Memristor-Based Phase-Lead Controller Circuit Design

Memristor-Based Phase-Lead Controller Circuit Design Tsung-Chih Lin1, Wei-Nan Liao1, and Valentina Emilia Balas2 1

Department of Electronic Engineering, Feng-Chia University, Taichung, Taiwan 2 Department of Automation and Applied Informatics, Aurel Vlaicu University of Arad, Arad, Romania {tclin,m0049793}@fcu.edu.tw, [email protected]

Abstract. Memristor predicted and described by L. O. Chua is a new electrical element. Memristor with variable resistance which changes automatically as its voltage changing can be used as a tunable parameter in control systems. Phase-lead controller is widely used in industrial process control applications. The conventional method for adjusting phase-lead controller parameters is based on the “trial and error” procedure which leads to an inconvenient and approximated design. In the meantime, once the parameters are adjusted, they are fixed during the control system operation. In this paper, memristor-based phase-lead (M-phase-lead) controller is presented. Based on memristor constitutive relations of the memristance and memductance, the proposed M-phase-lead controller is more flexible during the design process. The theoretical description of design scheme is presented and simulation example is given to show the effectiveness of the advocated design methodology. Keywords: Frequency-domain, Memristor, phase-lead.

1

Introduction

Memristor characterized by a relationship between the charge q(t) and the flux φ (t ) was predicted as the fourth circuit element and described by Leon O. Chua in 1971 [1]. In 2008, Stanley Williams and his Team at HP Information and Quantum System Laboratory announced the discovery of the missing circuit element memristor, acronym for memory resistor [4]. Recently, research on circuit based on memristor [3], [5]-[15] is becoming a focal topic for research because it will reduce power consumption to use memristors in computer by saving the time to reload data and could lead to the human-like learning. Improvement in the response of feedback tracking control systems and disturbance rejection can be achieved in the frequency domain. The introduction of a phase-lead controller named low-frequency attenuation controller results in an increase in the resonant frequency and a reduction in the settling time of the system. The great drawback of phase-lead controller is how to adjust parameters with many source of uncertainty in dynamic real-world environments. In industrial process control applications, the conventional method for adjusting phase-lead controller parameters is based on the “trial and error” procedure which leads to an inconvenient and approximated design. In the meantime, once the parameters are adjusted, they are fixed during the control system operation. Due to the resistance of a memristor can change V.E. Balas et al. (Eds.): Soft Computing Applications, AISC 195, pp. 309–318. springerlink.com © Springer-Verlag Berlin Heidelberg 2013

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adaptively with it voltage changing, a memristor (M) can be used a tunable gain. In this paper, we use a memristor to replace the resistor in the phase-lead controller circuit schematic. This paper is organized as follows. First, the circuit model of the conventional phase-lead controller is introduced in Section 2. The circuit model of the M-phased-lead controller is presented in Section 3. Simulation examples to demonstrate the performance of the proposed schemes are provided in Section 4. Section V gives the conclusion of the advocated design methodology.

2

The Circuit Model of the Conventional Phase-Lead Controller

This section presents the frequency-response characteristics by use of the cascade controller, phase-lead controller. The main objective of the controller is to reshape the frequency-response plot of the basic system by means of a controller in order to achieve the performance specifications. In designing phase-lead controller via Bode plots, we want to change phase diagram, increasing the phase margin to reduce the percent overshoot and increasing the gain crossover to realize a faster transient response. The circuit as shown in Fig. 1 is a phase-lead controller.

Fig. 1. Typical phase-lead controller circuit

The lead designation of this network is based on the steady-state sinusoidal response. The sinusoidal output E2 leads the sinusoidal input E1 . The circuit transfer function is

s E2 ( s ) QT = E1 ( s ) 1 + s T 1+

where T =

R 1 + R2 , R2 Q= 0 . Therefore, the memristance can be obtained as

M (q ) =

dϕ ( q)  a, = dq b,

q 1

Fig. 4. The constitutive relation of the charge-controlled memristor

(7)

Memristor-Based Phase-Lead Controller Circuit Design

313

Similarly, for the flux-controlled memristor, a monotonically increasing and piecewise-linear characteristic is assumed [4]. The memristor constitutive relation as shown in Fig. 5 can be expressed as

q(ϕ ) = gϕ + 0.5( f − g )( ϕ + 1 − ϕ − 1)

(8)

where f , g > 0 . Therefore, the memductance can be obtained as

W (ϕ ) =

dq (ϕ )  f , = dϕ  g ,

ϕ 1

(9)

Fig. 5. The constitutive relation of the flux-controlled memristor

Because the resistance of a memristor can change adaptively with its voltage changing, the resistor from phase-lead circuit is replaced with memristor as follows. The M-phase-lead by replacing resistor R2 with a flux-controlled memristor whose characteristic is given by Fig. 5 is shown in Fig. 6.

314


Fig. 6. The M-phase-lead circuit

The circuit transfer function (1) can be modified as

s 1+ E2 ( s) QT = E1 ( s) 1 + s T  1+    1 +     1+   1 +    where Q =

s ( s

ϕ 1

1 + R1 g ( ) R1C2

1 + RW (ϕ ) . RW (ϕ ) , 1 1 T= R C 1 + RW (ϕ ) 1 1 2

For circuit components chosen as R2 = 483Ω, C2 = 7 *10 −5 F , f = 300 and g = 800 , the Bode diagram of is shown in Fig. 7. By comparing Fig. 2 with Fig. 7, the proposed M-phase-lead controller is more flexible than the conventional phase-lead controller. During the control system operation, once the parameters are adjusted, they are fixed for conventional phase-lead


315

Fig. 7. Bode magnitude and phase plot for M-phase-lead circuit

controller, but there exist robust gain margin and phase margin intervals for M-phase-lead controller by adapting slopes of the constitutive relation of the charge-controlled memristor and flux-controlled memristor.

4

Simulation Example

In this section, the proposed M-phase-lead controller will be applied to an aircraft roll control system as shown in Fig. 8, where GM ( s ) and GA (s ) are transfer functions of motor and aircraft, respectively. The torque on the aileron generates a roll rate. The resulting roll angle is then controlled to meet prescribed design specifications through a feedback system as shown. The comparisons between M-phase-lead controller and conventional phase-lead controller are given to show the proposed M-phase-lead controller is more flexible.

Fig. 8. The aircraft roll control system

316


Example: M-Phase-Lead Controller Design

100 1 and G A ( s ) = , design a ( s + 100) ( s + 36) phase-lead controller to yield a 48 phase margin and velocity constant kv = 40 .

Given the transfer functions GM ( s ) =

First, K must be set at 1440 to meet the specification kv = 40 , yielding the open-loop transfer function as

G ( s) = where G ( s ) =

144000 s ( s + 36)( s + 100)

K GM ( s )GA ( s ) . In order to meet the phase margin requirement s

imposed by 48 , the conventional phase-lead controller can be designed as

Gc ( s ) =

2.38( s + 26.07) ( s + 61.78)

The circuit parameters of the phase-lead shown in Fig. 1 are chosen as R1 = 20Ω , R2 = 27.4Ω , C2 = 1.4*10−3 F . The forward transfer function of the compensated system can be obtained as

Gc ( s )G ( s ) = By replacing resistor

342720( s + 26.07) s ( s + 61.78)( s + 36)( s + 100)

R2 with a flux-controlled memristor, the M-phase-lead

controller can be expressed as

s + 142.9  1.25 s + 178.6 , ϕ < 1 Gc ( s) =  6 s + 7.143 , ϕ > 1  s + 42.86 Therefore, the compensated system’s forward transfer function can be described as

180000( s + 142.9)   s ( s + 178.6)( s + 36)( s + 100) , ϕ < 1  Gc ( s )G ( s ) =  864000( s + 7.143)  , ϕ >1  s ( s + 42.86)( s + 36)( s + 100) Fig. 9 summarizes the design and shows the effect of the compensation of the conventional phase-lead controller and the M-phase-lead controller.


317

Fig. 9. Bode plots for M-phase-lead controller and conventional phase-lead controller

Final results, obtained from the simulations and the actual frequency responses are shown in Table 1. Table 1. Characteristics of the Phase-lead and M-phase-lead compensated system

Parameter

Proposed

Uncompensated

Phase-lead

M-phased-lead

specification

system

compensated

compensated

system

system

40

40

40

40

Phase Margin

45

34

45

14 ~ 37

Gain-cross over

----

29.4 rad/sec

37.1 rad/sec

29 rad/sec ~

kv

frequency Closed-loop

73.6 rad/sec ----

49.99 rad/sec

68.47 rad/sec

bandwidth Gain margin

50.32 ~ 110.55 rad/sec

----

10.6 dB

11.4 dB

3.65 ~ 12 dB

318

5


Conclusion

Due to the fact that memristor with variable resistance which changes automatically as its voltage changing can be used as a tunable parameter in control systems, we have proposed a method for designing a memristor-based phase lead controller for industrial process control applications to overcome the limitation that parameters of the conventional phase-lead controller are fixed during the control system operation. An aircraft roll control system is fully illustrated to meet the prescribed specifications. The resulting M-phase-lead controller gives an adaptable specification band and it is more flexible during the design process.

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