Adaptive Speed Tracking Control Systemwith Adaptive Anti-Windup Compensator

International Journal of Engineering Practical Research (IJEPR) Volume 2 Issue 3, August 2013 www.seipub.org/ijepr

77

Adaptive Speed Tracking Control System with Adaptive Anti-Windup Compensator Itamiya, K.*1, Sawada, M.2 1Dept. of Electrical and Electronic Eng., National Defense Academy,10-20, Hashirimizu 1-Chome, Yokosuka, Japan 2Doctor Course of Equipment and Structural Eng., NDA, 10-20, Hashirimizu 1-Chome, Yokosuka, Japan *[email protected]; [email protected]

Abstract

Typical anti-windup controllers proposed so far did not necessarily guarantee a good control performance when the characteristic of the controlled object greatly fluctuated. In order to improve the anti-windup performance against the largeproperty fluctuation of controlled object, this paper proposes an adaptive speed tracking controller to which an adaptive anti-windup compensator is involved. The effectiveness of the proposal controller is shown by the theoretical analysis and experiment results.

Keywords

Adaptive Control; Anti-windup Compensator; Input Saturation; Speed Tracking Control

Introduction

The adaptive velocity control is known as an effective speed tracking control method for rotor systems with load change. However, a preferable control performance cannot be obtained when the controller has integrator property and the input of controlled object is saturated due to the restriction of actuator. It is because so-called windup phenomenon is generated in the controlled variable.

To solve such a problem, lots of methods [(Hanus, 1987), (Doyle, 1987), (Walgama, 1990), (Wada, 2002a, 2002b), (Watanabe, 1994, 1999), (Saeki, 1999), (Kiyama, 2001), (Grimm, 2003)]including the robust control theory for designing anti-windup compensator have been proposed. But the desired control performance may not be satisfied when the load change is unexpectedly large since the anti-windup property based on those design depends essentially upon the property of controlled object. Therefore, a new compensator that can adaptively achieve an optimal anti-windup control performance corresponding to the load change may be necessary.

On the other hand, conventional researches related to a control input saturation and adaptation context may

be classified into two categories. For example, Miller proposeda switching controller (Miller, 1993) and carriedout asymptotic error regulation for a control input constraint caused by constant disturbance signal when a reference signal is constant. It has solved the problem of saturation in the adaptive method but it is not a study dealing with the adaptive control system itself. Karason (Karason, 1994) and Yang (Yang, 2003) proposed adaptive control methods in the presence of input constraint. In the paper (Karason, 1994), the control input is switched from an original adaptive control signal into a certain signum function signal so that an input saturation is avoided quickly and tracking performance is asymptotically restored. The control hedging approach (Yang, 2003)has beenproposed in order to avoid instability of the adaptive output feedback control system caused by an input saturation as well a certain signum function signal is also applied.

However, without using any signum function methods, the adaptive control which can suppress a windup phenomenon caused by an input saturation is not almost proposed. Based on these backgrounds, here an adaptive speed control system has been put forward with adaptive anti-windup compensator and experiment has been done to verify the effectiveness of the proposed control method. Our method does not use any signum functions. A certain auxiliary signal in order to suppress the windup phenomenon is added adaptively to the original adaptive speed control input, depending on the amount of saturation and the current values of adjustable parameters. This signal will disappear automatically when there is no saturation, and the control input becomes the original pure adaptive speed control input. There is a novelty in our way in such a meaning.

The rest of the paper is organized as follows. System representation and statement of problem are presented in the next section. Then, ‘MRACS in Non-Saturation

www.seipub.org/ijepr International Journal of Engineering Practical Research (IJEPR) Volume 2 Issue 3, August 2013

78

Case’, a typical windup phenomenon generated in the adaptive speed control system is illustrated. A new adaptive speed control system with adaptive anti-windup compensator is proposed and the stability analysis and the optimal performance of the proposed control system are stated in the subsequent section. Also, experimental results in order to verify the usefulness are shown. The last section concludes the paper.

Notations: The following notations are applied; such as bold face style means a vector or a matrix and a symbol𝑠 denotes the Laplace operator. The symbol T expresses the transpose of a vector or a matrix. A definition symbol is expressed as ≔ and a symbol ∥⋅∥ means a Euqulidean norm of a vector. The symbol 𝐿2 and 𝐿∞ mean the set of square integrable functions and uniformly bounded functions, respectively.

Furthermore,

(𝐺(𝑠)[𝑣])(𝑡) ≔ �𝑡

0𝑔(𝑡 − 𝜏) 𝑣(𝜏) d𝜏

𝑥[𝑗](𝑡): = �((𝑠 + 𝑎m)𝑗[𝑥])(𝑡) for 𝑗 ≤ 0(𝐷 + 𝑎m)𝑗𝑥(𝑡) for 𝑗 > 0

where 𝐺(𝑠) is the Laplace transforms of 𝑔(𝑡), 𝑗 means integer, 𝑎m is a positive constant and 𝐷 is a time differential operator d/d𝑡.

System and Problem Statement

The controlled object considered here consists of a direct current motor, its load and a servo amplifier. It is assumed that the coulomb friction that occurs in contact side of rotation axis is compensated a priori by appropriate method. Furthermore, it is also assumed that the dynamics of servo amplifier can be negligible.

Then, the rotational speed of load in time 𝑡 or its sensor value, 𝑦(𝑡), can be expressed as the following mathematical model;

𝑦(𝑡) = �𝑏

𝑠 + 𝑎[𝑢]� (𝑡) (1)

where𝑎: = 𝜎/𝐽, 𝑏: = 𝑘/𝐽, 𝜎 > 0is the equivalent viscous friction coefficient,𝐽 > 0 is the equivalent moment of inertia, 𝑘 is the product of the servo amplifier gain and the conversion coefficient from voltage to torque, 𝑢(𝑡) is the actual applied voltage but is not manipulated variable. Also, it is assumed that 𝑦(0) = 0.

For any 𝑎m > 0 and 𝑝 > 0, (1) is equivalent to

�𝑠 + 𝑝

𝑠[𝑦]� (𝑡) = 𝜃T𝜁(𝑡) + �

𝑝𝑠

[𝑦]� (𝑡) (2)

𝜃: = [ 𝜃1, 𝜃2 ]T: = [ 𝑏, 𝑎m − 𝑎 ]T (3)

𝜁(𝑡): = � 𝑢[−1](𝑡), 𝑦[−1](𝑡) �T

(4)

The equation (2) represents a parameterization of emphasized 𝑦(𝑡) in the frequency domain that is lower than𝑝 [rad/s].

Assumptions:

The controlled object is assumed as follows;

A1) 𝜃 after the load changes is unknown.

A2) 𝑢(𝑡) has the saturation characteristic;

𝑢(𝑡) = 𝑔s(𝑢𝑐): = �𝑢max (𝑢c(𝑡) > 𝑢max)𝑢c(𝑡) (𝑢min ≤ 𝑢c(𝑡) ≤ 𝑢max) 𝑢min (𝑢c(𝑡) < 𝑢min)

(5)

where 𝑢𝑐(𝑡) is the manipulated variable (the output of controller) and it is assumed that 𝑢𝑚𝑖𝑛 < 0, 𝑢𝑚𝑎𝑥 > 0 without loss of generality.

Remark 1: The true 𝑢(𝑡) cannot be measured directly when it is saturated. However, it can be always obtained by inserting an artificial saturation element between a controller and the controlled object if the element has the I/O characteristic whose linear section (the width of non-saturated region) is narrower than or equal to that of saturation property of the controlled system. Then, the element output 𝑣(𝑡) is equivalent to 𝑢(𝑡) (see Fig.1). This element can be easily realized by an appropriate numerical calculation program.

FIG. 1 SYSTEM TO TAKE OUTSATURATEDSIGNAL 𝑣(𝑡)

In the following, it is assumed that 𝑢(𝑡) can be obtained by such a method.

MRACS in Non-Saturation Case

Control Input Synthesis

When 𝑢(𝑡) does not have saturation property, namely, 𝑢(𝑡) = 𝑢c(𝑡) , consider the following control law corresponding to (1).

�𝑠 + 𝑝

𝑠[𝑦m]� (𝑡) = 𝜃�T(𝑡) 𝜁(𝑡) + �

𝑝𝑠

[𝑦]� (𝑡) (6)

where 𝜃�(𝑡): = [𝜃�1(𝑡), 𝜃�2(𝑡)]T is the estimate of 𝜃 .The signal 𝑦m(𝑡) is the differentiable desired output


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corresponding to 𝑦(𝑡) and it may be generated by the output of following reference model;

𝑦m(𝑡): = �𝑎𝑚

𝑠 + 𝑎𝑚[𝑟]� (𝑡) (7)

where the design parameter 𝑎m > 0 is the desired bandwidth of control system, 𝑟(𝑡) is a piece-wise continuous and bounded reference signal.

The equation (6) can be translated into

𝑦m(𝑡) = 𝜃�T(𝑡)𝜁(𝑡) + �𝑝𝑠

[𝑦�]� (𝑡) (8)

where 𝑦�(𝑡): = 𝑦(𝑡) − 𝑦m(𝑡). Therefore, the control input is synthesized by

𝑢(𝑡) =𝑎𝑚𝑟(𝑡) − 𝜃�2𝑦(𝑡) − 𝜁𝑇(𝑡) 𝜃�̇(𝑡) − 𝑓(𝑡)

𝜃�1(𝑡)

; 𝑓(𝑡): = �𝑝 (𝑠 + 𝑎𝑚)

𝑠[𝑦�]� (𝑡) (9)

This can be obtained by solving (8) with respect to 𝑢(𝑡).

Also, (2) and (6) yield the following tracking error equations;

𝑦�(𝑡) = �𝑠

𝑠 + 𝑝�𝜃�T(𝑡) 𝜁(𝑡)�� (𝑡) (10)

which never depend upon 𝜃�̇ (𝑡) while 𝑦�(𝑡) depends on it when the so-called certainty equivalent control input is used.

Remark 2: From the above error equation, it is observed that the spectrum of 𝑦�(𝑡) in the frequency domain that islower than 𝑝[rad/s] attenuates due to the effect of the high-pass filter 𝑠/(𝑠 + 𝑝) in spite of 𝜃�(𝑡) ≠ 0.

Adaptive Law

Components of adjustable parameter 𝜃�(𝑡) in (9) are updated for all 𝑡 > 0 by according tothe following adaptive laws;

𝜃�̇(𝑡) = Γ(𝑡)�𝑞(𝑡) − 𝑅(𝑡)𝜃�(𝑡)� (11)

where

Γ(𝑡): = diag[ 𝛾1{𝜃�1(𝑡) − 𝜃1,min}/𝑁0(𝑡), 𝛾2/𝑁0(𝑡) ] (12)

𝑁0(𝑡): = �{𝜃�1(𝑡) − 𝜃1,min}2 + 1 (13)

𝑞(𝑡): = �𝜎

𝑠 + 𝜎[𝑦N𝜁N]� (𝑡) (14)

𝑅(𝑡): = �𝜎

𝑠 + 𝜎�𝜁N𝜁N

T�� (𝑡) (15)

𝑦N(𝑡): = 𝑦(𝑡)/𝑁(𝑡), 𝜁N(𝑡): = 𝜁(𝑡)/𝑁(𝑡) (16)

𝑁(𝑡): = �𝜌 + 𝜁T(𝑡) 𝜁(𝑡) (17)

𝛾1 > 0 and 𝛾2 > 0 are adaptive gains, 𝜃1,min is a known positive constant which satisfies 𝜃1 > 𝜃1,min . Initial estimates 𝜃�1(0) and 𝜃�2(0) are set so as to satisfy 𝜃1,min < 𝜃�1(0) < ∞ and |𝜃�2(0)| < ∞ respectively. 𝜎 > 0 and 𝜌 > 0 are design parameters. It is clear that Γ(⋅), 𝑦𝑁(⋅), 𝜁𝑁(⋅), 𝑞(⋅), 𝑅(⋅) ∈ 𝐿∞. The above update laws are essentially equivalent to the Kreisselmeier's adaptive law (Kreisselmeier, 1977) which minimizes the cost;

𝐽(𝑡): = �𝑡

0𝜎𝑒−𝜎(𝑡−𝜏)�𝑦N(𝜏) − 𝜃�T(𝑡) 𝜁N(𝜏)�2d𝜏 (18)

The trade-off can be taken by the design parameter 𝜎 > 0 ; small 𝜎 improves convergence property and large 𝜎 increases the sensitivity of estimation. The equation (11) is the smooth projection adaptive law (Akella,2005), (Tanahashi, 2007), (Sawada, 2013) which always guarantees 𝜃1,min < 𝜃�1(𝑡) and the existence of 𝜃�̇(𝑡) for all 𝑡 ≥ 0. Then, the following lemma holds.

Lemma 1 (Sawada and Itamiya, 2013): The adaptive control system which consists of (1), (9) and (11) is globally stable and 𝑦�(𝑡) converges to zero.

proof: Let 𝑉(𝜃�1, 𝜃�2) be

𝑉�𝜃�1, 𝜃�2� ≔1𝛾1

�−𝜃�1(𝑡) + ln �𝜃1 − 𝜃1,min

𝜃�1(𝑡) − 𝜃1,min�

𝜃1−𝜃1,min

�

+1

2𝛾2𝜃�2

2(𝑡) ; 𝜃�𝑖: = 𝜃𝑖 − 𝜃�𝑖(𝑡), 𝑖 = 1,2 (19)

which evaluates parameter estimation error. Obviously, it is a positive definite function since 𝑉(0, 0) = 0 and

𝜃�1(𝑡) − 𝜃1,min > �𝜃1 − 𝜃1,min��ln{ 𝜃�1(𝑡) − 𝜃1,min�

−ln{ 𝜃1(𝑡) − 𝜃1,min} + 1].

Then, the time derivative �̇�(𝜃�1, 𝜃�2) along the trajectory (11) becomes

�̇�(𝜃�1, 𝜃�2) = −𝐽(𝑡)/𝑁0(𝑡) ≤ 0 (20)

Since 𝐽(𝑡) can be represented as

𝐽(𝑡) = 𝜃�T(𝑡)𝑅(𝑡)𝜃�(𝑡) ; 𝜃�(𝑡): = 𝜃 − 𝜃�(𝑡) (21)

This fact leads to 𝜃� (⋅) ∈ 𝐿∞, hence, 𝑁0(⋅) ∈ 𝐿∞. Further-more, 𝐽(⋅) ∈ 𝐿1 from the integration of (20). This fact means 𝑅1/2(⋅)𝜃�(⋅) ∈ 𝐿2 . Therefore, 𝜃�̇(⋅) ∈ 𝐿∞ ∩ 𝐿2 since 𝜃�̇(𝑡) = Γ(𝑡)𝑅1/2(𝑡)𝑅1/2(𝑡)𝜃�(𝑡) . In addition, it can be shown that 𝑦�(⋅)/𝑁(⋅) ∈ 𝐿2 and 𝜃�1(𝑡) > 𝜃1,min


80

(Tanahashi, 2007), (Sawada, 2013). Theglobal stability and convergencepropertyof 𝑦�(𝑡) can be proven in the same way (Tanahashi, 2007) by using the above 𝐿 −stability properties with respect to adaptive loop and Bellman-Gronwall lemma (Ioannou,1996). Q. E. D.

Fig.2 shows simulation results (the dashed line is 𝑦m(𝑡) and the solid line means 𝑦(𝑡) when 𝑢(𝑡) =𝑔s(𝑢c) = 𝑢c(𝑡) (𝑢min = −∞, 𝑢max = ∞) where simu-lation parameters are set as Table 1.

TABLE1 SIMULATION PARAMETERS

𝑎 0.5 → 0.5/3 (𝑡 ≥ 38) 𝑔s(⋅) known

𝑏 1.0 → 1.0/3 (𝑡 ≥ 38) 𝛾1 10.0 𝑎m 1.0 𝛾2 10.0 𝑝 1.0 𝜌 0.01 𝜎 1.0 𝜃1,min 0.05

𝜃�1(0) 1.0 𝜃�2(0) 0.5 𝑟(𝑡) rectangular signal: − 1 → +1, period: 40

FIG. 2 SIMULATION RESULT IN NO SATURATION CASE

From this figure, it is understood that the control system has a preferable property shown by lemma 1.

On the other hand, the simulation result is shown in Fig.3 when the same adaptive controller is used but 𝑢(𝑡) has saturation property; 𝑢(𝑡) ≠ 𝑢c(𝑡), hence,

FIG. 3 SIMULATION RESULT IN SATURATION CASE

𝜁(𝑡) = [𝑢c[−1](𝑡), 𝑦[−1](𝑡)]T , 𝑢min = −1.5, 𝑢max = 1.5. In

the upper figure, the solid line means 𝑦(𝑡) and the dashed line is 𝑦m(𝑡). In the lower figure, the solid line is 𝑢(𝑡) and the dashed line means the output 𝑢c(𝑡) of adaptive controller. The saturation of 𝑢(𝑡) is observed after 𝑡 = 38 because the load has tripled compared with that at first. Then, 𝑦(𝑡) shows a typical windup phenomenon; large overshoot generated after delay of response. Thus, the properties of lemma 1 are not satisfied and undesirable transient response is generated when the integrator to generate 𝑓(𝑡) is used and the input 𝑢(𝑡) is saturated.

FIG. 4 BLOCK DIAGRAM OF PROPOSED CONTROL SYSTEM

A New Adaptive Speed Control System with Adaptive Anti-Windup Compensator

In order to improve the bad transient response of adaptive speed tracking control system when 𝑢(𝑡) saturates, a new adaptive controller has been put forward with adaptive anti-windup compensator as follows;

Control synthesis:

𝑢c(𝑡) ≔𝑎m 𝑟(𝑡) − 𝜃�2(𝑡) 𝑦(𝑡) − 𝜁𝑇(𝑡) 𝜃�̇(𝑡) − 𝑓𝑎(𝑡)

𝜃�1(𝑡) (22)

where 𝑓a(𝑡) is the signal which is adaptively synthesized in order to compensate the windup phenomenon and defined as follows;

𝑓a(𝑡): = 𝑝 𝑦�(𝑡) + 𝑧(𝑡) (23)

�̇�(𝑡) = 𝑎m𝑝 𝑦�(𝑡) + 𝑝 𝜃�1(𝑡) 𝑢�(𝑡) ; 𝑧(0) = 0 (24)

𝑢�(𝑡): = 𝑢c(𝑡) − 𝑢(𝑡) (25)

where 𝑦m(𝑡), 𝜁(𝑡) are defined as (7), (4) respectively and 𝜃�̇(𝑡), 𝜃�(𝑡) are updated by (11). Fig.4 shows the block diagram of the proposed control system.(22) is the control law based on so-called conditioning technique (Hanus, 1987) when 𝜃�(𝑡) = 𝜃.


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Stability Analysis and Optimal Performance of Control System

Theorem 1: All signals in the speed tracking control system that consists of the controlled object (1) and the proposed adaptive controller with adaptive anti-windup compensator (11)∼(17), (22)∼(25) are bounded. Furthermore, when 𝑢(𝑡) is saturated, the control system has the optimal control performance in the meaning that 𝐿2 gain from 𝑟(𝑡) to 𝑦� (𝑡) is minimized if the design parameter 𝑎m is large enough to satisfy such that

𝑎m ≥𝑏2

4𝑎+ 𝑎 (26)

and if 𝜃�(𝑡) converges to 𝜃. Then, the minimal 𝐿2 gain is 𝑎m/𝑏.

Proof: The proof when 𝑢(𝑡) is not saturated (𝑢�(𝑡) ≡ 0 for all 𝑡 ≥ 0) is the same as the proof of lemma 1 (see Tanahashi et al., 2007). Also, in saturation case, 𝑢(⋅) and 𝑦(⋅) are both bounded since the controlled object is stable system and −𝑢min ≤ 𝑢(𝑡) ≤ 𝑢max . Therefore, the boundedness of all signals in the control system can be shown by proving internal stability of the controller when 𝑢(𝑡) is saturated.

The following differential equation can be obtained by substituting (22) at the right of (24).

�̇�(𝑡) = −𝑝 𝑧(𝑡) + 𝑣(𝑡) (27)

𝑣(𝑡) ≔ 𝑝 (𝑎m − 𝑝)𝑦�(𝑡) + 𝑝 𝑎m𝑟(𝑡) − 𝑝 𝜃�T(𝑡) 𝜁[1](𝑡)

−𝑝 𝜃�̇T(𝑡) 𝜁(𝑡) (28)

So that, 𝑣(⋅) ∈ 𝐿∞ since 𝑦(⋅), 𝑟(⋅), 𝑢(⋅), 𝜁(⋅), 𝜁[1](⋅), 𝜃�̇(⋅) and 𝑦m(⋅) belong to 𝐿∞ . Hence, this fact leads to 𝑧(⋅) ∈ 𝐿∞ and 𝑢c(⋅) ∈ 𝐿∞ from (27) and (22).

Next, the optimal control performance property is proven as follows. The input-output property of actuator saturation may be written by

𝑢(𝑡) =12

{𝑢c(𝑡) − 𝑤s(𝑡)} (29)

𝑤s(𝑡): = Δs�𝑢c(⋅)� ⋅ 𝑢c(𝑡) (30)

−1 ≤ Δs�𝑢c(⋅)� < 1 (31)

where Δs(⋅) is memoryless nonlinear function (see the block diagram of Fig.5). Then, the proposed control system can be represented as the equivalent feedback loop in Fig.6.

FIG. 5 REPRESENTATION OF SATURATION USING

MULTIPLICATIVE UNCERTAINTY ΔS(⋅)

FIG. 6 FEEDBACK LOOP COMPOSED OF Δs(⋅) AND

GENERALIZED PLANT 𝐺(𝑠)

𝐺(𝑠) is a generalized plant thatis stable system and defined as

𝐺(𝑠) ≔ ��̇�(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑤(𝑡)𝑧(𝑡) = 𝐶𝑥(𝑡) + 𝐷𝑤(𝑡) (32)

where 𝑥(𝑡): = [𝑦(𝑡), 𝑧(𝑡), 𝑦m(𝑡)]T, 𝑤(𝑡): = [𝑤s(𝑡), 𝑟(𝑡)]T, 𝑧(𝑡): = [𝑢c(𝑡), 𝑦�(𝑡)]T,

𝐴: = �− 𝑝+𝑎m+𝑎

2− 1

2𝑝2

− 𝑝(𝑝−𝑎m−𝑎)2

− 𝑝2

𝑝(𝑝−2𝑎m)2

0 0 −𝑎m

�, 𝐵: = �− 𝑏

2𝑎m

2𝑝𝑏2

𝑝𝑎m2

0 𝑎m

�,

𝐶: = �− 𝑝+𝑎m−𝑎𝑏

− 1𝑏

𝑝𝑏

1 0 −1�, 𝐷: = �0 𝑎m

𝑏0 0

� (33)

Consider now

𝑊�𝑥(𝑡)�: = 𝑥T(𝑡)𝑃𝑥(𝑡) ; 𝑃T = 𝑃 > 0 (34)

Then, the time-derivative along the trajectory of (32) satisfies

�̇��𝑥(𝑡)� ≤ −∥ 𝑧(𝑡) ∥2+ 𝑤T(𝑡)𝛤𝑤(𝑡) (35)

𝛤: = diag{1, 𝛾2} (36)

for positive𝛾and 𝑃 such that

�𝑃𝐴 + 𝐴T𝑃 𝐶T 𝑃𝐵

𝐶 −𝐼 𝐷𝐵T𝑃 𝐷T −𝛤

� < 0 (37)

From Schur complement of block matrix, the above matrix inequality is equivalent to

⎩⎪⎨

⎪⎧𝛾2 > �

𝑎m

𝑏�

2

𝛾2 > −𝛽T𝑄−1𝛽 + �𝑎m

𝑏�

2

𝑄 < 0

(38)

where 𝛽: = 𝑃𝑏2 + 𝐶T𝑑 (39)

𝑄: = 𝑃𝐴 + 𝐴T𝑃 + 𝐶T𝐶 + 𝑃𝑏1𝑏1T𝑃 (40)


82

𝑏1: = [ −𝑏/2, 𝑝 𝑏/2, 0 ]T

𝑏2: = [ 𝑎m/2, 𝑝 𝑎m/2, 𝑎m ]T

𝑑: = [ 𝑎m/𝑏, 0 ]T� (41)

Therefore, some constant 𝜀 and 𝛾2 that exist may be described as the form;

𝛾2 = �𝑎m

𝑏�

2+ 𝜀2 (42)

Hence, the condition (37) is replaced into the existence condition of the positive definite and symmetric matrix solution 𝑃 for the Riccati equation;

𝐴T𝑃 + 𝑃𝐴 + 𝐶T𝐶 + (𝐵T𝑃 + 𝐷T𝐶)T

⋅ (�̄� − 𝐷T𝐷)−1(𝐵T𝑃 + 𝐷T𝐶) = 0 (43)

�̄�: = diag �1, �𝑎m

𝑏�

2+ 𝜀2� (44)

The necessary and sufficient condition for existence of solution 𝑃 of (44) is well known (Nishimura, 1996) as

(i) The Hamilton matrix 𝐻 does not have eigenvalue on the imaginary axis of the complex plane,

(ii) 𝐴 � is stable matrix and

(iii) ��̃�, 𝑅�� is controllable pair.

𝐻, �̃� and 𝑅� are defined as follows;

𝐻: = � �̃� 𝑅�𝑅�T

−𝑄�𝑄�T −�̃� � (45)

�̃�: = 𝐴T + 𝐶T𝐷(Γ̄ − 𝐷T𝐷)−1𝐵 T (46)

𝑄� : = (Γ̄ − 𝐷T𝐷)−12𝐵T = �

1 0

01𝜀

� 𝐵T (47)

𝑅�: = 𝐶T�𝐼 + 𝐷(Γ̄− 𝐷T𝐷)−1𝐷T�1/2

= 𝐶T ��𝜀2𝑏2 + 𝑎m2

𝜀2𝑏2 0

0 1

� (48)

When those three conditions are concretely calculated, the following relationship can be obtained.

1) 𝛾min = 𝑎m/𝑏 when 𝑎m ≥𝑏2

4𝑎+ 𝑎

2) 𝛾min =2𝑎

�4𝑎m𝑎 − 𝑏2⋅

𝑎m

𝑏 ( > 𝑎m/𝑏)

when 𝑏2

4𝑎< 𝑎m <

𝑏2

4𝑎+ 𝑎

where 𝛾min means the lower bound of 𝛾.

Also, 𝛾 satisfying those conditions does not exist when 0 < 𝑎m < 𝑏2/(4𝑎). Hence, the condition that 𝐿2 gain 𝛾 takes the minimum value is 𝑎m ≥ 𝑏2/(4𝑎) + 𝑎.

Q. E. D

Experiment Results

Simple numerical simulations and the experiments for a real DC motor were done to verify the effectiveness of the proposed method.

Fig.7 shows the numerical experiment results when the adaptive anti-windup compensation is applied to the adaptive velocity control of a DC motor in the same setting as Table 1 of the section, ‘MRACS in Non-Saturation Case.’

By the proposed method, it is verified that the overshoot is well suppressed and the control performance is improved in Fig.7 compared with Fig.3.

It is understood that the saturated time has shortened in the proposal method based onthe comparison of 𝑢(𝑡) in Fig.3 and Fig.7.

FIG. 7 SIMULATION RESULT WITH THE PROPOSED

CONTROLLER IN SATURATION CASE

Next, we will indicate the simple experimental results in which the proposed controller is applied to a real DC motor. In order to verify the effectiveness of the proposed controller, 4 experiments were done according to the presence of adaptation and anti-windup action.

In the beginning, we describe the outline of the DC motor for experiments and the experiment method.

The Industrial Emulator Model 220 (ECP) shown in Fig.8 was used to the actual control experiment.

Fig.9 shows the block diagram of the control experiment system. Loads of driving motor and the disc-like shape are connected with the timing belt. It can be thought that driving motor and the load are juxtaposed because this timing belt uses the one with a little slipping and expansion and contraction. The


83

FIG. 8 INDUSTRIAL EMULATOR MODEL 220 (ECP)

rotary encoder (resolution: 4000[pulse]/2 𝜋 [rad]) is installed in the load and the angular information can be acquired by it. Moreover, the disturbance motor is installed in the load through another timing belt, and, as a result, disturbance can be added. In the experiment described as follows, the controller is realized by DSP, and the control input is added to the driving motor through the servo amplifier. Moreover, rotation information is fed back from the rotary encoder installed in the load as angular information through the encoder processing part in DSP.

The real machine experiments for speed tracking control under great load change were done respectively when 4 different controllers were used, that is, (i) when the speed controller that does not have the functions for both adaptation and anti-windup is used, (ii) when a non-adaptive speed controller with anti-windup function of CT type is used, (iii) when a usual adaptive controller without anti-windup function is used, and (iv) when the proposed controller is used. Hence, it is an adaptive controller with adaptive anti-windup function.

The torque of disturbance motor was synthesized after 𝑡 = 55 [s] so that the moment of inertia of load increases by about five times compared with that at first. It was able to be achieved by using the information of angular acceleration on load. Table 3 shows the figure number corresponding to each experiment result.

FIG. 9 BLOCK DIAGRAM OF CONTROL EXPERIMENT SYSTEM

TABLE2 CONTROLLER AND FIGURE NUMBER

Controller type Figure number

(i) Fig. 10

(ii) Fig. 11

(iii) Fig. 12

(iv) Fig. 13

It is verified that the control input 𝑢(𝑡) is saturated after the load changes and the windup phenomena have occurred in the case that the speed controller does not have an anti-windup function (see Fig.10). Fig.11 shows the experimental result when the speed controller with an anti-windup mechanism is employed. The overshoots have not been completely removed though the windup phenomena are almost suppressed due to the fact that the anti-windup mechanism is designed based on the property of controlled object before load changes. It can be observed that big windup phenomena have been generated because control input, 𝑢(𝑡) , is saturated though the controller output, 𝑢c(𝑡) , is synthesized adaptively according to the load change in the adaptive speed control system without anti-windup function (Fig.12). On the other hand, the windup phenomena are effectively suppressed and the excellent time response, 𝑦(𝑡) , is obtained in the proposal method (see Fig.13) because the proposed adaptive anti-windup function has been added to the adaptive speed controller while the control input, 𝑢(𝑡), is saturated with the load change.

Conclusions

A new adaptive speed control system has been proposed with adaptive anti-windup compensator. The validity of the proposed system is shown by the stability analysis, the performance analysis and simple numerical simulations. In addition, it has also been applied to the real machine experiment and consequently, the effectiveness was confirmed. The reason why this method goes well might be in structural easiness of the controlled object. Enhancement of a complex system is future tasks. It seems that the proposed method can be regarded as the first step that solves the saturation problem of the control input in adaptive control systems.

ACKNOWLEDGMENT

The authors would like to thank the reviewers who have a rapid peer review.


84

FIG. 10 EXPERIMENT RESULTS (NON ADAPTIVE CONTROL

AND NON ANTI-WINDUP CONTROL

FIG. 11 EXPERIMENT RESULTS (NON ADAPTIVE CONTROL

AND ANTI-WINDUP CONTROL

FIG. 12 EXPERIMENT RESULTS (ADAPTIVE CONTROL AND

NON ANTI-WINDUP CONTROL

FIG. 13 EXPERIMENT RESULTS (ADAPTIVE CONTROL AND

ANTI-WINDUP CONTROL

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Keietsu Itamiyareceived his M. E. and D. E. degrees from the University of Tsukuba, Japan, in 1988 and 1991, respectively. In 1994, he joined the faculty of National Defense Academy, where he is currently an Associate Professor of the Department of Electrical and Electronic Engineering.

His research interests include control systems design. He is a member of SICE, ISCIE, IEEJ and IEEE CSS.

Masataka Sawadareceived his B.E. and M.E. degrees from National Defense Academy, Japan, in 1995, 2000, respectively. In 1995, he joined the Japan Air Self Defense Force. His research interests include adaptive control system design. He is currently doctor candidate of Equipment and Structural Eng.,

Graduate School of Science and Engineering, National Defense Academy. He is a student member of SICE.

Documents

Adaptive Speed Tracking Control Systemwith Adaptive Anti-Windup Compensator