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Design of adaptive fuzzy wavelet neural sliding mode controller for uncertain nonlinear systems
ISA Transactions 52 (2013) 342–350
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ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Research Article
Design of adaptive fuzzy wavelet neural sliding mode controller for uncertain nonlinear systems Maryam Shahriari kahkeshi n, Farid Sheikholeslam, Maryam Zekri Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan 8415683111, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 July 2012 Received in revised form 28 January 2013 Accepted 30 January 2013 Available online 27 February 2013 This paper was recommended for publication by Prof. A.B. Rad
This paper proposes novel adaptive fuzzy wavelet neural sliding mode controller (AFWN-SMC) for a class of uncertain nonlinear systems. The main contribution of this paper is to design smooth sliding mode control (SMC) for a class of high-order nonlinear systems while the structure of the system is unknown and no prior knowledge about uncertainty is available. The proposed scheme composed of an Adaptive Fuzzy Wavelet Neural Controller (AFWNC) to construct equivalent control term and an Adaptive Proportional-Integral (A-PI) controller for implementing switching term to provide smooth control input. Asymptotical stability of the closed loop system is guaranteed, using the Lyapunov direct method. To show the efficiency of the proposed scheme, some numerical examples are provided. To validate the results obtained by proposed approach, some other methods are adopted from the literature and applied for comparison. Simulation results show superiority and capability of the proposed controller to improve the steady state performance and transient response specifications by using less numbers of fuzzy rules and on-line adaptive parameters in comparison to other methods. Furthermore, control effort has considerably decreased and chattering phenomenon has been completely removed. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Fuzzy wavelet neural Smooth sliding mode control Adaptive control Chattering elimination
1. Introduction Control of uncertain nonlinear systems has been an important research in control literatures [1–7]. In practice, system can be uncertain in terms of its dynamic structure or its parameters. Lately, for handling unknown nonlinear systems many approximator based adaptive control techniques have attracted considerable attentions. Approximators such as neural networks (NNs), fuzzy systems (FSs) and wavelet functions, due to their inherent capability in function approximation, have been widely used to represent a model of unknown nonlinear systems for controller design [1,8–15]. Also approximators such as FSs have employed as a state observer to estimate the state variables where not all of the state variables of the nonlinear dynamic systems are available [16]. Although unknown dynamics of nonlinear system can be represented using mentioned approximators, but they can only represent model with limited accuracy, so there remain approximation errors between real system and approximated model. To compensate the approximation errors, disturbances and uncertainties, several control system design and analysis approaches have been proposed. Among the existing robust control approaches, SMC is an effective robust control approach for
n
Corresponding author. Tel.: þ98 311 3915484. E-mail address: [email protected] (M. Shahriari kahkeshi).
uncertain nonlinear systems [17]. Insensitivity and robustness of the SMC against the uncertainties and external disturbance and also its fast response and good transient performance have made it as a popular design tool for robust control of uncertain systems. Despite these feathers, it also suffers from large control chattering phenomenon. Chattering phenomenon results from high frequency control switching and may cause potentially harmful high frequency vibration of the controlled plant [18]. In order to reduce the chattering problem in SMC, a small boundary layer around the sliding surface is introduced in [17–19]. However this method can lead to solve the chattering problem but usually there exist a finite steady state error and so asymptotical convergence is lost. As another solution, adaptive fuzzy sliding mode control (AFSMC) [3,20–25], which is based on the integration of SMC and adaptive fuzzy control, has attracted attention of many researchers. The fundamental idea of AFSMC is to employ fuzzy models for approximating both the equivalent and switching control terms in the SMC to attenuate the chattering phenomenon. As another solution to eliminate chattering phenomenon and achieve zero steady state error, fuzzy controller was proposed in [26] which combine SMC and PI control term. Then some AFSMC literatures used PI control term to eliminate chattering phenomenon [27,28] but they suffer from difficulties associated with the FSs. Since in most of AFSMC, fixed structure is used to describe the unknown system and there is no precise method to design the premise part of fuzzy rules, so before
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M. Shahriari kahkeshi et al. / ISA Transactions 52 (2013) 342–350
designing a control law, one must determine the number and location of the basis and membership functions via trial and error. However for some complex unknown system with high input dimension, this method cannot be responded and often lead to unnecessarily large structure for plant representation. To overcome this shortcoming several approaches have been proposed for the generation of the IF–THEN rules [29–32]. Also it is critical to notice that approximators such as traditional FSs only presents localized approximation of the function and cannot capture all local and global behavior of the model. Also as the last point, it should be considered that handling problem with large dimension using AFSMC lead to considerable increment in number of fuzzy bases and cause model complexity. So to overcome the above mentioned problems, recently proposed FWNN in [33] is incorporated with adaptive schemes for control of unknown nonlinear systems [34–40]. In the considered network, integration of the time frequency localization of wavelets, inference properties of FSs to handle the complexity and learning capability of NNs led to simple structure, high approximation accuracy and good generalization capability. Recently, according to the mentioned properties, in [34] an adaptive FW nonsingular terminal sliding mode tracking controller was proposed for robotic systems. Different from the classical SMC, the terminal sliding-mode has a nonlinear sliding surface. Also in [35], AFWNC designed for a class of uncertain SISO nonlinear systems. Proposed scheme in [35] uses FWN to approximate feedback linearization control input such that the closed loop system stability guarantee but controller robustness against external disturbance is not discussed. Also in [36] HN reinforcement learning controller based on FWN was presented which adopts an actor critic reinforcement learning controls scheme. Also an observer based indirect self-structuring robust adaptive fuzzy wave-net controller for uncertain nonlinear system was designed in [37]. It uses fuzzy wave-net in [39] to approximate the system’s unknown structure and then HN tracking based algorithm is used to reduce the influence of the approximation error and external disturbance on the tracking error. In proposed self-structuring controller in [37] FWN structure is determined via trial and error and no straightforward approach is presented to determine network dimension for each problem regardless of input dimensions. As another application, author in [40] proposed an adaptive FWN synchronization controller to synchronize two nonlinear identical chaotic gyros. Proposed scheme composed of a neural controller which utilizes a FWN to online mimic an ideal controller and a fuzzy compensator to remove the chattering phenomena on conventional SMC. This paper proposes an AFWN-SMC for a class of unknown nonlinear systems. Proposed scheme includes two terms: AFWNC and A-PI controller. AFWNC is designed to construct an equivalent term of SMC which invokes FWNs to approximate the unknown dynamics of uncertain system, and an A-PI controller is employed as switching control term of SMC. Moreover PI type adaptive laws are developed by Lyapunov direct method to tune adjustable parameters in real time operation such that the system tracking error converges to zero asymptotically. Main feathers of the proposed scheme are:
Ability to design smooth SMC law with good transient
response characteristics and better steady state performance for a class of nonlinear systems while the structure of the system is unknown and no prior knowledge about uncertainty is available. Ability to determine network size, structure, number and location of basis functions, fuzzy rule and sub-WNN, automatically.
343
Providing less number of fuzzy rules and basis functions and
low computational complexity via constructing wavelet lattice and using OLS algorithm in design step. Ability to deal with high order problems regardless of the input dimensions. Since wavelets with different dilation values under fuzzy rules are invoked to capture local and global behaviors of unknown dynamics, FWN can provide both globalized and localized approximation of the function, so proposed scheme has good generalization capability and better performance.
To validate the results obtained by the proposed scheme, a classical SMC [18], AFSMC [23] and AFWNC [35] are adopted from the literatures and applied for comparison. Simulation results show that the tracking performance and transient response characteristics are improved significantly, while chattering phenomenon is completely eliminated also control effort; fuzzy rules and adjustable parameter are reduced. This paper is organized as follows. Problem statement is briefly explained in Section 2. In Section 3, brief description about FWN is presented and AFWNC is constructed. Design of the proposed scheme is presented in Section 4. To evaluate the performance and capabilities of the proposed approach, simulation results are given in Section 5. Finally, Section 6 concludes the paper.
2. Problem statement Consider a class of SISO nonlinear affine systems which are described by the following equation: xðnÞ ¼ f ðxÞ þgðxÞu þ dðtÞ y¼x
ð1Þ
where f ðxÞ and gðxÞ are two unknown nonlinear functions and d(t) denotes the unknown external disturbance, which is bounded, assume that D is the upper bound of the disturbance, i.e. ðn1Þ T _ 99d(t)99rD. Also x ¼ ½x, x,:::,x ¼ ½x1 ,x2 ,:::,xn T A Rn is the state vector of the system which is assumed to be available for measurement and uAR, yAR are the control input and output of the system, respectively. It is assumed that the nonlinear system (1) is controllable thus the input gain gðxÞ should be nonzero, in this paper we assume that gðxÞ 4 0. The control objective is to design a control law so that the system state xðtÞ can track a desired trajectory y d ðtÞ in the presence of model uncertainty and external disturbance. Let the tracking error is defined as (2). _ . .,eðn1Þ T e ¼ xy d ¼ ½e, e,.
ð2Þ
The control objective is to design the control input such that the state of the system x tracks any given desired trajectory y d in the presence of uncertainties and disturbances, so that tracking error (2) should converge to zero: lim 99e99-0
t-1
ð3Þ
The sliding surface is defined as follows [17]: d S ¼ ð þ cÞn1 e ¼ eðn1Þ þcn1 eðn2Þ þ . . . þc2 e_ þ c1 e ¼ c T e dt
ð4Þ
where c ¼ ½c1 ,c2 ,. . .,cn1 ,1T should be chosen such that the corresponding characteristic polynomial become Hurwitz. To achieve the classical SMC, the control strategy consists of two design steps: first, to force the system toward a desired dynamics in finite time and secondly to maintain the system on the sliding surface. Thus the overall control law, which consists of two terms, equivalent control term (ueq) and reaching control term (usw), can
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be represented as: un ¼ ueq g 1 ðxÞusw
ð5Þ
where ueq ¼ g 1 ðxÞ½f ðxÞ
n1 X
ci eðiÞ þydðnÞ dðtÞ,
usw ¼ Z sgnðSÞ
ð6Þ
i¼1
where Z is strictly positive constant. Since in real world application, external disturbance d(t) is usually unknown, the equivalent control input is modified to: ueq ¼ g 1 ðxÞ½f ðxÞ
n1 X
ci eðiÞ þydðnÞ
ð7Þ
i¼1
In practice, system is uncertain or its dynamics (i.e. nonlinear functions f ðxÞ, gðxÞ) are generally unknown, so the control input (7) is not applicable. Moreover, the discontinuous switching control term in (6) produce excessive chattering phenomenon which is undesirable, so it is hard task to implement control input (5). Thus, in this paper, to overcome mentioned challenges, AFWNC which involves two FWNs, to represent the model of the unknown systems, is constructed to approximate equivalent control term ueq in (7) and an A-PI control law is used as switching control term to eliminate chattering phenomenon completely and improve steady state performance. The following section outlines the details of AFWNC that is used in the proposed scheme.
the membership function, respectively. Also f^ i is the output of local model for rule Ri and is equal to the linear combination of a ðkÞ finite set of wavelets cM ,tk ðxÞ with the same dilation value. i ðkÞ Wavelets cM ,t k ðxÞ are expressed by the tensor product of 1-D i wavelet functions as follow:
ckMi ,t ¼ 2Mi =2 cðkÞ ð2Mi xt k Þ ¼
n Y
2Mi =2 c ð2Mi xi t kj Þ ðkÞ
ð10Þ
j¼1
In this paper, translated and dilated version of the Mexican hat wavelet function is considered. Thus, the cMi ,tk ðxÞ in Eq. (10) is j expressed by the following equation: 0 ! 0 !2 1 ! 1 k 2 xt kj xt kj 1 xt j @ A @ A exp ð11Þ c ¼ 1 2 Mi Mi Mi by applying TSK fuzzy inference mechanism and considering f^ i as an output of each sub-WNN (or each rule), the output of FWN f^ ðx9y f Þ can be expressed as: f^ ðx9y f Þ ¼
cf X
m^ i ðxÞf^ i ¼
i¼1
cf X
m^ i ðxÞyf i f i ðxÞ
ð12Þ
i¼1
Using a simplified notation, (12) can be rewritten as: T f^ ðx9y f Þ ¼ y f z f ðxÞ
^ i ðxÞ ¼ mi ðxÞ=ð where m
ð13Þ Pc
i¼1
mi ðxÞÞ, mi ðxÞ ¼ Pnj¼ 1 Aij ðxj Þ, cf is the numT
ber of rules to approximate function f ðxÞ and y f ¼ ½yf 1 , yf 2 ,:::, yf cf T is the adjustable parameter vector and z f ðxÞ ¼ ½zf 1 ðxÞ, zf 2 ðxÞ,:::,
zf cf ðxÞT is the fuzzy wavelet basis functions. By the similar
3. Adaptive fuzzy wavelet neural controller (AFWNC)
procedure, the function gðxÞ can be modeled as follows: In this paper, AFWNC is constructed to approximate the equivalent control term of SMC. It invokes FWNs to represent the model of the unknown dynamics of uncertain nonlinear system. For this, FWN is expressed as a series expansion of fuzzy wavelet basis functions to model the unknown dynamics (i.e., f ðxÞ, gðxÞ) of the equivalent control term (7) by tuning the parameters of the corresponding networks. Thus it is necessary to give the basic concepts of FWN here. FWN combines multiresolution analysis of wavelet transform and traditional TSK fuzzy system. In this network, constant or linear functions in the consequence part of TSK model are substituted with wavelet functions [33]. In FWN model each fuzzy rule corresponds to one sub-WNN which consists of wavelets with a specified dilation value. A typical FWN for approximating arbitrary nonlinear function f ðxÞ can be described by the following linguistic rule [35]: Rif : If x1 is Ai1 and x2 is Ai2 and ::: and xn is Ain , then f^ i ¼ yf i f i where fi is defined as f i ¼
PT f i
k¼1
wMf
i
ðkÞ ,t k cM f ,tk ðxÞ, i
ð8Þ
and yf i is free
adjustable parameter which should be updated online based on PI type adaptive laws. Also Rif is the ith rule (1rircf) and Mf i A Z is the dilation parameter for rule Rif . In this work, t kj is the translation value for the corresponding wavelet k and xj, (1rj rn) is the jth input variable of x and n is the input dimensions. Also T f i denotes the total number of wavelets for the ith rule, and Aij is the fuzzy set characterized by the following Gaussian-type membership function. Where Aij ðxj Þ denotes the membership function value of xj in Aij . 2
Aij ðxj Þ ¼ e9ðxj pj1 Þ=pj2 9pj3 ¼ eðððxj pj1 Þ=pj2 Þ i
i
i
i
Þpij3 =2
ð9Þ
0 r pij3 r 5, and pij1 represent the center of memberfunctions, pij2 and pij3 determine the width and the shape of
pij1 ,pij2 A R, ship
i
^ y g Þ ¼ y Tg z ðxÞ gðx9 g
ð14Þ
T
where y g ¼ ½yg1 , yg2 ,:::, ygcg T is the adjustable parameter vector and
z g ðxÞ ¼ ½z1 ðxÞ, z2 ðxÞ,:::, zcg ðxÞT is the fuzzy wavelet basis functions and cg is the number of rules which are applied to approximate function gðxÞ. Design procedure of the PI type adaptive laws to update the adjustable parameter vector in (13) and (14) is given in Section 4. In the network construction step, first a library of candidate wavelets are selected by constructing wavelet lattice [41] and then OLS algorithm [42] is used to purify them for choosing important and efficient wavelets for constructing sub-WNN, determining network structure and initial weights. Then based on dilation parameters of efficient wavelets, number of sub-WNN and fuzzy rules are determined. Also at the end of this step, premise and consequence parts of fuzzy rules are specified. Because of the efficient procedure of selecting wavelets employed in the OLS algorithm is not very sensitive to the input dimension so the dimension of the approximated function does not cause the bottleneck for constructing FWN. After assigning the network structure, its parameters (sub-WNN’s weights, translation parameters of efficient wavelets, membership functions parameters) are initialized according to [33]. Then two learning stages consist of extended Kalman filter (EKF) and recursive least square (RLS) estimator are utilized for network learning. All nonlinear parameters involving pijr and t kj where 1rjrn, 1rircf, r ¼1,2,3 and k ¼ T 1 ,T 2 ,. . .,T cf are learned using the EKF and linear parameter, sub-WNN’s weights wMi ,tk , are tuned using RLS estimator. Details of these algorithms can be found in [43,32]. Structure of FWN for function approximation, in AFWNC structure, is depicted in Fig. 1. As mentioned above, the result in (7) is realizable only while f ðxÞ and gðxÞ are well-known. However in practical application f ðxÞ and gðxÞ are unknown and ideal controller (7) cannot be implemented. So f ðxÞ and gðxÞ are replaced by FWN model (13)
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the control input is kept at D þ Z which is a constant saturated value, when the state is outside of the boundary layer. Therefore 9u^ SW ðS9y^ PI Þ9 ¼ D þ Z when 9S9 Z f. As a result, the A-PI control term can be written in the following form: 8 < y T z ðSÞ 9S9 o f PI PI u^ SW ðS9y PI Þ ¼ ð19Þ : ZþD 9S9 Z f Considering proposed AFWNC and applied A-PI control term, the proposed controller can be expressed as: h i 1 f^ ðx9y f Þc T e þyðnÞ u^ SW ðS9y PI Þ ð20Þ u¼ d ^ yg Þ gðx9 where y f , y g and y PI are adjustable parameter vectors. Therefore we have the following theorem: Theorem 1. If the control input (20) is applied to nonlinear system (1) and functions f^ , g^ and u^ SW in (20) are substituted with presented adaptive models in (13), (14) and (19), and adjustable free parameters vectors y f , y g and y PI are adapted by the adaptive laws in (21)–(23), then all signals of the closed loop system will be bounded and the tracking error will converge to zero asymptotically.
y_ f ¼ g1 Sx f ðxÞ Fig. 1. Structure of FWN for function approximation.
and (14). So AFWNC which is constructed to represent the equivalent control term (7) can be expressed as follow: h i 1 ð15Þ ueq ¼ f^ ðx9y f Þc T e þ ydðnÞ ^ yg Þ gðx9 ^ y g Þ can be represented as (13) and (14), where f^ ðx9y f Þ and gðx9 respectively.
8 > > < _y ¼ g2 Sz g ðxÞu g > > : 0
ð21Þ if ðy g A OÞ or ðy g A O and Sz g ðxÞu r0Þ
ð22Þ
otherwise
y_ PI ¼ g3 Sz PI ðSÞ
ð23Þ
where g1, g2 and g3 are the positive adaptation rates. Also adaptation for y g in (22) is constrained within the convex set O to satisfy: ^ y g Þ Z e 40g O ¼ fy g 9gðx9
4. Adaptive fuzzy wavelet neural sliding mode controller design According to Section 2, SMC-based controller has two terms as mentioned in (5) and (6). Since dynamics of nonlinear systems are unknown, so AFWNC is constructed to represent equivalent control term (in Section 3). Thus by adding switching control law to (15), the overall control u can be written as: h i 1 u¼ ð16Þ f^ ðx9y f Þc T e þ ydðnÞ usw ^ yg Þ gðx9 Moreover to avoid chattering phenomenon caused by switching term and to improve steady state performance, usw in (16) is replaced by the A-PI control action when the state is within the boundary layer, i.e. 9S9o f (f is the thickness of the boundary layer). Therefore, it is important to employ an A-PI control law with chattering elimination property as a switching control term (usw) before the design of adaptive laws to adjust adaptation parameters y f and y g . Thus the input and output of continues time A-PI controller can be expressed in the following form [27]: Z usw ðS9y PI Þ ¼ yp S þ yI Sdt ð17Þ where yp and yI are proportional and integral term gains, respectively. For the sake of convenience in presentation, uSW in (17) can be denoted as the following form: T u^ SW ðS9y PI Þ ¼ y PI z PI ðSÞ ð18Þ R where z PI ðSÞ ¼ ½S, Sdt A R2 is a regressive vector and y PI ¼ ½yp , yI A R2 is an adjustable parameters vector which should be adapted simultaneously along with the control procedure. Also
in this adaptive law, whenever y g A O
ð24Þ T we have y_ g ry g^ ¼ g
g2 Sz g ðxÞz Tg ðxÞu r 0 . This condition implies the vector y_ g points within 901 of ry g^ , so it points inside O or along the tangent plane of O at g
point y g . Hence y g will never leave O, i.e., y g ðtÞ A O 8 t Z 0 . The proof of this theorem is given in Appendix A. Block diagram of the proposed scheme is given in Fig. 2.
5. Simulation results In this section, the proposed approach is tested on a second-order nonlinear inverted pendulum system. The low dimension of the presented example does not alter the generality of the proposed algorithm for higher dimension. Two parts are derived. In the first part, the proposed controller is applied for stabilizing problem and in the second part; the performance of the proposed scheme is investigated in a tracking problem. To validate the results obtained by the proposed approach, a classical SMC, AFWNC and AFSMC are adopted from the literature and applied for comparison. 5.1. Study system The state equations of the inverted pendulum system can be expressed as the following equation: ( x_ 1 ¼ x2 ÞðmLx2 sinðx1 Þcosðx1 ÞÞ=m þ M cosðx1 Þ=m þ M ð25Þ þ L½4=3mcos x_ 2 ¼ gsinðx1L½4=3mcos 2 ðx Þ=m þ M 2 ðx Þ=m þ M u þ d 1
1
where x1 ¼ y and x2 ¼ y_ are the angular position and velocity of the pole, respectively. g ¼9.8 m/s2 denotes the acceleration due to
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Fig. 2. Block diagram of the proposed AFWN-SMC scheme.
gravity, M ¼1 kg and m ¼0.1 kg are the mass of cart and the mass of pole, respectively. Finally, L¼0.5 m is the half length of pole and d is an external disturbance.
5.2. Part A: Balancing the pole in the vertical position In this part, the objective is to generate an appropriate actuator force u to control the motion of the cart, such that the pole can be balanced in the vertical position. The sliding surface is chosen as S ¼e2 þ2e1 and external disturbance d(t) is assumed to be dðtÞ ¼ 1 þsinðp t=2Þ. Also, initial condition is selected as x 0 ¼ ½p=8,0. The implementation of the suggested approach is given below. In order to construct AFWNC to represent ueq, two FWNs are constructed to model unknown dynamics of the plant. Efficient wavelets are purified to construct each sub-WNN by OLS algorithm. According to the dilation parameters of the selected wavelets, five rules/sub-WNNs are chosen for constructing each network. According to dilation parameters of selected wavelets, the proposed controller which is used here contains 10 fuzzy rules with 10 membership functions which are assigned for each input variable and 10 adaptation parameters that tune the parameters of the controller, online. In order to find the effectiveness and superiority of the proposed method, the test results are compared with the results obtained by classical SMC and AFSMC. In the classical SMC method, the system is assumed to be known and control signal is computed according to (5). In the proposed AFSMC in [23], two FSs are used to approximate unknown dynamics of system and also other FS is utilized as a switching control term to attenuate chattering action. In this example, membership functions for system states are defined similar to [23] as follow:
mA1 ðxÞ ¼ mA3 ðxÞ ¼ mA4 ðxÞ ¼ mA6 ðxÞ ¼
1 , 1þ expð5ðx þ 2ÞÞ
mA2 ðxÞ ¼
1 1 þ expððx þ 1:5Þ2 Þ
,
1 1þ expððx þ 0:5Þ2 Þ 1 1þ expððx0:5Þ2 Þ
,
mA5 ðxÞ ¼
1 1 þ expððx1:5Þ2 Þ
,
1 1þ expð5ðx2Þ2 Þ
so there are, 72 rules to approximate the system functions f ðxÞ and gðxÞ. Also three fuzzy sets are defined for sliding surface S ¼ e_ þ 2e as follow:
mS1 ¼
1 , 1 þ expð5ðS þ3ÞÞ
mS2 ¼
1 1 þ expðS2 Þ
,
Fig. 3. State trajectories: (a) x1 ¼ y and (b) x2 ¼ y_ of mentioned approaches.
mS3 ¼
1 1 þexpð5ðS þ 3ÞÞ
So the switching control term usw is represented by FS and adaptive laws based on SMC are used to implement controller [23]. Fig. 3 illustrates the state trajectories of the system for initial condition x 0 . As shown in this figure, by using the proposed approach, the system states converge to zero rapidly as the system starts. Moreover the transient response specifications have improved significantly. It can be found that the system transient response for the proposed method has much less overshoot and settling time in comparison to the classic SMC and AFSMC. Fig. 4 shows the control signal of three approaches. As can be seen from Fig. 4, proposed scheme completely eliminate the chattering phenomenon. So the proposed approach has good disturbance rejection performance and chattering elimination. Also in comparison with other methods, control effort has decreased, considerably. The maximum value of the control effort, number of fuzzy rules and adaptation parameters obtained by three approaches are shown in Table 1. Also, in order to have a quantitative comparison of tracking error, mean square error (MSE) of three methods has been reported. According to the following table, the maximum value of control effort and number of fuzzy rules/
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347
Fig. 6. State trajectory and control signal with proposed AFMNC in [35] (five fuzzy rules, g ¼ 50).
Fig. 4. Control signal (part A).
Table 1 Maximum value of control signal, number of fuzzy rules and MSE criterion of all approaches in the stabilization problem. Criteria
Proposed scheme
Max u ðtÞ Fuzzy rules MSE
Classical SMC
AFSMC [23]
9.8
4.8
30
12 0.0049
– 0.0807
75 0.457
Fig. 7. State trajectory and control signal of mentioned approaches for x 0 ¼ ½p=6,0.
demonstrates that the proposed controller performs much better as compared to the other mentioned approaches. 5.3. Part B: Tracking a desired trajectory
Fig. 5. State trajectory and control signal with the proposed method, (five fuzzy rules for each function, g1 ¼ 0.05, g2 ¼0.01).
adaptation parameters is decreased significantly using the proposed scheme. Also results with proposed scheme are compared with the proposed AFWNC in [35] for initial condition x 0 ¼ ½p=4,0. Figs. 5 and 6 show the results of the proposed scheme and presented AFWNC in [35], respectively. As can be seen, fast response with AFWNC implies significant initial control energy during the transient response and also lead to undesirable transient response characteristic such as undershoot. Simulation result for another initial condition x 0 ¼ ½p=6,0 in the presence of disturbance dðtÞ ¼ sinðtÞ is given in Fig. 7. It
Let us consider the inverted pendulum system again. The control objective is to maintain the system to track the desired angle trajectory yd ¼ yd ¼ p=9ðcos t þ 0:6cos 3tÞ in the presence of disturbance d ¼ 0:2sinð0:25tÞ for initial condition x0 ¼[p/6,0]. The sliding surface is S ¼4e2 þe1. Constructed networks in part A are used to model the AFWNC for constructing equivalent control term. State trajectories of the system for three mentioned approaches are illustrated in Fig. 8. Also the control signal is shown in Fig. 9. Again, these responses are similar to the responses in Figs. 3 and 4 for the stabilizing problem, showing the robustness and superiority of the suggested scheme. As can be seen from Fig. 9, the chattering phenomenon, which is clearly obtained in classical SMC, is completely eliminated using the proposed scheme. Also control effort with the proposed scheme is smaller than other mentioned approaches. Moreover, the maximum value of control effort, number of fuzzy rules, adaptation parameters
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6. Conclusions In this paper, an AFWN-SMC is proposed for a class of uncertain nonlinear systems. Proposed controller composed of two terms, AFWNC and A-PI controller. AFWNC is used to construct equivalent control term, it invokes two FWNs to approximate unknown nonlinear dynamics of uncertain system and A-PI controller is designed as switching control term to eliminate chattering phenomenon, provides smooth SMC law and improves steady state performance. The main advantage of the proposed scheme is the design of the smooth sliding mode control for a class of high-order nonlinear systems while the structure of the system is unknown and no knowledge about the uncertainty is available. The obtained simulation results show that the proposed scheme is quite effective in control of unknown nonlinear SISO systems. They illustrate that the proposed scheme significantly improves the transient response specifications and steady state performance. Also it has a robust tracking performance and acceptable behavior despite external disturbance. Furthermore number of fuzzy rules and online adjustable parameters are reduced, chattering phenomenon is completely removed while amplitude of control effort is reduced, considerably.
Appendix A Fig. 8. Tracking state (a) x1 ¼ y and (b) x2 ¼ y_ of proposed scheme and other mentioned approaches.
Proof of the theorem Proof. Lets define the optimal parameters vector as follow: " # y nf ¼ arg min sup 9f^ 9ðx9y f Þf ðxÞ9 ðA:1Þ y f A Of x A Rn
"
#
^ y g ÞgðxÞ9 y ng ¼ arg min sup 9gðx9
ðA:2Þ
y g A Og x A Rn
"
y nPI ¼ arg min
# sup 9u^ SW ðS9y PI ÞuSW ðS9y PI Þ9
ðA:3Þ
y PI A OPI x A Rn
where Of, Og and OPI are the constraint sets for y f , y g and y PI , respectively. Also lets define the minimum approximation error as follow: ^ y ng ÞÞu o ¼ f ðxÞf^ ðx9y nf Þ þ ðgðxÞgðx9
ðA:4Þ
according to the lemma in [29], o is bounded; i.e. oALN. Now consider the following Lyapunov function: V¼
1 2 1 ~T~ 1 ~T~ 1 ~T ~ S þ y y þ y y þ y y 2 2g1 f f 2g2 g g 2g3 PI PI
ðA:5Þ
where S is sliding surface defined in (4) and y~ f , y~ g and y~ PI are defined as follows:
Fig. 9. Control signal (part B).
y~ f ¼ y nf y f y~ g ¼ y ng y g , y~ PI ¼ y nPI y PI Table 2 Maximum value of control signal, number of fuzzy rules and IAE criterion of all approaches in tracking problem. Criteria Max u ðtÞ Fuzzy rules IAE
Proposed scheme
Classical SMC
AFSMC [23]
1.4
8.1
4.9
10 0.0045
– 0.0585
72 0.0988
ðA:6Þ
So according to above definitions, following relations can be obtained: T n nT T f^ ðx9y f Þf^ ðx9y f Þ ¼ y f z f ðxÞy f z f ðxÞ ¼ y~ f z f ðxÞ
ðA:7Þ
T ^ y g Þ ¼ y ngT z ðxÞy Tg z ðxÞ ¼ y~ g z ðxÞ ^ y ng Þgðx9 gðx9 g g g
ðA:8Þ T
and integrated absolute error (IAE) obtained by three approaches are reported in Table 2. Once again, Table 2 shows that the maximum value of control effort, number of rules and adaptation parameters are decreased significantly using the proposed scheme.
n nT u^ SW ðx9y PI Þu^ SW ðx9y PI Þ ¼ y PI z PI ðSÞy PI z PI ðSÞ ¼ y~ PI z PI ðSÞ
The time derivative of V is: T T T T 1 _ _ þ 1 ðy_~ y~ þ y_~ y_~ Þ þ 1 ðy_~ y_~ þ y_~ y_~ Þ þSSÞ V_ ¼ ðSS f f g g 2 g1 f f g2 g g
ðA:9Þ
M. Shahriari kahkeshi et al. / ISA Transactions 52 (2013) 342–350
þ
1 _~ T _~ _T _ ðy PI y PI þ y~ PI y~ PI Þ
this case and above procedure is true. The proof of the theorem is thus established. &
g3
1 _T_ 1 _T_ 1 _T _ ¼ SS_ þ y~ f y~ f þ y~ g y~ g þ y~ PI y~ PI
g1
g2
g3
ðA:10Þ References
according to (4) and definition of approximation error in (A.4), we have: n ^ y ng Þgðx9 ^ y g ÞÞ uu^ SW ðS9y PI Þ þdðtÞ þ o S_ ¼ f^ ðx9y f Þf^ ðx9y f Þ þ ðgðx9
T
T
T
n ¼ y~ f z f ðxÞ þ y~ g z g ðxÞ u þ y~ PI z PI ðSÞu^ SW ðS9y PI Þ þdðtÞ þ o
ðA:11Þ
By substituting (A.11) into (A.10): 1 T_ 1 T_ 1 _T V_ ¼ SS_ þ y~ f y~ f þ y~ g y~ g þ y~ PI y~ PI
g1
g2
g3
T T T n ¼ Sðy~ f z f ðxÞ þ y~ g z g ðxÞu þ y~ PI z PI ðSÞu^ SW ðS9y~ PI Þ þ dðtÞ þ oÞ
þ
1 ~ T _~ 1 T_ 1 T _ y f y f þ y~ g y~ g þ y~ PI y~ PI
g1
g2
g3
T T T n ¼ Sy~ f z f ðxÞ þSy~ g z g ðxÞu þ Sy~ PI z PI ðSÞSu^ SW ðS9y PI Þ þ SdðtÞ þ So
þ
1 ~ T _~ 1 T_ 1 T _ y f y f þ y~ g y~ g þ y~ PI y~ PI
g1 T
g2
g3
T
T
r Sy~ f z f ðxÞ þ Sy~ g z g ðxÞu þ Sy~ PI z PI ðSÞSðD þ ZÞ þ SdðtÞ þ So þ
1 ~ T _~ 1 T_ 1 T _ y f y f þ y~ g y~ g þ y~ PI y~ PI
g1
g2
g3
1 T _ 1 T _ o y~ f ðy~ f þ g1 Sz f ðxÞÞ þ y~ g ðy~ g þ g2 Sz g ðxÞuÞ
g1
349
g2
1 T _ þ y~ PI ðy~ PI þ g3 Sz PI ðSÞÞjSjZ þSo
g3
ðA:12Þ
_ _ _ Considering (A.6), we have y~ f ¼ y_ f , y~ g ¼ y_ g , y~ PI ¼ y_ PI , also according to proposed adaptive laws in (21)–(23), so (A.12) can be written as: V_ r 9S9Z þSo
ðA:13Þ
As mentioned above oALN,9o9 o e, so V_ r 9S9o9S9Z r9S99o99S9Z r ðeZÞ9S9
ðA:14Þ
Since Z is determined by the designer in (19), so one can choose it such thate Z o0and it is concluded that V_ r0. So the proposed control scheme is stable (in the sense of Lyapunov) and for all initial conditions; all signals of system are bounded. Since it is proved that the all signals of systems are bounded so it is clear that entire variables of (A.12) are bounded, therefore S_ A L1 . To proof the asymptotic convergence of the tracking error, it should be shown that lim 9SðtÞ9 ¼ 0. Due to boundedness of all t-1 signals, so e is bounded, also according to (4), S is bounded, i. e., 9S9o g so SALN. Integrating both sides of (A.14), we have: Z t 1 9S9dt r ð9Vð0Þ9 þ 9VðtÞ9Þ ðA:15Þ Z e 0 then it is obtained that SAL1. Based on Barbalat lemma [44], lim 9SðtÞ9 ¼ 0, so lim 9eðtÞ9 ¼ 0 thus the system is stable and the t-1 t-1 error will converge to zero asymptotically. Note that in the case that y g A O and Sz g ðxÞu Z 0, the term T
y~ g Sz g ðxÞu in (A.12) cannot be eliminated since y_ g ¼ 0. Because of T T y g A O, so y~ g ry g g^ r0, (i.e., y~ g z g ðxÞ r 0) when y g A O . Since the T term y~ g Sz g ðxÞu r is non-positive. So Eq. (A.14) can be satisfied for
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Contents lists available at SciVerse ScienceDirect
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Research Article
Design of adaptive fuzzy wavelet neural sliding mode controller for uncertain nonlinear systems Maryam Shahriari kahkeshi n, Farid Sheikholeslam, Maryam Zekri Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan 8415683111, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 July 2012 Received in revised form 28 January 2013 Accepted 30 January 2013 Available online 27 February 2013 This paper was recommended for publication by Prof. A.B. Rad
This paper proposes novel adaptive fuzzy wavelet neural sliding mode controller (AFWN-SMC) for a class of uncertain nonlinear systems. The main contribution of this paper is to design smooth sliding mode control (SMC) for a class of high-order nonlinear systems while the structure of the system is unknown and no prior knowledge about uncertainty is available. The proposed scheme composed of an Adaptive Fuzzy Wavelet Neural Controller (AFWNC) to construct equivalent control term and an Adaptive Proportional-Integral (A-PI) controller for implementing switching term to provide smooth control input. Asymptotical stability of the closed loop system is guaranteed, using the Lyapunov direct method. To show the efficiency of the proposed scheme, some numerical examples are provided. To validate the results obtained by proposed approach, some other methods are adopted from the literature and applied for comparison. Simulation results show superiority and capability of the proposed controller to improve the steady state performance and transient response specifications by using less numbers of fuzzy rules and on-line adaptive parameters in comparison to other methods. Furthermore, control effort has considerably decreased and chattering phenomenon has been completely removed. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Fuzzy wavelet neural Smooth sliding mode control Adaptive control Chattering elimination
1. Introduction Control of uncertain nonlinear systems has been an important research in control literatures [1–7]. In practice, system can be uncertain in terms of its dynamic structure or its parameters. Lately, for handling unknown nonlinear systems many approximator based adaptive control techniques have attracted considerable attentions. Approximators such as neural networks (NNs), fuzzy systems (FSs) and wavelet functions, due to their inherent capability in function approximation, have been widely used to represent a model of unknown nonlinear systems for controller design [1,8–15]. Also approximators such as FSs have employed as a state observer to estimate the state variables where not all of the state variables of the nonlinear dynamic systems are available [16]. Although unknown dynamics of nonlinear system can be represented using mentioned approximators, but they can only represent model with limited accuracy, so there remain approximation errors between real system and approximated model. To compensate the approximation errors, disturbances and uncertainties, several control system design and analysis approaches have been proposed. Among the existing robust control approaches, SMC is an effective robust control approach for
n
Corresponding author. Tel.: þ98 311 3915484. E-mail address: [email protected] (M. Shahriari kahkeshi).
uncertain nonlinear systems [17]. Insensitivity and robustness of the SMC against the uncertainties and external disturbance and also its fast response and good transient performance have made it as a popular design tool for robust control of uncertain systems. Despite these feathers, it also suffers from large control chattering phenomenon. Chattering phenomenon results from high frequency control switching and may cause potentially harmful high frequency vibration of the controlled plant [18]. In order to reduce the chattering problem in SMC, a small boundary layer around the sliding surface is introduced in [17–19]. However this method can lead to solve the chattering problem but usually there exist a finite steady state error and so asymptotical convergence is lost. As another solution, adaptive fuzzy sliding mode control (AFSMC) [3,20–25], which is based on the integration of SMC and adaptive fuzzy control, has attracted attention of many researchers. The fundamental idea of AFSMC is to employ fuzzy models for approximating both the equivalent and switching control terms in the SMC to attenuate the chattering phenomenon. As another solution to eliminate chattering phenomenon and achieve zero steady state error, fuzzy controller was proposed in [26] which combine SMC and PI control term. Then some AFSMC literatures used PI control term to eliminate chattering phenomenon [27,28] but they suffer from difficulties associated with the FSs. Since in most of AFSMC, fixed structure is used to describe the unknown system and there is no precise method to design the premise part of fuzzy rules, so before
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.01.004
M. Shahriari kahkeshi et al. / ISA Transactions 52 (2013) 342–350
designing a control law, one must determine the number and location of the basis and membership functions via trial and error. However for some complex unknown system with high input dimension, this method cannot be responded and often lead to unnecessarily large structure for plant representation. To overcome this shortcoming several approaches have been proposed for the generation of the IF–THEN rules [29–32]. Also it is critical to notice that approximators such as traditional FSs only presents localized approximation of the function and cannot capture all local and global behavior of the model. Also as the last point, it should be considered that handling problem with large dimension using AFSMC lead to considerable increment in number of fuzzy bases and cause model complexity. So to overcome the above mentioned problems, recently proposed FWNN in [33] is incorporated with adaptive schemes for control of unknown nonlinear systems [34–40]. In the considered network, integration of the time frequency localization of wavelets, inference properties of FSs to handle the complexity and learning capability of NNs led to simple structure, high approximation accuracy and good generalization capability. Recently, according to the mentioned properties, in [34] an adaptive FW nonsingular terminal sliding mode tracking controller was proposed for robotic systems. Different from the classical SMC, the terminal sliding-mode has a nonlinear sliding surface. Also in [35], AFWNC designed for a class of uncertain SISO nonlinear systems. Proposed scheme in [35] uses FWN to approximate feedback linearization control input such that the closed loop system stability guarantee but controller robustness against external disturbance is not discussed. Also in [36] HN reinforcement learning controller based on FWN was presented which adopts an actor critic reinforcement learning controls scheme. Also an observer based indirect self-structuring robust adaptive fuzzy wave-net controller for uncertain nonlinear system was designed in [37]. It uses fuzzy wave-net in [39] to approximate the system’s unknown structure and then HN tracking based algorithm is used to reduce the influence of the approximation error and external disturbance on the tracking error. In proposed self-structuring controller in [37] FWN structure is determined via trial and error and no straightforward approach is presented to determine network dimension for each problem regardless of input dimensions. As another application, author in [40] proposed an adaptive FWN synchronization controller to synchronize two nonlinear identical chaotic gyros. Proposed scheme composed of a neural controller which utilizes a FWN to online mimic an ideal controller and a fuzzy compensator to remove the chattering phenomena on conventional SMC. This paper proposes an AFWN-SMC for a class of unknown nonlinear systems. Proposed scheme includes two terms: AFWNC and A-PI controller. AFWNC is designed to construct an equivalent term of SMC which invokes FWNs to approximate the unknown dynamics of uncertain system, and an A-PI controller is employed as switching control term of SMC. Moreover PI type adaptive laws are developed by Lyapunov direct method to tune adjustable parameters in real time operation such that the system tracking error converges to zero asymptotically. Main feathers of the proposed scheme are:
Ability to design smooth SMC law with good transient
response characteristics and better steady state performance for a class of nonlinear systems while the structure of the system is unknown and no prior knowledge about uncertainty is available. Ability to determine network size, structure, number and location of basis functions, fuzzy rule and sub-WNN, automatically.
343
Providing less number of fuzzy rules and basis functions and
low computational complexity via constructing wavelet lattice and using OLS algorithm in design step. Ability to deal with high order problems regardless of the input dimensions. Since wavelets with different dilation values under fuzzy rules are invoked to capture local and global behaviors of unknown dynamics, FWN can provide both globalized and localized approximation of the function, so proposed scheme has good generalization capability and better performance.
To validate the results obtained by the proposed scheme, a classical SMC [18], AFSMC [23] and AFWNC [35] are adopted from the literatures and applied for comparison. Simulation results show that the tracking performance and transient response characteristics are improved significantly, while chattering phenomenon is completely eliminated also control effort; fuzzy rules and adjustable parameter are reduced. This paper is organized as follows. Problem statement is briefly explained in Section 2. In Section 3, brief description about FWN is presented and AFWNC is constructed. Design of the proposed scheme is presented in Section 4. To evaluate the performance and capabilities of the proposed approach, simulation results are given in Section 5. Finally, Section 6 concludes the paper.
2. Problem statement Consider a class of SISO nonlinear affine systems which are described by the following equation: xðnÞ ¼ f ðxÞ þgðxÞu þ dðtÞ y¼x
ð1Þ
where f ðxÞ and gðxÞ are two unknown nonlinear functions and d(t) denotes the unknown external disturbance, which is bounded, assume that D is the upper bound of the disturbance, i.e. ðn1Þ T _ 99d(t)99rD. Also x ¼ ½x, x,:::,x ¼ ½x1 ,x2 ,:::,xn T A Rn is the state vector of the system which is assumed to be available for measurement and uAR, yAR are the control input and output of the system, respectively. It is assumed that the nonlinear system (1) is controllable thus the input gain gðxÞ should be nonzero, in this paper we assume that gðxÞ 4 0. The control objective is to design a control law so that the system state xðtÞ can track a desired trajectory y d ðtÞ in the presence of model uncertainty and external disturbance. Let the tracking error is defined as (2). _ . .,eðn1Þ T e ¼ xy d ¼ ½e, e,.
ð2Þ
The control objective is to design the control input such that the state of the system x tracks any given desired trajectory y d in the presence of uncertainties and disturbances, so that tracking error (2) should converge to zero: lim 99e99-0
t-1
ð3Þ
The sliding surface is defined as follows [17]: d S ¼ ð þ cÞn1 e ¼ eðn1Þ þcn1 eðn2Þ þ . . . þc2 e_ þ c1 e ¼ c T e dt
ð4Þ
where c ¼ ½c1 ,c2 ,. . .,cn1 ,1T should be chosen such that the corresponding characteristic polynomial become Hurwitz. To achieve the classical SMC, the control strategy consists of two design steps: first, to force the system toward a desired dynamics in finite time and secondly to maintain the system on the sliding surface. Thus the overall control law, which consists of two terms, equivalent control term (ueq) and reaching control term (usw), can
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M. Shahriari kahkeshi et al. / ISA Transactions 52 (2013) 342–350
be represented as: un ¼ ueq g 1 ðxÞusw
ð5Þ
where ueq ¼ g 1 ðxÞ½f ðxÞ
n1 X
ci eðiÞ þydðnÞ dðtÞ,
usw ¼ Z sgnðSÞ
ð6Þ
i¼1
where Z is strictly positive constant. Since in real world application, external disturbance d(t) is usually unknown, the equivalent control input is modified to: ueq ¼ g 1 ðxÞ½f ðxÞ
n1 X
ci eðiÞ þydðnÞ
ð7Þ
i¼1
In practice, system is uncertain or its dynamics (i.e. nonlinear functions f ðxÞ, gðxÞ) are generally unknown, so the control input (7) is not applicable. Moreover, the discontinuous switching control term in (6) produce excessive chattering phenomenon which is undesirable, so it is hard task to implement control input (5). Thus, in this paper, to overcome mentioned challenges, AFWNC which involves two FWNs, to represent the model of the unknown systems, is constructed to approximate equivalent control term ueq in (7) and an A-PI control law is used as switching control term to eliminate chattering phenomenon completely and improve steady state performance. The following section outlines the details of AFWNC that is used in the proposed scheme.
the membership function, respectively. Also f^ i is the output of local model for rule Ri and is equal to the linear combination of a ðkÞ finite set of wavelets cM ,tk ðxÞ with the same dilation value. i ðkÞ Wavelets cM ,t k ðxÞ are expressed by the tensor product of 1-D i wavelet functions as follow:
ckMi ,t ¼ 2Mi =2 cðkÞ ð2Mi xt k Þ ¼
n Y
2Mi =2 c ð2Mi xi t kj Þ ðkÞ
ð10Þ
j¼1
In this paper, translated and dilated version of the Mexican hat wavelet function is considered. Thus, the cMi ,tk ðxÞ in Eq. (10) is j expressed by the following equation: 0 ! 0 !2 1 ! 1 k 2 xt kj xt kj 1 xt j @ A @ A exp ð11Þ c ¼ 1 2 Mi Mi Mi by applying TSK fuzzy inference mechanism and considering f^ i as an output of each sub-WNN (or each rule), the output of FWN f^ ðx9y f Þ can be expressed as: f^ ðx9y f Þ ¼
cf X
m^ i ðxÞf^ i ¼
i¼1
cf X
m^ i ðxÞyf i f i ðxÞ
ð12Þ
i¼1
Using a simplified notation, (12) can be rewritten as: T f^ ðx9y f Þ ¼ y f z f ðxÞ
^ i ðxÞ ¼ mi ðxÞ=ð where m
ð13Þ Pc
i¼1
mi ðxÞÞ, mi ðxÞ ¼ Pnj¼ 1 Aij ðxj Þ, cf is the numT
ber of rules to approximate function f ðxÞ and y f ¼ ½yf 1 , yf 2 ,:::, yf cf T is the adjustable parameter vector and z f ðxÞ ¼ ½zf 1 ðxÞ, zf 2 ðxÞ,:::,
zf cf ðxÞT is the fuzzy wavelet basis functions. By the similar
3. Adaptive fuzzy wavelet neural controller (AFWNC)
procedure, the function gðxÞ can be modeled as follows: In this paper, AFWNC is constructed to approximate the equivalent control term of SMC. It invokes FWNs to represent the model of the unknown dynamics of uncertain nonlinear system. For this, FWN is expressed as a series expansion of fuzzy wavelet basis functions to model the unknown dynamics (i.e., f ðxÞ, gðxÞ) of the equivalent control term (7) by tuning the parameters of the corresponding networks. Thus it is necessary to give the basic concepts of FWN here. FWN combines multiresolution analysis of wavelet transform and traditional TSK fuzzy system. In this network, constant or linear functions in the consequence part of TSK model are substituted with wavelet functions [33]. In FWN model each fuzzy rule corresponds to one sub-WNN which consists of wavelets with a specified dilation value. A typical FWN for approximating arbitrary nonlinear function f ðxÞ can be described by the following linguistic rule [35]: Rif : If x1 is Ai1 and x2 is Ai2 and ::: and xn is Ain , then f^ i ¼ yf i f i where fi is defined as f i ¼
PT f i
k¼1
wMf
i
ðkÞ ,t k cM f ,tk ðxÞ, i
ð8Þ
and yf i is free
adjustable parameter which should be updated online based on PI type adaptive laws. Also Rif is the ith rule (1rircf) and Mf i A Z is the dilation parameter for rule Rif . In this work, t kj is the translation value for the corresponding wavelet k and xj, (1rj rn) is the jth input variable of x and n is the input dimensions. Also T f i denotes the total number of wavelets for the ith rule, and Aij is the fuzzy set characterized by the following Gaussian-type membership function. Where Aij ðxj Þ denotes the membership function value of xj in Aij . 2
Aij ðxj Þ ¼ e9ðxj pj1 Þ=pj2 9pj3 ¼ eðððxj pj1 Þ=pj2 Þ i
i
i
i
Þpij3 =2
ð9Þ
0 r pij3 r 5, and pij1 represent the center of memberfunctions, pij2 and pij3 determine the width and the shape of
pij1 ,pij2 A R, ship
i
^ y g Þ ¼ y Tg z ðxÞ gðx9 g
ð14Þ
T
where y g ¼ ½yg1 , yg2 ,:::, ygcg T is the adjustable parameter vector and
z g ðxÞ ¼ ½z1 ðxÞ, z2 ðxÞ,:::, zcg ðxÞT is the fuzzy wavelet basis functions and cg is the number of rules which are applied to approximate function gðxÞ. Design procedure of the PI type adaptive laws to update the adjustable parameter vector in (13) and (14) is given in Section 4. In the network construction step, first a library of candidate wavelets are selected by constructing wavelet lattice [41] and then OLS algorithm [42] is used to purify them for choosing important and efficient wavelets for constructing sub-WNN, determining network structure and initial weights. Then based on dilation parameters of efficient wavelets, number of sub-WNN and fuzzy rules are determined. Also at the end of this step, premise and consequence parts of fuzzy rules are specified. Because of the efficient procedure of selecting wavelets employed in the OLS algorithm is not very sensitive to the input dimension so the dimension of the approximated function does not cause the bottleneck for constructing FWN. After assigning the network structure, its parameters (sub-WNN’s weights, translation parameters of efficient wavelets, membership functions parameters) are initialized according to [33]. Then two learning stages consist of extended Kalman filter (EKF) and recursive least square (RLS) estimator are utilized for network learning. All nonlinear parameters involving pijr and t kj where 1rjrn, 1rircf, r ¼1,2,3 and k ¼ T 1 ,T 2 ,. . .,T cf are learned using the EKF and linear parameter, sub-WNN’s weights wMi ,tk , are tuned using RLS estimator. Details of these algorithms can be found in [43,32]. Structure of FWN for function approximation, in AFWNC structure, is depicted in Fig. 1. As mentioned above, the result in (7) is realizable only while f ðxÞ and gðxÞ are well-known. However in practical application f ðxÞ and gðxÞ are unknown and ideal controller (7) cannot be implemented. So f ðxÞ and gðxÞ are replaced by FWN model (13)
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the control input is kept at D þ Z which is a constant saturated value, when the state is outside of the boundary layer. Therefore 9u^ SW ðS9y^ PI Þ9 ¼ D þ Z when 9S9 Z f. As a result, the A-PI control term can be written in the following form: 8 < y T z ðSÞ 9S9 o f PI PI u^ SW ðS9y PI Þ ¼ ð19Þ : ZþD 9S9 Z f Considering proposed AFWNC and applied A-PI control term, the proposed controller can be expressed as: h i 1 f^ ðx9y f Þc T e þyðnÞ u^ SW ðS9y PI Þ ð20Þ u¼ d ^ yg Þ gðx9 where y f , y g and y PI are adjustable parameter vectors. Therefore we have the following theorem: Theorem 1. If the control input (20) is applied to nonlinear system (1) and functions f^ , g^ and u^ SW in (20) are substituted with presented adaptive models in (13), (14) and (19), and adjustable free parameters vectors y f , y g and y PI are adapted by the adaptive laws in (21)–(23), then all signals of the closed loop system will be bounded and the tracking error will converge to zero asymptotically.
y_ f ¼ g1 Sx f ðxÞ Fig. 1. Structure of FWN for function approximation.
and (14). So AFWNC which is constructed to represent the equivalent control term (7) can be expressed as follow: h i 1 ð15Þ ueq ¼ f^ ðx9y f Þc T e þ ydðnÞ ^ yg Þ gðx9 ^ y g Þ can be represented as (13) and (14), where f^ ðx9y f Þ and gðx9 respectively.
8 > > < _y ¼ g2 Sz g ðxÞu g > > : 0
ð21Þ if ðy g A OÞ or ðy g A O and Sz g ðxÞu r0Þ
ð22Þ
otherwise
y_ PI ¼ g3 Sz PI ðSÞ
ð23Þ
where g1, g2 and g3 are the positive adaptation rates. Also adaptation for y g in (22) is constrained within the convex set O to satisfy: ^ y g Þ Z e 40g O ¼ fy g 9gðx9
4. Adaptive fuzzy wavelet neural sliding mode controller design According to Section 2, SMC-based controller has two terms as mentioned in (5) and (6). Since dynamics of nonlinear systems are unknown, so AFWNC is constructed to represent equivalent control term (in Section 3). Thus by adding switching control law to (15), the overall control u can be written as: h i 1 u¼ ð16Þ f^ ðx9y f Þc T e þ ydðnÞ usw ^ yg Þ gðx9 Moreover to avoid chattering phenomenon caused by switching term and to improve steady state performance, usw in (16) is replaced by the A-PI control action when the state is within the boundary layer, i.e. 9S9o f (f is the thickness of the boundary layer). Therefore, it is important to employ an A-PI control law with chattering elimination property as a switching control term (usw) before the design of adaptive laws to adjust adaptation parameters y f and y g . Thus the input and output of continues time A-PI controller can be expressed in the following form [27]: Z usw ðS9y PI Þ ¼ yp S þ yI Sdt ð17Þ where yp and yI are proportional and integral term gains, respectively. For the sake of convenience in presentation, uSW in (17) can be denoted as the following form: T u^ SW ðS9y PI Þ ¼ y PI z PI ðSÞ ð18Þ R where z PI ðSÞ ¼ ½S, Sdt A R2 is a regressive vector and y PI ¼ ½yp , yI A R2 is an adjustable parameters vector which should be adapted simultaneously along with the control procedure. Also
in this adaptive law, whenever y g A O
ð24Þ T we have y_ g ry g^ ¼ g
g2 Sz g ðxÞz Tg ðxÞu r 0 . This condition implies the vector y_ g points within 901 of ry g^ , so it points inside O or along the tangent plane of O at g
point y g . Hence y g will never leave O, i.e., y g ðtÞ A O 8 t Z 0 . The proof of this theorem is given in Appendix A. Block diagram of the proposed scheme is given in Fig. 2.
5. Simulation results In this section, the proposed approach is tested on a second-order nonlinear inverted pendulum system. The low dimension of the presented example does not alter the generality of the proposed algorithm for higher dimension. Two parts are derived. In the first part, the proposed controller is applied for stabilizing problem and in the second part; the performance of the proposed scheme is investigated in a tracking problem. To validate the results obtained by the proposed approach, a classical SMC, AFWNC and AFSMC are adopted from the literature and applied for comparison. 5.1. Study system The state equations of the inverted pendulum system can be expressed as the following equation: ( x_ 1 ¼ x2 ÞðmLx2 sinðx1 Þcosðx1 ÞÞ=m þ M cosðx1 Þ=m þ M ð25Þ þ L½4=3mcos x_ 2 ¼ gsinðx1L½4=3mcos 2 ðx Þ=m þ M 2 ðx Þ=m þ M u þ d 1
1
where x1 ¼ y and x2 ¼ y_ are the angular position and velocity of the pole, respectively. g ¼9.8 m/s2 denotes the acceleration due to
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Fig. 2. Block diagram of the proposed AFWN-SMC scheme.
gravity, M ¼1 kg and m ¼0.1 kg are the mass of cart and the mass of pole, respectively. Finally, L¼0.5 m is the half length of pole and d is an external disturbance.
5.2. Part A: Balancing the pole in the vertical position In this part, the objective is to generate an appropriate actuator force u to control the motion of the cart, such that the pole can be balanced in the vertical position. The sliding surface is chosen as S ¼e2 þ2e1 and external disturbance d(t) is assumed to be dðtÞ ¼ 1 þsinðp t=2Þ. Also, initial condition is selected as x 0 ¼ ½p=8,0. The implementation of the suggested approach is given below. In order to construct AFWNC to represent ueq, two FWNs are constructed to model unknown dynamics of the plant. Efficient wavelets are purified to construct each sub-WNN by OLS algorithm. According to the dilation parameters of the selected wavelets, five rules/sub-WNNs are chosen for constructing each network. According to dilation parameters of selected wavelets, the proposed controller which is used here contains 10 fuzzy rules with 10 membership functions which are assigned for each input variable and 10 adaptation parameters that tune the parameters of the controller, online. In order to find the effectiveness and superiority of the proposed method, the test results are compared with the results obtained by classical SMC and AFSMC. In the classical SMC method, the system is assumed to be known and control signal is computed according to (5). In the proposed AFSMC in [23], two FSs are used to approximate unknown dynamics of system and also other FS is utilized as a switching control term to attenuate chattering action. In this example, membership functions for system states are defined similar to [23] as follow:
mA1 ðxÞ ¼ mA3 ðxÞ ¼ mA4 ðxÞ ¼ mA6 ðxÞ ¼
1 , 1þ expð5ðx þ 2ÞÞ
mA2 ðxÞ ¼
1 1 þ expððx þ 1:5Þ2 Þ
,
1 1þ expððx þ 0:5Þ2 Þ 1 1þ expððx0:5Þ2 Þ
,
mA5 ðxÞ ¼
1 1 þ expððx1:5Þ2 Þ
,
1 1þ expð5ðx2Þ2 Þ
so there are, 72 rules to approximate the system functions f ðxÞ and gðxÞ. Also three fuzzy sets are defined for sliding surface S ¼ e_ þ 2e as follow:
mS1 ¼
1 , 1 þ expð5ðS þ3ÞÞ
mS2 ¼
1 1 þ expðS2 Þ
,
Fig. 3. State trajectories: (a) x1 ¼ y and (b) x2 ¼ y_ of mentioned approaches.
mS3 ¼
1 1 þexpð5ðS þ 3ÞÞ
So the switching control term usw is represented by FS and adaptive laws based on SMC are used to implement controller [23]. Fig. 3 illustrates the state trajectories of the system for initial condition x 0 . As shown in this figure, by using the proposed approach, the system states converge to zero rapidly as the system starts. Moreover the transient response specifications have improved significantly. It can be found that the system transient response for the proposed method has much less overshoot and settling time in comparison to the classic SMC and AFSMC. Fig. 4 shows the control signal of three approaches. As can be seen from Fig. 4, proposed scheme completely eliminate the chattering phenomenon. So the proposed approach has good disturbance rejection performance and chattering elimination. Also in comparison with other methods, control effort has decreased, considerably. The maximum value of the control effort, number of fuzzy rules and adaptation parameters obtained by three approaches are shown in Table 1. Also, in order to have a quantitative comparison of tracking error, mean square error (MSE) of three methods has been reported. According to the following table, the maximum value of control effort and number of fuzzy rules/
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347
Fig. 6. State trajectory and control signal with proposed AFMNC in [35] (five fuzzy rules, g ¼ 50).
Fig. 4. Control signal (part A).
Table 1 Maximum value of control signal, number of fuzzy rules and MSE criterion of all approaches in the stabilization problem. Criteria
Proposed scheme
Max u ðtÞ Fuzzy rules MSE
Classical SMC
AFSMC [23]
9.8
4.8
30
12 0.0049
– 0.0807
75 0.457
Fig. 7. State trajectory and control signal of mentioned approaches for x 0 ¼ ½p=6,0.
demonstrates that the proposed controller performs much better as compared to the other mentioned approaches. 5.3. Part B: Tracking a desired trajectory
Fig. 5. State trajectory and control signal with the proposed method, (five fuzzy rules for each function, g1 ¼ 0.05, g2 ¼0.01).
adaptation parameters is decreased significantly using the proposed scheme. Also results with proposed scheme are compared with the proposed AFWNC in [35] for initial condition x 0 ¼ ½p=4,0. Figs. 5 and 6 show the results of the proposed scheme and presented AFWNC in [35], respectively. As can be seen, fast response with AFWNC implies significant initial control energy during the transient response and also lead to undesirable transient response characteristic such as undershoot. Simulation result for another initial condition x 0 ¼ ½p=6,0 in the presence of disturbance dðtÞ ¼ sinðtÞ is given in Fig. 7. It
Let us consider the inverted pendulum system again. The control objective is to maintain the system to track the desired angle trajectory yd ¼ yd ¼ p=9ðcos t þ 0:6cos 3tÞ in the presence of disturbance d ¼ 0:2sinð0:25tÞ for initial condition x0 ¼[p/6,0]. The sliding surface is S ¼4e2 þe1. Constructed networks in part A are used to model the AFWNC for constructing equivalent control term. State trajectories of the system for three mentioned approaches are illustrated in Fig. 8. Also the control signal is shown in Fig. 9. Again, these responses are similar to the responses in Figs. 3 and 4 for the stabilizing problem, showing the robustness and superiority of the suggested scheme. As can be seen from Fig. 9, the chattering phenomenon, which is clearly obtained in classical SMC, is completely eliminated using the proposed scheme. Also control effort with the proposed scheme is smaller than other mentioned approaches. Moreover, the maximum value of control effort, number of fuzzy rules, adaptation parameters
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6. Conclusions In this paper, an AFWN-SMC is proposed for a class of uncertain nonlinear systems. Proposed controller composed of two terms, AFWNC and A-PI controller. AFWNC is used to construct equivalent control term, it invokes two FWNs to approximate unknown nonlinear dynamics of uncertain system and A-PI controller is designed as switching control term to eliminate chattering phenomenon, provides smooth SMC law and improves steady state performance. The main advantage of the proposed scheme is the design of the smooth sliding mode control for a class of high-order nonlinear systems while the structure of the system is unknown and no knowledge about the uncertainty is available. The obtained simulation results show that the proposed scheme is quite effective in control of unknown nonlinear SISO systems. They illustrate that the proposed scheme significantly improves the transient response specifications and steady state performance. Also it has a robust tracking performance and acceptable behavior despite external disturbance. Furthermore number of fuzzy rules and online adjustable parameters are reduced, chattering phenomenon is completely removed while amplitude of control effort is reduced, considerably.
Appendix A Fig. 8. Tracking state (a) x1 ¼ y and (b) x2 ¼ y_ of proposed scheme and other mentioned approaches.
Proof of the theorem Proof. Lets define the optimal parameters vector as follow: " # y nf ¼ arg min sup 9f^ 9ðx9y f Þf ðxÞ9 ðA:1Þ y f A Of x A Rn
"
#
^ y g ÞgðxÞ9 y ng ¼ arg min sup 9gðx9
ðA:2Þ
y g A Og x A Rn
"
y nPI ¼ arg min
# sup 9u^ SW ðS9y PI ÞuSW ðS9y PI Þ9
ðA:3Þ
y PI A OPI x A Rn
where Of, Og and OPI are the constraint sets for y f , y g and y PI , respectively. Also lets define the minimum approximation error as follow: ^ y ng ÞÞu o ¼ f ðxÞf^ ðx9y nf Þ þ ðgðxÞgðx9
ðA:4Þ
according to the lemma in [29], o is bounded; i.e. oALN. Now consider the following Lyapunov function: V¼
1 2 1 ~T~ 1 ~T~ 1 ~T ~ S þ y y þ y y þ y y 2 2g1 f f 2g2 g g 2g3 PI PI
ðA:5Þ
where S is sliding surface defined in (4) and y~ f , y~ g and y~ PI are defined as follows:
Fig. 9. Control signal (part B).
y~ f ¼ y nf y f y~ g ¼ y ng y g , y~ PI ¼ y nPI y PI Table 2 Maximum value of control signal, number of fuzzy rules and IAE criterion of all approaches in tracking problem. Criteria Max u ðtÞ Fuzzy rules IAE
Proposed scheme
Classical SMC
AFSMC [23]
1.4
8.1
4.9
10 0.0045
– 0.0585
72 0.0988
ðA:6Þ
So according to above definitions, following relations can be obtained: T n nT T f^ ðx9y f Þf^ ðx9y f Þ ¼ y f z f ðxÞy f z f ðxÞ ¼ y~ f z f ðxÞ
ðA:7Þ
T ^ y g Þ ¼ y ngT z ðxÞy Tg z ðxÞ ¼ y~ g z ðxÞ ^ y ng Þgðx9 gðx9 g g g
ðA:8Þ T
and integrated absolute error (IAE) obtained by three approaches are reported in Table 2. Once again, Table 2 shows that the maximum value of control effort, number of rules and adaptation parameters are decreased significantly using the proposed scheme.
n nT u^ SW ðx9y PI Þu^ SW ðx9y PI Þ ¼ y PI z PI ðSÞy PI z PI ðSÞ ¼ y~ PI z PI ðSÞ
The time derivative of V is: T T T T 1 _ _ þ 1 ðy_~ y~ þ y_~ y_~ Þ þ 1 ðy_~ y_~ þ y_~ y_~ Þ þSSÞ V_ ¼ ðSS f f g g 2 g1 f f g2 g g
ðA:9Þ
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þ
1 _~ T _~ _T _ ðy PI y PI þ y~ PI y~ PI Þ
this case and above procedure is true. The proof of the theorem is thus established. &
g3
1 _T_ 1 _T_ 1 _T _ ¼ SS_ þ y~ f y~ f þ y~ g y~ g þ y~ PI y~ PI
g1
g2
g3
ðA:10Þ References
according to (4) and definition of approximation error in (A.4), we have: n ^ y ng Þgðx9 ^ y g ÞÞ uu^ SW ðS9y PI Þ þdðtÞ þ o S_ ¼ f^ ðx9y f Þf^ ðx9y f Þ þ ðgðx9
T
T
T
n ¼ y~ f z f ðxÞ þ y~ g z g ðxÞ u þ y~ PI z PI ðSÞu^ SW ðS9y PI Þ þdðtÞ þ o
ðA:11Þ
By substituting (A.11) into (A.10): 1 T_ 1 T_ 1 _T V_ ¼ SS_ þ y~ f y~ f þ y~ g y~ g þ y~ PI y~ PI
g1
g2
g3
T T T n ¼ Sðy~ f z f ðxÞ þ y~ g z g ðxÞu þ y~ PI z PI ðSÞu^ SW ðS9y~ PI Þ þ dðtÞ þ oÞ
þ
1 ~ T _~ 1 T_ 1 T _ y f y f þ y~ g y~ g þ y~ PI y~ PI
g1
g2
g3
T T T n ¼ Sy~ f z f ðxÞ þSy~ g z g ðxÞu þ Sy~ PI z PI ðSÞSu^ SW ðS9y PI Þ þ SdðtÞ þ So
þ
1 ~ T _~ 1 T_ 1 T _ y f y f þ y~ g y~ g þ y~ PI y~ PI
g1 T
g2
g3
T
T
r Sy~ f z f ðxÞ þ Sy~ g z g ðxÞu þ Sy~ PI z PI ðSÞSðD þ ZÞ þ SdðtÞ þ So þ
1 ~ T _~ 1 T_ 1 T _ y f y f þ y~ g y~ g þ y~ PI y~ PI
g1
g2
g3
1 T _ 1 T _ o y~ f ðy~ f þ g1 Sz f ðxÞÞ þ y~ g ðy~ g þ g2 Sz g ðxÞuÞ
g1
349
g2
1 T _ þ y~ PI ðy~ PI þ g3 Sz PI ðSÞÞjSjZ þSo
g3
ðA:12Þ
_ _ _ Considering (A.6), we have y~ f ¼ y_ f , y~ g ¼ y_ g , y~ PI ¼ y_ PI , also according to proposed adaptive laws in (21)–(23), so (A.12) can be written as: V_ r 9S9Z þSo
ðA:13Þ
As mentioned above oALN,9o9 o e, so V_ r 9S9o9S9Z r9S99o99S9Z r ðeZÞ9S9
ðA:14Þ
Since Z is determined by the designer in (19), so one can choose it such thate Z o0and it is concluded that V_ r0. So the proposed control scheme is stable (in the sense of Lyapunov) and for all initial conditions; all signals of system are bounded. Since it is proved that the all signals of systems are bounded so it is clear that entire variables of (A.12) are bounded, therefore S_ A L1 . To proof the asymptotic convergence of the tracking error, it should be shown that lim 9SðtÞ9 ¼ 0. Due to boundedness of all t-1 signals, so e is bounded, also according to (4), S is bounded, i. e., 9S9o g so SALN. Integrating both sides of (A.14), we have: Z t 1 9S9dt r ð9Vð0Þ9 þ 9VðtÞ9Þ ðA:15Þ Z e 0 then it is obtained that SAL1. Based on Barbalat lemma [44], lim 9SðtÞ9 ¼ 0, so lim 9eðtÞ9 ¼ 0 thus the system is stable and the t-1 t-1 error will converge to zero asymptotically. Note that in the case that y g A O and Sz g ðxÞu Z 0, the term T
y~ g Sz g ðxÞu in (A.12) cannot be eliminated since y_ g ¼ 0. Because of T T y g A O, so y~ g ry g g^ r0, (i.e., y~ g z g ðxÞ r 0) when y g A O . Since the T term y~ g Sz g ðxÞu r is non-positive. So Eq. (A.14) can be satisfied for
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