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Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties
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Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties Alireza Nasirin, Sing Kiong Nguang, Akshya Swain The Department of Electrical and Computer Engineering, The University of Auckland, Auckland, New Zealand Received 29 August 2012; received in revised form 26 November 2012; accepted 24 December 2012
Abstract This paper proposes an adaptive scheme of designing sliding mode control (SMC) for affine class of multi-input multi-output (MIMO) nonlinear systems with uncertainty in the systems dynamics and control distribution gain. The proposed adaptive SMC does not require any a priori knowledge of the uncertainty bounds and therefore offers significant advantages over the non-adaptive schemes of SMC design. The closed loop stability conditions are derived based on Lyapunov theory. The effectiveness of the proposed approach is demonstrated via simulations considering an example of a two-link robot manipulator and has been found to be satisfactory. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Sliding Mode Control has been recognized as one of the effective nonlinear robust control methods due to various advantages this offers which include its invariance to system uncertainties and external disturbances [1]. Essentially, the sliding mode control is a particular type of variable structure control and uses discontinuous control action to drive the state trajectory towards a specific hyper plane in the state space. The state trajectory is then maintained to slide on the specific hyper plane until the origin of the state space is reached [2]. The first step of SMC design is to select a sliding surface which models the desired performance in state space. Then the controller is designed such that the system state trajectories move towards the sliding surface and stay on it. Since its introduction in [3,4], this has gone through n
Corresponding author. Tel.: þ64 221923874. E-mail addresses: [email protected], [email protected] (A. Nasiri), [email protected] (S. Kiong Nguang), [email protected] (A. Swain). 0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.12.019 Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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various theoretical and practical developments and the related literature until 2011 are well documented in [2]. The effectiveness of various types of SMCs for both SISO and MIMO systems have been demonstrated by researchers in the past. For example, in [5–7], the authors have designed SMCs for both SISO and MIMO linear as well as nonlinear systems and applied these to variety of practical systems such as robot manipulators, aircrafts, underwater vehicles, space crafts, flexible space structures, electrical motors, power systems, and automotive engines, etc. [8–10]. Most of the SMCs for nonlinear systems, which have been reported, essentially focus on particular class of nonlinear systems such as Lipschitz class of nonlinear systems. Another class of systems which are of practical interest are switched hybrid systems, singular systems, Markovian systems, etc. In recent years various researchers have proposed SMCs for these systems e.g. in [11], the problems of stochastic stability have been considered; in [12] SMC for Markovian jump singular time-delay system has been investigated by considering the bounded L2 gain performance; in [8–10], the SMC has been applied for a switched stochastic systems and, in [13], a robust adaptive SMC has been designed for semi-active vehicle suspension system with parameter uncertainties and external disturbances with the linear model. The performance of SMCs is dependent on the knowledge about the uncertainty bounds of the system which are hard to know in practice. To overcome the problems associated with the unknown bounds, various adaptive techniques, including methods based on fuzzy logic, have been proposed in recent past [14–20]. However, the design of fuzzy based SMCs require some prior heuristic knowledge about the system and the number of fuzzy rules increases as the order and the complexity of the system increases. Recently in [20], adaptive SMCs have been designed for affine class of SISO nonlinear systems with uncertainty. However, the extension of this to MIMO systems is more complicated. In the present study an adaptive SMC has been designed for affine class of MIMO nonlinear systems, assuming that there exists uncertainty both in the system dynamics and in the control distribution gain. Various conditions of stability for the proposed adaptive SMC have been derived using Lyapunov approach and the effectiveness of the adaptive SMC is demonstrated considering an example of a two link robot manipulator. The rest of the paper is organized as follows. Section 2 briefly describes about the affine class of MIMO nonlinear systems and introduces few definitions which are subsequently being used in adaptive SMC design. A robust SMC for MIMO nonlinear systems is proposed in Section 3. In Section 4, design of the proposed adaptive SMCs is carried out. The effectiveness of the proposed design is illustrated by an example in Section 5 with conclusions in Section 6. 2. Problem formulation Consider a MIMO affine nonlinear system x_ ¼ AðxÞ þ
m X
bi ðxÞui
i¼1
y1 ¼ h1 ðxÞ ^ ym ¼ hm ðxÞ
ð1Þ
Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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where x 2 Rn is state vector, u 2 Rm represents control inputs, y 2 Rm stands for outputs, AðxÞ,b1 ðxÞ, . . . ,bim ðxÞ are smooth vector fields, and h1 ðxÞ, . . . ,hm ðxÞ are smooth functions, defined on the open set of Rn. A multivariable nonlinear system of the form (1) has a vector relative degree fr1 r2 . . . rm g at a point xo if Lbj LkA hi ðxÞ ¼ 0 81rjrm and krri 1 for all x in a neighborhood of Denote 2 r1 1 Lb1 LA h1 ðxÞ 6 r2 1 6 Lb1 LA h2 ðxÞ GðxÞ ¼ 6 6 4 rm 1 Lb1 LA hm ðxÞ
ð2Þ
xo .
Lbm LrA1 1 h1 ðxÞ
Lbm LrA2 1 h2 ðxÞ
Lbm LrAm 1 hm ðxÞ
3 7 h 7 7 ¼ g1 ðxÞ 7 5
g2 ðxÞ
gm ðxÞ
i
ð3Þ
where gi ðxÞ ¼ ½g1i ðxÞ g2i ðxÞ gmi ðxÞT
ð4Þ o
Note that GðxÞ is non-singular at x ¼ x . Using feedback linearization, the nonlinear system (1) can be transformed into following form [21]: yðrÞ ¼ fðxÞ þ GðxÞu where
2
3
ð5Þ 2
3 f1 ðxÞ 6 r2 7 6 7 6 LA h2 ðxÞ 7 6 f2 ðxÞ 7 6 7 6 7 fðxÞ ¼ 6 7¼6 7 4 5 4 5 rm LA hm ðxÞ fm ðxÞ LrA1 h1 ðxÞ
ð6Þ
and ðr2 Þ ðrm Þ T 1Þ yðrÞ ¼ ½yðr 1 ,y2 , . . . ,ym
ð7Þ
ðr2 Þ ðrm Þ T 1Þ x ¼ ½x1 ,x2 , . . . ,xn T ¼ ½y1 , . . . ,yðr 1 ,y2 , . . . ,y2 , . . . ,ym , . . . ,ym
ð8Þ
The relativePdegree of the system is assumed to be equal to the order of the system and is expressed as m i ¼ 1 ri ¼ n. This implies that the system does not have any zero dynamics. Let the desired output trajectory be given by yd ¼ ½yd1 yd2 ydm T
ð9Þ
The tracking error of the system is therefore expressed as e ¼ ½e1 e2 em T
ð10Þ
where ðri 1Þ i 1Þ T ei ¼ ½ei e_i eiðri 1Þ T ¼ ½ydi yi y_ di y_ i ydi yðr i
8i ¼ 1, . . . ,m
ð11Þ
Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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Define a set of sliding surfaces in the error space passing through the origin to represent a sliding manifold as follows: 3 2 ðr1 1Þ þ a11 e1ðr1 2Þ þ þ a1,ðr1 1Þ e1 e1 7 6 ðr2 1Þ 6 e2 þ a21 e2ðr2 2Þ þ þ a2,ðr2 1Þ e2 7 T 7 6 ð12Þ s ¼ ½s1 s2 sm ¼ 6 7 ^ 5 4 ðrm 1Þ m 2Þ em þ am1 eðr þ þ am,ðrm 1Þ em m The derivative of s is given by 2 ðr1 Þ 3 1 1Þ þ þ a1,ðr1 1Þ e_ 1 e1 þ a11 eðr 1 6 ðr2 Þ 7 2 1Þ 6 e2 þ a21 eðr þ þ a2,ðr2 1Þ e_ 2 7 T 2 6 7 s_ ¼ ½_s 1 s_ 2 s_ m ¼ 6 7 ^ 4 5 ðrm Þ ðrm 1Þ em þ am1 em þ þ am,ðrm 1Þ e_ m
ð13Þ
Using the relations in Eqs. (5) and (11), the derivative of the sliding manifold in Eq. (13) can be expressed as 3 2 rX 1 1 ðr1 iÞ a e 7 6 7 6 i ¼ 1 1i 1 7 6 7 6 r2 1 7 6 X ðr2 iÞ 7 6 a e 2i 2 7 6 ðrÞ ð14Þ s_ ¼ yd fðxÞGðxÞu þ 6 i ¼ 1 7 7 6 7 6 ^ 7 6 7 6 rm 1 7 6X m iÞ 5 4 ami eðr m i¼1
3. Design of robust sliding mode control This section proposes a Lyapunov based approach to design a robust sliding mode controller for a MIMO affine nonlinear system. Consider the MIMO nonlinear system described in Eq. (1). After suitable nonlinear transformation, the outputs of this system can be expressed as (5) yðrÞ ¼ fðxÞ þ GðxÞu where GðxÞ and fðxÞ are defined in Eqs. (3) and (6). Note that the functions fðxÞ and GðxÞ are dependent on the system parameters, and any uncertainties in the system parameters are reflected in them. Let the uncertainties in fðxÞ and GðxÞ be expressed as DfðxÞ ¼ fðxÞf 0 ðxÞ ¼ ½Df1 ðxÞ Df2 ðxÞ Dfm ðxÞT
ð15Þ
DGðxÞ ¼ GðxÞG0 ðxÞ
ð16Þ
and where f 0 ðxÞ and G0 ðxÞ represent nominal values of fðxÞ and GðxÞ. Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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Assumption 1. The nominal control distribution function matrix G0 ðxÞ is non-singular. Let ^ ^ 1 g^ 2 g^ m GðxÞ ¼ DGðxÞG1 0 ðxÞ ¼ ½g
ð17Þ
^ where the ith column of GðxÞ is given by g^ i ¼ ½G^ 1i ðxÞ G^ 2i ðxÞ G^ mi ðxÞT
ð18Þ
^ Note that GðxÞ is defined for simplifying various assumptions and theorems which are proposed in this study. Assumption 2. The uncertainties DfðxÞ and DGðx) satisfy the following: 8 JDfi ðxÞJoFi > < maxj ðjG^ ij ðxÞjÞok^ i and mk^ i o1 k^ i 40 8i ¼ 0, . . . ,m > : jG^ ðxÞjog^ ^ max,ii 40 ii max,ii o1 and g
ð19Þ
^ and G^ ii ðxÞ, where Fi, k^ i and g^ max,ii are the upper bounds of Dfi ðxÞ, the ith row of GðxÞ respectively. Using the notation introduced, the time derivative of sliding manifold in Eq. (14) is expressed as s_ ¼ C0 ðx,eÞDfðxÞðG0 ðxÞ þ DGðxÞÞu
ð20Þ
where 2
3
rX 1 1
1 iÞ a1i eðr 1 i¼1
7 6 7 6 7 6 7 6 r2 1 7 6 X ðr2 iÞ 7 6 a2i e2 7 6 ðrÞ T C0 ðx,eÞ ¼ ½c1 ðx,eÞ c2 ðx,eÞ cm ðx,eÞ ¼ yd f 0 ðxÞ þ 6 i ¼ 1 7 7 6 7 6 ^ 7 6 7 6 rm 1 7 6X ðrm iÞ 5 4 ami e
ð21Þ
m
i¼1
It is assumed that each element of C0 ðx,eÞ satisfies ci ðx,eÞrlðxÞ
8i ¼ 1, . . . ,m:
ð22Þ
Theorem 1. For the system in Eq. (5), the motion on the sliding manifold given in Eq. (12) is asymptotically stable, using the controller 1 1 u ¼ G1 0 ðxÞC0 ðx,eÞ þ G0 ðxÞ P signðsÞ þ G0 ðxÞ F0 ðx,eÞ U signðsÞ
ð23Þ
where P ¼ diagðr1 ,r2 , . . . ,rm Þ, U ¼ diagðg1 ,g2 , . . . ,gm Þ and F0 ðx,eÞ ¼ diagðjc1 ðx,eÞj, jc2 ðx,eÞj, . . . ,jcm ðx,eÞjÞ for i ¼ 1,2, . . . ,m. The different elements of matrix P and U satisfy Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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the following: 8 Fi > > r4 > < i 1g^
max,ii
ð24Þ
g^ max,ii > > > : gi 4 1mk^ i Proof. Consider the scalar semi-positive quadratic Lyapunov function V as 1 V ¼ ST S 2
ð25Þ
The time derivative of V along the sliding manifold S is given by V_ ¼
m X
s_ i si
ð26Þ
i¼1
where s_ i si ¼ si Dfi ðxÞsi ri signðsi Þsi jci ðx,eÞj gi signðsi Þsi
m X
G^ ij ðxÞ cj ðx,eÞsi G^ ii ðxÞri signðsi Þ
j¼1
si G^ ii ðxÞ jci ðx,eÞj gi signðsi Þ ¼ si Dfi ðxÞri jsi jG^ ii ðxÞri jsi jsi
m X
^ jci ðx,eÞj gi jsi j G^ ij ðxÞ cj ðx,eÞjci ðx,eÞj gi jsi jGðxÞ
j¼1
ð27Þ Using Assumption 2, and the inequalities of Eqs. (22) and (24), Eq. (27) becomes s_ i si oFi jsi jri jsi jg^ max,ii ri jsi j þ
m X
jG^ ij ðxÞj jcj ðx,seÞj jsi jjci ðx,eÞj gi jsi j
j¼1
þ g^ max,ii jci ðx,eÞj gi jsi j o½ri ð1g^ max,ii ÞFi :jsi j½gi ð1mk^ i Þg^ max,ii jlðxÞj:jsi jo0
ð28Þ
So, the motion on the sliding manifold is asymptotically stable and the output can therefore, track the desired reference. & Remark 1. To determine the control system parameters ri and gi , we need to know Fi, k^ i and g^ max,ii . However, in practice, it is often difficult to find them. 4. Design of robust adaptive sliding mode control In practice, finding the bounds for the uncertainties is difficult. Hence, an adaptive method is required to cope with this differently. Theorem 2. For the system in Eq. (5), the motion on the sliding manifold given in Eq. (12), using the controller given in Eq. (23), is asymptotically stable with the following Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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adaptive laws: ( ri ¼ kri jsi j gi ¼ kgi jsi j jlðxÞj
8i ¼ 1,2, . . . m
7
ð29Þ
where kri and kgi are rates of adaptation. Proof. Define r~ i and g~ i as ( r~ i ¼ ri0 r0 8i ¼ 1,2, . . . m g~ i ¼ gi0 gi where 8 Fi > > > ri0 41g^ <
max,ii
g^ max,ii > > > : gi0 4 1mk^ i
ð30Þ
8i ¼ 1,2, . . . m
ð31Þ
The constants, Fi, g^ max,ii and k^ i are defined in Eq. (19). Consider the Lyapunov function V as V¼
m m m 1g^ max,ii 2 1 X 1X 1X 1mk^ i 2 s2i þ r~ i þ g~ i 2i¼1 2 i ¼ 1 kri 2 i ¼ 1 kgi
ð32Þ
The time derivative of V along the manifold S is given by ! m m m ^i X X X ^ max,ii 1 g 1m k r~_ i r~ i þ V_ ¼ s_ i si þ g~_i g~ i k ri kgi i¼1 i¼1 i¼1 ! ! m m X X 1g^ max,ii _ 1mk^ i _ ¼ s_ i si þ v_ i g~ i g~ i ¼ r~ i r~ i þ kri kgi i¼1 i¼1 where
ð33Þ
! 1mk^ i _ g~ i g~ i kgi m X ^ ¼ si Dfi ðxÞri jsi jG^ ii ðxÞri jsi jsi jci ðx,eÞj gi jsi j G^ ij ðxÞ cj ðx,eÞjci ðx,eÞj gi jsi jGðxÞ
1g^ max,ii _ v_ i ¼ s_ i si þ r~ i r~ i þ kri
j¼1
ð1g^ max,ii Þ ðri0 r0 Þ jsi jð1mk^ i Þ ðgi0 gi Þ jsi j jlðxÞj
ð34Þ
Using Assumption 2, and the inequalities of Eqs. (22), (30), and (31), Eq. (34) becomes
1g^ max,ii _ v_ i ¼ s_ i si þ r~ i r~ i þ kri
! 1mk^ i _ g~ i g~ i kgi
oFi jsi jri jsi jg^ max,ii ri jsi j þ
m X
jG^ ij ðxÞj jcj ðx,eÞj jsi jjci ðx,eÞj gi jsi jG^ ii ðxÞ jci ðx,eÞj gi jsi j
j¼1
ð1g^ max,ii Þ ðri0 r0 Þ jsi jð1mk^ i Þ ðgi0 gi Þ jsi j jlðxÞj oFi jsi jr jsi j þ g^ r jsi j þ mk^ i jlðxÞj jsi jjlðxÞj g jsi j þ g^ i
max,ii i
i
max,ii
jlðxÞj gi jsi j
Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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ð1g^ max,ii Þ ðri0 r0 Þ jsi jð1mk^ i Þ ðgi0 gi Þ jsi j jlðxÞjo0
ð35Þ
So, the motion on the sliding manifold is asymptotically stable and the output can therefore, track the desired reference. & 5. Example In this section, the effectiveness of the proposed controller is illustrated via simulations by considering, an example of a two-link robot manipulator. 5.1. Modeling The dynamic equation of a two-link robot manipulator is given by [22] _ q_ þ GðqÞ ¼ t MðqÞq€ þ Cðq, qÞ
ð36Þ
_ are given, where the moment of inertia M(q) and coriolis centripetal forces Cðq, qÞ respectively, by " # ðm1 þ m2 Þl12 m2 l1 l2 ðs1 s2 þ c1 c2 Þ MðqÞ ¼ ð37Þ m2 l1 l2 ðs1 s2 þ c1 c2 Þ m2 l22 " 1
_ ¼ m2 l1 l2 ðc1 s2 Þ s1 c2 Cðq, qÞ
0 q_ 1
q_ 2 0
The gravitational force G(q) is expressed as " # ðm1 þ m2 Þl1 gs1 GðqÞ ¼ m2 l2 gs2
# ð38Þ
ð39Þ
where q ¼ ½q1 q2 T are the generalized coordinates. m1, m2 are masses of the link and l1 and l2 are lengths of the links. q1, q2 are the angular positions. t ¼ ½t1 t2 T is the vector of applied torques. The acceleration due to gravity is g ¼ 9:8 m=s2 . For convenience, define si ¼ sinðqi Þ, ci ¼ cosðqi Þ for i¼ 1,2. The state, input and output vectors are defined as x ¼ ½x1 x2 x3 x4 T ¼ ½q1 q_ 1 q2 q_ 2 T
ð40Þ
u ¼ ½u1 u2 T ¼ ½t1 t2 T
ð41Þ
y ¼ ½y1 y2 T ¼ ½q1 q2 T
ð42Þ
Using the definitions in Eqs. (40)–(42), the dynamics of two-link manipulator is re-expressed as x_ ¼ AðxÞ þ
2 X
bi ðxÞui
i¼1
y1 ¼ h1 ðxÞ ¼ x1 y2 ¼ h2 ðxÞ ¼ x3
ð43Þ
Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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where AðxÞ ¼ ½A1 ðxÞ A2 ðxÞ A3 ðxÞ A4 ðxÞT
ð44Þ
The various elements of AðxÞ i.e. A1 ðxÞ, . . . ,A4 ðxÞ are given by A1 ðxÞ ¼ x2
ð45Þ
ðs1 c3 c1 s3 Þ ½m2 l1 l2 ðs1 s3 þ c1 c3 Þx22 m2 l22 x24 l1 l2 ½ðm1 þ m2 Þm2 ðs1 s3 þ c1 c3 Þ2 1 ½ðm1 þ m2 Þl2 gs1 m2 l2 gs3 ðs1 s3 þ c1 c3 Þ þ l1 l2 ½ðm1 þ m2 Þm2 ðs1 s3 þ c1 c3 Þ2 ð46Þ
A2 ðxÞ ¼
A3 ðxÞ ¼ x4 A4 ðxÞ ¼
ð47Þ
ðs1 c3 c1 s3 Þ ½ðm1 þ m2 Þl1 x22 þ m2 l1 l2 ðs1 s3 þ c1 c3 Þx24 l1 l2 ½ðm1 þ m2 Þm2 ðs1 s3 þ c1 c3 Þ2 1 þ ½ðm1 þ m2 Þl1 gs1 ðs1 s3 þ c1 c3 Þ þ ðm1 þ m2 Þl1 gs3 l1 l2 ½ðm1 þ m2 Þm2 ðs1 s3 þ c1 c3 Þ2
ð48Þ
Fig. 1. Case 1: m1 ¼0.7 kg, m2 ¼0.8 kg, l1 ¼ 1.3 m, l2 ¼1.3 m and kri ¼ kgi ¼ 5 for i¼ 1,2. Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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and 2
b1i
3
6 b2i 7 6 7 bi ðxÞ ¼ 6 7 4 b3i 5 b4i
8i ¼ 1,2
ð49Þ
b11 ¼ b31 ¼ b12 ¼ b32 ¼ 0
ð50Þ
where
b21 ¼
b22 ¼
b41 ¼
m2 l22 m2 l12 l22 ½ðm1 þ m2 Þm2 ðs1 s3 þ c1 c3 Þ2
ð51Þ
m2 l1 l2 ðs1 s3 þ c1 c3 Þ þ m2 Þm2 ðs1 s3 þ c1 c3 Þ2
ð52Þ
m2 l1 l2 ðs1 s3 þ c1 c3 Þ þ m2 Þm2 ðs1 s3 þ c1 c3 Þ2
ð53Þ
m2 l12 l22 ½ðm1 m2 l12 l22 ½ðm1
Fig. 2. Case 2: m1 ¼1.1 kg, m2 ¼ 1.4 kg, l1 ¼ 0.9 m, l2 ¼0.8 m and kri ¼ kgi ¼ 5 for i¼ 1,2. Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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b42 ¼
ðm1 þ m2 Þl12 m2 l12 l22 ½ðm1 þ m2 Þm2 ðs1 s3 þ c1 c3 Þ2
11
ð54Þ
and si ¼ sinðxi Þ,
ci ¼ cosðxi Þ
for i ¼ 1,3
ð55Þ
Applying suitable nonlinear transformation as described in Section 2, the system of Eq. (43) can be expressed in the form of Eq. (5) and is given by y€ ¼ fðxÞ þ GðxÞu Using Eqs. (3) and (6), G(x) and f(x) will be as follows: " # " # G11 G12 b21 b22 GðxÞ ¼ ¼ G21 G22 b41 b42 fðxÞ ¼ ½f1 ðxÞ f2 ðxÞT ¼ ½A2 ðxÞ A4 ðxÞT
ð56Þ
ð57Þ
ð58Þ
Fig. 3. Case 1: m1 ¼0.7 kg, m2 ¼0.8 kg, l1 ¼ 1.3 m, l2 ¼1.3 m and kri ¼ kgi ¼ 50 for i¼ 1,2.
Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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5.2. Simulations For the system in Eq. (56) with two inputs and two outputs, define a sliding manifold with two surfaces (S ¼ ½s1 s2 T ). Following Eqs. (11) and (12), the tracking errors and the sliding surfaces of the manipulator are expressed as ( ei ¼ ydi yi 8i ¼ 1,2 ð59Þ si ¼ e_ i þ ai ei The function C0 ðx,eÞ of Eq. (21) for the manipulator is given by C0 ðx,eÞ ¼ ½c1 ðx,eÞ c2 ðx,eÞT
ð60Þ
where ci ¼ y€ di fi ðxÞ þ ai ei for i ¼ 1,2. The controller is derived as ui ¼ Gi1 ðxÞ1 c1 þ Gi2 ðxÞ1 c2 þ Gii ðxÞ1 ri signðsi Þ þ Gii ðxÞ1 gi jci jsignðsi Þ
8i ¼ 1,2 ð61Þ
Fig. 4. Case 2: m1 ¼ 1.1 kg, m2 ¼ 1.4 kg, l1 ¼0.9 m, l2 ¼ 0.8 m and kri ¼ kgi ¼ 50 for i¼1,2. Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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In this study, the desired output trajectories are selected as qd1 ¼ 0:01 sinð5t þ p=2Þ and qd2 ¼ 0:01 sinð5t þ p=2Þ. The parameters of the sliding manifold are set to a1 ¼ a2 ¼ 5. The nominal values of the system parameters considered in this example are m1 ¼ 1 kg, m2 ¼ 1 kg, l1 ¼ 1 m and l2 ¼ 1 m. The adaptive SMC is implemented for this system considering uncertainties in the masses m1 and m2 lengths l1 and l2 of the manipulator. These parameters are varied over a wide range in such that they satisfy Assumption 2 of Eq. (19). The results of the simulations, only for two of the cases, i.e. case 1 and case 2 are given. The parameters for the two cases are as follows:
Case 1: m1 ¼ 0.7 kg, m2 ¼ 0.8 kg, l1 ¼ 1.3 m and l2 ¼ 1.3 m. Case 2: m1 ¼ 1.1 kg, m2 ¼ 1.4 kg, l1 ¼ 0.9 m and l2 ¼ 0.8 m.
The adaptation rates kri and kgi of the SMC are varied over a wide range, e.g. between 1 and 100. The results of the simulation for kri ¼ kgi ¼ 5 and kri ¼ kgi ¼ 50 for i ¼ 1,2 are shown for each case in Figs. 1–4. Note that the performance of the adaptive SMC under various choices of adaptation rates is not significantly different from each other. From Figs. 1(c), 2(c), 3(c) and 4(c), it is observed that the outputs of the system q1 and q2 successfully track the desired signals. From plots of the errors shown in the part (b) of the figures show that the tracking errors converge to zero rapidly in all cases, although there exist some sort of chattering in the control signals. 6. Conclusions An adaptive SMC for affine class of MIMO nonlinear systems has successfully been designed based on Lyapunov approach. The adaptive nature of the controller does not require any knowledge about the uncertainty bounds and therefore offers significant advantages over non-adaptive SMCs. The stability conditions of the controller are derived and the performance of the proposed controller is demonstrated considering an example of a two-link robot manipulator, and has been shown to be satisfactory.
References [1] V. Utkin, Variable structure systems with sliding modes, IEEE Transactions on Automatic Control 22 (2) (1977) 212–222. [2] A. Pisano, E. Usai, Sliding mode control: a survey with applications in math, Mathematics and Computers in Simulation 81 (5) (2011) 954–979. [3] V. Utkin, Sliding modes and their application in variable structure systems, Translated from the Russian by A. Parnakh (Moscow: MIR), Automation and Remote Control 44 (1974) 1105. [4] A.F. Fillipov, Differential equations with discontinuous right-hand side, American Mathematical Society Transactions 2 (42) (1964) 199. [5] Y. Feng, L. Li, X. Han, Second-order nonsingular terminal sliding mode control of uncertain MIMO linear systems, in: 1st International Symposium on Systems and Control in Aerospace and Astronautics, 2006, ISSCAA 2006, 2006, pp. 6–1355. [6] E.M. Jafarov, Robust sliding mode controllers design techniques for stabilization of multivariable time-delay systems with parameter perturbations and external disturbances, International Journal of Systems Science 36 (7) (2005) 433–444. [7] S. Mondal, C. Mahanta, Nonlinear sliding surface based second order sliding mode controller for uncertain linear systems, Communications in Nonlinear Science and Numerical Simulation 16 (9) (2011) 3760–3769. Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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[8] L. Wu, W.X. Zheng, H. Gao, Dissipativity-based sliding mode control of switched stochastic systems, IEEE Transactions on Automatic Control (99) (2012) 1. [9] L. Wu, W.X. Zheng, Passivity-based sliding mode control of uncertain singular time-delay systems, Automatica 45 (9) (2009) 2120–2127. [10] L. Wu, D.W. Ho, Sliding mode control of singular stochastic hybrid systems, Automatica 46 (4) (2010) 779–783. [11] L. Wu, P. Shi, H. Gao, State estimation and sliding-mode control of Markovian jump singular systems, IEEE Transactions on Automatic Control 55 (5) (2010) 1213–1219. [12] L. Wu, X. Su, P. Shi, Sliding mode control with bounded L2 gain performance of Markovian jump singular time-delay systems, Automatica 48 (8) (2012) 1929–1933. [13] J. Fei, M. Xin, Robust adaptive sliding mode controller for semi-active vehicle suspension systems, International Journal of Innovative Computing, Information and Control 8 (1(B)) (2012) 691–700. [14] M. Roopaei, M.Z. Jahromi, Chattering-free fuzzy sliding mode control in MIMO uncertain systems, Nonlinear Analysis: Theory, Methods & Applications 71 (10) (2009) 4430–4437. [15] W. Lin, C.S. Chen, Robust adaptive sliding mode control using fuzzy modelling for a class of uncertain MIMO nonlinear systems, IEE Proceedings—Control Theory and Applications 149 (3) (2002) 193–201. [16] R. Palm, Robust control by fuzzy sliding mode, Automatica 30 (9) (1994) 1429–1437. [17] W.-S. Lin, C.-S. Chen, Adaptive fuzzy sliding mode controller design for nonlinear MIMO systems, in: 2001 IEEE International Conference on Systems, Man, and Cybernetics, vol. 4, 2001, pp. 2232–2238. [18] A. Shahraz, R.B. Boozarjomehry, A fuzzy sliding mode control approach for nonlinear chemical processes, Control Engineering Practice 17 (5) (2009) 541–550. [19] T.-C. Lin, S.-W. Chang, C.-H. Hsu, Robust adaptive fuzzy sliding mode control for a class of uncertain discrete-time nonlinear systems, International Journal of Innovative Computing, Information and Control 8 (1(A)) (2012) 347–359. [20] W. Tao, W. Xie, Y. Zhang, Sliding mode fault tolerant control dealing with modeling uncertainties and actuator faults, ISA Transactions 51 (3) (2012) 386–392. [21] A. Isidori, Nonlinear Control Systems, Springer-Verlag, 1989. [22] M. Manceur, N. Essounbouli, A. Harnzaoui, MIMO second order sliding mode fuzzy type-2 control, in: IEEE International Conference on Systems, Man, and Cybernetics (SMC), 2010, pp. 1812–1820.
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Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties Alireza Nasirin, Sing Kiong Nguang, Akshya Swain The Department of Electrical and Computer Engineering, The University of Auckland, Auckland, New Zealand Received 29 August 2012; received in revised form 26 November 2012; accepted 24 December 2012
Abstract This paper proposes an adaptive scheme of designing sliding mode control (SMC) for affine class of multi-input multi-output (MIMO) nonlinear systems with uncertainty in the systems dynamics and control distribution gain. The proposed adaptive SMC does not require any a priori knowledge of the uncertainty bounds and therefore offers significant advantages over the non-adaptive schemes of SMC design. The closed loop stability conditions are derived based on Lyapunov theory. The effectiveness of the proposed approach is demonstrated via simulations considering an example of a two-link robot manipulator and has been found to be satisfactory. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Sliding Mode Control has been recognized as one of the effective nonlinear robust control methods due to various advantages this offers which include its invariance to system uncertainties and external disturbances [1]. Essentially, the sliding mode control is a particular type of variable structure control and uses discontinuous control action to drive the state trajectory towards a specific hyper plane in the state space. The state trajectory is then maintained to slide on the specific hyper plane until the origin of the state space is reached [2]. The first step of SMC design is to select a sliding surface which models the desired performance in state space. Then the controller is designed such that the system state trajectories move towards the sliding surface and stay on it. Since its introduction in [3,4], this has gone through n
Corresponding author. Tel.: þ64 221923874. E-mail addresses: [email protected], [email protected] (A. Nasiri), [email protected] (S. Kiong Nguang), [email protected] (A. Swain). 0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.12.019 Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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various theoretical and practical developments and the related literature until 2011 are well documented in [2]. The effectiveness of various types of SMCs for both SISO and MIMO systems have been demonstrated by researchers in the past. For example, in [5–7], the authors have designed SMCs for both SISO and MIMO linear as well as nonlinear systems and applied these to variety of practical systems such as robot manipulators, aircrafts, underwater vehicles, space crafts, flexible space structures, electrical motors, power systems, and automotive engines, etc. [8–10]. Most of the SMCs for nonlinear systems, which have been reported, essentially focus on particular class of nonlinear systems such as Lipschitz class of nonlinear systems. Another class of systems which are of practical interest are switched hybrid systems, singular systems, Markovian systems, etc. In recent years various researchers have proposed SMCs for these systems e.g. in [11], the problems of stochastic stability have been considered; in [12] SMC for Markovian jump singular time-delay system has been investigated by considering the bounded L2 gain performance; in [8–10], the SMC has been applied for a switched stochastic systems and, in [13], a robust adaptive SMC has been designed for semi-active vehicle suspension system with parameter uncertainties and external disturbances with the linear model. The performance of SMCs is dependent on the knowledge about the uncertainty bounds of the system which are hard to know in practice. To overcome the problems associated with the unknown bounds, various adaptive techniques, including methods based on fuzzy logic, have been proposed in recent past [14–20]. However, the design of fuzzy based SMCs require some prior heuristic knowledge about the system and the number of fuzzy rules increases as the order and the complexity of the system increases. Recently in [20], adaptive SMCs have been designed for affine class of SISO nonlinear systems with uncertainty. However, the extension of this to MIMO systems is more complicated. In the present study an adaptive SMC has been designed for affine class of MIMO nonlinear systems, assuming that there exists uncertainty both in the system dynamics and in the control distribution gain. Various conditions of stability for the proposed adaptive SMC have been derived using Lyapunov approach and the effectiveness of the adaptive SMC is demonstrated considering an example of a two link robot manipulator. The rest of the paper is organized as follows. Section 2 briefly describes about the affine class of MIMO nonlinear systems and introduces few definitions which are subsequently being used in adaptive SMC design. A robust SMC for MIMO nonlinear systems is proposed in Section 3. In Section 4, design of the proposed adaptive SMCs is carried out. The effectiveness of the proposed design is illustrated by an example in Section 5 with conclusions in Section 6. 2. Problem formulation Consider a MIMO affine nonlinear system x_ ¼ AðxÞ þ
m X
bi ðxÞui
i¼1
y1 ¼ h1 ðxÞ ^ ym ¼ hm ðxÞ
ð1Þ
Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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where x 2 Rn is state vector, u 2 Rm represents control inputs, y 2 Rm stands for outputs, AðxÞ,b1 ðxÞ, . . . ,bim ðxÞ are smooth vector fields, and h1 ðxÞ, . . . ,hm ðxÞ are smooth functions, defined on the open set of Rn. A multivariable nonlinear system of the form (1) has a vector relative degree fr1 r2 . . . rm g at a point xo if Lbj LkA hi ðxÞ ¼ 0 81rjrm and krri 1 for all x in a neighborhood of Denote 2 r1 1 Lb1 LA h1 ðxÞ 6 r2 1 6 Lb1 LA h2 ðxÞ GðxÞ ¼ 6 6 4 rm 1 Lb1 LA hm ðxÞ
ð2Þ
xo .
Lbm LrA1 1 h1 ðxÞ
Lbm LrA2 1 h2 ðxÞ
Lbm LrAm 1 hm ðxÞ
3 7 h 7 7 ¼ g1 ðxÞ 7 5
g2 ðxÞ
gm ðxÞ
i
ð3Þ
where gi ðxÞ ¼ ½g1i ðxÞ g2i ðxÞ gmi ðxÞT
ð4Þ o
Note that GðxÞ is non-singular at x ¼ x . Using feedback linearization, the nonlinear system (1) can be transformed into following form [21]: yðrÞ ¼ fðxÞ þ GðxÞu where
2
3
ð5Þ 2
3 f1 ðxÞ 6 r2 7 6 7 6 LA h2 ðxÞ 7 6 f2 ðxÞ 7 6 7 6 7 fðxÞ ¼ 6 7¼6 7 4 5 4 5 rm LA hm ðxÞ fm ðxÞ LrA1 h1 ðxÞ
ð6Þ
and ðr2 Þ ðrm Þ T 1Þ yðrÞ ¼ ½yðr 1 ,y2 , . . . ,ym
ð7Þ
ðr2 Þ ðrm Þ T 1Þ x ¼ ½x1 ,x2 , . . . ,xn T ¼ ½y1 , . . . ,yðr 1 ,y2 , . . . ,y2 , . . . ,ym , . . . ,ym
ð8Þ
The relativePdegree of the system is assumed to be equal to the order of the system and is expressed as m i ¼ 1 ri ¼ n. This implies that the system does not have any zero dynamics. Let the desired output trajectory be given by yd ¼ ½yd1 yd2 ydm T
ð9Þ
The tracking error of the system is therefore expressed as e ¼ ½e1 e2 em T
ð10Þ
where ðri 1Þ i 1Þ T ei ¼ ½ei e_i eiðri 1Þ T ¼ ½ydi yi y_ di y_ i ydi yðr i
8i ¼ 1, . . . ,m
ð11Þ
Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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Define a set of sliding surfaces in the error space passing through the origin to represent a sliding manifold as follows: 3 2 ðr1 1Þ þ a11 e1ðr1 2Þ þ þ a1,ðr1 1Þ e1 e1 7 6 ðr2 1Þ 6 e2 þ a21 e2ðr2 2Þ þ þ a2,ðr2 1Þ e2 7 T 7 6 ð12Þ s ¼ ½s1 s2 sm ¼ 6 7 ^ 5 4 ðrm 1Þ m 2Þ em þ am1 eðr þ þ am,ðrm 1Þ em m The derivative of s is given by 2 ðr1 Þ 3 1 1Þ þ þ a1,ðr1 1Þ e_ 1 e1 þ a11 eðr 1 6 ðr2 Þ 7 2 1Þ 6 e2 þ a21 eðr þ þ a2,ðr2 1Þ e_ 2 7 T 2 6 7 s_ ¼ ½_s 1 s_ 2 s_ m ¼ 6 7 ^ 4 5 ðrm Þ ðrm 1Þ em þ am1 em þ þ am,ðrm 1Þ e_ m
ð13Þ
Using the relations in Eqs. (5) and (11), the derivative of the sliding manifold in Eq. (13) can be expressed as 3 2 rX 1 1 ðr1 iÞ a e 7 6 7 6 i ¼ 1 1i 1 7 6 7 6 r2 1 7 6 X ðr2 iÞ 7 6 a e 2i 2 7 6 ðrÞ ð14Þ s_ ¼ yd fðxÞGðxÞu þ 6 i ¼ 1 7 7 6 7 6 ^ 7 6 7 6 rm 1 7 6X m iÞ 5 4 ami eðr m i¼1
3. Design of robust sliding mode control This section proposes a Lyapunov based approach to design a robust sliding mode controller for a MIMO affine nonlinear system. Consider the MIMO nonlinear system described in Eq. (1). After suitable nonlinear transformation, the outputs of this system can be expressed as (5) yðrÞ ¼ fðxÞ þ GðxÞu where GðxÞ and fðxÞ are defined in Eqs. (3) and (6). Note that the functions fðxÞ and GðxÞ are dependent on the system parameters, and any uncertainties in the system parameters are reflected in them. Let the uncertainties in fðxÞ and GðxÞ be expressed as DfðxÞ ¼ fðxÞf 0 ðxÞ ¼ ½Df1 ðxÞ Df2 ðxÞ Dfm ðxÞT
ð15Þ
DGðxÞ ¼ GðxÞG0 ðxÞ
ð16Þ
and where f 0 ðxÞ and G0 ðxÞ represent nominal values of fðxÞ and GðxÞ. Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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Assumption 1. The nominal control distribution function matrix G0 ðxÞ is non-singular. Let ^ ^ 1 g^ 2 g^ m GðxÞ ¼ DGðxÞG1 0 ðxÞ ¼ ½g
ð17Þ
^ where the ith column of GðxÞ is given by g^ i ¼ ½G^ 1i ðxÞ G^ 2i ðxÞ G^ mi ðxÞT
ð18Þ
^ Note that GðxÞ is defined for simplifying various assumptions and theorems which are proposed in this study. Assumption 2. The uncertainties DfðxÞ and DGðx) satisfy the following: 8 JDfi ðxÞJoFi > < maxj ðjG^ ij ðxÞjÞok^ i and mk^ i o1 k^ i 40 8i ¼ 0, . . . ,m > : jG^ ðxÞjog^ ^ max,ii 40 ii max,ii o1 and g
ð19Þ
^ and G^ ii ðxÞ, where Fi, k^ i and g^ max,ii are the upper bounds of Dfi ðxÞ, the ith row of GðxÞ respectively. Using the notation introduced, the time derivative of sliding manifold in Eq. (14) is expressed as s_ ¼ C0 ðx,eÞDfðxÞðG0 ðxÞ þ DGðxÞÞu
ð20Þ
where 2
3
rX 1 1
1 iÞ a1i eðr 1 i¼1
7 6 7 6 7 6 7 6 r2 1 7 6 X ðr2 iÞ 7 6 a2i e2 7 6 ðrÞ T C0 ðx,eÞ ¼ ½c1 ðx,eÞ c2 ðx,eÞ cm ðx,eÞ ¼ yd f 0 ðxÞ þ 6 i ¼ 1 7 7 6 7 6 ^ 7 6 7 6 rm 1 7 6X ðrm iÞ 5 4 ami e
ð21Þ
m
i¼1
It is assumed that each element of C0 ðx,eÞ satisfies ci ðx,eÞrlðxÞ
8i ¼ 1, . . . ,m:
ð22Þ
Theorem 1. For the system in Eq. (5), the motion on the sliding manifold given in Eq. (12) is asymptotically stable, using the controller 1 1 u ¼ G1 0 ðxÞC0 ðx,eÞ þ G0 ðxÞ P signðsÞ þ G0 ðxÞ F0 ðx,eÞ U signðsÞ
ð23Þ
where P ¼ diagðr1 ,r2 , . . . ,rm Þ, U ¼ diagðg1 ,g2 , . . . ,gm Þ and F0 ðx,eÞ ¼ diagðjc1 ðx,eÞj, jc2 ðx,eÞj, . . . ,jcm ðx,eÞjÞ for i ¼ 1,2, . . . ,m. The different elements of matrix P and U satisfy Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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the following: 8 Fi > > r4 > < i 1g^
max,ii
ð24Þ
g^ max,ii > > > : gi 4 1mk^ i Proof. Consider the scalar semi-positive quadratic Lyapunov function V as 1 V ¼ ST S 2
ð25Þ
The time derivative of V along the sliding manifold S is given by V_ ¼
m X
s_ i si
ð26Þ
i¼1
where s_ i si ¼ si Dfi ðxÞsi ri signðsi Þsi jci ðx,eÞj gi signðsi Þsi
m X
G^ ij ðxÞ cj ðx,eÞsi G^ ii ðxÞri signðsi Þ
j¼1
si G^ ii ðxÞ jci ðx,eÞj gi signðsi Þ ¼ si Dfi ðxÞri jsi jG^ ii ðxÞri jsi jsi
m X
^ jci ðx,eÞj gi jsi j G^ ij ðxÞ cj ðx,eÞjci ðx,eÞj gi jsi jGðxÞ
j¼1
ð27Þ Using Assumption 2, and the inequalities of Eqs. (22) and (24), Eq. (27) becomes s_ i si oFi jsi jri jsi jg^ max,ii ri jsi j þ
m X
jG^ ij ðxÞj jcj ðx,seÞj jsi jjci ðx,eÞj gi jsi j
j¼1
þ g^ max,ii jci ðx,eÞj gi jsi j o½ri ð1g^ max,ii ÞFi :jsi j½gi ð1mk^ i Þg^ max,ii jlðxÞj:jsi jo0
ð28Þ
So, the motion on the sliding manifold is asymptotically stable and the output can therefore, track the desired reference. & Remark 1. To determine the control system parameters ri and gi , we need to know Fi, k^ i and g^ max,ii . However, in practice, it is often difficult to find them. 4. Design of robust adaptive sliding mode control In practice, finding the bounds for the uncertainties is difficult. Hence, an adaptive method is required to cope with this differently. Theorem 2. For the system in Eq. (5), the motion on the sliding manifold given in Eq. (12), using the controller given in Eq. (23), is asymptotically stable with the following Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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adaptive laws: ( ri ¼ kri jsi j gi ¼ kgi jsi j jlðxÞj
8i ¼ 1,2, . . . m
7
ð29Þ
where kri and kgi are rates of adaptation. Proof. Define r~ i and g~ i as ( r~ i ¼ ri0 r0 8i ¼ 1,2, . . . m g~ i ¼ gi0 gi where 8 Fi > > > ri0 41g^ <
max,ii
g^ max,ii > > > : gi0 4 1mk^ i
ð30Þ
8i ¼ 1,2, . . . m
ð31Þ
The constants, Fi, g^ max,ii and k^ i are defined in Eq. (19). Consider the Lyapunov function V as V¼
m m m 1g^ max,ii 2 1 X 1X 1X 1mk^ i 2 s2i þ r~ i þ g~ i 2i¼1 2 i ¼ 1 kri 2 i ¼ 1 kgi
ð32Þ
The time derivative of V along the manifold S is given by ! m m m ^i X X X ^ max,ii 1 g 1m k r~_ i r~ i þ V_ ¼ s_ i si þ g~_i g~ i k ri kgi i¼1 i¼1 i¼1 ! ! m m X X 1g^ max,ii _ 1mk^ i _ ¼ s_ i si þ v_ i g~ i g~ i ¼ r~ i r~ i þ kri kgi i¼1 i¼1 where
ð33Þ
! 1mk^ i _ g~ i g~ i kgi m X ^ ¼ si Dfi ðxÞri jsi jG^ ii ðxÞri jsi jsi jci ðx,eÞj gi jsi j G^ ij ðxÞ cj ðx,eÞjci ðx,eÞj gi jsi jGðxÞ
1g^ max,ii _ v_ i ¼ s_ i si þ r~ i r~ i þ kri
j¼1
ð1g^ max,ii Þ ðri0 r0 Þ jsi jð1mk^ i Þ ðgi0 gi Þ jsi j jlðxÞj
ð34Þ
Using Assumption 2, and the inequalities of Eqs. (22), (30), and (31), Eq. (34) becomes
1g^ max,ii _ v_ i ¼ s_ i si þ r~ i r~ i þ kri
! 1mk^ i _ g~ i g~ i kgi
oFi jsi jri jsi jg^ max,ii ri jsi j þ
m X
jG^ ij ðxÞj jcj ðx,eÞj jsi jjci ðx,eÞj gi jsi jG^ ii ðxÞ jci ðx,eÞj gi jsi j
j¼1
ð1g^ max,ii Þ ðri0 r0 Þ jsi jð1mk^ i Þ ðgi0 gi Þ jsi j jlðxÞj oFi jsi jr jsi j þ g^ r jsi j þ mk^ i jlðxÞj jsi jjlðxÞj g jsi j þ g^ i
max,ii i
i
max,ii
jlðxÞj gi jsi j
Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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ð1g^ max,ii Þ ðri0 r0 Þ jsi jð1mk^ i Þ ðgi0 gi Þ jsi j jlðxÞjo0
ð35Þ
So, the motion on the sliding manifold is asymptotically stable and the output can therefore, track the desired reference. & 5. Example In this section, the effectiveness of the proposed controller is illustrated via simulations by considering, an example of a two-link robot manipulator. 5.1. Modeling The dynamic equation of a two-link robot manipulator is given by [22] _ q_ þ GðqÞ ¼ t MðqÞq€ þ Cðq, qÞ
ð36Þ
_ are given, where the moment of inertia M(q) and coriolis centripetal forces Cðq, qÞ respectively, by " # ðm1 þ m2 Þl12 m2 l1 l2 ðs1 s2 þ c1 c2 Þ MðqÞ ¼ ð37Þ m2 l1 l2 ðs1 s2 þ c1 c2 Þ m2 l22 " 1
_ ¼ m2 l1 l2 ðc1 s2 Þ s1 c2 Cðq, qÞ
0 q_ 1
q_ 2 0
The gravitational force G(q) is expressed as " # ðm1 þ m2 Þl1 gs1 GðqÞ ¼ m2 l2 gs2
# ð38Þ
ð39Þ
where q ¼ ½q1 q2 T are the generalized coordinates. m1, m2 are masses of the link and l1 and l2 are lengths of the links. q1, q2 are the angular positions. t ¼ ½t1 t2 T is the vector of applied torques. The acceleration due to gravity is g ¼ 9:8 m=s2 . For convenience, define si ¼ sinðqi Þ, ci ¼ cosðqi Þ for i¼ 1,2. The state, input and output vectors are defined as x ¼ ½x1 x2 x3 x4 T ¼ ½q1 q_ 1 q2 q_ 2 T
ð40Þ
u ¼ ½u1 u2 T ¼ ½t1 t2 T
ð41Þ
y ¼ ½y1 y2 T ¼ ½q1 q2 T
ð42Þ
Using the definitions in Eqs. (40)–(42), the dynamics of two-link manipulator is re-expressed as x_ ¼ AðxÞ þ
2 X
bi ðxÞui
i¼1
y1 ¼ h1 ðxÞ ¼ x1 y2 ¼ h2 ðxÞ ¼ x3
ð43Þ
Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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where AðxÞ ¼ ½A1 ðxÞ A2 ðxÞ A3 ðxÞ A4 ðxÞT
ð44Þ
The various elements of AðxÞ i.e. A1 ðxÞ, . . . ,A4 ðxÞ are given by A1 ðxÞ ¼ x2
ð45Þ
ðs1 c3 c1 s3 Þ ½m2 l1 l2 ðs1 s3 þ c1 c3 Þx22 m2 l22 x24 l1 l2 ½ðm1 þ m2 Þm2 ðs1 s3 þ c1 c3 Þ2 1 ½ðm1 þ m2 Þl2 gs1 m2 l2 gs3 ðs1 s3 þ c1 c3 Þ þ l1 l2 ½ðm1 þ m2 Þm2 ðs1 s3 þ c1 c3 Þ2 ð46Þ
A2 ðxÞ ¼
A3 ðxÞ ¼ x4 A4 ðxÞ ¼
ð47Þ
ðs1 c3 c1 s3 Þ ½ðm1 þ m2 Þl1 x22 þ m2 l1 l2 ðs1 s3 þ c1 c3 Þx24 l1 l2 ½ðm1 þ m2 Þm2 ðs1 s3 þ c1 c3 Þ2 1 þ ½ðm1 þ m2 Þl1 gs1 ðs1 s3 þ c1 c3 Þ þ ðm1 þ m2 Þl1 gs3 l1 l2 ½ðm1 þ m2 Þm2 ðs1 s3 þ c1 c3 Þ2
ð48Þ
Fig. 1. Case 1: m1 ¼0.7 kg, m2 ¼0.8 kg, l1 ¼ 1.3 m, l2 ¼1.3 m and kri ¼ kgi ¼ 5 for i¼ 1,2. Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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and 2
b1i
3
6 b2i 7 6 7 bi ðxÞ ¼ 6 7 4 b3i 5 b4i
8i ¼ 1,2
ð49Þ
b11 ¼ b31 ¼ b12 ¼ b32 ¼ 0
ð50Þ
where
b21 ¼
b22 ¼
b41 ¼
m2 l22 m2 l12 l22 ½ðm1 þ m2 Þm2 ðs1 s3 þ c1 c3 Þ2
ð51Þ
m2 l1 l2 ðs1 s3 þ c1 c3 Þ þ m2 Þm2 ðs1 s3 þ c1 c3 Þ2
ð52Þ
m2 l1 l2 ðs1 s3 þ c1 c3 Þ þ m2 Þm2 ðs1 s3 þ c1 c3 Þ2
ð53Þ
m2 l12 l22 ½ðm1 m2 l12 l22 ½ðm1
Fig. 2. Case 2: m1 ¼1.1 kg, m2 ¼ 1.4 kg, l1 ¼ 0.9 m, l2 ¼0.8 m and kri ¼ kgi ¼ 5 for i¼ 1,2. Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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b42 ¼
ðm1 þ m2 Þl12 m2 l12 l22 ½ðm1 þ m2 Þm2 ðs1 s3 þ c1 c3 Þ2
11
ð54Þ
and si ¼ sinðxi Þ,
ci ¼ cosðxi Þ
for i ¼ 1,3
ð55Þ
Applying suitable nonlinear transformation as described in Section 2, the system of Eq. (43) can be expressed in the form of Eq. (5) and is given by y€ ¼ fðxÞ þ GðxÞu Using Eqs. (3) and (6), G(x) and f(x) will be as follows: " # " # G11 G12 b21 b22 GðxÞ ¼ ¼ G21 G22 b41 b42 fðxÞ ¼ ½f1 ðxÞ f2 ðxÞT ¼ ½A2 ðxÞ A4 ðxÞT
ð56Þ
ð57Þ
ð58Þ
Fig. 3. Case 1: m1 ¼0.7 kg, m2 ¼0.8 kg, l1 ¼ 1.3 m, l2 ¼1.3 m and kri ¼ kgi ¼ 50 for i¼ 1,2.
Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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5.2. Simulations For the system in Eq. (56) with two inputs and two outputs, define a sliding manifold with two surfaces (S ¼ ½s1 s2 T ). Following Eqs. (11) and (12), the tracking errors and the sliding surfaces of the manipulator are expressed as ( ei ¼ ydi yi 8i ¼ 1,2 ð59Þ si ¼ e_ i þ ai ei The function C0 ðx,eÞ of Eq. (21) for the manipulator is given by C0 ðx,eÞ ¼ ½c1 ðx,eÞ c2 ðx,eÞT
ð60Þ
where ci ¼ y€ di fi ðxÞ þ ai ei for i ¼ 1,2. The controller is derived as ui ¼ Gi1 ðxÞ1 c1 þ Gi2 ðxÞ1 c2 þ Gii ðxÞ1 ri signðsi Þ þ Gii ðxÞ1 gi jci jsignðsi Þ
8i ¼ 1,2 ð61Þ
Fig. 4. Case 2: m1 ¼ 1.1 kg, m2 ¼ 1.4 kg, l1 ¼0.9 m, l2 ¼ 0.8 m and kri ¼ kgi ¼ 50 for i¼1,2. Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019
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In this study, the desired output trajectories are selected as qd1 ¼ 0:01 sinð5t þ p=2Þ and qd2 ¼ 0:01 sinð5t þ p=2Þ. The parameters of the sliding manifold are set to a1 ¼ a2 ¼ 5. The nominal values of the system parameters considered in this example are m1 ¼ 1 kg, m2 ¼ 1 kg, l1 ¼ 1 m and l2 ¼ 1 m. The adaptive SMC is implemented for this system considering uncertainties in the masses m1 and m2 lengths l1 and l2 of the manipulator. These parameters are varied over a wide range in such that they satisfy Assumption 2 of Eq. (19). The results of the simulations, only for two of the cases, i.e. case 1 and case 2 are given. The parameters for the two cases are as follows:
Case 1: m1 ¼ 0.7 kg, m2 ¼ 0.8 kg, l1 ¼ 1.3 m and l2 ¼ 1.3 m. Case 2: m1 ¼ 1.1 kg, m2 ¼ 1.4 kg, l1 ¼ 0.9 m and l2 ¼ 0.8 m.
The adaptation rates kri and kgi of the SMC are varied over a wide range, e.g. between 1 and 100. The results of the simulation for kri ¼ kgi ¼ 5 and kri ¼ kgi ¼ 50 for i ¼ 1,2 are shown for each case in Figs. 1–4. Note that the performance of the adaptive SMC under various choices of adaptation rates is not significantly different from each other. From Figs. 1(c), 2(c), 3(c) and 4(c), it is observed that the outputs of the system q1 and q2 successfully track the desired signals. From plots of the errors shown in the part (b) of the figures show that the tracking errors converge to zero rapidly in all cases, although there exist some sort of chattering in the control signals. 6. Conclusions An adaptive SMC for affine class of MIMO nonlinear systems has successfully been designed based on Lyapunov approach. The adaptive nature of the controller does not require any knowledge about the uncertainty bounds and therefore offers significant advantages over non-adaptive SMCs. The stability conditions of the controller are derived and the performance of the proposed controller is demonstrated considering an example of a two-link robot manipulator, and has been shown to be satisfactory.
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Please cite this article as: A. Nasiri, et al., Adaptive sliding mode control for a class of MIMO nonlinear systems with uncertainties, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/ j.jfranklin.2012.12.019