[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 - A novel adaptive neural sliding mode control for systems with unknown dynamics

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  • Abstract In this paper, an adaptive neural sliding mode controller (ANSMC) is proposed as an asymptotically stable robust controller for a class of Control Affine Nonlinear Systems (CANSs) with unknown dynamics. In the proposed method a Control Affine Radial Basis function Network (CARBFN) is developed for online identification of CANSs. A recursive algorithm based on Extended Kalman Filter (EKF) is used for training of CARBFN to develop an adaptive model for CANSs with unknown and uncertain system dynamics to reduce the uncertainties to low values. Since the CARBFN model learns the system time-varying dynamics online, the ANSMC will compute an efficient control input adaptively. Due to high degree of robustness, the proposed controller can be widely used in real world applications. To demonstrate this efficiency, a robust control system is successfully designed for a chaotic Duffing forced oscillator system in the presence of unknown dynamics as well as the unknown oscillation disturbance which is not available for measurement

    I. INTRODUCTION he control of uncertain nonlinear systems is an important and challenging problem in designing real world control systems. To deals with this fact, robust

    control and adaptive control can be used. Adaptive control techniques are restricted to the parameterization of known model structures but of unknown parameters (parametric uncertainty). Consequently, when a poorly known dynamic model exists, adaptive control can not be proper [1].

    The robust variable structure controllers using a sliding mode control (SMC) can be successfully used for control of nonlinear systems operating under structural uncertainty conditions. The main characteristic of SMC is that it uses a high-speed switching control law to drive the system states from any initial state onto a surface in the state space (sliding surface), and to maintain the states on the surface for all subsequent time.

    Although SMC offer many properties, such as fast dynamic response, insensitivity to parametric variations and external disturbance rejection, however there are still some weaknesses. The main drawback of the standard sliding modes is mostly related to the so-called chattering caused by the high-frequency control switching. When the states are on the sliding surface, a very small disturbance forces the states to leave the surface. As these disturbances are always available (i.e. computational quantization error of digital controllers), the control law creates high frequency switching around sliding surface. The high frequency component of chattering is not only undesirable by itself but Manuscript received April 9, 2010. The Authors are with the Department of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran (Correspondence author: H. Modares, e-mail: Ha.modarres@stumail.um.ac.ir). (e-mail: rowhanimanesh@ieee.org, karimpor@ferdowsi.um.ac.ir).

    they can also excite unmodeled high-frequency plant dynamics which could result in unforeseen instability. To solve this problem several successful and efficient approaches have been proposed [2]-[4].

    Another important problem in SMC which is neglected in most of the control approaches is plant modeling problem. In most of the practical engineering problems, the model of plant is not available and the control engineer must identify the plant. SMC method is invalid when the model of plant is unknown. Recently, much research works have been done to use soft computing methodologies such as artificial neural networks and fuzzy systems in order to apply SMC to unknown dynamic models. Akbarzadeh-T and Shahnazi [2] combined SMC and PI control to attenuate chattering and applied fuzzy system to approximate the unknown disturbances as well as uncertain dynamics. Slotine et al. [3] and Spong et al. [5] theoretically considered some aspects of multivariable SMC. Combinations of fuzzy logic and neural networks with SMC were considered in [6]-[10]. Radial Basis Functions and fuzzy neural networks were proposed for adaptive and fixed sliding mode control of robotic manipulators in [11]-[13]. Sliding mode control of several specific applications based on different types of neural networks were considered in [14]-[18].

    Most of the previous works trained neural networks as well as fuzzy systems in an offline manner. However, some systems are highly time varying and if any change in system dynamic occurs, none of them can track these changes. Under these circumstances, an environmentally independent good performance scheme, i.e. an adaptive control strategy that provides satisfactory tracking performance, to robust against time-varying properties of system dynamics and external disturbances is required. Due to this, in this paper, we proposed an Adaptive Neural Sliding Mode Control (ANSMC) to deal with unknown time-varying dynamic systems. We developed a Control Affine Radial Basis Function Network (CARBFN) to learn the system time-varying dynamics online, based on Extended Kalman filter (EKF). The Radial Basis Function Network (RBFN) is chosen because of its simple structure, ease of implementation and fast training times [19]. The EKF is outlined for online training of RBFN, because the EKF can learn the process dynamics much faster than the steepest descent method [20], which is a significant issue in adaptive control. While the adaptive CARBFN model learns the time-varying dynamics online, the ANSMC will compute an optimal control variable based on the adaptive model and consequently reduces tracking errors of the transient state and increase the robustness of sliding mode control. The proposed ANSMC method can efficiently control the nonlinear system with unknown and time varying dynamics. Due to simplicity as well as robustness, the proposed

    A Novel Adaptive Neural Sliding Mode Control for Systems with Unknown Dynamics

    H. Modares, A. Rowhanimanesh, A. Karimpour

    T

    40

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

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

  • approach has high potential to employ in real world applications.

    The rest of paper is organized as follows: Next section describes conventional sliding mode control. Section 3 introduces the RBFN, briefly. Section 4 presents the EKF based RBFN training method. In section 5, the proposed ANSMC controller is introduced. Section 6 and 7 contain simulation results and conclusions, respectively.

    II. SLIDING MODE CONTROL A. Conventional SMC Consider nonlinear systems whose dynamical equations

    can be expressed in the canonical form [2], [3]: ),()(),(),()()( tXdtutXbtXftx n ++= (1)

    where nTn RtxtxtxtX = )]()...()([)( 21 is the state vector, ),( tXf and ),( tXb are two unknown bounded nonlinear

    functions, belongs to RR n , Ru and Ry are the input and output of the system and ),( tXd is an unknown bounded disturbance. The control objective is to design a controller which tracks a desired time varying state trajectory. Consider nTdnddd RtxtxtxtX = )]()...()([)( 21 is the desired trajectory vector in the state space. Then, the tracking error is defined as follows:

    TnTndn

    ndddtetetetxtx

    txtxtxtxtXtXtE)](),...,(),([])()(

    )...()()()([)()()()1()1()1(

    21

    =

    ==

    (2)

    The control goal considered in this paper is that for any given target )(tX d , a SMC is designed such that the resulting state response of the tracking error vector satisfies the following condition:

    0||)()(||lim||)(||lim = tXtXtE dtt (3)

    In the following, we present a brief introduction to classical SMC design method. In traditional SMC, the states of the controlled system are first guided to reside on a switching surface (i.e., the sliding surface) in state space. A time varying surface ),( tXs is defined in the state space by equating the variable ),( tXs defined below, to zero.

    ecececetXs nnn

    12)2(

    1)1( ...),( ++++=

    (4) If the coefficients 1,...,1, = nici are chosen such that

    the following polynomial is Hurwitz, then the error converges to zero once the states are on 0),( =tXs . That is, all the roots of the following characteristic polynomial describing the sliding surface have negative real parts with desirable pole placement:

    122

    11 ...)( cccP nn

    n ++++=

    (5) Thus the objective becomes to design a control law that forces the state trajectory to a sliding surface 0),( =tXs in finite time and to remain on this surface.

    Consider 2),( stXV = as a candidate Lyapunov function. For achieving the mentioned objective the control law must be chosen as follows:

    or 21 2 sssss

    dtd

    mm (6) where s is the absolute value of s, 0;m and we have:

    =

    +++=1

    1

    )()( ),()(),(),(n

    i

    nd

    ii xtXdtutXbtXfecs (7)

    Therefore, (6)

    sxtXdtutXbtXfecs mn

    i

    nd

    ii +++

    =

    1

    1

    )()( )),()(),(),(( (8)

    Solving the above inequality for )(tu , the SMC control law

    )(tu is obtained as follow [2], [3]:

    )]sgn(),(

    ),([),(

    1)(1

    1

    )()(

    stXd

    xtXfectXb

    tu

    b

    n

    i

    nd

    ii

    += =

    (9)

    where b is switching gain and sgn(.) is the sign function given by:

    =

    +

    =

    0if00if10if1

    )sgn(sss

    s (10)

    In practice ),( tXf , ),( tXb and ),( tXd are unknown and must be identified, furthermore ),( tXd may not be available for measurement.

    B. Adaptive Neural SMC In this paper, a Control Affine Radial Basis function Network (CAFNN) is developed for online identification of the class of Control Affine Nonlinear Systems (CANSs) described in Eq. (1). Then, a recursive algorithm based on Extended Kalman Filter (EKF) is used for training of CARBFN to develop an adaptive model for CANSs with unknown and time varying nonlinear functions ),( tXf and

    ),( tXb , to keep the identification errors to low values. Let we denote the approximations of functions ),( tXf and

    ),( tXb as f

    and b

    , respectively. Then, the adaptive neural sliding mode controller (ANSMC) control law

    )(tu is changed as follow:

    )]sgn(

    ),([1)(1

    1

    )()(

    s

    tXdxfecb

    tu

    b

    n

    i

    nd

    ii

    =

    +=

    (11)

    In the following we describe how we obtain f

    and b

    in the proposed method.

    III. RADIAL BASIS FUNCTION NETWORKS (RBFN) RBFN have received considerable applications, such as function approximation, prediction, recognition, etc. [21]-[23] due to its simple structure and fast convergence speed. RBFN has been traditionally used as a multidimensional

    41

  • interpolation technique implementing general mapping RRf m : according to [24]:

    ||)(||)(1

    01

    0 k

    M

    kkk

    M

    kk CXwwGwwXf +=+=

    ==

    (12)

    where is a nonlinear function, selected from a set of typical ones (usually a Gaussian function), ||.|| denotes the Euclidean norm, kw are the tap weights and mk RC are called RBFN centers. It is easy to see that the formula above is equivalent to a special form of a 2-layer perceptron. Fig 1 depicted RBFN structure. When constructing an RBFN, it is important to determine the network structure and the network parameters, i.e. centers and the weights of RBFN. The task of tuning the centers and the weights of RBFN is called training of the network. Several methods for training RBFN are introduced in the literature. Generally, training a neural network is a challenging nonlinear optimization problem. Various derivative-based methods have been used to train neural networks. Derivative-free methods have also been used to train neural networks. Derivative-free methods are more robust than derivative-based methods with respect to finding a global minimum. However, they typically tend to converge more slowly than derivative-based methods. Derivative-based methods have the advantage of fast convergence, but they tend to converge to local minima. In this paper we use EKF for online training of RBFN proposed in [25], because of its fast convergence speed and apply it for adaptive modeling of unknown dynamics system under control. In the following, we describe the proposed Control Affine RBFN (CARBFN) used in this paper to identify control affine nonlinear systems.

    IV. CONTROL AFFINE RBFN (CARBFN) A. The structure of CARBFNN

    Consider the nonlinear systems described in Eq. (1). We can use the following state space model for this system:

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

    )()()()(

    32

    21

    txtytXdtutXbtXftx

    txtxtxtx

    n=

    ++=

    =

    =

    #

    (13)

    where nn RtxtxtxtX = )](),...,(),([)( 21 is the state vector of the system which is assumed to be available for measurement and using a simple discrete time observer, an acceptable approximation of it can be achieved. ),( tXf and

    ),( tXb indicate general nonlinear functions whose models are time-varying as the nonlinear process evolves over time. To estimate the unknown functions ),( tXf and ),( tXb adaptively, a special neural network, a control affine RBFN (CARBFN) is proposed. The basic idea is based on control affine feedforward neural network proposed in [26]. CARBFN contains two parts: an inner RBF neural network and a synthesis layer with lateral connection. Fig. 2 shows

    the structure of CARBFN. The dashed frame in Fig. 2 is the inner network, which is a RBFN. This inner network only has as its inputs )](),...,(),([)( 21 kxkxkxkX n= where is the sample of X(t) at the k-th sampling time. the output of inner network are )),(( kkXf

    and )),(( kkXg . The system input,

    u(k), is applied as an input to the synthesis layer where it is multiplied to one of the inner network output, )),(( kkXg , and then the result is added to the other output of the inner network, )),(( kkXf

    , through the lateral connection. Hence,

    the synthesis layer calculates the final output of the CARBFN as )()),(()),(()( kukkXgkkXfkxn

    += where )),(( kkXf

    and )),(( kkXg are the two outputs of the inner

    network. There are no adjustable weights in this synthesis layer. Calculations in the CARBFN can be summarized as follows. Get the input vector of the inner network as:

    )](),...,(),([)( 21 txtxtxtX n= (14) Then, define the output of the inner network as:

    ,)]),((),),(([)),(( TkkXbkkXfWkXF

    = (15) where

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

    0 k

    M

    kk CkXwwkkXf +=

    =

    (16)

    In the synthesis layer, the output of CARBFN is calculated as

    )()),(()),(()( kukkXbkkXfkxn += (17)

    Fig. 1. Structure of RBFN

    Fig. 2. Structure of CARBFN

    42

  • V. ONLINE TRAINING OF CARBFN USING EKF

    A. Extended Kalman Filter (EKF) The EKF is an efficient learning algorithm for parameter

    estimation. In fact, EKF is extension of the Kalman filter algorithm in which the nonlinearity of the system dynamics is linearized around the most recent state estimate, before the Kalman filter is applied. The EKF algorithm for parameter estimation can be described as follows: Consider a nonlinear finite dimensional discrete time system of the form

    18)( ttt wf +=+ )(1 )19( ttt vhy += )(

    where the vector t is the state of the system at time t, tw is the process noise, ty is the observation vector, tv is the observation noise, and (.)f and (.)h are nonlinear vector functions of the state. Assume that the noise sequences }{ tw and }{ tv are Gaussian and independent from each other with

    (20) 0][][ == tt vEwE

    (21) [ ] = == stR stvvEstC tTstv 0.),( (22) [ ] = == stQ stwwEstC tTstw 0.),( (23) [ ] 0.),( == Tstwv vwEstC

    The problem addressed by the extended Kalman filter is to find an estimate 1+t

    of 1+t given ),...,0( kjy j = .

    If the nonlinearities in Eq. (1) are sufficiently smooth, we can expand them around the state estimate t

    using Taylor

    series to obtain: (24)

    tt

    ttt

    wfF

    =

    ),(

    (25) t

    t

    ttt

    whH

    =

    ),(

    Then, the system model is approximated as: (26) tttt wF +=+ 1 (27) tttt vHy +=

    It can be shown that the desired estimate t

    can be obtained by the recursion

    (28) )]([)( 11 += ttttt hyKf

    (29) 1][ += t

    Tttt

    Tttt RHPHHPK

    (30) tT

    ttT

    ttttt QFPHKPFP +=+ )(1 where tK is shown as the Kalman gain. tP is the covariance matrix of the state estimation error.

    B. On-line training of CARBFN In the following, online estimation of CARBFN

    parameters using EKF is described. Note that the inner network of CARBFN shown in Fig. 2 is an RBFN type

    network with n input and 2 output layer, where n is the order of CANS. So, CARBFN can train as a RBFN, since there is no adaptable weight in the synthesis layer.

    Consider the RBFN depicted in Fig. 1 with n inputs, l centers, and m outputs. We use y to denote the target vector for the RBFN outputs, and )( th

    to denote the actual

    outputs at the t-th iteration of the optimization algorithm. (31) T

    mMmM yyyyy ].........[ 1111= (32) T

    mMmMt yyyyh ].........[)( 1111

    = Note that the y and y vectors each consist of mM elements, where m is the dimension of the RBFN output and M is the number of training samples. In order to cast the optimization problem in a form suitable for Kalman filtering, we let the elements of the weight matrix W and the elements of the centers iC constitute the state of a nonlinear system, and we let the output of the RBFN constitute the output of the nonlinear system to which the Kalman filter is applied. The state of the nonlinear system can then be represented as

    (33) TTl

    TTm

    T vvww ]......[ 11= The vector thus consists of all (ml+nl) of RBFN parameters arranged in a linear array. The system model to which the Kalman filter can be applied is

    (34) ttt w+=+ 1 (35) ttt vhy += )(

    where )( th is the RBFN nonlinear mapping between its parameters and its outputs, tw and tv are artificially added noise processes. Now we can apply the Kalman recursion of Eqs. (21-23). (.)f is the identity mapping and ty is the target output of the RBF network. (Note that although ty is written as a function of the Kalman iteration number t, it is actually a constant.) )( th

    is the actual output of the RBFN

    given the RBFN parameters at the t-th iteration of the Kalman recursion. tH is the partial derivative of the RBFN output with respect to the RBFN parameters at the t-th iteration of the Kalman recursion. tF is the identity matrix (again, a constant even though it is written as a function of t). The Q and R matrices are tuning parameters which can be considered as the covariance matrices of the artificial noise processes tw and tv , respectively. Low level state noise is assumed for the states which represent parameters. This prevents the gain matrix of the EKF from becoming too small. As a consequence, changes in the dynamic system can be quickly detected also when the EKF is running for a long time. We recommend the readers to refer to [GG] for further details. The use of the Kalman filter results in better learning than conventional RBFN and faster learning than gradient descent [GG].

    VI. SIMULATION RESULTS In this section the proposed approach is applied to design a robust control system for Duffing forced oscillator system where the dynamics is unknown as well as the oscillation

    43

  • disturbance is not available for measurement. The Duffing forced oscillator exhibits behavior ranging from limit cycles to chaos due to its nonlinear dynamics. This system is described by the following 2nd order differential equation:

    (36) )()cos(3222

    tutxxdtdx

    dtxd

    +=++

    Let ,2,1,25.0,1.0 ==== ,1)0()0(1 == xx 0)0()0(2 == xx . For uncontrolled Duffing system (u(t)=0), the

    behavior of the system is chaotic for 2= as can seen from Fig. 3. Also the sensitivity of the output response to the initial condition is shown by comparing the output signals for 1)0( =x and 01.1)0( =x . Here, we suppose that the dynamics is unknown and furthermore )cos()( ttd = is not available for measurement but 5),(

  • Fig. 6. Output (solid) of controlled system for

    )pulse stepunit)(5.0)0( == txandx d

    VII. CONCLUSIONS In this paper, an adaptive neural sliding mode controller (ANSMC) is proposed for control affine nonlinear systems where the dynamics is unknown and the disturbances may not be available for measurement. A Control Affine Radial Basis function Network (CARBFN) is developed for online identification of this class. The proposed CARBFN converts the problem of online identifying the two nonlinear functions of control affine nonlinear systems into identifying an RBFN with two output neuron, adaptively, which can be easily solved by an Extended Kalman filter (EKF), adaptively. This efficiency is demonstrated in the simulation example where an ANSMC controller is successfully designed for the chaotic Duffing forced oscillator system in the presence of unknown dynamics as well as the oscillation disturbance which is not available for measurement.

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    [3] J.J. Slotine and L. Weiping., Applied Nonlinear Control, Prentice Hall, 1991.

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    [7] V. Mkrttchian and A. Lazaryan, Application of neural network in sliding mode control, Proc of IEEE Conf on Control Applications, 2000, 653-657.

    [8] L. Hongli, Z. Dianzhi and L. Wenhua., Sliding mode control of nonlinear system based on neural networks, Proc of 3rd World Congress on Intelligent Control and Automation, 2000, pp. 962-966.

    [9] M. Lee, A sliding mode controller with neural network and fuzzy logic, Proc of IEEE Conf on Neural Networks, 1997, pp. 2414-2417.

    [10] A. abanovi, K. Jezernik and M. Rodi., Neural network application in sliding mode control systems, IEEE Workshop on Variable Structure Systems, 1996, pp. 143-147.

    [11] Z. Chen, J. Zhang, Z. Wang and J. Zeng, Sliding Mode Control of Robot Manipulators Based on Neural Network Reaching Law, IEEE Conf on Control and Automation, 2007, pp. 370-373.

    [12] H. B. Xu, F. C. Sun and Z. Q. Sun, The adaptive sliding mode control based on a fuzzy neural network for manipulators, IEEE Conf on Systems, Man, and Cybernetics, 1996, pp. 1942-1946 vol.3.

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    [14] C.Jie, X. Cangsu and L. Shaomin, A sliding mode semi-active control for suspension based on neural network, 7th world congress on Intelligent Control and Automation, 2008, pp. 6143-6148.

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    [17] H. Zhao, H. Yu and W. Gu, Fuzzy Neural Network-Based Sliding Mode Control for Missile's Overload Control System, International Conf on Neural Networks and Brain, 2005, pp. 1786- 1790.

    [18] J.M. Carrasco, J.M. Quero, F.P. Ridao, M.A. Perales and L.G. Franquelo, Sliding mode control of a DC/DC PWM converter with PFC implemented by neural networks, IEEE Trans on Circuits and Systems I: Fundamental Theory and Applications, vol.. 44, pp.1997, 743-749, August 1997.

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