Hung-Yuan Chung* and Lon-Chen Hung

Transcription

1 54 Int. J. Computer Applications in echnology, Vol. 8, No. 4, 007 Design of an adaptive neural sliding-mode controller for seesaw systems Hung-Yuan Chung and Lon-Chen Hung Department of Electrical Engineering, National Central University, Jhong-Li, ao-yuan, 30, aiwan, ROC Corresponding author Abstract: In this paper, an Adaptive Neural Network Sliding-Mode Controller (ANNSMC) design approach is proposed. We present a novel adaptive method for a Neural Network (NN) system to approximate unknown non-linear continuous functions. he proposed scheme combines the benefits of the adaptive control, NN and Sliding-Mode Control (SMC) manner without precise system model information. It has online adjusting ability to cope with the unknown parametric and external disturbance by tuning the control parameters. he proposed ANNSMC approximator is then applied in the control of a seesaw system. Simulation results of the seesaw system show that the proposed controller is feasible and effective. Keywords: adaptive control; neural network; sliding-mode control; seesaw. Reference to this paper should be made as follows: Chung, H-Y. and Hung, L-C. (007) Design of an adaptive neural sliding-mode controller for seesaw systems, Int. J. Computer Applications in echnology, Vol. 8, No. 4, pp Biographical notes: Hung-Yuan Chung received the PhD Degree in Electrical Engineering from the National Cheng Kung University, ainan, aiwan, ROC, in 987. In August 987, he joined the Department of Electrical Engineering at the National Central University, Jhong-Li, aiwan as an Associate Professor. In August 99, he was promoted as Professor. His research and teaching interests include system theory and control, adaptive control, fuzzy control, neural network applications, and microcomputer-based control applications. Lon-Chen Hung joined the Department of Computer and Communication at the Diwan University, ainan, aiwan, ROC as an Assistant Professor in February 007. His research interests include sliding-mode control, fuzzy theory and control, and neural-network applications. Introduction Sliding-Mode Control (SMC) is an effective modern control to cope with non-linear systems that is well known for its robustness features. It has been developed and applied to closed-loop control systems for three decades (Utkin, 977; Humg et al., 993; Ackermann and Utkin, 998; Young et al., 999; Bartolini et al., 000; Michiels and Roose, 00; Edwards et al., 00; Sha et al., 00; Choi, 003; Edwards and Spurgeon, 003; Adamy and Flemming, 004; Barambones and Garrido, 004; Koshkouei et al., 005). It has been widely provided to control non-linear systems, especially systems that have model uncertainty and external disturbance (Humg et al., 993; Michiels and Roose, 00; Edwards and Spurgeon, 003; Barambones and Garrido, 004). Robustness is the best advantage of SMC. However, the main character of SMC is that it uses a high-speed hitting control law to drive the system states from any initial state onto a user-specified surface in the sliding surface, and to keep the states on the surface for all subsequent time (Michiels and Roose, 00). heoretically, it will gradually draw near the origin of a phase plane. hey have shown that the boundary layer can be reached in finite time and the ultimate boundedness of states is acquired asymptotically even though there exists some disturbance of dynamic uncertainties of the system. SMC provides a good performance in tracking of some non-linear systems (Sha et al., 00). Nevertheless, a notorious feature of the SMC approach is the discontinuity around the switching hyperplane that means some of the state variables being in oscillation. Hence, it has received considerable attention from the research community (Utkin, 977; Ackermann and Utkin, 998; Bartolini et al., 000). o reduce the chattering, some researchers have attempted to use a saturation function or a sigmoid function for replacing the sign function (Sundareshan and Askew, 997; Ertugrul and Copyright 007 Inderscience Enterprises Ltd.

2 Design of an adaptive neural sliding-mode controller for seesaw systems 55 Kaynak, 000). Moreover, the equivalent control is difficult to calculate. In order to avert the computational burden, a calculation method can be used to estimate the value of equivalent control. More recently, the use of intelligent methods based on fuzzy logic, NN and other methods adapted from artificial intelligence have also been indicated (Sundareshan and Askew, 997; Parma et al., 998, 999; Ertugrul and Kaynak, 000; Chang et al., 00; Wai, 003; Costa et al., 003; Chen and Peng, 004; Hussain and Ho, 004; Wai and Li, 004; Hussain and Yee, 004; Lin and Hsu, 005; ang et al., 006). Here the sliding surface variable, s, is employed as the controlled variable for designing a NN controller. his sliding variable, s, will be used as the input to estimate the control law. his can be used in non-linear time-varying system with unknown state variables (Sundareshan and Askew, 997; Parma et al., 998, 999; Ertugrul and Kaynak, 000; Huang et al., 00). Sundareshan and Askew (997) showed an NN-based scheme for adaptively approximate a variable structure controller to control a flexible manipulator arm; however, it needs the offline tuning with the back-propagation algorithm. Ertugrul and Kaynak (000) showed the parallel NN can be applied to attain a SMC for the direct drive scara-type robot system; however, it express NN based on the gradient descent method and apply offline tuning and the stability analysis of closed-loop is not yet to proceed. Parma and de Menezes used SMC to train NN as described for offline tuning (Parma et al., 998) and then for online tuning (Parma et al., 999) of Multi-Layer Perceptrons (MLPs); however, system stability is still not guaranteed by the sliding line. he ANNSMC approach is presented to solve the problem of robust control for a seesaw system in the existence of uncertainties and external disturbance. It will be seen that the NN-based control can overcome certain problems encountered in a conventional SMC approaches for non-linear systems, such as the assumption that the non-linearities are of the norm-bounded type. he sliding-mode used is a linear function of the current system states. he online learning algorithm of ANNSMC is determined in the Lyapunov sense; thus, the stability of the control system can be guaranteed. Furthermore, to relax the need for the uncertain bound in the hitting controller, an observation technique is provided to calculate the uncertain bound, so that the chattering phenomena of the control efforts can be relaxed. his controller ensures some properties, for instance the robust performance and stability properties. We show that the ANNSMC has the following advantages: it can well control non-linear systems with system parameters that are unknown the unknown model and extra disturbance can be dealt with through the NN approach, and therefore would attain robust control system performance though of existing uncertainties the ANNSMC can not only reinforce the robustness to parameter uncertainties but also reduce the chattering phenomenon in the conventional SMC. he rest of the paper is divided into five sections. In Section, the SMC design for uncertain systems is presented. In Section 3, a design for an ANNSMC is described. In Section 4, the proposed controller is used to control a seesaw system. Finally, we conclude with Section 5. Sliding-Mode Control (SMC) design for uncertain systems Consider a single input nth order non-linear system x = f( x) + g( x) u+ d () ( n ) where x = [ x x x ] is the state vector of the system, f( x ) = [ f( x) f( x) f ( )] n x is the vector whose elements are non-linear function x which are not exactly known, g( x ) = [ g( x) g( x) g ( )] n x is the vector whose elements are non-linear function x which are exactly known, u indicates the control input of the system, d indicates the unknown disturbance, respectively. he input gain matrix g(x) is assumed to be known and bounded, also its inverse g (x) exists for all x. Moreover, f(x) and d are also assumed to be bounded. he aim of a control system is to design a controller u such that the system output x(t) can reach a desired equilibrium point. For a non-linear system that the sliding surface can be chosen as a linear differential equation of order n as ( n ) ( n ) s = x + k x + + k x + k x () n 0 where k i are selected to provide the desired eigenvalue placement for the differential equation s = 0 to provide the desired drive of the system s trajectory to the equilibrium point. Besides, disturbance d definitely influences the phase trajectories. However, if these trajectories point in opposite directions in the neighbourhood of s = 0, the system will again drive in a sliding surface and, while this dominion holds, its motion will be invariant with regard to disturbance d. Utkin (977) presented a relay to handle in a sliding surface is equivalent to an operational amplifier with infinite gain. But if the relay is operating in a sliding surface, its input signal is discontinuous and its sign vibrant at infinite frequency. he states ensure that a sliding-mode exists on a sliding surface, and if this sliding surface can be approached in finite time, one has to satisfy the η-reachability condition as follows (Humg et al., 993; Utkin, 977; Young et al., 999; Michiels and Roose, 00): s s η s or s sgn( s) η (3) where η is a small positive constant. By taking the time derivative of both sides of equation (), we obtain: n ( i+ ) s = kx i + f + g i= ( x) ( x) (4)

3 56 H-Y. Chung and L-C. Hung and the control force u which ensures the sliding condition (3): x x (5) n ( i+ ) sgn( s) ki x + f( ) + g( ) u η. i= Assume the non-linear function f(x) and g(x) and the disturbance d can be exactly known, then the control law that satisfies equation (3) is given by k ( i+ ) u = g ( x) f( x ) ki x h sgn( s) (6) i= 0 where h is a positive number, sgn( ) is the sign function given by, s > 0 sgn( s) = 0, s = 0., s < 0 o reduce the chattering influence caused by the discontinuous control law, the boundary layer approach can be used (Young et al., 999; Bartolini et al., 000). In this approach, a boundary layer is added around the sliding surface. Within this layer, the control is chosen to be a continuous approximation of the switching function, i.e., the hitting term h sgn(s) is replaced in corrective term by h sat(s/φ) where sat is the saturation function, given by sgn( s/ Φ), if ( s/ Φ ) > sat( s / Φ ) = ( s/ Φ), if ( s/ Φ) and Φ is the boundary layer thickness. Hence, we have the hitting control law modified as k ( i+ ) u = g ( x) f( x) ke i h sat( s/ Φ) i= 0 = u + u. eq h Hence, in the sliding surface, an equivalent controller u eq will be k ( i+ ) ueq = g ( x) f( x) ke i (0) i= 0 and the hitting control u h will be u h = h g ( x) sat( s/ Φ) = M sat( s/ Φ) where the hitting gain M is a positive number. (7) (8) (9) () Remark : As mentioned above, the thickness of the boundary layer necessary to attain complete reduction of chattering is proportional to the amplitude of vibration, which is in turn related to the value of the hitting gain M. his acts on SMC where hitting control input u h strength is fixed. Furthermore, the ideal control law can not be acquired, as it is not possible to realise equation (9). o solve such a problem, a novel approach of the weight adaptation of the NN control is presented to calculate an equivalent control input u eq and a simple tune-bound estimation technique is also applied for relaxing the necessity of uncertainty bounds and cancelling the chattering phenomena. he tune-bound estimation is designed to regulate the upper bound on the uncertain term. he presentation of ANNSMC will be discussed in the Section 3. 3 Design of the proposed control In this section, we indicate how to describe an ANNSMC for receiving the equivalent control through weight adaptation. hen, we construct the hitting control to guarantee the system s stability. If the state trajectory can be forced to slide on the sliding surface, then a stable equivalent control system is attained. However, if the function f is unknown (for simplicity, we assume g is known), there is no way to yield equivalent control u eq. he NN is used as the non-linear approximator. If the state trajectory can be forced to slide on the sliding surface, then a stable equivalent control system is achieved. A set of NN base is applied to approximate in equation (9). Motivated by the principle of SMC, the control law consists of the following two parts; one is the calculated sliding component u ANNSMC constructed by a tuning technique. he impact of this term is to force the system state to slide on the sliding surface. Another is the hitting control u h that moves the states toward the sliding surface. 3. Basic NN approximation technique he simple two-layer NN to approximate a general smooth non-linear function on a compact set S R k. According to the NN approximation property, we have f ( x) = w σ( w x) + ε( x ) () where x = [, x,, x k ] is the input to NN, σ( ) is an active function, where w and w are defined as the collection of NN weights for the output and the hidden layer, respectively, and ε(x) is the NN approximation error. Briefly, let us define NN weight error w = w w, w = w w ( ^ indicates estimation value, indicates ideal value), the norm of a vector x is denoted by x = x x (ang et al., 006). 3. Sliding-Mode Control (SMC) based on Neural Network (NN) A single-hidden-layer NN with two layers of adjustable weights is shown in Figure. Following the notation used in Lin and Hsu (00), the output of this NN takes the form y = w σ ( w s) (3) where w and w are the input and y is output of the NN, respectively; σ( ) represents the hidden-layer activation function; w are the interconnection weights between the input and hidden layers; and w are the

4 Design of an adaptive neural sliding-mode controller for seesaw systems 57 interconnection weights between the hidden and output layers. his structure has one input, hidden-layer neurons, and one output. he activation function is considered as a hyperbolic tangent function (Ge et al., 999) s s e e σ () s =. s s e + e (4) and the hidden-layer output error is given as σ = σ ( w s) ( w s). (9) From the function σ( ) with parameter x, one may express its aylor series with another parameter σ() x = σ() x + σ () x x + O () x (0) Figure he single-hidden-layer NN with two layers where σ' is the Jacobian, and the last term indicates terms of order x. Consequently, σ = σ ( w s) w s+ O ( w s). () he control law for the ANNSMC system as indicated in Figure is assumed to have the following form: us (, w, w ) = uannsmc + uh () A main property of a NN regarding the feedback control goal is the universal function approximation property. A NN is capable of approximating any smooth function to any desired accuracy, provided that the number of hidden-layer neurons is sufficiently extensive. By the universal approximation theorem (Lin and Hsu, 00), there exist perfect weight vectors w and w such that where u ANNSMC is the approximate equivalent control, and the hitting control u h is designed to stabilise the states of the control system that have a compensating effect. Figure he structure of the ANNSMC system Θ= y( s, w, w ) +Φ= w σ ( w ) +Φ (5) where Φ is the approximation error, which generally decreases as the net size increases. For any choice of a positive number Φ p, one can find a feedforward NN such that Φ Φ p for all s. he ideal NN weights in vectors that are needed to best approximate a given non-linear function are difficult to resolve. In fact, they may not even be unique. However, all one needs to know for control aim is that, for a specified value of Φ p some ideal approximating NN weights exists. hen, a calculate of Θ can be given by Θ= y(, s w, w ) = w σ ( w s) (6) where w and w are the estimated values of the ideal NN weights w and w that are applied by some online weight tuning algorithms subsequently to be detailed. For notational convenience, σ( v s) = [ σ σ σ ] m are represented by ws ws ws ws e e e e i = = ws ws ws ws σ e + e e + e. (7) he estimation errors of the weights of NN are defined as w = w w, w = w w (8) Define the NN controller estimation error as u ANNSMC as u ANNSMC = ueq uannsmc = uannsmc + uannsmc = w ( w ) w ( w s) + = w σ( w ) + w σ + w σ +. σ s σ s (3) Applying the aylor series approximation equation (3) for σ, according to which the approximation error is u = w ( w ANNSMC σ s) + w [ σ ( w s) w s+o ( w s)] + + = w w + w w w w σ σ( s) σ ( s) s+ ε (7) where ε = w O ( w s)] + w σ + is supposed to be bounded by ε U, in which is the absolute value.

5 58 H-Y. Chung and L-C. Hung Remark : he SMC main difficulties are the need for partial system parameters, uncertainty bounds and that the chattering phenomenon occurs due to control efforts (Wai and Lee, 004). In order to cope with these problems, an ANNSMC combining a NN controller and a hitting controller is studied in this work. he NN control is used to learn the equivalent control law in the SMC due to the unknown model, and the hitting control is designed to suppress the controlled system dynamics on the sliding surface for all time. In this hitting controller, a simple tune-bound observation technique is also adopted for relaxing the necessary of uncertainty bounds in the SMC. he tune-bound estimation designed to modify the upper bound on the uncertain term, which has the benefit of simple structure, can ensure the states to be zero. In order to analyse the overall system stability, we introduce assumptions as follow: Assumption : g(x) in equation () is differentiable, i.e., d dx [ g ( x)] = D[ g ( x)] = D[ g ( x)] x dt dt where g g g Dg [ ( x)]=,,,. (5) x x x4 Assumption : he following inequality holds 0, s 0 s =. s Ψ, s > 0 (6) Assumption 3: here exists optimal values for the weight of NN such that g uannsmc ueq + s x + ε = U (7) x where the uncertainty bound U is a positive constant. his uncertainty bound can not be measured for mechanical models. herefore, a bound estimation is developed to calculate the bound of approximation error. Ξ= Ut () U (8) where Ut () is the estimated uncertainty bound. he adaptive laws will be developed to adjust the parameters w, w and U to estimate w, w and U, respectively. heorem : Considering the dynamic non-linear systems described by () with the sliding-mode (). If Assumptions -3 hold, for the neural network SMC law is designed as (), in which the adaptation laws of the NN controller with robustifying term are designed as (9), (30) and the hitting controller is designed as (3) with the adaptive bound calculation express in (3), then can guarantee the asymptotic stability of the close-loop system. he NN adaptive laws are given by w = w = ρ s σ( w s ) ρ s w (9) w = w = ρ s σ ( w s ) w ρ s w (30) u = U tanh( s/ Ψ ) (3) h U =Ξ= ρ s ρ s U (3) 3 3 where ρ, ρ and ρ 3 are positive constants. Moreover, the system states converge to the sliding surface asymptotically. Proof: Choose the Lyapunov function as s w w w w Ξ V = g ρ ρ ρ3 (33) where Ξ= Ut () U, s = s Ψ tanh( s/ Ψ ), Ψ is the boundary layer thickness, and ρ, ρ and ρ 3 are positive constants. he variation of this function (33) with respect to time is ss g V = + s x + w w + w w + ΞΞ g x ρ ρ ρ3 ss g = + s x + w w + w w + ΞU g x ρ ρ ρ3 η g s + s uannsmc + uh ueq + s x g x + ε +uannsmc uannsmc ) + ww + ww + ΞU ρ ρ ρ3 η g = s + s uannsmc ueq + s x + ε g x + s ( uannsmc uannsmc ) s uh + ww + ww + ΞE ρ ρ ρ3 η g = s + s uannsmc ueq + s x + ε g x + s ( w σ( ws ) + w σ ( ws ) w s uh ) + ww + ww + ΞU ρ ρ ρ3 η s + w w + s σ ( w + s ) s w g ρ + + w w s σ ( w s ) w s w ρ U +Ξ s s U ρ3 (34) By selecting appropriate values for Ψ, equations (9), (30) and (3) implies that V is negative semidefinite

6 Design of an adaptive neural sliding-mode controller for seesaw systems 59 V η s. (35) g Figure 3 he balancing mechanism of the inverted wedge If s Ψ, s = 0. hen V = 0, and V = 0. If s > Ψ and s = s has the same sign as s. By the algorithm, we have s s < 0. herefore V = s s < 0. hen for all t 0, V 0 holds. So it is a monotonous non-increasing function. Because V 0, limv exists, i.e., V( ) exists. hen s is t bounded and w and w are bounded too. Since a continuous function is bounded in the closure set, so x i is bounded, and s is bounded too, therefore s is uniform continuous, and then V = s s is uniform continuous. Since V(t) is bounded and lim Vt d V( ) V(0) t = exists, 0 then by the Barbalat lemma (Slotine and Li, 99), we have limv = 0, and obtain lim s = 0. t t he ANNSMC control law is summarised in equation (). he parameters vector w and w adjusted by equations (9) and (30). he aim is to construct an adaptive control methodology for unknown time-dependent non-linear plants without using an explicit model of the plant. We developed an adaptive NN to approximate the unknown parts of the system non-linear functions, so that the system uncertainties can be kept small and guarantee Lyapunov stability of the dynamic non-linear system. 4 Simulation results of the seesaw system In this section, we shall indicate that the ANNSMC design is applicable to the seesaw system as shown in Figure to verify the theoretical development. According to the basic physical concept, in the unstable system of a seesaw mechanism, if the vertical line along the centre of gravity of the inverted wedge is not passing through the fulcrum perpendicularly, then the inverted wedge will result in a torque and rotation until the stable state is reached. 4. Description of dynamical model he balancing mechanism of the seesaw is shown in Figure 3. he previous study gives the system model by using Lagrange s formulations based on the principle of balance of force and torque as follows. Nr ( θ + x) Nx θ Ngsin θ = u, I θ + N[ r( r θ + x) + x θ + xx θ] (36) Mgr sin θ Ng( r sinθ + x cos θ) = 0. he dynamic equation of the seesaw mechanism is given as follows: x+ Ngsin θ Bx = mx, ( Mgsin θ) h + Ngsin ( θ + φ ) ( x + h ) + uh µθ = Wθ (37) where N is the mass of the cart, M is the mass of the wedge, x is the distance of cart from the origin, θ the angle of inclination of the wedge, h is the height of wedge, h is the height of centre of mass of wedge, λ is the damping coefficient of the angle, B is the damping coefficient of the cart, φ is the angle that the wedge makes with the vertical line, W is the wedge inertia given by equation (37) M a W = + h. From Figure 3, it can be derived as follows W = ρc ( x + y )dxdy b ( a/ b) y ( )d d 0 ( a/ b) y a = ρabc + b 4. i ii (38) = ρc x + y x y (39) In the simulation, the parameters of seesaw system are used as follows W = 0.0, h = 0.36, h = 0.5, M =.6, N = 0.35, λ = 0., B = 0.4, g = 9.8, d = 0.5 sin(t). 4. he application algorithm he four input state variables of ANNSMC that consist of actual state, as x = [ θθ x x ]. he seesaw system given in equation (37) has four variables of states. he variables are the angle (θ) that the wedge makes with the vertical line, the change of the angle (θ ) that the wedge makes with vertical line, and the position of the cart from the origin (x), and the change of the angle ( x ).

7 60 H-Y. Chung and L-C. Hung Define the sliding surface able he initial states of seesaw system s = k + k + k x+ k x (40) θ θ 3 4. In addition, the coefficients of the sliding surface and the control gains of the ANNSMC are given in the following: k =, k =, k 3 = 3.5, k 4 = 3, ρ = 0., ρ = 0., ρ 3 = 0., Ψ = 0.5. he position of the cart x he angle of inverted wedge θ Figure 4 he initial states of the seesaw 5 centimetres 30.0 degrees he parameters of ANNSMC are appropriate selection under the condition of asymptotic stability, i.e., with the same sign of coefficients and that if t, then x(t) 0. All the gains in the ANNSMC are chosen to acquire the best transient control performance in simulation considering the need for stability, limitation of control forces and possible operating conditions. Since all weights of the NN are render initialised, the states are gradually reduced through online tuned manner of the ANNSMC that was developed in the sense of Lyapunov stability analysis, so that system-position stability of the closed-loop system can be ensured whether the uncertainties occur or not. In this case, the initial states are shown in able and in Figure 4. Figure 5 he position response of the seesaw system for SMC he simulation results of the SMC (Slotine and Li, 99) are shown in Figures 5 and 6 where the control forces are shown in Figure 7. he conventional SMC controller equation (6) can not be implemented because f is unknown exactly. Although favourable states responses can be derived by the SMC as in the simulation, the chattering phenomena in the control efforts result in imprecise states responses. Figure 6 he angle response of the seesaw system for SMC

8 Design of an adaptive neural sliding-mode controller for seesaw systems 6 Figure 7 he control force of the seesaw system for SMC he Fuzzy-Neural Network Sliding-Mode Control (FNNSMC) in which the inputs are sliding surface s and are fully connected between two layers. he Fuzzy-Neural Network (FNN) structure in Zeng (00) is adopted in this work and the connecting weights between the output and rule layers of the FNN are initialised with random number [0, ] and the parameters of Gaussian functions are initialised with random number [ 3, 3]. he FNN has two, six, nine, and one neuron(s) at the input, membership, rule, and output layers, respectively. he FNN control method is not only offline learning adjustment but also can not ensure the stability. he simulation results of the FNNSMC system are shown in Figures 8 and 9 with the composed control forces shown in Figure 0. he block diagram of the ANNSMC is depicted in Figure. he simulated results of the computed position control system are depicted in Figures and composed control forces shown in Figure 3. Figure 8 he position response of the seesaw system for FNNSMC

9 6 H-Y. Chung and L-C. Hung Figure 9 he angle response of the seesaw system for FNNSMC Figure 0 he control force of the seesaw system for FNNSMC Figure he position response of the seesaw system for ANNSMC

10 Design of an adaptive neural sliding-mode controller for seesaw systems 63 Figure he angle response of the seesaw system for ANNSMC Figure 3 he control force of the seesaw system for ANNSMC Based on what we want is to balance the angle of the inverted wedge and the position of the cart, we can give the proper commands for balancing control in different situations, and thus construct the controller of the system. Comparing the control law of the presented control method, SMC (Slotine and Li, 99) and FNNSMC (Zeng, 00) for the seesaw system, we observe that the performance of the proposed ANNSMC approach is better than SMC and FNNSMC and it is evident that the presented scheme has excellent control capability for handling imprecisely known processes despite existing simultaneously different situations. Remark 3: Unlike SMC discussed in Utkin (977), Slotine and Li (99), Ackermann and Utkin (998) and Young et al. (999), prior knowledge of the upper bound of the system uncertainties is not necessary. he ANNSMC is introduced to calculate the upper bound of the system uncertainties online and the tuning is then used as a parameter to ensure that impacts of the system uncertainties can be eliminated and asymptotic state convergence can be derived for a seesaw system. Remark 4: It can be seen from heorem that the weights of the NN are suitably adjusted in the Lyapunov sense. It is not essential for the weights of the NN to converge to their optimal values, and the values of the weights are adaptively raised until the sliding variable vector s converges to zero. hen the weights will become constant to maintain the state dynamics in the sliding-mode and ensure that the state asymptotically converges to zero. 5 Conclusions In this paper, a novel ANNSMC system has been presented with the idea of combining NN with SMC incorporating adaptive law. Furthermore, the ANNSMC system has been proposed to solve the position balance control problem for

11 64 H-Y. Chung and L-C. Hung highly non-linear and coupling, and the complete dynamic model is difficult to derive precisely. he chattering phenomena of the conventional SMC can be eliminated with the incorporation of the adaptive NN technique. he proposed control has both the ability to adjust the NN and to repeat the major merits of SMC, i.e., robustness. he design has been proved to ensure the closed-loop stability and it has been shown that the states can be made as small as desired. Form simulation results show the seesaw system would be reached at purpose and can be stabilised to the equilibrium. References Ackermann, J. and Utkin, V. (998) Sliding mode control design based on Ackermann s formula, Automatic Control, IEEE rans. on, Vol. 43, No., pp Adamy, J. and Flemming, A. (004) Soft variable-structure controls: a survey, Automatica, Vol. 40, No., pp Barambones, O. and Garrido, A.J. 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