A Smooth Second-Order Sliding Mode Controller for Relative Degree Two Systems S. Iqbal 1, C. Edwards 2, A. I. Bhatti 3 Abstract— A novel smooth second order sliding mode controller based on a modified “twisting” algorithm for relative degree two systems is discussed to provide robustness as well as accuracy. A robust exact observer is introduced in the closed loop system to reject the drift signals. A Lyapunov method combined with homogeneity concepts are used to prove finite time stability of the overall system. The method is verified through simulations on a benchmark example of a DC motor. I.

INTRODUCTION

Certain classes of model uncertainties and external disturbances can easily be rejected by Sliding Mode Control (SMC) (Utkin, 1992; Edwards and Spurgeon, 1998). This robustness and high accuracy has made this type of control a popular approach. In this technique, a constraint manifold is chosen ( 0) and the control law is derived to maintain the states on this surface. Traditionally this has been achieved through the implementation of a discontinuous feedback control law. The main drawback of this methodology is the resulting so-called chattering which may excite hidden dynamics in the system and may also be destructive for control actuators. Another disadvantage of traditional SMC is that it can only be applied to systems having relative degree one with respect to the switching function (i.e. the control signal has to appear explicitly in ). In the last decade, a number of methods have been proposed to avoid the chattering effect. Saturation approaches (Slotine and Li, 1991) and sigmoid approximations (Burton and Zinober, 1986) have been used to smooth the transition near the switching surface by changing the dynamics in a small vicinity of the discontinuity surface. Such approaches risk possible reductions in accuracy. The equivalent control (Utkin, 1992) involves the convex combination of control effort on both sides of the sliding surface. This method maintains system trajectories on the constraint surface 0, if the system dynamics are exactly known which is not always possible. Other approaches which do not compromise performance have recently be proposed. The terminal sliding mode method (Man et al, 1994) is built on a nonlinear switching manifold and can significantly improve the transient performance of the closed loop system. This technique does not satisfy Lipchitz conditions (Yu and Man, 2002) and has an unbounded right-hand side (Levant, 2003). In dynamical sliding modes (Sira-Ramirez, 1993) the surface is not only depen-

dent upon the states but also on the inputs of the system. This method adds additional dynamics to the compensated system and hence increases the overall complexity. Higher Order Sliding Modes (HOSM) (Emel'yanov et al, 1996; Levant, 1993; Fridman and Levant, 1996; Bartolini et al, 1998) is the most prominent and effective technique to remove chattering effects whilst preserving the important properties of traditional first order SMC. It also eliminates the relative degree restriction of conventional SMC. The main shortcoming of this method is its sensitivity to un-modeled fast dynamics (Shtessel et al, 2003). All the above methods reduce chattering from the control law but do not completely succeed in removing its effects. Switching is still embedded in these techniques. Smooth Second Order Sliding Mode Control (SSOMC) (Shtessel et al, 2007) using the super twisting algorithm has recently been proposed for relative degree one systems with respect to the switching function. In (Shtessel et al, 2007) an exact observer (Davila et al, 2005; Levant, 1998, 2003) is employed to compensate for the drift terms in the closed loop dynamics. In this way a chattering free control is achieved with robustness. This paper builds on the SSOMC framework for relative degree two systems with respect to the sliding manifold by using the “twisting” algorithm (Levant, 1993; Orlov, 2005). Whilst this control law is smooth, it may be sensitive to external and internal disturbances. To overcome this problem, as in (Shtessel et al, 2007), a robust exact observer is employed in the closed loop system to compensate for the drift term precisely. Finite time convergence of the overall system is established by using Lyapunov methods combined with a homogeneity-based approach (Levant, 2008; Orlov, 2005). The resulting overall system is finite time stable and robust against (sufficiently) smooth uncertain disturbances. The rest of the paper is structured as follows; a smooth “twisting” controller for relative degree two systems is discussed in Section II together with the higher order observer. A case study involving a DC motor is used to validate the proposed technique in Section III. Conclusions are drawn in Section IV. II. SMOOTH TWISTING 2SMC CONTROLLER Consider a Single Input Single Output (SISO) system, whose sliding variable dynamics with respect to the input is relative degree two: ,

1.

2.

3.

Sohail Iqbal, Department of Electronic Engineering, Mohammad Ali Jinnah University, Islamabad, Pakistan. Currently, he is a Visiting Research Fellow at University of Leicester, UK, (e-mail: [email protected]). Prof. Christopher Edwards, Control & Instrumentation Group, Engineering Dept, University of Leicester, UK, email: [email protected] Prof. Aamer Iqbal Bhatti, Control & Signal Processing Research Group, Department of Electronic Engineering, Mohammad Ali Jinnah University, Islamabad, Pakistan; (e-mail: [email protected]).

(2.1)

In (2.1) the switching surface , and , is a bounded and sufficiently smooth uncertain function. The control signal is required to provide smooth control action. As in (Shtessel et al, 2007) the aim is to cancel the drift term , using an estimator based on an exact observer. The control action u is then developed based on the σ-dynamics

(2.1) under the assumption system

,

0, i.e. for the nominal

Remark 1: The motion in the system (2.2) can be reasonably called a Second-Order Sliding Mode. Observer Structure

If the control law ⁄

| |

| |



where , 0 and 2 is employed, and , 0, then the closed loop system in state space form with variables and can be written as follows, ⁄

| |

| |



(2.2) Theorem 1: The system in (2.2) is globally uniformly finite time stable. Moreover the system generates a smooth second order sliding motion only at the origin.

Since the intended control law is smooth, the closed loop dynamics (2.1) will be sensitive to , . The overall closed system therefore requires a good estimate of the drift signal to cancel out its effects. Assume the states and are available and the control is Lebesgue-measurable. The function , is assumed to be unknown but 1 times differentiable with , , where 0 is a ‘Lipshitz constant’. Then the exact observer structure (Davila et al, 2005; Levant, 1998, 2003) can be introduced into the closed loop to compensate for the undesired disturbances. The proposed observer structure can be written as follows

Proof: The only possible equilibrium point for system (2.2) is the origin, i.e. σ1 = σ2 = 0. Consider as a candidate Lyapunov function ,

2

1

|

1 2



| |





|

which is differentiable, radially unbounded and positive definite. Its derivative with respect to time is | |



| |







|

, | |







(2.3)

Since (0, 0) is the only possible equilibrium point for the system in (2.2), applying LaSalle’s invariance principle (Slotine and Li, 1991) for smooth systems, the only possible trajectory of (2.2) on the invariant manifold 0 is 0. This shows that the system is globally uniformly asymptotically stable with respect to the origin. Moreover, it can be easily verified that system (2.2) is homogenous, and its homogeneity degree is equal to –1 by using the transformation ,

,

0

and the dilation

where :

,

|



(2.5)

and therefore | |

|



| | ,

|

This observer has a slightly different structure to the one in (Shtessel et al, 2007) because a relative degree two system is considered in (2.1), , . whereas in the work of (Shtessel et al, 2007) relative degree one structures are tackled. Lemma 1: Suppose the input signal u is bounded and Lebesgue-measurable then the following inequalities can be established in finite time for some positive constants and . | |

| ,



|

(2.4)

is define as

,

Then according to (Bhatt and Bernstein, 2000; Bacciotti and Rosier, 2001) the system (2.2) is globally uniformly finite time stable at the origin. This means that the trajectories of the system with smooth control belong to the surface 0 after a finite time interval. This proves the theorem. Theorem 1 relates to the control of a double integrator tem , whereas the real problem to be tackled is the control of system (2.1). This will be addressed by the use of an observer to estimate and ‘cancel’ , .

,

,

,

2, … ,

,

1, …

1 (2.6)

where the noise on

,

and



,



Proof: The proof is similar to (Levant, 2003; Shtessel et al, 2007). Define and , , ,

In absence of noise (i.e. when and (2.5),

0), using equations (2.1)

Control Law For the system in (2.1) consider the control law | |

2⁄

| |

2⁄

1

,

(2.9)

,

with 1, 2 and where is obtained from using the observer in (2.5) . The closed loop system is then given by

,

In the same way, it can be proven that

| | | |

,

⁄ ⁄

The idea is that in finite time, the term will cancel the drift signal , , and the dynamics above will become (2.2). The results of this section will now be formally stated:



Theorem 2: Suppose the drift term , in (2.1) is m-1 times differentiable and , , where L is the ‘Lipshitz constant’. Then the closed loop system (2.1), (2.5) and (2.9) is finite time stable.

|

Proof: The proof of this theorem is a consequence of Theorem 1 and Lemma 1, by introducing the following definitions,

From the mth order observer (2.5) it follows |

|



|

|

|



|

| ,

(2.7) The structure in (2.7) is similar to the exact differentiator from (Levant 1998; Levant 2003). For ease of implementation, the equations can be written in such a way that the derivatives on the right hand side of each equation are excluded (Levant, 2003; Shtessel et al, 2007). The resulting differential inclusion can be understood in the Flippov sense (Flippov, 1988). It is easy to see that (2.7) preserves the differential inclusion with respect to the dilation 0,

and

0, . .

and therefore the system is homogenous, and its homogeneity degree is equal to –1. Therefore 0 in finite time and the following exact inequalities are obtained (in finite time): ,

, , As shown in Lemma 1, the system (2.5) can easily be transformed into (2.7) by using above definitions. The resulting system is homogenous and its homogeneity degree is equal to –1 by using the dilation: 0,

and

Let the σ-dynamics in (2.1) with control law (2.9) and the observer (2.5). When exact measurements are available, the term z2 will be cancelled by , in finite time. After that the dynamics (2.2) will be established. By following Theorem 1, the system (2.2) is finite time stable. The rest of the proof is similar to Theorem 2 of (Levant 2005). This proves the theorem. III.

,

,

2, … (2.8)

More generally if noise is present i.e. 0, if σ and u belong ⁄ ⁄ to the inclusions , and , the bounds in (2.6) can be obtained using arguments similar to (Levant, 2003). This proves the Lemma 1. Remark 2: In particular, it can be seen that is a finite time estimate of the unknown drift term , . This fact will be exploited in the control law development. The main results of this paper will now be presented.

0, . .

SIMULATION EXAMPLE

In this section, the control technique is demonstrated on an industrial benchmark DC motor (Utkin et al, 1999) with the following dynamics

(3.1) In system (3.1), u is the input terminal voltage, and i are the states of the system and represent shaft speed and armature current respectively. The load torque is defined as

. All the parameters of the DC motor and their nominal values are listed in Table 1. Symbol

1

Values/Units

Inertia of the Motor Rotor and Load

0.001 Kgm2

Armature Resistance

0.5 Ω

Armature Inductance

1.0 mH

Back-EMF Constant

0.001 V/rad

Torque Constant

0.008 Nm/A

Coefficient of Viscous Friction

0.01 Nm s /rad

0.5 Speed (rad/sec)

Name

1.5

0

-0.5

-1

Table 1: The DC Motor Parameters

-1.5

Let be the reference shaft speed, and be the tracking error. Define and , then the error dynamical system using equation (3.1) can be represented as

0

5

10

15

20

25 30 Time (sec)

35

40

45

50

55

Figure 3.1: The Speed Response of DC Motor Figure 3.2 shows the resultant acceleration of the motor.

, 1

(3.2) where the function

,

0.8

is

0.6

,

(3.3) Here the drift term depends upon the reference speed and ⁄ 0 , load torque. The constants are defined as ⁄ 0, ⁄ 0 . To ensure existence of inequali, , where L is the ‘Lipshitz constant’, a second ty order low-pass pre-filter for reference signal is introduced.

Acceleration (rad/sec2)

0.4

0

0.2 0 -0.2 -0.4 -0.6 -0.8 -1

The smooth second order sliding mode control law chosen for the DC motor with controller’s parameters 3, 2, 5 and 6 is given as 1

| |

| | (3.4)

0

10

20

30 Time (sec)

40

50

60

Figure 3.2: The Acceleration Response of DC Motor Figure 3.3 exhibits the tracking of the drift term by the proposed observer. As shown by the figure, the observer precisely tracks the drift signal exactly after a certain (finite) time. 3

where z2 is the estimate of the drift term (3.3) with observer parameters 1.1 and 3.3. The signal is generated according to,

2

1



|

|

0



-1

(3.5) -2

Figure 3.1 shows the simulation results of the proposed smooth “twisting” controller for speed control of the DC motor. As illustrated, the speed transition time is about 4 seconds.

Drift Signal Observed Signal -3

0

5

10

15

20

25 30 Time (sec)

35

40

45

50

55

Figure 3.3: The Actual Drift Term and Observed

Figure 3.4 shows the motor current. Remember that the current is not explicitly controlled; its behavior is a result of acceleration control, since the current is determined through the second term of (3.1), i.e.

1.5

1

1 Speed (rad/sec)

0.5

0

-0.5 1

-1 0.5

Current (Amp)

-1.5

0

5

10

15

20

25

Time (sec)

0

Figure 3.6: The Sinusoidal Speed Reference Response -0.5

Figure 3.7 depicts the results of the proposed observer. The drift signal is followed within one second and well tracked thereafter.

-1

-1.5

2 0

5

10

15

20

25 30 Time (sec)

35

40

45

50

55

1.5

Figure 3.4: The Current response of the speed Controller for DC Motor

1

0.5

As depicted in Figure 3.4 the results are very good compared to a more conventional SMC design (see for example Utkin, 1999; Figure 10.5 pp 177).

0

-0.5

Figure 3.5 shows the control effort generated by the smooth real twisting controller to track the desired speed changes.

-1

-1.5

-2

3

Actual Drift Signal Observed Drift Signal 0

5

10

15

20

25

Tme (sec)

Figure 3.7: The Actual Drift Term and Observed for Sinusoidal Speed Reference

2

Control Effort (Volts)

1

Figure 3.8 shows the current waveform generated by the DC motor.

0

1.5 -1

1 -2

0

5

10

15

20

25 30 Time (sec)

35

40

45

50

55

Figure 3.5:The Control law for DC Motor

Current (Amp)

0.5 -3

0

-0.5

The validity of the proposed controller for a continuous sinusoidal waveform as the reference speed given to the DC motor is discussed below. Figure 3.6 shows sinusoidal speed control of the DC motor with the smooth 2SMC controller. The plots show that the signal is tracked precisely.

-1

-1.5

0

5

10

15

20

25

Time (sec)

Figure 3.8: The Current response for the Sinusoidal Speed Reference

Figure 3.9 shows the control effort generated by the controller to track the reference sinusoidal speed.

ACKNOWLEDGMENT This work was supported by Higher Education Commission (HEC) of Pakistan.

3

REFERENCES [1] Bacciotti, A. and L. Rosier, (2001). Liapunov Functions and Stability

2

Control Effort (Volts)

1

[2]

0

[3] -1

[4]

-2

-3

0

5

10

15

20

25

[5]

Time (sec)

Figure 3.9:The Control law for Sinusoidal Reference Speed Reference To verify the robustness of the proposed controller, certain parameters of the DC motor have been changed – specifically the viscous friction coefficient and the armature resistance are increased by 100% (unknown to the controller) during the simulation. These changes do not affect the performance of the controller significantly. Figure 3.10 demonstrates the speed response of the DC motor with purturbations in the parmeters. The two subplots at the bottom of the figure shows variation in friction and resitance respectively.

[6] [7]

[8] [9]

[10] [11]

1

[12]

0.8

0.6

S peed (rad/s ec)

0.4

[13]

0.2

0

[14]

-0.2

-0.4

-0.6

[15]

-0.8 With Perturbation Without Perturbation

-1 0

5

10

15

20

25

30

35

40

45

50

55

10

5

10

15

20

25

30

35

40

45

50

55

5

10

15

20

25

30

35

40

45

50

55

Friction

0.02 0.015

[16]

Res istance

0.01

0.8

[17]

0.6 0

Time (sec)

Figure 3.10:The Speed Response of DC motor with Perturbation in the Parameters.

[18]

[19]

IV.

CONCLUSIONS

A new smooth second order sliding mode control law for relative degree two systems using a modified “twisting” algorithm is discussed. A robust exact observer is used to compensate the drift term in the closed-loop dynamics. Finite time stability of the overall system is proven using a homogeneity-based approach.

[20] [21] [22]

in Control Theory, Lecture notes in control and information sciences (Vol. 267). New York: Springer. Bhatt, S. and D. Bernstein, (2000). Finite Time Stability of Continuous Autonomous Systems, SIAM Journal on Control and Optimization, Vol. 38(3), pp.751-766. Bartolini, G., A. Ferrara and E. Usai, (1998). Chattering avoidance by second-order sliding mode control, IEEE Transactions of Automat. Control, Vol. 43(2), pp. 241-246. Burton, J. A. and A. S. I. Zinober(1986), Continuous approximation of variable structure control, International Journal of Systems Science, Vol. 17(6) pp. 875 - 885 Davila, J., L. Fridman and A. Levant, (2005). Second-order slidingmode observer for mechanical systems. IEEE Transactions of Automatic Control, Vol. 50 (11), pp. 1785-1789. Edwards, C. and S. K. Spurgeon, (1998). Sliding Mode Control: Theory and Applications, Taylor & Francis Inc, PA. Emel'yanov, S. V., S. K. Korovin, and L.V. Levantovsky, (1996), High order sliding modes in control systems, Computational Mathematics and Modeling, Vol. 7(3), pp. 294–318. Filippov, A. F., (1988), Differential Equations with Discontinuous Right-Hand Side, Kluwer Dordrecht, The Netherlands. Fridman, L., and A. Levant, (1996), Sliding modes of higher order as a natural phenomenon in control theory, Lecture Notes in Control and Information Science, Robust Control via Variable Structure and Lyapunov Techniques, Springer-Verlag London, pp. 107–133 Levant, A., (1993), Sliding order and sliding accuracy in sliding mode control, International Journal of Control, Vol. 58 (6), pp. 1247–1263. Levant, A., (1998), Robust Exact Differentiation via Sliding Mode Technique, Automatica, Vol. 34(3), pp. 379 - 384 Levant, A. (2003). Higher-order sliding modes, differentiation and output-feedback control, International Journal of Control, 76 (9/10), pp. 924–941. Levant, A., (2005), Homogeneity approach to high-order sliding mode design, Automatica, Vol. 41(5), pp 823-830 Man, Z., A. P. Paplinski and H. Wu, (1994) , A Robust MIMO Terminal Sliding Mode Control Scheme for Rigid Robotic Manipulators. IEEE Transactions on Automatic Control, Vol. 39(12), pp. 2464–2469. Orlov, Y., (2005), Finite Time Stability and Robust Control synthesis of Uncertain Switched Systems, SIAM Journal of Control and Optimization, Vol. 43(4), pp. 1253-1271. Sira-Ramirez, H., (1993), On the Dynamical Sliding Mode Control of Nonlinear Systems, International Journal of Control, 57, 1039-1061. Shtessel, Y. B., I. A. Shkolnikov, and M. D. J. Brown, (2003), An Asymptotic Second-Order Smooth Sliding Mode Control, Asian Journal of Control, Vol. 5(4), pp. 498-504. Shtessel, Y. B., I. A. Shkolnikov, and A. Levant,(2007), Smooth second-order sliding modes: Missile guidance application, Automatica, Vol. 43(8), pp. 1470-1476. Slotine, J.-J. E., and Li, W., (1991), Applied Nonlinear Control, Prentice-Hall, London. Utkin, V. I., (1992), Sliding Modes in Optimization and Control Problems, Springer Verlag, New York. Utkin, V. I., J. Guldner and J. Shi (1999), Sliding Mode Control in Electromechanical Systems – Taylor and Francis London. Yu, X., and Z. Man, (2002), Variable Structure Systems with Terminal Sliding Modes, Lecture Notes in Control and Information Sciences, Variable Structure Systems: Towards the 21st Century, Springer Berlin / Heidelberg, Vol. 274, pp. 109-127.

A Smooth Second-Order Sliding Mode Controller for Relative Degree ...

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