Adaptive control for continuous-time systems with actuator and sensor hysteresis

Automatica 64 (2016) 196–207

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Adaptive control for continuous-time systems with actuator and sensor hysteresis✩ Xinkai Chen a,1 , Ying Feng b , Chun-Yi Su c a

Department of Electronic and Information Systems, Shibaura Institute of Technology, Saitama 337-8570, Japan

b

School of Automation Science and Engineering, South China University of Technology, Guangzhou, 510641, China

c

College of Mechanical Engineering and Automation, Huaqiao University, Xiamen, 361021, China2

article

info

Article history: Received 19 April 2014 Received in revised form 15 May 2015 Accepted 20 October 2015

Keywords: Hysteresis Adaptive control Actuator Sensor Prandtl–Ishlinskii model

abstract This paper discusses the model reference control for a continuous-time linear plant containing uncertain hysteresis in both actuator and sensor devices. The difficulty of controlling such systems lies in the fact that the hysteretic uncertainties exist in both the input and the output of the plant, the genuine input and genuine output of the plant are not available, while the ultimate goal is to control the plant output which may not be correctly measured. New adaptive control schemes with the actuator uncertainty and sensor uncertainty compensations are developed for linear plants with either known or unknown dynamics, where adaptive estimates of the genuine outputs of the plants are simultaneously generated. The global stability analysis of the closed-loop system becomes very complicated and challenging, and the proposed control laws ensure the uniform boundedness of all signals in the closed-loop system. The tracking error between the estimated plant output and the desired output is guaranteed to converge to zero asymptotically. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction In the past two decades, smart materials have been well and widely studied in the industrial and academic fields. Since most of the smart materials have a bidirectional property between electric field and mechanical force, they can be used as actuators and sensors. So far, smart material-based actuators and sensors have received much attention due to their excellent characteristics and promising applications. For example, piezo actuators (Croft, Shed, & Devasia, 2001) and magnetostrictive actuators (Natale, Velardi, & Visone, 2001; Tan & Baras, 2004) can be applied to ultra-high precision positioning to meet the requirements of nanometer resolution in displacement, high stiffness and rapid response; shape memory alloys (Gorbet, Morris, & Wang, 2001; Nespoli, Besseghini, Pittaccio, Villa, & Viscuso, 2010; Romanoa & Tannurib, 2009)

✩ The material in this paper was partially presented at the 50th IEEE Conference on Decision and Control and European Control Conference, December 12–15, 2011, Orlando, FL, USA. This paper was recommended for publication in revised form by Associate Editor Gang Tao under the direction of Editor Miroslav Krstic. E-mail addresses: [email protected] (X. Chen), [email protected] (Y. Feng), [email protected] (C.-Y. Su). 1 Tel.: +81 48 687 5805; fax: +81 48 687 5198.

2 On leave from Concordia University, Canada http://dx.doi.org/10.1016/j.automatica.2015.11.009 0005-1098/© 2015 Elsevier Ltd. All rights reserved.

can be applied to robotics and active vibration control systems, to meet the requirement of low energy consumption, long lifetime, etc; ionic polymer metal composite (lPMC) actuators (Chen & Tan, 2010, 2011) are expected to be used as artificial muscles to meet the requirement of low energy consumption, noiselessness, flexibility, lightness. On the other hand, piezo sensors (Yong, Fleming, & Moheimani, 2013) and magnetostrictive material-based sensors (Jia, Liu, Wang, Liu, & Ge, 2011) are used to meet the requirement of heavy load, rapid response, low power consumption, downsizing, better capability to adapt to harsh working environment,etc; IPMC sensors (Chen, Tan, Will, & Ziel, 2007; Ganley, Huang, Zhu, & Tan, 2011) are expected to meet the requirement of low energy consumption, lightness, medical applications, etc. However, it is reported that a non-smooth and nonmemoryless strong nonlinear phenomenon ‘‘hysteresis’’ exists between the input and output of all the smart material-based actuators and sensors (see Banks & Smith, 2000; Cross, Krasnosel’skii, & Pokrovskii, 2001; Drincic, Tan, & Bernstein, 2011; Hanawa, Wang, Deng, & Jiang, 2012; Mayergoyz, 1991; Moheimani & Goodwin, 2001; Natale et al., 2001; Seco, Martin, Pons, & Jimenez, 2004; Smith, 2005; Webb, Lagoudas, & Kurdila, 1998) and the corresponding hysteresis nonlinearities are usually unknown in practice. As a result, the outputs of the actuators and the inputs of the sensors become unknown signals in practice since the actuators are usually used to actuate the plants

X. Chen et al. / Automatica 64 (2016) 196–207

as an input directly and the sensors are usually used to measure the plant output directly. When the hysteretic nonlinearity exists in systems, the system usually exhibits undesirable inaccuracies or oscillations and even instability (Tao & Kokotovic, 1995a,b). For the systems driven by actuators with uncertain hysteresis, the control problem has received considerable attention recently Ren, Ge, Su, and Lee (2009); Tao and Kokotovic (1995a); Wang, Chen, Liu, Liu, and Lin (2014); Wang and Su (2006); Wen and Zhou (2007); Zhou, Wen, and Zhang (2004) and Zhou, Wen, and Li (2012). The common control approach pioneered by Tao and Kokotovic (1995a) is to construct an inverse hysteresis model to compensate the effect of the hysteresis. Essentially, the inversion problem depends on hysteresis modeling methods. Since the hysteresis in the smart material-based actuators are very complicated with multivalues and non-smooth features where the operator-based hysteresis models such as Prandtl–Ishlinskii model and Preisach model (Visintin, 1994) are usually applied, the hysteresis inversion then becomes very challenging. In order to mitigate this difficulty, another approach is proposed in Chen, Hisayama, and Su (2010) where an adaptive implicit inversion of the hysteresis is introduced and updated in accordance with the parameters in the controller which can stabilize the closed-loop system. When output signals measured by sensors with uncertain hysteresis are used in feedback control systems, the development of the controllers is another difficult task because the genuine output of the plant is not available and the eventual purpose is to control the output of the plant. The control techniques reported in the literature until now are surprisingly spare. A pioneer work is Tao and Kokotovic (1995b) where a special hysteretic uncertainty ‘‘backlash’’ is considered. By constructing an adaptive backlash inverse, the uniform boundedness of the signals in the closedloop system is assured. Another important work is Parlangeli and Corradini (2005) where the considered hysteresis can be described by ten parameters. By using the sliding mode robust control method, output zeroing is achieved. By considering the fact that smart material-based actuators and sensors are widely used currently, it is necessary to address the challenge for the control design of the system in presence of both actuator and sensor hysteresis. The difficulty of controlling this class of systems lies in the fact that the hysteretic uncertainties exist in both the input and the output of the plant, the genuine input and genuine output of the plant are not available, and the ultimate goal is to control the plant output which is not an available signal. About the regulation problem for the continuous-time systems in the presence of actuator and sensor uncertain hysteresis, a pioneer work is Parlangeli and Corradini (2005) where the considered hysteresis is relatively simple and can be characterized by ten parameters. It should be mentioned that the control for discretetime linear plant containing uncertain hysteresis in both actuator and sensor devices has been discussed in Tao (1996) and Chen and Ozaki (2010). However, the controller design for continuoustime system is not simply an extension of its counterpart, and the designs for continuous-time system and discrete-time system are based on different strategies. Furthermore, since rapid sampling of a continuous-time linear plant will always give rise to a non-minimum phase discrete-time linear system (Astrom & Wittenmark, 1997) and the results until now (Chen & Ozaki, 2010; Tao, 1996) are limited to minimum phase discrete-time plants, the continuous-time plant is adopted in this paper to avoid this difficulty. In this paper, the model reference control for a continuous-time linear plant containing uncertain hysteresis in actuator and sensor devices simultaneously is discussed, where Prandtl–Ishlinskii (PI) model is used to express the hysteresis nonlinearity. The adoption of PI model is based on the fact that it can describe the

197

Fig. 1. Block scheme of the considered system.

hysteresis existing in smart materials, especially the typical hysteresis behavior in piezo actuators (see Chen & Hisayama, 2008; Krejci & Kuhnen, 2001) and IPMC materials (Hanawa et al., 2012), and possesses the unique inverse property (Krejci & Kuhnen, 2001). In the developed scheme, new adaptive control schemes with the actuator uncertainty and sensor uncertainty compensations are developed for linear plants with either known or unknown dynamics, where adaptive estimates of the genuine outputs of the plants are simultaneously generated. Only the parameters directly needed in the formulation of the controller are adaptively estimated online. The proposed control laws ensure the uniform boundedness of all signals in the closed-loop systems. Furthermore, the tracking error between the estimated plant output and the desired output is guaranteed to converge to zero asymptotically. Generally, the zero convergence of the tracking error between the genuine plant output and the desired output cannot be guaranteed. Finally, the proposed algorithms are illustrated by simulation examples. The organization of the paper is as follows. Section 2 formulates the problem of model reference tracking in the presence of unknown actuator and sensor hysteresis nonlinearities and gives the Prandtl–Ishlinskii (PI) hysteresis model. In Section 3, adaptive control schemes are developed for plants with either known or unknown dynamics. In Section 4, simulation results are presented to illustrate the proposed algorithms. Section 5 concludes this paper. 2. Problem statement 2.1. System description Consider the adaptive control for the continuous-time plant driven by an actuator with hysteresis and measured by a sensor with hysteresis shown in Fig. 1. The considered system is described by y(t ) = G(s)[u](t ) = kp

Z (s) P (s)

[u](t ),

(1)

u(t ) = H1 [v](t ),

(2)

z (t ) = H2 [y](t ),

(3)

where y(t ) ∈ R is the genuine output of the linear plant which is also the input of the sensor, u(t ) ∈ R is the input of the linear plant which is also the output of the actuator, v(t ) ∈ R is the input to the actuator, z (t ) ∈ R is the output of the sensor which is also the measured output of the linear plant; H1 [·] and H2 [·] are the PI hysteresis operators whose structure will be given in the next subsection; G(s) is the transfer function of the linear plant, kp is the high frequency gain, P (s) and Z (s) are described by the following polynomials. P (s) = sn0 + pn0 −1 sn0 −1 + · · · + p1 s + p0 , Z (s) = s + zm−1 s m

m−1

+ · · · + z1 s + z0 ,

R denotes the space of real numbers.

(4) n0 > m .

(5)

198

X. Chen et al. / Automatica 64 (2016) 196–207

The control purpose is to drive the output y(t ) of the linear plant by using the measured output z (t ) to track the output ym (t ) of the reference model described by

From (13) and (15), v(t ) can also be expressed by the stop operator

Pm (s)[ym ](t ) = q(t ),

v(t ) =

(6)

where Pm (s) is a monic polynomial with degree n∗ = n0 − m, q(t ) is the input of the reference model. The following assumptions are made for the control system. A1: Z (s) is a stable polynomial. A2: The upper bound of the degree n0 of P (s) is known as n. A3: The sign of the plant high frequency gain kp is known. Without loss of generality, suppose it is positive. A4: The relative degree n∗ = n0 − m of the plant is known. Remark 1. It should be mentioned that u(t ) and y(t ) are not available signals. The available signals which can be used in the control design are v(t ) and z (t ) and their accumulated values. The existence of unknown hysteretic nonlinearities in both actuator and sensor makes the considered problem very challenging.

1 p0 +

Prandtl–Ishlinskii (PI) hysteresis operators will be introduced in the following. The basic element of the PI operator is the socalled stop operator (Brokate & Sprekels, 1996) and play operator with threshold r. For arbitrary piece-wise monotone function v(t ), define er : R → R and fr : R × R → R as er (v) = min(r , max(−r , v)),

(7)

fr (v, α0 ) = max(v − r , min(v + r , α0 )),

p(r )dr

0

u(t ) +

∞



p¯ (r )Er [u](t )dr .

(16)

0

For any initial value z−1 ∈ R, the hysteresis operator z (t ) = H2 [y](t ) in (3) is defined by z (t ) = s¯0 y(t ) −

∞



s¯(r )Fr [y](t )dr

(17)

0

where s¯0 is a positive constant, s¯(r ) is the  ∞ density function ∞ satisfying s¯(r ) ≥ 0 with 0 s¯(r )dr < s¯0 and 0 r s¯(r )dr < ∞. Similarly, for the operator H2 [·] defined in (17), there exist 1 a  ∞density function s(r ) ≥ 0 satisfying s(r ) ∈ L (0, ∞) and rs ( r ) dr < ∞ such that 0 y(t ) = s0 z (t ) +

∞



s(r )Fr [z ](t )dr

(18)

0

with s¯0 =

2.2. Hysteresis model

∞

y(t ) =

1 s0

and

∞ 0

s(r )dr = s¯ − ∞1 s¯(r )dr − s0 . Furthermore, 0 0

1 s¯0 −

∞ 0

s¯(r )dr

z (t ) −

∞



s(r )Er [z ](t )dr .

(19)

0

3. Adaptive control 3.1. Some preliminaries

(8)

where α0 ∈ R can be any value. For any initial value w−1 ∈ R and r ≥ 0, the stop operator Er [∗; w−1 ](t ) and the play operator Fr [∗; w−1 ](t ) are respectively defined as

To begin with, introduce an (n − 1)th order monic stable polynomial Λ(s) and define a(s) = [1, s, . . . , sn−2 ]T . Now, consider the polynomial equation

Er [v; w−1 ](0) = er (v(0) − w−1 ),

  θ1T a(s)P (s) + θ2T a(s) + θ20 Λ(s) kp Z (s)   = Λ(s) P (s) − θ3 kp Z (s)Pm (s) ,

(9)

Er [v; w−1 ](t ) = er (v(t ) − v(ti ) + Er [v; w−1 ](ti )),

(10)

Fr [v; w−1 ](0) = fr (v(0) − w−1 ),

(11)

Fr [v; w−1 ](t ) = fr (v, Fr [v; w−1 ](ti )),

(12)

for ti ≤ t ≤ ti+1 , where the function v(t ) is monotone for ti ≤ t ≤ ti+1 (Brokate & Sprekels, 1996; Visintin, 1994). It can be seen that Er [v; w−1 ](t ) + Fr [v; w−1 ](t ) = v(t ).

(13)

The stop and play operators are rate-independent which are mainly characterized by the threshold parameter r ≥ 0. For simplicity, denote Er [v; w−1 ](t ) and Fr [v; w−1 ](t ) by Er [v](t ) and Fr [v](t ) respectively in the following of this paper. The PI hysteresis operator u(t ) = H1 [v](t ) in (2) is defined by u(t ) = p0 v(t ) +

p(r )Fr [v](t )dr

(14)

0

where p0 is a positive constant, p(r ) is the density function which 1 is  ∞usually unknown, satisfying p(r ) ≥ 0 with p(r ) ∈ L (0, ∞) and rp(r )dr < ∞. 0 Now, let us consider the inverse operator of H1 [·]. The following lemma is cited (see Krejci & Kuhnen, 2001 and Chapter 2 in Brokate & Sprekels, 1996). Lemma 1. For the operator H1 [·] defined in (14), there exists a density function p¯ (r ) ≥ 0 satisfying p¯ (r ) ∈ L1 (0, ∞) and ∞ ¯ r p ( r )dr < ∞ such that 0

v(t ) = p¯ 0 u(t ) −

∞



p¯ (r )Fr [u](t )dr

(15)

0

with p¯ 0 =

1 p0

and

1 where θ1 ∈ Rn−1 , θ2 ∈ Rn−1 , θ20 ∈ R, and θ3 = k− p ∈ R are parameters which exist uniquely (see Lemma 5.1 in Tao, 2003). Operating both sides of (20) on y(t ) and applying (1) yields

  θ1T a(s)Z (s)[u](t ) + θ2T a(s) + θ20 Λ(s) Z (s)[y](t ) = Λ(s)Z (s)[u](t ) − θ3 Λ(s)Z (s)Pm (s)[y](t ).

∞ 0

p¯ (r )dr =

1 p0

−

 ∞1 . p0 + 0 p(r )dr

(21)

By observing that the polynomials Λ(s) and Z (s) are stable, the relation between the input and the output of the linear plant can thus be expressed as

∞



(20)

a(s)

[u](t ) + θ2T

a(s)

[y](t ) Λ(s) Λ(s) + θ20 y(t ) + θ3 Pm (s)[y](t ),

u(t ) = θ1T

(22)

where an exponential decaying term is omitted (Tao, 2003). Since u(t ) and y(t ) are not available signals, substituting their expressions (14) and (18) into (22) gives p0 v(t ) +

∞



p(r )Fr [v](t )dr 0

a(s) p(r )θ1T [Fr [v]](t )dr Λ(s) Λ(s) 0  ∞ a(s) a(s) + s0 θ2T [z ](t ) + s(r )θ2T [Fr [z ]](t )dr Λ(s) Λ(s) 0  ∞ + s0 θ20 z (t ) + θ20 s(r )Fr [z ](t )dr + θ3 Pm (s)[y](t ).

= p0 θ1T

a( s )

[v](t ) +

0



∞

(23)

X. Chen et al. / Automatica 64 (2016) 196–207

Therefore, it can be easily seen that, based on (23), the model reference control input can be derived by replacing Pm (s)[y](t ) with q(t ) if all the parameters are available. Remark 2. The reason of choosing the PI operator of the sensor side in the form of (17) is just for the purpose of making the signs of all the terms on the right hand side of (23) to be plus. Certainly, if it is chosen as the form in (14), the following formulations can be similarly constructed without any difficulty.

pˆ 00 (t )v(t ) +

∞

pˆ 0 (r , t )Fr [v](t )dr = W1 (t ),

(24)

0

where W1 (t ) is defined as ∞ a(s) [v](t ) + pˆ 0 (r , t )θ1T [Fr [v]](t )dr Λ(s) Λ (s)  ∞0 a(s) a(s) [z ](t ) + sˆ0 (r , t )θ2T [Fr [z ]](t )dr + θ2T Λ(s) Λ(s) 0  ∞ + θ20 z (t ) + sˆ0 (r , t )θ20 Fr [z ](t )dr + θ3 q(t ). (25)

W1 (t ) = pˆ 00 (t )θ1T

a(s)



0

Remark 3. It should be noted that W1 (t ) is an available signal since every term involving v(t ) contains a filter and the instantaneous value of v(t ) at time t is irrelevant. The signal v(t ) will be calculated piecewisely on the time intervals t ∈ [tj , tj+1 ] on which W1 (t ) is monotonically increasing or decreasing. Without loss of generality, suppose W1 (t ) is monotonically increasing on the interval tj ≤ t ≤ tj+1 . For each t ∈ [tj , tj+1 ], (1)

define a new variable V¯ µ(1) (t ) with V¯ 0 (t ) = v(tj ) and another new variable Wµ(1) (t ), where µ is a positive parameter (1)

V¯ µ(1) (t ) = V¯ 0 (t ) + µ, Wµ(1) (t ) = pˆ 00 (t )V¯ µ(1) (t ) +

∞ 0

pˆ 0 (r , t )Fr [V¯ µ(1) ](t )dr .

∞



sˆ0 (r , t )Fr [z ](t )dr

(28)

and the error as e1 (t ) = yˆ 1 (t ) − ym (t ).

(29)

Then, from (18), (23), (24) and (28), the variable e1 (t ) defined in (29) can be expressed as e1 (t ) = yˆ 1 (t ) − y(t ) + y(t ) − ym (t ) ∞



s˜0 (r , t )Fr [z ](t )dr +

= 0

kp  Pm (s)

∞



p˜ 0 (r , ·)η2 (r , ·)dr +

+ 0

p˜ 00 (·)η1 (·)

∞





s˜0 (r , ·)η3 (r , ·)dr (t ) (30) 0

with

η1 (t ) = −v(t ) + θ1T

a( s )

Λ(s)

[v](t ),

(31)

a(s) [Fr [v]](t ) − Fr [v](t ), Λ(s) a(s) η3 (r , t ) = θ2T [Fr [z ]](t ) + θ20 Fr [z ](t ) Λ(s)

η2 (r , t ) = θ1T

(32) (33)

where s0 = 1 and relation (6) are used, and the ‘‘tilde’’ variables (parameter errors) denote the differences between the ‘‘hat’’ variables (estimated parameters) and their corresponding genuine parameters. By observing (30), it can be seen that the adaptive algorithms of updating pˆ 00 (t ), pˆ 0 (r , t ) and sˆ0 (r , t ) cannot be directly constructed since there is a filter P 1(s) before the errors p˜ 00 (t ), m

p˜ 0 (r , t ) and s˜0 (r , t ) (see Tao, 2003). For this reason, define a new error ε1 (t ) as

ε1 (t ) = e1 (t ) + kp ξ1 (t )

(34)

with

  1 1 ξ1 (t ) = pˆ 00 (·) − pˆ 00 (·) [η1 ](t ) P (s) Pm (s)  ∞ m  1 1 + − pˆ 0 (r , ·) [η2 ](t )dr pˆ 0 (r , ·) Pm (s) Pm (s) 0 ∞   1 1 + sˆ0 (r , ·) − sˆ0 (r , ·) [η3 ](t )dr . Pm (s) Pm (s) 0

(35)

Substituting (30) into (34) yields (26)



To apply the adaptive control law derived from (24), it is necessary to develop algorithms to estimate the required parameters pˆ 00 (t ), pˆ 0 (r , t ), and sˆ0 (r , t ). Since the plant output y(t ) is not available, by referring (18), define the estimated plant output as

0

Since kp and the parameters in P (s) and Z (s) are known, the parameters θ1 , θ2 , θ20 , θ3 in (20) can be obtained. However, because the parameters p0 and s0 and the density functions in (14) and (18) are all unknown, the control scheme cannot be derived. To overcome this difficulty, the adaptive method will be used to estimate the unknown parameters needed in the control scheme. Without loss of generality, assume that s¯0 = 1, i.e. s0 = 1. Otherwise, by observing (1), (14) and (17), s¯0 p0 can be treated as p0 , s¯0 p(r ) can be treated as p(r ), and s0 s¯(r ) can be treated as s¯(r ). Suppose that the estimates of p0 , p(r ), and s(r ) are respectively expressed as pˆ 00 (t ), pˆ 0 (r , t ) and sˆ0 (r , t ) at instant t, where these estimates should be respectively determined such that pˆ 00 (t ) ≥ 0, pˆ 0 (r , t ) ≥ 0 and sˆ0 (r , t ) ≥ 0 for all r and t. With these estimates, by observing (23), the design task is to find a signal v(t ) as an input of the actuator so that the following equation holds



(1) (1) For t = 0, V¯ 0 (0) can be defined as V¯ 0 (0) = vmin , where vmin is the admissible minimum value of v(t ). In this paper, the calculated v(t ) is called the ‘‘implicit inversion’’ of W1 (t ).

yˆ 1 (t ) = z (t ) +

3.2. Adaptive control design for known G(s)

199

(27)

The value of v(t ) can be derived from the following algorithm. Step 1: Let µ increase from 0. Step 2: Calculate V¯ µ(1) (t ) and Wµ(1) (t ). If Wµ(1) (t ) < W1 (t ), then let µ increase continuously and go to Step 2; Otherwise, go to Step 3. Step 3: Stop the increasing of µ, memorize it as µ0 and let v(t ) = V¯ µ(10) (t ).

 ∞ ε1 (t ) = p˜ 00 (t )ζ1 (t ) + p˜ 0 (r , t )ζ2 (r , t )dr 0  ∞ + s˜0 (r , t )ζ3 (r , t )dr

(36)

0

with

ζ1 (t ) =

kp Pm (s)

[η1 ](t ),

ζ3 (r , t ) = Fr [z ](t ) +

ζ2 (r , t ) = kp

Pm (s)

[η3 ](t ).

kp Pm (s)

[η2 ](t ),

(37) (38)

200

X. Chen et al. / Automatica 64 (2016) 196–207

Remark 4. ξ1 (t ) is introduced so that ε1 (t ) can be in the form of (36) which can be used to construct the adaptive algorithms to update the parameters pˆ 00 (t ), pˆ 0 (r , t ) and sˆ0 (r , t ) (see Tao, 2003). Define m1 (t ) = 1 + ζ12 (t ) +



∞

|ζ2 (r , t )|dr

2

+

1

parameters belong to L2 ∩ L∞ . Lemma 3. For the estimated parameters pˆ 0(r , t ) and sˆ0 (r , t ), it gives ∞ ∞ ∞ ˆ ˆ s ( r , t )dr < ∞, 0 r pˆ 0 (r , t )dr < ∞ and p ( r , t ) dr < ∞ , 0 0 0 0

∞ 0

0

r sˆ0 (r , t )dr < ∞.

∞

∞



ε (t ) seen that √m1 (t ) ∈ L∞ and the time derivatives of all the estimated

|ζ3 (r , t )|dr

2

.

(39)

0

By observing the expression of ε1 (t ) in (36), the parameter adaptation law is chosen as

 ε1 ( t )   ζ1 (t ) if pˆ 00 (t ) > 0 −µ1   m1 (t )   ε1 ( t ) (40) p˙ˆ 00 (t ) = −µ1 ζ1 (t ) if pˆ 00 (t ) = 0 and  m 1 (t )    ε1 (t )ζ1 (t ) < 0   0 otherwise  ε ( t ) 1   ζ (r , t ) if pˆ 0 (r , t ) > 0 −µ2  3+α m (t ) 2  ( 1 + r )  1  ε1 ( t ) p˙ˆ 0 (r , t ) = −µ2 ζ2 (r , t ) if pˆ 0 (r , t ) = 0 and  ( 1 + r )3+α m1 (t )    ε1 (t )ζ2 (r , t ) < 0   0

otherwise

(41)

 ε1 (t )   ζ3 (r , t ) −µ3   (1 + r )3+α m1 (t )   ε1 (t ) s˙ˆ0 (r , t ) = −µ3 ζ3 (r , t )  ( 1 + r )3+α m1 (t )      0

if sˆ0 (r , t ) > 0 if sˆ0 (r , t ) = 0 and

ε1 (t )ζ3 (r , t ) < 0 otherwise

Proof. By observing 0 (1 + r )3+α p˜ 20 (r , t )dr < ∞ and < ∞, it can be easily checked that

∞ 0

rp(r )dr

∞



r pˆ 0 (r , t )dr 0

∞



r p˜ 0 (r , t )dr +

=

rp(r )dr 0

0

≤

∞





∞

r2

(1 + r )

0

∞

 dr 3+α

(1 + r )3+α p˜ 20 (r , t )dr

 12

0

∞

rp(r )dr

+ 0

< ∞.

(45)

∞

Based on this result, it can be easily proved that 0 pˆ 0 (r , t )dr <  ∞ boundedness of pˆ 0 (r , t ). Similarly, ∞∞by observing the uniform ˆ s ( r , t ) dr < ∞ and r sˆ0 (r , t )dr < ∞ can be proved. 0 0 0 Lemma 4. For each t, there exist a function sˆˆ0 (r , t ) ≥ 0 satisfying ∞ ∞ˆ sˆ0 (r , t )dr < ∞ and 0 r sˆˆ0 (r , t )dr < ∞ such that 0  ∞ z (t ) = yˆ 1 (t ) − sˆˆ0 (r , t )Fr [ˆy1 ](t )dr (46) 0    ∞  ∞ = 1− sˆˆ0 (r , t )dr yˆ1 (t ) + sˆˆ0 (r , t )Er [ˆy1 ](t )dr . 0

0

(47)

(42) where µi > 0 for i = 1, . . . , 3 and α > 0 are design parameters.

Furthermore, sˆˆ0 (r , t ) is uniformly bounded.

Remark 5. From (40)–(42), it can be seen that pˆ 00 (t ) ≥ 0, pˆ 0 (r , t ) ≥ 0 and sˆ0 (r , t ) ≥ 0 for all r and t. pˆ 00 (0), sˆ0 (r , 0) and pˆ 0 (r , 0) should be chosen such that pˆ 00 (0) > 0, pˆ 0 (r , 0) ≥ 0, ∞ ∞ sˆ0 (r , 0) ≥ 0, 0 < 0 r pˆ 0 (r , 0)dr < ∞ and 0 < 0 r sˆ0 (r , 0)dr < ∞.

Proof. From (28), the relationship (46) is obvious by using the results in Krejci and Kuhnen (2001) and Chapter 2 in Brokate

In the following, the stability of the system (1)–(3) controlled by the input derived from (24) will be analyzed. Lemma 2. The adaptive law (40)–(42) guarantees that all the estimated parameters belong to L∞ , the time derivatives of all the ε (t ) estimated parameters belong to L2 ∩ L∞ and √m1 (t ) ∈ L2 ∩ L∞ .

∞

Simultaneously, 0 (1 + r )3+α p˜ 20 (r , t )dr r )3+α s˜20 (r , t )dr < ∞.

1

< ∞ and

∞ 0

(1 +

Theorem 1. All the signals in the closed-loop system consisting of the system (1)–(3), reference model (6), controller derived from (24), and adaptive law (40)–(42) are bounded and the tracking error e1 (t ) = yˆ 1 (t ) − ym (t ) belongs to e1 (t ) ∈ L2 and limt →∞ e1 (t ) = 0. Furthermore, limt →∞ ε1 (t ) = 0 and the estimated parameters in (40)–(42) converge to certain values for a fixed r.

From (18) and (28), the genuine plant output tracking error y(t ) − ym (t ) is governed by

∞



Remark 6. Relationship (46) gives an inverse expression of relation (28).

Proof. The proof is given in Appendix A.

Proof. Consider the positive definite function 1 2 1 ˜ 00 (t ) + µ− V1 (t ) = µ− 1 p 2

and Sprekels (1996). The uniform boundedness of sˆˆ0 (r , t ) can be obtained from Lemma 2 and Krejci and Kuhnen (2001).

(1 + r )3+α p˜ 20 (r , t )dr 0

+ µ3

−1

∞



(1 + r )

˜ (r , t )dr .

3+α 2 s0

(43)

0

By taking the time derivative of V1 (t ) along the trajectory of (40)– (42), it gives V˙ 1 (t ) ≤ −

2ε12 (t ) m1 ( t )

≤ 0.

(44)

From (43) and (44), it gives that all the estimated parameters ε (t ) belong to L∞ and √m1 (t ) ∈ L2 . From (36) and (40)–(42), it can be 1

y(t ) − ym (t ) =

∞



s˜0 (r , t )Fr [z ](t )dr ,

(48)

0

which is mainly dominated by the estimation error s˜0 (r , t ) = sˆ0 (r , t ) − s(r ). Since the genuine plant output y(t ) cannot be measured, this result can be considered to be reasonable. 3.3. Adaptive control design for unknown G(s) Since kp and the parameters in P (s) and Z (s) are unknown, the parameters θ1 , θ2 , θ20 , θ3 in (20) cannot be obtained.

X. Chen et al. / Automatica 64 (2016) 196–207

Furthermore, since the parameters and the density functions in (14) and (18) are all unknown, the control scheme discussed in Section 3.1 cannot be derived. In this subsection, the adaptive method will be employed to estimate the unknown parameters needed in the control law design. In the following, without loss of generality, assume that s¯0 = p0 = 1. Otherwise, by observing (1), (14) and (17), kp p0 s¯0 can be treated as kp , p1 p(r ) can be treated as p(r ), and s0 s¯(r ) can be treated 0 as s¯(r ). Suppose that the estimates at instant t of p(r ), θ1 , p(r )θ1 , θ2 , s(r )θ2 , θ20 , θ20 s(r ) and θ3 are respectively denoted as pˆ (r , t ), θˆ1 (t ), pˆ 1 (r , t ), θˆ2 (t ), sˆ2 (r , t ), θˆ20 (t ), sˆ20 (r , t ) and θˆ3 (t ), where the estimate pˆ (r , t ) should be determined such that pˆ (r , t ) ≥ 0 for all r and t. With these estimates, by observing (23), the design task is to find a signal v(t ) as an input of the actuator so that the following equation holds

v(t ) +

− − − −

Thus, from (18), (23), (49), (51) and (52), the new error ε(t ) defined in (53) gives

ε(t ) =

∞



s˜(r , t )Fr [z ](t )dr −

pˆ (r , t )Fr [v](t )dr = W (t ),

(49)

+ θ˜1T (t )

W (t ) = θˆ1T (t )

[v](t ) +



∞

pˆ T1 (r , t )

a(s)

Pm (s) Λ(s)

∞



p˜ T1 (r , t )

+

+ θ˜2T (t ) [Fr [v]](t )dr

Λ(s) Λ(s) 0  ∞ a ( s ) a( s ) + θˆ2T (t ) [z ](t ) + sˆT2 (r , t ) [Fr [z ]](t )dr Λ(s) Λ (s)  ∞ 0 + θˆ20 (t )z (t ) + sˆ20 (r , t )Fr [z ](t )dr + θˆ3 (t )q(t ).

s˜T2 (r , t ) 0

(50)

The signal v(t ) (which is the implicit inversion of W (t )) satisfying (47) can be similarly derived by using algorithm stated in Section 3.2. To apply the adaptive control v(t ) derived from (49), it is necessary to develop algorithms to estimate the required parameters pˆ (r , t ), θˆ1 (t ), pˆ 1 (r , t ), θˆ2 (t ), sˆ2 (r , t ), θˆ20 (t ), sˆ20 (r , t ) and θˆ3 (t ). First of all, define the estimated plant output as yˆ (t ) = z (t ) +

∞



T + θ˜20 (t )

kp Pm (s)

sˆ(r , t )Fr [z ](t )dr ,

(51)

where sˆ(r , t ) is the estimate of s(r ) at instant t, sˆ(r , t ) should be determined such that sˆ(r , t ) ≥ 0 for all r and t. Then, define the error between the estimated output and the desired output as e(t ) = yˆ (t ) − ym (t ).

(52)

Similar to the discussions in Section 3.2, the error e(t ) can also not be used to construct the parameter estimation laws. In order to overcome this difficulty, define a new error ε(t ) as

ε(t ) = e(t ) + kˆ p (t )ξ (t )

(53)

1

pˆ (r , ·) − pˆ (r , ·)

1

 [Fr [v]](t )dr

Pm (s) Pm (s)  1 1  a( s ) − θˆ1T (·) − θˆ1T (·) [v](t ) Pm (s) Pm (s) Λ(s)  ∞ 1 1  a(s) − pˆ T1 (r , ·) − pˆ T1 (r , ·) [Fr [v]](t )dr Pm (s) Pm (s) Λ(s) 0  1 1  a( s ) − θˆ2T (·) − θˆ2T (·) [z ](t ) Pm (s) Pm (s) Λ(s) 0

kp Pm (s)

kp

Pm (s)

[Fr [v]](t )dr

[Fr [v]](t )dr

[z ](t ) a(s)

Pm (s) Λ(s)

[z ](t ) +

[Fr [z ]](t )dr ∞



s˜20 (r , t ) 0

kp Pm (s)

[Fr [z ]](t )dr

[q](t ) + k˜ p (t )ξ (t ),

(55)

where the ‘‘tilde’’ variables (parameter errors) denote the differences between the ‘‘hat’’ variables (estimated parameters) and their corresponding genuine parameters, and relationship (6) and the fact s0 = p0 = 1 are used. Define m0 ( t ) = 1 +

∞



|Fr [z ](t )|dr

2

+

∞

 0

   2  1     P (s) [Fr [v]](t ) dr m

 1 a( s ) 2 + [v](t ) Pm (s) Λ(s)    ∞  1 a(s) 2    dr + [ F [v]]( t )  P (s) Λ(s) r  0

m

 1 a( s ) 2 + [z ](t ) Pm (s) Λ(s)    ∞  1 a(s) 2    +  P (s) Λ(s) [Fr [z ]](t ) dr m 0    1 2  ∞  1 2    dr + [z ](t ) + [ F [ z ]]( t ) r   P (s) P (s) m

with ∞

+ θ˜3 (t )

a(s)

0

0



a(s)

kp

∞

 +

kp

kp

[v](t )

Pm (s) Λ(s)

Pm (s) Λ(s)

0

ξ (t ) =

a(s)

kp

0

a(s)

p˜ (r , t ) 0

0

where W (t ) is defined as

∞



0

∞



201

1 T 1  a(s) sˆ2 (r , ·) − sˆT2 (r , ·) [Fr [z ]](t )dr Pm (s) Pm (s) Λ(s) 0  1 1  T T [z ](t ) θˆ20 (·) − θˆ20 (·) Pm (s) Pm (s)  ∞ 1 1  sˆ20 (r , ·) − sˆ20 (r , ·) [Fr [z ]](t )dr Pm (s) Pm (s) 0  1 1  [q](t ) (54) θˆ3 (·) − θˆ3 (·) Pm (s) Pm (s) ∞



0

m

 1 2 + [q](t ) + ξ 2 (t ). Pm (s)

(56)

By observing the expression of ε(t ) in (55), the parameter adaptation law is chosen as

 ε(t )   Fr [v](t ) −γ1  ( 1 + r )3+α m0 (t )   ε(t ) s˙ˆ(r , t ) = −γ1 Fr [v](t )  ( 1 + r )3+α m0 (t )      0

if sˆ(r , t ) > 0 if sˆ(r , t ) = 0 and (57)

ε(t )Fr [v](t ) < 0 otherwise

202

X. Chen et al. / Automatica 64 (2016) 196–207

p˙ˆ (r , t )

Lemma 6. For the r, t)  ∞sˆ(r , t ) in (57) and pˆ(∞  ∞estimated parameters in (58), it gives 0 sˆ(r , t )dr < ∞, 0 pˆ (r , t )dr < ∞, 0 r sˆ

 1 ε(t )  [Fr [v]](t ) if pˆ (r , t ) > 0 γ2  3 +α   (1 + r ) m0 (t ) Pm (s)   ε(t ) 1  γ2 [Fr [v]](t ) if pˆ (r , t ) = 0 and = ( 1 + r )3+α m0 (t ) Pm (s)   1   [Fr [v]](t ) > 0 ε(t )   Pm (s)  0

(59) (60)

(63) (64)

Proof. The proof is given in Appendix B. (65) (66)

0

∞

(1 + r )3+α s˜2 (r , t )dr + γ6−1 k˜ 2p (t )  ∞  kp γ2−1 (1 + r )3+α p˜ 2 (r , t )dr + θ˜1T (t )Γ1−1 θ˜1 (t ) 0  ∞ p˜ T1 (r , t )Γ2−1 p˜ 1 (r , t )dr + θ˜2T (t )Γ3−1 θ˜2 (t ) 0  ∞ 2 s˜T2 (r , t )Γ4−1 s˜2 (r , t )dr + γ3−1 θ˜20 (t ) 0  ∞  γ4−1 s˜220 (r , t )dr + γ5−1 θ˜32 (t ) . (67) 0

By taking the time derivative of V (t ) along the trajectory of (57)– (66), it gives V˙ (t ) ≤ −

2ε 2 (t ) m0 (t )

y(t ) − ym (t ) =

∞



s˜(r , t )Fr [z ](t )dr ,

(71)

0

which is mainly dominated by the estimation error s˜(r , t ) = sˆ(r , t ) − s(r ). 4. Simulation results

y(t ) = G(s)[u](t ) = u(t ) = 0.5v(t ) + z (t ) = 0.9y(t ) −



2s + 3 [u](t ), (s − 1)(s − 4) ∞

 0∞

(72)

e−0.067(r −1) Fr [v](t )dr ,

(73)

0.03e−0.005(r −1) Fr [v](t )dr

(74)

2

2

0

0

+

From (18) and (51), the genuine plant output tracking error y(t ) − ym (t ) is governed by

In this section, consider the system described by

Proof. Consider the positive definite function

+

(70)

Theorem 2. All the signals in the closed-loop system consisting of the system (1)–(3), reference model (6), controller derived from (49), and adaptive law (57)–(66) are bounded and the tracking error e(t ) = yˆ (t ) − ym (t ) belongs to e(t ) ∈ L2 and limt →∞ e(t ) = 0. Furthermore, limt →∞ ε(t ) = 0 and the estimated parameters in (57)–(66) converge to certain values for a fixed r.

Lemma 5. The adaptive law (57)–(66) guarantees that all the estimated parameters belong to L∞ , the time derivatives of all the ε(t ) estimated parameters belong to L2 ∩ L∞ , and √m (t ) ∈ L2 ∩ L∞ .

+

sˆˆ(r , t )Er [ˆy](t )dr . 0

0

(62)

In the following, the stability of the system (1)–(3) controlled by the input derived from (49) will be analyzed.

+

∞

Proof. The proof is similar to that of Lemma 4.

r pˆ (r , 0)dr < ∞.

V (t ) = γ1−1

(69)

Furthermore, sˆˆ(r , t ) is uniformly bounded.

Remark 7. From (57) and (58), it can be seen that pˆ (r , t ) ≥ 0 and sˆ(r , t ) ≥ 0 for all r and t. sˆ(r , 0) andpˆ (r , 0) should be chosen ∞ such that sˆ(r , 0) ≥ 0, pˆ (r , 0) ≥ 0, 0 r sˆ(r , 0)dr < ∞ and



∞ˆ sˆ(r , t )dr ≤ ∞ such that 0  ∞ sˆˆ(r , t )Fr [ˆy](t )dr z (t ) = yˆ (t ) − 0   ∞ sˆˆ(r , t )dr )ˆy(t ) + = (1 −

(61)

where γi > 0 for i = 1, . . . , 6, Γj = ΓjT > 0, for j = 1, . . . , 4 and α are design parameters.

0

r pˆ (r , t )dr < ∞.

Lemma 7. For each t, there exists a function sˆˆ(r , t ) ≥ 0 satisfying

(58)

∞

0

Proof. The proof is similar to that of Lemma 3.

otherwise

ε(t ) 1 a(s) θ˙ˆ 1 (t ) = −Γ1 [v](t ), m0 (t ) Pm (s) Λ(s) ε(t ) 1 a(s) p˙ˆ 1 (r , t ) = −Γ2 [Fr [v]](t ), m0 (t ) Pm (s) Λ(s) ε(t ) 1 a(s) θ˙ˆ 2 (t ) = −Γ3 [z ](t ), m0 (t ) Pm (s) Λ(s) ε(t ) 1 a(s) s˙ˆ2 (r , t ) = −Γ4 [Fr [z ]](t ), m0 (t ) Pm (s) Λ(s) ε(t ) 1 θ˙ˆ 20 (t ) = −γ3 [z ](t ), m0 (t ) Pm (s) ε(t ) 1 s˙ˆ20 (r , t ) = −γ4 [Fr [z ]](t ), m0 (t ) Pm (s) ε(t ) 1 θ˙ˆ 3 (t ) = −γ5 [q](t ), m0 (t ) Pm (s) ε(t ) ˙ kˆ p (t ) = −γ6 ξ (t ), m0 (t )

∞

(r , t )dr < ∞ and

≤ 0.

(68)

Based on (67), (68), the expression of ε(t ) in (55) and the adaptive laws in (57)–(66), similar to the proof of Lemma 2, this lemma can be proved.

with z−1 = 0 and u−1 = 0. The parameters in (73) and (74) are unknown. The control purpose is to drive the output y(t ) of the above system to track the output ym (t ) of the reference model described by (s + 1)[ym ](t ) = q(t ) with q(t ) = 10 sin(2π t ). The desired output ym (t ) is shown in Fig. 2. In the simulation, the integral interval of the threshold parameter r is chosen as 0 ≤ r ≤ 20, the discretization level of it is 0.1; the sampling period is set to 0.001; the stable polynomial Λ(s) in (20) is chosen as Λ(s) = s + 3. The parameter α in (41), (42), (57) and (58) is chosen as α = 1. Simulation 1. Simulation results for known G(s). In this case, the parameters in (20) can be obtained as kp = 2, θ1 = 1.5, θ2 = −2.0, θ20 = −0.5, and θ3 = 0.5. The initial values of the estimated parameters are chosen as pˆ 00 (0) = 1.00, pˆ 0 (r , 0) = 0.01, and sˆ0 (r , 0) = 0.001. The adaptive gains in (40)– (42) are chosen as µ1 = 6, µ2 = 10, and µ3 = 1. The convergence of the estimated parameters pˆ 00 (t ), pˆ 0 (r , t ) and sˆ0 (r , t ) has been confirmed in the simulation study. The

X. Chen et al. / Automatica 64 (2016) 196–207

Fig. 2. The output ym (t ) of the reference model.

203

Fig. 5. The error yˆ 1 (t ) − ym (t ) for known G(s). Table 1 Design parameters and initial values for unknown G(s). Design parameters

Initial values

γ1 = 1 γ2 = 15 Γ1 = 350 Γ2 = 15 Γ3 = 350 Γ4 = 15 γ3 = 350 γ4 = 15 γ5 = 350 γ6 = 50

sˆ(r , 0) = 0.001 pˆ (r , 0) = 0.01 θˆ1 (0) = 1 pˆ 1 (r , 0) = 0.01 θˆ2 (0) = 1 sˆ2 (r , 0) = 0.001 θˆ20 (0) = 1 sˆ20 (r , 0) = 0.001 θˆ3 (0) = 1 kˆ p (0) = 1

Fig. 3. The control input v(t ) for known G(s).

Fig. 6. The control input v(t ) for unknown G(s).

Fig. 4. The genuine plant output tracking error y(t ) − ym (t ) for known G(s).

control input is shown in Fig. 3. The genuine output tracking error is shown in Fig. 4. It can be seen that the tracking error of y(t )− ym (t ) still remains. The tracking error between the estimated output yˆ 1 (t ) and ym (t ) is shown in Fig. 5. It can be seen that, as guaranteed by Theorem 1, zero tracking of the error yˆ 1 (t ) − ym (t ) can be obtained. Simulation 2. Simulation results for unknown G(s). The chosen design parameters and initial values in (57)–(66) are shown in Table 1. The convergence of the estimated parameters in (57)–(66) has been confirmed via simulations. The control input is shown in Fig. 6. The genuine output tracking error is shown in Fig. 7. It can be seen that the tracking error of y(t )− ym (t ) still remains. The tracking error between the estimated output yˆ (t ) and ym (t ) is shown in Fig. 8. It can be seen that, as guaranteed by Theorem 2, zero tracking of the error yˆ (t ) − ym (t ) can be obtained.

Fig. 7. The genuine plant output tracking error y(t ) − ym (t ) for unknown G(s).

Remark 8. The proposed adaptive method is an overparameterization scheme (e.g. p(r )θ1 , s(r )θ2 , s(r )θ20 ). Since the goal of this paper is to show the controller design strategy in a simple setting that reveals its essential features, the overparameterization issue will be further investigated in the future

204

X. Chen et al. / Automatica 64 (2016) 196–207

By collecting the bounded terms in the right hand side of the inequality (A.3) together, it can be seen that there exists a uniformly bounded function λ1 (t ) such that 1

f1 (t ) ≤ λ1 (t ) +

+

s + a0  ∞ 1 0

s + a0

1

[|kp ξ1 |](t ) +

s + a0

[|¯ε1 ζ2 |](t )dr +



∞

[|¯ε1 ζ1 |](t ) 1 s + a0

0

[|¯ε1 ζ3 |](t )dr . (A.5)

An upper bound will be derived for the right hand side of (A.5) in the following. Now, introduce two filters defined as ∗

Fig. 8. The error yˆ (t ) − ym (t ) for unknown G(s).

sK1 (s) = 1 − K (s),

developments, similar to the development of the backstepping control technique. 5. Conclusions

K (s) =

an

(A.6)

(s + a)n∗

where a > 0 is a parameter which will be determined. With Z (s) G(s) = kp P (s) , using −a0 K1 (s) + (s + a0 )K1 (s) = 1 − K (s), it yields g (t ) + a0 K1 (s)[g ](t ) − K1 (s)[u](t ) = K (s)G−1 (s)

This paper has discussed the model reference adaptive control for the continuous-time linear system in the presence of both actuator and sensor hysteresis, where the hysteresis is described by Prandtl–Ishlinskii model. New adaptive control schemes with actuator uncertainty and sensor uncertainty compensations are developed for linear plants with either known or unknown dynamics, where adaptive estimates of the genuine outputs of the plants are simultaneously generated. The proposed control laws ensure the uniform boundedness of all signals in the closed-loop systems. The tracking error between the estimated plant output and the desired output is guaranteed to converge to zero asymptotically, whereas the genuine output tracking error of the plant may not be able to be proved to approach to zero asymptotically.

[y](t ). (A.7)

By using Fr [v](t ) = v(t ) − Er [v](t ) (relation (13)), observing Er [.](t ) ≤ r and employing Lemma 3, it can be seen that there exists a uniformly bounded signal λ2 (t ) such that the input relation in (24) can be rewritten as a(s) a(s) [v](t ) + sˆ(t )θ2T [z ](t ) Λ(s) Λ(s) + θ20 sˆ(t )z (t ) + λ2 (t )

pˆ (t )v(t ) = pˆ (t )θ1T

(A.8)

with pˆ (t ) = pˆ 00 (t ) +

Acknowledgments

1 s + a0

∞



pˆ 0 (r , t )dr ,

sˆ(t ) = 1 +

0

∞



sˆ0 (r , t )dr . 0

(A.9) This work is partially supported by the Grants-in-Aid for Scientific Research of Japan Society for the Promotion of Science (JSPS) (No. C-24560553; No. C-15K06152; No. 14032011-000073), Funds for Natural Science Foundation of China (NSFC) (No. 61228301; No. U1201244; No. 613111143), the Program of Pearl River Young Talents of Science and Technology in Guangzhou (2013J2200100), and State Key Laboratory of Synthetical Automation for Process Industries (Northeastern University), China. Appendix A. Proof of Theorem 1

By substituting (16) into (A.8), dividing the both sides with (t ) pˆ∞ , observing Er [.](t ) ≤ r, and using Lemma 1 and p(r )dr p + 0

a(s) a(s) [u](t ) + ν(t )θ2T [z ](t ) Λ(s) Λ(s) + θ20 ν(t )z (t ) + λ3 (t ).

u(t ) = θ1T

yˆ 1 (t ) = ym (t ) + ε1 (t ) − kp ξ1 (t ).

(A.1)



Introduce two fictitious signals 1 s + a0

[|ˆy1 |](t ),

g (t ) =

1 s + a0

[u](t ),

(A.2)

f1 ( t ) ≤

s + a0

[|ym |](t ) +

1 s + a0

[|ε1 |](t ) +

1 s + a0

[|kp ξ1 |](t ). (A.3)

By the definition of m1 (t ), it ca be easily checked that



|ε1 (t )| ≤ |¯ε1 (t )| 1 + |ζ1 (t )| +  ∞  + |ζ3 (r , t )|dr ,

|ζ2 (r , t )|dr 0

(A.4)

0

ε (t ) where ε¯ 1 (t ) is defined as ε¯ 1 (t ) = √m1 (t ) ∈ L2 ∩ L∞ . 1

a(s)

(s + a0 )



[g ](t ) Λ(s)   a(s) = K1 (s)ν(·)θ2T + K1 (s)θ20 ν(·) [z ](t ) Λ(s) 1 s + a0

[y](t ).

(A.11)

Then, by substituting (19) into (A.11) and observing Er [.](t ) ≤ r, it can be seen that there exists a uniformly bounded signal λ4 (t ) such that



∞





1 + K1 (s) a0 − θ1T

+ K1 (s)[λ3 ](t ) + K (s)G−1 (s)

where a0 is a positive constant. Thus, 1

(A.10)

Thus, operating by K1 (s) on the both sides of (A.10) and substituting it into (A.7) gives

From (29) and (34), it gives

f1 ( t ) =

0

Lemma 4, it can be seen that there exist a uniformly bounded function ν(t ) and a uniformly bounded signal λ3 (t ) such that



1 + K1 (s) a0 − θ1T

a(s)

(s + a0 )



[g ](t ) Λ(s)  a(s) = K1 (s)ν(·)θ2T + K1 (s)θ20 ν(·) Λ(s) 1 1  ∞ + K (s)G−1 (s) [z ](t ) + λ4 (t ). s + a0 s¯0 − 0 s¯(r )dr

(A.12)

X. Chen et al. / Automatica 64 (2016) 196–207

Let a > 0 in K (s) and K1 (s) be constant and large enough such that





a(s)

1 + K1 (s) a0 − θ1T Λ(s) (s + a0 )

−1

is stable and proper (see Tao,

2003). Then, (A.12) implies that g (t ) = T1 (s, ·)[z ](t ) + λ5 (t ),

(A.13)

where T1 (s, ·) is a stable and strictive proper operator, and λ5 (t ) is a uniformly bounded signal. Thus, by Lemma 1, there exist a stable and strictive proper operator T2 (s, t ) and a uniformly bounded signal λ6 (t ) such that

(s + a0 )−1 [v](t ) = T2 (s, ·)[z ](t ) + λ6 (t ).

(A.14)

Since Lemma 4 implies that the boundedness of (s + a0 )−1 [z ](t ) depends on the boundedness of f1 (t ), and in turn, the boundedness of (s + a0 )−1 [y](t ) depends on the boundedness of f1 (t ) from Lemma 2, it can be  ∞seen that the boundeness of the terms 1 [|¯ε1 ζ1 |](t ) and 0 s+1a [|¯ε1 ζ2 |](t )dr in the right hand side of s+a 0

0

inequality (A.5) depends on the boundedness of (s + a0 )−1 [f1 ](t ). Then, relation (A.14) implies that the boundedness of g (t ) depends on the boundedness of f1 (t ). Since relation (22) means that the boundedness of y(t ) depends on the boundedness of g (t ) and (s + a0 )−1 [y](t ) (which is obvious by dividing both sides with θ3 Pm (s)), it can be concluded that the boundedness of y(t ) depends on the boundedness of f1 (t ). Thus, the boundedness of z (t ) also depends on the boundedness of f1 (t ). ∞ Therefore, the boundeness of the term 0 s+1a [|¯ε1 ζ3 |](t )dr in 0 the right hand side of inequality (A.5) depends on the boundedness of (s + a0 )−1 [f1 ](t ). Now, let us estimate the upperbound of the term s+1a [|kp ξ1 |](t ) 0 in the right hand side of inequality (A.5). For an arbitrary signal ∞ 2 ∞ δ(t ) and a ∗signal σ (t )∗ ∈ L with σ˙ (t ) ∈ L ∩ L , by denoting Pm (s) = sn + pn∗ −1 sn −1 + · · · + p1 s + p0 , it gives

σ (t )Pm−1 (s)[δ](t ) − Pm−1 (s)[σ δ](t ) n∗ −1

  1 s + p n ∗ −1 s + · · · + p2 s + p1 σ˙ [δ] (t ) = Pm (s) Pm (s)   n∗ −2 n∗ −3 s + p n ∗ −1 s + · · · + p2 s + σ˙ [δ] (t ) Pm (s) Pm (s)   n ∗ −2 s + p n ∗ −1 s + ··· + σ˙ [δ] (t ) Pm (s) Pm (s)   ∗ +

1

Pm (s)

σ˙

s n −1

Pm (s)

[δ] (t ).

t



e−α3 (t −τ ) x1 (τ )f2 (τ )dτ

|Fr [δ](t2 ) − Fr [δ](t1 )| ≤ sup |δ(t ) − δ(t1 )|

(A.18)

t1 ≤ t ≤ t2

for arbitrary 0 ≤ t1 ≤ t2 . Then, by (17) and (A.18), it can be seen that z (t ) is uniformly continuous. By applying (A.18) again, it yields that Fr [z ](t ) is uniformly continuous. Thus, from (36), it can be concluded that ε1 (t ) is uniformly continuous. Therefore, limt →∞ ε1 (t ) = 0, and in turn, limt →∞ e1 (t ) = 0 as from Barbalat Lemma. This completes the proof. Appendix B. Proof of Theorem 2 From (52) and (53), it gives yˆ (t ) = ym (t ) + ε(t ) − kˆ p (t )ξ (t ).

(B.1)

For the fictitious signal defined as 1

f (t ) =

s + a0

[|yˆ |](t ),

(B.2)

it yields 1 s + a0

[|ym |](t ) +

1 s + a0

[|ε|](t ) +

1 s + a0

[|kˆ p ξ |](t ). (B.3)

     1   |ε(t )| ≤ |¯ε (t )| 1 + ζ4 (t ) + ζ5 (t ) +  [q](t ) + |ξ (t )| (B.4) P m ( s) with

ζ4 ( t ) = (A.15)

(A.16)

for some x0 (t ) ∈ L∞ , x1 (t ) ∈ L2 ∩ L∞ with x1 (t ) ≥ 0, T3 (s, ·) being a stable and strict proper operator, T4 (s, ·) being a stable and proper operator with a non-negative impulse response function. By introducing f2 (t ) = T4 (s, ·)[f1 ](t ), operating T4 (s, ·) on both sides of (A.16), noting that T4 (s, ·) has a non-negative impulse response function, it gives f2 (t ) ≤ α1 + α2

p˙ˆ 0 (r , t ) ∈ L2 in (41), s˙ˆ0 (r , t ) ∈ L2 in (42). By the definition of ξ1 (t ) and (A.15), it can be seen that ξ˙1 (t ) ∈ L∞ and ξ1 (t ) ∈ L2 . Therefore, limt →∞ ξ1 (t ) = 0 as from Barbalat Lemma. By using u(t ) ∈ L∞ and y(t ) ∈ L∞ , it gives y˙ (t ) ∈ L∞ from (22) (which is obvious by dividing both sides of (22) with θ3 Pm (s) and then multiplying both sides with s). Thus, y(t ) is uniformly continuous. For the play operator and a signal δ(t ), it holds (Theorem 2.3.2 in Brokate and Sprekels (1996))

By the definition of m0 (t ), it can be easily derived that

By Lemma 2, (A.15) and the abovementioned results, it can be seen that the boundedness of ξ1 (t ) depends on the boundedness of f1 (t ). By combining the above results, from (A.5), it gives f1 (t ) ≤ x0 (t ) + T3 (s, ·)[x1 T4 (s, ·)[f1 ]](t )

in (16). Therefore, y(t ) ∈ L∞ , yˆ 1 (t ) ∈ L∞ , ε1 (t ) ∈ L∞ , e(t ) ∈ L∞ , m1 (t ) ∈ L∞ . ε (t ) Furthermore, by using |¯ε1 (t )| = √m1 (t ) ∈ L2 ∩ L∞ and m1 (t ) ∈ 1 L∞ , it can be seen that ε1 (t ) ∈ L2 . Thus, p˙ˆ 00 (t ) ∈ L2 in (40),

f (t ) ≤

n ∗ −2

205

(A.17)

     1   1 a(s)      [ F [v]]( t ) dr + [v]( t )   P (s) Λ(s)   P (s) r m m 0   ∞  1 a( s )    +  P (s) Λ(s) [Fr [v]](t ) dr m 0 

∞

and

ζ5 ( t ) =

   1 a(s)  |Fr [z ](t )|dr +  [z ](t ) Pm (s) Λ(s) 0   ∞  1 a( s )    +  P (s) Λ(s) [Fr [z ]](t ) dr m 0   ∞  1  1   dr +| [z ](t )| + [ F [ z ]]( t ) r   Pm (s) Pm (s) 0 

∞

(B.6)

ε(t ) where ε¯ (t ) is defined as ε¯ (t ) = √m (t ) ∈ L2 ∩ L∞ . 0

By collecting the bounded terms in the right hand side of the inequality (B.3) together, it can be seen that there exists a uniformly bounded function β1 (t ) such that

0

for some α1 , α2 , α3 > 0. By observing x1 (t ) ∈ L2 ∩ L∞ and applying the well-known Groneall-Bellman Lemma to (A.17), it can be concluded that f2 (t ) ∈ L∞ , and in turn, f1 (t ) ∈ L∞ in (A.16), z (t ) ∈ L∞ , g (t ) ∈ L∞ in (A.14), u(t ) ∈ L∞ in (A.10), v(t ) ∈ L∞

(B.5)

f (t ) ≤ β1 (t ) +

+

1 s + a0

1 s + a0

[|kˆ p ξ | + |¯ε ξ |](t )

[|¯ε ζ4 |](t ) +

1 s + a0

[|¯ε ζ5 |](t ).

(B.7)

206

X. Chen et al. / Automatica 64 (2016) 196–207

An upper bound will be derived for the right hand side of (B.7) in the following. By using Fr [v](t ) = v(t ) − Er [v](t ) (relation (13)), observing Er [.](t ) ≤ r and employing Lemma 6, it can be seen that there exists a uniformly bounded signal β2 (t ) such that the input relation in (49) can be rewritten as



∞



pˆ (r , t )dr



v(t )  ∞ a(s) a(s) T ˆ = θ1 (t ) pˆ T1 (r , t )dr [v](t ) + [v](t ) Λ(s) Λ (s) 0  ∞ a(s) a(s) sˆT2 (r , t )dr + θˆ2T (t ) [z ](t ) + [z ](t ) Λ(s) Λ(s) 0  ∞ sˆ20 (r , t )drz (t ) + β2 (t ). + θˆ20 (t )z (t ) +

1+

0

(B.8)

0

By substituting (16) into (B.8), dividing the both sides with ∞ 1+ 0 pˆ (r ,t )dr ∞ , p0 + 0 p(r )dr

observing Er [.](t ) ≤ r, and using Lemma 6, it can

be seen that there exist two uniformly bounded vectors ν2 (t ) and ν3 (t ), a uniformly bounded function ν4 (t ), and a uniformly bounded signal β3 (t ) such that u(t ) = ν2T (t )

a(s)

[u](t ) + ν3T (t )

Λ(s) + ν4 (t )z (t ) + β3 (t ).

a( s )

Λ(s)

[z ](t ) (B.9)

Based on (B.9), similar to the procedure of obtaining (A.11), it can be concluded that there exists a uniformly bounded signal β4 (t ) such that    a(s) (s + a0 ) [g ](t ) 1 + K1 (s) a0 − ν2T (·) Λ(s)  a ( s ) = K1 (s)ν3T (·) + K1 (s)ν4 (·) Λ(s)

+

1 s¯0 −

∞ 0

s¯(r )dr

K (s)G−1 (s)

1 s + a0

 [z ](t ) + β4 (t ).

(B.10)

Let a > 0 in K (s) and K1 (s) be constant and large enough such that





−1

a(s)

1 + K1 (s) a0 − ν2T (·) Λ(s) (s + a0 )

is stable and proper (Tao,

2003). Then, (B.10) implies that g (t ) = T5 (s, ·)[z ](t ) + β5 (t ),

(B.11)

where T5 (s, t ) is a stable and strictive proper operator, and β5 (t ) is a uniformly bounded signal. Thus, by Lemma 1, there exist a stable and strictive proper operator T6 (s, t ) and a uniformly bounded signal β6 (t ) such that 1 s + a0

[v](t ) = T6 (s, ·)[z ](t ) + β6 (t ).

(B.12)

Thus, similar to the proof of Theorem 1, it can be seen that the boundeness of the terms s+1a [|¯ε ζ4 |](t ) in the right hand side of 0

inequality (B.7) depends on the boundedness of s+1a [f ](t ). 0 Similarly, it can also be proved that the boundeness of the term 1 [|¯ε ζ5 |](t ) in the right hand side of inequality (B.7) depends on s+a 0

the boundedness of s+1a [f ](t ). 0 Similarly, by Lemma 6, (A.15) and the abovementioned results, it can be proved that the boundedness of ξ (t ) depends on the boundedness of f (t ). By combining the above results, from (B.7), it gives f (t ) ≤ x0 (t ) + T7 (s, ·)[x2 T8 (s, ·)[f ]](t )

(B.13)

for some x0 (t ) ∈ L , x2 (t ) ∈ L ∩ L with x2 (t ) ≥ 0, T7 (s, ·) being a stable and strict proper operator, T8 (s, ·) being a stable and proper ∞

2

∞

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Xinkai Chen received his Ph.D. degree in engineering from Nagoya University, Japan, in 1999. He is currently a professor in the Department of Electronic and Information Systems, Shibaura Institute of Technology, Japan. His research interests include adaptive control, smart materials, hysteresis, sliding mode control, machine vision, and observer. Dr. Chen has served as an Associate Editor of several journals, including IEEE Transactions on Automatic Control, IEEE Transactions on Control Systems Technology, IEEE/ASME Transactions on Mechatronics, European Journal of Control, etc. He has also severed for international conferences as organizing committee members including Program Chairs, Program Co-Chairs, etc.

207 Ying Feng received her B.S. and M.S. degrees in electrical engineering from Zhejiang University, China, in 2000 and 2003, respectively, and Ph.D. degree in control engineering from South China University of Technology, China, in 2006. She joined the School of Automation Science and Engineering in South China University of Technology, China since 2011. Her research interests include adaptive robust control, smart materials actuators control and mechatronics.

Chun-Yi Su received the Ph.D. degree in control engineering from the South China University of Technology, Guangzhou, China, in 1990. He joined Concordia University, Montreal, QC, Canada, in 1998, after a seven-year stint with the University of Victoria, Victoria, BC, Canada. He is with the College of Mechanical Engineering and Automation, Huaqiao University, on leave from Concordia University, Canada. His current research interests include the application of automatic control theory to mechanical systems. He is particularly interested in control of systems involving hysteresis nonlinearities. He has authored or co-authored over 300 publications in journals, book chapters, and conference proceedings. Dr. Su has served as an Associate Editor of the IEEE Transactions on Automatic Control, the IEEE Transactions on Control Systems Technology, and the Journal of Control Theory and Applications. He has been on the Editorial Board of 18 journals, including the IFAC Journal of Control Engineering Practice and Mechatronics. He served for many conferences as an Organizing Committee Member, including the General Co-Chair of the IEEE International Conference on Mechatronics and Automation in 2012, and the Program Chair of the IEEE Conference on Control Applications in 2007.