Article AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 13 (2020) 314–320 Contents lists available at http://qu.edu.iq Al-Qadisiyah Journal for Engineering Sciences Journal homepage: http://qu.edu.iq/journaleng/index.php/JQES * Corresponding author. E-mail address: mujtaba_abdulkadhim@mustaqbal-college.edu.iq (Mujtaba A. Flayyih) https://doi.org/10.30772/qjes.v13i4.724 2411-7773/© 2021 University of Al-Qadisiyah. All rights reserved. This work is licensed under a Creative Commons Attribution 4.0 International License. Design of Hydraulic Exciter Using Nonstandard Backstepping with Integral Sliding Mode Mujtaba A. Flayyih a , Mohsin N. Hamzah b , Jafar M. Hassan c a Al-Mustaqbal University College, Hillah, Iraq. b University of Technology, Baghdad, Iraq. c Al-Turath University College, Baghdad, Iraq. A R T I C L E I N F O Article history: Received 10 November 2020 Received in revised form 31 December 2020 Accepted 31 December 2020 Keywords: Hydraulic simulator, Quarter car, Suspension system, Sliding Mode control. A B S T R A C T The hydraulic system plays an important role in the design of many mechanical devices due to its high power to weight ratio. In this work, a theoretical study of a hydraulic stimulator was performed to conduct suspension tests in a quarter car in the laboratory. Due to high nonlinearity in the hydraulic systems, a robust controller based upon integral sliding mode (ISMC) with nonstandard backstepping was used to perform the tracking process of the hydraulic actuator to the required road profiles. The controller design divided into two parts: the first part deals with the generation of the ideal control force which satisfied all the tracking requirements by using ISMC. The second part of the controller is the replacing of the ideal control force with the actual hydraulic force with all parameters by using nonstandard backstepping control. The effectiveness of the exciter control system was performed by using two road profiles i.e. Bump road profile and random road profile. © 2020 University of Al-Qadisiyah. All rights reserved. 1. Introduction Active vibration control is one of the most advanced technologies to improve functionality and safety of mechanical structures [1], [2]. Hydraulic systems have been widely used in a large number of fields applications to perform the activity of these vibration systems due to their small size to power ratios and their ability to apply very large force and torque. Hydraulic systems play an important role in transportation, earth moving equipment, aircraft and industry machinery with heavy duty applications [3], [4] . The hydraulic system is highly nonlinear system due to different parameters such pressure flow characteristic, variations of control volumes and associated stiffness, etc. Merritt [5]. From the control point of view, the hydraulic actuator have two main functions: the first one is the force tracking control Niksefat et al. [6] and the second one was the position control Ali et al. [7]. In this work, the hydraulic system designed as an exciter for testing the quarter car suspension system model in the laboratory i.e. for generation road profiles for the system. Therefore, the literature focused on the positioning control of the hydraulic system . There are many researches used the linearized dynamic equation of the hydraulic servo-system such as Eong L. et al. [8], Dean H. et al. [9] and others. This linear system is not suitable to give real results when applying control methods because of physical of the hydraulic system which is highly nonlinear. Another trends of studies used linear control methods such as G. P. Liu et al. [10], Bobrow et al. [11] etc. G. P. Liu et al. [10] performed analytical and experimental study for two hydraulic actuators to track the position of the system by using an http://qu.edu.iq/ mailto:mujtaba_abdulkadhim@mustaqbal-college.edu.iq https://doi.org/10.30772/qjes.v13i http://creativecommons.org/licenses/by/4.0/ https://en.wikipedia.org/wiki/Hillah https://en.wikipedia.org/wiki/Iraq javascript:; javascript:; MUJTABA A. FLAYYIH , MOHSIN N. HAMZAH , JAFAR M. HASSAN /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 13 (2020) 314–320 315 optimal-tuning nonlinear PID control strategy of two hydraulic actuators to track the position of the system. In the same context, Bobrow et al. [11] adopted the optimal linear quadratic control to solve the problem of a servo single rod hydraulic cylinder system. These linear control methods such as PID controller, pole placement, optimal controls etc. which are not sensitive for disturbances and nonlinearities of the hydraulic system. On the other hand, Xiuping Yuan et al.[12] performed an analytical study of the single rod hydraulic actuator and made certain the structure of the Fuzzy controller, input or output variable, the quantitative variable, the proportional variable and fuzzy reasoning theorem based on the manipulative experiment. Backstepping control method is one of the robust control methods and used by many researchers such as Prut Nakkarat et al. [13] and others. They employed an experimental and theoretical study of the electrohydraulic actuator. The researchers derived directly the dynamic equation based upon force variables instead of position variables. They used proportional integral observer to estimate the force rate and pressure. Sliding mode controller is the one of the well algorithms used in hydraulic systems since it has its outstanding characteristics in compensating for the nonlinearities and disturbances of electromechanical systems, therefore many of researchers applied these methods on electrohydraulic systems, for example Y Liu et al. [14], Kai Guo et al. [15] and others. Y Liu et al. [14] designed a standard sliding mode controller for the electrohydulic system connected to the flexible load, i.e. spring and mass. They introduced first-order weighted error function as sliding surface. the experimental results presented the effectiveness of the position control of this system. While, Kai Guo et al. [15] performed a theoretical and experimental study for force tracking of the electrohydraulic actuator. They used cascade controller based on an extended disturbance observer to track desired position trajectory for electro-hydraulic single-rod actuators in the presence of both external disturbances and parameter uncertainties. The researchers used sliding mode controller to compensate the disturbance estimation error estimated. In the same context, they used the backstepping techniques with Lapenove stability for tracking the actuator pressure. they presented that the proposed controller gives excellent tracking performance in the presence of the disturbances and nonlinearities. The main objective of the present work is to design and implement an ISMC with non-standard backstepping for the exciter system with hydraulic actuator. The first step in this controller is to design an ideal control force to perform the required single using ISMC. The current ISMC is an update to the conventional sliding mode, on which the reaching phase was eliminated by initiating the system state at sliding surface, therefore, the robustness is per formed from the first instant. The feature of the ISMC is that it keeps the order of the system where the uncertainties and perturbations rejected from the system model and make the system as an ideal model with known nominal parameters. In addition, ISMC can also be used as a perturbation’s estimator, which solves one of the main drawbacks of the sliding mode control, which is the chattering problem Utkin et al. [16]. The second step is the inclusion of the nonlinear single- rod hydraulic actuator as controller force by using the non- standard backstepping which based on the sliding mode control to perform stability. Also, the observer sliding mode control was used as a differentiator of the ideal control force in the backstepping step due to the nonlinearities and the uncertainties appears in the derivative of the ideal control force. 2. Mathmatical Modelling Mathematical model is defined as a set of equations which describe the dynamic characteristic of the system there are differed way to represent the model depending on the application of the system and circumstances, one mathematical model may be better suited than the other models for specific. The exciter model system used in this paper consists of mass (m) which tracked the required path generated by the actuator, Fig. (1). Therefore, the mathematic model presented as: 𝑚�̈� = 𝑢 − 𝑚𝑔 − 𝑐�̇� (1) Where 𝑧 is the displacement of the mass and u is the ideal control force, g is the acceleration of gravity and c is the damping coefficient of the actuator. Figure 1. schematic of the model The force which be used to generate path of the system is produced by hydraulic actuator. Therefore, it is necessary to explain the dynamics equation of the actuator and servo valve, firstly the actuator dynamics equation is: 𝐹ℎ = 𝑃1 𝐴1 + 𝑃2𝐴2 (2) Where 𝐹ℎis the actuator hydraulic force, 𝑃1 is pressure at the chamber (1) of the cylinder, 𝑃2is pressure at the second chamber of the cylinder, 𝐴1 is the cross sectional area of the chamber (1) while 𝐴2 the cross sectional area of the chamber (2). The derivation of P1 & P2 will be started from the continuity equation through the cylinder which be written as in [5], [17] 𝑉1 𝛽 𝑃1̇ = −𝐴(�̇�) + 𝑄1 − 𝑄𝐿1 − 𝑄𝐿𝐸1 𝑉2 𝛽 𝑃2̇ = 𝐴(�̇�) − 𝑄2 + 𝑄𝐿1 − 𝑄𝐿𝐸2 (3) Where 𝑉1 = 𝑉𝑂1 + 𝐴 𝑧 , 𝑉2 = 𝑉𝑂2 + 𝐴 𝑧 are the total volume of the chambers, 𝑉𝑂1 and 𝑉𝑂2 are the extended and retract chamber volumes when (𝑧) equal to zero, 𝛽 is the effective bulk modules of the hydraulic oil, 𝑄𝐿1is the internal flow leakage of the cylinder, 𝑄𝐿𝐸1 and 𝑄𝐿𝐸2 are the external flow leakage of the chambers, 𝑄1 is the supply flow rate to the forward chambers and 𝑄2 is the return flow rate from the return chamber. The forward and return flow rates can be described as [7]: 𝑄1 = 𝑘𝑣 𝑥𝑣{𝑠𝑔 (𝑥𝑣)(𝑃𝑆 − 𝑃1) + 𝑠𝑔 (− 𝑥𝑣)(𝑃1 − 𝑃𝑡 )} 𝑄2 = 𝑘𝑣 𝑥𝑣{𝑠𝑔 (𝑥𝑣)(𝑃2 − 𝑃𝑎 ) + 𝑠𝑔 (− 𝑥𝑣)(𝑃𝑠 − 𝑃2)} (4) Define function 316 MUJTABA A. FLAYYIH , MOHSIN N. HAMZAH , JAFAR M. HASSAN /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 13 (2020) 314–320 𝑆𝑔(∗) = { 1 𝑖𝑓 ∗ ≥ 0 0 𝑖𝑓 ∗ < 0 (5) Where 𝑘𝑣 is the servo valve flow coefficient, 𝑥𝑣 is the displacement of the servo valve, 𝑃𝑆 is the supply pressure and 𝑃𝑡 is the pressure of the tank. Practically while the hydraulic actuator is running, the servo valve displacement 𝑥𝑣 can be handled using the input voltage 𝑢ℎ that corresponds to the different required forces. The servo valve dynamic equation can be approximate as Fialho et al. [18] 𝑥�̇� = 1 𝜏 (−𝑥𝑣 + 𝑢ℎ) (6) Where 𝜏 is the time constant of the servo-valve. It is being showed that the dynamic of the servo-valve is much faster than the actuator dynamic when the time constant value is very small. Thus, the equation (5) can be simplified as an algebraic equation −𝑥𝑣 + 𝑢ℎ = 0 Deshpande et al. [19]. Therefore, the equation (3) can be rewritten as: 𝑄1 = 𝑢ℎ 𝐻1 𝑄2 = 𝑢ℎ 𝐻2 (7) 𝐻1 = 𝑘𝑣 {𝑠𝑔(𝑢ℎ)(𝑃𝑆 − 𝑃1 ) + 𝑠𝑔 (− 𝑢ℎ)(𝑃1 − 𝑃𝑡 )} and 𝐻2 = 𝑘𝑣 {𝑠𝑔 (𝑢ℎ)(𝑃2 − 𝑃𝑎) + 𝑠𝑔 (− 𝑢ℎ)(𝑃𝑠 − 𝑃2)}. The hydraulic force of the actuator simplified as: 𝐹ℎ̇ = 𝑃1 ̇ 𝐴1 − 𝑃2 ̇ 𝐴2 𝐹ℎ̇ = 𝛽 𝑉1 (−𝐴1(�̇�) + 𝑢ℎ 𝐻1) 𝐴1 − 𝛽 𝑉2 (𝐴2 (�̇�) − 𝑢ℎ 𝐻1) 𝐴2 𝐹ℎ̇ = −(�̇�) { 𝛽 𝐴1 2 𝑉1 + 𝛽 𝐴2 2 𝑉2 } + 𝑢ℎ { 𝛽 𝐻1 𝐴1 𝑉1 + 𝛽 𝐻2 𝐴2 𝑉2 } (8) The states variable of the dynamic system are written as: 𝑥1̇= 𝑧 �̇�2 = ( 𝑎1 𝑥3 − 𝑔 + 𝑎1 𝑥2) �̇�3 = −(𝑥2) 𝑎3 + 𝑢ℎ𝑎4 (9) Where 𝑎1 = 1 𝑚 , 𝑎2 = 𝑐 𝑚 , 𝑎3 = { 𝛽 𝐴1 2 𝑉1 + 𝛽 𝐴2 2 𝑉2 } and 𝑎4 = { 𝛽 𝐻1 𝐴1 𝑉1 + 𝛽 𝐻2 𝐴2 𝑉2 } Where 𝑥1, 𝑥2 represents the displacements and velocity of the moving mass and 𝑥3 represents the dynamic equation of the pressure difference of the single rod hydraulic actuator. 3. Controller Design 3.1. ISMC of the ideal control force The main objective of the control system in this work is tracking process i.e. the mass follows of the required path which is supplied to control system as an input signal. Therefore, the first step in the control design is to drive an error function between the required path and the input required signal as follows: 𝑒(𝑡) = 𝑥1 − 𝑥𝑟 (10) Where the 𝑥1 required displacement and the 𝑥𝑟 is the ideal signal supplied to the system. Now let the state variables of the equation (12) as: 𝑒(𝑡) = 𝑥1 − 𝑥𝑟 = 𝑒1 (𝑡) 𝑒1̇(𝑡) = 𝑥1̇ − �̇�𝑟 = 𝑒2(𝑡) �̇�2(𝑡) = �̇�2 − �̈�𝑟 (11) Therefore, the equation (13) presents as: 𝑒1̇(𝑡) = 𝑥2 − �̇�𝑟 �̇�2(𝑡) = ( 𝑎1 𝑢 − 𝑔 + 𝑎2 𝑥2) − �̈�𝑟 (12) The electrohydraulic system includes several types of uncertainties and perturbations arise from variations of the system parameters such as variation of mass dynamics and the variations of hydraulic system, therefore the dynamic system of the electrohydraulic is rewritten as: 𝑒1̇(𝑡) = 𝑥2 − �̇�𝑟 �̇�2(𝑡) = ( 𝑎1 𝑢 − 𝑔 + 𝑎2 𝑥2) − �̈�𝑟 + 𝑑(𝑒, 𝑡) + ∆𝑎1 𝑢 + ∆𝑎2 𝑥2 (13) Where 𝑒𝑖 , ( 𝑖 = 1 𝑡𝑜 ) ∈ 𝑅 2 , 𝑎1 ∈ 𝑅 1 , 𝑢 ∈ 𝑅, 𝑑(𝑒, 𝑡) ∈ 𝑅 , 𝑑(𝑒, 𝑡) unmodeled dynamics and the nonsmoothed nonlinearities in the system and ∆𝑎1 𝑢, 𝑎𝑛𝑑 ∆𝑎2 𝑥2 represents uncertain parameters of the system dynamic. In the same context, let the design control law is: 𝑢 = 𝑢𝑛 + 𝑢𝑠 (16) Where the 𝑢𝑛 𝑎𝑛𝑑 𝑢𝑠 represented the nominal and discontinuous controls of the system respectively. The nominal control used to stabilize the nominal system dynamics with the desired characteristics Such as the dynamics of the nominal system become as: 𝑒1̇(𝑡) = 𝑥2 − �̇�𝑟 �̇�2(𝑡) = ( 𝑎1 𝑢𝑛 − 𝑔 + 𝑎2 𝑥2) − �̈�𝑟 (14) While the discontinuous control 𝑢𝑠 worked to rejected the perturbation terms and the nonlinearities of the dynamic system in equation (15). After that, the design of the integral sliding mode and the perturbation terms can be formulating as follows: 𝑒1̇(𝑡) = 𝑥2 − �̇�𝑟 �̇�2(𝑡) = ( 𝑎1 𝑢𝑛 + 𝑎1 𝑢𝑛 − 𝑔 + 𝑎2 𝑥2) − �̈�𝑟 + 𝛿(𝑒, 𝑢) (15) Where 𝛿(𝑒, 𝑢) = 𝑑(𝑒, 𝑡) + ∆𝑎1 𝑢 + ∆𝑎2 𝑥2 is the perturbation term and represented the parameters variations, unmodeled dynamics, non- smooth nonlinearities and external disturbances and it assumed to be obtain matching condition of the dynamic system Kim et al. [20]. 𝛿(𝑒, 𝑢) = 𝑎3 �̂�(𝑒, 𝑢) (16) The design procedure of the ISMC begins with the definition of the sliding variable𝑠(𝑥): 𝑠(𝑥) = 𝑠𝑜(𝑒) + 𝑧 (17) Where 𝑠(𝑒), 𝑠𝑜 (𝑒) 𝑎𝑛𝑑 𝑧 ∈ 𝑅 1. In the same context, the sliding variable contains two parts. The first one 𝑠𝑜 (𝑒) will be designed similar to the conventional sliding mode i.e. as a linear combination of the system states. The term of the 𝑧 represented the integral term and determined as below. From the sliding mode control theory [20], the derivation from the switching surfaces 𝑠 and its time derivative should be having an opposite sign in the nearby area of the switching surface 𝑠 = 0, i.e. lim 𝑠→0 +�̇� < 0 𝑎𝑛𝑑 lim 𝑠→0 −�̇� < 0 Or s �̇� < 0 (18) Elimination of the reaching phase which is distinguishing property of the ISMC achieved by choosing the initial condition 𝑧 same that the initial condition of the sliding mode control 𝑠 which is zero. This mean that the dynamic of the system is in sliding surface from first instantaneity by selecting 𝑧(0) = −𝑠𝑜 (0). In this design the 𝑠𝑜 (𝑒) is equal to the state variable 𝑒2, therefore, �̇� is differentiated as below: �̇� = 𝑒2̇ + �̇� = ( 𝑎1 𝑢𝑠 + 𝑎1 𝑢𝑛 − 𝑔 + 𝑎2 𝑥2) − �̈�𝑟 + (19) MUJTABA A. FLAYYIH , MOHSIN N. HAMZAH , JAFAR M. HASSAN /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 13 (2020) 314–320 317 𝛿(𝑒, 𝑢) + �̇� The integral term will be chosen similar to the derivative of the ISMC in [1] i.e. �̇� = ( 𝑎1 𝑢𝑛 − 𝑔 + 𝑎2 𝑥2) − �̈�𝑟 (20) Thus �̇� becomes: �̇� = 𝑎1 𝑢𝑠 + 𝛿(𝑒, 𝑢) (21) And accordingly, the sliding condition becomes: 𝑠 ̇𝑠 = 𝑠 (𝑎1 𝑢𝑠 + 𝛿(𝑒, 𝑢)) (22) By selecting the conventional sliding mode control 𝑢𝑠 = −𝜌(𝑥) 𝑠𝑖𝑔𝑛(𝑠) (23) Then the equation (24) becomes: 𝑠 ̇𝑠 = 𝑠 (𝑎1 (−𝜌(𝑒) 𝑠𝑖𝑔𝑛(𝑠)) + 𝛿(𝑒, 𝑢)) Since s*𝑠𝑖𝑔𝑛(𝑠)=|𝑠| then 𝑠 ̇𝑠 = − 𝑎1 𝜌(𝑒) |𝑠| + 𝑠 𝛿(𝑒, 𝑢)) 𝑠 ̇𝑠 ≤ − 𝑎1 𝜌(𝑒) |𝑠| + |𝑠| |𝛿(𝑒, 𝑢)| 𝑠 ̇𝑠 ≤ |𝑠| {− 𝑎1 𝜌(𝑒) + |𝛿(𝑒, 𝑢)|} 𝑠 ̇𝑠 ≤ −|𝑠| {𝑎1 𝜌(𝑒) − |𝛿(𝑒, 𝑢)|} 𝑠 ̇𝑠 ≤ −𝑎1 |𝑠| { 𝜌(𝑒) − |𝛿(𝑒, 𝑢)| 𝑎1 } (24) It was assumed that 𝑎1 > 0 . The discontinue gain 𝜌(𝑥) that will perform the inequality in the equation (27) to ensure the right-hand side of this equation to be negative value i.e.: 𝜌𝑜 (𝑥) > | 𝛿(𝑥, 𝑢)| 𝑔2 Or 𝜌𝑜 (𝑥) = 𝑘𝑜 + | 𝛿(𝑥,𝑢)| 𝑎4 , where the 𝑘𝑜 > 0 (25) by referring to the equation (24), the discontinuity in the right-hand side due to the 𝑢𝑠 will needed the finite time T to reach to the origin and then the system dynamic become in the sliding motion. Based on the the dynamic system will be achieved by using the equivalent control which performed as follows: when 𝑠(𝑡) = 0, ∀ 𝑡 ≤ 𝑇, [21]also the �̇�(𝑡) = 0 and with 𝛿(𝑥, 𝑢) satisfied the matching condition in equation (19), the equivalent control can be calculated from the equation (24) as shown: 0 = 𝑢𝑠 + 𝛿(𝑥, 𝑢) ⇒ [𝑏 𝑢𝑠 ]𝑒𝑞= - 𝛿(𝑥, 𝑢) (26) By substituting the equivalent control of the equation (29) in to the control law of the ISMC with perturbations i.e. equation (18), the system is reduced to the nominal form as in equation (17) with the dimension equal to n. For this reason, the ISMC is considered as full order sliding mode control because of the dimension of equations (15) & (17) are equal. Since the remaining system is only the nominal system as in equation (10) the next step in the design of this control method is the determination of the nominal part of the system: 𝑒1̇(𝑡) = 𝑥2 − �̇�𝑟 �̇�2(𝑡) = ( 𝑎1 𝑢𝑛 − 𝑔 + 𝑎2 𝑥2) − �̈�𝑟 (27) Therefore, the control design of the nominal system is: 𝑢𝑛 = 1 𝑎1 (−𝑛1𝑒1 − 𝑛2 𝑒2 + 𝑔 + 𝑎2 𝑥2 − �̈�𝑟 ) (28) As the result the nominal system dynamic is: 𝑒2̇ = −𝑛1𝑒1 − 𝑛2 𝑒2 , (29) Where 𝑛1 & 𝑛2 are assigned based on the required system characteristic. 3.2. Backstepping of the hydraulic force Backstepping control is a well method to the control problems nonlinear systems such as active suspension system. A virtual control assumed to follow the intermediate variable, which performed to achieve the objectives of the control design. The hydraulic actuator force 𝐹ℎ considered virtual control force to track the ideal control 𝑢 that designed by ISMC to perform the position tracking of the mass by the electrohydraulic system. The error between the virtual variable 𝐹ℎ and the active control element 𝑢 will be approach gradually to zero, thus: 𝑒𝑒(𝑡) = 𝐹ℎ − 𝑢 (30) Where 𝑒𝑒(𝑡) is the error function between the ideal and virtual function. To derive the control law that will make the value s(𝑡) goes to zero, a nonstandard Backstepping [22] is utilized here. First we have needed to derive the error function dynamics error function by differentiating equation (32) 𝑒�̇�(𝑡) = 𝐹ℎ̇ − �̇� = �̇�3 − �̇� = −(𝑥2) 𝑎3 + 𝑢ℎ𝑎4 − �̇� (31) The hydraulic system has many unmodelled parameters, no smoothed nonlinearities and uncertain parameters, therefore the equation (34) is rewritten as: 𝑒�̇�(𝑡) = −(𝑥2) 𝑎3 + 𝑢ℎ𝑎4 − �̇� + 𝑑(𝑥, 𝑡) (32) Where 𝑑(𝑥, 𝑡) unmodeled dynamics the nonsmoothed nonlinearities system and represents uncertain parameters of the system dynamic. Let the design control law is: 𝑢ℎ = 𝑢ℎ 𝑛 + 𝑢ℎ 𝑑 (33) Where the 𝑢ℎ 𝑛 𝑎𝑛𝑑 𝑢ℎ 𝑑 represented the nominal and discontinuous controls of the system respectively. The nominal control used to stabilize the nominal system dynamics with the desired characteristics Such as the dynamics of the nominal system becomes as: 𝑒�̇�(𝑡) = −(𝑥2) 𝑎3 + 𝑢ℎ 𝑛𝑎4 − �̇� (34) In the other hand, the discontinuous part used to reject the uncertainties in the system model i.e. the discontinuous control is performed as: �̇�(𝑡) = 𝑢ℎ 𝑑 𝑎4 + 𝑑(𝑥, 𝑡) (35) Let the 𝑢ℎ 𝑑 = −𝜌𝑜 (𝑥) 𝑠𝑖𝑔𝑛(𝑒𝑒(𝑡)) 𝑒(𝑡) �̇�(𝑡) = 𝑒𝑒(𝑡) (−𝜌𝑜 𝑠𝑖𝑔𝑛(𝑒𝑒(𝑡)) 𝑎4 + 𝑑(𝑥, 𝑡)) = −𝜌𝑜 (𝑥)|𝑒𝑒(𝑡)| 𝑎4+ |𝑑(𝑥, 𝑡)| 𝑒𝑒(𝑡) ≤ −𝜌𝑜 (𝑥)|𝑒𝑒(𝑡)| 𝑎4+ |𝑑(𝑥, 𝑡)| |𝑒𝑒(𝑡)| ≤ |𝑒(𝑡) |𝑎4 {−𝜌𝑜 (𝑥) + |𝑑(𝑥,𝑡)| 𝑎4 } (36) It was assumed that 𝑔2 > 0 . The discontinue gain 𝜌𝑜 (𝑥) that will perform the inequality in the equation (19) to ensure the right-hand side of this equation to be negative value i.e.: 𝜌𝑜 (𝑥) > | 𝛿(𝑥, 𝑢)| 𝑔2 Or 𝜌𝑜 (𝑥) = 𝑘𝑜 + | 𝛿(𝑥,𝑢)| 𝑎4 , where the 𝑘𝑜 > 0 (37) 3.3. State Differentiator Design 318 MUJTABA A. FLAYYIH , MOHSIN N. HAMZAH , JAFAR M. HASSAN /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 13 (2020) 314–320 In Eq. (31), the proposed control law assumes ideal control force 𝑢 and its derivative are available. The first state u is available since it determined from the first step of the control design, i.e. ISMC. The second state �̇� is the time derivative of u. Hence, we need to obtain it using an observer. A robust sliding mode differentiator (SMD) is proposed here to get �̇� by knowing u only. Kokotović et al. [23] give the sliding mode differentiator; 𝜎 = 𝑢 − 𝜂 �̇� = 𝛼 ∗ 𝑡𝑎𝑛−1(𝛾 𝑢) 𝜏𝑣̇ + 𝑣 = 𝛼 ∗ 𝑡𝑎𝑛−1(𝛾 𝑢) } (38) Where 𝜎 is the SMD variable, 𝛼 and 𝜌 are differentiator parameters. The third equation in (39) 𝜏�̇� + 𝑣 = 𝛼 ∗ 𝑡𝑎𝑛−1(𝛾 𝑢) is a low pass filter (LPF) with time constant 𝜏, where the output of the LPF, 𝑣, is the estimated derivative 𝑢. According to Deshpande et al. [19], the bound on the steady state estimation error is given by: |𝑣(𝑡) − �̇� | ≤ 2 𝜏 𝜌 𝑡𝑎𝑛 ( 𝜋 2𝛼 ℎ) (39) Where ℎ = sup𝑡 |𝑢| 4. Result and discussion As mentioned above, the hydraulic system exciter in this work used to simulate the road disturbances on the quarter car suspension system. Therefore, two cases of road profiles studied to perform this purpose Mishary [24] Fig. (2): Case 1: Bump Road Profile The mathematical model of this profile is Fig. (2 a), [18] 𝑥𝑟 = { 𝑑𝑜(1 − cos(𝜔𝑟 𝑡)), 𝑡 ≤ 1 0 , 𝑡 > 1 Where 𝑑𝑜 represents the peak amplitude and 𝜔𝑟 a constant frequency in the disturbance model which depends on the car velocity and on the width of the disturbance on the road, which are set as. 𝑑𝑜 = 0.1 𝑚 and 𝜔𝑟 = 2𝜋 𝑟𝑎𝑑. Case 2: Random Road Profile The mathematical model of this profile is Fig. (2 b), [19] 𝑥𝑟 = 0.05 cos(2𝜋𝑡) sin (0.06 𝜋 𝑡) Table 1. the model parameters of the active suspension Parameter Value parameter value 𝑚 290 kg 𝑐 1000 N m/sec 𝑘𝑣 1e-9 𝛽 1.4e9 A 0.001311 𝑚2 𝑃𝑠 10342502 𝑁/𝑚 2 𝑉0 0.01311 𝑚 3 (a) (b) Figure 2. the simulated road disturbance (m) Fig. (2) presents the vertical displacement of the quarter car mass and the required profile for two cases. It shows the effectiveness of the proposed control method of the single rod hydraulic actuator to track the required profile with high accuracy. Since the proposed control method in this work is divided into two steps, it is necessary to show the features of the ISMC and nonstandard backsteeping to perform tracking process. For ISMC, i.e. generation of the ideal control force, the sliding variable is the main parameter should be discussed. Fig. (3) shows that the sliding variable (𝑠) equals to zero from the first instant i.e. 𝑠 = 0 for 𝑡 ≥ 0. This behavior of sliding surface (𝑠) which begins and continues with zero along simulations time presented the effectiveness of the of ISMC over the conventional sliding mode control which reaches the stability over reaching phase not from the first instant. (a) (b) Figure 2. Vertical displacement of the hydraulic actuator vs required displacement MUJTABA A. FLAYYIH , MOHSIN N. HAMZAH , JAFAR M. HASSAN /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES 13 (2020) 314–320 319 (a) (b) Figure 3. sliding surfaces As mentioned in the mathematical modeling, the hydraulic model is added to the system by nonstandard backstepping with sliding mode. Fig. (5) presented the ideal control force which was designed to satisfy the suspension system requirements with actual control force i.e. hydraulic force. This Figure shows the high convergence between them over all simulation time which indicated the powerful of the nonstandard backstepping for the inclusion of the hydraulic force in the overall system model. The controller design ends with determination of the voltage which used as input to the servovalve. The servovalve adjusts the flow rate and pressure to the actuator chamber. Fig. (6) shows the voltage curve of the two cases of the road profiles. (a) (b) Figure 4. hydraulic force vs. ideal control force (a) (b) Figure 5. input voltage of the hydraulic actuator 5. Conclusions This work, represents a design of simulator for testing the quarter car active suspension system with generation different road profiles to simulate the actual road. The following conclusion have been drawn: The robust ISMC gives better results in the design of virtual control force which performs all simulator requirements. 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