Design of a Memristor-Based 2-DOF PI Controller and Testing of Its Temperature Profile Tracking in a Heat Flow System

In this study, a memristor-based 2-DOF PI controller (2-DOF Mem-PI) was designed for the temperature profile tracking control of a model of heat flow experiment (HFE) setup. A simulation study is presented in which the performance of the designed controller is compared with a standard 2-DOF PI controller. Compared to 1-DOF control structures, 2-DOF controllers that include an extra adjustable parameter perform better in terms of response to disturbances and improving the transient response of the system. In addition, memristor-based controllers (Mem-PI and 2-DOF Mem-PI) and standard controllers (PI and 2-DOF PI) were compared and it was determined that because of the variable memristance value, the control structures containing memristors showed an adaptive feature. The simulation results demonstrated the success of the proposed controller in temperature profile reference tracking and showed the memristor to be applicable in nonlinear control structures.

control theory issues such as the operating environment, The associate editor coordinating the review of this manuscript and approving it for publication was Qi Zhou. simulation results to demonstrate the practical and simple 30 nature of the proposed new tuning rules [2]. Al-Saggaf et al. 31 proposed a new model-based analytical design for fractional-32 order controllers. The efficiency and performance of this 33 proposed method was demonstrated by simulations and on 34 a heat flow platform experimentally [3]. In another study, 35 a control design approach for input saturated linear systems 36 was presented. Two practical applications in both a heat flow 37 system and a liquid flow system were presented to verify 38 the effectiveness of the proposed control design method. The 39 effectiveness of the proposed approach was demonstrated by 40 experimental results [4]. Hernandez-Perez et al. investigated 41 the stabilization problem of the class of highly unstable 42 time-delayed systems. In the control structure they proposed, 43 a new control law called the PIf controller, consisting of 44 a standard PI controller and a first-order low-pass filter, 45 was used. The performance of the proposed controller was 46 tested in a thermal flow assembly equipped with a time delay 47 and a recycling path, and it was reported that an adequate 48 performance had been achieved [5]. In order to obtain a 49 generally acceptable process response, Nath et al. proposed 50 ever, researchers have realized many models and circuits by 79 working on circuits that act as memristors [19], [20] presented.

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In this study, a novel memristor-based 2-DOF PI controller

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The heat flow experimental (HFE) setup produced by 108 Quanser, which enables simulation and experimental studies 109 related to the control of a heat flow system, is shown in 110 Figure 1. The air mass heated by the coil is transported 111 through a channel with three temperature sensors at fixed 112 distances using a fan whose speed can be measured.

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Although it is difficult to obtain a thermodynamic model 114 of the system, the state variables of this system can be deter-115 mined as follows: where V h and V b are the blower and heater voltages, respec-118 tively. Similarly, T n and T a are the n th sensor and ambi-119 ent temperature, respectively, and x n indicates the distance 120 between the heater and the n th sensor.

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The first-order transfer function model of the system, 122 as follows, is sufficient to design a temperature controller.
where τ n and K n are the time constant and steady state gain 125 for the n th sensor, respectively. The voltage-to-temperature 126 transfer function of the HFE system is as follows [36]: In practical applications, time delay and dead time occurs 129 in real systems [37], [38]. In the handbook presented by 130 Quanser, dead time is not included as in the equation given 131 in (3).

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In 1971, Leon Chua claimed that there should be a fourth 134 basic circuit element in addition to the resistor, inductor, and 135 capacitor. He stated that it is a power-consuming element 136 such as a resistor, that it cannot be modeled by other circuit 137 elements, and that its value can be expressed by the ratio of 138 voltage to current depending on the load [17]. The memristor, 139 which was always thought of as a mathematical concept, was 140 produced by an HP research team about 36 years later [18]. 141 where ω is the state variable of the device, M represents 146 the memristance, R ON indicates low resistance states, R OFF 147 indicates high resistance states, and D represent the total 148 length of the TiO 2 memristor, respectively.

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The memristance value, which is directly dependent on the 150 change of the ω value, is defined as: Here, µ v is the mobility of the memristor [18]. The physical 153 structure of the memristor is given in Figure 2.  The control signal u(s) indicated in the block diagram in 165 Figure 3 can be expressed as follows: If this equation is rearranged, is obtained, where e(s) is the error signal specified in (6).

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In the integrator circuit shown in Figure 4, the resistance 174 of the memristor is automatically changed with the voltage 175 applied to it, so that an adjustable gain can be obtained [39]. 176 The mathematical expression of the memristor-based inte-177 grator circuit can be written as follows: where C is capacitance and M represents the memristance. If the integration process in the 2-DOF PI control structure 182 given in Figure 3 is carried out with the memristor-based 183 integrator circuit given in Figure 4, a memristor-based 2-184 DOF PI controller structure is obtained. A block diagram 185 of the temperature profile tracking control of the HFE with 186 the memristor-based 2-DOF PI (2-DOF Mem-PI) approach 187 is given in Figure 5.

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Using (4), (6), (9) and (10), the control signal u(s) in the 189 proposed control structure can be written as follows:  characteristic of the standard PI controller described in [38] 205 are given in (15) and (16), respectively.   (15) and (17), it is 218 stated that for p = 2, two adjustable controller parametersK p 219 andT I and triple dominant poleŝ 3 values can be obtained as 220 indicated below:  In Figure 7, both controllers are seen to perform similarly 245 in terms of time to reach the reference. However, the 2-DOF 246 Mem-PI controller was more successful than the 2-DOF PI 247 controller in terms of eliminating the error early on. 248 Figure 8 gives the control signals of the 2-DOF Mem-PI 249 controller and the 2-DOF PI controller. Both control signals 250 have a similar form, but the 2-DOF Mem-PI control signal is 251 smoother than that of the 2-DOF PI.

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Considering the previous study of [28], it was thought 253 to examine the relationship between memristor-based con-254 trollers and classical controllers, and for this purpose, the 255 temperature control of HFE for the same reference signal was 256 tested with the standard 1-DOF PI, 1-DOF Mem-PI, standard 257 2-DOF PI, and 2-DOF Mem-PI controllers. The reference 258 temperature profile tracking simulation results of these four 259 controllers are given in Figure 9.   The graphic in Figure 9 shows that, regardless of whether   eliminated the error more quickly than the standard control 266 structures (PI and 2-DOF PI). This is more clearly seen where 267 the error signals are given in Figure 10.  value. Depending on the voltage change of the error signal 327 at the controller input, the resistance value of the memristor 328 takes a value between the R ON and R OFF values. Therefore, 329 an adaptation effect occurs in the system.

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Future studies are planned to demonstrate the effectiveness 331 of this simulation study, to test the proposed controller struc-332 ture on a real system with a dead-time included system model, 333 and to analyze the stability, sensitivity and frequency stability 334 of the system.