Observer based terminal sliding mode control

13th IFAC Symposium on Large Scale Complex Systems: Theory and Applications July 7-10, 2013. Shanghai, China

Observer based terminal sliding mode control Dongya Zhao*, Ge Zhao*, Shaoyuan Li**, Quanmin Zhu*** *College of Chemical Engineering, China University of Petroleum, Qingdao, 266555, China (Tel: 86-13698651078; e-mail: [email protected]; [email protected]). ** Department of Automation, Shanghai Jiaotong University, Shanghai 200240, China (e-mail: [email protected]) *** Bristol Institute of Technology, University of the West of England, Frenchy Campus Coldharbour Lane, Bristol BS16 1QY, UK, ([email protected]) Abstract: A new observer based terminal sliding mode control (TSMC) approach is proposed for a class of second order nonlinear systems which can be used to robotic manipulators, space crafts, satellites and so on. By using equivalent output injection sliding mode observer (SMO) method, the new designed TSMC can achieve finite-time stability. Corresponding stability analysis is presented. An indispensable illustrative example is bench tested to validate the effectiveness of the proposed approach. Keywords: Sliding mode observer, sliding mode control, output feedback. Edwards, Spurgeon 2009) have been proposed recently. However, they require infinite-time to converge system states to equilibrium point. The proposed approach can achieve finite-time stability.

1. INTRODUCTION TSMC has some advantageous properties, such as fast convergence, strong robustness and high precision and so on (Li, Du, Lin 2011; Du, Li, Qian 2011; Zhao, Li, Gao, Zhu 2009; Zhao, Li, Gao 2009). It has attracted extensive attentions. The design procedure of TSMC is to design a feedback control law which can make system states reach to a pre-described terminal sliding mode (TSM) in finite-time, then the system states will converge to their equilibrium points along TSM in finite-time. Most of existing TSMC require full state feedback. However, full state may not be available in many cases (Daly, Schwartz 2006; Islam, Liu 2010). For example, in mechanical control systems, velocity signals cannot be obtained in general or is contaminated by measure noise. The main objective of this paper is to develop a framework for observed based TSMC.

The rest of this paper is organized as follows. In Section 2, the problem is formulated. In Section 3, observer based TSMC is designed with corresponding stability analysis. In Section 4, an illustrative example is presented to validate the effectiveness of the proposed approach. Finally, in Section 5, some concluding remarks are given. 2. PROBLEM FORMULATION Consider the following second order nonlinear system:

x1 = x2

x2 = f 0 ( x ) + g 0 ( x ) ( u + d ( t , x ) )

Note that the asymptotically stable observer cannot be used to TSMC, because it will make the TSMC lost the finite-time stability which is one of the main superior properties over the linear sliding mode control. How to design an observer based TSMC is challenging and interesting.

y = x1 where x = [ x1 , x2 ] , T

f ( x ) : R 2 → R1 is nonlinear term,

g ( x ) : R 2 → R1 , u ∈ R1 is control input, d ( t , x ) ∈ R 2 is lumped system uncertainty caused by modeling error and external disturbance, y ∈ R1 is system output. Suppose the desired trajectory r is smooth, i.e., r and r are bounded and exist. Tracking error and its derivative can be defined as follows:

The equivalent output injection SMO has simple structure and strong robustness (Haskara, Ozguner, Utkin 1998). By using it, a new observer based TSMC can be developed for a class of second order nonlinear uncertain systems which can stand for a large number of real systems, such as mechanical system. The proposed approach can estimate system states in finite-time and stabilize them to equilibrium points in finitetime. By combining the equivalent output injection SMO and TSMC together, the proposed approach can achieve finitetime stability. Then the proposed approach may provide an alternative high-precision robust control strategy. It should be mentioned that some asymptotical stability sliding mode observer (Bandyopadhyay, Gandhi, Kurode 2009; Xie 2007) or output feedback sliding mode controller (Choi 2008; Silva, 978-3-902823-39-7/2013 © IFAC

(1)

e1 = x1 − r e2 = x2 − r

(2)

The tracking error dynamic equation is written as follows:

e1 = e2

e2 = f 0 ( x ) + g0 ( x ) ( u + d ( t , x ) ) −  r 178

(3)

10.3182/20130708-3-CN-2036.00035

IFAC LSS 2013 July 7-10, 2013. Shanghai, China

The control objective of this paper is to design an output feedback TSMC by using equivalent output injection SMO method and TSMC principle.

In light of eˆ1 and eˆ2 , output feedback non-singular TSMC is designed as:

(

u1 = g 0−1 ( xˆ ) − f 0 ( xˆ ) − α −1 β −1 eˆ2

3. MATH OBSERVER BASED TERMINAL SLIDING MODE CONTROL Some assumptions are made before the controller design (Daly, Wang 2009): Assumption 1 Dynamic system (1) dose not have a finite escape time.

2 −α

)

sgn ( eˆ2 ) +  r

(8)

u2 = − g 0−1 ( xˆ ) Kl sgn ( sˆ )

(9)

u = u1 + u2

(10)

where Kl ∈ R1 is a positive number.

Assumption 3 The term g 0 ( x ) is bounded and invertible.

Theorem 1 Consider estimated tracking error dynamic equation (6), subjected to output feedback TSMC control law (8)-(10), estimated systems errors eˆ1 and eˆ2 will converge to 0 in finite-time, according to the principle of equivalent output injection SMO, system tracking errors e1 and e2 will converge to 0 in finite-time.

Assumption 4 The lumped uncertain term d ( t , x ) ≤ d 0 ( t , x ) ,

Proof: Consider the Lyapunov function for system estimation error dynamic equation (6) as follows.

Assumption 2 The control input u ( t ) belongs to the

extended Lp space, denoted as L p in this study. Any truncation of u ( t ) to a finite time interval is bounded.

d 0 ( t , x ) ∈ R1 is bounded and local Lipschitz continuous in states.

V=

Illumined by literatures (Haskara, Ozguner, Utkin 1998; Daly, Wang 2009), the following equivalent output injection SMO can be designed: xˆ1 = xˆ2 + γ 1 sgn ( y − xˆ1 ) xˆ2 = f 0 ( xˆ ) + g 0 ( xˆ ) u + γ 2 sgn ( x2 − xˆ2 )

1 2 sˆ 2

(11)

By differentiating V with respect to time along equation (6), it yields: V = sˆ ( eˆ2 + γ 1 sgn ( y − xˆ1 )

(4)

+αβ eˆ2

where xˆ1 and xˆ2 are estimations of x1 and x2 , respectively,

α −1

( f ( xˆ ) + g ( xˆ ) u + γ 0

0

2

sgn ( x2 − xˆ2 ) −  r)

)

(12)

Substitute control law (8) and （9） into equation (12):

xˆ = [ xˆ1 , xˆ2 ] , x2 = xˆ2 + ( γ 1 sgn ( y − xˆ1 ) )eq , ( γ 1 sgn ( y − xˆ1 ) )eq T

α −1 V = −αβ eˆ2 K l sˆ + sˆγ 1 sgn ( y − xˆ1 )

is the equivalent output injection, which can be obtained by passing the signal γ 1 sgn ( y − xˆ1 ) through a low pass filter,

+ sˆαβ eˆ2

(

α −1

≤ − αβ eˆ2

γ 2 sgn ( x2 − xˆ2 )

α −1

(13)

K l − γ 1 − αβ eˆ2

α −1

)

γ 2 sˆ

the details can be found in literature (Haskara, Ozguner, Utkin 1998). By using the injection SMO, state x1 and x2 can be reconstructed in finite-time. Then, the observer based TSMC is designed by using their estimations.

According to Assumption 1-4 and considering r and its α −1 α −1 derivatives are bounded, αβ eˆ2 ≥ 0 . Let k = αβ eˆ2 ,

Estimated tracking error and its derivative is defined as:

then inequality (13) can be written as:

eˆ1 = xˆ1 − r eˆ2 = xˆ2 − r

(

V ≤ −ε sˆ

(6)

α

(15)

According to sliding condition (Slotine, Li 1991), the system states will reach to the pre-described TSM sˆ in finite-time t = sˆ0 ε , sˆ0 = sˆ ( 0 ) . In light of TSM definition, estimation

Since only estimated system sates can be obtained, the following TSM surface is actually used in the controller design: sˆ = eˆ1 + β eˆ2 sgn ( eˆ2 )

(14)

If choose appropriate large Kl to make kK l > γ 1 + k γ 2 + ε , ε > 0 is a positive number, one can get:

The estimated tracking error dynamic equation can be written as:

eˆ1 = eˆ2 + γ 1 sgn ( y − xˆ1 ) eˆ2 = f 0 ( xˆ ) + g 0 ( xˆ ) u + γ 2 sgn ( x2 − xˆ2 ) −  r

)

V ≤ − kK l − γ 1 − k γ 2 sˆ

(5)

errors eˆ1 and eˆ2 will converge to zeros in finite-time. Note that in inequality (13), eˆ2 = 0 may hinder the reaching ability of TSM. However, it will be seen that eˆ2 = 0 is not an

(7)

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IFAC LSS 2013 July 7-10, 2013. Shanghai, China

attractor in the reaching phase. Substituting control law (8) and (9) into (6), it yields:

eˆ2 = −α −1 β −1 eˆ2

2 −α

sgn ( eˆ2 )

e1 = e2 e2 = −α −1 β −1 e2

(16)

+γ 2 sgn ( x2 − xˆ2 ) − K l sgn ( sˆ )

2 −α

sgn ( e2 )

(21)

According to the definition of non-singular TSM as t ≥ t + T , T = 1 β −1 α (1 − 1 α ) e1 ( t0 )

In the case of eˆ2 = 0 and sˆ ≠ 0 , there must be:

1−1 α

+ t0 , e1 = 0 and

e2 = 0 . □

eˆ2 = γ 2 sgn ( x2 − xˆ2 ) − K l sgn ( sˆ ) ≠ 0

(17)

4. ILLUSTRATIVE EXAMPLE

Therefore, the finite-time reaching ability of TSM is still satisfied (Yu, Yu, Shirinzadeh, Man 2005).

An illustrative example was demonstrated in this section. Consider a second order nonlinear uncertain system:

The system states on the sliding mode surface will be analyzed as follows. On the TSM, the behavior of system states is dominated by equivalent control law (Boiko, Fridman 2005). Make the derivative of the TSM sˆ = 0 :

x1 = x2

sˆ = eˆ2 + γ 1 sgn ( y − xˆ1 ) +αβ eˆ2

α −1

( f ( xˆ ) + g ( xˆ ) u + γ

2

sgn ( x2 − xˆ2 ) −  r)

x2 = x1 + x22 + u + d y = x1 It is obvious that f ( x ) = x1 + x22 , g ( x ) = 1 . The lumped

(18)

system uncertainty d = sin ( π t ) , the desired trajectory was

chosen as r = π 2 − 1.9 + sin ( t ) + sin ( 2t ) , initial values were

The equivalent control law can be obtained: ueq = − g

−1

(

( xˆ ) ( f ( xˆ ) − r + γ 2 sgn ( x2 − xˆ2 )

+α −1 β −1 eˆ2

2 −α

sgn ( eˆ2 ) + eˆ2

1−α

γ 1 sgn ( x1 )

selected as x1 ( 0 ) = 1 , x2 ( 0 ) = 1 , xˆ1 ( 0 ) = 0 , xˆ2 ( 0 ) = 0 . The

))

equivalent value term ( γ 1 sgn ( y − xˆ1 ) )eq on sliding surface

(19)

was obtained by passing the signal γ 1 sgn ( y − xˆ1 ) through a first order low pass filter with a bandwidth of 100 Hz.

Substitute equivalent control (19) into error dynamic equation (6), it yields:

The simulation was implemented in Simulink using a fixed step forth order Runge-Kutta solver with a sample period of T = 0.00001s . The controller’s parameters were selected as: γ 1 = 10 , γ 2 = 15 , α = 1.5 , β = 1 , K l = 36 .

e1 = e2 e2 = f ( x ) + g ( x ) d −  r− +γ 2 sgn ( x2 − xˆ2 )

(

+α −1 β −1 eˆ2

2 −α

g ( x) ( f ( xˆ ) − r g ( xˆ )

sgn ( eˆ2 ) + eˆ2

1−α

γ 1 sgn ( x1 )

Fig. 1 and Fig. 2 show the estimated system states and their real values subjected to the proposed approach, that is, output feedback TSMC. Dashed line is the estimated state. Solid line is the real state. Fig. 3 shows tracking performance of system output. Dashed line is the desired trajectory. Solid line is the real trajectory. From these figures, it can be seen that estimated states converge to their real values in finite-time and system output converge to its desired trajectory in finitetime. Fig. 5 shows control input, which is discontinuous and bounded due to discontinuous reaching law u2 .

(20)

))

The above dynamic equation stands for tracking error dynamics on TSM surface, i.e. sˆ = 0 . Note that according to Assumption 1-4 and the desired trajectory r is bounded and smooth, estimated tracking errors eˆ1 and eˆ2 remain in the set L . In this way, any finite truncation of the tracking error p

subjected to the equivalent control will be bounded. In terms of the principle of equivalent output injection SMO, if the gains γ 1 and γ 2 are chosen appropriately, the estimated states will converge to the real states in finite-time regardless of the stability of real system. If t2 < t , estimated system states will converge to real system states before they reach TSM. If t2 > t , estimated system states will reach to TSM before they converge to real system states. During the terminal sliding mode, as t > t2 , the observer error will converge to 0 regardless of real system behavior. When x1 = 0 and x2 = 0 , system error dynamic equation (23) can be rewritten as:

Fig. 1. Estimated state xˆ1 and real system state x1 of output feedback TSMC

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IFAC LSS 2013 July 7-10, 2013. Shanghai, China

Fig. 4. Control input of output feedback TSMC

Zhao, D., Li, S., Gao, F., Zhu, Q. (2009). Robust adaptive terminal sliding mode-based synchronised position control for multiple motion axes systems. IET Control Theory & Applications, 3(1), 136-150. Zhao, D., Li, S., Gao, F. (2009). Terminal sliding mode control for robotic manipulators. International Journal of Control, 82(10), 1804-1813. Daly, M. J., Schwartz, H. M. (2006). Experimental results for output feedback adaptive robot control. Robotica, 24(6), 727-738. Islam, S., Liu, P. X. (2010). Output feedback sliding mode control for robot manipulators. Robotica, 28(7), 975-987, 2010. Haskara, I., Ozguner, U., Utkin, V. (1998). On sliding mode observers via equivalent control approach. International Journal of Control, 71(6), 1051-1067. Bandyopadhyay, B., Gandhi, P. S., Kurode, S. (2009). Sliding mode observer based sliding mode controller for slosh-free motion through PID scheme. IEEE Transactions on Industrial Electronics, 56(9), 3432-3442. Xie, W.-F. (2007). Sliding-mode-observer-based adaptive control for servo actuator with friction. IEEE Transactions on Industrial Electronics, 54(3) 1517-1527. Choi, H. H., Sliding-mode output feedback control design. IEEE Transactions on Industrial Electronics, 55(11), 4047-4054. Silva, J. M. A.-D., Edwards, C., Spurgeon, S. K. (2009). Sliding-mode output-feedback control based on LMIs for plants with mismatched uncertainties. IEEE Transactions on Industrial Electronics, 56(9), 3675-3683. Daly, J. M., Wang, D. W. L. (2009). Output feedback sliding mode control in the presence of unknown disturbance. Systems & Control Letters, 58(3), 188-193. Slotine, J.-J. E., Li, W. (1991). Applied nonlinear control, Prentice Hall, Englewood Cliffs, New Jersey. Yu, S., Yu, X., Shirinzadeh, B., Man, Z. (2005). Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica, 41(11), 1957-1964. Boiko, I., Fridman, L. (2005). Analysis of chattering in continuous sliding-mode controllers. IEEE Transactions on Automatic Control, 50(9), 1442-1446, 2005.

5. CONCLUSIONS

ACKNOWLEDGEMENT

This paper has focused on the observer based terminal sliding mode controller design. In light of equivalent output injection SMO method, a novel TSMC is developed. By stability analysis and simulation, the effectiveness of the proposed approach has been validated. The future work will try to use this method to solve some practical issue.

This work is partially supported by the National Nature Science Foundation of China under Grant 61004080, 61273188, Shandong Provincial Natural Science Foundation under Grant ZR2011FM003, the Fundamental Research Funds for the Central Universities of China, Development of key technologies project of Qingdao Economic and Technological Development Zone under Grant 2011-2-52, Taishan Scholar Construction Engineering Special funding.

Fig. 2. Estimated state xˆ2 and real system state x2 of output feedback TSMC

Fig. 3. Tracking performance of output feedback TSMC

REFERENCES Li, S., Du, H., Lin, X. (2011). Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics. Automatica, 47(8), 1706-1712. Du, H., Li, S., Qian, C. (2011). Finite-time attitude tracking control of spacecraft with application to attitude synchronization. IEEE Transactions on Automatic Control, 56(11), 2711-2717.

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