Sliding mode output feedback control based on tracking error observer with disturbance estimator.

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Research Article

Sliding mode output feedback control based on tracking error observer with disturbance estimator Lingfei Xiao a,n, Yue Zhu b a Jiangsu Province Key Laboratory of Aerospace Power Systems, College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China b College of Engineering, Nanjing Agricultural University, Nanjing, China

art ic l e i nf o

a b s t r a c t

Article history: Received 9 December 2013 Received in revised form 17 March 2014 Accepted 3 April 2014 This paper was recommended for publication by Dr. Jeff Pieper

For a class of systems who suffers from disturbances, an original output feedback sliding mode control method is presented based on a novel tracking error observer with disturbance estimator. The mathematical models of the systems are not required to be with high accuracy, and the disturbances can be vanishing or nonvanishing, while the bounds of disturbances are unknown. By constructing a differential sliding surface and employing reaching law approach, a sliding mode controller is obtained. On the basis of an extended disturbance estimator, a creative tracking error observer is produced. By using the observation of tracking error and the estimation of disturbance, the sliding mode controller is implementable. It is proved that the disturbance estimation error and tracking observation error are bounded, the sliding surface is reachable and the closed-loop system is robustly stable. The simulations on a servomotor positioning system and a five-degree-of-freedom active magnetic bearings system verify the effect of the proposed method. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Sliding mode output feedback control Tracking error observer Extended disturbance estimator Differential sliding surface Five-DOF AMBs system Servo motor

1. Introduction Sliding mode control (SMC) is an effective control method with strong robustness [1–4]. For over fifty years, SMC has been widely studied and extensively employed in many applications [5–14]. Usually, there are two main design steps in SMC scheme. The first step is to select an appropriate sliding surface, which is with desired performance and can be achieved. The second step is to organize a satisfying control law, so that the system states can be forced to reach the sliding surface in a finite amount of time and remain on the sliding surface after achieving. In [5], a sliding mode controller design scheme was proposed to compensate for the nonlinear effects of the active magnetic bearings system with flexible rotor. Based on the sophisticated mathematical models of a class of control non-affine nonlinear systems, [6] designed a sliding mode controller by using a sequence of linear time-varying systems to approximate the nonlinear systems. Lin et al. [7] gave an intelligent double integral sliding mode controller on the basis of neural network, and enough data was required in order to train neural network. Yang et al. [8] developed a SMC approach for systems with mismatched uncertainties via a nonlinear disturbance observer, which required

n

Corresponding author. Tel.: þ 86 15005170182. E-mail addresses: [email protected] (L. Xiao), [email protected] (Y. Zhu).

the disturbances to be vanishing. Ginoya et al. [9] carried forward the result in [8] by using an extended disturbance observer. It is worth to point out that the results in [5–9] were given on the basis of full states feedback. However, in lots of practical applications, the full states information is not always available, merely the outputs are obtainable. Therefore, many researchers have paid attention to the output feedback control issue. In [10], a static output feedback SMC for uncertain linear systems was designed. A dynamic output feedback sliding mode control algorithm was presented in [11] by introducing an additional dynamics into the design of the sliding surface. Bartolini and Punta [12] considered the sliding mode output feedback stabilization of uncertain systems by designing a full-order observer. A second-order sliding-mode observer was used in [13] to realize the output feedback trajectory-tracking control for an uncertain DC servomechanism system. All of results in [10–13] were on the assumption that the bounds of disturbances are known. Godbole et al. [14] gave the performance analysis of generalized extended state observer in handling fast-varying sinusoidal disturbances, but it only considered the constant control gain in the systems. Therefore, in this paper, for a class of systems who suffers from vanishing or nonvanishing disturbances, under the condition that the bound of disturbance is unknown and states are not fully available, a novel output feedback tracking control method is presented on the basis of sliding mode control strategy.

http://dx.doi.org/10.1016/j.isatra.2014.04.001 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Xiao L, Zhu Y. Sliding mode output feedback control based on tracking error observer with disturbance estimator. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.04.001i

L. Xiao, Y. Zhu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

The primary contributions of this paper are as follows: (1) The mathematical model of the system can be only of moderate accuracy, either linear or nonlinear, and the control gain in the system can be a function of output not required to be a constant. (2) The disturbance may be vanishing or nonvanishing, and the bound of disturbance is unknown. (3) An original tracking error observer with disturbance estimator is created to obtain the observation of tracking error, and a novel sliding mode output feedback controller on the basis of a differential sliding surface is constructed. (4) It is proved that the disturbance estimation error and tracking observation error are bounded, the sliding surface is reachable and the closed-loop system is robustly stable. The remainder of this paper is organized as follows. Section 2 is the problem formulation, in which the considered system with disturbances is given. Section 3 produces the main results, including the design of differential sliding surface, the sliding mode controller, the tracking error observer, the analysis of the convergence of observation error and the robustness of the closedloop system. The block diagram of the proposed sliding mode output feedback control based on tracking error observer with disturbance estimator method (SMOFCEDO) is given in Section 3 as well. Simulation on a servomotor positioning system and a fivedegree-of-freedom (five-DOF) active magnetic bearings (AMBs) system is shown in Section 4. Section 5 draws the conclusions of this paper.

Remark 2.1. As shown in (7), the zero/first/second order derivatives of disturbance should be bounded by μ, but it is not required to know μ. Remark 2.2. Clearly, the model (1) is simple that means the exact mathematical model of the system is not required, the system can be linear or nonlinear. Based on (1), it will show that the method presented by us in the following is not very sensitive to the accuracy of mathematical model. Besides, it is obvious that the control gain b(y) in the system (1) is a function of output, b(y) is not required to be a constant. Remark 2.3. Throughout this paper, only one degree-of-freedom (DOF) system (1) is considered in the following. However, all of results can be extended to higher DOF systems conveniently. It is worth to point out that coupling terms among each DOF are included in d(t) when the systems are of higher DOF. In order to show the favorable expansibility of our method, simulations on a one-DOF servomotor positioning system and a five-DOF active magnetic bearings system are given in Section 4. Note: In the following formulas, for the sake of conciseness, the independent variables of functions are neglected in most of the cases when there is no confusion, such as f(z) is replaced by f, except when it is necessary to show independent variables in corresponding functions distinctly so to avoid misunderstanding, for example, f ðz^ Þ is different from f(z).

2. Problem formulation 3. Main results Consider a single input single output dynamical system described by y€ ¼ aðyÞ þ bðyÞu þ dðtÞ

ð1Þ

where y is the measurable output, u is the control signal. a(y), b(y) are known linear or nonlinear functions with respect to y. d(t) is the lumped disturbance, including external disturbance, parameter uncertainties, unmodelled dynamics and so on. Let e ¼ y yr be the tracking error for y, with yr being reference trajectory. Then e_ ¼ y_ y_ r , and according to (1), gives e€ ¼ y€ y€ r ¼ ðaðeÞ þbðeÞu þdÞ y€ r

ð2Þ

_ , (2) can be written into state-space Define η ¼ ½η1 ; η2 ¼ ½e; e form as " # η2 0 η_ ¼ þ d ð3Þ aðeÞ þbðeÞu y€ r 1 T

T

Let Bd ¼ ½0; 1T and " #

ψ9

η2

aðeÞ þ bðeÞu y€ r

" ¼

ψ1 ψ2

# ð4Þ

3.1. Differential sliding surface design

ð5Þ

s0 ¼ sη

ð6Þ

where s ¼ ½s1 ; 1, s1 4 0 should be chosen to guarantee that the state vector η on the sliding surface S 0 ¼ fηjs0 ðηÞ ¼ 0g has satisfying performance. Secondly, introduce a new sliding mode function as

Firstly, define sliding mode function for system (3) as

then system (3) turns to

η_ ¼ ψ þ Bd d The corresponding nominal system of (5) is

η_ ¼ ψ

Assumption 2.1. y_ is unmeasurable, bðyÞ a 0, yr is known and its … first/second/third order derivatives y_ r , y€ r , y r are known as well. Assumption 2.2. The lumped disturbance d(t) is continuous and satisfies dj d ð7Þ j r μ for j ¼ 0; 1; 2 dt where

μ is a positive scalar.

The aim of this paper is to realize that the tracking error system (13) is robustly stable, under the circumstances that only output y is measurable, while the bound of disturbance d(t) is unknown, and d(t) can either be vanishing disturbance or nonvanishing disturbance. Because SMC is an effective control method with strong robustness, SMC is employed to complete the design objective. According to SMC theory [4], there are two steps to complete the control system design. Firstly, a sliding surface with desired performance is required; secondly, a suitable control law is needed to drive system states to the sliding surface and remain on it thereafter. In this section, a differential sliding surface is produced at first, then an integral form of controller is proposed. Next, consider that the lumped disturbance d is unknown and y_ is unmeasurable, a novel tracking error observer with extended disturbance estimator is developed. As a result, the controller can be implemented. Then, it is proved that the disturbance estimation error and tracking observation error are bounded, the sliding surface is reachable and the closed-loop system is robustly stable.

ð8Þ

s ¼ sψ þ s2 s0

ð9Þ

where s2 4 0. Therefore, the sliding surface is S ¼ fηjsðηÞ ¼ 0g. Substitute (8) into (9) yields " # " # s ¼ ½s1 ; 1

ψ1 η1 ψ 2 þ s2 ½s1 ; 1 η2

¼ s1 ψ 1 þ ψ 2 þ s2 s1 η1 þ s2 η2

ð10Þ



3

On the sliding surface S, consider the nominal dynamic (6), there exist η_ 1 ¼ η2 ¼ ψ 1 , η€ 1 ¼ η_ 2 ¼ ψ 2 , then (10) turns to

similar approximation methods which probably result in the decreasing of robustness.

s ¼ η_ 2 þ ðs1 þ s2 Þη2 þ s2 s1 η1 ¼ η€ 1 þ ðs1 þ s2 Þη_ 1 þ s2 s1 η1

3.3. Tracking error observer design

ð11Þ

Therefore, one can select suitable s1 and s2 to realize that the state vector η has favorable dynamics on the sliding surface S. Remark 3.1. Compared with traditional integral sliding surface Rt sISMC ¼ η_ 1 þc1 η1 þ c2 0 η1 dτ , (11) is similar to the derivative of sISMC with c1 ¼ s1 þ s2 and c2 ¼ s2 s1 . Therefore, the sliding surface (9) is named “differential sliding surface” in this paper.

Let f ðz; y€ r ; vðzÞÞ ¼ Aðz; y€ r Þ þ Bv, then the system (13) turns to z_ ¼ f ðz; y€ r ; vðzÞÞ þBda d

ð20Þ

and the output of the system can be written as yðzÞ ¼ ½1; 0; 0z 9 hðzÞ Therefore, constructing the following tracking error observer: z^_ ¼ f ðz^ ; y€ r ; vðz^ ÞÞ þ P o ðy hðz^ ÞÞ þ Bda d^

3.2. Sliding mode controller design

ð21Þ

31

Consider system (3), introduce the following augmented system: 2 3 2 3 2 3 ψ1 0 0 6 7 6 7 6 7 z_ ¼ 4 ψ 2 5 þ 4 0 5v þ 4 1 5d ð12Þ 0 1 0 _ _ uT , v ¼ u. where z ¼ ½e; e; Let Aðz; y€ r Þ ¼ ½ψ 1 ; ψ 2 ; 0T , B ¼ ½0; 0; 1T , Bda ¼ ½0; 1; 0T , rewrite system (12) to z_ ¼ Aðz; y€ r Þ þBv þ Bda d

ð13Þ

According to (4), (10) can be rewritten to

2

where " # d^ ξ^ ¼ ^ ; d_

" p¼

p1 p2

# ;

" q¼

q1

"

#

q2

;

Q¼

q1

1

q2

0

#

^ _ p is the internal state of the where d_ is the estimations of d, estimator, q1 4 0 and q2 4 0 are the user selectable positive constants.

s ¼ s1 η2 þ ðaðeÞ þ bðeÞu y€ r Þ þ s2 s1 η1 þ s2 η2 2 3 e 6_7 e ¼ ½s2 s1 ; s2 þ s1 ; bðeÞ4 5 þ ðaðeÞ y€ r Þ u ¼ lðeÞz þ ðaðeÞ y€ r Þ

ð14Þ

where lðeÞ 9 ½s2 s1 ; s2 þ s1 ; bðeÞ. Hence, because of (13), the derivative of (14) is …

s_ ¼ lðeÞ0 z þ lðeÞz_ þ aðeÞ0 y r … ¼ lðeÞðAðz; y€ r Þ þ Bv þBda dÞ þ ðlðeÞ0 z þ aðeÞ0 y r Þ

where P o A R is a constant designable matrix. In the following, an extended disturbance estimator is created to obtain d^ in order to complete the tracing error observer design. _ T and design an extended disturbance estimator for Let ξ ¼ ½d; d ξ as 8 < ξ^ ¼ p þ qη 2 ð22Þ : p_ ¼ qψ þ Q ξ^

3.4. Stability analysis 3.4.1. Stability analysis of extended disturbance estimator According to (22), based on (3) and (4), the derivative of ξ^ is yielded as _

…

¼ lðeÞAðz; y€ r Þ þ lðeÞBv þ lðeÞBda d þ ðlðeÞ0 z þ aðeÞ0 y r Þ

ð15Þ

…

Since lðeÞB ¼ bðeÞ, lðeÞBda ¼ s1 þ s2 , let ϕðz; y€ r ; y r Þ ¼ lðeÞAðz; y€ r Þ þ … lðeÞ0 z þ aðeÞ0 y r , (15) can be simplified to …

s_ ¼ bðeÞv þ ðs1 þ s2 Þd þ ϕðz; y€ r ; y r Þ

ð16Þ

Following the reaching law approach [2–4], the controller can be chosen as …

vðzÞ ¼ bðeÞ 1 ½ qv s δv sgnðsÞ ϕðz; y€ r ; y r Þ ðs1 þ s2 Þd

ð17Þ

where qv 4 0 and δv 40 are reaching law parameters. However, to implement the control law, it needs the values of _ To obtain the estimation of d and y, _ a the uncertainty d and y. novel tracking error observer with an extended disturbance estimator will be developed in Section 3.3. Using the estimations of y, y_ and d into (17), the resulting controller takes the form as … ^ ^ 1 ½ qv sðz^ Þ δv sgnðsðz^ ÞÞ ϕðz^ ; y€ r ; y r Þ ðs1 þ s2 Þd vðz^ Þ ¼ bðeÞ

ð18Þ

^ z^ , d^ are the estimations of e, z, d, respectively. where e, Hence, controller (18) is designated as an observer based output feedback controller and then it is implementable. Therefore, according to (12), the control signal uðz^ Þ is Z t uðz^ Þ ¼ vðz^ Þ dτ ð19Þ 0

Remark 3.2. From (19), it is clear that the influence of sign function sgnðÞ is smoothed by the integral effect. Such an approach is not the same as using saturation function satðÞ or

ξ^ ¼ p_ þ qη_ 2 ¼ qψ 2 þ Q ξ^ þ qðψ 2 þ dÞ ¼ Q ξ^ þ qd Define the estimation error as ξ~ ¼ ξ ξ^ and let βξ ¼ ½0; 1T , then " # _ ~ξ_ ¼ ξ_ ξ^_ ¼ d ðQ ξ^ þqdÞ d€ " # ! q 0 0 € 0 1 d d ^þ 1 d Q þ ¼ ξ q2 0 d_ 1 0 0 d_ ¼ Q ξ~ þ β ξ d€

ð23Þ

For (22), one can choose suitable q1 and q2 to make the eigenvalues of Q are in the left half phase (LHP) with satisfying performance, and due to (23), it is always possible to find a positive definite symmetric matrix P ξ such that Q T Pξ þ Pξ Q ¼ Nξ

ð24Þ

for any given positive definite matrix N ξ . Therefore, suppose λξmin denotes the minimum eigenvalue of Nξ , and select Lyapunov T function as V ξ ¼ ξ~ P ξ ξ~ , then T _ _T V_ ξ ¼ ξ~ P ξ ξ~ þ ξ~ P ξ ξ~

€ T P ξ~ þ ξ~ T P ðQ ξ~ þ β dÞ € ¼ ðQ ξ~ þ β ξ dÞ ξ ξ ξ T T € T P ξ~ þ ξ~ T P ðβ dÞ € ¼ ðξ~ Q T P ξ ξ~ þ ξ~ P ξ Q ξ~ Þ þ½ðβξ dÞ ξ ξ ξ T T ¼ ξ~ ðQ T P ξ þ P ξ Q Þξ~ þ 2ξ~ P ξ β ξ d€



4

Because of (7) and (24), there exists

r

T V_ ξ r ξ~ N ξ ξ~ þ 2 J P ξ βξ J J ξ~ J μ r J ξ~ J ðλ J ξ~ J 2 J P β J μÞ

ξmin

ξ ξ

λ2No max 2 γ κ ξ

λNo max

ð25Þ

¼

1

where γ ξ ¼ λξmin ð2 J P ξ β ξ J μÞ. Therefore, the norm of the extended disturbance estimation error ξ~ is ultimately bounded and the bounds can be lowered by choosing q1, q2, P ξ and N ξ appropriately.

ð27Þ

κ

λNo max

λ κ V þ γ κ ξ λNo max o

2 No max 2

Clearly, when λNo max and γ ξ are small, or if κ is large, V_ o o 0 can be guaranteed. Because λNo max is the maximum eigenvalue of No and No can be chosen by the user, meanwhile γ ξ can be lowered by choosing q1, q2, P ξ and Nξ appropriately, it is convenient to realize that the error of observation z~ converges to a small bound of zero. This completes the proof. □

3.4.2. Stability of the tracking error observer ^ e_ e_^ ; u u ^ T ¼ ½e; ~ e_~ ; u ~ T is the error of Define z~ ¼ z z^ ¼ ½e e; T observation, select Lyapunov function as V o ¼ z~ N o z~ , where 33 No A R is a symmetric positive matrix, whose eigenvalues are between λNomin and λNomax , with λNomin and λNomax being known positive constants, λNomin r λNomax .

3.4.3. Reachability of the sliding surface Theorem 2. Under output feedback sliding mode controller (18), the sliding surface S specified by (9) is reachable, and the closed-loop system (13) is robustly stable, if δv is chosen as

Theorem 1. For system (13), under output feedback sliding mode controller (18), with tracking error observer (21) and extended disturbance estimator (22), the error of observation z~ converges to a bound of zero, if there exists a positive constant κ such that the condition (26) is held. N o ðf z P o hz Þ þ ½N o ðf z P o hz ÞT r 2κ I

λNo max

Vo þ

On the basis of (26) and (27), it gives ! λ2No max 2 2κ κ _ Vor γξ Vo þ Vo þ

Hence, after a sufficiently long time, the norm of the estimation error is bounded by J ξ~ J r γ ξ

κ

δv ¼ ðs1 þ s2 Þγ ξ

ð28Þ

Proof. According to sliding mode function (9) and output feedback sliding mode controller (18), on the basis of (16), it gives

ð26Þ

where I A R33 is an identity matrix, fz and hz are evaluated at ðϵo ðζ Þ; vðz^ ÞÞ and ϵo ðζ Þ ¼ ζ z þð1 ζ Þz^ .

s_ ¼ ð qv s δv sgnðsÞ ðs1 þ s2 Þd^ ϕÞ þ ðs1 þ s2 Þd þ ϕ ¼ q s δ sgnðsÞ þ ðs þ s Þd~ v

v

1

2

Proof. According to (20) and (21), the derivative of Vo is T T V_ o ¼ ðz~ N o z~ Þ0 ¼ 2z~ No ðz_ z^_ Þ T ^ ¼ 2z~ N o ½f ðz; y€ r ; vðzÞÞ þ Bda d ðf ðz^ ; y€ r ; vðz^ ÞÞ þ P o ðy hðz^ ÞÞ þBda dÞ T T ¼ 2z~ N o f½f ðz; y€ r ; vðzÞÞ f ðz^ ; y€ r ; vðz^ ÞÞ P o ðy hðz^ ÞÞg þ 2z~ N o Bda d~ "Z # 1 T T ðf P o hz Þ dζ z~ þ 2z~ N o B d~ ¼ 2z~ N o

Z ¼ 0

0

1

Obviously, under the condition (28), when s 4 0, it yields ~ s_ ¼ qv s ðs1 þ s2 Þγ ξ þ ðs1 þ s2 Þd~ ¼ qv s þ ðs1 þ s2 Þð γ ξ þ dÞ r qv s þðs1 þ s2 Þð γ ξ þ γ ξ Þ ¼ qv s

o0

da

z

while s o 0, there exists

T T z~ Lo z~ dζ þ 2z~ N o Bda d~

~ s_ ¼ qv s þ ðs1 þ s2 Þγ ξ þ ðs1 þ s2 Þd~ ¼ qv s þ ðs1 þ s2 Þðγ ξ þ dÞ Z qv s þðs1 þ s2 Þðγ ξ γ ξ Þ ¼ qv s

where Lo 9 No ðf z P o hz Þ þ ½N o ðf z P o hz ÞT . By standard computations and because of (25), there exists 2z~ N o Bda d~ r T

40

κ ~ T ~ λNo max ~ T N ðB dÞ ~ z No z þ ðBda dÞ o da κ λNo max

v zˆ u zˆ

b eˆ t 0

ð29Þ

1

ð30Þ

Select Lyapunov function as V c ¼ 12 s2 , then the derivative of Vc is V_ c ¼ ss_ . According to (29) and (30), it is clear that V_ c o 0 is held.

Controller

qv s zˆ

1

2

sgn s zˆ

zˆ, yr , yr

1

2

dˆ

u zˆ , v zˆ

System

y

a y

b y u d

v zˆ d

y Observer

zˆ Reference

y r , y r , y r , yr

s zˆ

zˆ

Sliding mode function

s0 s

f zˆ, yr , v zˆ ˆ p q 2 p q 2 Qˆ eˆ, eˆ, uˆ

T

,ˆ

Po y h zˆ

ˆ dˆ , d

Bd dˆ

T

zˆ

s

2 0

zˆ, dˆ

Fig. 1. Block diagram of the proposed SMOFCEDO method.



Therefore, the sliding surface S is reachable. Because the dynamic of sliding mode is specified in advance by sliding mode function (9), according to SMC theory [4], once the sliding surface S is reachable, the closed-loop system (13) is robustly stable as a result. This completes the proof. □

0.1 SMOFCEDO ISMCSMO

0.05 0 e (rad)

The block diagram of the proposed sliding mode output feedback control based on the tracking error observer with disturbance estimator method (SMOFCEDO) is given by Fig. 1.

5

−0.05 −0.1 −0.15 −0.2

4. Simulation

−0.25

In this section, a servomotor positioning system and a fivedegree-of-freedom active magnetic bearings system are considered. In order to make comparison with the presented SMOFCEDO method, integral sliding mode control method with sliding mode observer (ISMCSMO) is applied to the two systems as well.

0

1

2

3

4

5

6

Time (Sec)

Fig. 3. Tracking error (e). 0.04 SMOFCEDO ISMCSMO

4.1. Servomotor positioning system

0.02

Consider the dynamic of servomotor positioning system [15] b J

1 J

ð31Þ

e˜ (rad)

k J

θ€ ¼ s θ s θ_ ds sgnðθ_ Þ þ τ

0

where measurable output θ is the motor rotation angle, control signal τ is the motor torque, J ¼ 4:0 10 4 kg m2 is the total inertia, ks ¼ 0:17 N m=rad is the spring constant, bs is the unknown viscous friction coefficient, ds is the unknown dry friction coefficient. The reference model is chosen as " # " #" # " # 0 1 0 θm θ_ m ¼ þ ð32Þ € ω2 c ω2 2ζωn θ_ m

n

θm (rad)

1 0.8 0.6 0.4 0.2 0 1

2

3

4

5

6

θ˙m (rad/s)

2 1.5 1 0.5 0 0

1

2

−0.06 −0.08

0

1

2

3

4

5

6

~ Fig. 4. Observation error ðeÞ.

n

where ωn ¼ 5 rad=s, ζ ¼1 and the command c ¼1 rad. The trajectories of θm ; θ_ m are shown in Fig. 2. In order to do simulation, assume that bs ¼ 0:01 N m s=rad and ds ¼ 0:1 s 1 . By comparing (31) with (1), according to (32) and (2), one can ^ d ¼ ðbs =JÞθ_ ds sgnðθ_ Þ. find that e ¼ θ θm , e~ ¼ e e, Select s1 ¼ 150, s2 ¼ 10, qv ¼100, γ ξ ¼ 0:15, P o ¼ ½8; 1; 0T , q ¼ ½2000; 1T . In the simulation, the sampling time is T s ¼ 0:005 s. In order to make comparison, integral sliding mode control with sliding mode observer (ISMCSMO) is applied to the servomotor positioning system (31) as well. Choose integral sliding Rt surface sISMC ¼ η_ 1 þ 150η1 þ10 0 η1 dτ, reaching law is s_ ISMC ¼ 100sISMC þ δISMC satðsISMC =0:02Þ, δISMC ¼ 0:15 ð150 þ 10Þ, slid^ ing mode observer is z^ ¼ f ðz^ ; vISMC ðz^ ÞÞ þ ½8; 1; 0T satððe eÞ=0:02Þ. The simulation results are shown in Figs. 3–6. As shown in Fig. 3, the tracking error e converges faster, has lower valley value and possesses better tracking precision under

0

−0.04

3 Time (Sec)

Fig. 2. Reference ðθm ; θ_ m Þ.

4

5

6

0.18 0.16 0.14 0.12 (N.m)

θm

−0.02

0.2

0.1

0.15

0.08

0.1

0.06

0.05

0.04

0

0.02

−0.05

0 −0.02

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 SMOFCEDO ISMCSMO

0

1

2

3 Time (Sec)

4

5

6

Fig. 5. Control signal (u).

10 0 −10 −20 −30 −40 −50

10 0 −10 −20 −30 −40 −50

0.2 0 −0.2

0

1

2

3

5.5

5.6

5.7

4

5.8

5.9

6

5

6

0.2 0 −0.2

0

1

2

3

5.5

4

5.6

5.7

5.8

5.9

5

6

6

Time (Sec)

~ Fig. 6. Disturbance (d) and disturbance estimation error ðdÞ.



6

the presented SMOFCEDO method, compared with ISMCSMO method. From Fig. 4, one can see that the maximum absolute value of observation error e~ is larger and e~ converges slower under ISMCSMO method than those of under the proposed SMOFCEDO method. Fig. 5 illustrates that the control signal u is smoother at the beginning, and converges faster, has lower steady value in SMOFCEDO method. Fig. 6 shows that the disturbance is nonvanishing and at the end of simulation, disturbance estimation error d~ converges in a bound of zero. Relative to the maximum value of d, such a bound is very small. Besides, numerical comparison between SMOFCEDO and ISMCSMO methods about tracking error (e) and observation error ~ is given in Table 1. Obviously, Table 1 illustrates that all values (e) are smaller under SMOFCEDO method than those of with ISMCSMO method. Therefore, the presented SMOFCEDO method is effective and has better performance than ISMCSMO method. 4.2. Five degree-of-freedom active magnetic bearings system The fully suspended five degree-of-freedom (DOF) active magnetic bearings (AMBs) system considered in this subsection is composed of four radial DOF controlled by two identical radial AMBs (RAMBs) and one axial DOF controlled by a thrust AMB (TAMB) [7]. The Schematic view of AMBs system is depicted in Fig. 7. Two RAMBs including left and right RAMBs and one TAMB are fixed on the platform to suspend and regulate the rotor in the radial and axial DOF, respectively. The rotor is assumed to be a rigid and symmetric body, it is at or deviated from the center in X –Y-axes. The nominal air gaps of the RAMBs in X –Y-axes are denoted by variables X b and Y b . The deviations from the nominal air gaps of the left RAMB are denoted by variables X 1 and Y 1 . θX , θY and θZ denote the pitch, yaw and spin angles displacements around the X -, Y-, Z-axes of the rotor. The rotational speed of the rotor is denoted by ω ¼ θ_ Z . la, lb and lc represent the distances from the center of gravity to the left RAMB, right RAMB and external disturbances, respectively, in which l ¼ la þlb . m is the mass of the rotor, g is the gravity constant. f dX ,

f dY and f dZ are the external disturbance forces of the rotor corresponding to the X -, Y-, Z-axes. Detailed introduction of AMBs system can be found in [6,7,16–20], the following basic concepts are adapted from them. The dynamic model of the five-DOF AMBs system can be shown as M X€ þ GX_ ¼ KF þ ED þCg

ð33Þ T

where X ¼ ½X 1 ; X 2 ; Y 1 ; Y 2 ; Z is the state vector, F ¼ ½F X 1 ; F X 2 ; F Y 1 ; F Y 2 ; F Z T is the electromagnetic force vector, D ¼ ½f dX ; f dY ; f dZ T is the external disturbance vector, with f dX ¼ mω2 r sin ðωtÞ, f dY ¼ mω2 r cos ðωtÞ. M ¼ diag½1; 1; 1; 1; 1 is the mass matrix, C ¼ ½0; 0; 1; 1; 0T is the gravity vector. G, K and E are the gyroscope, electromagnetic force and external disturbance matrices, respectively. The details of F, G, K, E and some parameter definitions can be found in the Appendix. Because the electromagnetic force F is of strong nonlinearity, many researches are based on Taylor's expansions of F, see [5,7,21] and references therein. For example, Taylor's expansions of F X 1 with respect to its nominal operating position ðX 1 ¼ 0; iX 1 ¼ 0Þ, the nonlinear electromagnetic force F X 1 ¼ kX 1 ½ðib þ iX 1 Þ2 =ðX b X 1 Þ2 ðib iX 1 Þ2 =ðX b þ X 1 Þ2 can be represented by F X 1 ¼ krp X 1 þ kri iX 1 þ oX 1 with krp and kri being the position and current stiffness parameters for the RAMB, respectively, and oX 1 is the infinitesimal of higher order of F X 1 . It is noted that since the coils in X axis and Yaxis are circulated by the same bias currents ib and the nominal air gaps in X axis and Yaxis are also the same, that is X b ¼ Y b , the position and current stiffness parameters krp and kri obtained from the Yaxis are the same as the ones obtained from the X axis. However, the TAMB is different from RAMBs, the position and current stiffness parameters obtained from the Zaxis are represented by ktp and kti. Hence, by using Taylor's expansions of F X 1 and compared (33) with (1), yields y€ ¼ aðyÞ þ bðyÞu þd where 2

Table 1 ~ comparisons between SMOFCEDO and Tracking error (e) and observation error (e) ISMCSMO methods (servomotor positioning system). Variables

SMOFCEDO

ISMCSMO

Maximal absolute value of tracking error (e) Steady-state value of tracking error (e) ~ Maximal absolute value of observation error ðeÞ ~ Steady-state value of observation error ðeÞ

0.1212 0.0027 0.0709 0.0000

0.2032 0.0109 0.0737 0.0006

y ¼ X;

2

3 kri β1 6k β 7 6 ri 3 7 6 7 7 bðyÞ ¼ 6 6 kri β1 7; 6k β 7 4 ri 3 5 kti β 4

2

iX 1 6i 6 X2 6 u¼6 6 iY 1 6i 4 Y2 iZ

3 7 7 7 7 7 7 5

The detailed description of β i ði ¼ 1; 2; 3; 4Þ and d is given in the Appendix. In the simulation, the system parameters of five-DOF AMBs system are as follows. The transverse mass moment of inertia of

c

1

1

3 krp β 1 X 1 6k β X 7 6 rp 3 2 7 6 7 7 aðyÞ ¼ 6 6 krp β 1 Y 1 7; 6k β Y 7 rp 4 3 2 5 ktp β 4 Z

2

c

2

fd

fd fd

Unbalanced Disk

AMB 1

AMB 2

mg

la

lb lc Fig. 7. Schematic view of AMBs system.


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rotor about X –Y-axes is J ¼ 4:004 10 2 kg m2 . The polar mass moment of inertia of rotor about Zaxis is J z ¼ 6:565 10 4 kg m2 . The mass eccentricity of disk is r ¼ 1:0 10 5 m, m ¼2.56478 kg, la ¼ 0:16 m, lb ¼ 0:19 m, lc ¼ 0:263 m, X b ¼ 0:4 10 3 m, Z b ¼ 0:5 10 3 m, kri ¼ 80 N=A, krp ¼ 2:2 105 N=m, kti ¼ 40 N=A, ktp ¼ 3:6 104 N=m, kX ¼ 2:9091 10 6 , kZ ¼ 5:5556 10 6 , g ¼9.81 kg/m2.

The bias voltages for RAMBs and TAMB are set as 1.8 V and 2.2 V, respectively. The scaling of the input voltage and the output current of the power amplifier is 0.5 A/V. It is expected that the rotor is regulated perfectly to yr ¼ ½0; 0; 0; 0; 0T . Suppose the sampling time is 0.001 s, the rotor is operated at a constant speed ω ¼ 4800 rpm, disturbance f dZ ¼ 0:05 2 0:38 9:81, initial value is yð0Þ ¼ ½0:15; 0:15; 0:05; 0:05; 0:10T mm.



8

For the presented SMOFCEDO method, select sliding surface parameters as r1 ¼ ½8; 50; 100; 180; 100, r2 ¼ ½1000; 1000; 1200; 1200; 285, choose γ ξ ¼ ½0:9; 0:9; 0:9; 0:9; 0:9, qv ¼ ½50; 50; 30; 30; 50 in the controller, 2 3 103 103 103 103 103 6 7 P o ¼ 4 3 106 3 106 3 106 3 106 3 106 5 0

0

0

0

0

in the tracking error observer, 600 600 650 650 500 q¼ 1 1 1 1 1 in the extended disturbance estimator. To make comparison, the ISMCSMO method is employed to the five-DOF AMBs system as well. The designable parameters are chosen according to those in [7]. However, because [7] did not consider the output feedback control issue, some of the para-

4

x 10−4

meters are modified in order to keep the output feedback closedloop control system stable. Those parameters are c1 ¼ ½140; 140; 140; 140; 140, c 2 ¼ ½1:65; 1:65; 1:65; 1:65; 1:65 in integral sliding mode function, δISMC ¼ ½2:2; 2:2; 9:42; 9:42; 3:25 in the controller, and the observer parameter P oISMC is chosen as the same as that in SMOFCEDO method. In order to reduce chattering in ISMC controller, sgnðsISMC Þ is replaced by satðsISMC =0:02Þ in each DOF. The simulation results are shown in Figs. 8–15. Figs. 8, 10, 12 and 14 are under SMOFCEDO method and Figs. 9, 11, 13 and 15 are with ISMCSMO method. In Figs. 10 and 11, the red points are initial positions. By comparing with ISMCSMO method, it is obvious that under SMOFCEDO method: (1) The states in all of X axis, Yaxis, Zaxis are smoother and with higher precision. (2) The X –Y circles are better. (3) The deviation of observation errors on X axis, Yaxis is smaller. (4) The input signals on X axis and Yaxis are smoother, the fluctuation of current on the X axis is smaller.

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Besides, the rotor positions, rotor observation errors and rotor input currents are given numerically in Tables 2, 3 and 4, respectively. Consider that the rotor is a kind of dynamic balance systems, the mean values and variance values of relative variables



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Table 2 Rotor positions comparison between SMOFCEDO and ISMCSMO methods. Variables Steady-state Steady-state Steady-state Steady-state Steady-state Steady-state Steady-state Steady-state Steady-state

3

mean value of rotor position ðX 1 Þ (10 ) mean value of rotor position ðX 2 Þ (10 3) mean value of rotor position ðY 1 Þ (10 3) mean value of rotor position ðY 2 Þ (10 3) value of rotor position ðZÞ (10 3) variance value of rotor position ðX 1 Þ (10 7) variance value of rotor position ðX 2 Þ (10 7) variance value of rotor position ðY 1 Þ (10 7) variance value of rotor position ðY 2 Þ (10 7)

SMOFCEDO

ISMCSMO

0.0008 0.0010 0.1034 0.0687 0.0073 0.0078 0.0247 0.0071 0.0201

0.0018 0.0065 0.1770 0.1644 0.0158 0.0202 0.1244 0.0051 0.0206


L. Xiao, Y. Zhu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Table 3 Rotor observation errors comparison between SMOFCEDO and ISMCSMO methods. Variables

SMOFCEDO

ISMCSMO

Steady-state mean value of observation error ðX~ 1 Þ (10 7) Steady-state mean value of observation error ðX~ 2 Þ (10 7)

0.0667

0.4127

0.1525

1.7312

Steady-state mean value of observation error ðY~ 1 Þ (10 7) Steady-state mean value of observation error ðY~ 2 Þ (10 7) Steady-state value of observation error ðZ~ Þ (10 7) Steady-state variance value of observation error ðX~ 1 Þ (10 11)

0.0546

0.0470

0.1867

0.0989

0.0325 0.2186

0.0015 0.4704

Steady-state variance value of observation error ðX~ 2 Þ (10 11) Steady-state variance value of observation error ðY~ 1 Þ (10 11) Steady-state variance value of observation error ðY~ 2 Þ ð10 11 Þ

0.6490

2.3807

0.1994

0.4812

0.5534

1.9758

Table 4 Rotor input currents comparison between SMOFCEDO and ISMCSMO methods. Variables Steady-state Steady-state Steady-state Steady-state Steady-state Steady-state Steady-state Steady-state Steady-state Steady-state

mean value of current ðiX 1 Þ mean value of current ðiX 2 Þ mean value of current ðiY 1 Þ mean value of current ðiY 2 Þ mean value of current ðiZ Þ variance value of current ðiX 1 Þ variance value of current ðiX 2 Þ variance value of current ðiY 1 Þ variance value of current ðiY 2 Þ variance value of current ðiZ Þ

SMOFCEDO

ISMCSMO

0.0015 0.0030 0.4448 0.3351 0.0198 0.0190 0.0511 0.0227 0.0544 0.0012

0.0033 0.0193 0.5907 0.5150 0.0385 0.0200 0.1071 0.0057 0.0243 0.0000

are given in Tables 2–4. It is supposed that the steady-state begins at the 0.3 s. Table 2 illustrates that all of rotor position values under SMOFCEDO method are smaller than those in ISMCSMO method. Table 3 shows that except the mean value of observation errors in Y~ axis and Z~ axis, the rest of the values are smaller under SMOFCEDO method than those in ISMCSMO method. Among those values, the mean value of observation error X~ 2 , the variance value of observation errors X~ 2 and Y~ 2 reduce greatly under SMOFCEDO method. Table 4 indicates that except the variance value of observation errors in Y~ axis and Z~ axis, the rest of the values are smaller under SMOFCEDO method than those in ISMCSMO method. Therefore, the presented SMOFCEDO method has better performance on the whole.

5. Conclusions A novel output feedback tracking control method for a class of systems who suffers from disturbances is presented in this paper. The mathematical model of the system is not required to be with high accuracy, and the disturbances can be vanishing or nonvanishing without asking for their bounds to be known. On the basis of sliding mode control approach, a differential sliding surface is specified at first, and then a sliding mode controller is obtained by using reaching law approach. Consider that only outputs are measurable, an original tracking error observer is created on the basis of an extended disturbance estimator. Based on the

11

observation of tracking error and the estimation of disturbance, the sliding mode controller is implementable. The convergence of observation error, the reachability of sliding surface and the robust stability of closed-loop system are proved. Simulation results of a servomotor positioning system and a five-degree-of-freedom active magnetic bearing system show that the proposed sliding mode output feedback tracking control method has better performance than traditional integral sliding mode control with sliding mode observer method, and all of results presented in this paper can be extended to higher DOF systems conveniently.

Acknowledgments The authors would like to thank Professor Qing-Chang Zhong for his precious suggestions on this paper. And the authors also thank the support of the Fundamental Research Funds for the Central Universities (NJ20140022) and Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

Appendix " 2

ðib þiX 1 Þ2

ðib iX 1 Þ2

#3

6 kX 1 7 6 ðX b X 1 Þ2 ðX b þX 1 Þ2 7 6 " #7 6 7 2 2 6 7 6 kX ðib þiX 2 Þ ðib iX 2 Þ 7 6 2 7 ðX b X 2 Þ2 ðX b þX 2 Þ2 7 6 6 7 " # 6 7 ðib þ iY 1 Þ2 ðib iY 1 Þ2 7 6 7 k F ¼6 6 Y 1 ðY Y Þ2 ðY þ Y Þ2 7 1 1 b b 6 7 " # 6 7 2 2 6 7 ðib þ iY 2 Þ ðib iY 2 Þ 6 7 6 kY 2 7 2 2 6 7 ðY b Y 2 Þ ðY b þ Y 2 Þ 6 7 " # 6 7 2 2 6 7 ði þ i Þ ði i Þ Z Z b 4 kZ 5 b 2 2 ðZ b ZÞ ðZ b þ ZÞ where ib is the bias current, kX 1 , kX 2 , kY 1 , kY 2 , kZ are electromagnet parameters related to the corresponding AMB structure, core materials and temperature: 2 3 2 γ1 0 0 3 β1 β2 0 0 0 6β 7 6 7 0 0 7 6 2 β3 0 6 γ2 0 0 7 6 7 6 7 6 0 γ 1 0 7; 0 β1 β2 0 7 K ¼6 6 0 7; E ¼ 6 7 6 0 7 6 7 0 β2 β3 0 5 4 4 0 γ2 0 5 0 0 γ3 0 0 0 0 β4 2

0 6 0 6 6 G¼6 6 α1 6 4 α2 0 2

0 0

α1

α1 α2

α1

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0

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07 7 7 07 7; 7 05 0

α1 Y_ 1 þ α1 Y_ 2 þ krp β 2 X 2 þ kri β2 iX 2 þ oX 1 þ γ 1 f dX α2 Y_ 1 α2 Y_ 2 þ krp β X 1 þkri β iX þ oX þ γ f d

3

6 7 6 7 2 2 1 2 X 2 6 7 6 α X_ α X_ þk β Y þ k β i þ o þ γ f g 7 d¼6 1 1 7 rp 2 2 Y1 1 2 ri 2 Y 2 1 dY 6 7 6 α X_ þ α X_ þ k β Y þ k β i þ o þ γ f g 7 rp 2 1 Y2 2 1 2 2 ri 2 Y 1 4 5 2 dY γ 3 f dZ

α1 ¼ la J z ω=Jl, α2 ¼ lb J z ω=Jl, where β2 ¼ 1=m la lb =J, β3 ¼ 1=m þ l2b =J, β4 ¼ 1=m, γ 2 ¼ 1=m þ lb lc =J, γ 3 ¼ 1=m.

β1 ¼ 1=m þ a2 =J, γ 1 ¼ 1=m la lc =J,


12


References [1] Utkin VI. Variable structure systems with sliding modes. IEEE Trans Autom Control 1977;22:212–22. [2] Gao WB, Hung JC. Variable structure control of nonlinear systems a new approach. IEEE Trans Ind Electron 1993;40:45–55. [3] Hung JY, Gao WB, Hung JC. Variable structure control: a survey. IEEE Trans Ind Electron 1993;40:2–22. [4] Gao WB. Theoretical foundation for variable structure control. Beijing: China Science and Technology Press; 1998. [5] Jang Ming-Jyi, Chen Chieh-Li, Tsao Yi-Ming. Sliding mode control for active magnetic bearing system with flexible rotor. J Frankl Inst 2005;342: 401–19. [6] Serdar Tombul G, Banks Stephen P, Akturk Nizami. Sliding mode control for a class of non-affine nonlinear systems. Nonlinear Anal 2009;71:1589–97. [7] Lin FJ, Chen SY, Huang MS. Intelligent double integral sliding-mode control for five-degree-of-freedom active magnetic bearing system. IET Control Theory Appl 2010;5(11):1287–303. [8] Yang Jun, Li Shihua, Yu Xinghuo. Sliding-mode control for systems with mismatched uncertainties via a disturbance observer. IEEE Trans Ind Electron 2013;60(1):160–9. [9] Ginoya Divyesh, Shendge PD, Phadke SB. Sliding mode control for mismatched uncertain systems using an extended disturbance observer. IEEE Trans Ind Electron 2014;61(4):1983–92. [10] Zhang Jinhui, Xia Yuanqing. Design of static output feedback sliding mode control for uncertain linear systems. IEEE Trans Ind Electron 2010;57 (6):2161–70.

[11] Chang Jeang-Lin. Dynamic output feedback sliding mode control for uncertain mechanical systems without velocity measurements. ISA Trans 2010;49: 229–34. [12] Bartolini Giorgio, Punta Elisabetta. Sliding mode output-feedback stabilization of uncertain nonlinear nonaffine systems. Automatica 2012;48:3106–13. [13] Aguilar-Ibanez Carlos, Garrido-Moctezuma Ruben, Davila Jorge. Output feedback trajectory stabilization of the uncertainty DC servomechanism system. ISA Trans 2012;51:801–7. [14] Godbole Ashwini A, Kolhe Jaywant P, Talole Sanjay E. Performance analysis of generalized extended state observer in tackling sinusoidal disturbances. IEEE Trans Control Syst Technol 2013;21:2212–23. [15] Zhong QC, Rees D. Control of uncertain LTI systems based on an uncertainty and disturbance estimator. J Dyn Syst Meas Control 2004;126:905–10. [16] Schweitzer G, Bleuler H, Traxler A. Active magnetic bearings: basics, properties, and applications of active magnetic bearings. Zurich, Switzerland: VDF Hochschulverlag; 1994. [17] Schweitzer G. Active magnetic bearings—chances and limitations. In: Proceedings of the 8th international symposium on magnetic bearings. 2002. p. 1–14. [18] Sivrioglu S, Nonami K. Sliding mode control with time-varying hyperplane for AMB systems. IEEE/ASME Trans Mechatron 1998;3(1):51–9. [19] Sivrioglu S. Adaptive control of nonlinear zero-bias current magnetic bearing system. Nonlinear Dyn 2007;48:175–84. [20] Sivrioglu S. Adaptive backstepping for switching control active magnetic bearing system with vibrating base. IET Control Theory Appl 2007;1 (4):1054–9. [21] Chen Syuan-Yi, Lin Faa-Jeng. Robust nonsingular terminal sliding-mode control for nonlinear magnetic bearing system. IEEE Trans Control Syst Technol 2011;19(3):636–43.


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Error-Based Observer of a Charge Couple Device Tracking Loop for Fast Steering Mirror.

Robust Sliding Mode Control Based on GA Optimization and CMAC Compensation for Lower Limb Exoskeleton.

Robust model reference adaptive output feedback tracking for uncertain linear systems with actuator fault based on reinforced dead-zone modification.

Neural-Based Adaptive Output-Feedback Control for a Class of Nonstrict-Feedback Stochastic Nonlinear Systems.

Network-Based Output Tracking Control for a Class of T-S Fuzzy Systems That Can Not Be Stabilized by Nondelayed Output Feedback Controllers.

Online fault detection of permanent magnet demagnetization for IPMSMs by nonsingular fast terminal-sliding-mode observer.

Optimal second order sliding mode control for linear uncertain systems.

Second order sliding mode control for a quadrotor UAV.

Optimal second order sliding mode control for nonlinear uncertain systems.

Observer-Based Adaptive Neural Network Control for Nonlinear Systems in Nonstrict-Feedback Form.

A swarm intelligence-based tuning method for the Sliding Mode Generalized Predictive Control.

Tensor product model transformation based adaptive integral-sliding mode controller: equivalent control method.

Second-order sliding mode control for DFIG-based wind turbines fault ride-through capability enhancement.

Continuous uniformly finite time exact disturbance observer based control for fixed-time stabilization of nonlinear systems with mismatched disturbances.