[IEEE 2010 Third International Workshop on Advanced Computational Intelligence (IWACI) - Suzhou, China (2010.08.25-2010.08.27)] Third International Workshop on Advanced Computational Intelligence - Optimal sliding mode design for nonlinear systems

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  • AbstractThis paper designs a time-varying nonlinearswitching manifold in an optimal fashion for a class of nonlinear systems by developing the successive approximation approach of differential equation into infinite-time horizon. Based on the reaching law approach, we obtain a control input which drivesthe systems state trajectories to reach the nonlinear sliding surface. The stability of the nonlinear sliding mode is analyzed.The convergence velocity of every state trajectory on the idealsliding surface can be adjusted through choosing quadratic performance index. Simulation example is employed to test the validity of the proposed design algorithm..

    I. INTRODUCTIONHE control with sliding mode (SMC), which is commonly known as sliding mode control, is a nonlinear control strategy that is well known for its fast response, good

    transient performance, easy realization and insensitiveness to the matching parameter variations and external disturbances. Many approaches had been proposed for the design of sliding surface. Normally, it was designed as linear sliding surface[2, 3]. A necessary and sufficient condition for the existence of a linear sliding surface depending on outputs and compensator states was derived [2]. Based on a singular system approach and linear matrix inequality, a sufficient condition which guarantees the existence of linear switching surface and the stochastic stability of sliding mode dynamics was given in [3]. Pole assignment and optimal quadratic methods were presented in many researches (see, for example, [4] and [5]). To remove the reaching phase problems and eliminate chatting, a special integral sliding surface was introduced in [6] and [7]. [8] proposed a coupled sliding-mode control method for the periodic orbit generation and the robust exponential orbital stabilization of inverted-pendulum systems. More recently, combinations of fuzzy control and VSC approaches had achieved superior performance [9], [10]. Differing from the classical first-order sliding mode, a new second-order sliding mode control had been proposed [11], [12].

    For a class of nonlinear systems that could be recursively approximated as linear time-varying systems, a sliding modecontrol and corresponding time-varying sliding surfaces had been designed for each approximated system so that a given optimization criterion was minimized [13]. For a general

    R. Dong is with the Department of Mathematics, Henan Institute of Science and Technology, Xinxiang 453003, China. (Tel: +86 373 3040767;E-mail: drui@163.com.)M.-Q. Fan is with Department of Electrical Engineering and Information

    Technology, Shandong University of Science and Technology, Jinan, 250031, China. (E-mail: fmq@sdust.edu.cn.)

    Y.-R. Guo is with the Department of Mathematics, Henan Institute of Science and Technology, Xinxiang 453003, China. (yr640323@163.com)

    nonlinear system, Tang [14] proposed the successive approximation approach (SAA) of optimal control in finite horizon. By using the finite-step iterations, a suboptimal control law was obtained.

    In this paper, the SAA is developed into infinite-time horizon. A new sliding mode is designed based on the optimal control for nonlinear systems. We first treat some statevariables as virtual control input and give a quadratic performance index in infinite-time horizon. Using the SAA of differential equation theory, we obtain an optimal nonlinear switching manifold. The stability of the sliding mode is assured by the optimal switching surface chosen. Convergence velocity of the state trajectories of the closed-loop system in idea sliding surface is assured by minimizing the quadratic performance index. Finally, a simulation example is employed to test the validity of this approach in control design.

    II. PROBLEM STATEMENTConsider the following nonlinear system

    1 1 1 1 2 2

    2 3 1 2 1 2

    1 10 2 20

    ( ) ( ) ( ) ( ),( ) ( , ) ( , ) ( ),(0) , (0) ,

    x t Ax t f x Cf xx t f x x B x x u tx x x x

    = + += +

    = =

    (1)

    where ( ) inix t R , 1 2( ) [ ( ) ( )]T T Tx t x t x t= is the state vector,

    2( ) nu t R is the control vector, 1 1n nA R and 1 2n nC R

    are real constant matrices, 0ix are the known initial vectors

    ( 1,2)i = 1 2 3 1 2(0) 0, (0) 0, (0,0) 0,det ( , ) 0.f f f B x x= = =

    Assumption 1: The pair ),( CA is completely

    controllable.

    Assumption 2: There exist some positive constant such that

    11 1 1 1 1 1 1 1 1( ) ( ) || ||, ,

    nf x f y x y x y R 2

    where denotes some appropriate vector norm.

    Assumption 3: The matrix 2 2 2( )df x dx is nonsingular.

    Assumption 4: The function 2 2( )f x has unique

    equilibrium point; that is 2 2( ) 0f x = if and only if 2 ( ) 0x t = .

    Optimal Sliding Mode Design for Nonlinear SystemsRui Dong, Mingqu Fan, Yunrui Guo

    T

    414

    Third International Workshop on Advanced Computational Intelligence August 25-27, 2010 - Suzhou, Jiangsu, China

    978-1-4244-6337-4/10/$26.00 @2010 IEEE

  • Considering system (1), choose the following quadratic

    performance index:

    1 1 2 2 2 20

    1 [ ( ) ( ) ( ) ( )]2

    T TJ x t Qx t f x Rf x dt

    = + (3)

    where 1 1 2 2Q ,n n n nR R R are positive-definite matrices.

    The optimal nonlinear sliding mode design problem is to

    find a virtual optimal control law 2 2( )f x which minimizes

    the quadratic performance index (3) subject to the dynamic

    equality constraint (1). According to the necessary conditions

    for the optimality, we can obtain the following nonlinear

    two-point boundary value problem

    11 1 1

    1 1 1 1 2 2

    1 10

    ( ) ( ) ( ) ( (x )) ( ),

    ( ) ( ) ( ) ( ), 0,( ) 0, (0) ,

    Txt Qx t A t f t

    x t Ax t f x Cf x tx x

    = + +

    = + + > = =

    (4)

    where 11 1 1 1 1(x ) ( )Txf df x dx= . The virtual optimal control

    law can be expressed as

    2 2( ) ( ) 0TRf x C t+ = (5)

    and the controller can be obtained based on the sliding

    reaching law as following

    ( , ) ( )s s t Ks Esign s= = (6)

    where 2 21 2 1 2

    { , , , }, { , , , }n nK diag k k k E diag = = ,

    with 2, 0, ( 1, 2, , )i ik i n > = ,

    21 2( ) [ ( ), ( ), , ( )] ,Tnsign s sign s sign s sign s=

    1 0( ) 0 0

    1 0

    i

    i i

    i

    ssign s s

    s

    >= =

    =

    (7)

    where nz R is the state vector, 0z is the initial state vector,

    0

    1: ( )n nth C R R R which satisfies the Lipschitz conditions

    on 0

    ntR R , 0 0( , )tR t= ,

    n nG R and all eigenvalues of G satisfy

    Re( ( )) 0, 1,2, ,i G i n < = (8)Lemma 1: The vector function sequence { }( ) ( )kz t is

    represented by the expression

    0

    (0)0

    ( ) ( 1)0 0 0

    ( )0 0

    ( ) 0,

    ( ) ( ) ( ) ( ( ), ) ,

    ( ) , 1,2,

    tk k

    t

    k

    z t t t

    z t t t x t r h z r r dr t t

    z t z k

    =

    = + >

    = =

    (9)

    where ( ) exp( )t Gt = is state-transition matrix with respect

    to the matrix G . Then the sequence { }( ) ( )kz t uniformly converges to the solution of system (7) for any 0( , )t t .Proof. Because h is Lipschitz in z on

    0

    ntR R , there

    exist some positive constants and such that

    1 2 1 2

    ( , )

    ( , ) ( , )

    h z t z

    h z t h z t z z

    (10)

    And based on (8), one can obtain 0( ) ,t M t t (11)

    where M is a positive constant, denotes any appropriatevector or matrix norm. We first consider the situation of

    0[ , ]t t T , where T is a constant. For any j and k ,considering { }( ) ( )kz t as a sequence and using the arguments as in the proof of Lemma 1 of [14], we obtain

    ( ) ( ) 1 1 00

    12 1

    0 00

    0

    ( )( ) ( )

    !( )

    exp( ( )),( 1)!

    [ , ], 0,1, 2, .

    ik jk j k i i

    i kk k k

    T tz t z t z M

    iz M T t

    M T tk

    t t T k

    ++ +

    = +

    + +

    + =

    (12)

    Inequality (12) implies( ) ( )

    0lim ( ) ( ) 0, 0, [ , ]k j k

    kz t z t j t t T+

    = > (13)

    Now we consider the case of T . Choosing 2k T= , from inequality (12) we get

    2 2 2

    ( ) ( )

    ( 2) 10 0

    0

    ( 2)0

    2

    20

    ( ) ( )

    ( )exp( ( ))

    ( 1)!

    exp( )( )!

    ( ) exp( )( 2)!

    k j k

    k k k

    T T T

    TT

    z t z t

    z M T tM T t

    k

    z M TMT

    T

    Mz MT

    +

    + +

    +

    +

    (14)

    From (13)14 and the following fact

    415

  • 1( )lim 0( 2)!

    T

    T

    MT

    =

    (15)

    we can see ( ) ( )lim ( ) ( ) 0,k j kk

    z t z t+

    = 00, [ , )j t t > .

    Thus sequence { }( ) ( )kz t is uniformly convergent.

    Now we consider sequence ( ){ ( )}kz t . For any j and k ,

    from (10) we get( ) ( ) ( ) ( )

    min( ) ( ) ( ( ) ) ( ) ( )k j k k j kz t z t G z t z t + + + (16)

    Obviously, sequence { }( ) ( )kz t is uniformly convergent.

    Thus the limit of the sequence { }( ) ( )kz t is the solution of system (7).

    Remark 1: In practical control process, any system cant run

    endlessly. If T is large enough, we may consider T .

    We can adjust the iteration times according to the control time

    T and the demand for precision.

    IV. OPTIMAL SMC DESIGNING PROCESSLet

    1( ) ( ) ( ), (0, )t Px t g t t = + (17)where 1 1n nP R is a positive-definite matrix to be found.

    1(t) ng R is the thk adjoint vector. Calculating the derivative to the both sides of (17), we get

    11

    11 1

    ( ) ( ) ( )

    (t) ( ( )) ( )

    T

    T

    t PA PCR C P x tPCR C g Pf x t g t

    =

    + +

    (18)

    Substituting (17) into the first equation of (4) and comparing with (18), one can obtain

    1

    1 11

    1 1 1 1 1

    ( ) ( ) ( ) ( )( ( )) ( ) ( ( ( )))( ( ) ( )) 0

    T T T T

    x

    PA A P PCR C P Q x t PCR C g t A g tPf x t g t f x t Px t g t

    + + +

    + + + + (19)

    From equation (19) we obtain the following Riccati matrix equation:

    1 0T TPA A P PCR C P Q+ + = (20)According Assumption 1, the unique positive-definite matrix solution P exists. Comparing the third equation of (4) with (19) and (20), we can obtain the following adjoint vector differential equation

    1

    11 1

    1 1 1

    ( ) ( ) ( ) ( ( ))( ( ( )))( ( ) ( ))

    ( ) 0.

    T T

    x

    g t A CR C P g t Pf x tf x t Px t g t

    g

    = +

    =

    (21)

    Construct the following sequence(1,0)1(1, ) 1 (1, ) (1, 1)1 1 1 1(1, )1 10

    ( ) 0

    ( ) ( ) ( ) ( ( ))

    (0) , 1,2,

    k T k k

    k

    x tx t A CR C P x t f x tx x k

    =

    = +

    = =

    (22)

    For the thk iteration, (1, 1)1 ( )kx t is known and (22) is a

    nonhomogeneous linear vector differential equation, so we can obtain its solution (1, )1 ( )

    kx t . According to the optimal control theory, one can get all eigenvalues of the matrix

    1( )T TA CR C P satisfy 1Re[ ( )] 0Ti A CR C P < , 11, 2, ,i n= . From Assumption 2 we know 1 1( )f x satisfies

    the Lipschitz conditions. Based on Lemma 1, the solution sequence { }(1, )1 ( )kx t of system (22) uniformly converges to the solution of the following system

    11 1 1 1

    1 10

    (t) ( ) ( ) ( ), 0(0)

    Tx A CR C P x t f x tx x

    = + >

    =

    (23)

    Let 1(* )(1, )1 1lim ( ) ( )k

    kx t x t

    = and construct a sequence as

    1 1 1*11

    (1,0)

    (1, ) 1 (1, )

    * * * ( 1)1 1 1 11

    (1, )

    ( ) 0( ) ( ) ( )

    ( ) ( ( ))( ( ) ( ))

    ( ) 0

    k T T k

    kx

    k

    g tg t A CR C P g tPf x f x Px t g t

    g

    =

    =

    +

    =

    (24)

    For the kth iteration,1 1 1

    *11

    * * * ( 1)1 1 1 11( ) ( ( ))( ( ) ( ))k

    xPf x f x Px t g t + is known, so we

    can obtain its solution sequence (1, ) ( )kg t . Let 1( ) exp[( ) ]T Tt CR C P A t = . There exists a positive

    constant 1M such that 1( ) , [ , )t r M r t + . For same reason, the sequence { }(1, ) ( )kg t uniformly converges to the solution of the following system

    1

    1 1*11

    *11 1

    * *1 11

    ( ) ( ) ( ) ( )

    ( ( ))( ( ) ( ))

    ( ) 0

    T T

    x

    g t A CR C P g t Pf xf x Px t g t

    g

    =

    +

    =

    (25)

    Letting 1(* )(1, )lim ( ) ( )kk

    g t g t

    = and solving the following

    sequence

    1

    (2,0)1(2, ) (2, ) (2, 1)1 1 1 1

    *1 (2, )1

    (2, )1 10

    ( ) 0

    (t) ( ) ( )

    ( ( ) ( )), 0

    (0) , 1,2,

    k k k

    T k

    k

    x tx Ax t f xCR C Px t g t tx x k

    =

    = +

    + >

    = =

    (26)

    we get a solution sequence (2, )1{ ( )}kx t . Define 2(* )1 ( )x t as

    2(* ) (2, )1 1( ) lim ( )

    k

    kx t x t

    = . Construct the following sequences:

    0

    1

    *( ,0)1( , ) ( , ) ( , 1)1 1 1 1

    *1 ( , )1

    ( , )1 10

    ( ) 0, ( ) 0

    (t) ( ) ( )

    ( ( ) ( )), 0

    (0) , , 1,2,

    h

    h

    h k h k h k

    T h k

    h k

    x t g tx Ax t f x

    CR C Px t g t tx x k h

    = =

    = +

    + >

    = =

    (27)

    *1

    ( ,0)

    *( , ) 1 ( , )1 1

    * * ( , 1)1 11

    ( , )

    ( ) 0

    ( ) ( ) ( ) ( )

    ( ( ))( ( ) ( ))

    ( ) 0, , 1,2,

    h

    h hp

    h

    h k T T h k

    h k

    x

    h k

    g tg t A CR C P g t Pf x

    f x Px t g t

    g k h

    =

    =

    +

    = =

    (28)

    416

  • Repeating the foregoing iterations we can obtain sequences ( , )1{ ( )}h kx t and ( , ){ ( )}h kg t from (27) and (28) respectively.According the foregoing discussion, we have the

    following:Theorem 1: Define (* )(*)1 1( ) lim ( )hhx t x t= and

    (* )(*) ( ) lim ( )hh

    g t g t

    = . Then (*)1 ( )x t and (*) ( )g t are the

    solutions of the nonlinear TPBV problem (4). Thus the corresponding ( , )H K th nonlinear switching manifold sequence satisfies

    ( ) ( , ) ( , )2 2 1( ) ( ( ) ( )) 0H T H K H KRf x C Px t g t+ + = (29)

    and the optimal sliding surface is ( , )

    2 2 1( ) ( ( ) lim lim ( )) 0T H K

    H Kf x C Px t g t

    + + = (30)

    In fact, we cannot obtain optimal nonlinear switching manifold (30). We may find a suboptimal nonlinear switching manifold in practical applications by replacing

    ( , )lim lim ( )H KH K

    g t

    with ( , ) ( )H Kg t in (30). So, the ( , )H K th

    order suboptimal nonlinear switching manifold can be rewritten as:

    ( , )2 2 1( ) ( ) ( ) 0

    T T H KRf x C Px t C g t+ + = (31)Algorithm 1. (Find a suboptimal nonlinear switching

    manifold)Step 1: Solve the positive-definite matrix P from Riccati

    matrix equation (20). Let (1,0)1 ( ) 0x t = and 1k h= = . Give two positive real constants and .Step 2: Obtain the kth state vector (1, )1 ( )

    kx t from (27).

    Step 3: If (1, ) (1, 1)1 1( ) ( )k kx t x t < , let

    (* ) (1, )1 1( ) , 1h

    kx t x k= = ; else replace k by 1+k then go to step 2.Step 4: Let ( ,0) ( ) 0,hg t = and then obtain the kth adjoint

    vector ( , ) ( )h kg t from (28).

    Step 5: If ( , ) ( , 1)( ) ( )h k h kg t g t < , let (* ) ( , )( ) , 1h h kg t g k= = ; else replace k by 1+k then go to

    step 4. Step 6: Letting H h= , calculate ( )2 2( )

    Hf x from (29).Calculate ( )HJ from

    ( ) ( ) ( )1 1 2 2 2 20

    1 [ ( ) ( ) ( ( )) ( )]2

    H T H T HJ x t Qx t f x Rf x dt

    = + (32)Step 7: If ( ) ( 1) ( )( ) ,H H HJ J J < then stop and output

    the suboptimal control law ( )2 2( )Hf x .

    Step 8: Find the kth approximation ( , )1 ( )h kx t of the state

    vector 1( )x t from the state equation27.

    Step 9: If ( , ) ( , 1)1 1( ) ( )h k h kx t x t < , let

    (* ) ( , )1 1( ) ( ), 1h

    h kx t x t k= = ; else replace k by 1+k then go to step 8. Step 10: Replace h by 1h + then go to step 4.

    V. CONTROL LAW DESIGN AND STABILITY ANALYSISChoose the following switching function:

    ( , )2 2 1( ) ( ( ) ( ))

    T H KS Rf x C Px t g t= + + (33)The following control is employed:

    {

    ( )

    }

    1

    2 21 2 1

    2

    ( , )1 1 2 2 1

    2 23 1 2 2 2

    2( , ) ( , )

    ( )( ) ( , ) ( ) ( )

    ( ) ( ) ( ( ) ( ))

    ( ), ( ) ( )

    ( ) ( )

    TT T

    T T H K

    TT

    T H K T H K

    df xu t R B x x C PA KC P x tdx

    C Pf x Esign Rf x C Px t g t

    df xR f x x C PC KR f xdx

    KC g t C g t

    = + +

    + + + +

    + + +

    + (34)

    Theorem 2: Consider system (1) with control (34). Assume that the switching manifold is chosen as (33). Then the state trajectories of system (1) can reach switching surface 0s =in finite time.Theorem 3: The closed-loop system (1) with the control

    (34) is asymptotically stable.Proof: In quadratic performance index (3), Q and R are

    positive-definite matrices, so 11 2( )nx t L ; that is,

    1 10( ) ( )Tx t x t dt

    < (35)

    2 2 2 20( ) ( )Tf x f x dt

    < . (36)

    According to Assumption 2, 11 1 2( )nf x L , so we can get

    11 2( )

    nx t L from the first equation of system (1). Based onSchwarz inequality, we obtain 1 1 1( ) ( )

    Tx t x t L ; that is,1 12 2

    1 1 1 1 1 10 0 0( ) ( ) ( ) ( ) ( ) ( )T T Tx t x t dt x t x t dt x t x t dt

    < (37)

    It is obvious that

    1 1 1 11( ) ( ) ( ( ) ( ))2

    T Tdx t x t x t x tdt

    = . (38)

    Integrating (38) from 0 to , we obtain

    1 1 1 1 10 100

    1 1( ) ( ) ( ) ( )2 2

    tT T Tx t x t x t x t dt x x= + (39)So, there exists a constant c such that 1 1lim ( ) ( )Tt x t x t c = . Because 11 2( )

    nx t L , we get 0c = that is 1lim ( ) 0t x t = .

    From (5) and the third equation of (4) we get 2 2lim ( ) 0t f x = .

    Based on Assumption 4, we obtain 2lim ( ) 0t x t = .

    VI. SIMULATIONConsider nonlinear system1 with

    417

  • 11 21 11 121 2 1 1

    12 22 12 11

    11 213 1 2 2 2

    22 12 22

    11

    12 2 2

    2

    sin( )0 1, , , ( ) ,

    sin( )-1 1

    0 1 1 0( ) , , ( , ) ,

    1 0 1

    2(0) -1 1(0) 0 , ( )(0) 0

    x x x xA x x f x

    x x x x

    x xB x C f x x

    x x x

    xx f xx

    + = = = = +

    = = = +

    = =

    21

    22

    121

    1

    x

    x

    e

    e

    + +

    The matrices Q , R , K and E in quadratic cost function (3) and in sliding reachability condition (6) arechosen as

    3 0 0.01 0 1 0, , .

    0 6 0 0.01 0 1K E Q R = = = =

    Choosing 0.01, = we can get (1) 3.583,J =

    (2) 2.8273J = , (3) 2.3033,J = (4) 2.2004J = . Choosing

    0.05 = , we obtain the relative error of the performance

    index values satisfies (4) (3) (4)( )J J J < . It indicates

    that the 4th nonlinear switching surface(4,3)

    2 2 1( ) ( ) ( ) 0T TRf x C Px t C g t+ + = is close enough to the

    optimal solution 2 2 1( ) ( ) ( ) 0T TRf x C Px t C g t+ + = . So we

    obtain the switching function defined as equation (33). Based

    on reaching law approach (6), a control law that forces the

    system state to reach the sliding surface from initial system

    state within finite time is obtained. Simulation results are

    shown as following

    0 1 2 3 4 5 6 7-1

    -0.5

    0

    0.5

    t

    x11

    k=1k=2k=3k=4

    Fig. 1. State variable 11x of the closed-loop in the thk iteration

    0 1 2 3 4 5 6 7-0.1

    0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    t

    x12

    k=1k=2k=3k=4

    Fig. 2. State variable 12x of the closed-loop in the thk iteration

    0 1 2 3 4 5 6 7-2

    -1.5

    -1

    -0.5

    0

    0.5

    t

    x21

    k=1k=2k=3k=4

    Fig. 3. State variable 21x of the closed-loop in the thk iteration

    0 1 2 3 4 5 6 7-0.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    1.2

    t

    x22

    k=1k=2k=3k=4

    Fig. 4. State variable 22x of the closed-loop in the thk iteration

    Fig. 1-Fig. 4 clearly show that the state variables of the

    closed-loop system are asymptotically stable. Furthermore, it

    is obvious that the more iterative steps, the higher control

    precision.

    418

  • When 2.08t , the state variables 11 12 21( ), ( ), ( )x t x t x t and

    22 ( )x t of the closed-loop system reach the sliding surface

    from their initial states and remain on it. Choose the

    following quadratic performance index:

    ( ) ( ) ( )1 1 2 2 2 22.08

    1 [ ( ) ( ) ( ( )) ( )] .2

    N T N T NJ x t Qx t f x Rf x dt

    = +

    We get (1) (2)1.1520, 0.9026,J J= =(3) (4)0.9014, 0.9006J J= = . It means that the sliding mode

    designed can ensure the state variables of the closed-loop system converge to zero fast on idea sliding surface.

    ACKNOWLEDGMENTThis work is supported by National Nature Science

    Foundation under Grant 60574023 and Key Natural Science Foundation of Shandong Province under Grant Z2005G01.

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