control the current, velocity or position. In a practical implementation, some physical characteristics of the motor can vary during operation resulting in parametric variations on the system model [10,11]. In the context of tracking, the basic idea of backstepping is to make the loop system equivalent to stable cascade subsystems in the sense of Lyapunov, which gives it qualities robustness and asymptotic overall stability of the tracking 123 error. For a large class of systems, this technique is a systematic and recursive method of synthesis of nonlinear control laws. Thus, at each step of the process, a virtual command is generated to ensure the convergence of the first-order subsystems characterizing the continuation of trajectories towards their equilibrium states (zero tracking errors in the deterministic and non-perturbed case) [12–14]. This technique allows the synthesis of robust control law despite some misunderstanding of the parameters of the system and some perturbations [15,16]. The objective of this study was to apply control techniques, it is the PI, and backstepping adaptive backstepping on a DC motor powered by a chopper. The contribution is composed of five parts. The first is devoted to the presentation and modelling of the DC motor and speed control methods of DC Motor. The second part is reserved for motor control by the PI controller. In the third part, we give the principle and the general algorithm of backstepping followed by an application of our model. Then, the fourth we will apply the command by adaptive backstepping. Finally, simulation results are present and conclusions are drawn from the last part of this article. 2 Modelling of a DC motor Many researchers have addressed the control of electrical machines and specifically the control of DC motors [17– 19]. Several authors have tackled the control problem of electrical machines operating in the field-weakening region. For example, Asghar et al. [20], Zambada [21], Krim et al. [22], Seibel et al. [23] dealt with the control of induction motors operating in the field-weakening region. Tan et al. [24] explored the field-weakening control of a permanent magnet synchronous motor (PMSM). Hesari [25] studied design and implementation of maximum solar power tracking system using photovoltaic panels, and Bakshi and Bakshi [26] proposed a field-weakening speed control system for a selfregulated synchronous motor. Nouira et al. [27] studied a contribution to the design and the installation of an universal platform of a wind emulator using a DC motor. The equations of a DC machine for continuous permanent magnet arrangement can be fulfilled by the equations of the field voltages and frame, as well as the equation connects torque with speed. The equations of the motor assuming a simple model of the load are given as follows [28–30]: di a + L m i f ωr dt di f Vf = Rf i f + L f dt Va = Ra i a + L a (1) (2) Electr Eng Cm Cm (R + R h1 ) (R + R h2 ) r2 r1 r0 f3 Crn Cr r3 f2 f1 R h1 = 0 r rn Fig. 1 Torque-speed curve of a DC motor at different armature resistance Torque and rotor speed are combined with the following relationship: Ce = L m i f i a = J dωr + Fc ωr + Cr dt (3) r Fig. 2 Torque-speed curve of a DC motor at different fluxes • the reduction in the rigidity of the mechanical characteristic • the decrease in overload capacity • large power losses by the Joule effect 3.2 Speed control by the flux changes The parameters and nominal conditions of the DC machine are chosen as given in Table 1, in Appendix. If the drop of resistive voltage is neglected before, the voltage (Va >> Ra Ia ) expression of the speed becomes as given in Eq. 5. [26,28]: 3 Speed control methods of a DC motor The motor operating point is fixed by the intersection of the curves Cm (n) and Cr (n) as shown in Fig. 1. The speed control is to move the characteristic Cm (n) so as to obtain a different operating point. The adjustment process may be mechanical or electrical. The mechanical process is a change of the speed by means of a gear that changes the ratio. Electric processes are for three variables dependent on speed, namely the flux φf , the armature voltage Va and the resistance of the armature circuit [24]. 3.1 Speed control by insertion of a resistance into armature circuit This method is performed by using an adjustable resistor inserted in the armature circuit. It is a very simple method allows adjustment in the direction to decrease from its nominal speed. The speed–torque characteristic of separately excited DC motor (or shunt) can be expressed by the following equation: Ra Va − Ia = ωro − ω ωr = K φf K φf (4) The voltage and flux are fixed at their nominal values, and by this way, it can reduce the speed by increasing the armature resistance by a rheostat connected in series with the armature circuit. It presents the following disadvantages [31,32]: ωr = Va K φf (5) As depicted in Fig. 2, the no-load speed is inversely proportional to the flux φf (or current If ), and the tangent of the rate equation (Eq. 5) is inversely proportional to the square of the flux φf . It is also known that the torque directly proportional to the flux and the armature current Ce = K φf Ia . 3.3 Control of armature voltage This method consists of speed control by adjusting the armature voltage and it is very effective, stable and easy to implement. As seen in Fig. 3, the only variable controlled in here is the armature voltage of the DC motor based on the speed Eq. 6. [24–28]: ωr = Va Ra − i a = ωro − ωr K φf K φf (6) When the voltage Va is reduced, the no-load speed ω ro is reduced too. In addition, for the same value of load torque and flux φf , the armature voltage does not affect the drop in speed ωr . The tangent of the speed–torque characteristics is Ra 2 that independent of the armature voltage. Therefore, (K φf ) it is possible to obtain different values of the voltage induces a family of the non-deformed parallel characteristics. Speed control of a DC motor can be achieved either by varying 123 Electr Eng u2<u1 Cm u1 < u max u max Cn Fig. 5 Speed control block r Fig. 3 Torque–speed curve of a DC motor at different armature voltages a plane relative to the poles of the inner loop. According to the mechanical equation, [29,33,34] the following definition can be written: P ωr = Ce fc + J · s (7) Block diagram of a DC motor speed control is shown in Fig. 5: To evaluate the performance of the speed control of a DC motor, we performed the numerical simulations with the following conditions. • running the motor at no-load up to 100 rad/s • applying a torque value with nominal load torque as a value of 29.2 Nm at t = 5 s Fig. 4 Closed loop speed control block diagram of a DC motor Figure 6 shows that: the voltage across of the armature circuit or by varying the excitation flux. 4 Control of a DC motor by analogical PI The speed control of a separately excited DC motor fed by a chopper works like all other converters as depicted in Fig. 4. The difference between the two configurations is existence in an internal loop of the current. The reason of the difference is the particular characteristics of the power stage of the chopper. The current and speed loops will be examined, and its characteristics are explained. There are two control structures as the speed and the position of motors with a current fed by chopper. Figures 4 and 5 illustrate the configurations that have a single outer loop and an inner loop in which a controlled current value is compared with the current value of the armature, and the error is processed through a control circuit [8,29]. 4.1 Speed control The best outer loop is to control the speed or the position (size having the slowest dynamics). For this reason, the poles laid down for the outer loop is closer to the origin of roots 123 • The speed follows the reference with a slight overshoot and a response time of 0.95 s • The current is limited to the permissible value and increases by increasing the load • Good rejection of perturbation 4.2 Position control The position controller determines the reference speed and the reference current, thus maintaining at the same speed. Considering that, the dynamic speed is faster than its position, and assuming that the speed reaches its reference value, then the open loop transfer function is written as below [8,29]: θ 1 = ωr s (8) The controller is chosen as a proportional action. The block diagram of the position control loop is depicted in Fig. 7. The closed loop transfer function can be written as follows: kθ θ = ∗ θ 1 + kθ s (9) Electr Eng Fig. 6 Simulation results of adjusting current and speed of a DC motor * e k * r 1 s Fig. 8 Simulation results of adjustment in the current position of DC motor 5 Backstepping control technique of a separately excited DC motor Fig. 7 Block diagram of position control 5.1 Speed control To evaluate the performance of the control position of a DC motor, we performed the numerical simulations with the following conditions as shown in Fig. 8. In this section, we propose the backstepping control technique based on the command presented in the fourth section. This method will also be applied for the motor control centre and speed control with current limiting [29,31,35–37]. The main objective of the control law is that the rotor speed ωr is pursuing a reference signal ωref . Now we go back to the steps of backstepping: • applying a 10◦ position level • applying a torque to the nominal load 29.2 Nm at t = 5 s. Figure 8 highlights that the adjustment by PI controller gives satisfactory results as follows: • the position of the rotor reaches the reference position with small oscillations; • the current is limited to the permissible value; • good rejection of perturbation. • Step 1: The aim is to control the rotor speed and the amplitude of the induced current, so that the following errors can be defined: e1 = ωr∗ − ωr and ė1 = ω̇r∗ − ω̇r (10) 123 Electr Eng If it is the control action, using a single Lyapunov function V1 = 21 e12 , and its temporal derivative is given as below: V̇1 = e1 ė1   V̇1 = e1 ω̇r∗ − (L m i f i a − f c ωr − Cr )/J (12)   fc cr ∗ k1 e1 + ω̇r + + ωr = L m if J J (13) (11) So that the reference current i a∗ is chosen as: i a∗ J • Step 2: We define another error signal between the reference current and the armature current i a : e2 = i a∗ − i a , and its derivative : ė2 = i̇ a∗ − i̇ a (14) If this error and its derivative are replaced in to the Lyapunov function, then we obtain Eq. 15.; V2 = 1 2 1 2 e + e , its derivative is: V̇2 = e1 ė1 + e2 ė2 2 1 2 2 (15) V̇2 = e1 [ω̇r − (L m i f i a − f c ωr − Cr )/J ]   +e2 i̇ a∗ − (va − Ra i a − L m i f ωr ) /L a (16) Fig. 9 Current and speed of a DC motor obtained from the simulation with backstepping control The armature voltage Va can be written as follows: ⎡ k1 e1 + k2 e2 + wr∗ + i a∗ + Va∗ = L a ⎣  f + Jc + LLmaif ωr + cJr  Ra La − L m if J ia ⎤ ⎦ 5.2 Position control (17) To illustrate the influence of backstepping controller on the system performance, the same tests are repeated with the PI controller. According to the simulation results, following conclusions can be given: • improvement in total system performance with the insertion of backstepping controller compared to conventional PI controller • the speed reaches its set point value along with practically zero overshoot as depicted in Fig. 9 • total rejection of perturbation • the current is limited to its permissible value as illustrated in Fig. 9 123 In this section, we have proposed the continuing problem of a DC motor using Backstepping control method. The proposed control method does not only stabilize the motor but also causes the speed of tracking error to converge asymptotically to zero [24]. The model of the DC motor can be reformulated as follows [8] : ⎧ ⎪ ⎨ θ̇1 = ωr ω̇r = − Jf ωr + 1J Ce − 1J Cr ⎪ ⎩ Ċ = − kφ 2 ω − L C + kφ u e L r R e (18) L This model is used to design the backstepping algorithm in order to achieve the objective of stability and pursuit the position. For this, we followed the following steps: Electr Eng • Step 1: At this point, if the desired reference torque is chosen as: First, we consider the speed as a control variable. Let us define the signal position error as: e1 = θ − θ ∗ and its derivative ė1 = θ̇ − θ̇ ∗ = ωr − θ̇ ∗ 1 2 V̇ = e1 ė1  e and its derivative 1 = e1 ωr − θ̇ ∗ 2 1 • Step 3: Now the final signal error is obtained as: (20) The derivative of the Lyapunov function can be rewritten as:   V̇1 = −k1 e12 + e1 k1 e1 + ωr − θ̇ ∗ (21) e3 = Ce − Ce∗ and its derivative : ė3 = Ċe − Ċe∗ + J (1 + k1 + k2 ) θ̈ ∗ + J θ ∗   R fc − + Ce (1 + k1 + k2 ) − J L    kφ fc + U − Ċr + C r − 1 + k1 + k2 + J L (22) (29) If Va is considered as a control variable, the third Lyapunov function V3 is chosen as: • Step 2: At the second step, it is tried to converge the speed signal to its reference. Therefore, the signal error is redefined: e2 = (28) ė3 = −J (e2 − k1 e1 ) (k1 + k1 k2 )   kφ f2 + ωr − f c (1 + k1 + k2 ) + c − J L At that point, the desired reference speed can be selected as follows: ωr∗ = −k1 e1 + θ̇ ∗ (27) (19) If ωr is the control variable, using a single Lyapunov function: V1 =   Ce∗ = J (1 + k1 + k2 ) e2 − k12 e1 − θ̈ ∗ f c ωr + Cr V3 = ωr − ωr∗ = ωr + k1 e1 − θ̇ ∗ (23) 1 2 e and its derivative : V̇3 = e3 ė3 2 3 (30) So the Lyapunov function can be rewritten as: And Eq. (23) can be expressed as: ė1 = e2 − k1 e1 and its derivative is given follows : V̇3 = e3 (ė3 + k3 e3 ) − k3 e3 ė2 = k1 ė1 − θ̈ ∗ + ω̇r At this point, the desired reference voltage can be selected: = k1 (e2 − k1 e1 ) + 1 1 f ωr − Ce + Cr − θ̈ ∗ J J J (24) If Ce is considered as the virtual control variable, we can use the Lyapunov function V2 like: 1 2 1 2 e + e and its derivative: 2 1 2 2 V̇2 = e1 ė1 + e2 ė2  fc = e1 [e2 − k1 e1 ] + e2 −k12 e1 + k2 e2 − θ̈ ∗ + ωr J  1 1 (25) − Ce + Cr J J V2 = The Lyapunov function can be rewritten as: V̇2 = −k1 e12 + e1 (k1 e1 + e2 − k1 e1 ) − k2 e22  fc +e2 k2 e2 − k12 e1 + k1 e2 − θ̈ ∗ + ωr J  1 1 − Ce + Cr J J (31) Va∗ = −k3 e3 + J (e2 − k1 e1 ) (k1 + k1 k2 )   kφ f c2 − − ωr − f c (1 + k1 + k2 ) + J L − J (1 + k1 + k2 ) θ̈ ∗ − J θ ∗   R fc − − Ce (1 + k1 + k2 ) − J L    fc + −Ċr − C r − 1 + k1 + k2 + J (32) To evaluate the performance of position control of a DC motor using the Backstepping technique, we performed numerical simulations under the following conditions as illustrated in Fig. 10: • applying a 10◦ position level. • applying a torque rated load 29.2 Nm at t = 5 s. (26) Figure 10, shows that the control performance of backstepping gives satisfactory results as below: 123 Electr Eng ẋ = f N (x) + f  (ẋ) + gf (x)u f (34) where: T T    T x = i a i f ω , ga = L1a 0 0 , gf = 0 L1f 0 , ⎤ ⎡ ⎤ ⎡ 1 a − R L a (−RaN i a − E) L a ia ⎥ ⎢ ⎥ ⎢ f f N (x) = ⎣ − RLfN i f ⎦ , f  (x) = ⎣ − R L f if ⎦ f 1 − 1J Cr J (C e − f c ω − C rN ) The objective is to develop a nonlinear adaptive controller, which can stabilize and track the desired value (reference) ωref and interval ωr ≥ ωrN , compensating for parametric uncertainties resistance of the armature and the inductor, and reject the load torque. To obtain these, the system controls the motor speed ωr and the E defined as: Fig. 10 Simulation results of position and armature current of a backstepped DC motor e E = E − E ref (35) eω = ω − ωref (36) where ωref is the reference speed and E ref represents the set point weaken the flow, which is chosen between 0.85 to 0.95 of the rated armature voltage [13,37,39]. We define the following change of coordinates: z 1 = h 1 (x) = E • the position reaches the desired value • the current is limited to the permissible value • a total rejection of perturbation z 2 = h 2 (x) = ω z 3 = ℓfN h(x) = ∇h · f (x) 6 Adaptive backstepping control technique of a separately excited DC motor In this section, a proposed adaptive backstepping control based on the cascade setting discussed in the first section of this chapter and the work presented in [33,35] will be applied for the speed control of the DC motor with an estimate of the armature resistance, inductor resistance and load torque Cc observer. This method is inspired by the research presented by Jianguo Zhou and Zuo Z. Liu [33,35]: where the notations ℓf h(x) = ∇h · f (x) is ℓif h(x) = ℓf (ℓi−1 f h) are used for the Lie derivation of the function h(x) along a vector field f (x) and its iterative form, respectively. This change of coordinates is one-to-one in s =  x ∈ R 3 : i f = 0 and ω = 0}. Using the coordinate transformation above, the motor model (22) becomes as below: ż 1 = ℓ fN h 1 (x) + ℓ f h 1 (x) + ℓga h 1 (x)Va + ℓgf h 1 (x)Vf ż 2 = z 3 + ℓ f h 2 (x) ż 3 = ℓ2fN h 1 (x) + ℓ f ℓfN h 2 (x) +ℓga ℓfN h 2 (x)Va + ℓgf ℓfN h 2 (x)V f Ra = Ra − RaN Rf = Rf − RfN Cr = Cr − CrN (33) where: K RfN E + i f (Ce − f c ω − CrN ), Lf J   T Rf r ωi f i f , ℓ f h 1 = K − L f − C J ℓfN h 1 = − Unknown deviations from the parameters of its nominal values are RaN , RfN , and CrN . The values of Ra and Rf change due to ohmic heating (increase in temperature during operation), while the value of Cr is practically unknown. Whereas the two control variables Va and Vf , the motor model can be reformulated as: 123 (37) 1 Cr (Ce − f c ω − CrN ), ℓ f h 2 = − , J J K RfN ℓ2fN h 2 = i f (−RaN i a − E) − Ce J La J Lf ℓfN h 2 = (38) Electr Eng fc (Ce − f c ω − CrN ) J2  1 Rf BCr a −K R ℓf ℓfN h 2 = La + Lf J J K ℓga h 1 = 0, ℓgf h 1 = ω, ℓga ℓf h 2 = Ls K ℓgf ℓf h 2 = ia . J Lf θ3 = −  ia if 1 K if , J La T  1 a −K R La + J Rf Lf BCr J  , ζ2 (x) = 1 After the backstepping design [19], we define the new variables , ē1 = e1 ē2 = e2 ē3 = e3 − α V̂a = ℓga h 1 (x)Va + ℓgf h 1 (x)Vf V̂f = ℓga ℓ fN h 2 (x)Va + ℓgf ℓ fN h 2 (x)Vf (39) (44) where α is for stabilizing control e2 , and α = −k2 ē2 − θ̂2 ζ2 (x) Then, Eq. (38) becomes: = −k2 ē2 − θ̂2 ⎡ ⎤ ⎡ ⎤ ⎡ ℓfN h 1 (x) + ℓf h 1 (x) ż 1 1 ⎣ ż 2 ⎦ = ⎣ z 3 + ℓf h 2 (x) ⎦ + ⎣0 ℓ2fN h 2 (x) + ℓf ℓfN h 2 (x) ż 3 0 ⎤ 0 V̂a 0⎦ V̂f 1  (40) Taking the derivative of (32), we have ē˙1 = ℓfN h 1 + V̄a + θ1 ζ1 (x) = ℓfN h 1 + V̄a + θ̂1 ζ1 (x) − θ̃1 ζ1 (x) ˙ē2 = ē3 + α + ℓ f h 2 Define a linear reference model, as: ż m = Am z m + Bm u (45) (41) = ē3 − k2 ē2 − θ̃2 ˙ē3 = ℓ2f h 2 + V̄f + θ̂3 ζ3 (x) − θ̃3 ζ3 (3) − α̇ N = ℓ2f N h 2 + V̄ f + θ̂3 ζ3 (x) − θ̃3 ζ3 (3) − k2 ē˙2 − θ̂˙2 with,  T  T z m = z 1m z 2m z 3m , etu = E ref 0 ωref , ⎡ ⎡ ⎤ ⎤ −a1m 0 a1m 0 0 ⎦ , Bm = ⎣ 0 0 ⎦ Am = ⎣ 0 0 1 0 −a2m −a3m 0 a2m (46) where : ∂α ˙ ∂α ˙ θ̂2 ē2 + ∂ ē2 ∂ θ̂2 = −k2 ē˙2 − θ̂˙ α̇ = where a1m , a2m and a3m are constant design parameters that can specify the responses of the dynamic system. The tracking error e is defined as: (47) and the parameter errors : θ̃1 = θ̂1 − θ1 ⎤ ⎤ ⎡ z 1 − z 1m e1 e = ⎣ e2 ⎦ = ⎣ z 2 − z 2m ⎦ z 3 − z 3m e3 ⎡ θ̃2 = θ̂2 − θ2 (42) So, its dynamic error is: ė = A(x) + A(x) + B V̄ where     V̄a V̂a + a1m z 1m − a1m E ref = V̄f V̂f + a2m z 2m + a3m z 3m − a2m ωref ⎡ ⎤ ℓf h 1 (x) ⎦, A(x) = ⎣ e3 2 ℓf h 2 (x) ⎤ ⎡ ⎡ ⎤ ⎡ ⎤ ℓf h 1 (x) θ1 ζ1 (x) 10 A(x) = ⎣ ℓf h 2 (x) ⎦ = ⎣ θ2 ζ2 (x) ⎦ , B = ⎣ 0 0 ⎦ 01 ℓf lfN h 2 (x) θ3 ζ3 (x)    T C r Cr f θ1 = K − R , θ2 = − , ζ1 (x) = ω · i f i f , Lf − J J V̄ = (43) θ̃3 = θ̂3 − θ3 (48) Then, we define the following Lyapunov function:   1 2 1 2 1 2 1 2 2 2 ē + ē2 + ē3 + θ̃1 + θ̃2 + θ̃3 V = 2 1 γ1 γ2 γ3 (49) Taking the derivative of this function with respect to time, and replacing ē˙1 , ē˙2 , and ē˙3 with Eq. (38), we obtain: V̇ = ē1 (ℓ f N h 1 + V̄a + θ̂1 ζ1 − θ̃1 ζ1 ) + ē2 (ē3 − k2 ē2 − θ̃2 ) + ē (ℓ2 h + V̄ + θ̂ ζ − θ̃ ζ − k ē˙ − θ̂˙ ) 3 fN 2 f 3 3 3 3 1 1 1 + θ̃1 θ̃˙1 + θ̃2 θ̃˙2 + θ̃3 θ̃˙3 γ1 γ2 γ3 2 2 2 (50) To do, V̇ ≤ −k1 ē12 − k2 ē22 − k3 ē32 ≤ 0 (51) 123 Electr Eng • application of a nominal load torque 29.2 Nm at time t = 5 s and t = 10 s and t = 15 s. According to the simulation results, we note following conclusions: • improved overall system performance with the inclusion of adaptive backstepping controller compared to conventional PI and backstepping • when starting, the speed reaches its set value with virtually no overshoot • total rejection of disturbance • the current is limited with its allowable value 7 Conclusions In this paper, we have modelled a separately excited DC motor. The mathematical model was developed for the DC motor with separate excitation in the form of linear differential equations. These equations describing the dynamic characteristics of the motor were solved using block diagram and then simulated. Thereafter, the DC motor was run by a PI speed control. After that, simulation for speed adjustment by the backstepping and adaptive backstepping techniques was developed. The simulation results under the MATLAB Simulink environment proved the efficiency and improvement of the system performance of this PI controller. Fig. 11 Simulation results of position by adaptive backstepping control Appendix We design the following control law: Table 1 Motor driver system parameters Parameter Per unit values V̄a = −ℓfN h 1 − θ̂1 ζ1 − k1 ē1 Ra 0.6 Ω V̄f = −ℓfN h 2 − ē2 − k3 ē3 − θ̂3 ζ3 − k2 (ē3 − k2 ē2 ) − θ̂˙2 Rf 240 Ω La 0.012 H Lf 120 H Lm 1.8 H J 1 Kg m 2 Fc 5 × 10−5 Nms (52) With adaptive laws: θ̂˙1 = γ1 ē1 ζ1 θ̂˙2 = γ2 (ē2 + k2 ē3 )θ̂˙3 = γ3 ē3 ζ3 (53) To illustrate the influence of the adaptive backstepping based controller on system performance, we performed numerical simulations under the following conditions as depicted in Fig. 11. • application of a speed step of 100 rad / s • increase in reference speed 123 References 1. Zribi M , Al-Zamel A (2007) Field-weakening nonlinear control of a separately excited DC motor, Hindawi Publishing Corporation mathematical problems in engineering, Vol 1 2. Koreboina VB, Narasimharaju BL, Kumar DMV (2016) Performance evaluation of switched reluctance motor PWM control in Electr Eng 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. PV-fed water pump system. Int J Renew Energy Res (IJRER) 6(3):941–950 Rinderknecht F, Hellstern H, Schier M (2011) Electric wheel hub motor for aircraft application. Int J Renew Energy Res (IJRER) 1(4):298–305 Oguz Y, Harun O, Yabanova I, Oguz E, Kırkbaş A (2012) Efficiency analysis of isolated wind-photovoltaic hybrid power system with battery storage for laboratory general illumination for education purposes. Int J Renew Energy Res (IJRER). 2(3):440–445 Talavaru P, Nagaraj N, Kishore Kumar Reddy V (2014) Microcontroller based closed loop speed and position control of DC motor. Int J Eng Adv Technol (IJEAT) ISSN 3(5):2249–8958 Das HS, Dey A, Tan CW, Yatim A (2016) Feasibility analysis of standalone PV/wind/battery hybrid energy system for rural Bangladesh. Int J Renew Energy Res (IJRER). 6(2):402–412 Alphonse I, Thilagar H, Singh FB (2012) Design of Solar Powered BLDC Motor Driven Electric Vehicle. Int J Renew Energy Res (IJRER) 2(3):456–462 Kadwane S (2006) Converter based DC motor speed control using TMS320LF2407A DSK 1st IEEE conference on industrial electronics and applications, pp 1–5, Print ISBN: 0-7803-9513-1, 26 May 2006 Lekhchine S, Bahi T, Soufi Y (2013) Direct torque control for double star induction motor. Int J Renew Energy Res (IJRER) 3(1):121–125 Scott J, Round WH (2009) Speed control with low armature loss for very small sensor less brushed DC motors. IEEE Trans Ind Electron 56(4):1223–1229 Mirosevic M, Maljković Z, Pavlinović D (2011) The dynamics of diesel-generator unit in isolated electrical network. Int J Renew Energy Res (IJRER) 1(3):126–133 Gruber P, Balemi S (2010) Overview of non-linear control methods, 28 May 2010 Hamida MA, Ezzat M, Glumineau A, De Leon et J, Boisliveau R (2012) “ Commande par Backstepping avec action intégrale pour la MSAP : Tests expérimentaux ” , Conférence Internationale Francophone d’Automatique (CIFA2012), Grenoble, France. 4–6 Juillet, pp 973–978 Emna M, Kheder A (2016) An adaptive backstepping flux observer for two nonlinear control strategies applied to WGS based on PMSG. Int J Renew Energy Res (IJRER) 6(3):914–929 Ye XD (2012) Nonlinear adaptive control by switching linear controllers. Syst Control Lett 61(4):617621 Wang JH (2008) Nonlinear control of voltage PWM source rectifier, 1st edn. China Machine Press, Beijing, pp 10–12 Bodson M, Chiasson J (1998) Differential-geometric methods for control of electric motors. Int J Robust Nonlinear Control 8(11):923–954 Elmaguiri O, Giri F (2007) Digital backstepping control of induction motors. In: IEEE international symposium on industrial, electronics, pp 221–226 Wahsh SA, Shazly JH, Hassan AY (2017) Effect of rotor speed on the thermal model of AFIR permanent magnet synchronous motor. Int J Renew Energy Res (IJRER) 7(1):21–25 Asghar MSJ (2003) Speed control of wound rotor induction motors by AC regulator based optimum voltage control. In: Power electronics and drive systems, the fifth international conference, pp 1037–1040 Zambada J (2007) Field-oriented control for motors, Motor Design.com 22. Krim S, Gdaim S, Mtibaa A, Mimouni MF (2016) FPGA contribution in photovoltaic pumping systems: models of MPPT and DTC-SVM algorithms. Int J Renew Energy Res (IJRER) 6(3):866– 879 23. Seibel BJ, Rowan TM, Kerkman RJ (1997) Field-oriented control of an induction machine in the field-weakening region with DC-link and load disturbance rejection. IEEE Trans Ind Appl 33(6):1578– 1584 24. Tan Y, Chang J, Tan H (2003) Adaptive backstepping control and friction compensation for AC servo with inertia and load uncertainties. IEEE Trans Ind Electron 50(5):944–952 25. Hesari S (2016) Design and implementation of maximum solar power tracking system using photovoltaic panels. Int J Renew Energy Res (IJRER) 6(4):1221–1226 26. Bakshi V, Bakshi U (2009) Electrical machines. Technical Publications, Pune, India 27. Nouira I, Khedher A, Bouallegue A (2012) A contribution to the design and the installation of an universal platform of a wind emulator using a DC motor. Int J Renew Energy Res (IJRER) 2(4):797–804 28. Krause PC, O’Connell TS, Wasynczuk O, Hasan M (2017) Introduction to electric power and drive systems. In: IEEE press series on power engineering, ISBN: 1119214254/ 9781119214250, Wiley 29. Hughes A (2009) Electrical motors and drives: fundamentals, types and applications, 3rd edn. Newnes Publications, Elsevier, Oxford, UK 30. Lekhchine S, Bahi T, Soufi Y (2012) Comparative performances study of a variable speed electrical drive. Int J Renew Energy Res (IJRER) 2(4):591–595 31. Chaouch S, Nait-Said MS (2006) Backstepping control design for position and speed tracking of DC motors. Asian J Inf Technol 5(12):1367–1372 32. Farshbaf RA, Azizian MR, Shazdeh S, Mokhberdoran A (2016) Modelling of a new configuration for DFIGs using T-type converters and a predictive control strategy in wind energy conversion systems. Int J Renew Energy Res (IJRER) 6(3):975–986 33. Zhou J , Wang Y, Zhou R (2000) Adaptive backstepping control of separately excited DC motor with uncertainties PowerCon 2000. In: 2000 International conference on power system technology. Proceedings (Cat. No.00EX409), Perth, vol 1, pp. 91–96 34. Tsotoulidis SN, Safacas AN (2011) Analysis of a drive system in a fuel cell and battery powered electric vehicle. Int J Renew Energy Res (IJRER) 1(3):140–151 35. Liu ZZ, Luo FL, Rashid MH (2003) Adaptive MIMO backstepping controller for high-performance DC motor field weakening. Electr Power Compon Syst 31:913–924 36. Mishra SK, Purwar S, Kishor N (2016) An optimal and non-linear speed control of OWC wave energy plant with wells turbine and DFIG. Int J Renew Energy Res (IJRER) 6(3):995–1006 37. Nehrir MH, Fatehi F (1996) Tracking control of DC motors via input-output linearization. Electr Mach Power Syst 24(3):237–247 38. Khedher A, Khemiri N, Mimouni MF (2012) Wind energy conversion system using DFIG controlled by backstepping and sliding mode strategies. Int J Renew Energy Res (IJRER) 2(3):421–430 39. Elgammal A, Sharaf AM (2012) Dynamic self adjusting FACTSswitched filter compensation schemes for wind-smart grid interface systems. Int J Renew Energy Res (IJRER) 2(1):103–111 123