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Recurrent Wavelet Network Based Control for a Class of Delayed Uncertain Nonlinear Systems Subjected to Hysteresis Nonlinearity
| Content Provider | Semantic Scholar |
|---|---|
| Author | Puntambekar, Sachin Agrawal, N. K. |
| Copyright Year | 2016 |
| Abstract | The research presents recurrent wavelet based adaptive control (RWNC) scheme for uncertain nonlinearity based delayed systems with hysteresis. For the estimation of uncertainties present in the system and hysteresis nonlinearity effect used Self recurrent wavelet neural network (SRWNN). Deployment of hysteresis phenomenon in proposed work used Prandtl-Ishlinskii (PI) model. Proposed Controller consists of wavelet based feedback system and a robust control mechanism. The simulation study carried out for stability analysis of the proposed controller using Lyapunov-Krasovskii function and verifies the effectiveness of theoretical development. KeywordsAdaptive control; self recurrent wavelet neural networks; time delay systems; Lyapunov Krasovskii functional; hysteresis. INTRODUCTION The performance of the control system decreases due to the time delay in system dynamics. Several real time systems like aircraft, Process Control System, Manufacturing System and Communication System represent an unstable behavior due to time delay. Infinite Dimensional functional differential equation based control system are more sensitive as compared to the finite dimensional differential equation, in the presence of time delay. This is one of the major challenges in controller design and opens a new branch of research in controller design. The control strategies [3-5] are based on the application of Lyapunov-Krasovskii functional augmented with some linear matrix inequality techniques. Hysteresis is a commonly observed nonlinearity in physical systems, especially smart material based actuators exhibits this phenomenon. Hysteresis can be defined as a multi-valued nonlinear relationship between two variables.[6] In the system dynamic presence of hysteresis in actuators often degrades the system performance and inserts nonsmooth nonlinear dynamics in the system that can lead to undesirable inaccuracies and even instability. Due to the complexity reflected by the behavior of hysteresis affected systems, the classical control tools have been proven inefficient for the control and the regulation of such systems and it demands the augmentation of the conventional tools with some adaptive dynamics to mitigate the effect of hysteresis [2, 3]. Several researchers have addressed the issue of designing of control schemes for hysteresis based systems. Most of these schemes relay on an accurate model of hysteresis nonlinearity. Preissach model, Prandtl-Ishlinskii model is commonly used mathematical models to explain the hysteresis dynamics [6]. Various Issue has been addressed [7-9] related to controller design for delay based nonlinear systems subjected to hysteresis nonlinearity. Feedback linearization based control schemes have been proven highly effective for the control of nonlinear systems. However, a major drawback associated with these techniques is that effective implementation of these techniques needs completely known systems dynamics [10]. This requirement often confines the applicability of these techniques to a very narrow class of nonlinear systems. A common way to relax these constraints is to augment the classic controller with some adaptive tuner like neural networks, which results in some adaptive control scheme for effectively dealing withmathematical inaccuracies or model uncertainties. A class of Wavelet neural networks (WNN), called a feedforward neural networks in which wavelet as an activation function, having the approximation capability of neural network and localized features of wavelet functions. The performance of WNN provides better outcomes as compared to conventional neural network in terms of convergence rate, space learning capability and frequency localization functions [12-15]. Complexity of control problems is solved by WNN having the capability of convergence and recurrence [17-21]. The applicability of WNN gets restricted only to static environmental conditions owing to its feed-forward architecture. To enhance the performance of WNN under dynamic environmental conditions can be done by including recurrence characteristics in form of architectures namely output recurrent WNN (ORWNN) or self recurrent WNN (SRWNN). Application of SRWNN as a system identification tool has been cited in [19-21]. Figure 1 shows the basic diagram of the proposed system. Figure 1: Basic diagram of proposed work International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 1 (2016) pp 530-535 © Research India Publications. http://www.ripublication.com 531 In this work the proposed adaptive controller (RWNC) controls the functioning of hysteresis based delayed uncertain nonlinear system (HDUNS). RWNC is based on SRWNN with the capability of system identification as well as hysteresis compensator. The stability of HDUNS computes by Lyapunov-Krasovskii function in RWNC [22, 23]. Paper organized as follows: Section II explains system preliminaries. Section III describes system formulation. Section IV deal with controller designing issues. Section V analyzed Convergence aspects of RWNC. The effectiveness of RWNC is illustrated through a simulation in section VI while work concluded in section VII. SYSTEM PRELIMINERIES A) Hysteresis Nonlinearity There exist several models that represent the hysteresis behavior without giving any physical insight the system like Preissach model, Prandtl-Ishlinskii model, etc. This work utilizes Prandtl-Ishlinskii model to model the hysteresis nonlinearity present in the actuators[6,8]. Prandtl-Ishlinskii (PI) model Prandtl-Ishlinskii model is set up by composition of play and stop operators. Suppose [ ] r E u are basic elastic plastic elements or stop operators for all [ ] 0, r R ∈ .For a given density function ( ) p r , the model can be expressed as[6,8] ( ) ( ) ( )( ) 0 R r w t p r E u t dr = ∫ (1) Density function ( ) p r is supposed to be satisfied the following condition ( ) 0 p r ≥ with ( ) 0 rp r dr ∞ < ∞ ∫ Thus, with such density function, this operator maps ( ) 0 , C t ∞ into ( ) 0 , C t ∞ i.e. Lipschitz continuous inputs will yield Lipschitz continuous outputs. In terms of play operator ( ) r F u the PI model can be expressed as ( ) ( ) ( ) ( ) ( ) 0 0 R r w t p u t p r F u t dr = − ∫ (2) where ( ) 0 0 R p p r dr = ∫ is a constant, which depends on the density function. Equation(2), reflects that the hysteresis nonlinearity can be considered as the combination of a linear term and the integral term which accounts for the hysteresis. In this work hysteresis model defined by equation (2) is utilized for controller design, the hysteresis part of the (2) is treated as the disturbance and is approximated by using a wavelet network. B) Self Recurrent Wavelet Neural Network Regression analysis of function approximation, obtained from mother wavelet function using translation and dilation, is done by wavelet network [12]. Figure2. shows Feed-Forward WNN with self feedback wavelon layer as a modified version of SRWNN. Self feedback wave-lon layer enables the wavelet network (SRWNN) to store past information, high degree approximation of dynamic nonlinearities and more convenient for adaptive control as compared with conventional. Figure 2: Self Recurrent Wavelet Neural Network Output of an 'n' dimensional SRWNN with 'm' wavelet nodes can be expressed as ( ) ( ) ( ) ( ) ( ) ( ) 1 , , , ( ( ), ( )) , m i i i i i i i t t x t f x t x t t w t c t θ τ τ = φ − ⎛ ⎞ − = α φ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ∑ (3) where i φ is the th i wavelet node given by ( ) 1 , , , , ( , , , , ) n i i i i i ij ij ij ij ij j x w c x w c θ θ = φ φ = φ φ ∏ (4) where ij φ is the th i wavelet node with th j wavelon. ( ) n x t R ∈ shows input vector, with where τ is the point delay encountered in state vector. 1 2 [ , , , ] i i i in φ = φ φ φ K represents the feedback argument for the th i wavelet node. The Network's previous information stored in the vector 1 2 [ , , , ] i i i in φ = φ φ φ K . 1 2 [ , , , ] i i i in θ θ θ θ = K shows weighted vector , 1 2 [ , , , ] i i i in w w w w = K and 1 2 [ , , , ] i i i in c c c c = K represents are dilate and translate vectors for feedback input. This input vectors applied on th i wavelet node which represents as follows1 1 1 2 2 2 [ , , , ] T i i i i n in in x x x θ θ θ + φ + φ + φ K . In matrix form (3) can be rewritten as ( ) , , , T f x w c α θ = φ , φ (5) where [ ] , ,..., and , ..., T T mxn mxn m m w w w w R c c c c R 1 2 1 2, ⎡ ⎤ = ∈ = ∈ ⎣ ⎦ are dilation and translation parameters respectively ; International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 1 (2016) pp 530-535 © Research India Publications. http://www.ripublication.com 532 [ ] , , .., T m m R 1 2 α = α α α ∈ and [ ] 1 2 , ,.., T nxm m R θ θ θ θ = ∈ are the output and feedback weights respectively. [ ] 1 2 , . ., T n xm m R φ = φ φ φ ∈ shows feedback, input vector of SRWNN. With ideal wavelet approximator, let * f be the optimal function approximation * * * T f f = + Δ = α φ + Δ (6) where ( ) * * * * , , , x w c θ φ = φ ,φ and the optimal parameter vectors are * , , w ∗ α * * , c θ of , , w c θ α, respectively, and the approximation error denoted by Δ with assumption ∗ Δ ≤ Δ ,where ∗ Δ is a positive constant. The optimal parameter vectors for paramount approximation are difficult to determine. The definition of estimation functions is as followsˆ ˆ ˆ T f = α φ (7) where ( ) ˆ ˆ ˆ ˆ , , , , x w c θ φ = φ φ and ˆ ˆ ˆ ˆ , , , w c θ α are the estimates of * * * , , , w c θ ∗ α respectively. Defining the estimation error as * ˆ ˆ ˆ ˆ T T T f f f f f = − = − + Δ = α φ + α φ + α φ + Δ % % % % (8) where ˆ ˆ ∗ ∗ α = α − α, φ = φ − φ % % With the appropriate number of nodes, the estimation error f% can be reduced to arbitrarily small value on the compact set so that the bound f% ≤ m f% holds for all x ∈R . As the wavelet network representation is nonlinear in parameter which respect to certain adjustable parameters, to facilitate the derivation of tuning laws a Taylor expansion based following part |
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