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Optimal second order sliding mode control for nonlinear uncertain systems Madhulika Das n, Chitralekha Mahanta Department of Electronics and Electrical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India

art ic l e i nf o

a b s t r a c t

Article history: Received 6 January 2014 Received in revised form 26 February 2014 Accepted 17 March 2014 This paper was recommended for publication by Dr. Jeff Pieper

In this paper, a chattering free optimal second order sliding mode control (OSOSMC) method is proposed to stabilize nonlinear systems affected by uncertainties. The nonlinear optimal control strategy is based on the control Lyapunov function (CLF). For ensuring robustness of the optimal controller in the presence of parametric uncertainty and external disturbances, a sliding mode control scheme is realized by combining an integral and a terminal sliding surface. The resulting second order sliding mode can effectively reduce chattering in the control input. Simulation results confirm the supremacy of the proposed optimal second order sliding mode control over some existing sliding mode controllers in controlling nonlinear systems affected by uncertainty. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Optimal control Control Lyapunov function Integral sliding mode Terminal sliding mode Chattering mitigation

1. Introduction Optimal control is one of the most important branches in modern control theory and linear quadratic regular (LQR) is a well established optimal control technique for linear systems. The LQR control problem is solved by using its associated Riccati equation [1]. However, designing an optimal controller for the nonlinear system is still a challenging task. The optimal LQR problem for the nonlinear system often requires solving a nonlinear twopoint boundary value (TPBV) problem whose analytical solution does not exist in general [2,3]. One of the main approaches for designing optimal controllers for nonlinear systems is to solve the Hamilton–Jacobi–Bellman (HJB) partial differential equation [4]. Unfortunately, in general, the HJB partial differential equation for nonlinear systems cannot be solved analytically and it is computationally intensive due to its high dimension. In recent practice, the state dependent Riccati equation [5,6] has been used to find the solution of the optimal control problem for nonlinear systems. However, this method is applicable to some special class of nonlinear systems only. For designing optimal controller for nonlinear systems, control Lyapunov function (CLF) [7–11] has been effectively used. Unlike the Lyapunov function, the CLF is defined for systems with inputs having no specified feedback law. Sontag [12]

n

Corresponding author. Tel.: þ 91 7896172212. E-mail addresses: [email protected] (M. Das), [email protected] (C. Mahanta).

showed that if the CLF could be found for the nonlinear system, there would exist a feedback controller to make the system asymptotically stable. Moreover, every CLF solves the HJB equation associated with a meaningful cost. So, if it is possible to find the CLF for a nonlinear system, it is also possible to find the optimal control law without actually solving the HJB equation. Although being a cornerstone of the modern control theory, an optimal control technique has a major limitation that it is usually based on the precise mathematical model of the system under consideration. So, the exact definition of the system is required for designing the optimal controller. If the system is affected by any uncertainty caused by parameter change or external disturbance during operation, the optimal controller is most likely to fail. However, in reality, all physical systems are affected by system uncertainties due to modeling error, parameter variation or external disturbances. The performance criterion, which is optimized based on the nominal system, deviates from the optimal value under such uncertain conditions and may even drive the system towards instability. Hence the optimal controller must be robust to guarantee undegraded performance in the presence of uncertainty and disturbance. In the field of robust control, sliding mode has gained wide popularity as a successful control strategy for nonlinear uncertain systems. The sliding mode control (SMC) [13–20] was proposed in the beginning of 1970 and it has received growing attention in the control community due to its simplicity and inherent robustness towards the matched uncertainty. The SMC comprises a discontinuous control input that drives the controlled system onto a

http://dx.doi.org/10.1016/j.isatra.2014.03.013 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Das M, Mahanta C. Optimal second order sliding mode control for nonlinear uncertain systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.013i

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specified sliding surface. Once the system is on the sliding surface, it becomes immune to matched uncertainties. A typical SMC is conservatively designed for the worst case scenario in which the system uncertainties and external disturbances are dominant. In such cases, stability is the main concern which requires a high control gain to tackle the unknown parametric variations and external disturbance. But during nominal operation of the uncertain system, stability is not the only concern of controller design. Criteria like minimization of the control input and faster convergence to the equilibrium state also need attention in the nominal operation phase. Hence designing an optimal control strategy to drive the system when it is far away from the equilibrium where its nominal part is dominant is an important issue. On the other hand, close to the equilibrium, when uncertainties become the dominant factor, the SMC fits in as an ideal controller to ensure robustness. For this dual purpose, optimal control strategy has of late been integrated with the SMC giving rise to optimal sliding mode control (OSMC). Active research is continuing in the field of optimal sliding mode control (OSMC) in continuous as well as discrete domains [21–25]. Zhou et al. [26,27] proposed a time optimal sliding mode controller for the hard disk drive. Evolution of the OSMC is not limited to linear uncertain systems only. Attempts have been made to develop OSMCs for nonlinear uncertain systems with matched uncertainty [28–32]. Basin et al. [33] proposed an OSMC for time varying uncertain systems. In the OSMC, the optimal control and the SMC strategies are efficiently combined by designing an integral sliding surface [34–36] where an integral term is incorporated into the sliding manifold. This ensures that the system trajectories start in the sliding manifold itself and the reaching phase is totally eliminated. Hence the controlled system becomes invariant towards the matched uncertainty right from the beginning. Although the integral sliding mode (ISM) controller guarantees robustness against system uncertainty, the crucial part is the finite but high frequency switching of the control signal known as chattering which are harmful undesired oscillations. Commonly, a linear hyperplane is used to design the switching manifold in the SMC. But the main disadvantage of the linear manifold is that the system states in the sliding mode do not converge to zero in finite time. To achieve finite time convergence of the system, the terminal sliding mode [37–40] has been developed. In recent years, the terminal sliding mode controller has become very popular as it ensures finite time stability of the sliding surface. To overcome the singularity problem associated with terminal SMCs, nonsingular terminal sliding mode control (NTSMC) was proposed in [41,42]. Unfortunately, the terminal sliding mode control features the same drawback of chattering as in the case of conventional SMCs. The implementation of the OSMC is difficult due to the chattering present in the control input [28–32] which leads to premature wear and tear or even breakdown of the controlled system. A good number of methods have been proposed in the past to overcome chattering in the SMCs [43–49]. One way to reduce chattering is to use the boundary layer technique where the sign function is replaced with a saturation function and the sliding function converges and remains within the sliding layer [43–45]. However, this method degrades robustness of the controller. In [50], online estimation of the equivalent control is used to reduce the discontinuous control component. The most recent and interesting approach evolved for elimination of chattering is the higher order sliding mode (HOSM) methodology [42,46–49] which eliminates chattering but retains the key features of the sliding mode. In higher order sliding mode, the control action appears explicitly in the higher derivatives of the sliding variable. However, the main constraint in implementation of higher order sliding modes is the high information demand. Among higher order sliding mode controllers, second order sliding mode (SOSM)

controllers [51] are widely used because of their simplicity and low information demand. This paper proposes an optimal control strategy for nonlinear uncertain systems based on the second order sliding mode control. The design procedure has two major steps. In the first step, an optimal controller is designed for the nominal operating region based on the control Lyapunov function (CLF). In the second step, a second order sliding surface is designed by combining an integral sliding mode with a terminal sliding mode. The resulting control law which contains the discontinuous signum function is the derivative of the applied control. The actual control is obtained by integrating the derivative control and hence it becomes continuous and smooth devoid of the harmful high frequency chattering. The main advantage of the proposed optimal sliding mode control scheme is the robustness imparted to the optimal controller by the sliding mode strategy for which it is able to maintain its optimal performance even in the presence of uncertainty caused by system parameter change and external disturbance. Chattering mitigation in the control input is another important achievement of the proposed control method. The outline of this paper is as follows. Section 2 describes the nonlinear uncertain system and defines the control problem. The design procedure for the optimal controller is discussed in Section 3. The second order sliding surface design and the resulting controller are explained in Section 4. Effectiveness of the proposed controller is demonstrated in Section 5 by performing simulation studies. Conclusions are drawn in Section 6.

2. System description and problem formulation Let us consider the following nonlinear uncertain system: x_ ¼ f ðxÞ þ Δf ðxÞ þ ðgðxÞ þ ΔgðxÞÞu þ dðx; tÞ

ð1Þ

where x A Rn is the state vector and u A R1 is the control input. Further, f(x), g(x) are the nominal parts of the system and Δf ðxÞ, ΔgðxÞ represent the system uncertainty. Moreover, dðx; tÞ represents external disturbance affecting the system. Assumption. It is assumed that system uncertainties Δf ðxÞ, ΔgðxÞ and external disturbance dðx; tÞ satisfy the matched condition which means that these are in the range space of the input matrix. Now, it may be written that

Δf ðxÞ þ ΔgðxÞu þdðx; tÞ ¼ gðxÞdm ðx; u; tÞ

ð2Þ

where the known upper bound of dm ðx; u; tÞ is dmax and is given by dmax ¼ supjdm ðx; u; tÞj

ð3Þ

The objective of the proposed control scheme is to design a chattering free optimal sliding mode controller for the nonlinear uncertain system (1). The design of the optimal sliding mode controller is followed in two steps, viz. (i) designing the optimal controller for the nominal nonlinear system and (ii) designing a sliding mode controller to tackle the uncertainty affecting the system. So the control input u can be expressed as u ¼ u1 þ u2

ð4Þ

where u1 is the optimal control law to stabilize the nominal system and u2 is the sliding mode control used to keep the system onto the sliding surface to ensure robustness in the presence of uncertainties.

Please cite this article as: Das M, Mahanta C. Optimal second order sliding mode control for nonlinear uncertain systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.013i

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3. Optimal control for the nominal system

defined as [7]

Neglecting the uncertainties and external disturbance, the system defined in (1) can be written as

u ¼  12 R  1 Lg V 

x_ ¼ f ðxÞ þ gðxÞu1

ð5Þ

and the performance index J chosen to optimize the control input u1 is defined as Z 1 ðlðxÞ þ uT1 Ru1 Þ dt ð6Þ J¼

3

ð13Þ

If the level curve of V agrees with the shape of V  , then Sontag's formula (9) produces the optimal controller. Even though, in general, V  is not the same as V. So, a scaler function λ is considered such that V  ¼ λV. Now the optimal controller is given by u ¼  12 R  1 ðλLg VÞT

ð14Þ

0

Here l(x) is a continuously differentiable, positive semidefinite function, R A R1 is positive definite and ½f ; l is zero state detectable, with the desired solution being a state feedback control law. It is to be noted that the existence of a Lyapunov function for the nonlinear system (5) is a necessary and sufficient condition for determining its stability. One way to stabilize a nonlinear system is to select a Lyapunov function V first and then try to find a feedback control u1 that makes V_ negative definite. The Lyapunov stability criterion finds stability of dynamic systems without inputs and it has been typically applied to closed loop control systems. But the idea of the control Lyapunov function (CLF) [7] based controller is to define a Lyapunov candidate for the open loop system and then design a feedback loop that makes the Lyapunov derivative negative. Hence, if it is possible to find the CLF, then it is also possible to find a stabilizing control law u1. A Lyapunov function for the system defined in (5) is a positive definite, radially unbounded function V(x) and the derivative of the Lyapunov function is given by V_ ¼ Lf V þ Lg Vu1

ð8Þ

By using standard converse theorem it is inferred that if (5) is stabilizable, then there exists a CLF. On the other hand, if there exists a CLF for the system (5), then there also exists an asymptotically stabilizing controller which stabilizes the system (5). Sontag [12] proposed a CLF based controller as given below: 8 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 > > aðxÞ þ aðxÞ2 þ lðxÞbðxÞT R  1 bðxÞ > < 4 5bðxÞT for bðxÞ a 0 u1 ¼ bðxÞbðxÞT > > > :0 for bðxÞ ¼ 0 ð9Þ where aðxÞ ¼ Lf V;

bðxÞ ¼ Lg V

lðxÞ þ λLf ðV Þ 

λ2 4

Lg VR  1 ðLg ðVÞÞT ¼ 0

ð15Þ

Now, by solving the above equation (15) and using (10), λ is found as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 aðxÞ þ aðxÞ2 þ lðxÞbðxÞT R  1 bðxÞ A ð16Þ λ ¼ 2@ R  1 bðxÞbðxÞT Substituting the value of λ in (14), controller u is obtained as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 aðxÞ þ aðxÞ2 þ lðxÞbðxÞT R  1 bðxÞ 5bðxÞT ð17Þ u ¼  4 bðxÞbðxÞT

ð7Þ

where L represents the Lie derivative operator. Now, V(x) is a CLF if 8 x a 0, Lg V ¼ 0⟹Lf V o 0:

Moreover, λ can be determined by substituting V  ¼ λV in the HJB equation defined in (11) as

ð10Þ

So, the control input u is defined as follows: 8 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 T 1 > > > 4aðxÞ þ aðxÞ þlðxÞbðxÞ R bðxÞ 5 < bðxÞT for bðxÞ a 0  u ¼ bðxÞbðxÞT > > > : 0 for bðxÞ ¼ 0 ð18Þ The optimal controller u is exactly the same as the controller u1 found by using Sontag's formula (9). So, it is proved that the control effort u1 minimizes the performance index (6). Proof 2 (Stabilization of the nominal system). Let us consider (7) again and replace u1 there by using (9). Utilizing the property of the CLF i.e. Lf VðxÞ o 0 when Lg VðxÞ ¼ 0, it can be resolved that V_ o 0 for u1 ¼ 0 and for u1 a 0, (7) is obtained by using (10) as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V_ ¼  aðxÞ2 þ lðxÞbðxÞT R  1 bðxÞ o  j aðxÞ2 þ lðxÞbðxÞT R  1 bðxÞj o0 ð19Þ

Theorem 1. The control law u1 defined in (9) stabilizes the nominal nonlinear system defined in (5) by minimizing the performance index (6). Proof 1 (Minimization of the performance index). In order to find the optimal stabilizing controller for a nonlinear system, the Hamilton Jacobi Bellman (HJB) equation [52] needs to be solved. The HJB equation [52] is given by lðxÞ þ Lf V   14 Lg V  R  1 ðxÞðLg V  ÞT ¼ 0

ð11Þ

where V  is the solution of the HJB equation and is commonly referred to as a value function defined as Z 1 V  ¼ inf ðlðxÞ þuT1 Ru1 Þ dτ ð12Þ u1

t

If there exists a continuously differentiable, positive definite solution of the HJB equation (11), then the optimal controller is

Hence, it is proved that the control input defined in (9) minimizes the performance index (6) and stabilizes the nominal nonlinear system (5). Remark 1. In (6), the weighing matrix R needs to be chosen suitably to find the optimal control law ensuring the desired performance. Remark 2. It is easy to find the CLF for two dimensional systems by using analytical methods. However, if the system dimension is higher, it is difficult to find the CLF analytically. Remark 3. For applying the CLF based optimal controller, full knowledge about the system being considered is a necessary prerequisite. But if the system is affected by uncertainty in the neighborhood of the equilibrium state, the optimal controller loses its effectiveness. An efficient way to inhibit this limitation is to integrate the optimal controller with the sliding mode control (SMC) to ensure robustness.

Please cite this article as: Das M, Mahanta C. Optimal second order sliding mode control for nonlinear uncertain systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.013i

M. Das, C. Mahanta / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4

4. Integral terminal sliding mode control In order to tackle uncertainties affecting the nonlinear system (1), an integral sliding mode controller is proposed. In contrast to the conventional SMC, the system motion in integral sliding mode control (ISMC) has dimension equal to that of the state space. The optimal control strategy described in the earlier section can be easily incorporated into an integral sliding surface based SMC [23] as explained below. Let us consider an integral sliding surface defined as   Z t s ¼ G x  x0  x_ nom dt ¼ 0 ð20Þ 0

where G is the design parameter chosen such that Gg(x) is invertible, x0 is the initial state and xnom is the nominal part of the uncertain system (1) neglecting uncertainties and external disturbance. Hence, s_ ¼ G½x_  x_ nom 

ð21Þ

Then, by using (1) and (2), it follows that ð22Þ

As the reaching phase is eliminated in the ISMC, the sliding mode exists from the very beginning. In the conventional ISMC, the switching control u2 is designed based on the η-reachability condition [16] defined as s_ o  ρ sgnðsÞ

where η1 4 0 and ε1 4 0. Taking the first time derivative of the terminal sliding surface (26) yields   α α β _ 2  α=β € s_ ¼ s_ þ δ s_ α=β  1 s€ ¼ δ s_ α=β  1 þs ð33Þ s

β

β

δα

For the design parameters α; β satisfying (28) and (29), it can be shown [40] that s_ α=β  1 4 0

for

s_ α=β  1 ¼ 0

only for

s_ a 0 s_ ¼ 0

ð34Þ

in (33) can be Further, from (27), (28) and (34), δðα=βÞs replaced by η2 4 0 for s_ a 0. Hence (33) can be written as   β _ 2  α= β € s s_ ¼ η2 þs ð35Þ

δα

Substituting the value of s_ from (35), (32) can be expressed as   β _ 2  α=β € s η2 þ s ¼  η1 sgnðsÞ  ε1 s or;

δα β _ 2  α=β € s þ s ¼  η sgnðsÞ  εs ð36Þ δα where η ¼ η1 =η2 4 0 and ε ¼ ε1 =η2 4 0. Then (36) can be rewritten as

ð24Þ

s€ ¼  η sgnðsÞ  εs 

s40 s¼0

ð32Þ

ð23Þ

where ρ 4 0 and sgn(s) is defined as 8 > < 1; sgnðsÞ ¼ 0; > :  1;

s_ ¼  η1 sgnðsÞ  ε1 s

_ α=β  1

s_ ¼ G½f ðxÞ þgðxÞðu1 þ u2 Þ þ gðxÞdm ðx; u; tÞ  f ðxÞ gðxÞu1  ¼ G½gðxÞu2 þ gðxÞdm ðx; u; tÞ

Moreover, the terminal sliding surface brings the system onto the integral sliding surface in finite time. The above strategy of using a terminal sliding mode based on an integral sliding surface gives rise to a second order SMC. As s reaches zero in finite time, both s and s_ also reach zero in finite time. Using the constant plus proportional reaching law [53] for the terminal sliding surface s gives rise to

so0

β _ 2  α=β s δα

ð37Þ

Differentiating (31) with time gives rise to Using (22) and (23)> yields u2 o  ðGgðxÞÞ  1 ½ρ sgnðsÞ þ GgðxÞdm ðx; u; tÞ

ð25Þ

As the switching control u2 is influenced by the sign function of the sliding surface, chattering is prevalent in the control input. To eliminate chattering in the ISMC, a nonsingular terminal sliding manifold s [40,42] is used where

s ¼ s þ δs_ α=β and

ð26Þ

δ is the switching gain chosen such that

δ40

ð27Þ

Here α, β are selected in such a way that these satisfy the following conditions:

α; β A f2n þ 1 : n is an integerg

ð28Þ

and

α β

1 o o2

ð29Þ

In (26) above, s is the integral sliding surface designed as   Z t s ¼ G x ð30Þ x_ nom dt 0

_ _ s€ ¼ G½gðxÞu_ 2 þ gðxÞu 2 þζ

ð38Þ

Hence from (37) and (38), the switching control law is designed as follows:   Z t β _ 2  α=β _ ðGgðxÞÞ  1 þ GgðxÞu þ η sgnð s Þ þ εs ð39Þ s dτ u2 ¼  2

δα

0

where the design parameter is chosen in such a way that jGζ_ j o η [49]. Theorem 2. The nonlinear system (1) can asymptotically converge to zero if the second order sliding manifold is chosen as in (26) and the control law is designed as u ¼ u1 þ u2 where 8 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 > > aðxÞ þ aðxÞ2 þ lðxÞbðxÞT R  1 bðxÞ > < 4 5bðxÞT u1 ¼ bðxÞbðxÞT > > > :0

for bðxÞ a 0 for bðxÞ ¼ 0

and Z

t

u2 ¼  0

ðGgðxÞÞ  1





β _ 2  α=β _ þ GgðxÞu s 2 þ η sgnðsÞ þ εs dτ δα

ð40Þ

Then, using (1) and (2), the first time derivative of the sliding surface (30) is expressed as follows:

Proof (Convergence of the sliding surface). Let us consider the Lyapunov function as

s_ ¼ G½gðxÞu2 þ gðxÞdm ðx; u; tÞ ¼ G½gðxÞu2 þ ζ 

V 1 ¼ 12 s2

ð31Þ

where gðxÞdm ðx; u; tÞ ¼ ζ . It is observed from (30) that knowledge about the initial state of the system is not required to design the integral sliding surface.

ð41Þ

Taking time derivative of (41) yields V_ 1 ¼ ss_

ð42Þ

Please cite this article as: Das M, Mahanta C. Optimal second order sliding mode control for nonlinear uncertain systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.013i

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Using (33) and (38) in (42) yields   αδ V_ 1 ¼ s s_ þ s_ α=β  1 s€ β   αδ _Þ _ þ G ζ ¼ s s_ þ s_ α=β  1 ðGgðxÞu_ 2 þ GgðxÞu 2

ð43Þ

From (40), it is obtained that   β _ 2  α=β _ þGgðxÞu s u_ 2 ¼  ðGgðxÞÞ  1 2 þ η sgnðsÞ þ εs

ð44Þ

5

u1

xnom OPTIMAL

β

δα

So (43) can now be written as   αδ β _ V_ 1 ¼ s s_ þ s_ α=β  1  s_ 2  α=β  GgðxÞu 2 δα β  _ _  η sgnðsÞ  εs þ GgðxÞu 2 þ Gζ   αδ _ α=β  1 s ð  η sgnðsÞ  εs þ Gζ_ Þ ¼s

SYSTEM

x

CONTROLLER

u1

u2 TERMINAL

s

SLIDING SURFACE

INTEGRAL SLIDING SURFACE (s )

(σ)

Fig. 1. Block diagram of the proposed optimal second order sliding mode controller.

β αδ _ α=β  1 ½  jsjη  εs2 þ sGζ_  ¼ s β αδ r s_ α=β  1 ½ jsjη  s2 ε þ jsjjGζ_ j β

The block diagram of the proposed optimal second order sliding mode control strategy is shown in Fig. 1. ð45Þ 5. Simulation results

As δ 4 0 and 1 o α=β o 2, it follows that s_ α=β  1 4 0 for any s_ a 0 and s_ α=β  1 ¼ 0 only for s_ ¼ 0. Also since jGζ_ jo η, (45) is finally obtained as V_ 1 o 0

ð46Þ

Thus convergence of the terminal sliding manifold s to zero is proved for any initial condition. Suppose in time tr, s reaches zero from sð0Þ a 0 and s ¼ 0 8 t 4 t r . So, once s reaches zero, it remains at zero and based on (45), s will converge to zero in finite time ts. The total time required from sð0Þ a 0 to sðt s Þ can be obtained as follows: _ α=β

¼0 s þ δs 1 s ¼  s_ α=β or

ð47Þ

δ

As α, β are chosen according to (28) and (29), above Eq. (47) can be written as 1

δ β =α or or

sβ=α ¼  s_ ds dt ¼  β=α s δβ=α Z ts Z β=α dt ¼  δ tr

or

ts  tr ¼ 

α

αβ

δ

sðt r Þ

β=α

ds sβ=α

½sðt s Þ

ðα  βÞ=α

 sðt r Þ

ðα  βÞ=α



ð48Þ

Since sðt s Þ ¼ 0 at t ¼ t s , above Eq. (48) yields ts ¼ tr þ

α

αβ

δ

β=α

jsðt r Þj

ðα  βÞ=α

5.1. Example 1: Second order nonlinear system A second order nonlinear uncertain system [54] is considered as x_ 1 ¼ x2 x_ 2 ¼  ðx21 þ 1:5x2 Þ þ u ð0:3 sin ðtÞx21 þ 0:2 cos ðtÞx2 Þ þ dðtÞ

ð49Þ

Hence, it is confirmed that the integral sliding surface s converges to zero in finite time. Hence it follows that states of the system converge to the equilibrium state asymptotically. Remark 4. The parameter ε defined in (36) is one of the parameters determining the convergence rate of the sliding surface. It is clear that a large value of ε will force the system states to converge to the origin with a high speed. However, it will require a very high control input but in reality the input is always limited within a fixed value. Thus the parameter ε cannot be selected to be too large. In practice, a compromise has to be made between the response speed and the control input.

ð50Þ

where d(t) is a random noise of 0 mean and 0.5 variance. The above second order nonlinear system (50) can be expressed as (1) where f ðxÞ ¼ ½x2  x21  1:5x2 T , gðxÞ ¼ ½0 1T , Δf ðxÞ ¼ ½0  0:3 sin ðtÞx21  0:2 cos ðtÞx2 T and ΔgðxÞ ¼ 0. The initial state xð0Þ ¼ ½1:5 0T . The performance index J is defined as   Z 1  1 0 x þ uT 10u dt xT J¼ 0 1 0 The control Lyapunov function is chosen as   5:8 5:51 V ¼ xT x: 5:51 5:51

1

sðt s Þ

The proposed optimal second order sliding mode controller is applied to stabilize second and third order nonlinear systems affected by matched uncertainties as described below.

ð51Þ

ð52Þ

Design parameters of the proposed optimal second order sliding mode controller (OSOSMC) (40) are chosen as follows: G ¼ ½0 1;

α ¼ 5;

β ¼ 3;

δ ¼ 0:5;

η ¼ 0:6 and ε ¼ 0:2:

The proposed optimal second order sliding mode controller (OSOSMC) is applied to stabilize the above system (50). The results obtained by applying the proposed optimal second order sliding mode controller are compared with those obtained by using the adaptive sliding mode controller designed by Kuo et al. [54]. The states and the control inputs obtained by applying the proposed optimal second order sliding mode control (OSOSMC) and the adaptive sliding mode control (ASMC) proposed by Kuo et al. [54] are shown in Figs. 2 and 3 respectively. It is observed from Figs. 2 and 3 that both the controllers stabilize the considered nonlinear uncertain system to the equilibrium state at the same rate although the control input in the case of the proposed OSOSMC contains lesser chattering than that of Kuo et al. [54]. To measure the smoothness of the control input, the total variance (TV) [55] of the control input is computed. TV is

Please cite this article as: Das M, Mahanta C. Optimal second order sliding mode control for nonlinear uncertain systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.013i

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6

1.5

x1 with control law proposed by Kuo et al. [54] x2with control law proposed by Kuo et al.[54]

1

x1 with proposed OSOSMC

Method

Total variation (TV)

Control energy

ASMC [54] Proposed OSOSMC

3.65 1.52

1.13 0.60

x2 with proposed OSOSMC

0.5 States

Table 1 Control inputs using the ASMC [54] and the proposed OSOSMC.

0 0.5 1.5

x1

1 0

1

2

3

4

5

6

7

8

9

10

x2

x3

1

Time (sec) States

0.5

Fig. 2. States obtained by applying the proposed OSOSMC and ASMC by Kuo et al. [54].

0 0.5 1

0.6 Control law proposed by Kuo et al. [54]

1.5

Proposed OSOSMC

0.4

0

2

4

6

8

10

12

14

16

18

20

Time (sec) Control input

0.2

Fig. 4. States obtained by applying the proposed OSOSMC.

0 0.2

x1

x2

x3

0 0.05 6

0

1

2

3

4

5

6

7

1 6.05 6.1 6.15 6.2

8

9

0.5

10

Time (sec)

Fig. 3. Control inputs obtained by applying the proposed OSOSMC and ASMC by Kuo et al. [54].

States

0.4 0.6

1.5

0.05

0 0.5 1

calculated as follows:

1.5

n

TV ¼ ∑ jui þ 1 ðtÞ  ui ðtÞj

0

2

Let us consider a third order nonlinear uncertain system [56] given below: x_ 1 ¼ x2 x_ 2 ¼ x3 x_ 3 ¼  6x1  2:92x2 1:2x3 þ x21 þu þ dðtÞ

ð54Þ

where d(t) is a random noise of 0 mean and 0.5 variance. The above second order nonlinear system (54) can be expressed as  T (1) where f ðxÞ ¼ x2 x3  6x1 2:92x2  1:2x3 þ x21 , gðxÞ ¼ ½0 0 1T , Δf ðxÞ ¼ 0 and ΔgðxÞ ¼ 0. The initial state xð0Þ ¼ ½  1 1 0T . The performance index J is defined as

J¼ 0

10

12

14

16

18

20

Time(sec)

The control Lyapunov function is chosen as 2 3 5:000 0:166 0:176 6 7 VðtÞ ¼ xT 4 0:166 20:720 1:820 5x 0:176 1:820 5:850

0

0:1

6 TB 4x @ 0 0

0 0:1 0

0

1

3

7 C 0 Ax þ uT 2:8u5dt 0:1

ð55Þ

ð56Þ

Design parameters of the proposed optimal second order sliding mode controller (OSOSMC) (40) are chosen as follows: G ¼ ½0 0 1;

5.2. Example 2: Third order nonlinear system

2

8

Fig. 5. States obtained by applying GSMC by Liu et al. [56].

where n is the number of samples. Also, the energy of the control input u(t) is calculated by using the 2-norm method. Table 1 shows the TV and the 2-norm of the control input calculated for the period from 0 to 10 s with a sampling time of 0.1 s. It is clear from Table 1 that the proposed OSOSMC is able to produce a smoother control input with substantial reduction in the control effort than that in the case of the ASMC proposed by Kuo et al. [54].

1

6

ð53Þ

i¼1

Z

4

α ¼ 7;

β ¼ 5;

δ ¼ 5;

η ¼ 0:1 and ε ¼ 0:2:

The proposed optimal second order sliding mode controller (OSOSMC) is applied to stabilize the above nonlinear uncertain system (54). The results obtained by applying the proposed optimal second order sliding mode controller are compared with those obtained by using the global sliding mode controller (GSMC) designed by Liu et al. [56]. The states obtained by applying the proposed OSOSMC and the GSMC [56] are shown in Figs. 4 and 5 respectively. The control inputs obtained by using the proposed OSOSMC and the GSMC [56] are compared in Fig. 6. It is observed from Figs. 4–6 that both the controllers stabilize the considered nonlinear uncertain system to the equilibrium state at the same rate with control inputs containing almost equal level of chattering. Table 2 compares the total variation (TV) and the 2-norm of the control inputs for the proposed OSOSMC and the GSMC [56] calculated for the period from 0 to 20 s with a sampling time of 0.1 s. From Table 2 it is obvious that in both the methods, the control inputs are almost equally smooth. However, the control energy spent in the case of the proposed OSOSMC is substantially lower in comparison to that in the GSMC [56].

Please cite this article as: Das M, Mahanta C. Optimal second order sliding mode control for nonlinear uncertain systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.013i

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Proposed OSOSMC Control law proposed by Liu et al.[56]

Control input

0 0.1

5 0 0.1 17

10

17.5

18

15 0

2

4

6

8

10

12

14

16

18

20

Time (sec)

Fig. 6. Control inputs obtained by applying the proposed OSOSMC and GSMC by Liu et al. [56].

Table 2 Control inputs using the GSMC [56] and the proposed OSOSMC. Method

Total variation (TV)

Control energy

GSMC [56] Proposed OSOSMC

14.28 14.96

16.07 9.35

6. Conclusion This paper proposes an optimal second order sliding mode controller (OSOSMC) for nonlinear systems with matched uncertainty. Control Lyapunov function is used to design the optimal controller for the nominal system. The optimal control law is combined with the sliding mode control by designing an integral sliding surface for ensuring robustness. As a conventional sliding mode controller is severely affected by chattering, the proposed optimal sliding mode controller is made second order by adding a terminal sliding surface. The terminal sliding mode ensures finite time stability of the sliding surface. Simulation results confirm that the proposed optimal second order sliding mode control assures stability of the nonlinear uncertain system at the same speed with some existing sliding mode control strategies but spends substantially lower control energy than those controllers. Chattering reduction is another major advantage of the proposed optimal second order sliding mode controller. References [1] Lewis FL, Vrabie DL, Syrmos VL. Optimal control. New York: Wiley; 2012. [2] Tang G, Sun H, Pang H. Approximately optimal tracking control for discrete time-delay systems with disturbances. Prog Nat Sci 2008;18(2):225–31. [3] Tang G-Y. Suboptimal control for nonlinear systems: a successive approximation approach. Syst Control Lett 2005;54(5):429–34. [4] Kirk DE. Optimal control theory an introduction. New York: Prentice Hall; 1970. [5] Pittner J, Simaan M. State-dependent Riccati equation approach for optimal control of a tandem cold metal rolling process. IEEE Trans Ind Appl 2006;42(3): 836–43. [6] Nekoo SR. Nonlinear closed loop optimal control: a modified state-dependent Riccati equation. ISA Trans 2013;52(2):285–90. [7] Pahlevaninezhad M, Das P, Drobnik J, Moschopoulos G, Jain P, Bakhshai A. A nonlinear optimal control approach based on the control-Lyapunov function for an AC/DC converter used in electric vehicles. IEEE Trans Ind Inf 2012;8(3): 596–614. [8] Pukdeboon C, Zinober A. Control Lyapunov function optimal sliding mode controllers for attitude tracking of spacecraft. J Frankl Inst 2012;349(2): 456–75. [9] Freeman RA, Kokotovic PV. Inverse optimality in robust stabilization. SIAM J Control Optim 1996;34(4):1365–91. [10] Zhang J, Han Z, Huang J. Homogeneous feedback design of differential inclusions based on control Lyapunov functions. Commun Nonlinear Sci Numer Simul 2013;18(10):2790–800.

7

[11] Zhang J, Han Z, Huang J. Homogeneous feedback control of nonlinear systems based on control Lyapunov functions. Asian J Control 2013;15(6):1–9. [12] Sontag ED. A universal construction of Artstein's theorem on nonlinear stabilization. Syst Control Lett 1989;13(2):117–23. [13] Utkin V. Variable structure systems with sliding modes. IEEE Trans Autom Control 1977;22(2):212–22. [14] Utkin V. Sliding modes in control and optimization. Berlin: Springer-Verlag; 1992. [15] Utkin V. Sliding mode control in dynamic systems. In: IEEE proceedings on decision and control, vol. 3; 1993. p. 2446–51. [16] Edwards C, Spurgeon SK. Sliding mode control, theory and applications. London, UK: Taylor and Francis; 1998. [17] Edwards C, Spurgeon SK, Patton RJ. Sliding mode observers for fault detection and isolation. Automatica 2000;36(4):541–53. [18] Wai R-J, Chang L-J. Adaptive stabilizing and tracking control for a nonlinear inverted-pendulum system via sliding-mode technique. IEEE Trans Ind Electron 2006;53(2):674–92. [19] Wang J-J. Stabilization and tracking control of X–Z inverted pendulum with sliding-mode control. ISA Trans 2012;51(6):763–70. [20] Basin M, Rodriguez-Ramirez P. Sliding mode controller design for linear systems with unmeasured states. J Frankl Inst 2012;349(4):1337–49. [21] Young KD, Özgüner Ü. Sliding-mode design for robust linear optimal control. Automatica 1997;33(7):1313–23. [22] Xu R, Özgüner Ü. Optimal sliding mode control for linear systems. In: IEEE proceedings on variable structure systems; 2006. p. 143–8. [23] Pang HP, Tang G-Y. Global robust optimal sliding mode control for a class of uncertain linear systems. In: IEEE proceedings on control and decision; 2008. p. 3509–12. [24] Janardhanan S, Kariwala V. Multirate-output-feedback-based LQ-optimal discrete-time sliding mode control. IEEE Trans Autom Control 2008;53(1): 367–73. [25] Yuankai L, Zhongliang J, Shiqiang H. Dynamic optimal sliding-mode control for six-DOF follow-up robust tracking of active satellite. Acta 2011;69(7–8): 559–70. [26] Zhang DQ, Guo G. Discrete-time sliding mode proximate time optimal seek control of hard disk drives. In: IEEE proceedings on control theory and applications, vol. 147; 2000. p. 440–6. [27] Zhou J, Zhou R, Wang Y, Guo G. Improved proximate time-optimal slidingmode control of hard disk drives. In: IEEE proceedings on control theory and applications, vol. 148; 2001. p. 516–22. [28] Pang H, Wang L. Global robust optimal sliding mode control for a class of affine nonlinear systems with uncertainties based on SDRE. In: IEEE proceedings on computer science and engineering, vol. 2; 2009. p. 276–80. [29] Dong R, Tang G, Guo Y. Optimal sliding mode control for uncertain systems with time delay. In: IEEE proceedings on control and decision; 2010. p. 2814–9. [30] Dong R, Gao H-W, Pan Q-X. Optimal sliding mode control for nonlinear systems with uncertainties. In: IEEE proceedings on control and decision; 2011. p. 2098–103. [31] Baradaran-nia M, Alizadeh G, Khanmohammadi S, Azar BF. Optimal sliding mode control of single degree-of-freedom hysteretic structural system. Commun Nonlinear Sci Numer Simul 2012;17(11):4455–66. [32] Xu J-X. A quasi-optimal sliding mode control scheme based on control Lyapunov function. J Frankl Inst 2012;349(4):1445–58. [33] Basin M, Rodriguez-Ramirez P, Ferrara A, Calderon-Alvarez D. Sliding mode optimal control for linear systems. J Frankl Inst 2012;349(4):1350–63. [34] Baik IC, Kim KH, Youn MJ. Robust nonlinear speed control of PM synchronous motor using boundary layer integral sliding mode control technique. IEEE Trans Control Syst Technol 2000;8(1):47–54. [35] Niu Y, Ho DWC, Lam J. Robust integral sliding mode control for uncertain stochastic systems with time-varying delay. Automatica 2005;41(5):873–80. [36] Castanos F, Fridman L. Analysis and design of integral sliding manifolds for systems with unmatched perturbations. IEEE Trans Autom Control 2006;51(5):853–8. [37] Zhihong M, Paplinski A, Wu H. A robust MIMO terminal sliding mode control scheme for rigid robotic manipulators. IEEE Trans Autom Control 1994;39(12): 2464–9. [38] Yu X, Zhihong M. Multi-input uncertain linear systems with terminal slidingmode control. Automatica 1998;34(3):389–92. [39] Yu X, Zhihong M. Fast terminal sliding-mode control design for nonlinear dynamical systems. IEEE Trans Circuits Syst I: Fundam Theory Appl 2002;49 (2):261–4. [40] Komurcugil H. Adaptive terminal sliding-mode control strategy for DC–DC buck converters. ISA Trans 2012;51(6):673–81. [41] Feng Y, Yu X, Man Z. Non-singular terminal sliding mode control of rigid manipulators. Automatica 2002;38(12):2159–67. [42] Chen S-Y, Lin F-J. Robust nonsingular terminal sliding-mode control for nonlinear magnetic bearing system. IEEE Trans Control Syst Technol 2011;19(3): 636–43. [43] Su W-C, Drakunov S, Özgüner Ü. An O(T2) boundary layer in sliding mode for sampled-data systems. IEEE Trans Autom Control 2000;45(3):482–5. [44] Huang Y, Kuo T. Robust position control of DC servomechanism with output measurement noise. J Electr Eng 2006;88(3):223–8. [45] Huang Y-J, Kuo T-C, Chang S-H. Adaptive sliding-mode control for nonlinear systems with uncertain parameters. IEEE Trans Syst Man Cybern, Part B: Cybern 2008;38(2):534–9. [46] Laghrouche S, Plestan F, Glumineau A. Higher order sliding mode control based on integral sliding mode. Automatica 2007;43(3):531–7.

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[47] Defoort M, Floquet T, Kokosy A, Perruquetti W. A novel higher order sliding mode control scheme. Syst Control Lett 2009;58(2):102–8. [48] Defoort M, Nollet F, Floquet T, Perruquetti W. A third-order sliding-mode controller for a stepper motor. IEEE Trans Ind Electron 2009;56(9):3337–46. [49] Wang Y, Zhang X, Yuan X, Liu G. Position-sensorless hybrid sliding-mode control of electric vehicles with brushless DC motor. IEEE Trans Veh Technol 2011;60(2):421–32. [50] Slotine J-JE, Li W. Applied nonlinear control. Englewood Cliffs, New Jersey: Prentice-Hall; 1991. [51] Mondal S, Mahanta C. Nonlinear sliding surface based second order sliding mode controller for uncertain linear systems. Commun Nonlinear Sci Numer Simul 2011;16(9):3760–9.

[52] Freeman RA, Kokotović PV. Robust nonlinear control design. Boston: Birkhaüser; 1996. [53] Hung J, Gao W, Hung J. Variable structure control: a survey. IEEE Trans Ind Electron 1993;40(1):2–22. [54] Kuo T-C, Huang Y-J, Chang S-H. Sliding mode control with self-tuning law for uncertain nonlinear systems. ISA Trans 2008;47(2):171–8. [55] Skogestad S. Simple analytic rules for model reduction and PID controller tuning. J Process Control 2003;13(4):291–309. [56] Liu L, Han Z, Li W. Global sliding mode control and application in chaotic systems. J Nonlinear Dyn 2009;56(1–2):193–8.

Please cite this article as: Das M, Mahanta C. Optimal second order sliding mode control for nonlinear uncertain systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.013i